Theoretical Physics and Astrophysics 0080230679, 0080230660

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V. Ginzburg

THEORETICAL PHYSICS AND ASTROPHYSICS

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VOLUME 99

THEORETICAL PHYSICS AND ASTROPHYSICS

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THEORETICAL PHYSICS AND ASTROPHYSICS

by

V .L . GINZBURG P. N . Lebedev Physical Institute, Academy of Sciences of the USSR, Moscow

Translated by D . T E R

H A A R

Oxford University

PERG A M O N PRESS OXFORD • NEW YORK • TORONTO • SYDNEY • PARIS • FRANKFURT

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FRANCE FEDERAL REPUBLIC OF GERMANY

Copyright © 1979 Pergamon Press Ltd. A ll Rights Reserved. S o part o f this publication may be reproduced, stored in a retrieval system or transmitted in any fo rm or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing fro m the publishers.

First edition 1979 British Library Cataloguing in Publication Data Ginzburg, Vilalii Lazarevich Theoretical physics and asirophysics. (International series in natural philosophy; vol. 99). 1. Astrophysics 2. Physics 1. Title 11. Series 523.01 QB461 78^1009 ISBN 0-08-023067-9(hardcover) ISBN 0-08-023066-0(llexicover)

In order to make this volume available as economically and as rapidly as possible the author's typescript has been reproduced in its original form . This method un­ fortunately has its typographical limitations but it is hoped that they in no way distract the reader.

Printed and bound at William Clowes & Sons Limited Beccles and London

CONTENTS

PREFACE TO THE ENGLISH EDITION PREFACE TO THE RUSSIAN EDITION CHAPTER

I.

THE HAMILTONIAN APPROACH TO ELECTRODYNAMICS

CHAPTER

II.

RADIATION REACTION

CHAPTER

III.

UNIFORMLY ACCELERATED CHARGE

CHAPTER

IV.

RADIATION OF A MOVING PARTICLE

CHAPTER

V.

SYNCHROTRON RADIATION

CHAPTER

VI.

ELECTRODYNAMICS OF A CONTINUOUS MEDIUM

CHAPTER

VII.

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION

CHAPTER

VIII.

ON SUPERLUMINAL RADIATION SOURCES

CHAPTER

IX.

REABSORPTION AND RADIATIVE TRANSFER

CHAPTER

X.

ELECTRODYNAMICS OF MEDIA WITH SPATIAL DISPERSION

CHAPTER

XI.

DIELECTRIC PERMITTIVITY AND WAVE PROPAGATION IN A PLASMA

CHAPTER

XII.

THE ENERGY-MOMENTUM TENSOR IN MACROSCOPIC ELECTRODYNAMICS

CHAPTER

XIII.

FLUCTUATIONS AND VAN DER WAALS FORCES

CHAPTER

XIV.

SCATTERING OF WAVES IN A MEDIUM

CHAPTER

XV.

COSMIC RAY ASTROPHYSICS

CHAPTER

XVI.

X-RAY ASTRONOMY

CHAPTER

XVII.

GAMMA ASTRONOMY REFERENCES INDEX

v

PREFACE TO THE ENGLISH EDITION

As any author I am glad to see my book appear in English, a language accessible to all physicists nowadays.

However, it is hardly necessary to

wirte a special preface to say this or to mention that in the English edition a number of errors have been corrected.

The idea that such a preface might be

necessary arose when I received from Professor ter Haar the preliminary list of

references for

the English

references to Ginzburg.

edition and

noticed the

large number

of

Of course, the same references also appear in the

Russian edition, but there they do not appear together in one lot, and are also not so obvious in the text.

The reason is that in the Russian edition,

as is the usage with us, references are indicated by a number, without mention­ ing the name of the authors.

As a result my name never appears in the text,

while in the English translation one meets it very often!

Such a situation is

unusual for a book meant for students or, at any rate, one which is not a mono­ graph, and it may cause some unpleasant observations.

I want therefore to add

the following to what I said in the preface to the Russian edition.

I am, in

general, not a teacher and I do not like to lecture, especially not when the same material must be presented more than once.

At the same time, for the

first time lectures one needs a large amount of preparation.

In such a situa­

tion and if

of

I have

the possibility

to choose

the subject

completely freely I have given some lectures just once; written in preparing such a course.

the lectures

the present book was

When choosing the material and wishing to

share with my audience or my readers what appeared to me to be interesting and important I widely used my own papers and review articles. result was a very one-sided picture.

Of course, the

However, as there was no pretension of

priority claims to that I had chosen the most interesting problems, there was hardly any reason for reproaching me of immodesty.

A different question is

whether the book is useful and deserving attention, that that is, of course, up to the reader to decide. In conclusion I use this possibility to express my wannest thanks to Professor ter Haar for undertaking the arduous task of translation and also for a number of helpful remarks. Moscow, April, 1978

V.L. Ginzburg vii

PREFACE TO THE RUSSIAN EDITION

There are many textbooks of theoretical physics among which the manyvolume work of Landau and Lifshitz is the best known and most outstanding* It is impossible, however, to deal with all problems in such a course.

Moreover,

even the problems which are considered can usually not be looked at from dif­ ferent points of view.

At the same time, depending on their peculiar abilities,

the nature of their training, natural inclinations, and so on, different people often prefer different approaches, arguments, examples, and proofs. A natural possibility to satisfy an existing need is, clearly, to publish dif­ ferent textbooks and, in particular, different supplementary textbooks which are devoted to separate problems, aspects, and methods rather than to a syste­ matic exposition of a topic.

Such supplementary texts differ in principle

from the systematic ones in that the choice of material to a large extent is not predetermined.

One can say the same about the style and nature of the

exposition whereas in a systematic textbook one must impose very rigid restric­ tions with respect notation, and so on.

to conciseness,

content of

technical methods, unified

The present book is just such a supplementary text which

is devoted to a few problems in theoretical physics and astrophysics.

From

the table of contents it is imnediately clear, but we may also state it that, in general, we are dealing with problems which are in one way or other connec­ ted with electrodynamics. In order not to violate this trend, even though it was not rigorously expressed, we left outside the limits of the book

a number

of problems

of

the general

theory of relativity and of statistical physics which, in the opinion of the author, also should be the subject of supplementary courses of a similar kind. The basis of the exposition was a lecture course for students in the physics and astrophysics departments of the Moscow Physico-Technical Institute.

These

lectures were not meant to replace a systematic course and had just the charac­ ter of ’capita selecta' taking into account the interests of the department and not least the interests and capabilities of the author.

Of course, we are not

x

Preface to the Russian edition

saying that the problems which the author at a particular moment in the p a g f e was occupied with are more important or more interesting than many other It is simply the case that merely presenting material with which he is well familiar,

’more or less moved by the s p i J B 1

the author may perhaps hope to s u p p K

ment existing texts and monographs without checking whether he is rewriting to some extent duplicating them. As to the nature of the exposition one should note that we are dealing here not with lectures which are written out but with a special text specially pre­ pared for these lectures, in which rather often we have also included material which is not very suitable and in fact was not used for the lectures themselves (that is, for the oral presentation).

In this respect the book is in style

close to a monograph or a review article which finds reflection also in the rather large number of references to the literature.

As amongst those there

is a large number of references to work by the author, we emphasize that this, like the choice of material, is completely unconnected with any pretentions, but caused by the tendency, already mentioned, to touch only upon very familiar problems which were dealt with in detail in the papers referred to; moreover, a whole number of such papers were used directly in the text. We note, finally, that the book is definitely not intended for people with a mathematical inclination — such as ’pure*

theoretical physicists.

The excep­

tionally large role played by mathematics in theoretical physics is completely unquestionable and natural, but aiming at mathematical generality and rigour is by far not always justified — one must pay for this.

It is generally known,

in particular, that most new physical results have been obtained by relatively simple means while the 'mathematization' occurred only in the later stages. At any rate, physics, and not mathematics is the main point of theoretical physics. An exposition of theoretical problem with a 'general physics' bias is at least as permissible as the nowadays more widely propagated tendency to mathematical perfection. One may hope that this book will turn out to be useful for graduate students and also for post-doctoral and research workers. In conclusion I use this opportunity to thank all who read the manuscript or parts of it and whose remarks contributed to improvements in the text.

Moscow,

July,

1974

V.L. Ginzburg

Chapter I THE HAMILTONIAN APPROACH TO ELECTRODYNAMICS The Hamiltonian method in classical electrodynamics in vacuo■ Quantization. Photons and pseudophotons. Does a uniform moving electron radiate? We shall in what follows widely apply the so-called Hamiltonian method for the interpretation of a whole range of electrodynamical problems.

When we

use this method electrodynamics is formulated in a way which is strongly remin­ iscent of mechanics.

The transition from classical to quantum mechanical elec­

trodynamics is thus in the Hamiltonian framework completely analogous to the transition from classical, Newtonian mechanics to non-relativistic quantum mechanics.

Nowadays much more sophisticated methods are predominant in quantum

electrodynamics and in general in quantum field theory and there are strong arguments for using them.

However, the use of the Hamiltonian method is still

completely justified for the elucidation of a large number of physical aspects; this is, for instance, also done by Heitler (19£7) in his book.

Moreover, we

shall in what follows apply the Hamiltonian method mainly to classical electro­ dynamics, both in vacuo and in a medium* Before introducing the Hamiltonian method we shall give the main equations and relations and we shall do that in considerable detail for future convenience. The usual form of the Maxwell equations in vacuo is:^ n n 4tt 3E curl H = — pv + , div E = 4irp ,

0

.1)

curl E = - — c 3t div H * 0 . Here H

is the magnetic field strength, E

the electrical field strength, p the

charge density, V the velocity of the charges, and vacuo. charge e

We

assume

for

the

sake

of

c

the velocity of light in

simplicity that there

is

, at position ■*£(t), in the electromagnetic field.

+ We U8e everywhere in this book c g s units.

a single point

In that case

THEORETICAL PHYSICS AND ASTROPHYSICS

2

the charge density is given by a 6-function p - e6

(r - r (t)) .

(1.2)

It is well known that Eqs.(l.l) can be reduced to the equations for the elec­ tromagnetic potentials

A

and which are connected with the fields

E and

H by the relations

1 3A E - - —

- grad

,

H = curl A

.

(1.3)

The third and fourth of Kqs•(].]) are automatically satisfied by virtue of (1.3), as can be verified by substitution. From the first and second of Eqs.(l.l) we get, using (1.3) and the identity curl curl A

= - V 2A + grad d i v A

,

(1,4)

equations for the potentials A and $ :

^

0

■ erad ( e l f + div A )“ - T

p V' 1

+ - 4 ~ d i v A - - 4wp C dt

The set of Eqs.(1.5) determines the potentials A and $.

.>

(1.5)

The fields E and H

can be found using Eqs.(1.3) It is well known that the vector potential A and the scalar potential not uniquely determined.

A' = A + g r a d x where

X

,

(1.6)

^ |- ,

i s an arbitrary function of the coordinates and the time.

called a gauge transformation.

This is

One can easily show that the fields E and H

do not change under a gauge transformation. A* and

are

Indeed, we can change to new potentials:

just as well as in terms of A and

They can be expressed in terms of ;

one can verify this by sub­

stituting (1.6) into (1.3). The fact that the definition of the potentials is not unique enables us to impose upon A and

an additional condition.

This condition can be chosen in

such a way that the form of Eqs,(1.5) becomes as simple as possible. For instance, we can impose as such a condition the relation: divA + !

■

0 .

(1.7)

C dt

This is a relativistically invariant condition which i9 called the Lorentz condition, and the resulting gauge is called the Lorentz gauge.

It can be

3

THE HAMILTONIAN APPROACH TO ELECTRODYNAMICS written in the form

0 ,

(1.7a)

3x where we have assumed simulation over repeated indexes* a 9 will be done every­ We U9e here and elsewhere in thi9 book (apart from

where in what follows*

Chapter 12) the notation of Landau and Lifshitz (1975)*

We refer to that text

for the definition of four-vectors, for the difference between covariant and components and for the summation convention* One 9ees easily that if condition (1.7) is satisfied, the Maxwell equations take the following form:

QA i ( 7* - ? & ) * □*

-

T ” >

--

Atfp

(1«8)

/ i \ E (v* “ - r r r J * N c* 3t*'

•

One should not think that the condition (1.7) and the 9et of Eq 9 .(1.8) deter­ mine A and

4) completely.

We can still perform a gauge transformation of the

form (1.6), where in the present case X mu 9t satisfy the homogeneous equation □ X = 0.

The fields E and H remain invariant under the transformation.

Splitting the field into longitudinal and transverse components i 9 important, especially in the Hamiltonian framework.

We 9plit the vectors E and H

into

components E = E where

div Etr * 0

I

+ E

H = H

(1.9)

tr

and, by virtue of (l.l),divHtr - div M ■ 0.

We demand that the vector potential A describe only the transverse field; thi9 means that we impose on it the condition divA = 0 , instead of the additional condition (1.7).

(1.10) Sometimes

the

potential which

satisfies the condition (1.10) is denoted by A tr . If condition (1.10) i9 satisfied, the Eqs.(1.5) for A and

V2 * - 47Tp , c z 3tz

c

take the form

(1.11)

c

at

We 9ee that we have obtained the 'static 1 Poisson equation for the potential If

p i 9 the charge density (1.2) of a point charge, the solution i 9 the well

known one:

4

THEORETICAL PHYSICS AND ASTROPHYSICS

♦ • |. -v; ks:f where

r^(t)

is the position of the charge at time

A now describes only the transverse field. Coulomb gauged.

The potentials

t.

The vector potential

The gauge (].I0) is called the

A and are here determined apart from a

gauge function x(r*t) which satisfies the condition

V 2x ■ 0 .

We now evaluate the energy of the electromagnetic field,

• - W

* *

o .u ,

We substitute here the expressions for the fields E and

H in the form (1.9);

it is clear that in the case of the Coulomb gauge (1.10) we have 1 aA

Et r =-I 17 * Er ' « tad*-

(K15)

Substituting (1.9) and (1.15) into (1.14) we get

* = H J (Et r +«2)d ,' + 8 i K d3' + i R

r ‘E£>di r.

One shows easily that for a closed system, when the field 'at infinity1 vanishes, the last integral equals zero. is thus the sum of

The total energy of the electromagnetic field

the energies

of

the

transverse

and of the longitudinal

fields. If there are several point charges in the field, the energy of the longitudi­ nal field is simply the energy of the Coulomb interaction between the charges, that is,

\

=

wKdlr I - I

i *j

1J

(K,6)

The self energy of the point charges is infinite and is here, of course, neg­ lected.

It is important to note that the longitudinal part of the electromag­

netic field is not quantized.

Only the transverse field is quantized (see

Heitler 1947 and the discussion later in this chapter). 'f'lt is obvious that it is possible to introduce the Coulomb gauge — and to use Eqs.(l.ll) and (1.12) which are connected with it. It is therefore rather interesting that as long as thirty years ago when classical electrodynamics was already fully ’grown-up' the Hamiltonian method was usually developed on the basis of Eq.(1.8), although it led to complications (see, for instance, the first edition of Heitler*s book — the best one for its time — which ap­ peared in 1936); another indication of the unpopularity of the Coulomb gauge in the past can be seen from the fact that a paper devoted to that gauge was published as late as 1939 (Ginzburg, 1939c).

5

THE HAMILTONIAN APPROACH TO ELECTRODYNAMICS As the energy of the field of a point charge is infinite, one is often led to at least in an intermediate stage — that the charge is 1smeared out*

assume over

a region of radius

rQ .

in that case

^ e z/rQ .

The electrostatic

(classical) electron radius, defined by the relation r = e 2/mc2, where m

e

and

re ■ 2.8 * 10 ^ cm.

are the observed electron charge and mass, is equal to

We shall not be concerned here with the problems connected with the electro­ magnetic mass of the electron — or of other particles — or whether it is a point charge, and so on. Proceeding now along the path which will lead to the Hamiltonian formalism for electrodynamics we expand the vector potential

A

of the transverse electro­

magnetic field in a Fourier series,

A(r,t) =

Y j q^(t) 7 ^ c

The numerical coefficient \ f 4tt c

exp{i(k^«r )| .

is a normalization factor.

is a unit vector, that is, e^ = 1;

vector

(1.17)

The polarization

for the sake of simplicity we

assume here and henceforth that the vectors

are real.

In order that we

may apply the expansion (1.17) we must imagine the electromagnetic field to be enclosed in a large, cubic 'box'.

One can verify that the dimensions of this

'box* do not enter in any of the expressions for physically observable quanti­ ties.

We shall therefore everywhere put the size of this fboxf to be equal to

unity: L = L 3 = 1. The vector potential expansion (1.17) that is, the polarization

A

is a real quantity; q_^ = q* .

vector

it therefore follows from the

As the field is transverse,

of the harmonic with number

is at right angles to the wavevector

(e^»k^)=0,

that

X of the potential

of this harmonic.

To each direction

of k. there correspond two vectors e, . We should therefore introduce one X r A more index which can take two values, or, in other words, distinguish the vec­ tors e,

A1

and e,

.

A2

We shall not do this in what follows, in order to simplify

the notation, but we shall sum over the polarizations, when necessary, in the final expressions; we shall assume in those cases that

=

We can realize also a different expansion of the vector potential, namely,

A'

X,i

where the index

i

can take on only the values

I and

2 , while

THEORETICAL PHYSICS AND ASTROPHYSICS

6 - /§ir c e x cos(kx»r) , One sees easily that the functions

A^

- /BFce^eiiUk^-r) .

and

(1.19)

in which we have expanded in

(1.18), are mutually orthogonal, that is, .T(Au

-

V

d3r “ 4* cZ V i j

-

(l-20)

where the integral is over the volume of the 'box1. We assume the field to be enclosed in a 'box1 with specularly reflecting walla; the components of the wavevector quantity

n^ , n

, n^

2tt/L, where L

are here integers;

the directions of

.

the summation in (1.18) is over a hemisphere of

Apparently, what we have just said contradicts the ear­

lier statement that the size L equal to tion:

1.

must thus be integral multiples of the

is the linear size of the 'box', that is

of the 'box* is unimportant and can be put

However, one can easily check that there is here no contradic­

for sufficiently large values of L

this quantity does not occur in the

final results. It is clear from (1.18) that the transverse electromagnetic field is completely determined, if we give the set of quantities form a denumerable infinite set.

q^(t).

The quantities

q^(t)

The field is thus through (1.8) represented

as a system of an infinite, though denumerable, number of degrees of freedom. Let us see how one can express the energy of the electromagnetic field in terms of the quantities

q^(t)

which we can properly call the field coordinates.

We are interested in the energy *tr = If A

J

E

tr

+H

(1 .2 1 )

8tt

is given in the form (1.18), we can use Eqs.(1.3) and (1.15) to deter­

mine the fields Etr and H ; integral (1.21).

we can square them and substitute them into the

We then get *tr 4

L

•

(,-22)

a 7i

We have introduced here the notation PAi = ^Ai where the dot indicates differentiation with respect to the time. Eq.(I.22) we used the orthogonality condition (1.20).

(1.23) In deriving

the h a m i l i t o n i a n a p p r o a c h

to

7

electrodynamics

Each term in the sum (1.22) is the energy of a classical oscillator of frequency Therefore, (1.22) is the sum of the energies of separate oscillators, which we call the field oscillators. If all

^Xi^**^

0-18) are known, we can determine the energy of the trans­

verse electromagnetic field. of the

The problem is thus reduced to the determination

q^.(t).

To find the equations for the

q ^(t) we substitute the expansion (1.18) into

Eq»(l.l2) for the transverse vector potential. resulting equation by

the following equations for the

h i + uxq\ i

Multiplying both sides of the

and integrating over the volume of the ’box' we get q . .(t):

Ai

cos(k^*r(t)) f ( v -A xi(r (t))) = e /§i(ex*v (O) { sin(k^T (t))

(1.24)

This is the equation for an oscillator when there is an exciting force present; for

i=l

we choose

We derived Eq.(1.24)

cos(k^-r) and for

sin(k^*r).

in the assumption that there was a single point charge

(electron with charge the field.

i = 2:

e ;

see ( 1.2)) moving with a velocity

v(t) present in

The generalization to the case of several charges is obvious.

All relations considered here can be written completely analogously to the Hamiltonian equations of classical mechanics:

3 M (p.q) 3q

. _ 3 w (p»q) * q 3p

(1.25)

where

#(p,q)

is the Hamiltonian function of the mechanical system, and

and p

are, respectively, the generalized coordinates and momenta.

Our problem now is to find such a function tions of motion such as (1.25) from it. q^i

N(p^.,q^)

q

that we can obtain equa­

It is clear that Eqs.(1.24) for the

for the case of a free field - a field without charges - that is, the

equations

h i

+ ^ x h i ’ 0 ’

(1-26)

can be written in Hamiltonian form, if

*where

M

» „ - T

I

A.i

(1.33)

is the operator of the particle momentum which satisfies the commuta­

tion relations (r = {xj}, j = 1 , 2 , 3 ;

note that there is no summation over

j

here) p.x.-x.p. =- ill, j j r j

(1.34)

and is given by the expression (1.35)

P = - ihV . If the particle is in an electromagnetic field, p - (eA/c) and correspondingly

p

p

in (1.32) is replaced by

in (1.33) by

p A = - i h V - ~ A c c

.

The state of the system is determined by the wavefunction

T(r,t); the change

of the wavefunction with time is described by the Schrodinger equation

ihf- = iw •

d-36)

The wavefunctions of stationary states have the form y (r ,t) = exp(-iE t/fc.) ^ (r) 1 n n n where the

(r) are independent of

state, or it" quantum number). of a stationary state,

that is,

t

(n

(1.37)

is the number of the stationary

The square of the modulus of the wavefunction the

probability

that

a particle will be ob­

served in a given point in space, is independent of the time.

Substituting

+We are, of course, talking about the particles, as electrodynamics in vacuo (or from the quantum theoretical viewpoint the theory of spin-I particles with zero rest mass) is always a relativistic theory.

,0

THEORETICAL PHYSICS AND ASTROPHYSICS

(1.37) into (1.36) and dividing by

exp(-iE„t/h)

we have

W 'l'n (r) “ En'l'n(r) '

0.38)

Let ua consider a one-dimensional harmonic oscillator with unit mass.

It is

well known that the Hamiltonian of such an oscillator has the form M = ip2 where

q=q

-

0.39)

is the coordinate and o>0 the angular frequency of the oscillator.

The energy of the n th

stationary state is equal to n = 0 , 1 , 2 ....

(1.39a)

^ ( 9 ) ” cn exp (- qz/2q*) Hn (q/qo) ,

(1.39b)

En =

(n+ |)

;

and the wavefunction has the form

where factor.

q 0 = / h / u 0 , Hn (x )

is a Hermite polynomial, and

C

n

a normalization

In particular. = --- 7 7 exp(-qz/ 2qj) . (q0^U)4

(1.39c)

The matrix elements of the coordinate, q . , and of the momentum, p , , corres* rnn' 9 ponding to transitions from a state with quantum number n to a state with quantum number n

n' J n ± 1, and when n'* n ± 1

vanish unless

/ h(n + I) ^ V 2u #

J *nn (t)|a -0, we can use a similar

For instance, the matrix element is

in second approximation given by /(2)

Y

*nn 'W n fk

(1.74)

Equation (1.73) determines the probability for a transition to only a single final state with energy E

.

We are usually interested in the transition to

any of all possible states, that is, m

the integral (1.75)

J|b (t)|2 p(E )dE J1n 1 no i In this equation

p(E

)dE

is the number of final states —

no no sumed to be 'densely1 distributed - in the energy range from

As

which are as to

E ^ + dE^

t tends to infinity, the integral (1.75) equals (see Eq.(1.84) or for de­

tails, Heitler's book (1947)),

¥ i » ' i u ko

(1.76)

•

and the probability for a transition per unit time is thus given by the formula

W = iJ|bn(t)|2 P(Eno)dEno

(1.77)

|U'|2 p(Efc0).

Note that the transition occurs only if there are states E^ q which are arbitrarily close to Efco ;

this was reflected in (1.76).

It is clear from what

, one nn and (1.40) and understand by q

we have said earlier that when one evaluates the matrix elements must use Eqs.(1.65), and also (1.18), (1.19), the operators q ^

(for details see Heitler's book (1947)).

Using perturbation theory in such a simple or in a somewhat more complicated form enables us to find the answers to a whole set of problems in radiation theory (Heitler, 1947;

Berestetskii, Lifshitz and Pitaevskii,

1971).

However,

a wider application of perturbation theory encounters considerable difficulties which is formally reflected in the appearance of divergent (infinite) expres­ sions.

The appearance of divergent expressions is connected with the assump­

tion that the electron is a point particle, with the fact that the field has an infinite number of degrees of freedom, and so on.

Some of those difficul­

ties are not caused by the quantization and are classical in nature.

It is

sufficient to remember that the electrostatic energy of a point charge is infinite.

Even in classical electrodynamics one had learned to avoid such

difficulties.

In particular, one used for this the mass 'renormalization'

19

Tgg HAMILTONIAN APPROACH TO ELECTRODYNAMICS method^*

In quantum electrodynamics one also renormalizes the charge of the

particle and the situation altogether becomes more complicated.

The study of

the corresponding field of problems was for a long time in the centre of the attention of theoretical physics.

As a result a great deal of progress was

made and the infinities in quantum electrodynamics are practically made h a r m ­ less’.

A framework was developed in which one is able to find the answers to

problems which may arise and, in particular, in which one can take extremely minute radiative effects into account (Heitler,

1947;

Berestetskii, Lifshitz

and Pitaevskii, 1971). The present text is not concerned with any of these problems, although the material discussed a moment ago may, we hope, be useful for understanding the physical basis of quantum electrodynamics.

It was only important for us in

the plan for our further exposition to formulate the Hamiltonian method in classical electrodynamics and to get acquainted with the most elementary as­ pects of quantum electrodynamics. It is interesting that the Hamiltonian method has in the past hardly ever been applied in classical electrodynamics; changed to quantum electrodynamics. ’feedback’ afterwards.

the method became popular only when one However, as so often happens, there was a

To be precise, it became clear that the Hamiltonian

method is very convenient also for a number of classical problems, especially when there is a medium present (see Chapters 6 and 7).

Recently, when many

problems turned out to be already solved, when there appeared new and more complicated problems and, when moreover, a number of powerful mathematical methods, such as the diagram technique or the Green function method, were de­ veloped and started to be widely applied, the Hamiltonian method dropped out of sight both in the quantum and in the classical radiation theory.

We are,

however, convinced that the Hamiltonian method nevertheless retains the advan­ tage of clarity, simplicity, and rather large universality which makes its exposition and use very expedient, at least, for pedagogical purposes. As an illustration we use the Hamiltonian formalism to discuss the problem of the radiation by an oscillator — a harmonically oscillating charge. mine the field we must find the quantities

q^

To deter­

in the expansion (1.18), while

^ As far as we known the term 'mass renormalization* itself only appeared when the corresponding operations were carried out in quantum electrodynamics. This led to the assumption that the renormalization method was a product of quantum theory but this is, to say the least, incorrect (see Chapter 2 and Kramers 1944).

THEORETICAL PHYSICS AND ASTROPHYSICS 20 the equations of motion for these quantities have the fora (1.24)

J cos(k^ • r) , + u*qu

(e, 'V (t) ) 1 sin(k^ • r) ,

-e^F

’Ai where

r -r(t)

is the position vector of the emitting charge

e;

0.24) in the case

of an oscillator we have

r (t) =v = v cos a) t = a ti) cos oj t

r ( t ) - a o sinu>ot The argument

0

0

0 0

0.78)

0

(k^r(t)) in (1.24) is small, if o

0.79)

277 A0

where

is the wavelength of the emitted radiation.

Let us accept condition

(1.79), that is, we shall assume that the amplitude of the oscillations of the charge is much smaller than the wavelength of the radiation; valid for a non-relativistic oscillator, as the velocity A 0 -2ttc/w0.

this is always

v 0 =ajQa 0 « c, while

In that case (k^* r) in (1.24) is much less than unity and we are

justified in putting

cos(k^-r) = l,

sin(k^* r) * 0.

Therefore

■ 0 and for

we get the equation = e(e^-v0) ^8?cosa)0t .

qAJ +

(1.80)

= 0, when

rr U O

The solution of this equation which satisfies the boundary condition

bA

r

COS OJ^t]

V 1 CM2 As

a

result

of

all these

simple

calculations

we

get

an

expression for the

energy emitted by the oscillator per unit time into a solid angle

dH

tr dt

a oaio

tr

sin 20 dft , 8

where

dft:

0.85)

ttc 3

0 is the angle between the direction of the oscillations a 0 and the

wavevector

k Q , where

k 0 = a)0/c.

The emission which we have just determined — that is, that part of the field which increases proportional to the time

t — arises when the frequency of the

’force 1 which occurs on the right-hand side of Eq.(1.24) i9 equal to the eigenfrequency of the field oscillator

(D^ = ck^.

In this respect the harmonically

oscillating charge is completely typical even though in the dipole approxima­ tion (condition ( 1.79)) which we considered there is emission at only one fre­ quency, o)0 .

We note that in the quantum theory of radiation the situation is

completely analogous in the perturbation theory framework (compare (1.82) and (1.73);

for details see Heitler,

It is convenient

at

this stage

1947).

to

elucidate

a

few important aspects which

usually are kept out of sight by discussing the somewhat rhetorical question: can a uniformly moving electron radiate?

theoretical physics

and astrophysics

22 The standard, one might say automatic, answer to that question x. negatxve. I„ actual fact, however, one can make many reservations;

many of them are non­

trivial and have far from always been taken into consideration- whxch has led to paradoxes and mistakes. Firstly, one must be precise about the frame of reference in which the elec­ tron - of course, it need not be an electron, but can be any charge, which we call 'electron' for the sake of convenience - moves uniformly, that is, with a constant velocity v .

Usually, unless

a statement to the contrary is made

explicitly, one is dealing with motion in inertial frames of reference.

The

initial field equations were written down just in such frames and we were deal­ ing only with such frames of reference.

It is clear that if the electron moves

uniformly in a non-inertial frame of reference, it is accelerated relatively to an inertial system and will radiate. Secondly, one considers

uniform motion

in vacuo

and

not

in

a medium.

An

electron moving uniformly in a medium can emit both Cherenkov and transition radiation (see Chapters 6 and 7). Thirdly, one assumes that the electron velocity v < c

(the velocity of light

in vacuo). Often this condition is considered to be more or less trivial, but that is not the case.

The requirement of relativistic invariance does not at

all lead to the condition v < c

and, in particular, Eqs.(1.1) are completely

valid (and relativistically invariant) also when v > c .

It is true that a

particle of rest mass m can not be accelerated to a velocity v ^ c ,

as can

readily be seen from the expression for the particle energy C = mc2/(I - v 2/c2)^. However, this does not exclude the possibility to consider particles (tachyons) which always move with a velocity v > c mc 2/(v2/c2 - 1)®,

and an energy £ = imc2/(l - v 2/c2)^ -

The difficulties which in reality arise when one considers

motions with velocities

v>c

are connected with the possible violations of the

causality principle: it was just this fact and not the violation of relativis­ tic invariance which led to the requirement that v < c Pauli, 1958).

(see Einstein, 1907;

It is therefore probable that tachyons, which recently have been

discussed extensively in the physical literature, can not exist.

However,

sources of radiation (even if not separate particles) moving with velocities v >c

all the same do exist.

We shall discuss them in Chapter 8 .

After this discussion we shall state the problem more precisely: can an elec­ tron moving in vacuo in an inertial frame of reference with a constant velocity v < c radiate?

One can give at least four arguments that an electron under such

conditions does not radiate.

the

23

HAMILTONIAN APPROACH TO ELECTRODYNAMICS

The first argument, and in a certain sense the most consistent one, is connec— ted with the solution of the field Eqs.(l.l) for the case when v « constant. Such a solution (see, e.g., Heitler,

1947;

Pauli,

1938;

Landau and Lifshitz,

1975 and Chapter 3) indicates that the radiation field, that is, the field which decreases as

1/R and produces an energy flux at infinity, does not ap­

pear in the case under discussion;

at the same time it also becomes clear

that radiation would appear if v > c • The second argument does not require any calculations.

We can change to a

frame of reference in which the electron is at rest, which can always be done when v = constant, v < c .

In such a frame there is clearly no radiation:

electron is all the time at rest .

the

However, if we change from one inertial

frame to another no radiation can appear, and thus there is none either for v = constant.

A well known weakness of this argument is connected with the

fact that it can apparently also be applied to the case v > c that there is also no radiation in that case. must in the case

v>c

emit Cherenkov radiation even in vacuo (this is, inci­

dentally, one of the difficulties in the theory of tachyons; 8).

The solution of

and thus ‘prove*

On the other hand, a charge

see also Chapter

the ensuing paradox is that it is impossibletorealize

frame of reference in which the electron is at rest

a

when v > c .

The third argument is connected with the use of the energy and momentum conser­ vation laws. formulation £«

The simplest way, although not necessary, is to use a quantum for

(m2c l* + c2p2)^

the discussion.

If we

consider

a particle

with

energy

and momentum p we can verify that the energy and momentum

conservation laws do not allow such a particle to emit a photon with energy hu) and momentum h k , k = oj/c.

In fact we shall use such an argument for discussing

in Chapter 7 the conditions for radiation in a medium.

Incidentally, in the

framework of such an

approach radiation in vacuo is only impossiblewhen

for tachyons with an

energy

£=

(-m2c** + c 2p 2)^

v < c;

the velocity is

^ In fact, it was presupposed here that the electron will be at rest at all times in some frame of reference: we need therefore give a more detailed discussion, such as, for instance, the following one. Let us assume that the electron (a free charged particle) moves with a velocity v®constant, v < c , and does not radiate. We can then show that such a solution is com­ patible with the field equations when we change over to an inertial frame of reference in which the electron is at rest, its field is the electro­ static field, and in which there is no radiation. In fact, this argument differs from the preceding one only in that the solution of the field equations for a particle at rest are simpler than for a moving particle and may be assumed to be well known.

THEORETICAL PHYSICS AND ASTROPHYSICS

24

5-2L.

v 3p

/c V -

> C b

V

and the conservation law9 do not prevent the emission of photons by a uniformly moving particle. The fourth argument is connected with the use of the Hamiltonian method.

r (t) =vt

a uniformly moving electron

For

and (k^- r ) = (k^ *v)t, that is, the fre­

quency (k^*v) occurs on the right-hand side of the equations of motion for the field oscillators (see (1.24)). field oscillator in vacuo is v y^x = yQi cos fit , y z-axis).

The

= y ^ sin fit, with the

radiation

power

p = — (ii)2 = ----— 3c3 3c3 Under those circumstances just equal to (2.23).

ft = 2[il^A|ij] / 3c3

L

The term

then (2.23)

and the work done, (ft* S ) , is

in (2.22) is conservative and, clearly,

. L = Mm

is

4

that is, ft is some angular momentum of electromagnetic provenance. It m follows from the derivation of Eq,(2.2l), in complete analogy with the case of the charge

(see (2.14)), that we must assume in Eq.(2.22) that the rotational

frequency of the magnetic moment

fi «: u

I

(4.16)

'm e 2 /

The radiation is then, as we mentioned earlier and as is clear from (4.15), concentrated within angles o)^ u)(0) 'v (jjQ (£/mc2)2 .

0 ^ m c 2/ £

and

has

a characteristic

frequency

The total emitted energy is proportional to the factor

e**£2/m‘*= e**cl*/m2(l - v 2/c2), that is, it depends for a given charge both on the total energy

£

and on the rest mass

m - or on m

and the velocity

same time the acceleration depends merely on £ (see (4.9)).

v .

At the

THEORETICAL PHYSICS AND ASTROPHYSlCf

60

We shall in Chapter 5 also discuss the radiation by a charge which moves with relativistic velocity in a magnetic field.

Now, however, we shall discuss the

radiation force acting on relativistic particles. We have already

in

Chapter 3 derived

the

including the radiation force (see (3.11),

relativistic (3.12),

equation

of

motion

(3.19) and (3.20)).

It is,

however, convenient, as in the non-relativistic case, to consider at once the approximate nature of the formula for the radiation force, expressing it in terms of the field strength.

This was done by Landau and Lifzhitz (1975, §76),

but it is convenient to do it again here, starting from the equation of motion du'*’

- 'ie _ik

“ -dT = I F

.

\

-

i g5

-•

..i 2e2 (/ jd22u ~3c~ 3c \'"ds^ ds2

i

* ’

8 *

.

„ _ . i k d “k')

U

(4.17)

U 1-r) *

To a first approximation

“ Substituting

duL e _ik -dT= cF \ d 2uL/ds2

d 2u I ;

ds2

e —

I

8Flk — T V dx

. e 2 „ik „ + Z ^ T F fu

from (A. 18) into the expression for

,, ,os

I u

•

(4*18)

g1 we can write

(4.17) in the form me

du1 e _ik — = — F u. ds C K 2e 3 J8Flk f3F

I

e

£ V >

i 1

Fkiu ‘k£- +^ mcz VFw k£ u A / V,F "

V 'm2 u J

ii.

3mca 1l 3x£ T T -k” V ~ mcz ^ F ~

k

e

/

(4.19) We can in the ultra-relativistic case everywhere except in expressions such as y = { l - v 2/c2} ^ one and as v

put

v = c.

The main radiative term in (A. 19) is thus the last

c we can write the three-dimensional equation of motion in the

form_______________________________________________________________________________ ^ For the sake of convenience we remind ourselves here of the connection between any four-dimensional force g L and the corresponding threedimensional force We have (f -v)

,

gL - (g°.g) = */( 1 - v 2/c2)

f c/(l - v 2/c2)

where dp1 du1 i Tds T * ®c -T-ds = g

. *

dp ■ f dt

tt

)

RADIATION

61

of a m o v i n g particle

/(7- v 2/c2) where we have in the last expression for f tion of v

chosen the x-axis along the direc­

in order to write down explicitly the components of the field.

fact that for

y » 1

the force f

is parallel to v

The

is also clear from Eq.

(3.12) but that equation is, however, less convenient than (A.20). We now want to make a few remarks which are important and are sometimes not taken into consideration.

In the relativistic case the radiation is mainly in

the forward direction, in the direction of the velocity.

The recoil, or reac­

tion, must thus be in the backward direction, as directed radiation carries momentum.

This is also demonstrated by the fact that the force f

parallel to v .

is anti­

It seems that from this it also follows that the radiation

reaction will lead to a decrease of the velocity components in all directions. Such a conclusion is, however, in general false. Let us as an example consider the motion in a constant and uniform magnetic field

H 0.

As we shall in what follows repeatedly be interested in this case

we shall dwell upon it in somewhat greater detail for the case of a particle of charge Ze and mass M . The equation of motion then has the form (we neglect for the time being the reaction of the radiation) (4.21)

One can easily integrate this equation and it then becomes clear that the particle moves along a spiral: its velocity is v= V|j + v^, where

= constant

t For application of Eq.(A.21) this proviso is, strictly speaking, insuffi­ cient and we must also note that we shall also assume the charge Ze and the mass M to be constant. Of course, such an assumption is practically always implied, but it is not necessarily true, as for ions the charge and “ass are not constant quantities when we take into account nuclear fission and/or the *stripping off* of atomic electrons.

THEORETICAL PHYSICS AND ASTROPHYSICS

62 is

the

constant velocity along

which we take the

z-axis),

the

field (along

while

is the velo­

city component in the xy-plane, that is, at right angles to H 0

cos oj£t , v ±

v . __ » ±,x vn = v cos

* “H

Me

Here

X

x ,

we have

m - v^ sin oijjt , V|| • constant.

V± = v sin X » v 2 = v 2 + v|

(4.22)

v sin X

Me “h ~ t

rH =

(I)

is the angle between v

and

H 0, and

r^

the radius of the circle which the projection of

Fig. 4.5 The quantities vif v,, and x xy-plane,

(Fig.4.5);

the radius vector of the particle describes in the

we shall, as usual, call this radius the radius of curvature, but it

should not be confused with r*

, the radius of curvature of the trajectory in

space of the particle which is equal to r* = H

rH+ > ? * m c 2/ £ .

X)

In fact, however, an expres­

has a general character and its appearance is

not necessarily linked with the assumption about the 'needle1 nature

of

the

radiation or with the possibility to divide it into separate pulses (for de­ tails see Ginzburg, Sazonov, and Syrovatskii,

1968; Ginzburg and Syrovatskii,

1969). The radiation spectrum of an ultra-relativistic electron consists thus in the wave zone of harmonics of the frequency (5.11) sin2X By itself this fact is not very important when we

bear

in mind

that

in all

cases of interest to us the harmonics are not resolved and we are dealing with a continuous spectrum.

However, the change in the interval between the pulses

shows up not only in the spectrum,

but

also

in

all

characteristics

of

the

radiation field, in particular in its intensity registered at the point of observation.

Indeed, let the electron lose through radiation in each revolu­

tion (over a time

T = 2tt/oj*) an energy

as is clear from (4.37) and (4.39).

A G = ftT, where

ft = (2e**H2/3m2c 3)(G/mc2)

By virtue of what we have said earlier it

is then clear that this energy reaches an 'observer' which is positioned on a fixed sphere at a distance

R

from the electron over a time

T r and, hence,

the average observed power of the radiation (total energy flux) will be equal to AG _ ftT _ T

T

ft

(5.12)

sin2X

At first sight it might look as if we have here a contradiction of the energy conservation law.

The electron loses an energy ft per unit time.

The whole

of this energy changes to radiation and must, surely, be equal to the total flux of radiation through the sphere under consideration. as follows:

One often proceeds

one evaluates the radiative losses suffered by the particle and

equates them to the total flux of radiation.

In a stationary case and for an

emitter with a fixed centre of mass one can, indeed, proceed in this way.

Hoi

ever, in general, as we reminded ourselves in Chapter 3, the work performed by

76

THEORETICAL PHYSICS AND ASTROPHYSICS

the emitter per unit time (the power of the losses ft) is equal to the total flux through a surface plus the change in the field energy (d/dt) / [ (E2 ♦ H 2)/8ir]d,r in the volume enclosed in that surface.

In the case in which we are interested

the region of space occupied by the radiation and lying between the moving electron and the surface which is fixed in space on which the observation is performed decreases all the time.

At the same time the energy enclosed in

that region also decreases and hence the power of the observed radiation P is larger than the power of the losses the power of the losses

ft.

All the same, in a number of papers

ft has been used when going over to spectral quanti­

ties for the intensity.

Such an approach can of course not

lead

to a

correct expression for the intensity of the radiation fixed at some non-moving surface, if the motion of the emitter is taken into account.

However, if the

radiating particles are in a fixed volume (e.g., the shell of a supernova) or, to be more precise, if the distribution function of the radiating particles does not change with time, the intensity of the radiation of the assembly of particles is the same as the spectral power of the losses.

This conclusion

is evident from the energy conservation law and is, of course, confirmed by direct calculations (see Ginzburg, Sazonov, and Syrovatskii, 1966;

Ginzburg

and Syrovatskii, 1969, and below). We have assumed

that

the

fact

that

the whole of

this essentially entirely

elementary problem had not been elucidated for such a long time and had led to the use of formulae which were either not completely or not always correct^*, justifies such a detailed exposition and, in fact, a repeated return to the appropriate remarks. The elementary considerations and formulae given here provide us with a picture which is qualitatively completely clear and they also enable us to give esti­ mates for the characteristic features, such as the intensity, spectrum, and polarization, of the synchrotron radiation for concrete cases.

For obtaining

quantitative formulae we need, on the other hand, rather cumbersome calcula­ tions.

These calculations are performed on the basis of the well known formu­

lae for the retarded potentials and can be found,

^

for instance,

in the paper

This refers, in particular, to several formulae given by Ginzburg and Syrovatskii (1964a,1966a). Fortunately, in that case these formulae were applied only to cases (or in conditions) where the corresponding differences between the expressions for the intensity can be neglected (see below and Ginzburg, Sazonov, and Syrovatskii, 1968; and Ginzburg and Syrovatskii, 1969).

SYNCHROTRON RADIATION

77

by Ginzburg, Sazonov, and Syrovatakii (196B; see also Ginzburg and Syrovatakii 1969);

for the case of circular motion, that ia, when

s i n X * I, a number of

calculations can be found in 575 of Landau and Lifehitz’a book (1975)*.

We

shall therefore restrict outselves here to giving some final results and also to discussing their application in astrophysics.

Bearing in mind auch applica­

tions we digress somewhat from the nature of exposition which we have adopted on

the

whole

in

the

present

book;

that is,

in

agreement with the work by

Ginzburg and Syrovatskii (1964a,1966a) we shall give a rather large number of formulae which are, in fact, auxiliary in nature or convenient for calculations. Of course, readers who are not interested in applications of the synchrotron theory can omit these details.

The next part of the present chapter (up to and

including Eq.(5.66)) is thus in a well understood sense auxiliary in nature. We can expand the field of the synchrotron emission of a particle in a Fourier series in the overtones of the frequency has a period

T'.

In other words,

the

^

= 2tt/T' = (i)*/sin2X , as the motion

radiation field

at

large distances

from the charge can be written in the form

I

Re

n= 1

ri(i)(--til,w= En eXp L n\ c 'J n

(i)* = -- — , 1 1 sin2X

n« 0,1,2,3

... (5.13)

In the ultra-relativistic case considered we have for an electron, up to terms £ 3 = ( m c 2/ e )3

of order

2 e oi E

— — T T - {(C2 + l/'2) k j (g ) 1, +i ■ as before

S < 2 >

V 3e H

.llii (J-Y

J

4 7T me

(5.15a)

^mc2 •

We show in Fig.5.4 the angular distribution of the radiation fluxes

.(2 )

p

0)

and

We have chosen as scale unit along the vertical axis the coefficient

(3 e 3 H/4 7T2 R2 me 2 £) (v/vc) 2 for

> P ^ ^ .

From the above it is clear that we should use Eqs.(5.2l) to (5.23) rather than Eqs.(5.26) and (5.27) for synchrotron radiation sources which move with rela­ tivistic velocities. velocity

Vr

In

that

case,

if

the source as

a whole

moves

with

a

along the line of sight in the direction of the 'observer', the

intensity of the radiation is increased by a factor

(I - V T/ c ) ~ l

as compared

to a fixed source with the same electron distribution function (see Ginzburg, Sazonov, and Syrovatskii, 1968; Ginzburg and Syrovatskii,

1969).

Recently one

has begun to ascertain that synchrotron radiation sources under cosmic condi­ tions may have relativistic velocities.

For instance, in the case of explosions

of galactic nuclei leading to the formation of radio-galaxies, the emitting radio-clouds' probably move in a number of cases with velocities comparable

THEORETICAL PHYSICS AND ASTROPHYSICS

62 to the velocity of light*

It is possible that shells, jets, and outbursts

moving with relativistic velocities also exist in the case of other objects, in the first place in quasars.

For that reason there is undoubtedly interest

not only in stationary, but also in non-stationary (relativistic) synchrotron, and

in general

Syrovatskii,

magneto-breras

radiation sources

(Ginzburg,

Sazonov,

and

1968; Ginzburg and Syrovatskii, 1969; Rees, 1967; Ryle and Longair,

1967; Ozernoy and Sazonov,

1969).

In what follows we shall, however, concen­

trate our attention on stationary (or more precisely, quasi-stationary) sources for which we can use Eqs.(5.26) and (5.27) in the range (5.1). Before going any further, we shall consider the quantities which characterize radiation as this problem is usually not touched upon in electrodynamics text­ books . Any radiation flux is characterized not only by its frequency dependence, but also, in general, by four independent parameters, for instance, the position of the principal axes of the polarization ellipse, the intensities along the two principal directions, and the sense of rotation of the electric vector. It is, however, convenient to use for these parameters the Stokes parameters (see, for instance, Ginzburg and Syrovatskii,

1966a; Gardner and Whiteoak,

1966; Chandrasekhar,

For the radiation by a single

1960; Shurcliff,

1962).

Ie 9 Qe * ue » and

particle these parameters

ve

can be expressed in terms of

the radiation flux densities with respect to the two main directions of polari­ zation

and

p ^ ^ , and also in terms of

tan 8 , the ratio of the minor

to the major axis of the ellipse of the oscillations of the electric vector (see (5.17)), and the angle

X between some arbitrary fixed direction in the

plane of the figure and the major axis of this ellipse (that is, the direction at right angles to the projection of

H onto the plane of the figure)^.

The

corresponding equations are the following ones: >(1) ♦ o

( p < » - Py2))e°s 2 X , (5.28)

0) * "

Like flux

(P v

•nd p.(2 ) p,( I ) v density per unit

PJ2)) s i n 2 X

,

\

(p^1* - Py2*)tan 2 6 .

the Stokes parameters have the dimensions of an energy frequency

range;

the

index

e

indicates

that

these

parameters refer to the radiation of a single electron.

The angle X is reckoned clockwise and is clearly defined in the interval Oi The notation x i® introduced here in order not to confuse the angle X with the angle X between V and H .

83

SYNCHROTRON RADIATION The Stokes parameters possess two impor­ tant advantages:

they can be

directly

measured and they are additive for inde­ pendent (incoherent) that is, phases

radiation over

(Ginzburg

radiation

fluxes with random

which

and

fluxes,

one

can

Syrovatskii,

average

1966a).

Experimentally the Stokes parameters can be determined

by the usual methods for Fig.5.5. Definition of the Stokes parameters.

studying polarized radiation (Shurcliff 1962), a phase

namely,

by means of introducing

difference

the projections

of

£

between

one

We introduce an additional retard­ ing phase £ to the direction s2 relative to the oscillations in the perpendicular direction Sj, The angle 6 determines the plane of the position of the analyzer. The measured radiation flux is directed towards the observer.

of

the electric vector

oscillations in the wave (for instance, along the direction

Sj

the other projection,

in Fig.5.5) and the

perpendicular direction (s2

one onto the in Fig.5.5).

The subsequent analysis reduces to establishing the dependence of the intensity of the resulting radiation on the position of the analyzer

which

projection of the oscillations onto some arbitrary direction s

selects the

(see Fig.5.5).

If we denote the angle in the plane of the figure between s1 and s

by

intensity of the radiation which leaves

following

function of

£

and

6

the

analyzer

will

(see, for instance, Chandrasekhar,

= J {*e + Qe cos 2 6 + (Ue cos e By choosing the retarding phases

£

be

the

6 , the

1960):

sin n) sin 2 6}

.

and the position of the analyzer

(5.29) 6 appro­

priately we can measure the values of all the Stokes parameters. We note that the first Stokes parameter

Ig

determines the total radiative

flux density (or the intensity in the case of spatially distributed sources; vide infra) while the degree of polarization

II and the angle

X

are given by

the equations

/(Q *

+ V 2)

n - - * _ _ £ - - - - - - - - - - - - - - - - - - (5.30) e and U tan 2 x

e

Qe

(5.31)

THEORETICAL PHYSICS AND ASTROPHYSICS

84 Ve chooee from the two values of the angle

x ( ° ^ X 0, and the

By definition the angle

X

then

characterizes the direction in the plane of the figure in which the intensity of the polarized component is a maximum, and is reckoned clockwise from a chosen direction (in the case considered from the direction no elliptic (circular) polarization

Ve « 0

Sj).

If there is

and

I - I . max min I + I . max mm

n

Let us now consider the emission by a system of independently moving particles. Let

N ( £, R , T ) d C d 3 R d 2ft^

d 3 R = R 2 d R d 2ft

be

from

particles in

system in the direction

Iv S K v . k ) Ie (v,S , R ,

x»^)

k

a

volume element

£, £ + d £ , and velocities

separated electrons

Stokes parameters are additive,

dR

of

d 2ft^ around the direction

considered the emission

over

number

with an energy within the range

within the solid angle

Here

the

T .

As in the conditions

is

incoherent

so

that

the

the intensity of the radiation from such a

of observation is equal to^

= / Ie (v,£, R ,

x, K| (*v> •

As this function is an odd function its integral over all hence, V(v,k) =» 0.

\p vanishes and,

The radiation from a system of electrons thus turns out to

be linearly polarized.

This result is valid up to terms of order

it can easily be understood,

if we remember that the sign of

m c 2/C

and

ip determines the

sense of rotation of the electric vector in the wave emitted by a single elec­ tron.

As the power of the radiation (see (5.21) or (5.26) and (5.27)) is

independent of the sign of

^

while the distribution of the particles over

directions of motion in the limit of very small angles tically constant,

|i|/| ^ m e 2/C

is prac­

the contributions to the radiation in a given direction from

particles with positive and negative

ip will be the same, and the polarization

will be linear. An appreciable elliptic polarization could in the ultra-relativistic case occur only

if

the

velocity

distribution

of

the

electrons

is strongly anisotropic.

For this it is necessary that the distribution changes considerably within the limits of a very small angle of observation.

|i|/|~ m c 2/fc , and moreover, just in the direction

If, moreover, we take into account possible fluctuations in

the directions of the magnetic field,

it is clear that one needs very special

conditions for the realization of such a possibility (pulsars are of particular interest in this respect). We now give expressions for the intensity and polarization of the radiation in some concrete cases which are of particular importance for astronomical appli­ cations • If all electrons have the same energy (mono-energetic spectrum) and the magne­ tic field is uniform,

it follows from (5.35) that the intensity is equal to

SYNCHROTRON RADIATION

1.0 0 1 where

87

= me*

J

N e (k) H s i n X - f K s(n) dn = N (k) p(v) , vc V /Vc a

Ne (k) =*/ N e (R , k ) d R

(5.37)

is the number of electrons along the line of flight

with velocities in the direction towards the observer, per unit solid angle. The degree of polarization can be seen from (5.30) and (5.36) to equal

X

K*(V/V )

, when

2

---- a____ £__ OD

Vc (5.38)

,1

I Ki(n) an v/vc 3

V ^

when

3 vr

V »

Vc

As in the approximation considered the integration over the angular distribu­ tion of the electrons is equivalent to an integration of the radiative power of a single electron over all directions, Eq.(5.37) differs only by the factor Ne(k)

from the spectral distribution of the power of the total radiation (in

all directions) of a single electron: 0D

p(v) -

1/3

e3Hs?inX

mc F (^)

=

/

V^C/

\)/v

c

K4 (n) dn

J

Ks (n)dn

v/v_

= /j F(V ' (5‘39) H1

/V

me2

^

We show in Fig.5.6 the function F(x) = OD x / Ks(ri)dri which reflects the specx ? tral distribution of the emitted power,

3

0.92 -

and its values together with the values of the function

Fp(x ) = x K | ( x )

are

given in one of the appendixes of the book by Ginzburg and Syrovatskii (1964a)

0 0.29

4 (it is clear from (5.38) that the polari­

Fig.5.6. Spectral distribution of the power of the total radiation (in all directions) of a charged particle moving in a magnetic field (see (5.39)).

zation

n = Fp (x)/F(x)).

the maximum

in

the

We

note

spectrum of

that the

synchrotron emission of a single elec­ tron occurs at the frequency

eH

vm ~ °-29 VC = 0,07

- '-3>< '°6 Hi (-&7 ) ‘ (5.40a)

= 1.8 x I018 H J £ (erg)) 2 The frequency v

is here in Hz .

4 . 6 x IQ"6

(e(eV))2 .

THEORETICAL PHYSICS AND ASTROPHYSICS

88

For the maximum frequency (5.40a) the spectral density of the power of the total radiation of a single electron is equal to e 3H p = p(v m m

= 0.29

vr) w 1.6 ---r- = 2. 16 x I0” 22 H erg s ^ H z " 1 L

mc2

(5.40b)

l

If we forget about the numerical coefficient,

that is, look only at order of

magnitude estimates we can easily obtain this last relation as follows.

It

is clear from qualitative considerations or from Fig.5.6 that the width of the radiation spectrum of an electron is

Av'v V c ^ ( e H ^ / m c K C / m e 2) 2 .

The total

power of the synchrotron radiation,

taking into account the above remarks, may

be put equal to the losses

It is clear that the average spectral den­

(5.24).

sity of the power of the radiation is 7(7) ~

-■ £

12 / V‘3ttb, c * v ' (5.42)

09

SYNCHROTRON RADIATION where

T(x)

is the gamma function.

We then get (see (5.35)) the following

expression for the intensity of the radiation from a system of electrons with the energy spectrum (5,41) in a uniform magnetic field M:

V k>

/3 y+ I

r/3 y - n r /3Y±ii''l e3 ( 3e ~^(Y 0 ' 12 ) ' 12 ' me2 '2 itm^c5/ (5.43)

* K (k)(HsinX)J(Y+l) v~*(Y_1) , where

Ke (k)

is the coefficient in (5.41).

We assume that the electron distribution can be taken to be uniform and iso­ tropic, that is N(C, R ,k)

= 4 ¥ Ne ( e >.

where Ne ( £ ) d e

= Ke

e"Yde

(5.44)

is the number of electron per unit volume with arbitrary directions of motion and energies in the range

e , e +de.

In that case

Ke = 4¥ KeL where

Ke

is the coefficient in (5.44) and L

region along the line of sight.

-

(5-45>

the extension of the emitting

We note that in the general case

Ke (k)

de­

pends on the angle X between the direction of the magnetic field and the line of sight. In the case of a uniform field the degree of polarization depends solely on the index Y of the spectrum (5.41) and one can check from (5.30) and (5.42) that it is equal to Y+ I ^

tt

which amounts to 75% for

Y“ 3

,

and to 69% for

(5.46)

Y®2.

Equations (5.43) and (5.46) are, in general, unsuitable for applications to the synchrotron radiation of cosmic electrons, as the radiation observed is gathered from a large region of space with the magnetic field being oriented differently in different parts.

We rather assume that along the line of sight

the magnetic field is randomly directed.

In that case there is no polariza­

tion of the radiation and we can easily find the intensity by averaging (5.43) over all directions of the magnetic field.

Since

J

i 2L

\

(sinx) i(Y + °

Bin X dX

rq(Y

+ 5))

2 r(j(Y

+ 7))

’

(5.47)

THEORETICAL PHYSICS AND ASTROPHYSICS

90 this averaging

leads

to

the

following

expression

for

the

intensity

of

the

radiation for the case of a uniform and isotropic electron distribution with the energy spectrum (5.41) in a random magnetic field:

Iv S I - . v me

'4

7T m

»l(,,l,u

e •

t v-!(Y- ° e

(5.48) = 1.35 x 10 “ a(y) U , H !

(Y+1 ^ 6.26

1>

X

erg cm sterad s Hz

Here

1^

is the coefficient in (5.44) corresponding to unit volume, by

we must understand the average value of that quantity in the emitting region, and

a(Y)

is a coefficient depending on the index Y

2H

y

-D

a(Y) =

■ ? r ( * f r ) r( T

of the energy spectrum :

K

x

)

(5.49)

84

I

H

WH = KH W c.r. W

(5.67)

= (H2/8tt)V

is

the

total magnetic

is the total energy of relativistic particles (cosmic

rays and electrons) in the radio-emitting nebula. The radio-astronomical data allow us to judge only how many electrons there are in the source and what their energy is; all relativistic particles

Wc

to determine the total energy of

we must also establish a relation between it

and the energy of the relativistic electrons

We .

There are at the moment no

reliable methods for estimating the fraction W e of the total energy W c r (see, however, below) and as a second essential assumption one usually takes the energy of all cosmic rays in the source to be simply proportional to the energy of the relativistic electrons : wc.r. “ KeW e where

) »

grad

(6.35)

div (e grad ) = - 4tt ^ 6(r — r^Ct)) , (6.36) L where we are considering point charges to fix the ideas (the radius vector of the charge is

r^(t) * {x^ ,

%z ^ } 9 and

■ vJ

.

Equation (6.36) has the same form a6 in electrostatics and we can write down its solution as follows: e. i

^

(6.37)

>/£ 1C 2£ 3 / { ( x - x i)2/el + ( y - y i)z/C2 + ( z - z i > 2/e3}

One can al6o 6how easily that the energy of the field in the whole of space equals (provided the field it6elf vanishes at infinity in the required manner)

* = K

^

( E ’D +

1,2)1 d3f

= *tr + h

’

(6.38)

\ = i i r | (* v 4, , 7 *) 5,3r • Here

-

j

e .e. ■■ \ L ij / T [ e ^ / { ( x i - x j )z /e, + ( y j - y j ) 2 / e 2 + ( z i - z j ) 2 / e 3}

I

(6.39)

is the sum of the in6tantaneou6 Coulomb interactions energies of the charges (we neglect the self-energy of the charges).

The energy

H tr

is the analogue

of the transverse field energy and can also be written in the form

*tr = - 8irc } (D tr * jiT ) d

-

e

D tr “ " 7

JA .

r + 8 tf { (curl A)

d r

’ (6.40)

3C

U = ~ U

' d l v D tr “ 0 *

We now expand in series:

- X h u ( t ) A u (r) , q*t ( t ) A > ) l . (6.41) c - jA - ^

K t ( t > c At = 0 We impose two more conditions upon the a ^

(note that there is no summation

in the first of these conditions over the repeated index (e a n ’a n )

1) : (6.43)

(E a xi-a A2^

1

The fir6t of these is a normalization condition, and the second one corresponds to a choice of polarizations which corresponds to the normal waves; is clear from (6.42) and (6.43) that the vectors

I = 1 and for

planar (both for

K

* A * ) d 3 r = 4 7 T c 2 (a,0 *a

+ “u qU

2

AJL

v Ail um) ,0, )

for which

, p^ -

E - Hoj , where

0 and

.

(6.51)

) for the normal

waves which can propagate in the medium in a given direction;

when we neglect

the spatial dispersion and strictly longitudinal waves, there are two such waves, that is, i * 1,2).

This result generalizes (6.25) and was mentioned

earlier. A very important and characteristic feature of the electrodynamics in a medium is the possibility that emis­ sion may occur even for a uniformly moving charge. We must mention here two effects:

the Vavilov-Cherenkov

effect (Cherenkov radiation; Tamm and Frank, 1937; 1960;

Jelley,

1958;

see

Ginzburg 1940a, Bolotovskii,

1960, 1962; Landau and Lifshitz, 1960, Zrelov, 1968 and Ter-Mikaelyan, 1972; and

transition

Ginzburg,

radiation (Frank and

1945; Ginzburg and Frank,

1946; Garibyan,

I960, 1970;

and Tsytovich, 1974a, 1978; cussed in Chapter 7.

Fig. 6.1

Bass and Yakovenko, Ginzburg, 1975b).^

The Cherenkov cone

1965; Tamoykin, 1972; Ginzburg Transition

radiation

is

dis­

Now, however, we shall discuss Cherenkov radiation which

occurs in a uniform medium when the condition c v > v. ph n(w)

(6.52)

is satisfied, that is, provided the particle velocity v - constant is larger than the phase velocity of waves in the medium, Vph = c/n(oj).

^

In that case

The literature dealing with Cherenkov and transition radiation is huge (see the reviews by Jelley (1958); Ginzburg (I960); Bolotovskii (I960, 1962); Bass and Yakovenko (1965); Zrelov (1968); Tamoykin (1972); and Ter-Mikaelyan (1972); where one can find both original results and bibliographies.

THEORETICAL PHYSICS AND ASTROPHYSICS

116 a wave of frequency

(i) is emitted flt an angle

0 where 0

and

k

0 Q is the angle between

determine the orientation of the vector

and

k

v

while the

with respect to

the synmetry axes of the medium (in the case of a moving medium which is iso­ tropic in its rest frame the velocity of the medium symmetry axis).

U

plays the role of the

We restrict ourselves for the sake of simplicity in what

follows to an isotropic medium. The condition (6.53) for Cherenkov emission can be obtained not only from interference considerations or as the result of evaluating the radiation field (of course, in that case the emission condition follows automatically from the formulae)^ but also from the energy and momentum conservation laws (see Chapter 7) or as a resonance condition (Ginzburg,

1939a,b).

This last one occurs when

we solve the Cherenkov radiation problem by the Hamiltonian method, and we shall now turn to this. substituting

in

them

We shall start from the equations in the form (6.21),

for

the

case

of

a uniformly moving charge the radius

Strictly speaking, in order to obtain the condition (6.53) one need not go beyond the original expression (4.13) for the Lienard-Wiechert poten­ tials, generalized to the case where there is a medium by replacing c by vph = c/n (of course, this should be done only where c plays the role of tne wave velocity and not where it is a coefficient of the current density, for instance, on the right-hand side of Eqs. (1.8) or (6.11)). As the re­ sult from such a substitution we get, for instance, A=ev/{cr[l - (vn/c) cos0']}, and it is clear that in the wave zone, where 0; *0, the poten­ tial has a singularity just at the angle 0 ® 0 O , where cos 0 O = c/nv, (One should not confuse the angle 0 between k and v with the angle 0 which together with the angle v ptl = c/n (see condition (6.52)).

the necessity to find a positive value for the frequency

From

(D it is immediately

clear that we must take the absolute value of the denominator in (6.59) when ^ n ( w ) cos 0 > 1 .

(6.61)

The range of angles satisfying condition (6.61) is called the region of the anomalous or 1super-light* Doppler effect. for the super-light velocities (6.52).

Of course, that region exists only

If, on the other hand

-n(rn) cos 0 < I , we have the normal Doppler effect.

(6.62)

The regions of the normal and the anoma­

lous Doppler effect are separated by the Cherenkov cone (Fig.6.2).

Under real

conditions, the Doppler effect in a medium is rather complicated (Frank, 1942, 1959) as dispersion must be taken into account (the

(D-dependence of

n ) . We

restrict ourselves here to the remark that if one neglects dispersion, the frequency u)(0 a 0 O) would become infinite on the Cherenkov cone itself where (v/c) ncos 0 «1

(see (6.53)).

In fact, as

(D + ®

the refractive index

n(w) -► 1

THEORETICAL PHYSICS AMD ASTROPHYSICS

120

and strictly speaking there is no Cherenkov emission whatever on the Cherenkov cone (we assume that

v2 -

(6.74)

(2ttv ) 2 » U ) 2

3H.

P V m c 2/

or for the main part of the synchrotron emission (frequencies

V ~ \>c) , pro­

vided v

4 ecN — .--3H±

» c

In the interstellar medium, where

N «

N(cm” 3) 20 -------- Hz . Hi (Oe) 1 cm” 3

and

H± ~ 1 0

(6.75)

6

to

10” 5 O e

condi­

tion (6.75) is well satisfied for most frequency ranges used in radio-astronomy In denser objects

(some galaxies,

tion (6.75) is more rigid.

shells around supernovae,

reckon with the possibility that this condition is violated, factors such as, for instance, (see Chapter 9).

and so on) condi­

At sufficiently long wavelengths one must always provided other

reabsorption do not play an even greater role

Chapter VII CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION Cherenkov and Doppler effect from the quantal point of view. Radiation reaction in a medium. Cherenkov emission and absorp­ tion in an isotropic and magneto-active plasma, Cherenkov emission by dipoles. Emission in channels and gaps. Application of the reciprocity theorem. Transition radiation.+ When one analyzes various problems connected with the emission, absorp­ tion,

and

Systems*

amplification

of

electromagnetic

waves

when

charges

or

other

(such as atoms, bunches, or antennae) move in a medium, elementary

quantal pictures have turned out to be very fruitful.

It is important that

this is the case also when the problem is essentially classical so that the final results within the approximations used will be independent of the quan­ tum constant

R .

As the starting point for our use of quantum-mechanical considerations we take the concept of quanta, or photons in a medium, with an energy

R1) ; p 0fl - ® » 0 i l / / O - v J fl/c2) into (7.1) and (7.2) end we then get ss the condition for emission without a change in the internal state (Ginzburg,

cos 0 n = , N 0 n(d))v o \

1940a)

2mc^

2 (mc/n) (vQ cos 0 0 - c/n) (7.3)

fid) (1 - v 2/c2)* (1 - l/nz7 where When

0O

is the angle between

fiu)/mc2 < I

emission,

v q

snd

k .

this condition changes to the classical condition (6.53) for

ss would be expected (when

fio)Anc2 « I, the 'recoil' connected with

the emission of s quantum is relatively small).^ course,

that emission is possible (that is,

It is clear from (7.3), of

c o a 0 o 0) only when the

It is clear from (7.3) that if we want to be more precise the condition for classical behaviour must be written in somewhat different form, namely,

{fia>(n2 - l ) / 2 m c 2 } / [ l - v 2/ c 2] « I .

127

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION motion is with a velocity exceeding the speed of light, that is, when the inequality

v Qn / c > !

holds (see (6.52)).

When the result doeB not contain

fi a quantum-mechanics 1 calculation has only

a methodological value, but it often turns out to be more convenient.

Essen­

tially this reduces to using the conservation laws which have a wider range of applicability in the sense that they can be used also without invoking quantum mechanics.

Let us, in fact, assume that it follows from the classical theory

of an electromagnetic field in a medium that the connection between the energy W be

and the total momentum G

of the radiation in the medium is found to

G = (Mn/c) s.+

Moreover, if the changes in energy and momentum are sufficiently small we have for free motion of the charge

AC = (v* Ap), since

Because we have assumed the change between

v 0 and

Vj

AC

to be small we need not distinguish

and we denote the velocity of the source

v fl « Vj

From (7.4) and the conservation laws (7.1) and (7.2) we get, replacing by

v.

fio)

H, AC = M - (w - Ap) =

or

by

(s -w) ,

cos 0O * c/nv, that is, we get the Cherenkov condition (6.53).

is simpler to introduce at once the quanta

However, it

fid) and this is a completely natural

thing to do not only in the quantal, but also in the classical case.

We shall

proceed in this way. If we are considering the motion of a 'system9 rather than a 9structureless9 particle so that its internal energy can change, we have e o " / { (m + m Q )2c** + c2p 2} , where (m + m 0)c2 - me 2 + w Q ( m + m ^ c 2 -mc2 + w 1 R uj^ > 0

is

the

total

= /{(m + m j V energy

in

+ c2p 2} ,

the

the total energy in the upper state.

lower

state

and

Clearly, w^ - w fl ■

is the energy difference of the two levels of the system (atom, ...)

considered here.

It will be Bhown in Chapter 12 that the momentum of the field equala Gem “ tt/nc, while the momentum of the force imparted to the dielec­ tric in the radiation, G'm '- (n2 - 1) ■ W (nz - l)/nc. The total momentum lost to the emitter ia thus G - G e m + G^m ) ■ Wn/c.

THEORETICAL PHYSICS AND ASTROPHYSICS

128

If we now apply the conservation laws (7.1) and (7.2) with exactly the Doppler Eq.(6.59) with terms of order

a)00

fi(o/mc2 ^ )

ys g*t

If, however, we do not neglect

ha)/mc2, we find, as in the case of the Cherenkov emission

(see (7.3)), a somewhat more complicated expression (Ginzburg and Frank, 1947a). In practice, however, we can restrict ourselves to the usual form (6.59) of the formula for the Doppler effect.

In a quantal calculation, moreover,

another important fact is clarified which completely escapes attention in a classical derivation of Eq.(6.59);

namely, in the region of the normal Doppler

effect, that is, when (see (6.62)) — n(o)) cos 0 < 1 , c

(7.5)

the emission corresponds to a transition of the system from an upper level with energy

w x to a lower level with energy

wQ

(the direction of the transi­

tion is determined by the condition that the energy of the quantum emitted must be positive, that is, formally from the requirement the quantum is emitted inside the Cherenkov cone,

a) >0).

If, however,

that is, if we are dealing

with the anomalous Doppler effect and (see (6.61)) “ n(o)) cos 0 > 1 , c

(7.6)

the emission of the quantum is accompanied by a transition of the system from the lower level

w0

to the upper level

w1 .

The energy of the quantum, and

also the energy exciting the radiating system, is then derived from the kinetic energy of its translational motion. It is clear from this example that in quantum theory, in contrast to classical theory, when one finds the conditions for radiation themselves one determines at the same time the direction of the process, that ia, whether the transition is up or down.

It is just this fact which together with the possibility of

such a simple calculation of the induced emission (vide infra) makes quantal calculations so valuable for obtaining conditions for radiation, the condition for amplification (instability) of waves in beams, and so on. If

the

system

(v/c)n1

there ia also in stationary states a non-vanishing proba­

bility to find the system in level 1 and it radiates all the time both normal

CHERENKOV EFFECT* DOPPLER EFFECT, TRANSITION RADIATION and anomalous Doppler waves.

129

The population of the levels

0 and

1 , and also

the intensity of the emission of normal and anomalous waves, are clearly deter­ mined by the ratio of the total probabilities for the emission of these waves. For

a

system

with

many

levels

(Ginzburg

and

Fain, 1959)

the

emission of

anomalous Doppler waves while the system makes a transition upwards leads to the

possibility

instance,

of

the

amplification

of

’transverse

oscillations’ and, for

to the ionization of an atom.

To be more exact, there are here two cases (Ginzburg and Eidman, 1959b).

In

the first case the average energy of the transverse oscillations of the system decreases while it moves. functions

with

different,

This means that for a wavepacket consisting of wavebut

approximately

equal,

energies (we think, for

instance, of an electron moving along a magnetic field) the centre of mass of the packet in the energy scale diminishes.

The difference between subluminal

and 6uperluminal motions lies in this case in the rate at which the average energy changes and also in the nature of the spreading of the packet.

Thus,

when the system moves subluminally states with energies larger than those represented in the initial spectrum of the packet never become occupied.

If

the system moves superluminally, however, there is

to

a finite

probability

find the system (we assume, of course, that we have an ensemble of systems) in any, however highly excited, level which can be

reached with

condition

(7.6)

being valid, notwithstanding the fact that the average energy decreases. In

the

second

of

the

above mentioned cases the system is unstable even ’on

average’, that is, it6 average energy (and we are talking here about the energy of

the oscillations, that is, the

energy

of the excitations) increases with

time and, moreover, the nature of the spreading of the packet is changed. Elucidation of with which of the possibilities we are dealing requires actual calculations

of

calculation is, whatever

its

the

transition

generally

advantages,

probabilities.

speaking,

superfluous

In for

that

respect

a classical

a

quantal

system,

and it is natural to use the classical theory of

radiation, as we shall do below. We now note that quantum considerations like those given are nevertheless use­ ful also for an analysis of the problems, already mentioned, of the absorption and amplification of waves in particle beams (in the case of wave amplification the beam, in fact, becomes unstable). instability

infra).

It is, moreover, clear that when beams of ’systems’ with two or more

levels move

of

In this way one easily obtains criteria

for the

a particle beam moving in an

superluminally, as

isotropic

plasma (vide

a rule, amplification (negative absorption)

130

THEORETICAL PHYSICS AND ASTROPHYSICS

must occur rather than absorption (re-absorption) of anomalous Doppler vavee (Zheleznyskov,

1959).

This is connected with the fact that when a quantum

corresponding to the region of anomalous Doppler waves sngle

0 < 0O

(that is, flying at

to the velocity of the system) is absorbed,

an

the system will make

a transition not upwards, as in the normal effect, but downwards.^

On the

other hand, the upwards transition of the system corresponds now to induced emission which in the region of the normal effect corresponds to a downwards transition of the system.

Therefore, if all systems (stom6, electrons in a

magnetic field) in a beam moving 6uperluminally are, for instance, in the lower level, normal Doppler waves, emitted by one of the systems, will be absorbed in this beam, and the anomalous Doppler waves will be amplified: they will along their path transfer other systems upwards through induced emission,

that i6 with the emission of yet another anomalous Doppler quantum.

We shall in Chapter 9 dwell on the use of the Einstein coefficients for the spontaneous emission and absorption probabilities.

However, it i6 already now

convenient to use these coefficients in order to make what has been said above somewhat more quantitative. If both the upper and lower levels

1 and

0

are populated,

the coefficient

for absorption in the beam of normal Doppler waves is equal to (see Ginzburg and Zheleznyakov, Zheleznyskov,

1959a, 1965;

1959, 1970;

HI

Ginzburg, Zheleznyakov and Eidman,

Ginzburg,

8tt3c 2N (Na/N. - 1)

w _ = - —I dzt = a J ------- — l)

16 the 6ame as the condition which one can obtain by solving

the classical problem of the instability of an electron beam in a magnetic field (Zheleznyakov, 1959). The instability of electron beams noted here is, in particular, possible in a magneto-active plasma and is of interest for the theory of sporadic radio-emission of the Sun (Ginzburg and Zheleznyakov, 1959b; Zheleznyakov,

1958b,

1970).

The condition (6.53) for Cherenkov emission has in the classical case an inter­ ference character, and it is therefore universal for any kind of wave — of course.

132

THEORETICAL PHYSICS AND AS TROPHY* IC#

replacing the phase velocity of light, c/n(w), by the phase velocity the waves considered which may be sound waves, capillary waves,

...

Vp^

of

The same

is also true for the results given here which are obtained by using the energy and momentum

conservation

approach to them.

laws

employing

either

a

quantal

or

a

classical

In that esse the quantal approach, that is, the introduction

of quanta, is appreciably simpler, not only for light, but also for (longitu­ dinal) plasms waves

and sound.

quantum (phonon) is equal to where

u

In the latter case the energy of a sound

E = fid) and its momentum to p = fik 3 (E/u) S ,

is the sound velocity;

for sound the dispersion is usually unimpor­

tant and one need not distinguish between the phase and group velocities. course, as in electrodynamics,

Of

in the case of supersonic motion the emitting

acoustic system will make an 'upwards*

transition — that is, will be excited —

in the region of the anomalous Doppler effect and thus to a certain extent 'be amplified'

(Tamm,

1959).

Let us now consider an interesting feature connected with the fact that the directions of the phase and the group velocities are not the same, as may be the case in an anisotropic medium or when one takes into account spatial digpersion (see Agranovich and Ginzburg, velocity

1966).

If the projection of the group

du)/dk on the direction at right angles to the velocity of the parti­

cle, that is, the quantity

dco/dkr , where

kr

is the component of

angles to V , is negative, no energy, apparently, energy is absorbed by it.

1959).

velocity,

at right

leaves the emitter, but

However, under similar circumstances one must use

the advanced rather than retarded potentials 1957,

k

If we choose the vector

k

(Mandel'shtam,

1947; Pafamov,

always to be directed along the phase

this vector will in the case when

daj/dkr < 0

in the Cherenkov and

Doppler waves be directed towards the particle trajectory, while the energy — as should be the case — will go away from the trajectory.

In the case of

Cherenkov radiation the difference between the cases

daj/dkr > 0

and

is clear from Fig.7.1.

dco/dkr < 0

determined as

The angle

0 Q is in the case

dco/dkr < 0

before by the Cherenkov condition (6.53) as is clear from the choice made for the direction of servation laws

k

from interference considerations, and also from the con­

(7.1) and (7.2).

The latter statement follows from the fact

The quanta of plasma waves are often called plssmons. If we under­ stand by photons in a medium the quanta of any electromagnetic field where we are, strictly speaking, dealing with a free field, that is, a field when there are no charges or currents — plasmons are a parti cular case of photons in a medium.

133

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION chat

we

have

used

plane

waves of Che form exp {i(k • r ) - iut} for which Che momentum of Che corresponding quantum is equal Co when we

(Hwn/c)(k/k);

use

this

form of

waves there is no difference whatever in the directions of

k

in Figs.7 . la and 7.1b,

as

in

waves

terms the

of

the

plane

disposition

of

the wavefronts is the same in both cases — we consider fronts

with

k

the vector

lying on the Cherenkov cone. Equation

(6.59)

for

the

Doppler effect for the case dui/dkj. < 0 valid.

also

remains

However, the physi­

cal difference between the two

cases

is,

of

course,

Fig. 7.1 Cherenkov radiation when du)/dkr > 0 and when dw/dkr < 0. kr is the component of the wavevector at right angles to the particle velocity v

very important and is con­ nected with the different directions

of

velocity.

In fact, in an isotropic medium in the normal case (see Fig.7.la)

the

group

the group velocity is parallel to k . 7.1b the group velocity the particle velocity.

However, in the case depicted in Fig.

dtii/dk is directed at an obtuse angle

01=

tt-

0

to

Under such circumstances the Cherenkov radiation will,

when particles pass through a plate of finite thickness, emerge from the back surface of the plate, and will also be refracted in an unusual way at that surface (this is clear from M a n d e l s h t a m ’s work (1947)). For electrons moving with superluminal velocities in a plasma or a retarding medium in the presence of a magnetic field and also in similar cases of oscil­ latory motion of electrons, usually only the classical region is of interest — the quantum numbers corresponding to the transverse motion are large.

Under

such circumstances the problem of the emission of waves and of the damping or amplification

of

the

transverse

oscillations

of

the

electrons

may

and

in

134

THEORETICAL PHYSICS AND ASTROPHYSICS

practice must be solved by means of classical calculations.

The corresponding

calculations reduce essentially to the evaluation of the radiation reaction force when the charge moves in the medium.

Let us consider that problem in a

somewhat wider context. As the presence of the medium may radically change the character of the elec­ tromagnetic waves radiated by the moving particle (see Chapter 6) it is clear that the radiation reaction force in the medium is also changed, and sometimes very considerably.

For

instance,

an

oscillator

isotropic plasma with index of refraction radiate

at

all

if

with

a

frequency

n * { l - 4 TTNeVmU)* f2

u)2 = 4irNe2/m > to2 , when

e = n 2 < 0;

u) in an

does not

in a magneto-active

plasma there is in the non-relativistic approximation no radiation by an elec­ tron revolving in a magnetic field Ginzburg and Zheleznyakov,1958a;

H0

with a frequency

Ginzburg 1970b).

u ) y « e H 0/mc (see

In both these cases the

radiation force, of course, vanishes, while in vacuo it equals

f = (2e2/3c3)r.

On the other hand, in the case of uniform motion in a medium there is, if for some frequencies the velocity

v>c/n(u>),

the Cherenkov radiation force

which produces per unit time an amount of work equal to

(fr • v) = -dW/dt. Cn

It

is thus clear from (6.58) that f

rh

2 c v

I ['v 2c 2 (io) J L c/n(u>) < v

(7.9)

du)

In the light what we have said so far the problem naturally arises of how to evaluate the radiation reaction force for arbitrary motion of a charge in an arbitrary medium. attention.

In the past this problem has not particularly attracted

The point is, apparently,

that the radiation force when a charge

moves in a medium is usually appreciably smaller than the braking force con­ nected with ionization losses. which might be called radiative,

For example,

losses due to Cherenkov emission

are even in a transparent, but dense medium

only a small fraction of the total losses.

The position is, generally speak­

ing, not changed for the case of non-uniform motion of a charge.

There are,

however, interesting and practically important cases when taking the radiation forces into account when a charge moves in a medium is very important — motion in a magneto-active plasma, motion in channels, gaps, and close to the surface of a medium. We give here a scheme to evaluate the radiation force in a medium (Ginzburg and Eidman, selves.

1959b) where, as elsewhere, we shall not be afraid to repeat our­

For a point charge with charge density

p-efi(r-R)

the equations

135

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION for the field and the equations of motion have the form

curl H = —

C

(7.10)

, div H - 0 ,

mv

[v a Ho]} +

{E . + I

✓ 1 -v2/c2

+ e f{E(r)+ Here

f div D • 4u efi(r-R) ,

Cot

I 3H

curl E _d_ dt

ev 6(r-R) + - ^

(v = R = dR/dt), E0 and H0 are the

R(t) is the position of the charge

external fields, and

(7.ID

c [v aH (r )]| £(r-R) d3

E and H the fields produced by the charge itself (for

the sake of simplicity we assume the medium to be non-magnetic). The only effective method of solving the problem for an arbitrary medium is by expanding the fields in terms of normal plane waves, that is, using the method which we have called the Hamiltonian one.

a g(w) Eg(u)) , 0,6 = 1 , 2 , 3 ;

V w) F

As a result we have

1 c

A = /A tt

3A 3t

V ; H = curl A ,

V

X

c

X,j=i,2

c l ‘ >i

exp

"y

lCa 6 (u>) (aX j K j ^ x

[i(kx * r)]

’

3A _ Q + c.c.

E

(7.12)

(7.13)

aS 3x. 3

where condition (7.13) is chosen for the sake of convenience, c.c. indicates the complex conjugate quantity, summation is understood to be performed over indexes which occur twice, and the argument Fourier components;

shows that we are talking about

the real fields are equal to

E ® E + E* , and so on. and the a^.

w

In Eqs.(7.!2) and (7.13)

D = D + D* = D + c.c., n^j

is the refractive index

are the complex polarization vectors corresponding to the

jt^1

normal wave. The equations for the potentials, obtained from (7.10), have the form —

a 2A, -

I

V2A - grad div A -

(7.12), and (7.13)

I

+ C.C.

#B - - ea&

j

« _ i l evfi(r-R) ,

cje c 32 — ^ + c.c. :1- A ire6(r- R) “ B 3xo3x 0

(7.IA) ,

136

THEORETICAL PHYSICS AND ASTROPHYSUJf

where

is the unit vector along the

a-axis

and j

« ev 6(r - R )

current density corresponding to the particle considered;

ifl the

the somewhat dif­

ferent notation and definition of various quantities as compared to those used in Chapter 6 is because of convenience and in order to refer the reader to the papers by Eidman and the author (Ginzburg and Eidmanf 1959b; Eidman,

I960)

where the calculations are given in detail using the same notation. Substitution of the expansion (7.12) into (7.1 A) produces a set of oscillator equations for the field amplitudes

q> . which can be integrated in an elemenAJ It is necessary to substitute the fields obtained this way into the

tary way.

equation of motion (7.11). d d t

As a result we get

VX

2 / j

- V 2/C 2

r r

ra i c/v

we

get

from

for the latter case. in

(7.15)

an a

isotropic

formula

for

Moreover,

medium

with

the braking

a

This

for the case of a retractive

force

(7.9)

index due

to

Cherenkov radiation. Ginzburg and Eidman (1959b) considered the superluminal motion of an oscilla­ tor.

In an isotropic medium we have for the case of an oscillator vibrating

parallel to the translational velocity R * {0,0,v q t + R 0 sinftt} ;

V

yQ

= {o,O,v0 +

cos &t} , v ^ = R Qft ; (7,16)

a i “ { l* 0 * 0 } , a 2=, {0, cos0 , - sin 0},

k = {o,ksin0,kcos0},

and in what follow we shall be dealing only with the dipole approximation, that is, the case when k R 0 > £ n(u>) R 0 « ! .

(7.17)

Under those circumstances we get the following expression for the work done by the radiation field on the particle:

137

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION T

T

e

J ( - f rad 0

) dt « v

oJ o

y

(7.18)

dt + v rad,z ~

) o

J

e 2R2T < ------\ Ac se 0

T f

*80n(a)) cos 0 < I

du)

U)3 [lL 0 n

1

+

(7.19)

0 Qn((i)) cos 0 > 1 u

02i 0

where

Q

Cl) =

(7.20)

1 - B.n(u>) cos 6| ’ B°

c

*

If the dispersion law has a ’step-function* character, that is.

n(to) = n = constant, for

u)
ti)m ,

(7.21)

one can write the result (7.19) in the form e^RJjnT

sin30 d0 (7.22)

A |1 - 0Q ncos 0 |5

4c where for the anomalous Doppler effect

0 < 0 < arc cos

1+ -------

and for the normal Doppler effect 1 -fl/u) arc cos ------Son The quantity T

U ^ (dW/dt) dt = - A > 0

by the particle.

Expression

< 0
1

!—

(, ,a ) !l

0 2n 2 (u)) '

(LgVld.}

L P»n ( u ) )'

'• *

d(A)

w/ J

. *

(7.23)

138

THEORETICAL PHYSICS AND ASTROPHY8ICl

A

_ e R oflltnT f f 4c3

sin30 dO

f

sin30 d6

1

I J O - ^ n c o s e ) 1* ^

’ (7.24)

G = arc cos J — 1 1 + — ) , ®on \ %J

6 = arc cos -jp— (1 . 6„n V w m/

The radiation propagating outside the Cherenkov cone, which corresponds to the second integral in (7.23) or (7.24), leads thus to a damping of the oscilla­ tions while the radiation inside this cone (the anomalous Doppler effect), corresponding to the first integral in (7.23) or (7.24), amplifies the oscil­ lations.^

This result is in complete agreement with the quantunmechanical

considerations (vide supra).

One sees easily that the second integral in

(7.23) is larger than the first one, and that the same is true in (7.24). Hence it follows that the vibrations of the oscillator are always damped in an isotropic medium and

A^-*0

only if

0 on(tu) -*00 in an appreciable region of

integration. Ginzburg and Eidman (1959b; Eidman, 1960) considered also the motion of an oscillator vibrating at right angles to its translational velocity helical motion of a charge in a magnetic field.

v 0 and the

They showed that as in the

preceding case in an isotropic medium the vibrations are always damped (this result does not necessarily follow for other radiating systems such as, for instance, a sufficiently long antenna). It is convenient for the calculation of some of the features of the super­ luminal motion of charges in anisotropic media to consider the motion of an oscillator along the optic axis of a uni-axial non-gyrotropic crystal with the electron assumed to be oscillating in the same direction. R

m {0,0,vQt + R 0 sinftt} , k = {0,k sin0, k cos 0}

,

a i “ {0, cos 0 + Kj sin 0, - sin 0 + Kj cos 0} , a 2 *{l,O,0} (n* - e ) cos 0 l 1 sin 0

. I -1

where

n

. 2. 2_ _ sin 0 . cos 0 +

, kR0

In that case

,

(7.25)

.

1

is the refractive index for the extra-ordinary wave which in the

If the work A , or part of it, is positive, it corresponds to an amplification of the oscillations, as A^ is the work done by the radiation force on the particle.

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION given case is just the one which is radiated.

139

The quantity

Kj

is the ratio

of the components of the electric field strength vector in the extra-ordinary wave which are parallel to and at right angles to the vector

ax

vector itself is parallel to the polarization vector

k ; the electria

(see Chapter 6).

We can now obtain the expressions which correspond to Eqns.(7.19) and (7.23) e2RjT A

e 2 (u>) sin2 0|l - (cot 0 / n ^ Q n x/ i 6) |~l

f

- ----- — J to3 — ---------------------------------------------- dw , P0L j + L 2 [e^utfsin2© + C || cos2©]2

f

A^

e2R 2ftT e e 2 (a>) sin20| 1-(cot ©/njXBnj/B©) I” 1 = ------ J u>2 -------------------------------------------4 c 3P 0 ' [ci (u))sin20 + C|| (u))cos2©]2

+

C 2 (u)) sin2© | \ - (cot 0 / n J O n j / a O ) I"1 ----------------------------------[ci (w)sin20 + C|| (u>)cos2© ] 2

f J(D2 L2

The integration domains

Lj

and

du>

. d(i)r • J

(7.26)

L 2 are here determined by the Doppler rela­

tions 1 - B on(u,0) cos 0 = n/w for normal Doppler frequencies

(domain

Lj)

(7.27)

and

P on(o),0) cos 0 - 1 = ft/w for the anomalous Doppler frequencies (domain integrals in Eqn. (7.26) for

A^

(7.28)

L 2) .

One sees easily that both

are always positive.

radiation at the normal Doppler frequencies ponds to a damping of the oscillations,

This means that the

(first integral in (7.26)) corres­

while

the

radiation

at

anomalous

Doppler frequencies corresponds to an amplification of the oscillations. We must note that such a subdivision is somewhat arbitrary and that, of course, only the force which is the difference of the two integrals has a physical meaning. In contrast to the isotropic case, in the present problem the oscillations may not only be damped, but they can also be amplified — we are, of course, speaking about the sign of the whole of the work parts.

For instance, let

^>0;

in that case

Eqn.(7.25) for

€\\

and

nj(8J+ “

A^

and not just about its

be frequency-independent and let

€|j< 0,

at an angle determined by the condition (see

n*) e i 8in20® + e llcos20« = 0

(7.29)

In such a medium the extra-ordinary waves can propagate at an angle |0 |
0 > 0^

Moreover,

n 2 < 0,

n 2 is a minimum and equal to

we can always choose where

we have already

and Che waves cannot props*

£ ± when

£|| such that the Cherenkov angle

0on c o s 0 Q = 1.

0-0.

0 Q is larger than

0 Q corresponds to a value

is positive.

1 - 0 O n a cos 0 - 1 +

0 < J 7T , and in

tt < 0 < 0^ — normal Doppler

In the backward direction — for

waves are emitted, but now A

0^,

n 2 < 0 — and only anoma­

lous Doppler waves are emitted in the forward direction — for 0 < 0^ •

>I

Clearly, under such conditions Cherenkov radiation is

totally absent — the angle

fact for

If now

nj |cos 0|

and the total work

One can check this by using (7.27) and (7.28) to change in

(7.26) to an integration over

0,

as a result of which we get for the case

considered

e2Q“RjT

t

9

4c 30#

1

i

e f,

n3(0)sin30d0

9

[ 0 ,^(0) c o s 0 - I]1*

i

n[(0') sin30' d0' e

|,

[B#i»j(0') co* 6' + 1]"

G = arc tan/|C|| |/e ± , where

0' »ir-0.

Here

A^>0

(7.30)

by virtue of the fact that the first integral in

(7.30) is always larger than the second one.

In the case considered the oscil­

lations are thus amplified. Eidman (1960) has studied the problem of the motion of charges in a magnetized plasma and showed that in that case under well-defined conditions the oscilla­ tions are amplified or, to be more precise, that spiral lines along which the particle moves in the magnetic field are ’unwound1.

Amplification thus occurs,

for instance, when eH„ 4 TfNe 2 4 5 = 00 . 2 —r « I * , 0L U = — 9, 2/a)£ = 10.

0 Q * 0.99

and

- 0.01

If on the other hand, we the transverse motion of the

particle is damped. When the oscillations are amplified the energy of the translational motion (in this case the motion along the field) changes into energy in the transverse motion. tion

As a result the translational velocity

necessarily

velocity

c/nmax

ceases

when

the

velocity

in the given medium.

v Q diminishes and amplifies-

v # reaches

the minimum

light

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION

141

The difference in the sign of the force acting upon the oscillatory motion of the particle in the case of the normal and anomalous Doppler radiation clearly agrees completely with the conclusion reached by using the conservation laws (vide supra).

We have already stressed that in the isotropic case that dif­

ference leads to a weakening of the f r i c t i o n 1 or even its virtual vanishing but it cannot cause amplification of the vibrations of the oscillator;

the

quantal amplification of the vibrations which is connected with the spreading of the packet in 'energy space' occurs, of course, in the case of superluminal emission, also in an isotropic medium.

Amplification of the oscillations is

possible in an anisotropic medium and, in particular, in a magnetized plasma. It is totally obvious that the instability of 'superluminal' particle beams which occurs in the classical approximation even in the case of an isotropic medium is closely connected with the radiation reaction of a single particle considered here. We note also that we have here considered a medium at equilibrium or, at any rate, such a medium that the normal waves in it are absorbed when damping (conductivity) is taken into account, that is, that their amplitude diminishes as they propagate through the medium. In media with a negative conductivity — which are sometimes called inverted media — the normal waves are amplified (maser effect) and the problem of the radiation reaction needs special consideration (Ginzburg and Eidman, Gavrilov and Kolomenskii,

1971,

1972).

1963;

In that case the oscillations can be

amplified even for subluminal motion (vc/n). In an isotropic plasma, that is when there is no external magnetic field

H 0,

we have for transverse waves

4ttN e 2

2

,2

(the phase velocity of the waves

< 1

m(u)2 + v 2 ) eff Vp^=c/n>c),

and, hence, Cherenkov radiation

is impossible (see, however. Chapter 8; we assumed above that

v < c).

However,

when we take into account thermal motion in an isotropic plasma, longitudinal

theoretical

142 plasma waves can propagate^ Ginzburg,

1970b and Chapter 11) 2i.2

1 -0Jp/u)

k T 4 TTNe2 ft2 - --- . 0) = ------m 1 1 2 ’ P me

3 KT kg

e

AND AS TROPHY m i

which have a refractive index equal to (see

2 _ c k

Here

physics

and

m

are the electron charge and mass,

Boltzmann's constant, and

T

N

(7,31)

is the electron density,

the absolute temperature.

Equation. (7.31) is

equivalent to the dispersion equation a)2 *

P

k»T B 1.2 m

+ 3

and leads to the following expressions for the phase and group velocities:

_ a) _ _c_ = / 3J Kk nT/m B1/m V ph

" k " "3 ■3

/( ,l ., '1 -„,2 -ll)Z/ )2/ Z P 3k„T r3knT

dk

gr

mo)

L

m

’

V

(7.32)

a)2/ J

The plasma waves form one of the three normal wave branches in a plasma which appear on equal footing.

The phase velocity of the plasma waves can be less

than the speed of light in vacuo

c

and hence the Cherenkov effect can occur

for those waves for 'normal' motion of the source (particle) with a velocity v 0

fs (v)

has a positive derivative.

but as the result of a special investigation. a

stability

criterion

tive — because

of

the

is

The same condition

is obtained also classically (Bohm and Gross, 1949),

not

less

greater

but

in

complexity

The quantum method for obtaining a

of

certain the

sense

much

problem — for

more effec­

the

above-

mentioned case of the motion of a beam of charged particles in a magnetized plasma;

in this case we must take into account the change in the velocity

components of the particle at right angles to the magnetic field, or in quan­ tum terms, take into account transitions between energy levels for the motion at right angles to the field, which is quantized (Zheleznyakov,

1959, 1970).

To illustrate what we have said we shall consider for the sake of simplicity the emission in the direction of the velocity particles;

in the general case the function

v

by a one-dimensional beam of

fs (Vk),

where

Vjta v c o s 0

Velocity component of the particles in the beam along the wavevector emitted waves, plays the role of kind

fs (v).

is the

k of the

For a distribution function of the

143

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION

(v r - v 0 cos 0 >’} • 2kBTs shown in Fig.7.2, the coefficient

const x exp ■

fs

in region II, where y 0

df g(v^)/dv^ < 0, while

in region I where

dfs (v^)/dv^ >0.

By virtue of the Cherenkov condition v c o s 0 o = v k = c/n3(d))

and waves emitted

by particles with different values of

V^

(in particular, values corresponding to the regions Fig. 7.2 * , \ fs (vk)

The distribution function * ., . . for particles m a beam

I

lity, even if part of

Vfc .

and

II

in Fig.7.2) have

different frequencies and hence cannot , cancel one another and guarantee stabi" y c .

THEORETICAL PHYSICS AND ASTROPHYSICf

146

- ^ - + ( v 7 r )fl + - ( E . V w )f()= 0 (we have neglected here collisions, function,

that

is,

Fourier transforms,

in

the

case

, f = f 0 + f! . |£0 | » | f , l

and

of

f Q(v)

ia the zeroth-order distribution

equilibrium

that is, substituting

(7.34)

a

Maxwellian

distribution).

f x (v , r , t ) = g(v) exp {i (k • r) - i w t} ,

then lead to the equation i { w - (k» »)} fj = | ( E *7w ) f # .

If

0)/ (k*v) , we can divide by

sion for

fj ;

{(i)-(k»v) }

if we then substitute

—

we get a dispersion relation connecting c2k2/u)2 = n2 2 3 , where

and obtain a well defined expres-

f 1 into the field equation

1 8 2E curl curl E + - ^

in the form

’ 't = e / w f i d 3 y >

(i) with

nj^2>3

k ; we can write that equation

are the refractive indexes used

above for waves of the appropriate type — transverse waves dinal waves (n3). {(a)- (k*v)}

(6.35)

If, however, 0) = (k*v ) ,

(n1>2) an^ (a ■ 1, 2, ... ;

we have neglected here the Doppler ahift of

the frequency for the sake of simplicity).

However, when there is a plasma,

the nature of the radiation — its intensity, directionality, and polarization — is changed and apart from the frequencies

sw£

radiation may occur with a

continuous 6pectrum which is clearly Cherenkov radiation (when the particle roves strictly along the field magneto-brems radiation is completely missing). Moreover, when the particle moves, for instance, along a circle in the plane at right angles to the field

H Q only discrete frequencies

areemitted,

that is, we are only dealing with magneto-bremsstrahlung, to use the termino­ logy employed here,

However, it is clear physically that also in that case

the radiation spectrum is practically continuous provided the radius of the circle is sufficiently large and

g/mc2 »1,

and it6 nature is in the appro­

priate frequency range close to the spectrum of the Cherenkov radiation.

In

view of what we have said the only consistent method is in the general case to consider magneto-brems and Cherenkov radiation and absorption in a unified way (Eidman, 1958, 1962). Let us dwell upon the defennination of the frequencies emitted (and absorbed) in a magnetized plasma in somewhat more detail. equation of the field amplitudes

where

**Xj + wXj ^Xj = ^

ir .

= czk£/n£j , while

R(t)

To do this we write down the

which we introduce earlier (see (7.12)),

(vayexpf-i^.R)} and v = d R / d t

(7.37)

i f(t)

are the radius vector and

velocity of the radiating particle. Equation (7.37) is obtained by substituting the expansion (7.12) into Eqn.(7.l4) for the vector potential, multiplying that equation by integrating over space. ’force*

f(t)

If we disregard

a

constant

a *^ exp {- i (k^• r ) } and

factor

the

form

of

the

in (7.37) is at once clear, as j ( V 8 X j ) exp { - i O V ' M d ’ r = e(v-8*^) exp { - i(kx * R)}

when je * e v 6 ( r - R)

(see (7.14)).

Equation (7.37) has a solution for

q^

which grows with time which corres­

ponds to radiation at only the frequencies spectrum of the ’force* f(t). we have

R *vt

the ’force’ f .

which are represented in the

If, for instance, the electron moves uniformly,

and only the frequency

d)= (k *v)

occurs in the spectrum of

The condition for radiation thus takes the form (i)^ . »Cd* (k*v)

148

THEORETICAL PHYSICS AND ASTROPHYSICI

that is, we get at once the Cherenkov condition (7.36) as we mentioned

already

in Chapter 6.

H0

For an electron in a magnetic field

along the z-axis we have

R = | r 0 cosuigt , R 0 sinui|jt , v ztj ,

W*

v1 CM“£t , v z} •

vi =

. (7.38)

f (t) * const x (- a*v \ x ± x

sin(i)*t + a* v, cos 0)* t + a* v ) * H y l H z z/

exp | - i ^ k R e sin a sinO)*t+kvzt cos a^| ,

where, for the sake of simplicity, we have put between

k

H0

and

(the

z-axis).

kx = 0,

and

a

is the angle

Using the expansion of a plane wave in

Bessel functions, + °o exp

i k^ R 0 sin a sin 0 *, (i) =

. s-Ei1)) d'r '

(1) where

j

n

2)

•

is changed.

Such

a formulation is a very general one — but, apparently, not generally accepted — and it must be sorted out. When a source moves in vacuo it emits electromagnetic waves either when its velocity

v>c

or, when it is accelerated,

v/c changes with time.

that is, formally when the parameter

It is just this last possibility which is usually con­

sidered in the theory of radiation, since a separate particle, apart from the particularly hypothetical tachyons, only can have a velocity consider in the next chapter sources moving with a velocity in a medium the parameter

v/Vp^ =» vn(aj)/c

properties

e^.(a),k)).

of

a

linear

medium

v/c

in the general case the electro­

are

characterized

by

the

tensor

Now, the superluminal regime, when vn/c > 1, is, firs tly, realized

without any particular difficulty, and radiation occurs even when (Cherenkov effect).

that independently of the value of the parameter

accelerated, when

vn/c = const

Secondly, it is clear by analogy with the vacuum case

it changes with time.

and/or time.

(we shall

plays the role of the parameter

(we are now considering a transparent medium; magnetic

v < c

v > c) . For motion

vn/c

radiation occurs when

But this is possible not only when the particle is

dv/dt^O,

but also in the case when

n changes in space

Indeed, the radiation is determined by the change in n

at the

position of the particle (radiator) or close to its position in the zone where the wave is formed.

In other words, the value

r£(t) is the position vector of the charge;

neglect now dispersion and the possibility that tory

r^(t)

n

t and

on

itself, but near it. r

is important, where

n

changes not on the trajec­

Such radiation, for which the dependence of

is ’responsible* is called transition radiation.

in its ’pure form* If, however,

n(t,r.(t))

for the sake of simplicity we

Of course,

transition radiation occurs only when v ■ const,

v > c/n

and/or

dv/dt i* 0,

v < c/n.

transition radiation occurs in combina­

tion with Cherenkov radiation and/or bremsstrahlung*

Later on, in order not

^ Bolotovskii (1960, 1962) and Zrelov (1968) have reviewed the results of solving problems in the theory of Cherenkov radiation when boundaries are present.

159

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION

to introduce unnecessary complica­ tions we assume that v = const , v
1

3

CM

2ei l[i +

>1 An [ , ------- V _ 1 -,} J L (l -v2/cz)w2J j J

id 2

P

J

or approximately U

TTc

, U)

in —

0)

u)

;

C

W

je ____ _ (

6 ttc

0) »

0)

o j **

J

e (i) = =-- r = (i) C [ l - v 2/c2P P Mc

U=K W

The

transition

radiation

in

the

doj

forward

Lit 3c

_L _ Mc,2

(7.68)

direction

lies

thus

fox

relativistic particles mainly in the X-ray range (for dense media to

I016 s” 1) and the characteristic frequency

and the total emitted energy is clear from

U

(7.67)

U)c ~

ultra~

1015

(1015 to 1016) £ / M c 2 s ~ l

increases linearly with increasing C / M c 2 . It

(7.67) and (7.68) that in the main in the case which is now under

consideration photons with energies

Fill) ~

= Fl(i)p (C/Mc2

are emitted, while

on average their number when the particle crosses the medium-vacuum boundary is equal to

U/Fiii) ~ e2/Fic »

1/137.

The application of transition radiation

for registering high-energy particles is therefore only realistic when one uses many boundaries, for instance, in the case of a set of plates (see

1960,1970; al.,

Bass and Yakovenko,

1965;

Arutyunyan et al.,

Garibyan,

1971, 1972; Wang et

1972; Ter-Mikaelyan, 1972; Alikhanyan et al., 1972; Cherry et al., 1972)..

For plates,

and

in

general

for

an

inhomogeneous

medium,

the

ratio

of

the

163

CHERENKOV EFFECT, DOPPLER EFFECT, TRANSITION RADIATION

Fig.7,5 where

t=0 thickness

SL

The zone

the radiation is formed

t

of

the

plate — or

another

length which

characterizes

the

inhomo-

geneity of the medium — to the zone where the radiation is formed plays an essential role. v « c/n

For a radiator at rest and when it moves with a velocity

the radiation is formed in a region of size

phase of the wave

I ~ c/n(j = X/2tt — and the

= kJl = (2tt/X)& changes just over such a distance by an

amount of the order of unity.

In the case of Cherenkov radiation when a change

moves in a channel or a gap with a characteristic size above that when

a/X « I

a

we also verified

the radiation remains the same as in a continuous

medium, that is, it is formed in a region with dimensions of the order of

X.

However, for fast moving sources the zone where the radiation is formed is, generally speaking, not at all equal to the length Cherenkov radiation the zone

t

X/2tt;

for the case of

of the formation is of the order of

X/2tt only

in the direction at right angles to the velocity which is just so important in the case of channels and gaps.

To find the zone

A, where the radiation is

v

formed for the case of a source moving with a velocity at an angle

0

the point

and the phase of the wave emitted by it in the direction of

equal to

A ^.

to

V

, we consider Fig.7.5.

The zone of formation

I

Let at time

and emitting waves

phase 7T

the source be at k

be

is defined as that distance along the

trajectory of the source (distance between the points

by

t= 0

A

and

B)

for which the

c

distances

ft e 10s s“ *,

already for

R>3km.

As the simplest model or example of motion with a superluminal velocity we can mention a light pulse of plane waves which is obliquely incident upon some plane dividing boundary (screen) (Frank, Fig*8.2 Incidence of a pulse on a plane screen

1942),

If we denote the

angle of incidence of the wave

THEORETICAL PHYSICS AND ASTROPHYSICS

176 on the screen by

y

(clearly, y

is the angle between the wavevector

normal to the screen; Fig.8.2) the that

is,

the

spot

of

light

on

intersection

the

screen,

k

and tha

of the pulse with the screen*

moves

along

the

screen with

a

velocity v = ---— — , n 2 sinY * where

n1 >1

(8.5)

is the index of refraction of the medium above the screen, which

for the sake of simplicity we assume to be non-dispersive — in fact,

it is

important for us only that the velocity of the light pulse is taken to equal c/n1.

Clearly, the velocity of the light spot, or to be more precise, the

streak of light, can always be made larger than incidence

¥,

and in vacuo it exceeds

c by changing the angle of

c in general for all angles

y

as in

that case

v =

’

(8-6)

A beam of electrons moving along the normal to the front of the beam with a velocity

u r Q and the boundary condition E * f($-ftt) at the surface of the cylinder r - r 0 . In the system of coordinates which rotates around the z-axis with velocity ft the field is thus a static one.

177

ON SUPERLUMINAL RADIATION SOURCES

I

Kp[is{ (ft/c)r + - ftt} ]

(0.8)

/F

s= 1

The constant phase surface is determined by the equation — r + - ftt c r

(8.9)

const

(8 .10 )

r = const Equation (8.10) is the equation of a spiral. radius

R

For a cylindrical screen of

at a large distance the equal phase surface intersects the screen

along the generatrix of the cylinder for which

(8 .11 )

R = const + c

*) ’

V where the angle

0 which determines the generatrix under consideration changes

with time according to

d0/dt = ft.

In other words, the intersection (spot)

moves along the screen with a velocity

r =R

v = R

dt

(8 .12)

= J2R .

We have thus in a more formal way ob­ tained the obvious — or, at any rate well known — result (8.4). tant that the angle

It is impor­

Y between the equal

phase surface and the screen is given by the condition (Fig.8.3) tan Y =

Fig.8.3 Spot from a rotat­ ing source (lighthouse) on a spherical or cylindrical screen

dr R d

c ftR

(8.13)

For small angles

Y we have, of course,

tan Y «

and

sin'i1 « t

agreement with

(8.6).

v « c/sinY,

in

In other words,

the large velocity is, as we noted earlier, caused (for instance, when

v » c) by the angle

V

between the wavefront and the screen being small.

We have here in fact not made any assumptions about the nature of the field and only — and this for the sake of simplicity — assumed that its speed of propagation was equal to v>c

c.

Hence it is clear that spots with velocities

can be obtained not only in the case of electromagnetic waves, but also

in that of gravitational waves.

Using the ray treatment we are led to the

THEORETICAL PHYSICS AND ASTROPHYSICS

178

possibility that we can have spots moving with any velocities both for neutri­ nos (velocity

c)

and for any other kind of particle (velocity

fact that the appearance of a velocity

v>c

u < c)

The

for spots does not contradict the

theory of relativity cannot cause any shadows of doubt.

It is sufficient to

say that this result is obtained for completely realistic examples, for instance, when a pulse of light or of electrons impinges on a screen (see Fig. 8.2).

Nevertheless, as a supplement we note that the application of the

velocity of light for synchronizing watches, which is normally used in an exposition of the theory of relativity is, firstly, not a unique method, but only one of a number of possible methods.

Secondly, such a method is, indeed,

in most cases the most convenient and appropriate one, not because the velocity of light is the maximum possible speed, but because it is universal — that is, it is the same for all inertial systems of reference (of course, provided one chooses identical rulers and clocks in all systems). theless speaks of the velocity of light in vacuo

Finally, when one never­

c as the maximum possible

speed, one has in mind the velocity with which perturbations, interactions, or ’signals’ can be transferred.

Such a statement is, indeed, valid — at least

in the framework of the theory of relativity and in the whole of physics known to us.

Light

and other spots which we dis­ cussed

do not

violate

that

statement, although they can move with velocities

v > c,

that is, they cannot be used to transfer velocity

a signal with a

v > c.

Indeed, let

us consider a pulse (of light or of electrons) the inter­ section of which with a screen (spot) moves along the screen along the x-axis with a velo­ city

v > c and reaches the

points 1 and 2 with coordinates

Fig.8.4 Intersection of a pulse and a plane screen

When a rotating source emits particles with a velocity u the trajectories of such particles are: r * r + u ( t - t ), s ftt0, whence r » r Q + u(t-$/fl), where t Q is the time of emission.

ON SUPERLUMINAL RADIATION SOURCES and if

x2

at time

v = u/sin ¥ > c,

that is,

tj

and

t2

179

(Fig.8,4).

the events

1 and

(x2 - X j ) 2 > c (t2 - t j ) 2 .

Clearly, x 2 -

and

3 with coordinates

The perturbation ('cut') which is 'plotted1 tj

turns out to be at time

x 3 = Xj + u sin ¥ (t2 ~ t ) ,

(x3 - x ^ 2 + y 2 = u 2 (t^ - t ^) 2 < c 2 (t2 - t i)2 .

ever, not hit the point 2. is. of course,

and

2 are separated by a spatial interval

in the point 1 on the moving pulse at time the point

+ v (t2 - 1 2)

t2 at

y 3 = u ( t 2 - t 1) c o s V

This perturbation does, how­

Nonetheless, the super luminal velocity of the spot

in no sense an apparent one;

it is just as real as any other

velocity of a macroscopic structure or body.

Therefore we emphasize that the

superluminal velocities of spots are of a different nature frcm the apparent velocities such as because

we

are

u ± max (see (8.3)); the velocity

dealing

with

the

observation

at

u ± max

a

which are emitted at different times (vide supra).

given

can exceed time

c

t of

signals

The retardation caused by

the fact that the speed of propagation of light is finite is then important. Taking retardation into account also affects appreciably the behaviour of spot when they are observed

at

some point.

We shall restrict ourselves here to the simplest example of

a

light spot which

moves with a constant velocity a plane screen point

0'

and is

(Fig.8.5).

understand

here

v

observed

along in

the

By observation we

the

reception

of the

light emitted by the spot when it hits the screen, that is, as the result of scattering, or due to luminescence of the screen when it is illuminated.

If

v ^ c, the spot will be observed in the 'normal' way, as a spot moving on the screen from top to bottom.

Fig. 8.5 Observation of a spot on a screen

Let us now

assume that

v -*■ ® , that is, that the

whole

of

track

the

instantaneously.

spot

is

outlined

In that case the spot

will first of all be noted in the point 0

which is closest to

screen).

0'

(the

straight

After that the observer sees,

from the point

0

line

in opposite directions.

also observe two spots.

00'

clearly, When

is

two

perpendicular spots

c have the form

Ae Yae exp { - K e z - i ( u ) t - 6 e - k ez)} d u +

Au) +

A Y exp {- K z - i ( u ) t - 6 - k z)}dai J o ao v L o o o J A oj

.

If we now form the tensor D D* from these components and also evaluate the d * O' p derivative J ^ ( Da D g) an||(v,e)Ne ( £ ) d e

One can easily calculate the emissivity (9.28) for a power-law spectrum, using Eqns,(9.19) and (5.42).

We restrict ourselves here to the remark that when

there is no reabsorption it is clear from (9.25) that we get for the selfradiation of a uniform source of size (o) _ .•

II

’

L

l (o)

c

I (o)

C||

= 3y + 5 2

(9.29)

When we take reabsorption into account we get by integrating Eqn.(9.27) with the condition that at the start of the layer (where e

I± = —

I .= 0 1, II

e ||

[> “ e x P ( - U 1 z ) ]

For a thin layer (source of size

\

X!0) _ Ei . 3 V ^ 5

I„

L( o )

II

z ■ 0)

2

»

X || =

L)

t 1 “ e x P (“ ^||z ) ]

(9.30)

« 1 and „

n

V V

1 !! _ „ n 1!

_ Y + 1 (9.31) v +i

REABSORPTION

and

radiative

For a thick layer

u

.. L » 1 * II

and

.Vi r ll Of course, the index

,

e llM i

207

transfer

(9 .3 2 ) V

3y + 8

those parts of expressions y

3

n 1 !!!

6y + 13

(9.31) and (9.32) which do not contain

have a general value and refer not solely to a power-law spectrum.

We remind ourselves that when we use a power-lav spectrum in our calculations we assume that

y >y

(see Chapter 5).

Assume that the magnetic field along the line of sight is on average random in direction.

We further assume that when waves propagate in such a field

their polarization does not change when the direction of the field changes (this occurs when the geometric optics approximation is inapplicable to des­ cribe the polarization of the normal waves due to the fact that conditions such as

X|dne>0/dz| « |ne - n Q |, which we mentioned earlier, are not satisfied;

for details see Zheleznyakov,

1970, §24).

In such cases the anisotropy of the

absorption vanishes for the propagation of waves in a random field and waves with any polarization will be absorbed identically with some absorption coeffi­ cient

y ,

For a given angle

x

the average absorption coefficient is equal to

l (W1 + >J ||) = U(x> To obtain field

H

y,

that is, the average of

y(x)

over the angles

x between the

and the line of sight (the velocity of the emitting electrons)

it is

natural to evaluate the expression TT

•I

U(X) sinx dx = Jy K _ ™ 8

7 m 8 W

L (9 B ext) = €-((*), -B ext)

non- symmetric (in

which may occur in antiferromagnetics

ELECTRODYNAMICS OP MEDIA WITH SPATIAL DISPERSION

223

when there is no external magnetic field present;

in antiferromagnetics B eJft

is the statistically averaged magnetization in a given micro-volume of the crystal, which vanishes only on the average taken over the whole elementary cell of the magnetic structure of the crystal). The definition of a gyrotropic medium as a medium with a non-symmetric tensor

c. .(u i k)

is, of course, somewhat formal.

It will, however, become clear from

what follows that it is just the non-symmetric which leads to those peculiar features

(for

feature

instance,

of

the

the

rotation

tensor of

the

plane of polarization in normal waves when there is no absorption) which dis­ tinguish gyrotropic from non-gyro tropic media. It is often convenient to split the tensor parts

Re £ . . and ij

Ime.. ij

into its real and imaginary

and also into two Hermitean tensors

c '.

*J

and t f! . XJ

£. . = Re e. . + i Ime. . , ij iJ iJ €. . = £' . IJ 1J

i e" ij

(10.14)

€?. IJ

(fi'T Ji

(10.15)

(e'M v Ji'

where the asterisks, as in other cases, indicate complex conjugates; we note the Hermitean conductivity

that one sometimes introduces instead of ij tensor

,

a . . , defined by lj

4 7T a . . --- iJ-. (1)

IJ

(10.16)

It is true that one also uses the complex conductivity tensor a.. * a /. + i a.,/. = -iuj(e..-6..)/47r ; ij ij M v ij iJ in that case the quantity

should occur in (10.16).

When one neglects spatial dispersion and when there is no constant magnetic field present we find clearly from (10.12) and (10.13) that the tensor

^ (a))

is symmetric

e. .(u3) = £.. (u>) *j

ao.i7)

ji­

Both in this case and in the more general case (10.11) we have clearly Re £. . = £.'.

iJ

i-J

It follows from requirements which that the function £^j(u) » k)

and

are

Im£. . = £ r.f. .

ij

J-J

connected with the causality principle

in a medium which is in equilibrium (or at least

which is stable) does not have any singularities in the upper half-plane and on the real axis of the complex variable

U) .

stand

and

a whole

number

of

relations

for

Using this fact one can under­ properties

of

the

functions

THEORETICAL PHYSICS AMD ASTROPHYSICS

224 e^(o),k).

The

connecting

most

important

Re£.j((D,k)

with

ones

of

those

Ime^j(a),k) .

are

the

dispersion relations

Taking spatial dispersion into

account introduces here little what is new — it usually reduces to the same relations as for the function parameter.

j (u>) , but including the vavevector k

as a

We shall therefore not dwell upon the dispersion relations — this

is done in detail by Landau and Lifshitz (I960) for the case without spatial dispersion in an application to an isotropic medium.

The generalization to

the case of an anisotropic medium and a medium with spatial dispersion can be found in

§ 1 of the book by Agranovich and Ginzburg (1966).

An important part of the electrodynamics of continuous media is devoted to the study of the propagation of electromagnetic waves produced by sources outside the medium considered. just that kind of

In particular, in crystal optics one studies usually

problem

and most often one is dealing with even a narrower

problem — the propagation of plane monochromatic waves in which the electric field is of the form c _ c i (k •r) - iu)t 4 c - co e * here nates

(10.18)

E 0 is a complex vector which is independent of the space and time coordi­ r

and

t, while

k

and u) are the wavevector and frequency.

One should bear in mind that Eqn.(10.l8) with

E 0 = constant

is not the most

general one — sometimes one must consider also a field of the type (10.18), but with

E 0 = Ea>( k * r ) ^ E a> = constant.

However, this necessity arises only very

rarely (case of singular axes in crystals of the lowest crystal systems and a few other cases;

see Agranovich and Ginzburg,

1966, §2).

We shall therefore

restrict ourselves in what follows to expressions of the form (10.18). Solutions of the type (10.18) satisfy the homogeneous electromagnetic field equation, that is, Eqns.(lO.l) without the external (given) currents and charges j

and

Pe x t * only if k

and

u

are interrelated.

This connection

is given by the dispersion equation and allows us, for example,

to express k

in terms of u) :

k Here tion,

n = n+itc k

(u >s )s

is the complex index of refraction,

the index of absorption (p-2u)k:/c

terms of intensity), and plane waves in which

s

(10.19) n

is the index of refrac­

is the absorption coefficient in

a real unit vector (we consider now only uniform

k = k ! + i k 2 , k x and

k 2 being collinear; moreover,

in

accordance with the statement of the problem which one encounters in optics we have taken the frequency

u) to be real).

The dispersion equation determines

ELECTRODYNAMICS OF MEDIA WITH SPATIAL DISPERSION th* function

n

225

in t e n s of the coefficients which occur in the field equations,

that is, in terms of the permittivity tensor there correspond several values other solution — a normal wave.

and to each value of o> and k

n = n^ where the index

I

corresponds to one or

Normal waves (for given (d and I , but different

Jt) differ also in their polarization, that is, in the vector

E 0^

in (10.18)

which is determined (apart from a multiplying factor) from the field equations. The problem of crystal optics^ consists formally speaking, thus in the first instance of a study of the functions

n^(o),s) and E 0^(d),s).

In turn all

information about these functions is contained, if we do not mention the field equations, in the complex dielectric permittivity tensor e^(a),k) .

If we in

this case neglect spatial dispersion, that is, assume that € ^ - € .j (a>) , the problem reduces to what is usually called 'classical' crystal optics.

In this

way classical crystal optics is, of course, contained in crystal optics with spatial dispersion as a special (or limiting) case. In the optics of practically non-absorbing or weakly absorbing crystals the spatial dispersion is weak in that sense that its magnitude is determined, as we already mentioned, by the small parameter ka ~

A

AQ

«

1 .

(10.20)

This is just the reason why one can usually neglect spatial dispersion in crystal optics, unless we are dealing with qualitatively new effects (gyrotropy, optical anisotropy of cubic crystals, the appearance of additional normal waves, a non-vanishing group velocity for longitudinal waves, and so on).

Moreover,

strongly

and

the presence of a small parameter enables us to simplify

to make

more

precise

the

study

of

the

effect

of

spatial

dispersion. Taking spatial dispersion into account in crystal optics and in general in electrodynamics is not something principally new and it can be traced back to the last century.

However, it is only in the last 15 to 20 years that crystal

optics with spatial dispersion became a more or less independent subject of study.

At the same time, and even more strikingly, taking spatial dispersion

into account became an organic constituent of part of modern electrodynamics of continuous media, of plasma physics, solid state theory, theory of metals, and so on.

The modern structure and position of crystal optics, even if we

^ We use here this term for the sake of simplicity also when we are deal­ ing with a more general case, namely, the propagation of waves in an arbitrary (in general, anisotropic) medium.

THEORETICAL PHYSICS AND ASTROPHY0ICI

226

forget about plasma physics, must thereby be based upon electrodynamics takln| spatial dispersion into account.

In other words, one should start from the

relation (10.5) and obtain a whole series of general results, and after that go over as a particular (albeit very important) case to the exposition of classical crystal optics.

However, in the optical literature one proceeds,

as a rule, in the old-fashioned way — one develops to begin with, and often even exclusively,

the classical crystal optics.

This is just the reason why

it is appropriate to discuss in the present book crystal optics taking spatial dispersion into account. We now turn to finding all normal electromagnetic waves of the type (10.18) in an infinite uniform medium characterized by the tensor satisfy Eqns.(lO.l) with

Jext;~ 0

and

Pe x t =0

k).

Such waves

from which we get the wave

equation 1 32D curl curl E + —j — T = 0 .

(10.21)

C 31

For the plane waves (10.18) Eqn . (10.21) takes the form

h

(i)

{kz E - k (k -E)}

(10.22)

.

Substituting here the relation (10.5) we find j ^ 4 e . . ( < o , k ) - k 2 6.. + k. k.l E. = 0 .

l c 2 i]

iJ

i

jJ

This algebraic set of equations has a non-trivial solution determinant vanishes,

(10.23)

J

E^0

provided its

that is, provided

c2

o .

e. .(u> , k) - k 2 6. . + k. k. u 1 j

(10.24)

Equation (10.24) is often called the dispersion equation — it establishes a relation between b) and k

for normal waves, and its solution can be written

in the form “t “

' A - 1, 2. 3, ...

(10.25)

or in the form (10.19). that is, expressing k

in terms of

k

.

where the index

. 5* - - j r -

b) , (,0 , S ) : r2

(«ij *i s j ) 5 M

One can also

(«ij 8i 8 j ) *Tck - ) = e(a>) E(a)) As in a longitudinal wave by definition E||-Ek/k, in it D - 0 and the field

E

(see (10.22))

is non-vanishing only when condition (10.30) is satisfied.

When, however, spatial dispersion is taken into account Eqn.(10.28) gives for longitudinal waves a dispersion relation U|| - a>[|(k). Waves with

E^ t 0

play the main role in crystal optics, as just these waves

are most strongly excited by light.

When spatial dispersion is taken into

account Eqn.(10.27) can for these waves have in some spectral ranges not two, but a larger number of solutions.

However, even in the case when new solutions

('additional waves') do not turn up, spatial dispersion leads to a number of new effects amongst which natural optical activity (gyrotropy), and also the optical anisotropy of cubic crystals (it

is well

known

that

when

spatial

THEORETICAL PHYSICS AND ASTROPHYSICS

220

dispersion is neglected cubic crystals are optically isotropic) are the most important ones.

All these effects were already mentioned and are discussed

below, and for their analysis in a phenomenological theory it is necessary to know how the tensor

e^(u>,k)

depends on k

for small

k.

Before we turn to that problem, we make a few remarks. bear in mind that when we consider the k-dependence

For instance, we must

of

e^j (a) * k)

remember that Eqn.(l0.5), which connects the quantities the same value of k , was

obtained

medium is spatially uniform.

earlier

under

the

D

and

we must

E

for one and

assumption

that

the

However, crystals are in actual fact not spatially

uniform media as, for instance, the lattice sites are not equivalent to other points.

Use of the tensor

e^j(u>,k) , which was introduced under the assump­

tion of a uniform medium (after statistical averaging), for applications to crystals must therefore clearly be limited. Agranovich and Ginzburg, use of Eqn.(10.5)

An analysis of this problem (see

1966) enables us to reach

the

conclusion that the

in crystals is justified provided the wavevector

k

is small

compared to the basis vectors of the elementary cell of the reciprocal lattice, that is, provided inequalities,

k

1/a or

\ » a,

where

the meaning of which is

a

obvious

is the lattice constant.

These

already from purely qualitative

considerations, are clearly satisfied in the optical range of wavelengths where

a/X ~ 10~3 .

We shall therefore in what follows use the tensor

e£j(k)

without restrictions in crystal optics. Another and completely independent problem about the conditions of the applic­ ability of the material Eqn. (10.5)

when

spatial

dispersion

is

taken

into

account arises when we change from a crystal of infinite extent which is, of course, an idealization (which, however, was used when we changed from (10.3) to (10.5) and (10.6)) to finite size crystals.

As E q n . (10.3) is an integral

relation the presence of boundaries of the crystal have been taken into account in it and the exact boundary conditions for the fields are retained.

If the

point r

is far from the crystal surface at a distance appreciably longer than

the size

R

of the neighbourhood which makes the main contribution to the A

value of the induction, the kernel

A

becomes equal to the kernel

(10.6) which is used for an infinite crystal.

c^j

in

For such points the electric

field in the form (10.4) clearly leads to the induction

D ( r , t ) = D ( a , , k ) e i (k ’ r ) - i&)t which is also in the form of a plane wave and the relation between the ampli“ tudes of

E

and

D

is just given by Eqn,(10.5).

It therefore follows from

electrodynamics

of media with

spatial

229

dispersion

what we have said that one can use the material equation in the form (10*5) provided the thickness of the crystal is large compared to the magnitude of

R

is usually of the order of

a

where

R. a

In dielectrics

is the lattice

constant. One should, moreover, note that when we consider additional waves and at the same time use the tensor

££j(u>,k) , and not the general integral relation

(10.3), the boundary conditions (10.2) are insufficient.

One can in principle

obtain additional boundary conditions (10.3), but usually they are introduced less rigorously from different considerations (see Agranovich and Ginzburg, 1966, §10).

One should in connection of the problem of the boundary condi­

tions in the electrodynamics of media with spatial dispersion make yet another remark which one should bear in mind also when there are not additional waves present.

The question is the necessity to take into account the possible role

of higher derivatives when writing down the actual form of the boundary condi­ tions (10.2).

Let us elucidate this using the equation div D « 0

as an example.

To obtain the boundary condition at the boundary between the media 1 and 2 we take a limit — we integrate the equation 3D 3D 3D * + -JL + *

div D .

dx

dy

dz

along a direction normal to the diffuse separating boundary. direction as the

z-axis

Taking that

and taking the limit of a sharp boundary we get the

condi tion

^2z ~ Di2 “ where we have put £ = £1

€2^2z“

0

ei Eiz *

(10.31)

,

D ■ £ E when changing to the second equation, and have taken

in the first and

£ = e2

D = £E +

in the second medium. curl E + curl (6j

where, of course, in a uniform medium

j

Let us now assume that

E),

(10.32)

D = e E + (i )/u)i] - p k 2

(10.42)

235

ELECTRODYNAMICS OF MEDIA WIfH SPATIAL DISPERSION (the index indexes

i

of 0)^ has, of course, nothing whatever to do with the tensor

i, j, ... ).

the behaviour of

This kind of expression sometimes approximately describes

e(u) , k) in the vicinity of the frequency of a quadrupole

absorption line. As long as the term p k 2 in (10.42) is unimportant we are dealing with an expansion of the form (10.40).

However, in the general case -pk2

[e(io , k) -e(oj)]'

Pkz

which corresponds to neither (10.40) nor (10.41).

One can, however, easily

generalize Eq n . (10.42) for any crystal in the spirit of the phenomenological expansion (10.40): E . .(a))

e.j(w , k)

ij ' 7

+

Y..fl(u>)k0 ij

+ a..n (o> , k) k. k ,

7 I

ljJhn

7 Jim*

(10.43) = CijJlm(a,) + in ijftmn(a,)kn + ^ i j fcnnp^ kn kp * Similarly we can replace

Y

i

n

(10.40) by

^j£(w,k),

and so o n -

For

a non-gyrotropic cubic crystal Eqns.(10.43) and (10.42) are equivalent where p(w) k knk = ijto 1 m

6. . [ (

- yk2

If we generalize E q n . (10.42) somewhat and write it in the form a. . k 0 k e..(U ,k) = e..(M ) + ---------- u i 2 L _ L j n ---- ----- , 1J

1J

(10.44)

[(u-UjJ/a) ] + iv + ^ Jllnk jl 'cm

at resonance, that is, as w -►(!). and v = 0, the tensor e..(u),k) becomes as * i lj dependent, in general, on s =k/k, that is, it turns out to be a non-

k-^0

analytical function of k . the analytical behaviour of even at resonance.

However, in actual fact we have always E^.(u3,k) as function of

k

as k ** 0

v^0

and

is retained,

Therefore, the necessity for an expansion such as (10.43)

can arise only in the vicinity of a resonance and when we neglect damping. When we take damping into account, however,

the use of an expansion such as

(10.43) can be justified in general only when we study the effects of spatial dispersion to higher order. The tensors occurring in Eqns.(10.39) to (10.41) satisfy a number of relations following from the general symmetry properties of the tensor we discussed earlier.

€ ^ ( u ) , k ) which

For instance, by virtue of (10.10)

e. .(w) - E..(u)) , Y. .Q(oj) - - Y..n(w) , a. .0 (w) - a..0 (a>) ,

ij

7

j i ' 7 * ijJI' 7

j i l t ' 7 * ij£mx 7

jixm

(10.45)

THEORETICAL PHYSICS AND ASTROPHYSICS

236

Moreover, one can always choose the tensors that

a...

1j£m

=a..

and

ljmJc

a ^j^m

and

^ijJim *n 8UC^ a vaV

B..rt = B. . „ (we shall assume in what follows that we

ljJem

have made such a choice).

ijm£

We remind ourselves also that the magnetic induc­

tion of the external field

® ext

is always, unless a statement to the contrary

is made, assumed to be zero. When there is a centre of symmetry or in general for a non-gyro tropic medium it follows from (10.12) that

6. When there is no absorption and when k and hence the tensor

.

ljL

Yijit= 0

=0

(10.46) Ime. .(u) , k) = 0 xj v is Hermitean. By virtue of

is real,

e^((i>,k) = R e e „ ( u > , k )

the tensor

(10.45) in this case, all the tensors

* EIj(o,) * Yiji(to) * 6ijH(0,) * are real.

* and Bij*m(o,)

We mentioned earlier dipole and quadrupole absorption lines.

In

fact, near dipole lines it is usually sufficient to use the expansions (10.39), (10.40), or (10.41) while Eqns.(10.42) to (10.44) can be met with in first instance in the case of quadrupole lines.

We must at the same time emphasize

that the above mentioned expansions in series

k

are not multipole expansions.

Moreover, the use of Eqns.(10.39) to (10.41) is not limited at all to the region of some lines.

We must also bear in mind that in an arbitrary optic­

ally anisotropic medium and for an arbitrary direction of propagation of light the appearance of resonance lines (or when we neglect absorption — the appearance of a pole in the function n 2 = n 2) ance growth of the components of

e^(u ) , k )

is not connected with the .reson­ as is already clear from the

formulae of classical crystal optics (see, for instance, (10.27)). The tensors

E..(u),k),

eij(w)

E ”1(u),k)

and, of course,

*eij(w) * Yij* ’ 6iji ’

the tensors

and

Bij*m

which occur in (10.40) and (10.41) can be significantly simplified when the crystal has symnetry elements.

For instance, when there is a centre of sym­

metry

Y . . . - 6 . . # a 0 (see (10.46)) and in an isotropic, non-gyro tropic medium ij£ ij* has only two independent components so that in that case the tensor a..„ l jilm (see also Chapter 11)

, k) * e(oo)

♦ « 1(u) ( 6 „ + s ^ ) k 2 ♦ ljJ• lm 0.i j .J h. n (vi d )7 n 2 s I 0s m

tensor

C^Ou.k)

to diagonal

(I0.51a)

(10.51b)

form

THEORETICAL PHYSICS AND ASTROPHYSICS

238 by suitably choosing principal axes.^ axes

is

e..(u));

not

the

same

as

that

of

For arbitrary

I or

as

that

s the direction of these

of

the

in those cases where the axes of the tensor

axes

e. .( id)

of

the

tensor

are fixed (that

is, when there is no degeneracy, which occurs in cubic and uni-axial crystals) the axes of the tensor e^((D , k ) fact that the s-dependent

are close to the axes of

((d)

j

due to the

terms in (10.50) and (10.51) are small.

In crystal optics with spatial dispersion there is, of course, great interest in those principal axes of

E£j((D,k)

which have the same direction as s.

For rhombic crystals the x- , y- , z-axes are such axes. vector s

If, for instance, the

is along the x-axis, the principal values of the tensor

e£j(co,k)

are equal to (see here and below Table III in Agranovich and Ginxburg,

C

1

2

= t

eXX

(ui, k) = '

XX X

= e

yy

() and denote the principal values of that tensor by and

I/n22 .

ties of order

k2

The components

and

for this choice of axes.

turn out to be small quanti­ The condition that the determinant

of the 9et (10.54) must vanish gives an equation which enables us to determine

THEORETICAL PHYSICS AND ASTROPHYSICS

242 the possible values of

n2 .

If we drop terms of order

and so on.

k 3 , k"

this equation has the form

GHiO&-*s) ■5;

(10.55)

223 (u , k) 4 " 2

If we put in Eqn. (10.55) 6 123 * 0

and

k = 0, its solutions

n 2 #2 = n 2 #12

of course, the same as the solutions of the Fresnel equation with If, however, 6 123 4 0 tion of s

and

even when

are>

€ £ j “ c £j(^)

k ^ 0, E q n . (10.55) determines for a given direc­ = E^(co)

and

= e“y(u)) not two ^u t »

general,

several values of the index of refraction (Eqn.(10.55) is for non-longitudinal waves in the given approximation the same as (10.27), as should be the case). The analysis of Eqn . (10.55) is particularly simple for media which are optic­ ally isotropic when spatial dispersion is neglected. For an isotropic medium

E ^ ( od , 0) = e^y(o) » °) = l/e(u>).

(10.55) that in that case for an isotropic medium with lect the k-dependence of

the quantities

£xx , c ”y , and

instead of (10.55) the following equation to determine v2 , x2 ( 1 I V U)2 2 2

W

" e(u)j

"

C2

This equation has clearly three roots for n2 , n2 , n2 .

One sees easily from 6 l23 / 0

>2 3

6 123

one can neg­

so that we get

n (10.56)

’

n

n 2 , that is, we find three values

It may turn out that all three roots of Eqn.(10.56) correspond

to relative long wavelengths and it is then admissible to consider all three solutions in the framework of a macroscopic approach. vicinity of resonance (that is, when lect the (o-dependence

of

6 123

« u>£ , where

check

the validity of

the

detailed discussion of the results Agranovich and Ginzburg (1966).

n 2 (to)

condition

can be

for all three solutions and an/XQ«

found

in

I

(see

§6.3

of

(10.20)). the book

6 123

and

€(to)

A by

We shall therefore here only give a few re­

sults for the case where we neglect the absorption of the waves. ties

to be

the solution of Eqn . (10.56) then allows us to

find in the resonance region the functions to

in the

e(to^) -*■*) one can neg­

and one may assume the function e(u))

known in that frequency range;

thus

In particular,

The quanti­

are then real and in the region to « to- we can put 2 A to? (10.57)

where

£ “ (to-to. )/to. , A - 2ir Neff e*/m(o? ; here

and mass of a free electron,

N eff/N

e

and

m

are the charge

is the oscillator strength, where

the total number of electrons per unit volume and

Neff

N

that part of them

which 'effectively* determines the optical properties of the medium in the

is

243

ELECTRODYNAMICS OF MEDIA WITH SPATIAL DISPERSION

sp ectral

region considered.

The behaviour of the functions

n(£)

obtained by solving Eqn.(10.56) for this case is shown in Fig. 10.1. An interest­ ing feature of the dispersion curves shown there is that to the right of the turning point

£m = (um - Ci)£>/u>£

there

exists only a single real solution whereas to the left of it there are three real solutions.

We note that

the multiple root (that is, the turn­ ing point) corresponds to the frequency godi

which satisfies the equation

e K i ) = t 2*

&i 2 >)

*•

do.58)

The solution of E q n . (10.55) is even simpler in the case of an isotropic Fig.10.1 The refractive index n t , n 2 , n 3 as function of E = (go - u^) /gu£

non-gyrotropic medium.

As in that

case we have not only the equation

near the reson­

ance frequency co^ for the case of a gyrotropic, but iso­

e^i(a), 0) = e ^ ( w » o) - l/e(u>), but also

6123

tropic and non-absorbing medium

Eqn. (10.55) becomes

j_ _ j _ _ w

n2

a Z2

(10.59)

C (go)

This equation is obtained from (10.55) by writing l + Bk2 e“ 1 (oi, k ) = e -1 (a), k ) = xx ' yy e(w) Equation (10.59) determines two values of the refractive index

n2

1*2

which

correspond to the same polarization of light, and i ‘1 2' n’ where go

3' = ( go2/ c 2) 3 .

V 2e(oo)6' When

3' veff) '

For an unlimited applicability of the classical theory it is also necessary that the electron gas stays non-degenerate;

this is equivalent to the require­

ment T > T 0 ~ fi2 N * / m k B . The meaning of the degeneracy temperature temperature the energy

TQ

is the following:

(11.19) at this

kfiT 0 is of the order of the zero-point energy

connected with the localization of an electron in a volume of order

r 3 ~ 1/N.

In most cases which one encounters (although not by any means always) the con­ ditions (11.16) to (11.19) are satisfied and it turns out that the restrictions connected with the fact that so far we have neglected spatial dispersion are

255

THEORETICAL PHYSICS AND ASTROPHYSICS more important.

By its very nature spatial dispersion (see Chapter 10) is

a wave with frequency

unimportant only when the field (say, the field of

production of polarization

a

'response*

of

the medium, for instance, the production of a

P as a result of the field E.

When we take thermal motion into

account an electron traverses in a plasma during a period

£~

tv

(u

X ■ 27r/k) changes little over a distance corresponding to the

and wavelength

/kgT/m.

It

follows

spatial dispersion, provided

from what we have £


f. - f. 1

1,00

(v)

01.25)

We have assumed here that the electron and ion temperatures are the same. must, of course, be the case for complete equilibrium. . f 9ee footnote on next page

This

One must, however,

DIELECTRIC PERMITTIVITY AND WAVE PROPAGATION IN A PLASMA bear

in

mind

momentum) in

that a

the

exchange

of momentum

1970b).

respect

to

m/M ^ I0” 3

is small;

(a

for details

One sometimes therefore considers a non-isothermal

plasma with the Maxwellian distributions

(11.25) but with different electron

and ion temperatures equal to, respectively, Te »

(relaxation with

plasma proceeds appreciably faster than energy exchange

consequence of the fact that the parameter see Ginzburg,

257

T£

and

T^

(in some cases even

Ti).

If we are interested in problems relating to the realm of the linear theory (in particular, be small^

linear electrodynamics) we may assume the electric field

E

to

and, in accordance with that consider the change in the distribution

function as a perturbation.

In other words we shall look for the distribution

function in the form

f = f00(v) +

v)

,

|f'|«foo>

(11.26)

where we choose the Maxwell distribution (11.25) for the function try to find the tensor

C.j( , k )

E tr = k 2 E tr

which leads to condition (11.51).

01.53)

THEORETICAL PHYSICS AND ASTROPHYSICS

264 By virtue of (11.54) and (11.2)

e'

n

-

4TTCJ k

2n K

ID

n

(1K55)

K

where the square root inside the braces is always assumed to be positive (for instance, for We must for

a*0 C(id)

and

Zr < 0

J |c7 |= - J

this root equals

).

in (11.54) take for transverse waves in the plasma Eqn.

(11.6) whence we get simple formulae in the limiting cases. For instance, when |c'| 1 1 »

^ 0)

(11.56)

,

we have if

e' >

o ,

r n«/e'

=

1 ■ - r

P 2 -1 effJ

’

(11.57) 2ttq

p

u /e ' if

e' < 0 ,

n «

eff

4

2o)(o)2 + v 2f f ) / e '

ttN

e^

p

(l)2 V r r p eff

2 tto

uA-e')

2u(u2 + v2ff)A - e') ,

^-cr

(11.58)

“J _ ,1* = [ (DZ + v| ff -1

If, however,

|e'|

a)

(11.59)

*

we have * [— tV e« f L2(i)((d2 + V 2ff)J According to (11.57) the refractive index sions is small, so that

id2

n < 1 , vph = c / n > c

n < l ; when the number of colli­

5.64 x 10* /N

(11.61)

and Cherenkov radiation and thus absorption by sepa­

rate particles is impossible.

Hence it is clear that the whole of the damping

is connected solely with collisions

(we have

and are not damped in the medium as

v ef f “>'^»

remark which follows).

(11.60)

» v 2 __ , this case is in practice realized when erf

(i) > a) P If

.

in mind

waves

which

propagate

see condition (11.61) and the

The result (11,41) does not contradict what we have

just said, as it is obtained in the non-relativistic approximation and using a Maxwellian velocity distribution.

In this distribution there are formally

dielectric

permittivity

and

wave

also particles with velocities

propagation

v>c

in a p l a s m a

265

and this leads to the appearance of an

exponentially weak damping of transverse waves in (11.41).

A relativistic

calculation leads, of course, to a complete absence of collisionless damping of transverse waves in a plasma. full, as the expression for

Nonetheless, Eqn.(11.41) was written out in

e tr

is suicable not only for considering normal

waves (we emphasize this important fact yet again;

see Chapter 10).

Moreover,

for transverse waves the factor exp (-o)Z/2k2 v£) = exp (- c2/2n2 (u)v2 ) is so negligibly small that the corresponding damping is all the same practic­ ally equal to zero. For transverse waves in an isotropic plasma taking spatial dispersion into account therefore does not play a role and Eqns.(11.57) are adequate and they are widely used in radio-astronomy and the theory of propagation of radio-waves in the ionosphere. Even in the case of weak true absorption, that is, under condition (11.56), the field in the plasma may be strongly damped - this occurs when that case, say, when

where

E q is the field at

e'uip)

or

non-transparent (the same plasma for (i) e « 1 we have for the low-frequency to

T e , but with particles of mass

isothermal M.

sound

in

a gas

with

When collisions are neglected

longitudinal waves are always strongly damped in an isothermal plasma.

The

presence of collisions and, especially, a large number of neutral particles (that is, when we change to a weakly ionized plasma) changes the picture in the region of very long wavelengths (such that the mean free path v /veff compared to the wavelength, that is,

kv « v efj).

Is small

In that case we are dealing

already with waves related to ordinary sound which may be weakly damped. We now turn to a consideration of a magnetized plasma and, to be precise, a uniform plasma in a uniform and constant magnetic field

H 0.

In the framework of the elementary theory taking the magnetic field account reduces to adding the Lorentz force

to Eqn.(ll.5),

as

H 0 into

a result

of

which we have n,rn + m V eff*rn = e E o e’1(0t + f [?n A H o] •

(11.69)

Solving this equation and the analogous equation for the ions we find r f

and

(i) , and after that the current i - •

At the same time, by definition, the two expressions we find

n= 1 j^ = - (ito/4 tt) (e^^ - 6 ^ ) Ej

and by comparing

We have elsewhere (Ginzburg, 1970b,

§ 10)

given the corresponding calculations in detail and here we shall merely quote the results:

DIELECTRIC permittivity and wave propagation in

c xx = e yy

= 1

»

,

\0)2 +

plasma

a

269

)

J a)2 - a)0)u + H

eff'

W^ 10* i V eff>

(

io{((o + iveff)2-ug} C Z2

e

= 1 ----p----E---tO ((0 + i v _r ) eff'

--e Xy

(11.70)

= i yX

(o(&) + o)H + i V ef£)((o-a)H + i v ef£)

®xz “ ^zx

®yz

®zy ~ ® '

It is important that we have chosen here a right-handed Cartesian system of coordinates with the

0)^

z-axis

for electrons (charge >IH .

along the field

e < 0)

eH

--- « 1.76 x I07 H.

me “»*

me

H Q . Moreover, the gyro-frequency

is taken to be positive: s

0 ~

-l ^ 2 tic 1.07 X 10" 1 ' ----------------- cm. “■ = ’ x»

"H

H a “o

=

Of course, the role of the ions has been neglected in (11.70).

,,, (11.71)

While this can

always be done when there is no magnetic field by neglecting terms of order N^/m

as compared to Ne/m, in a magnetized plasma the role of ions is usually

unimportant only provided

le IH, = 1 . 7 6 x l07 5 H ( s-*

to » ftR = where

(11.72)

ft^ is the ion gyrofrequency.

Inequality (11.72) turns sometimes out to be insufficient and the role of the ions may be important even at higher frequencies.

For instance, when waves

propagate across the field— when they propagate at an angle H Q , which is equal to \ 7T , the to » V

” V M/m

effect

of

ions

can

be

a

between k

neglected

only

and when

ftH .

Waves which can be considered neglecting the effect of the ions are called high-frequency waves.

Waves and, in general, fields with a frequency to «

ftR

01.73)

we shall call low-frequency waves and fields. The reason why the magnetic field

H 0 at low frequencies (satisfying condition

(11.73)) radically changes the 'response1 of the plasma to an external field E ■ E

q

e 10J

is clear already from the equations of motion.

the elementary theory to (11.69) and the equation

These reduce in

THEORETICAL PHYSICS AHD ASTROPHYSICS

270

+ Mvf** eff fn

=~ e E #e i

(i) where the term proportional to V e^

~ —UL'n

" t

•

» OJjl which is often met with under cosmic

conditions. For transverse propagation (a = Jir) we get from (11.87)

5S-

) 0) - a>H - u>*

K

In the ordinary wave (2) the polarization vector (the vector along H 0 and it is from this clear that as

n2

in an isotropic plasma.

n^

(11.90)

E ) is directed

for this wave has the same value

In the extra-ordinary wave (1) the vector

E

describes an ellipse in a plane at right angles to H 0 , that is, with respect

27;

DIELECTRIC PERMITTIVITY AND WAVE PROPAGATION IN A PLASMA to k

it has both longitudinal and transverse components (we remind ourselves

once again that in the case (11*90) the vector k the field

itself is perpendicular to

H Q).

If we speak of the diagrams of the functions the general case,

they

are

n 1 2 ( 1.

(DZ

n

V'hr

= ®

1(1 o >

II

o

ni>2

,

“2-

(i)2 “•to2 1® H

' ,

for

u< 1 ,

“ h ""!, cos2 a - u)2w

for

u> 1 .

(11.91)

u)? ico- cuA n cos2a

u-l u cos2 a -1

v = 0, the root v l0

(11.92)

is, of course, fictitious when

For the same reason when u > I , but u c o s 2a < 1, the function

has ho poles,

As an illustration we have given in Figs. 11.3 and 11.4 a few graphs of the functions n 2

(v)

for different values of u

and a .

We have used as vari-

able the quantity v=a)2/cu2; this may turn out to be artificial.

In fact,

such graphs are very convenient when we are dealing with explanations for the way n2 , depends on the electron density N other graphs, for instance, of

(indeed, v = 4 ttN e2/mo)2).

Some

n2 2 ((u/iup) , turn out to be also convenient;

an example of these is given in Fig.11.5.

As there is a region of small w-

values in this graph it is necessary to remind ourselves once again that Eqn. (11.05) and the subsequent ones refer to the high-frequency case w When the frequency decreases the effect of the ions starts to become discerni­ ble and yet another branch of oscillations appears.

We restrict ourselves

here by giving Fig. 11.6 in which we show the functions n 2 2 (oj) , taking ions into account. We discuss yet one more important limiting case

Up » w 2 , (i)2 » 03^ , a)2 4c o)^cos2a

, a) >

(11.93)

,

or v > 1 , v > u , u cosIa > 1. Under the conditions (11.93) we have according to (11.87)

A cos a

-2 c2k 2 V n 2 * — g— w —■ ■■ u /u cos

a

“p A h c N |e | ■ — ■ j — * uHOjj cos a uiHpCosa

p

(11.94)

THEORETICAL PHY8ICS AMD ASTROPHTSlci

dielectric

permittivity

and wave

propagation

in a plasma

fcollisionless plasma taking ions into account. We assume that ^ (o^ and the values of the roots and the poles of the functions n* _ (ai) are indicated approximately. 19 *

279

THEORETICAL PHYSICS AMD ASTROPHYSXft

280 or

c2 k 2

c H q k 2 cob a

cob a

03 «

OK95) 4 ttN |e |

Under the conditions considered here the wave (this wave is damped, as

1 does not propagate at all

n2 >2 < 0) and for the wave

2

the value

n 2 * 1•

One

encounters such waves, for instance, in the Earth's magnetosphere ('atmospheric whistlers') and in solid state plasmas

('helical waves'

in metals which are in

a magnetic field). Apart from everything else, less legitimate v ^ * tjj/k.

If

the

the neglecting of the spatial dispersion is the

smaller

the

phase

velocity

of

the

waves

considered

v ^ ^ v T , the effects of spatial dispersion are large.

It is

clear in this connection that spatial dispersion in the first instance must be taken into account in the region of the poles of the functions are obtained for a cold plasma. ’helical waves'

nj 2 (ui) which

For the ’atmospheric whistlers' and the

the role of the spatial dispersion is also larger than for

waves with a large phase velocity because of the condition

n2

» 1 which

192

means that

Vp^ = c/nx 2

is much smaller.

Finally, one more remark: only a uniform plasma.

a reminder that in the foregoing we have considered

Meanwhile under actual conditions a plasma is always

non-uniform — either there are boundaries or the plasma properties (such as the electron density) change from point to point.

A lot of attention is there­

fore paid to the propagation of waves in a non-uniform plasma.

Of large value

for plasma physics is also the discussion of non-equilibrium plasmas (for instance, a cold plasma penetrated by electron or ion beams),

the study of

various non-linear phenomena and effects, relativistic considerations, and so on. dered

We have not mentioned plasma physics in a wider context and only consi­ calculations

propagation in it;

of

the

dielectric

permittivity

of

a

plasma

and

wave

we have thus in the present chapter only touched upon a

very small part of that field of problems.

Chapter XII TH E ENERGY-MOMENTUM TENSOR IN MACROSCOPIC ELECTRODYNAMICS The energy-momentum tensor in macroscopic electrodynamics. Applications of the energy and momentum conservation laws to the radiation of electromagnetic wave (photons) in a medium. Forces acting on the medium. We have already discussed in Chapter 10 the Poynting theorem and the energy conservation law in macroscopic electrodynamics (as applied to a simple model medium).

Agranovich and Ginzburg (1966) considered the energy conserva­

tion law for the case where an arbitrary temporal and spatial dispersion was taken into account within wide

limits

and

discussed in general in the literature. the momentum

conservation

law

this

has

electrodynamics.

of

the momentum

of

the

often

been

There

are

Firstly, one encounters the momentum of the

electromagnetic field appreciably more seldom than the energy. problem

also

It is impossible to say the same about

in macroscopic

apparently two reasons for this.

problem

field

in

a medium

has

Secondly, the

to a certain extent

turned out to be confused — it is connected with the choice for the expression for the energy-momentum tensor of the electromagnetic field in a medium.

This

problem has been discussed for more than 60 years up to the most recent times (see Brevik, Ginzburg,

1970a,b;

1973c;

Miller, 1972;

Walker and Lahoz,

Skobel'tsyn, 1975;

1973; Robinson,

Ginzburg and Ugarov,

1973,

1975;

1976; and the

literature cited there; we use the last named paper in what follows).

It is

therefore opportune to give a discussion of the momentum conservation law in macroscopic electrodynamics. In order to avoid complications which are not directly related to the problems which we wish to explain we shall consider a nonmagnetic, dispersionless medium at rest.

The field equations then have the form, well known to the

reader, but it is convenient to write them down once more (B = H, independent of

U) and

and

e

is

k ): 9E 9t ’

(12.1)

ic 1 3H curl E -------

>

(12.2)

div e E = A tt p

>

(12.3a)

div H

•

(12.3b)

curl H = — j + — c c

c 9t

=

0 281

282

THEORETICAL PHYSICS AND ASTROPHYSICS

E

We take the scalar product of Eqn.(12.1) with H.

and that of Eqn.(12.2) with

Subtracting the equations obtained from one another and using the identity

(E • curlH) - (H *curl

E) - - div [E a H ] , we get 8 7 ^ - ( e E 2 + H 2 ) = - (j . E ) - d i v S ,

(12.4)

that is* we obtain Poynting's theorem which in the present case we can without particular complications interpret as the energy conservation law (the energy density is

w *

(e E 2 + H 2)/8tt and the energy flux density is S - (c/4ir) [ Ea H]).+

We now take the vector product of Eqn.(12.1) and

H

and of Eqn.(12.2) and e E

and adding the expressions obtained we find {tH A curl H 1 + e tE We add now the expression

a

cun

E ]} = - - [i a H ] - ^

£

[ E a H] .

-pE to both sides of this equation and on the left

hand side use Eqn.(12.3a) to change it to - E (div

e E )/4ir .

As a result we

find

{[Ha curl H] ♦ «[E A curl E] - Ediv £ e |+

^

[E

a

H] *

■ - { p E . i [ i A H ] , t z J . J r [E A H ]}.

(,2.5,

On the right hand side we have here the Lorentz force density fL =

PE + I [jA H]

and the volume force density

which is sometimes called the Abraham force.

The minus sign on the right hand

side of (12.5) is connected with the fact that the sum f L + f A

is the force on

the medium, while Eqn.(12.5) determines the balance of the forces and momentum as referring to the field, where

»A - zfe C6 A n]

S =

8 tt

A H]= c2gA , (12.10)

--S

ik fi

•

fa =

( i,k= 1, 2, 3, 4;

fa + f a

*

(12.11)

“ 7 « ’ E> *

a , 0 = 1 , 2 , 3;

x„ = ict)

The tensor (12.10) is the energy-momentum tensor suggested by Abraham for a uniform medium at rest; for a moving medium this tensor looks somewhat more complicated

(vide infra).

The energy-momentum tensor used by Minkovski under the same assumptions as in the case (12.10) and (12.11) has the form

^

,

9M =

aH] = e g A

,

(12.12)

ax(M) f- . £

= P Ea + ^

i A H Ja

’

.

(12.13)

It is completely obvious that at least from a formal point of view the conser­ vation laws (12.11) and (12.13) different splitting up

into

are

terms

identical — they differ merely of

the

same

sum.

in

To be precise,

the if we

THEORETICAL PHYSICS AMD AfTR0Ptfr«M|

264

transfer the Abraham force (12.6) from the right-hand to the left-hand aide af Eqn.(12.ll) and combine it with we get at once the expreaaiam /w\ 3T^k y/9x^ and we can take for the energy-momentum tensor the Minkovski tens sc . Such ambiguity in the choice for an expression for the energy-momentum tensor is nonetheless surprising in that it is very general in nature and occurs eves in field theory in vacuo (see, for instance, Landau and Lifshitz, 1975* S 32). Moreover, the field in a medium is not a closed system — only the system con­ sisting of the field and the medium is 'closed* and the medium is characterised by

its

own

energy-momentum

tensor

.

3Tik /9x^ = 0 valid for the total tensor

T ik ~

There

is

a

conservation

law

+ T ik’m ^ * where

is the energy-momentum tensor of the field (for instance, the tensor (12.10)); however the tensor T ju is not uniquely defined and even less so are its parts (med) (e.m.) T ik and T ik . The force density is altogether a different thing; it is, at least in principle, a unique and measurable quantity.

In this connection

the fate of the ’controversy’ about the Abraham and Minkovski tensors will ultimately be solved as a result of the choice of an expression for the force. The Abraham force (12.6) is genetically connected with the force due to the magnetic field (Lorentz force) acting upon the displacement current.

It is in

reality impossible to question this force notwithstanding that it has as yet not been reliably measured directly.^ 'in favour’ of the Abraham tensor.

In that way the problem would be solved

Skobel’tsyn (1973) has also shown in detail

that the various expressions one meets with in the literature which contradict the choice of the Abraham tensor are without a foundation, and this was done also by Brevik (1970a,b).

We restrict ourselves here to one of the arguments

in favour of choosing Minkovski's tensor rather than Abraham's. when one

This is that

chooses the Minkovski tensor one finds for the case of a quasi-

monochromatic wave train in any system of reference for the flux of the field energy in a transparent medium

vgr

t*ie 8r °up velocity.

S * wv

, where w is the energy density and gr M This just analogous to the relation Oa g ■ - ga vgr,B

for the Minkovski tensor (see Agranovich and Ginzburg, literature cited there).

1966,

§ 3.2, and the

When, on the other hand, one chooses the Abraham

tensor such relations do not hold and this is taken somehow to be a disadvan­ tage or difficulty.

Indeed, as Skobel'tsyn (1973) proved especially and in

^ The appropriate possibilities have been discussed by Brevik (1970a,b) and Skobel'tsym (1973). Undoubtedly, for one's 'peace of mind' it would be justified to measure the Abraham force. This has been done relatively recently (Walker and Lahoz, 1975a,b) and the validity of Eqn.(12.6) was confirmed, albeit only with a relatively low accuracy.

t h e e n e r g y -m o m e n t u m

tensor

in m a c r o s c o p i c

electrodynamics

285

detail, this is all again connected with the presence of the volume force when we use the Abraham tensor. the medium and the relation fied,

In a moving medium this force does work on

S = w v gr

therefore can and should not be satis­

The situation is here completely analogous to the one for a medium at

rest^ when the relation S * w V gr

is violated when there is absorption and,

in general, when there are some energy sources or sinks in the medium. applying this to the momentum density flux already

to

the

Ca8 ” ” 8a vgr,8

case

of

a

When

Ea vgr g what we have said refers

transparent medium

at

rest,

as

the

can hold only when there are not volume forces.

relation This last

requirement is at once satisfied by the Minkovski tensor (we assume that there (M) / 3x^ - 0 .

are no charges or currents) for which

All we have said allows us to take the Abraham tensor to be the *correct’ one, but it seems to us that one can label the Minkovski tensor as being ’incorrect’ only in a somewhat formal approach.

In reality in most situations the results

obtained using the Abraham and Minkovski tensors are completely identical. This makes it possible not only to use in appropriate cases the Minkovski tensor but even to consider its application to be fully suitable, if in that way one reaches a certain simplification. One should therefore hardly label (M) the Minkovski tensor T ^ to be ’erroneous’; rather it is an auxiliary con­ cept which can be used fully.

This does not imply any loss of ’prestige’ for

the more fundamental and, when it is convenient, ’truer’ energy-momentum tensor (A) of an electromagnetic field in a medium, T} -

■ i i ? { [ ' • AH- i * t E- A _M -A S 9 = TcF » 9 wL k ,(A) “A t 1 = g L = cn k i G,(M)

where

,(A,M) Gv *

nwL k^ “ M. = g L = c k

LJ 2 K = ^1— = r L 2 2

j-

W

[ (J2) do> = J v 0) | —'CD

R 2 C 2 Ul2 + (LCu>2 - I)2 A

A

Ui2 C 2 LRf (u) , T) d(D (13.12) R 2 C2 u>2 + (LCo)2 - I)2

the resulting circuit is completely analogous to an undamped harmonic

oscillator described by the equation mx + kx = 0 with eigenfrequency l/vEc

ai^ = A / m =

(see (13.1))-. However, it is well known that for an oscillator its aver­

age energy at a temperature W

T

is given by (it is unnecessary to elucidate this)

= U + K = 2U = 2K = j m x 2 + j k x 2 = - - + \ L J 2 = fiu). x Ho), coth ---2 i 2kfiT

fill). 1 * \ ^0)^+ — exp [ffoi/kgT] - 1 Of course,

the internal

energy of the resistance

R

(13.13)

itself has here not been

taken into consideration. We note further that for small = l/^^LC

R

the integrals (13.11) and (13.12) have a

steep maximum at

Ui =

and we can therefore put (strictly speaking

this is valid as

R-*0 ; a - C R / )

ao

f U = f(u),T)

adn ---------

{ a 2n 2 +

(n - D -

a n 2 dn K = f (u ,T) | ti a 2n 2 + (n* D-

= -7Tf(u,T) . (13.14)

The integrals occurring here are evaluated exactly (most simply using the resi­ due theorem), but to obtain the result it is sufficient to take into account that as a + O, both integrals reduce to

J a dn/[a2 + 4(n - I)2] « \ 7T. Comparing

290

THEORETICAL PHYSICS AND ASTROPHYSfeCt

(13.13) and (13.14) and bearing in mind that the frequency we find for

is arbitrary*

f(tu,T) f(»,T) -

coth

•O J . 1 S

B Finally, we get from (13.9) Nyquist's formula

COth 2 ^

= f (? nU> + exp (5l'/TBi)'- l) ’

ao e2 ="

|

R(a>) aj coth 5 7 % da) *

o

03.16)

B

In the classical case when

Ru) k^T

and unvarying L

cerned (that is,

and

U + J k ^ T ) whereas

C

being satisfied) for sufficiently

becomes classical as far as U

K + 0.

is con­

We only need remind ourselves that

the requirement of quasi-stationarity of the circuit, which we started from, imposes well known limitations on the quantities

L, C,

and

R (in particular,

it is impossible simply to let R-*00 as in that case the circuit turns out to be open and formally

(£2

■+00 , although

(J2 )^-* 0).

We do not know, however,

whether such limitations affect the analysis of the corresponding actual prob­ lems at all or seriously interfere with the transition to the limiting cases fi/RC « kBT and fiR/L » kBT

(see (13.23)).

The cause of the above-noted situation — the different behaviour of K — is completely obvious.

U

and

A classical system may retain its characteristic

features even for strong damping — it 'remains itself'.

For instance, the

oscillations of a pendulum (oscillator) are very strongly damped when the viscosity of the medium surrounding it increases, but it remains all the same a pendulum.

If, on the other hand, we have a quantum harmonic oscillator

with frequency

, this system has, when there is no damping, energy levels

which lie at a distance (as the result, say,

from one another.

of

collisions

or

levels spread out and finally overlap.

of It

When the damping increases

interaction is

also

with

radiation)

perfectly

obvious

the that

when the overlap of the levels is large the system possesses a clearly pro­ nounced continuous spectrum and has little in comnon with the quantum harmonic oscillator.

Also, in various quantum systems, depending on the nature of

their energy spectrum,

the average total energy and the average kinetic or

potential energy are, in general, completely different. The harmonic oscillator plays in physics an exceptionally large role by no means solely because such a system is often met with (pendulum, molecular oscillations, and so on).

It is even more important that an extra-ordinarily

wide class of problems, connected with the discussion of small (linear) pertur­ bations

and

waves

in media with

'distributed constants', that

is,

in tne

FLUCTUATIONS AND VAN DER WAALS FORCES

301

electrodynamics of continuous media, in acoustics, and so on, reduces to some extent to the oscillator problem.

This

is

also true of the electromagnetic

field in vacuo — the Hamiltonian method expounded in Chapters 1 and 6 for electrodynamics in vacuo

or

in a medium

is

a clear

illustration of

this.

Besides, the expansion into waves, by no means only plane waves, goes beyond the limits of the Hamiltonian method and has, as we have said, an extremely wide range of applications.

It is in this connection at once clear that the

study of electrical fluctuations in an electric circuit which was made above can be generalized not only to discrete systems (mechanical oscillator, dis­ crete chains, and so on) but also to continuous media. Landau and Lifshitz (I960)

or

Rytov (1966)

for

While referring to

details we

shall

give

here

merely a few expressions which relate to the fluctuations of an electromag­ netic field in a medium. We can indicate the presence of fluctuations by writing the relation between D

and

E

in the framework of linear electrodynamics in the form

D^( il>, r ) * J

(to , r , r' ) Ej (to , r ' ) d 3 r' + K. (&>, r ) .

(13.24)

We have here already made the transition to the Fourier components with res­ pect to to, and otherwise this relation differs from (10.3) only through the addition of a fluctuating electric induction K(co,r) the appearance of fluctuations in D

which takes into account

even when there is no average field

When there are no external sources the basic field equations for

E

and

E .

B

take the form

(13.25)

The relation (13.24) clearly takes into account the possibility of spatial dispersion and we can therefore without losing generality assume that (see Chapter 10).

B«H

When we neglect spatial dispersion we can (and sometimes

must) introduce the magnetic permeability and in that case we must when con­ sidering fluctuations introduce also a fluctuating magnetic induction L(a>) (see Landau and Lifshitz, 1960, §90). According to the fluctuation-dissipation theorem we have in equilibrium

K . ( u , r ) Kj

(w ,

r' )

E

(K. (r ) K ( r ' ) ) ^

(13.26)

302

THEORETICAL PHYSICS AND ASTROPHYSICS

where the bar indicates statistical averaging. sion and put

B ”H

( - £ - £f “

- Loi2 - i (i)R + 1/C However, the frequencies

. R.

It

are the eigenfrequencies for the circuit (and

this is the auxiliary circuit) Lq ♦ f - R q + J - 0 , which differs from (13.1) by the replacement of sion we explain once more that the solutions frequency (13.41)).

(13.41)

R

by

(oi/u)1) R (to avoid confu­

exp [- ia)j (w) t]

in which

the

oi, as in (13.40), is considered to be a parameter, satisfy Eqn. In terms of the frequencies

form

oij (u>) we can write Eqn. (13.39) in the

+«

T «

w

w(u),T)

((d ,T)

du)J(u>)

a> doi + 012 (a)) - oi2

(llj((!)) - (1)Z

doi.

du

(13.42)

One can easily check directly that Eqns.(13.39) and (13.42) are identically the same. Of course, in the case of a circuit nothing whatever is changed, but in the case of generalizations (for instance, oscillations in a gap) it is necessary to perform some calculations starting from Eqn. (13.42) and using, mainly, the frequencies 01^(11)) which are the analogues of the frequency method is well known (see Vainshtein, Ginzburg, 1975),

1957, §5 100 to 102

(u) .

The last

and Barash and

Let the whole system be immersed in some auxiliary resonator

with perfectly conducting walls and consider the frequency u meter, while the eigenfrequencies of the resonator

w a (ai)

to be a para­

are determined from

the homogeneous field equations i(13.43)

iu) (ui) a H d)a ((»)(“ *r> c

*

is the linear operator occurring in (13.24) and where we have omitted

the tensor indexes

i,j

to simplify the notation.

that the auxiliary resonator is the same

as

the

It is clear from (13.43) actual

one

(that

is,

the

system considered) when u)a * ai and when there are no external and fluctuating

311

FLUCTUATIONS AND VAN DER WAALS FORCES sources.

The eigenfunctions

and

(qj) (u>, r)

of the auxiliary

resonator have a number of properties (such as orthogonality) which enable us to find particularly simply the induced solutions which depend on K(u>,r) for the actual problem which is described by Eqns.(13.25).

The average internal

energy of the system is in this case simply of the form +® w(u>, T) w(u>, T)

s -4 l

I ;

a) dai +

2

—oo ~a

(a)) - u)2

» « I «;

2(a.)-a.2

which immediately generalizes Eqn.(13,42). clear that as

e " -»■ 0

d(i)2 (u») a du>, (13.44) du»

This last fact makes it already

(or R-*-0) Eqn.(13.44) goes over into (13.36) for W

(the same is, of course, also true for

3).

Moreover, Eqn.(13.44) is valid

also for absorbing media and it is clear from (13.43) that it can be used also when ve take into account anisotropy and spatial dispersion.

When applying

it to the gap problem (see Fig.13.3) for an isotropic medium without spatial dispersion one can check (Barash and Ginzburg, 1972, 1975; Barash,1975a,b; Barash, 1976)

that from Eqn. (13.44) or, to be more precise, from the analogous

expression for the free energy 3

Eqn.(13.33) follows.^

When solving yet more

complicated problems, for instance, when the media 1 and 2 are anisotropic (Barash, 1978) the advantage of the method described here of an expansion in eigenfrequencies becomes even more striking.

One may think that just such an

approach would dominate when one solves a whole

number

about molecular media under conditions

of

related

forces

different

problems

for

various

geometrical

or when calculating the

free energy in an absorbing medium, and so on. In conclusion we

shall discuss yet

another electrodynamic problem — that of

the

fluctuation effect

of

fluctuating voltages in a resonator on electrons

passing

through

(Ginzburg and Fain, 1957a,b).

it

Fig. 13.4

The problem of the

flight of electrons through an empty resonator

As T-*0 the free energy becomes the same as the internal energy. In the problem of the van der Waals forces, however, one is usually interested only in the case as T + 0 (see Lifshitz, 1956; Landau and Lifshitz,I960; Abrikosov, Gor'kov, and Dzyaloshinskii, 1965). This is the reason why we did not write down the expression for 3 .

312 Let

THEORETICAL PHYSICS AND ASTROPHYSICS a

non-relativiatic

resonator at time

t*0

electron with

initial

and leave it at time

t

energy

K 0 - J mv*

enter

with energy K

the The

field in the resonator is assumed to be uniform and in the direction of the electron velocity — along the x-axis

(such a case corresponds to a resonator

of a well defined shape, for instance, such as shown schematically in Fig. 13.4).

If E - Ej cos a)t + (E2 + E q ) sino)t ,

we have mx = eE , vT = x(t) = v 0 + ^ |ej sin wt + (E2 + E q ) (1 - cos on) We

shall

E = E2 = 0

further assume and

that

Ej

E 2 = E 2 = V 2 / I , where

and

I

E2

are

random

J

.

variables

(13.45) so

that

is the thickness of the resonator (the

path traversed by the electrons in the resonator) and V 2 is the mean square of the fluctuating voltage at the 'plates1 of the resonator. order

e2 we then have (K0 Kn =

f4Vl

2 idor

Up to terms of

m v 2) 1

sin2 - cut + E 2 (1 - cos

2evo E 0 sin2 | u»t ,

ojt

U 2 (13.46)

(a k t )2 =

k 2 - ( k t )2 = ( k t

The term of order

- k 0)2 - ( k t -

k

o)2 = * C v » CD2

Z

• .1

I 2 S11* 2A ) and e(u) ) 5 e s we may put € ,,(we) = e,,(o)s) = 0. As far as ft = coe -

is

concerned,

it

is,

are real quantities, the

in general,

scattering not

possible

wave to

u)e

that is, that of

frequency

neglect

its

absorption. If the wave with frequency

ft propagates freely in the given medium its disper­

sion relation has the form 2 2

— 2— = (n+ iic)2 * e(ft) - e' (ft) + ie"(ft) .

(14.43)

The introduction of the term 'real exciton' is connected with the fact that one can also consider other excitons, such as Coulomb and mechanical exci­ tons (see Agranovich and Ginzburg, 1966, 1971; Agronovich, 1968). We emphasize also that the terminology in this field is not established and one should bear this in mind when getting acquainted with the literature. The 'three-photon' process discussed here (we are dealing with the 'interaction' of three waves or three photons in a medium with frequen­ cies U)e , o>s , and ft) is possible only in a medium without a symmetry centre, but amongst such media there are also non-gyrotropic crystals of the class T)6.j).

332

THEORETICAL PHYSICS AND ASTROPHYSICS

This equation is, of course, the usual expression connecting

ft and

q for the

propagation of transverse electromagnetic waves in an isotropic medium. virtue of

(14.43)

the

normal (free) waves

propagating

in

the

By

medium

in an

arbitrary direction (z-direction) are the following E = E # exp

K z - i(«t - - nz)| ,

« = {ie' + [(i e ' ) 2 + ( i e " ) * ] * } * .

(14.44)

K-j-ie' +taer +ae")2]*}*. As a result of the presence of absorption (that is, when

e 7/(ft) ^ 0) the normal

waves (polaritons) are absorbed and, for instance, if the frequency the wavevector

q

in the normal waves is complex.

However, when light is scat­

tered with the formation of polaritons, those have real (14.32).

This apparent contradiction^

tering is a forced process^

and

that

ft is real,

ft and

q by virtue of

is removed when we remember that scat­ the

dispersion

Eqn. (14.43)

does

not

apply to the polaritons formed in the scattering involving the formation of polaritons when we neglect absorption.

When absorption is taken into account,

however, it is not a free polariton, but some polariton-like wave which is formed.

This does not hinder, of course, the use of combinational scattering

of light to study polaritons.

The situation is in this respect analogous to

that discussed earlier for the case of Rayleigh scattering in liquids.

To be

precise, we get for the scattering involving the formation of polaritons a formula for the line width determine

also

polaritons.

I(ft,q) in which occur the same parameters which

the propagation

of

the

normal

electromagnetic

waves — the

For further details and the formula itself for the scattering

The fact that one found here some difficulties is clear from a number of papers cited by Agranovich and Ginzburg (1972). For instance, in one of them an attempt was made to connect ft and q in the maximum of the com­ binational scattering line through the relation c 2q 2 /ft2 = n 2 ; in another paper the relation c2q2/ft2 ■ e '(ft) is discussed. In both cases this was done in order to have a real quantity on the right-hand side of the dis­ persion relation. Such an approach does not lead to agreement with observations and is mainly incorrect in essence as ft and q referring to a polariton which is formed in a scattering process are not at all related through a dispersion relation. ^ We have here in mind any scattering process, including spontaneous ones and not only the so-called induced scattering arising when waves of a large intensity are scattered (Bloembergen, 1967; Fabelinskii, 1968; Sushchinskii, 1969; Gorelik and Sushchinskii, 1969; Starunov and Fabelinskii,

1970).

SCATTERING OF WAVES IN A MEDIUM

333

line width we refer to the paper by Agranovich and Ginzburg (1972).

We note

here merely the fact that in the paper referred to no random 'forces1 f(r,t) are introduced,

the consideration of which is particularly convenient for the

classical approach to the scattering problem.

Moreover,

in the equation for

the polariton field in that paper there occurs explicitly a 'force1 which takes into account the effect on the medium of the electric fields of the incident and the scattered waves.

Such an approach which is equivalent to considering

the energy of the interaction of the incident and the scattered waves with the sound or exciton wave which is formed (absorbed) as a result of the scattering is natural in those cases when it is necessary or expedient to evaluate the intensity in the framework of quantum theory. The examples given above may be thought to demonstrate the specific points of the scattering line width problem as compared to the discussion of the width of an absorption line of light or sound which is determined by the homogeneous equations for the propagation of the appropriate waves. polariton

absorption

frequency

line

formed

when

in

the

crystal

a

example, free

is absorbed (of course, one must vary the frequency

occurrence of a line). tion index

is

For

« m e 2 ).

The case of relativistic elec­

trons and arbitrary frequencies of the radiation is discussed in Chapter 16# In the electric field of the wave

E * E cos [(k* r) - u»t)] ,

(14.45)

and we find, when we use the equation of motion ® r = and neglect in (14.45)

the phase

eE

(k • r)

acquires a velocity

(14.46)

as compared to

w t , that the electron

^^ V H r

---— — sin ait + v . ma) u

In order that not only the velocity velocity be non-relativistic,

(14.47)

v Q without a field, but also the induced

the condition eEa

— 77 me a)

« •

(U.48)

must be satisfied, and we shall here assume that to be the case. Just because we have assumed that

v ~

[v^ + ( e E 0/ m w ) 2] « c

(14.46) where we have neglected the Lorentz force ~[v A H], this is so, if also

HQ ~ EQ.

To be precise,

For a plane wave in vacuo, of course,

but in a medium with refractive index

n

we have

relative role of the magnetic field increases. guides where for well-defined oscillations also possibly observe the inequality the effect of the Lorentz force. behaviour

we can use Eqn.

HQ »

HQ ®nEQ

and when

H Q= E0 , n

1 the

The same may happen in wave­

(modes), or in some points, one can EQ .

At any rate, we neglect here

We note that the condition for classical

R w « m e 2 automatically guarantees also the possibility to neglect

the radiative friction force ^

as we did in

be more precise, when v < c/n) the phase replacing the field

(14.45) by

(14.46).

Finally, when

kr ~ u m v t / c «oit

v < c

(to

which justifies

E * E Q cos cut .

Taking what we have said into account we start from (14.46) and can restrict ourselves

to the dipole approximation, where

The condition c /re ,

where

Rw

me2

is equivalent to the inequality

re » e 2/mc2 - 2.82 * 1CT1 3 cm

Ru) < c/(R / m c ) «c

is the classical electron radius.

Under these conditions the radiative friction force is very weak Chapter 2).

(see

335

SCATTERING OF WAVES IN A MEDIUM cu0 r ------ - cos wt moiz

,

.. .. e* e p = er = — E = — m m

E

cos cot .

(14.49)

u

Hence, using E q n . (6.28) we get the time-average of the intensity of the radia­ tion scattered into the solid angle

d 2ft :

1 = (j^) E“ 5* Sin^ = Io ^ Sin2^ = Io d° ’

(1«.50)

as the intensity of the incident radiation is T = ££ 7^ = t?2 Io 4 tt 8 tt o (as in (14.5) the angle between wave is denoted by Clearly,

E 0 and the wavevector

k

of the scattered

lj; in contrast to Eqn. (6.28) where we used the notation

0).

the total cross-section for scattering is r a =

f l d 2ft d a = — ---- = 1 tt r 2

J

(14.51)

io

8 tt / g2 \2 = *| tt r 2 = -j- ( 2 ) = 6.65 * 10~25 cm 2

(the cross-section

Thomson cross-section).

For unpolarized light

course, as before

(here

0 = 0^

0

is called the

da = \ r 2 (l + cos20) but, of

is the scattering angle).

It is clear from the calculation given here that the refractive index of the medium

n

in which the scattering by a free electron takes place drops out of

the expressions for start

at

known

formulae

once

da

from for

(we

the the

note

that

expression

it

for

instantaneous

p

is

more

in

convenient

(14.49)

intensity

I ®

and

in vacuo

to

use

the

(p 2 / 4 ttc 3) sin2i/;

to well and

J l d 2fi = (2/3c3) (p) 2 ). It is clear from (14.7) and (14.51)

that the extinction coefficient for a gas

of independently scattering electrons of density

N

equals

h = aT N .

(14.52)

Assuming the gas to be ideal and evaluating the fluctuations

6e

for a gas of

free electrons we get the same result from Eqn . (14.10) if we bear in mind that in that case

4 7TNe2

c = 1

muT

de p°V

and 8N

mw2

1

0 T

kBT N

The effect of the ions is here, clearly, completely negligible, not only in the expression for words

we

assume

e

that

but also in that for the compressibility the

directly, or indirectly.

ions

do

not

contribute

to

the

0^ .

In other

scattering

either

336

THEORETICAL PHYSICS AND ASTROPHYSICS

When can one proceed in this way 7 from the ions to 6 ^ oi »

k vT,e,i

and

Of course, first of all the contribution

must be small*

In an isotropic plasma it is small when

oi » v ef £ (see (11.40));

only the inequality

u> >

in a rarefied,

k v T , v T ■ V k^T/m

isotropic plasma

is important.

For transverse

waves this condition is always satisfied, but it is not at all sufficient for taking the fluctuations that are important in the problem to be independent. Indeed, fluctuations are only independent in volumes at distances from one another

r » £, where

£

is the correlation radius.

gas the mean free path plays the role of Debye radius

r^

£

In an uncharged (neutral)

and in a rarefied plasma it is the

(see (11.14)).

On the other hand, the phases of the scattered waves differ less that for scattering volumes which are smaller than the wavelength

tt

X = X Q/n .

only For

larger volumes or larger distances between the scattering volumes the phase of the scattered waves is the whole time 'disturbed1 due to the thermal motion, that is, the time-dependence of the fluctuations.

For the scattered light

which is not resolved into a spectrum (or even for a sufficiently wide spec­ tral band) scattering by volumes at distances is clear that for

X » £

r » £

is incoherent.

Hence it

or, to be more precise, when (see (14.33);

0

is the

scattering angle)

x

2 tt

q

« £

(14.53)

,

2 sinje

the correlation of the fluctuations at distances of the order of play

a

role

in

the

sense

that

the

phase

of

the

scattered

£ does not

waves

is

thus

'disturbed1 at appreciably smaller distances, at which the gas behaves as being nearly ideal.^ We have noted and used this fact already earlier in applications to a gas of neutral particles.

For a plasma, however,

the scattering by single electrons

only, neglecting the role of the ions completely, is therefore allowed only, provided (see (11.14)) X

7rc

2 sinJ 8

nu) sinj 0

(

< rr

kBT

\*

4.9

8 irNe2'

(

T(°K) (14.54)

cm

N(cm-3)/

This statement is obvious in the case of a neutral gas — at distances less than the mean free path

£

the particles do not interact at all.

In a gaseous plasma which we are considering the interaction between particles is also weak by virtue of the condition also guarantees that the inequality N"1

rD > NT^

e 2 N 5 d E “ - 4 F -

Of course, the use of the total energy

E = E^+Mc2

< 1 5 '7 >

is convenient only in the

relativistic region, but it is just that one which is studied for the cosmic rays at the Earth.

For soft cosmic and sub-cosmic rays one used more often

the kinetic energy E ^ . the total energy nucleon

£ = E/A

E

Moreover, it is convenient to use for nuclei not only

or the kinetic energy E^ , but also the total energy per

or the kinetic energy per nucleon

£jt = Ejc/A,

where A

the atomic weight or, to be precise, the mass number of the nucleus.

is

Finally,

Eqns.(15.2) and (15.4) have been written down assuming the particle distribu­ tion to be isotropic because the cosmic rays at the earth, if we eliminate the effect of the Earth's magnetic field, are to a high degree isotropic.

The degree of anisotropy of the cosmic rays is defined as follows:

Jmax

Jmin

(15.8)

Jmax - Jmin . : where

J and J • are, respectively, the maximum and minimum cosmic ray DoX mm intensities as far as direction is concerned (we have assumed here that J(0)

has only one maximum, say, in the direction

0=0;

a dependence such as j(0) = J Q + J 1 Cos0, so that

in other words, we assume 6 = J 1/J()).

The degree of

anisotropy of the cosmic rays has not yet been established reliably; cosmic rays with energies larger than 100 GeV ^ the value

6 < 10 ** -

for all The aniso­

tropy can thus be shown up only as the result of special studies, while in all other cases we can with justice assume the cosmic rays to be completely iso­ tropic (we remind ourselves once more that we have assumed that the effect of the Earth's magnetic field has been eliminated).

The problem of the study of

the primary cosmic rays, that is, cosmic rays outside the atmosphere or taking its effect into account, is thus in fact the determination of the functions J^(E)

for all components of the cosmic rays — for the protons and nuclei

This means that J Q = J Q (E > I00 GeV) and Jj = J x (E > 100 GeV) ; for larger E and especially for E > I016 to 1017 eV the degree of anisotropy may be larger than indicated.

(the

347

COSMIC RAY ASTROPHYSICS proton-nuclear component) and for the electron-positron component.

However,

the fraction of positrons in the electron-positron component is very small for E > 1 GeV

(a few per cent) and they have as yet not been reliably distinguished

from the electrons.

This is also not done in the overwhelming majority of

cases and one measures the intensity

J e (E)

of the whole electron-positron

component which is simply called the electron component.

In the case of the

proton-nuclear component the separation of the nuclei into charges, separation of isotopes,

is far from always done;

ders

ray

the

total

cosmic

intensity

Jc>r

proton— nuclear component as the fraction

(E)

of

one often, or

let alone

therefore, consi­ practice

electrons

(that

is,

the

their total ratio

J (E)/J (E) ) is of the order of one per cent, and, moreover, electrons are e c .r. relatively easily separated. We

do

not

plan

to give

here

a

detailed

introduction

Ginzburg and Syrovatskii,

1964a, 1966b, 1973; Hayakawa,

Ginzburg,

1973;

1970a;

Hobart,

Prilutskii, and Rozental',

Ginzburg

and

to

1969;

Ptuskin,

cosmic

rays (see

Weekes,

1976a,b;

1969;

Ozernoi,

1978), and we restrict ourselves to only a few

remarks including an indication of some characteristic values of such quanti­ ties as

J

and

w.

At the Earth (outside the influence of the Ear t h ’s magnetic

field) we can as a guideline take for all cosmic rays the following values:^

J ~ 0.2

c.r.

I

particles

0.3

c m z .s .sterad 4 ttJ ----- io

c.r.

to

~

~

10-12

c.r. 4 tt

— jo particles £------ =--1 A r

10 -3

,

,

(15.9)

erg c m 2.s .sterad

From Table 15.1 one can reach some idea about the chemical composition of the relativistic cosmic rays

(€ = E/A > 2.5 GeV/nucleon), where we have split the

nuclei into the traditionally used groups

(for instance,

the L-group contains

We must bear in mind that the energy spectrum of the cosmic rays at the Earth has a maximum corresponding to an energy Ek ~ 2 5 0 MeV for protons. OD The cited values

/ J(E) dE and similar integrals therefore Ek - 100 MeV converge. However, their magnitude changes with the solar activity cycle because the contribution from slower particles changes and it is thus to seme extent sensitive to the lower integration limit.

348 the

THEORETICAL PHYSICS AND ASTROPHYSICS Li,

Be,

and

B

nuclei).

One should bear in mind that the errors in the

values in this table are not less than tens of per cents — the data change when new material is accumulated and when more refined experimental methods are used.

The last allow us in a number of cases to obtain information not

only about groups

of

nuclei,

but

also

about

recently relatively very rare nuclei with have been observed,

separate

Z £ 83

nuclei.

Moreover,

(that is, heavier than lead)

the quantity of which in cosmic rays is 8 orders of magni­

tude less than of all nuclei of group

H.

Group of nuclei

Atomic Number

Intensity J(> € = 2.5 GeV/Nucleon) par t i d e s/m2.s .sterad.

P

1

Abundance relative to H group nuclei in cosmic rays

average in the Universe

1300

650

3000 to 7000 250 to 1000

a

2

94

L

3 to 5

2.0

47 1

M

6.7

3.3

H

6 to 9 £ 10

2.0

1

VH

£ 20

0.5

WH

^ 30

~ 1C ~ **

io -5 2.5 to 10 1 0.05

0.26

~ 10-1*

- 10"**

The most important feature characterizing

the composition of the cosmic rays

which is clear from Table 15.1 consists in the presence of a rather consider­ able flux of nature.

L

nuclei, notwithstanding their negligible amount on average in

This peculiarity confirmed also separately for

and also for other rare nuclei

(for instance the 3He

Li, Be,

nucleus),

and

B

nuclei,

indicates the

appreciable role played by the transformation of the chemical composition of the cosmic rays when they propagate in interstellar space and, possibly, also in the sources (that is, in the regions where the cosmic rays originate or, in other words, are accelerated). The energy dependence (energy spectrum) of the cosmic rays is comnonly written in the form JA (>e) = I

V

€ ) d € = KA € ~ (Y~ l) ' JA (e) "

,

(15.10)

€ where, as already stated, indicates

that

atomic number J(E)

we

are

€ * E/A

dealing

is the energy per nucleon and the index

with

nuclei

(or group of nuclei)

A; moreover, we have introduced similar quantities

for all cosmic rays.

of

A

average

J(>E)

and

349

COSMIC RAY ASTROPHYSICS In practice the spectrum is not a power-law, that is, the index y dependent.

is energy

However, and this is rather important, in a very wide range of

energies the approximation of the spectrum in the form (15.10) turns out to be a good one.

For instance, in the energy range

according to a number of data value

y=2.7

or y = 2.6

y=2.7±0.2.

to be the best.

2x 109 eV < E< 3 x 1015 eV

Apparently we can now assume the To give some ideas we give, as an

example, the following cosmic ray spectrum in the range 1010 eV < E < 1015 eV :

/E(eV)\-1.6 V 109 / In the low-energy region E^ < 109 to

— particles— ^ y = 2.62 ± 0.05 . (15.11) cm .s.sterad

10l 0 eV the index y changes and

spectrum depends strongly on the level of solar activity. cuss that region.

the

We shall not dis­

We note only that the very important problan (in the frame­

work of theories about the origin of the cosmic rays)

of

the shape of the

spectrum in the low-energy region and, in particular, of the presence of a maximum in the energy spectrum far from the Sun (beyond the limits of the solar system)

has

not yet

~ 100 MeV/nucleon cosmic rays.

been elucidated.

Apparently,

down

to energies

there is not yet a maximum in the spectrum of the galactic

At an energy E ~ 1015 eV

there is a more or less steep bend or,

at any rate, a change in the spectrum and for E > 1 0 15eV Eqn.(15.11) does not hold, and the following spectrum is closer to reality JOE)

- (2.0t 0.8) \ i015/

7 “ ^ , cm .s.sterad

(15.12)

y = 3.2 ± 0.2 ; according to other data the factor (2.0±0.8) in (15.12) must be replaced by (3.74 ±0.20) and (15.12)

y - 3.16 ±0.1.

When

E ~ 1 0 1 5 eV

the spectra (15.11) and

agree ('join') within the limits of the attainable accuracy, as should

be the case.

Possibly, the spectrum for

assumption has not yet been proved. ray spectrum (for

E > 1 0 10 eV;

the differential spectrum;

E > 1O10 eV flattens again, but this

We show in Fig.15.1 the integral cosmic

we remind ourselves that y is the index of

see (15.10)).

The maximum energy of observed cosmic rays is of the order of

102° to 1021 eV.

In that region the cosmic ray spectrum must, in general, 1steepen' suddenly as a result of the considerable losses which cosmic rays undergo at such large energies when they interact with the radiation which is present in the inter­ stellar and intergalactic space.

However, so far such a 'steepening' has not

350

theoretical

been

physics

observed

and

astrophysics

and the problem of the

spectrum and thus of the origin of the cosmic rays with very high energies is

T3

still open.

a

The values of

Y

given here refer to

all cosmic rays, but until recently it £ o \ in a> o

was assumed that the chemical composi­ tion of the cosmic rays, at least up to energies of 100 to 1000 GeV energy-independent.

o

a

was

And it was also

assumed in this way that in that region the

index

y «

all groups

of

2.7

referred

nuclei.

In

also

1972

to

there

appeared indications that the chemical

E, eV

composition

already

in

the

energy

range up to

100 GeV/nucleon depended,

albeit weakly, on the energy — we are

Fig. 15.1 The integral cosmic ray spectrum at the Earth. y is the power index for the differential spectrum; for a power-law integral

referring here to the decrease in the

spectrum

Li, Be, and

J(>E) = const. E

fraction of secondary nuclei

)

(such as

B) which are formed as the

result of fragmentation of heavier nuclei when the energy increases. also possible that for nuclei of the H

group (mainly iron nuclei)

is somewhat smaller than for those of the M

group

(C, N, 0, and

F

It is

the index y nuclei). The

corresponding data are discussed by Ptuskin (1974) and Juliusson (1974). the

region

of

still

€ > I011 eV/nucleon)

higher

energies

E > 1012 eV

(or,

to be

more

In

precise,

the changes in the chemical composition may turn out to

be more substantial, but there are as yet no reliable data about this. The electron component of the cosmic rays has been studied in less detail than the proton-nuclear component.

The spectrum in the energy range up to 1 GeV

ia

particularly sensitive to processes on the Sun and in the solar system and it is itself rather complex here.

For

E = Ee > 1 GeV

the power-law approxima­

tion fits already better and, for instance, in the range trum is a power-law spectrum with index to other data

y = 3.0 ± 0.2).

in the energy region

no

Ee > 50 to 100 GeV

information

(to be true, according

At lower energies the index

although it is possible that y « 3 is practically

y * 2.7 ± 0.1

5 < E < 50 GeV the spec­

decreases and

the data so far contradict each other,

up to energies

about

y

the

electrons

E^ ~ 500 to 1000 GeV (there at

even

higher energies)*

351

COSMIC RAY ASTROPHYSICS To give some ideas we give Che following differencial electron spectunn: ,_2 J (E) = 1.27 x 10

-(2.7 ±0.1)

electrons cmz .s.sterad.GeV '

(15.13)

5 < E < 300 GeV , where the electron energy is measured in GeV;

10-2

J (> E) » e

^

K7

,

5x

hence

10’

1010 eV

Cosmic rays with energies

and, moreover, the field is

E > 1012 eV

and, in practice also for

can clearly not be contained in the solar system.

Bearing in mind,

however, that the shape of the cosmic ray spectrum at the Earth changes little

352

THEORETICAL PHYSICS AND ASTROPHYSICS

up Co energies

E ~ 1015 eV,

ve have all reasons Co assume

chac chis spectrum

(for E > 10*0 Co 1011 eV) is characCerisCic aC lease for ChaC region of Che Galaxy which adjoins Che solar system.

However, Che condicions in Che neigh*>

bourhood of Che Sun is, apparently, very Cypical for huge pares of the Galaxy. In Chis connecCion radio-astronomy data about Che non-thermal radio-emission with a continuous spectrum which c>(jisc ~ 107 yr.

In the galactic model with a halo

the occupied volume is the cosmic ray halo (v^alo ~ tO68 cm 3) and the charac­ teristic lifetime time).

Tc r

halo ~ 10* yr (we give below an estimate of the life­

Hence and from the values (15.18) for

W£

it is clear that to

retain a quasi-stationary regime the cosmic ray sources in the Galaxy in both models mentioned must emit (accelerate) cosmic rays with a power of the order

Uc r

Both supernovae

W W . ~ „c.-r ->hal° ~ Tc.-r.v>dlac ~ 10“° - 101*1 erg/e . c.r.,halo c.r.,disc and

outbursts

of

the

galactic

nucleus

are

able

(15.20)

to inject

cosmic rays with such a power.

The choice between galactic models has not yet been made;

it is, for instance,

connected with the measurement of the characteristic cosmic ray lifetime T _ J c.r. and to do this there are several possibilities (Ginzburg and Ptuskin, 1976a,b). The same could be achieved by solving the problem of the existence or not of a radio-halo.

The

elucidation of the role of some sources or other of cosmic

In most and even in all supernovae there is probably a rotating magne­ tized neutron star — a pulsar (see, for instance, Ginzburg, 1971). Particles may be accelerated in the supernova outburst itself, in the supernova shell, and close to the pulsar. The acceleration of parti­ cles by pulsars is thus one of three possibilities, and its share is insufficiently clear.

356

THEORETICAL PHYSICS AND ASTROPHYSICS

raya is also possible by various methods.

As

an

example

we

note

that

the

galactic nucleus cannot serve as a source for the electron component of the cosmic rays in the high-energy region

E ^ I

to

lOGeV.

The fact is that on

the path from the centre of the Galaxy to the solar system high-energy elec* trons lose most of their energy as a result of synchrotron and Compton losses (see Chapter 4).^ We shall restrict ourselves here to the remarks we have made about the origin of the cosmic rays as

our

main

aim

is

to

illustrate

a

number

of

physical

processes and mechanisms which are of interest in high-energy astrophysics. The following processes and mechanisms must be analyzed. The processes for the acceleration of the proton-nuclear and electron compo­ nents of the cosmic rays in various cosmic conditions and various regions (stellar outbursts,

turbulent plasma in supernova shells, acceleration in

interstellar space, acceleration near pulsars, acceleration in solar flares, and so on)• Energy loss mechanisms of various kinds of fast particles.

Transformation of

nuclei in collisions. Diffusion and isotropization mechanisms for cosmic rays, in particular taking plasma effects into account. Processes and mechanisms for the generation (production) by cosmic and subcosmic rays of photons with various energies and their application to radioastronomy, optical. X-ray, and gamma-astronomy. and scattering

of

photons

in

all

wavebands

The problem of the absorption

also

belongs

to

this

class

problems.

We have given here the state of the problem of the origin of cosmic rays as it was in 1975-76. As this problem is studied on a very wide front (which is also true of X-ray and gamma-astronomy) the situation changes comparatively fast and some remarks and estimates in Chapters !5 to 17 turned out to be obsolete at the time of the publication of the English edition of the present book. We must, however emphasize once again that the aim of these chapters is not to give a survey of the state of highenergy astrophysics, but mainly to discuss some physical problems rele­ vant to this topic. Recent data in cosmic rays and gamma-astronomy are contained in the proceedings of the International Conference on Cosmic Rays which occur every year (see Plovdiv, 1977 for the picture in 1977). In our opinion the most important achievement of recent days has been the confirmation of the galactic model with a halo (Ginzburg, 1978; see also Ginzburg and Ptuskin, 1976a,b).

of

357

COSMIC RAY ASTROPHYSICS Furthermore, there is, clearly, the problem of constructing a quantitative

theory for the origin of cosmic rays in the Galaxy, taking into account losses, diffusion, transformation of the chemical composition, and so on.

To do this

one must, of course, give a more detailed model (make precise the region filled by the cosmic rays, the distribution of the sources, the parameters of the interstellar space, and so on).

The state of the problem is such that the

’trial and error1 method is unavoidable — one must work out various models and choose the best of them by comparison with the observational data (see Ginzburg and Syrovatskii,

1964a, 1966b, 1973; Ginzburg, 1969a, 1975a; Bulanov, Dogel',

and Syrovatskii,

1972a,b;

Ptuskin,

1972, 1974;

Ginzburg and Ptuskin,

1976a,b).

It is completely obvious that the whole of the corresponding set of problems is extremely extensive.

We have already mentioned this in relation to synchro­

tron emission (see Chapters 5 and 9).

In the foregoing we have also touched

upon several other processes which are of interest in astrophysics, but the largest part of the problems enumerated has not yet been elucidated.

Unfor­

tunately, it is totally impossible to do this in any detail or fully in the framework of the present book.

Below (in this chapter and in Chapters 16 and

17) we restrict our discussion to that of energy losses, to an exposition of the general scheme of cosmic ray diffusion taking the transformation of their chemical ccmposition into account and (in the case of the electron component) taking losses into account;

after that we shall consider mechanisms for pro­

ducing X-rays and gamna-rays and make a few remarks about related problems. We shall not consider particle acceleration mechanisms under cosmic conditions (see Ginzburg and Syrovatskii, Toptygin, 1973; Tsytovich,

1964a; Dorman,

1972; Kaplan and Tsytovich, 1973;

1977).

When charged particles pass through matter there occur several processes which together are usually called 'ionization energy losses’.

If we

assume

the

motion of the particle given (to be precise, uniform and rectilinear) and neglect changes in the mass and charge of the particle due to nuclear trans­ formations and decay or ’stripping' of orbital electrons (we have here in mind the motion of an atomic nucleus) the ionization of the atoms in the medium, their excitation, and Cherenkov radiation all contribute to the ionization losses. It is true that a sharp division of the action of a charged particle on a medium into those three kinds of processes is not always possible, especially not in a dense medium.

Moreover, in the case of a plasma one must speak about

358

THEORETICAL PHYSICS AND ASTROPHYSICS

the transfer the

of

ionization

cules.

energy to the electrons and ions in the plasma and not about and

excitation

proceeding in

a

gas of neutral atoms or mole-*

For sufficiently slow particles charge transfer play6 a role.

When

there is a beam of particles present rather than separate particles collective effects may occur — beam instabilities and so on. tion of

6-electrons

The problems of the forma­

(recoil electrons) in a medium and of multiple scattering

when particles pass through a given layer of a substance problem of ionization losses.

are

bordering on the

Sometimes one must also take into account fluc­

tuations in ionization losses and a spread in mean free paths. Even from this short list it is clear how wide-ranging the problems of ioniza­ tion losses are;

one might well devote a special set of lectures to them.

Of

course, there is an extensive literature devoted to this field, but we restrict ourselves to referring to Bohr’s classical paper (1948 ; see also Heitler, 1947; Landau and Lifshitz, 1969).

1960, 1977;

Ginzburg and Syrovatskii,

1964a; Hayakawa,

In what follows we only give a number of formulae which one must use

for calculating ionization losses in a gas or plasma, and we make a few remarks on that account. The basis for the calculation of ionization losses for fast particles is the formula (sometimes called the Bethe-Bloch formula)

(15.21) where v

N

is the electron density in the matter,

m

the electrom mass, 0*v/c,

the velocity of the fast particle considered with charge

ionization energy of the atoms in the medium,

Wmax

ferred by the particle to the atomic electrons, and ’density effect*.

Z e , A the average

the maximum energy trans­ f

a correction for the

The basis for obtaining Eqn.(15.21) is, indeed,

the classi­

cal Rutherford formula which determined the cross-section for the scattering of a particle of charge rest of charge

Z e , mass

e, mass m

M

and initial velocity

(interaction energy

Z e 2/r).

v

by a particle at

In the collision the

particle initially at rest (to be precise, an electron) acquires some energy W

and the incident particle loses the same energy (elastic collision).

cross-section, expressed in terms of

The

W, equals

d a - 2ir

mv2 W2 (the derivation is given by Landau and Lifshitz inappropriate to repeat it).

(1976, 5 19) and it is therefore

For the energy lost by the incident particle we

359

COSMIC RAY ASTROPHYSICS

have dE

2 ttZ 2 e**

Ju

in

do

W,min

After multiplying by the electron density N we are from this led to a formula like (15.21) and the real problem is to give a more precise expression for the logarithmic factor taking relativistic effects into account (so far we have clearly used the non-relativistic formula), as well as the binding of the elec­ trons in the atoms, and so on. The qualitative meaning of the logarithmic term in (15.21) becomes clear if we take into account that it essentially has the form

J

where

p

const, x in (p

irmax

/p . )•

min'*

is the impact parameter.

For close collisions (p ~ Pm in) ^-electrons

are formed with an energy reaching

On the other hand, the contribution

from distant collisions (p ~ pm a x ) increases as

An [ 1/ (l - f52 )] = in (E/Mc2)2

by virtue of the compression of the field of the particle as

v->-c (for that

reason the Fourier component of the field with a given frequency

u) ~ c9/fi

corresponds for increasing energy, roughly speaking to ever further distances) However, when

pmax

decreases there are between the particle and the electron

to which the energy is transferred more and more particles in the medium*

The

latter screen the field of the particle and ceteris paribus this screening is, of course, the larger the denser the medium.

The effect of the screening (or

the ’density effect’) is just taken into account by the term f

in (15.21).

In the ultra-relativistic case (for more details see below) the term

f has a

universal character c92

) + fLn

f = £n (1 -

4irNe2 m

+

As a result Eqn.(15.21) takes the form 2ttNZ2 e h j 0 me2 The independence of the given

m

1 n

expression

medium (apart from the electron density

c

^maa

(15.22)

2TTNtl2 e 2 for

f

of

the

properties

of

the

N) is connected with the fact that

in the cage discussed of rather high energies the properties of the medium at high frequencies are important, and then we have for any medium

e = 1

wp 2 = ui

4lTNe2

1 ----2— mu)

*

The ionization losses of ultrarelativietic electrons (E » m e 2 ) in atomic hydrogen are, according to (15.21) equal to

THEORETICAL

360

PHYSICS AND ASTROPHYSICS

me

1 . 5 3 x 110s 0 s 13 In = 1.53*

+ 18.el eV.cm2/g

(15.23)

me (we have given here the values of the ionization losses in various units for the sake of convenience when one wants to use them in different cases). In (15.23) energy

N

is the density of hydrogen at eV/s . J Wmav = J E ;

(15.24)

moreover, we have intro­

duced a more precise numerical value of the logarithmic factor in agreement with Tsytovich*s calculations (1962b): -j

in (15.24);

we have replaced the

-1

in (15.22) by

of course, this refinement is not particularly important.

Equations (15.23) and (15.24) usually give results which do not differ as to order of magnitude.

For instance, for

N = 0.1 cm“ 3

losses (15.24) are twice those of (15.23).

and

E = 5 * l O 0 eV

the

The losses (15.24) proceed forming

6-electrons (that is, transferring energy to the plasma electrons) and through Cherenkov radiation of plasma waves.

The necessity to apply Eqn.(15.22) for

a plasma, which takes into account the density effect, is completely clear from what we have said earlier:

for a rarefied plasma

e = 1 - w 2/u>2

for all

frequencies and it is just using that expression which led to (15.22).

We assume the plasma to be isotropic, that is, there to be no magnetic field; we know that under those circumstances a particle cannot emit transverse Cherenkov waves in the plasma. We emphasize also that we understand by plasma waves longitudinal waves which can propagate not only in a plasma, but also in any medium provided €(w)=0. The peculiarity of a plasma in this respect is merely the weak damping of sufficiently long-wavelength plasma waves. In a condensed medium an appreciable part of the ionization losses can also be connected just with the generation of plasma waves.

362

THEORETICAL PHYSICS AND ASTROPHYSICS

Equation

(15.-3)

refers

to

the

however, the condition E > m e 2

case

of

ultra-relativistic

electrons.

If,

is not satisfied and, in particular for non-

relativistic electrons (but with a velocity

v » v a , where

v&

is the velocity

of the atomic electrons;

in the case of hydrogen this means that the kinetic

energy of the electron

= E - me 2 » 15 eV), one can use Eqn.(15.21), replacing

Wmax

by

J Efc, for calculations with errors not exceeding a few per cent.

For particles of total energy E

and mass

M » m = 9.1 * 10“ 28 g, that is, for

mesons9 protons, and nuclei, Eqn.(15.21) leads to the following results.

Let

(15.25)

E < — Me2 m The maximum energy transferred to the electron is then equal to

W

max

= 2mv2 f-^) W v

.

(15.26)

When condition (15.25) is satisfied Eqn.(15.21) gives for losses in atomic hydrogen

(

k V kfiT/m « a)p where

(this means that

rD = V (kBT/0TTNe2)

k r D - k V( k fiT/8 irNe2) «

is the Debye radius).

convenience we have repeated here what was said in Chapter

11.

1 , or

A ■

For the sake of

367

COSMIC RAY ASTROPHYSICS

In the independent particle approximation (a sufficiently rarefied bean) each particle in the beam mov e s , scatters, and radiates independently of the other particles.

In that

case

one

can,

in general, consider

bremsstrahlung due to the particles in the parent plasma

the

scattering

and

as the result of

binary collisions; the same applies also to the formation of

6-electrons

and

the part of the ionization losses which is connected with it which is due to close collisions.

However, we have already emphasized that Cherenkov radia­

tion is essentially a collective effect. index

n± < 1

and,

hence, is

the phase

V p ^ 1 = c / n j^>c.

It

cos 0 = c/n(o)) v

(see (6.53)) cannot

particle velocity and emitted wave).

On

clear

0

the

the

that

the

angle

other

In the case (15.38) the refractive

velocity

condition

be

the for

transverse Cherenkov

satisfied when

between

hand,

of

v

and

n
2 c2 ly I ^ ----1 max u>v2 p

(15.47)

U)

In the classical approach to the problem one uses a kinetic equation for the distribution function

fg

and one chooses as the initial distribution in the

beam, for instance, the distribution (15.36). frequency u ^ m ' - i y

One then determines, say, the

for a wave with real vavevector

k (or complex wavevector

k for real u>), As a result one obtains, of course, the same result (15.44) or, more precisely,

(15.45).

Just this identity of results indicates the com­

plete equivalence of the classical and quantal approaches in the problem under discussion (see, for instance, Ginzburg and Zheleznyakov, 1969a, 1965 in this connection) where we have in mind when talking about the quantal approach the method using the Einstein coefficients for the transition probabilities^. The region of applicability of this method is restricted, in particular, in connec­ tion with the condition

|y| « u> ~ u^.

However, in

the

region where

it

is

applicable the Einstein coefficient method is very fruitful as we have already demonstrated in Chapter 9. It is clear from what we have said that when there is a particle beam present in the plasma (average velocity of the particles in the beam V B ^ VT = V kgT/m) this beam is unstable — longitudinal (plasma) waves in it grow. rate

y

is proportional to the particle density

and also use the fact that

a>*

a Ns ) •

Ng

The growth

in the beam (see (15.45)

From this it follows already that in a

beam of a sufficiently low density the growth of the waves due to negative absorption of Cherenkov waves is rather small (small over a time characteriz­ ing the process; small along the whole path of the beam, and so on); on the other hand,

the collective effect (instability) in beams may in completely

realistic cases be very important.

As a result the energy losses in a beam

and its spreading out may proceed much faster (or along a shorter path) than for separate particles.

The solution of the problem about the losses and

scattering (isotropization) of a beam is rather complicated as it is here not possible to limit ourselves to the linear approximation and we must develop a

Only in this sense or a similar one could the quantal and classical approaches confront one another. If, on the other hand, we have in mind the very possibility of solving any classical problem using the equations from quantum theory, that possibility is obvious as classi­ cal mechanics and classical electrodynamics are limiting cases of the appropriate quantum-mechanical constructions.

372

THEORETICAL PHYSICS AND ASTROPHY8lC|

non-linear theory (see Kaplan and T9ytovich, 1973;

Tsytovich, 1977;

and the

literature cited there). Why have we dwelled on the instability of beams in a chapter devoted to cosmic rays ?

At first sight this looks the stranger where we have earlier empha­

sized the isotropy of the cosmic rays due to which there are absolutely no conditions for the appearance of a beam instability.

Moreover, because of

the extreme rarefaction of the cosmic plasma (electron density N ^ 1 cm*3 in interstellar space and N ^ I0” 5 cm~3 in intergalactic space) plasma effects should turn out to be completely unimportant in cosmic ray astrophysics. However, this last argument must of course not be taken to be serious as abso­ lute values of the density

N

and of other quantities cannot play a role — one

should compare them with the appropriate values which are important for the processes considered.

As to the isotropy of the cosmic rays, one of the most

important problems is to establish its cause, and also to elucidate the condi­ tions under which there is no isotropy.

An analysis of plasma effects in

cosmic ray astrophysics is thus, indeed, necessary.

Moreover, there is no

doubt that these effects can be very important. Let us, for instance, consider the 1outflow* of cosmic rays from a region with a magnetic field H x in which the cosmic rays are isotropic into a surrounding region with a magnetic field

H 2 0; moreover, for nuclei with small source densities (especially

nuclei of the group

L,

that is, for

Li, Be, and

B)

we can assume that

q£ = 0.

The set (15.62) is algebraic and can be solved rather simply; all difficulties are connected with insufficient information about the cross-sections

and

and even more with insufficiently accurate data about the chanical (and even more the isotopic) composition present the uniform model

tion of cosmic rays with energies 7 g/cm2

(see Ptuskin,

of

the

cosmic

rays

at

the

Earth.

At

(15.62) describes fairly well the chemical composi­

1972;

Ginzburg

^ 1 to and

2GeV/nucleon for

Syrovatskii,

1973;

x^~x « 5 Ginzburg

to

and

Ptuskin, 1976a,b). Thus T where

N

c.r.

x pc

X

4 X 106

cMN ^

N

(15,63) yr '

is the gas density (of nuclei with an average mass

the region occupied by the cosmic rays.

M ~ 2 x 10“ 2U g) in

In the disc model the average value

364

THEORETICAL PHYSICS AND A S T H O m S i O i

of

N ~ 0.3

to

I cm~3

and in the model with a halo

N ~

I

to

3 x \ Q~2 c*'3;

f

a result the corresponding values do not contradict the estimates (15*55). More important is something else — the chemical composition is first of all determined by the thickness Tc r

x

and it is therefore impossible to find the time

from data about the chemical composition.

It

is

true

that

in more

refined models which take diffusion into account the dependence of the chemi­ cal composition on Tc r#

determined by the diffusion coefficient

D£(E)

is

more important, but the accuracy of all data is still insufficient to solve the pr o b l m .

More reliable for success promises to be another method — taking

into account the role of radio-active nuclei in the cosmic ray composition (the best known example is the T *

2.2xio6 E/Mc2 yr,

10Be

nucleus for which the average lifetime

where the factor

vistic slowing-down of the time).

E/Mc2

takes into account the relati­

The possibility of radio-active decay is

not taken into account in (15.62) and if one does so for radio-active nuclei, there should be on the left-hand side the sum ing the density of radio-active nuclei

(N£/x£) + (Nj/cpT£).

in comparison with the densities of

a number of stable nuclei one can in principle find both cpi.

and thereby determine

Tc r

£

Determin­

X£ = c p T c r

and the average gas density

p

^£

and

in the

region occupied by these cosmic rays (for details see Ginzburg and ptuskin, 1976a,b; Plovdiv, 1977;

Ginzburg,

1978).

We now make more precise the transfer Eqns. (15.51) for an application to elec­ trons and positrons. N i = N e(r, t , E)

In that case we must, clearly, assume that in (15.51)

or separately

Ne_

(electrons) and

N e+

(positrons).

Simpli“

fications occur when we assume that the problem is stationary (we drop the derivative

3Ne/9t)

and when we neglect the 'catastrophic ' energy losses.

We

then have - d i v (De 7 N e ) + ^

(be (E) Ne ) =

Qe (r , E) .

(15.64)

The characteristic nuclear lifetime for the proton component of the cosmic rays is (see (15.60)) E nucl ~

3 x 1015

|(dE/dt)nucl|

Even for the gaseous disc with

N ~

r ~

(15.60a) 3

I cm- 3 , the time

T nuci ~

I08 yr »

Tc.r.,disc * and for N ~ 10- 2 cm-3 we have already Tm c l ~ I010 yr » Tc.r. halo ~ * to 3 * 1 0 * y r . Hence it follows that the cosmic ray lifetime T - is* indeed, determined by their leaving the Galaxy and not c .r . by the losses (this is, in general, not the case for the rather heavy nuclei;

see

Table 15.2).

305

COSMIC RAY ASTROPHYSICS The tern

Qe

must

take

into

account

the

appearance

of

electrons

and

(or)

positrons not only as a result of their acceleration, but also due to various decays of unstable particles (y

» and so on)

which

collisions with cosmic rays (we can refer here to positron pairs produced by gamna rays)*

are

formed

6-electrons

in nuclear

and electron-

In the equation like (15.64) for the

positrons we must also introduce a term accounting for their annihilation. The most important difference arising when we consider the electron component of the cosmic rays as compared to the proton-nuclear component consists in the necessity to take, in general, the energy losses of the electrons into account. The variable

E

in Eqn.(15.64) therefore does not remain a parameter as was

the case in Eqns.(15.59) and (15.62). Integrating Eqn.(15.64) enables us to find the electron spectrum N e (r,t,E). Knowing this spectrum for the whole Galaxy we can evaluate the intensity of the synchrotron radio-emission received on Earth.

The same holds, of course,

also for the radio-emission from supernova shells, radio-galaxies, and so on. We

shall

not

give

Syrovatskii, 1964a; and Ptuskin,

here

the

corresponding

calculations

(see Ginzburg

Bulanov, Dogel*, and Syrovatskii, 1972a,b;

and

and Ginzburg

1976a,b) restricting ourselves to the physical processes which

must be taken into account.

The acceleration of the electrons in the sources

and the generation of ’secondary1 electrons and positrons by the proton-nuclear component of the cosmic rays deter-nines the power of the sources

Q^(r,E). The

electrons lose energy as a result of ionization, brems (radiative), magnetobrems, and Compton losses; these all contribute to the coefficient

be (E)

in

(15.64). We have already considered the ionization losses (see (15.23)), the magnetobrems losses were discussed in Chapter 4, and the brems and Compton losses will be considered in Chapter 16.

It is, however, expedient to give all these

losses for ultra-relativistic electrons in one place.

Ionization losses.

In atomic hydrogen

(15.23a) In an ionized gas 2TTNe“( „ _

/dE^ W j .

■

me

V

m 2c 2E

s\

4irNe2 fi2

V

' 05.24)

386

THEORETICAL PHYSICS AND ASTROPHYSlCf

Brens (radiative) losses. “ (If)

” '0- I S N a E eV/s

In the atomic interstellar gas (see ( 16.4g]| -

5.1 * 10” 10

Na

eV/8 -

me

In a fully ionized gas (see (16.46)) -

“ 7 X ICT11 tlfjln

\dt/r

\

(N * N

3

+ 0.36^ —

me2

eV/t

/ me2

is the nuclear or electron density),

Magneto-brems and Compton losses.

■

'v/* •

The part of this expression which corresponds to the magneto-brems losses is proportional to H 2 and is obtained from Eqn.(4.39), assuming that the direc­ tion of the magnetic field changes randomly so that

H2 « | H2.

The second

part of expression (15.65) corresponds to Compton losses in an isotropic radia­ tion field with energy density measured in energies

erg/cm3),

Wp^

(the quantity

H 2/87T+Wpj1

is in (15.65)

and we have here considered only the region of electron

E 10’ eV

|£.n N | < 15

the brems losses

The ratio of the magneto-brems and Compton losses to the brems losses (16.46) equals (dE/dt) m C ,r

m

+ (dE/d t)

C^i!0:(£+

(dE/dt)r

In the Galaxy (in the disc)

N

\ 8 tt

H 2/8tt ~ 10"12 erg/cm3 and

(for the black-body background radiation of temperature

\

E

_

(15.67)

P*V m e 2 Wp^ ~

10“ 12 erg/cm3

T*2.7°K

alone we

4 x 10“ * 3 erg/cm ; in the Galaxy, especially in the disc there are "Ph also many optical photons emitted by the stars). Therefore in the gas disc have

(for

N ~

1 c m " 3)

nmC r ~ 3 * 10 5 E/mc2 £ I

for

E

1010 eV, and in the

387

COSMIC RAY ASTROPHYSICS halo (with

N ~

10~2 cm“ 3)

f ^ I

for

E

^ I 0 fl eV.

Even in the radio­

disc, let alone in the halo, the main role is therefore played for the electron component in the most interesting energy range magneto-brems and Compton losses.

E ^ 108

to

l09 eV

by the

We must add to this that the brems losses

in fact belong to the 'catastrophic1 losses — they are mainly accompanied by the emission of photons of energies 'goes out of play'.

flu ~ E.

As a result the electron simply

The average characteristic time

Tr

for such losses (see

(16.48) below) equals E Tr ---------I (dE/dt) Even when

N ~

electrons

move

I cm“ 3 the time about

in

the

do not play

a

role.

(15.68)

r

Tf ~ 3 x 107 yr, which is less than the time the gas

already so large (compared to

I 0 15

---- S N

disc.

Tc r. ^ *

When to

N ~

IO” 2 cm” 3

3 * I O 0 yr)

the time

Tr

is

t*ie brems losses

For evaluating the electron spectrum in the Galaxy one

therefore usually takes into account only the rnagneto-brems and Compton losses. With this we conclude the present chapter which was devoted to a few problems in cosmic ray astrophysics.

Here, in contrast to other chapters we paid con­

siderable attention to descriptive, diminished

the

space devoted

to

essentially astrophysical, material which

theoretical problems.

We

did this

as

the

corresponding astrophysical information is practically completely absent in general physics and theoretical physics courses.

However,

if one does not use

it and does not take it into account, one cannot deal at all with cosmic ray astrophysics

and

one

could especially apply material features.

remains

is merely left with purely physical results which one to

cosmic rays.

undetermined

and,

in

In that case, however, the main,

loses

the choice of

all astrophysical

We, on the other hand, wanted to retain the astrophysical aspects.

This tendency will also be seen, although not quite so strongly, two chapters.

in the next

Chapter XVI X-RAY ASTRONOMY Processes leading to the formation of X-rays and gamma-r ay s • Definition of the quantities used in X-ray and gamma-astronomy. X-ray brems-emission by a non-relativistic gas (plasma). Bremsstrahlung by relativistic electrons and brems (radiative) energy losses. Scattering of relativistic electrons by photons (inverse Compton effect) . Compton energy losses. X-ray synchrotron emission. Remarks about the comparison between theory and observations.

X-ray and ganma rays 'by themselves'

(that is, without considering

their interaction with matter) differ not only merely in wavelength but are also 'neighbours' in the electromagnetic wave spectrum.

It is therefore

expedient to start the discussion of the processes leading to the appearance of cosmic X-ray and gamma-radiation without splitting the bands in more detail. We shall therefore first of all list the processes which lead to the production of both X-rays and gamma rays. It is true that it is opportune to note beforehand that the mean free path of the absorption coefficient of even hard gamma-rays, let alone those of the softer photons, does not exceed approximately

100g/cm2 .

Hence it is clear

that the cosmic gamma- and X-ray radiation which reaches the Earth cannot come from regions with an extra-ordinarily high density, for instance, from the interior of neutron stars.

It is from this also clear that the photon emis­

sion and absorption processes which we shall encounter in X-ray and gamnaastronomy have, so to speak, the usual character of those in atomic or nuclear physics.

In other words, we do not need to consider here some new, not yet

known emission or absorption mechanisms.

The specific features which arise in

X-ray and ganma-astronomy are connected in the first place with the fact that

Customarily we shall call photons with energies 1 0 0 < E ^ < 1 0 5 eV (wavelength A « 12400/EX (eV) A

approximately between 0.1 and 100 A) X-rays.

radiation emitted by atomic nuclei is usually called Ex = E y < 1 0 5 eV.

y-rays

However,

even when

We have denoted the energy of X-ray and gamna-photons by

Ex and Ey , but sometimes we shall understand by hard photon (in the X and y ranges).

389

Ey the energy of any

390

THEORETICAL PHYSICS AND ASTROPHYSICS

in the laboratory one is usually dealing with the scattering of hard photons by slow electrons, while in the Universe the scattering of high-energy elec­ trons by optical and radio-photons plays a larger role.

There are, of course,

also other peculiarities but in all known cases they refer to an actual problmn or to parameters which characterize the problem and not to the matter itself of the elementary processes discussed.

Hence,

when

we

are

talking

about

elementary processes which are important for X-ray and gamma astronomy we may assume that the picture is rather obvious. The following processes lead to the production of X-ray and gamma photons: 1.

Bremsstrahlung of electrons and positrons apart from certain exceptions

(we shall in what follows not mention positrons separately). We have here in mind collisions of electrons with various nuclei and also with other electrons in which both the incident and the scattered electron have a continuous spectrum while, apart from recoil, the scattering particle does not change its state. A particle with kinetic energy

E^

can emit a brems photon only with an energy

Ev v s Ek (for the sake of simplicity we have in mind only collisions with T rather heavy particles at rest; when recoil is taken into account

E„

< Eu).

From this it is clear that non-relativistic electrons can produce only X-rays as the result of the brems mechanism. gamma photons.

Relativistic electrons can also give

The intensity of the bremsstrahlung at nuclei is for relativis­

tic protons or nuclei of mass energy by a factor

M

less than for electrons with the same total

(M/m)2 > 3 . 4 * JO6 •

There is thus every reason to limit

the discussion to bremsstrahlung by electrons. One can also take the radia, . + ± + tion accompanying the appearance of electrons and positrons in the tt -►u -► e decay or that arising in the formation of

6-electrons

(recoil electrons) to

be bremsstrahlung. We do not intend in what follows to dwell upon all aspects of the theory of bremsstrahlung. equilibrium

We

shall

consider

non-relativistic

plasma

only and

two

cases:

bremsstrahlung

bremsstrahlung

of

of

an

relativistic

electrons. 2.

Recombination and characteristic X-ray radiation occurring when an elec­

tron makes a transition from a level in the continuous spectrum to an atomic level or when it goes from one atomic level to another. In a6trophysical terminology we are dealing, respectively, with free-bound

39*

X-RAY ASTRONOMY and bound-bound electron transitions , vhereas brernestrahlung corresponds to free-free transitions.

Type 2 processes will be touched upon only in passing

in the present chapter.

3.

Compton scattering of relativistic electrons by X-ray, optical, and

radio-photons. We have already mentioned this process before (see, for instance, Chapter 15). Various relations are possible between the energies of the incident and scat­ tered particles, but we shall consider only the case E > E y » € ph , where

E, Ey, and

€ph

(16.1)

are the energies (in the *laboratory* frame, that is,

the frame fixed in the Earth or the Galaxy) of the primary electron, the scat­ tered y-

or X-ray-photon

the

energy

primary

Gph

photon,

refers

to

respectively;

the

region

of

the

in

the

majority of

cases

(Epk ~ 1 eV)

or the thermal black-body background radio-radiation (€ph ~ ! 0 ~ 3 e\

X~

the

and

optical

I mm) , but we are also interested in the scattering of electrons by cosmic

X-rays (in that case

€p^ ~ 102

to

lO^eV

and

y-rays

are formed as the result

of the scattering) and by radio-photons, say, of synchrotron origin case

€ph ~ 10“ 5

to

10-7 eV,

X~

IOcm

to

10 m

and

(in that

X-ray and optical photons

are produced by the scattering). The scattering of relativistic protons and nuclei by photons is appreciably less efficient (in comparison with the scattering of electrons, there occurs a factor it. 4.

(m/M)2 ^ 3 x I0“ 7 ) and in practically all known cases we can neglect

Compton radiation will be considered in the present chapter. Synchrotron radiation.

For

the

sake

of

convenience

characteristic frequency

vm

we

write

down

once

again

Eqn. (5.40a)

emitted by an electron of energy

E » me2

for

the

in a

magnetic field v = 1.2 X 10* H, ( - V ) “ A ' m e 2'

= A . 6 x i o " 6 H, (E(eV))2 Hz . 1

Hence one sees easily that in fields a frequency

X

^0.1 I

Vm ~ )018 s” 1

{Xm

^ 10~3 0e

= c/v^ ~ 3 I )

(y-rays) is emitted only when

(16.2)

radiation is emitted with

for

E > lO^eV.

E ^ 1013 eV;

emission with

In galaxies, radio­

galaxies, and most supernova shells synchrotron X-ray and gaama-radiation can thus arise only if there are electrons of very high energies present.

ThX6

fact clearly limits the possibility of applying the synchrotron mechanism to hard photons.

It is sufficient to 6ay that electrons of energy

E ^ 1019 eV

392

THEORETICAL PHYSICS AND ASTROPHYSICS

in a field

Hj^ ~

10-3 Oe

lose half their energy in a time = 5 . 1 x 1 0 ® m e 2/H 3 E s < 1 yr

T

ID

(see (4.42)).

1

In stars, quasars

some regions of supernova shells rather strong magnetic fields.

(close to their nucleus) and, possibly, in (particularly near pulsars)

there may exist

Clearly, under such conditions X-ray synchro­

tron radiation may be produced already by electrons of lower energies. instance, for

H ± ~ 10 2 Oe

with energies

\>m ~

the frequency

For

1O10 s ' 1 occurs for electrons

E ~ 5 x lo 10 eV ; in that case, however, Tm ~

I s.

From what we have said it seems that the appearance of cosmic synchrotron X-ray

and

gamma-radiation

(forgetting

about

pulsars,

or,

vicinity of compact sources — white dwarfs, neutron stars, relatively improbable.

Nonetheless,

in

general, the

’black holes’) is

this conclusion is in fact rather condi­

tional as there are circumstances under which the acceleration of electrons or their injection into an extended region with a rather strong field can be very efficient.

As an example we may mention the Crab Nebula for which the X-ray

emission has a synchrotron nature (as was shown relatively recently by measur­ ing the polarization of the radiation).

The pulsar PSR0531 which is situated

in the nebula plays here (directly or indirectly) the role of an efficient electron injector.

Cosmic synchrotron X-ray emission is thus observed and,

undoubtedly, plays an important role (and one should note that as data are accumulated

this

role

becomes

ever

more

important).

We

shall

return

to

synchrotron X-ray emission later on. 5.

Decay of neutral pions into two y-photons

The rest energy of a only by cosmic rays. collisions

(p

tt°

equals

c2 = 135 MeV

(tt0 -*y + y ).

and, hence, the

Their production occurs mainly in

is a proton, and

a

tt°

are produced

p-p, p-0t , and

a-p

a helium nucleus).

However, at sufficiently high energies

tt°

’s can also be produced through

photo-production when cosmic rays collide with (radio- , optical, or X-ray) photons which occur in space.

The general expression for the threshold

for photoproduction of particles of rest mass E » Me2 , rest

mass

M = AMp ) collides

the form^ E .

min

t

ZM + m 2M ----- s m c H * cph

with

m^

photons ..o Me

"Ph,0

E^iti

when a nucleus (total energy

2 e ph

See footnote on next page

of

energy

Cp^

has

(16.3)

X-RAY ASTRONOMY

393

where

2M+m €ph,0

7T

(16.4)

W m TTC

2M

is the threshold for pion photoproduction at a nucleus of mass tt°

photoproduction threshold

€ph,0

^or

nuc^eons

rest

E . » min When

€ph ~ 10"3 eV

€ p^ ~

1 eV

at rest. The

is approximately

l50MeV and, hence, the energy of cosmic protons which generate instance, on optical phonons with

M

tt°

'

s

,

for

must exceed an energy

150 M„ c2/2€_k MeV ~ 1017 eV. P P«

(background radiation)

~ 102° eV.

It is just such

photoproduction processes which will, in general, lead to the 'cut-off' of the cosmic ray spectrum for

E ^ 1019

to

1020 eV.

Gamna-rays are, of course formed not only in the

tt°

decay but also in several

other decay processes and we shall discuss these processes in Chapter 17. 6.

Electron and positron annihilation (e+ + e -►y + y).

There are always some positrons in the universe, as they are formed in the ir+ -*y+ -»-e+

decay and in a number of other processes.

the annihilation of relativistic,

or

at

any

One must distinguish

rate fast, positrons in flight

It is well known that one obtains expressions for the threshold for the production of particles from energy and momentum conservation laws. Now­ adays the corresponding calculations are normally simplified by using four-dimensional vectors. For instant, we obtain Eqn.(16.3) by denoting the appropriate four-vectors for the incident particle, for the pion, and for the photon by p. - {p,iE/c}

, 7K = {w.iE^/c}

p ? = p 2 - E 2/c2 = - M 2 c2

,

it?

, k £ = {k , i€ph /c} ,

=- m ^ c 2

, k? = 0 .

(in a different notation, for instance, p1 = (E/c,p); see Landau and Lifshitz, 1975). The energy-momentum conservation law for the photoproduction of a pion has the form k^ + P\ i - P2,i + ^i* Squaring this relation and using the above notation we get M 2c2 +2k.pj ^ * (p^ . + iri )2 At the threshold of photoproduction, however, we have

(pj i + *^i)2 -

- (M + m 7f)2c2 , as we can use for the evaluation of this quantity any frame of reference, and in the centre of mass frame at the threshold for the production the particle and the pion are at rest. Moreover, if the photon and the particle collide head-on 2k.p. = - 2 ( € ph/c)|P l | - 2 e p h E/c2 « - A e ^ E j / c 2 , where the last expression refers to the case Ej » M e 2. We thus get at once Eqn.(16.3) for Ej - Emin * For photoproduction on a particle at rest 2kj p£ = - 2 €p^ M and we get Eqn.(16.4) for €p h * € p h > Q .

39 A

THEORETICAL PHYSICS AND ASTROPHYSICS

and the annihilation of (slow) positrons at rest.

In the first case ganma-

rays are formed with a continuous, or at least a very wide, spectrum.

In the

second case (annihilation of positrons which have been stopped) the ganmaradiation is monochromatic ( E y * m c 2 = 0.51 MeV)

and because of this feature it

can, in principle, be distinguished above the background of the continuous spectrum. The gamma-radiation arising from the annihilation of anti-protons by protons or of any other particles by their anti-particles can practically play no role unless one talks about regions where matter and anti-matter come into contact. We do not think that such a possibility has a great probability and at any rate there are no sufficiently well defined indications for its realization (for details see Stecker, 1971; 7.

Stecker and Trombka, 1973).

Nuclear gamma-rays arising from radiative transitions in atomic nuclei.

In stellar atmospheres and in outbursts (such as supernova outbursts) nuclei are excited in nuclear reactions and as a result of collisions with fast parti­ cles, and this can lead to gamma-radiation.

The exciting agent in interstellar

and intergalactic space are the cosmic and subcosmic rays.

It is important to

emphasize that the spectrum of the nuclear gamma-radiation may be either con­ tinuous, or discrete (we refer lines).

here

to

the presence

of more

or

less

sharp

The latter case occurs for nuclear reactions in which slow particles

(nuclei in interstellar space) are excited.

However, if a nucleus which is

part of the cosmic rays is excited in some collision, it usually has a large velocity and when we take the contributions from cosmic rays with different energies into account, its gamma-radiation produces a continuous spectrum. We now remind the reader of some basic definitions and notations

(we follow

here and in other places in this chapter Ginzburg and Syrovatskii, Ginzburg,

1965 and

1969b).

In observations one measures one of the following quantities: the intensity

Jy (Ey)

and the flux

Fy (Ey)

in photon number, or

the intensity

Iy (Ey)

and the flux

$>y (Ey)

in energy;

(16.5)

The intensities and fluxes given here are differential quantities; for instance,

395

X-RAY ASTROPHYSICS Jy (Ey) dEy

is the number of photons with energies in the range

Ey ,

Ey + dEy

crossing unit area (normal to the photon momentum) per unit time and per unit solid angle.

The corresponding integral quantities have the form

y > y = J y y dE-I » ey

IY (>EY ) = f

dEY "

ey

v>v - iV? 4> (>E ) Y Y

Let in a volume element time q(Ey) dEy d V d 2Q

J

EY JY (EY } dEY *

ey

(16.6)

*JEf'vew °•

-I

* (E') dE' . Y Y Y

dV

a source of X-rays or gamma-rays produce per unit

photons moving off into an element of solid angle

with energies within the range

Ey , Ey + dEy . The quantity

the emittance (emissive power)

in photon number.

used earlier (see, for instance, obvious relation

which we

q(Ey)

q(Ey) = Cv /h2v .

d 2ft,

is called

The emittance

(5.52)) is connected with

Ey q(Ey)dEy = £v dv , whence

q(Ey)

through the

If the emission

is isotropic, it is convenient to use also the emittance in all directions * TTSy (16.7)

q(Ey) = 4 TT q (Ey) = h 2v In gamma-astronomy

one

uses

mainly

the

astronomy the application of the emittance quantities)

is not less wide-spread.

q(Ey)

emittance ev

If the

whereas

in X-ray

(and in general the energy-scale

X-rays

or gamna-rays are formed

by cosmic rays (or by any other particles) with an isotropic intensity we have

„

q(Ey) dEy = 4irq (Ey) dEy = 4irN(r ) dEy

Here

N(r)

J(E) ,

J

cr(Ey , E) J(E) dE .

(16.0)

is the density of the atoms (or, say, electrons, soft photons, and

so on) in the source and CT(Ey ,E) dEy = dEy j

a(Ey , E , ( i ' ) d V

(16.9)

is the cross-section, integrated over the angles at which the photons leave, for the production of photons (with energies in the range particles of energy Let

the

source

be

Ey,

E y + dEy)

by

E. at

a

distance

R

from

the

observer.

In

that

case

the

396

THEORETICAL PHYSICS AND ASTROPHYSICS

radiation flux from the source into a solid angle

d 2ft equals

I* q(E ) dFy (Ey) - Jy (Ey ) d 2n - d 2n

I

f4

--- R 2dR - d 2fl 1 q(Ey ) dR

0 and

C0O

L f

Jy (Ey ) “

J

(16.10)

0

j

q(Ey ) d R - N ( L )

where

a (Ey , E) J(E) dE ,

(16.11)

e N(L) =

N(R) dR

(16.12

0 is the number of atoms (or other particles with which the cosmic rays which produce the gamma-radiation collide) along the line of sight; soft photons

in the case of

L Nph(D

along the line of sight.

-

|

V

dR

In (16.11) we have assumed the cosmic ray intensity

to be constant along the whole path

L.

One can easily drop that restriction.

In the case of gamma-rays one often calls

Jy(Ey)

and

Jy(> Ey)

the differen­

tial and integral gamma-ray energy spectra. For discrete sources

(especially when their dimensions are small) one normally

uses the following expressions for the flux : r Fy (Ey) = J

J

T

J q ( E Y) d 3 r N Jy (Ey ) d 2fi = ------^-----«-5-

n

r2

R

a ( ¥ y , E) J(E) dE ,

(16.13)

Ey

where the integration is over the solid angle under which the source is seen; the source is at a distance

R

from the observer;

Ny = R2 |

N(L) d 2S2 *a

J

in (16.13)

N( r ) d 3r

(16.14)

Q is the total number of particles (or soft photons)

in the source.

We now turn to a discussion of the mechanisms of X-ray emission and we start with X-ray bremsstrahlung of a hot non-relativistic gas (plasma). A hot gas which is partially or completely ionized is a source of brems-, recombination, and line-spectrum (characteristic) X-ray emission.

At suffi­

ciently high temperatures (we shall make clear what this means in what follows) bremsstrahlung plays the main role.

Moreover,

if we are dealing with a hydro­

gen or hydrogen-helium plasma, one does not consider the line-spectrum X-ray emission at all.

X-ray bremsstrahlung is observed in X-ray ’stars' and also

in the solar spectrum.

397

COSMIC RAY ASTROPHYSICS

Ve give below the basic formulae with which one is dealing when discussing X** ray bremsstrahlung (for a more general approach see Heitler, 1947; Salpeter, 1957;

Hayakawa, 1969;

Lifshitz, and Pitaevskii,

Blumenthal and Gould, 1970;

1971).

Bethe and

Berestetskii,

Apart from for X-ray astronomy problems,

this kind of radiation is also of interest for thermo-nuclear investigations and in research about the use of a hot plasma in laboratories as a powerful source of X-rays. For sufficiently fast, but still non-relativistic electrons one may assume that the following conditions hold: e2Z/fiv « As

I

, jmv2 « me2 .

(16.15)

e2/tlc = 1/137 the first condition (16.15) will, of course, not be satisfied

for very heavy elements, but we have in mind the case of light elements. total energy emitted in a single collision equals (see Heitler, f W = I E cr(EY , E) dE J ^

16 Z 2 E6 / = --------= -y-a r 2 Z 2 m e 2 3 m c 3 fi

where, as before, we have denoted the photon energy by Brems (radiative)

1947, 525)

,

and

(16.16)

re = e2/mc2 .

losses of a single electron per unit time are equal to

16 Z 2 e 6Nav ( = W N v = -----------r a 3 me 3 fi where

E^

The

= 2 . 5 * 1 0” 3 3 Z 2 N v erg/s ,

is the density of nuclei in the medium.

When

E ~ me2

and

(16.17)

v « c

Eqn.(16.17) and Eqn . (16.46) given below for the relativistic region give for Z = 1 approximately the same value

- (g^)r w 8 N fle 6/mc2 fl .

We note that in the

non-relativistic approximation the radiation from electron-electron collisions is appreciably weaker than for electron-proton collisions.

The point is that

when identical particles collide there is no dipole radiation due to the momen­ tum conservation law, while the quadrupole radiation is weaker than the dipole radiation by a factor of the order of For an equilibrium plasma range

v , v + dv

the

density

equals

dN = N(v) dv = 4 ttN Q

(v/c)2 . of

electrons

with

velocities

in

the

3 ^

j)

V * GXP (~ 2 ^

t)

dv = N •

(16.18)

The total power of the radiation from unit volume of the plasma is thus

4ire

- f

" J

mv

M

|dt|

1

d v 3 dv

2 k T; 32 /2 Z 2 e*N N(k„T/m)i a o

3/

it me5

fi

(16.19)

398

THEORETICAL PHYSICS AND ASTROPHYSICS

The factor

4 tt

is used here in order that the integral emittance

is defined per unit solid angle.

£ ■ / C^d

Due to the quasi-neutrality condition, which

is usually well satisfied, we have for a completely ionized plasma of one kind of atoms

N * ZNa .

Thus, we have from (16.19) for a hydrogen plasma

Atte - 1.57 X I0“ 27 N 2 /T erg/cm3.s , where the temperature is measured in degrees absolute

(16.20)

(T has the meaning of

the electron temperature); clearly, T (°K) = (kfl/1.6 * lO'12)^1 T(eV) = 1.6 x \ q " t(eV) and N

is the electron density in cm“ 3 .

Somewhat more exact calculations

which take into account electron-electron collisions and relativistic correc­ tions lead to the expression 4 ire = 1.6 x 1CT2 7 N 2 /T (l + 4 . 4 * 1 0 " 10 T) .

(16.21)

Equations (16.19) and (16.20) ate applicable only when conditions (16.15) hold which for

Z = 1 give

v »

3 x 10®

cm/s, or,

T ~ 2X- » 3k B

-5— ~ |05 °K . n 2kB

(16.22)

On the other hand, from the condition that relativistic corrections be small we have T «

2 £ i ~ 1010 °K . fcB

Of course, not only the integral emittance sivity

, but also the differential emis-

e v , introduced earlier, is of interest. 03

f

|

J o

0(Ey , E) v(E) N(E) dE =

e

Y'

00

= h2

V dv

«r r- / 2 hv/m

Ey = hv - ha) is the photon energy and

thermal equilibrium N(v) dv The cross-section

a(Ey,E)

By definition

a>

od

e = | Ev dv - | dEy Ey o 0

where

e

(16.23)

a(hv, E) vN(v) dv ,

E= £mv2

the electron energy; in

is given by Eqn.(16.I8). depends only weakly on

E

(see

Heitler, 1947;

Blumenthal and Gould, 1970) and to a first approximation we can put a(Ey,E) » con6t/Ey = const/v , whence E^ - const x exp (- hv/kflT) . One can easily determine the constant from the condition

399

X-RAY ASTRONOMY

€ dv = e J0

v

and ue have thus 7.7 * IQ'38 N 2 ex

/ _hv_\

erg (16.24)

4ir/T

C X P \ kBT'

an 3.s.Hz

We now discuss the other limiting case of small energies (low temperatures) when — » fiv

1

.

(16.25)

When condition (16.25) holds one can perform the calculations classically. Indeed, the electron wavelength to which

the

electron

Ze2/rm £n = \ m v 2

approaches

and is equal to

inequality (16.25) holds classically.

A = h/mv = 2tt fi/mv while the smallest distance the

nucleus

is

rm £n =■ 2 Ze2/mv2.

rm £n »

A/tt

found It

from

is

the

clear

condition

that when

and one can describe the electron motion

One can also describe the radiation classically, but one nni6t in

the integration over frequencies take a quantal element into consideration — one integrates only up to the frequency velocity before the emission.^

V = m v 2/2h,

where v

is the electron

Landau and Lifshitz (1975, §70) give a detailed

account of the classical evaluation of the radiation of a particle moving in a Coulomb field.

We are here interested in the quantity

J

dW = where charge

W(p)

W(p) 2irp dp ,

is the energy emitted in the range

e passes a nucleus of charge

Ze

u) , to + dto

at a distance

particle which we assume to be an electron is equal to recoil of the nucleus).

p m;

when a particle of (the mass of the we neglect the

If

Ze we have

tlv

16 TTZ2e* dli)

-

dW =

(16.26)

^that is.

co »

=

3 /3 v 2 m 2 c 3

tlco/

32 TT2Z 2e6

:— :— r dv

(16.27)

3/3 j / j v ‘ m"c'

One can, of course, obtain the same formula quantum-mechanic ally, but in this case the classical calculation is completely sufficient (the condition (16.25) is at once the condition for the applicability of the quasi-classical approxi­ mation for a Coulomb field).

To be more precise, a classical consideration i6 applicable provided hv = flu < J m v 2 ; in a number of cases one can use the classical formulae approximately also when hv £ J m v 2 .

400

THEORETICAL PHYSICS AND ASTROPHYSICS

The total energy emitted in the collision is equal to hvmax * I “ v*

l6Tr2Z2e s W

|

(16.28)

dW 3 /3 m e 3 h

The inaccuracy of the initial expression (16.27) caused by condition (16.26) is unimportant when we integrate over the frequency.

More important is the

restriction

connected

up

0) * m v 2/2fi

(vide infra).

with

the use

of

Eqn. (16.27)

to

the

frequency

For a hydrogen plasma with a Maxwellian distribu­

tion we have , _ 4 ire » ^

* 2h v/m f dW m 2 ( m -r— N 4tt I------- y dv \2tt k«T /

Jo

3

v a exp

32 TT /2tt N 2 exp (- hv/k T) = 4 tT£

(

mv2 \ - ----- ) dv = \ 2knT /

h exp (- hv/k T) D k T B1

3 / 3 n ^ ( k „ T ) ^ c* o erg = 6.8 x IO- 5 # — exp (- — ) /T V kgT/ koT' c m . B . H z

(16.29)

In deriving this formula we used the fact that a photon of energy emitted only by an electron of energy

J m v 2 > hv.

hv

can be

The integral emittance €

is equal to 16/JirN2 e 6 (kgT/m)* 4ire

= | 47re^dv = | W dN 3 /3 me 3 t!

=

I. 4 2 x !0“ 27 N 2 /T

erSc m 3 .s

.

(16.30)

The value (16.30) differs from (16.20) merely by a factor 1.57/1.46*1.1 which in applications to astrophysical problems can usually be assumed to be equal to unity.

Therefore, although one should use Eqns.(16.29) and (16.30) in the

temperature range (see (16.25) with

Z = 1)

^ 3kg

io5

one can in fact always use Eqns. (16.20), (16.21), precise,

the

use

of

(16.20),

(16.31)

*i2k 2- B t and (16.24).

(16.21), and (16.24)

at

all

To be more

temperatures

is

allowable as long as we are talking about accuracies of the order of tens of percents (however,

(16.24) is inaccurate when

hv/kgT « 1 ).

In the high-

temperature region given by (16.22) and (16.23) E q n . (16.21) is clearly very accurate.

Even at high temperatures Eqn.(16.24) is rather accurate only when

401

X-RAY ASTRONOMY hv/k^T > 1 (when vide infra).

hv/k^T E) FdEy , where

If we are dealing with a 'bare1

nucleus, that is, with scattering by a Coulomb centre,

•i = In

the

scattering

by

nuclear charge, and as

(16,40)

atoms

the

a result

electrons the

in

the

cross-section

atomic

shells

changes.

The

screen

the

amount

of

screening is determined by the parameter

r _ Rc me2 Ey „-i 5 IT — E - Ey Z * The meaning of this parameter becomes clear if we take into account that when an electron is scattered by a nucleus the latter acquires a momentum where

r

Ap — Tl/r,

is the effective distance at which the electron passed the nucleus

(for details see, for instance, Heitler, 1947, §25).

Moreover, it follows

from the conservation laws that

Ap

1 Ell 2 E

^ me E - Ey

and thus r

_R_

E (E ~ EY>

mc

m e 2 Ey

On the other hand, in the statistical model of the atom the radius of the atom a ~ aQ Z £

where

a 0 * n2/me2 ■ (fi/mc) (e2/Rc)*1 ■ 5.3* 10“9 cm.

introduced a moment ago is clearly of the order of the ratio

The parameter a/r.

The

THEORETICAL PHYSICS AND ASTROPHYSICS

406

harder the emitted photon, the closer the electron must pass the nucleus and the less the screening will be; (16.40) are valid.

in that case, if the parameter

£ ^

1 » Eqns,

Softer photons are anitted when the electron passes at

appreciably larger distances from the nucleus.

When

£«

1 the screening is

large and for heavy atoms we have «

« to (191 * Z ~ ’)

, $2 = - f t n (191

It is clear from Eqns. (16.39) to (16.41) that the a weak

one

and

that the Ey-dependence

j .

E-dependence

of

is given by the factor

remark is especially valid for complete screening,

(16.41)).

(16.41) o(E^ , E) is 1/Ey

(this

Moreover, Eqn.

(16.41) is inaccurate for light elements. If we are talking about errors which are not less than a few percent the crosssection

for

bremsstrahlung

under

conditions

of

complete

screening

can be

written in the form a ( E Y ,E)dEY

MdE « --- i

,

(16.42)

r Y where M;

tr

is the radiative unit length (in

g/cm2)

in a gas of atoms of mass

under the conditions of (16.41)

cr

= 4 — Z 2 M -1 Hn (191 xZ~* ) tic

and for hydrogen (Z = 1 , M = 1.67 * 10~2U g) we would have

tr » 7 3 g / c m 2 .

In

fact, because of the inaccuracy of Eqn.(16.41) for light elements the value of t^

for hydrogen is somewhat lower even when we take into account the contri­

bution from electron-electron collisions. and Pomanskii,

Detailed calculations

1964) lead to the following values for

tc ~43.3, tN =38.6, tQ = 34.6, tFe = 13.9 g/cm2 r

tr : ^ = 62.8, tjje = 93.l,

(we have dropped here the index

and replaced it by the symbol for the relevant element).

interstellar medium (about 90% H put

M = 2 x l 0 ~ 2lbrems(E Y ) = I dR

where

is

gamna-rays and

the

intensity

Na (R)

1

1M R ) a r (EY .E) J e (E,R) d E ,

of

the

electron

component

which

produces

the

is the density of atoms in the interstellar medium.

Using Eqn . 0 6 . 4 2 ) and assuming that the intensity line of sight we get

(16.43)

^

o

Je

ia constant along the

x- ray astronomy

407 MN(L)

J e ( >EY)

tr

Ey

J Y,brems^EY^ where

M(L) = MN(L)

1.5 X )0 -2 M(L)

J e (>Ey ) —

—~ ,

5

06.44)

EY

i s the mass of gas along the l i n e of s ig h t ( i n g/cm2)

and Je ( >Ey ) = }

Je (E)dE .

®r In u n-ioniz e d hydrogen with Ey ^ E the parameter £ * 102 mc2/E and £ 4C 1 f o r E » 5 * l 0 7 eV; under those c onditions one can use Eqns. (16.42) and (16.44). In a f u l l y ionized medium i t screening.

Indeed,

r D * V (kgT/STrNe2) order of

I 0 3 cm.

in

th a t

is

p ractically

case

always p e rm is s ib le

the screening r a d iu s

which i s , f o r in s ta n c e , fo r In t h a t example

E/mc2 ~ 3 * 1013, t h a t i s ,

is

to n e g le c t

the Debye ra d iu s

T ^ I O 1* °K, N ~0.1 cm' 3 of the

r ^ ~ r ~ (H/mc) (E/mc2) (vide supra) only when

E ~ I 0 19 eV.

In an ionized gas we have th e r e f o r e

u s u a lly r « r ^ , and screening i s unim portant.

When th e r e i s no screening one

must under the c o n d itio n s (16.38) use E q n s .(1 6 .3 9 ), (16.40), and (16.43).

We

note t h a t the brem sstrahlung i s then not taken i n to account which a r i s e s in the c o l l i s i o n s of the in c id e n t e le c tr o n w ith the atomic e le c tr o n s (and in g e n e ra l with the e le c tr o n s in the medium, fo r in s ta n c e , in a plasma).

To a

rough approximation one can take the e f f e c t of the e l e c t r o n - e l e c t r o n c o l l i ­ sions on the c r o s s - s e c t i o n (16.39) i n to account by r e p la c in g th e f a c t o r Z2 by Z(Z + 1 ) .

The meaning of th a t s u b s t i t u t i o n c o n s i s t s c l e a r l y i n th e f a c t th a t

the c r o s s - s e c t i o n o ( E y , E )

i s fo r e l e c t r o n - e l e c t r o n c o l l i s i o n s approximately

the same as f o r e l e c tr o n - p r o to n c o l l i s i o n s ; moreover, we have, of co u rse , used the f a c t th a t th e r e a r e Z e le c tr o n s in the atom.

We emphasize t h a t t h i s

approximate way of taking the e l e c t r o n - e l e c t r o n c o l l i s i o n s i n t o account i s in te g r a te d over the a n g le s. i n te g r a l c r o s s - s e c t i o n

J u s t because of t h i s the c o n tr i b u t i o n to the

o(Ey,E)

from pro c e sse s connected w ith t r a n s f e r r i n g a

large momentum to the atomic e l e c t r o n tu r n s out to be i n s i g n i f i c a n t . value

tr =

66

The

g/cm 2 given above f o r the i n t e r s t e l l a r medium was obtained

taking in to account the brem sstrahlung in e l e c t r o n - e l e c t r o n c o l l i s i o n s (see Dovzhenko and Pomanskii, 1964).

Below we sh a ll a ls o take the e l e c t r o n - e l e c t r o n

c o l l i s i o n in to account by r e p la c in g Z2 by Z (Z + 1 ) choice of the va lue of

or by the a p p r o p r ia te

tr .

Due to brem sstrahlung th e e l e c t r o n s lo se energy — th e corresponding lo s s e s a r e , as we have a lr e a d y mentioned, c a l l e d brems or r a d i a t i v e l o s s e s .

We

emphasize th a t r a d i a t i v e lo s s e s occur mainly i n l a r g e amounts ( t h a t i s , they belong to the c l a s s of

'c a ta s tr o p h ic 1 losses;

see Chapter 15).

I t i s , for

THEORETICAL PHYSICS AMD ASTBOPHYSICl

408

in s ta n c e , c l e a r from (16.42) t h a t th e t r a n s f e r r e d e nergy i s E EY a(Ey , E) dEy c o n s t . dEy ~ c o n s t . E , o t h a t i s , i t i s determ ined by th e e m issio n of photons of energy

J

J

r a d i a t i v e lo s s e s t h e r e f o r e f l u c t u a t e w i l d l y .

Ey ~ E.

The

We r e s t r i c t o u r s e l v e s , however,

to e v a lu a tin g the average lo s s e s per u n i t p a th l e n g t h E - ( £ ) r = 1 Na EY a ( E Y, E) d EY .

(16.45)

0

where

Na

i s the d e n s i t y of atoms and where we have used the f a c t t h a t the

cro ss-sectio n a(Ey,E)

i s norm alized to u n i t e l e c t r o n f l u x (moreover, by

v i r t u e of (16.38) we have r e p la c e d the upper l i m i t of the i n t e g r a l E); f o r the u l t r a - r e l a t i v i s t i c e l e c t r o n s c o n s id e r e d h e r e th e l o s s e s per u n it c =

3

x

time a r e o b ta in e d

1 0 10

simply by m u lt i p l y i n g

the v a lu e

E - m c 2 by dE (^ -)

(16.45)

by

cm /s.

In a completely ionized gas (plasma) or when there is no screening we have from (1 6 .3 9 ),

(1 6 .4 0 ), and (16.45)

( E y ) =

l

J

is the number of electrons with energies

qe (E)dE

E > Ey

produced per sec along a

path length L along the line of sight while Ey is some average value.

Actual

estimates for the Galaxy indicate that the intensity Jy pr ‘ €Ph * €P h , 2 = V (1 - v /c ^ J

] - (v /c ) cos

02

+ (eph/E ) ( l - cos

0

)

= * ( 6 ph . E .

0

, ,

0

,

0

)

If

E >

ey

(16.54) (16.55)

» e ph

we have approximately €ph (1 ” (v/c) cos ^

0

j]

1 - (v /c ) cos 0 2 + (Cph/E) (1 - ( v /c ) cos 0 j ) cos 0 2

(16.56)

I t i s convenient to w r i t e the c r o s s - s e c t i o n f o r the s c a t t e r i n g of u n p o la riz e d p a r t i c l e s in i n v a r i a n t form (see B e r e s t e t s k i i , L i f s h i t z , and P i t a e v s k i i , 1971)

(16.57) where d 2

i s an element of s o l i d a ngle c orresponding to th e d i r e c t i o n of

k 2 ( th e s o l i d a ngle ft

used below r e f e r s to th e d i r e c t i o n of k x ) .

If in the initial state the electron is at rest (that is, p * from (16.54) and (16.57) the well known expressions

Pl m 0)

ve get

413

X-RAY ASTRONOMY

Ph_________

i + (eph/mc2)(i -

ar d2fi2 - i f - i i y f i l . y c

2

* w v

\ y

(16.58)

e)

cob

( ^ l + ^ L - 8i n 2e ) d 2fi 2 * V ey €ph )

In the n o n - r e l a t i v i s t i c l i m i t when in (16.59)

Cp^^m c2

(16.59)

we can put

V S h

( t h i s i s , of c o u rse , a ls o c le a r from (16.58)) and then a„dz Q

I + c o s 20 ) d 2 J2

(16.60)


l- 3 x 10"*0 e

According to (1 6 .8 1 ) , the energy of the 5

x | 0 9 eV.

In the same f i e l d

v 2 ~ I 0 11* to

I 0 15

H, „ - H. , i •2 l,i Hz (X2 ■ 0.3 to 3 pm) can

42 J

X-RAY ASTRONOMY

be em itted only V ~ I 0 ie Hz and, an energy

by e l e c t r o n s

of

e n e r g ie s

E2 ~ 5 * 1 0 12 eV.

For X-rays

hence, f o r an unchanged magnetic f i e l d e l e c t r o n s must have

E 2 ~ 3 x 10 1** eV.

We must b e a r i n mind t h a t sy n c h ro tro n l o s s e s a re p r o p o r tio n a l to H2 E2

(se e

(A.39)) and th e r e f o r e p a r t i c l e s w ith very high e n e r g ie s or when they move in a strong f i e l d a r e slowed down f a s t .

An e s tim a te of the energy and of the ' l i f e ­

time' in a magnetic f i e l d can c o n v e n ie n tly be found by u sin g E q n s.(4 .4 1 ) and (4 .4 2 ).

We can then e x p re ss in E q n .(4.42) the e l e c t r o n energy i n terms of the

c h a r a c t e r i s t i c frequency of i t s r a d i a t i o n (16.81) and, th u s , o b ta in immediately a r e l a t i o n between the observed frequency and the c h a r a c t e r i s t i c l i f e t i m e (th e time over which the energy reduces by a f a c t o r two) of the a o i t t i n g e l e c t r o n s : 5 x 10 s me 2 _ 5 .5 * 1 0 n s » ------j— s « H H* v2 Here H± i s measured in Oersted and v in H ertz.

„ 1 .8 * 1 0 " 3 — j— y r .

(19.83)

The time Tffl expressed in

terms of the frequency h a s , of c o u r s e , a somewhat a r b i t r a r y c h a r a c te r as we have chosen f o r \J the frequency c orresponding to th e maximum in the r a d i a t i o n spectrum of mono-energetic e l e c t r o n s . In a f i e l d

H± “ 3 x 10~6 0e the time Tm f o r e l e c t r o n s w ith e n e r g ie s

5 X 1012, and 3 x 10ll*eV For our Galaxy and,

i s , r e s p e c t i v e l y , 2 * 1 0 a , 2 * 10 5 ,

in g e n e r a l ,

for

normal g a la x ie s

5 * 10 9 ,

and 3 x 10 3 y e a r .

f o r which the v a lu e

H = 3 X 10” 6 Oe may be assumed to be t y p i c a l a c h a r a c t e r i s t i c time Tm of the order of

10s

year and, even more, one of th e o rd e r of

103

year i s very s h o r t

and i t i s t h e r e f o r e n a t u r a l t h a t th e o p t i c a l and X-ray s y n c h ro tro n r a d i a t i o n w i l l be weak.

The p o s i t i o n can change only when th e r e i s a powerful i n j e c t i o n

of high-energy e l e c t r o n s in t o th e i n t e r s t e l l a r space from some kind of so u rc e s, fo r in s ta n c e , from supernova s h e l l s . As we have i n d i c a t e d , th e o p t i c a l and X-ray sy n c h ro tro n r a d i a t i o n i s com pletely described by the form ulae g iv e n e a r l i e r (see Chapter 5; c o n d itio n (16.80) i s assumed to be s a t i s f i e d ) .

Moreover, t h e r e occurs even a s i m p l i f i c a t i o n which

i s connected w ith the f a c t t h a t a t high fr e q u e n c ie s one can n e g le c t the f a c t th a t the r e f r a c t i v e index

n(u)

d i f f e r s from u n i t y , as w e ll as the r e a b s o r p ­

tio n and the r o t a t i o n of the p o l a r i z a t i o n p la n e in the cosmic plasma.

One

must, however, s o l e l y take i n t o account the a b s o r p tio n of the r a d i a t i o n on i t s path to the E arth or i n the source i t s e l f (by g a s, d u s t ) . For the sake of convenience we s h a l l n e v e r t h e l e s s g iv e a few e x p re ssio n s which

422

THEORETICAL PHYSICS AND ASTROPHYSICS

are u se fu l for c a lc u la tio n s .

In the X-ray r e g i o n and sometimes a l s o i n the

o p t i c a l r e g io n one o f t e n u se s in s te a d of the energy f l u x th e p a r t i c l e nufltiber (photon) f l u x or i n t e n s i t y which we have denoted by Fv and J v .

The t r a n s i ­

t i o n i s c l e a r l y obtained by d iv id in g the e n e r g y - e x p r e s s i o n s by the photon energy hv .

The photon number i n t e n s i t y i s thus a c c o rd in g to (5.48) equal to J(V) -

- 3.26 x 1 0 - » a(Y) LKe H! ( Y + l ) * 0 \{(Y + O ^ 6.26 x l p 1lBJ

photons 2

cm . s . s t e r a d . H z o r , i f we change from the frequency v to the photon energy

®

(16.84)

^ p h = ^v » exPr eased

i n eV, d\> ■ J W ) d€ ph

J ‘V

0.79 a(y) LK.H i (Y ♦ >)

/jJL5|Ji20iy ^ V oh ' ph here

L

«

i s measured m

V— 1

cm, Ke in e r g 1

’

.cm

,

P i ™ ™ --- , cm2 . s.sterad.M eV

(16.,5)

, H in Oe, and €ph in

eV.

S im ila r ly , th e photon f lu x from a d i s c r e t e source ( s e e ( 5 .5 9 ) ) e quals F(v) =

V $(v) It v K e "H1(Y+,) = 3.26 x I 0 - ‘ S a (v) ---- - ------------ x hV n2

^ 6.26 0 1Bj 6 .2 6 xx )lO18^

o r , in terms of th e photon energy

R2

(16.86)

€p^ : hv * 4.14 x i o 1S v eV ,

V Ke HJ(Y+D 0.79 aCy)

1Y photons cm2 .s.H z ’

/2 .5 9 x io'*\^ ^ +*^ photons V c / cm2 .s.e V c ph.

(16.87)

Moreover, i f we can assume the e l e c t r o n spectrum to be the same in the whole of the so u rc e , i t i s convenient t o use the follow ing e x p r e s s io n f o r the r a t i o of the r a d i a t i v e f lu x e s a r d i f f e r e n t fr e q u e n c ie s

v { and v 2 (see (5.59) or

(1 6 .8 6 )) W W

vj

" v7 v v

/-v ^ K

y- D

(16.88)

V v

We have assumed h e re t h a t the r a d i a t i o n a t the frequency of volume Vx in which the magnetic f i e l d s tr e n g th i s the frequency v 2 comes from a volume V2 w ith a f i e l d

a r i s e s i n a source

Hx and the r a d i a t i o n a t H2.

In t h a t c a s e , i f

we a r e d e a lin g w ith r a d i a t i o n by e l e c t r o n s w ith the sane energy E2 ~ E 1 , th e

423

X-RAY ASTRONOMY fre q u e n c ies Vj and

a re r e l a t e d through the Eqn. (16.82) , w hile the r a t i o

of the f lu x e s equals * 22

Vr V2 a t

The

from the whole source w i l l

then be determined byJ the f lu x r a t i o from the volumes V,1 and2

.

Such a

s i t u a t i o n can o c c u r, f o r i n s t a n c e , in the case of nebulae i n the c e n t r a l p a r t of which th e r e i s a r e g io n (say, surrounding a p u ls a r ) w ith a ve ry str o n g mag­ n e tic f i e l d . A characteristic feature of the synchrotron radiation is the fact that in an

ordered f i e l d t h i s r a d i a t i o n i s p o la r iz e d to a la r g e de gre e . For i n s t a n c e , f o r a power-law e l e c t r o n spectrum Ne (E) = Ke E—Y i n a uniform f i e l d the degree of p o l a r i z a t i o n equals (se e ( 5 .4 6 ) ) ^max no =

*min

*max + ^min

1117

(16.90)

Y+ 7

As we have m entioned, t h e r e a re no d e p o l a r i z a t i o n f a c t o r s in the X-ray r e g io n which are caused by th e pre se n c e of a medium (v a r io u s forms o f Faraday r o t a ­ tio n ).

The degree of p o l a r i z a t i o n fo r a given index y r e f l e c t s thus only the

degree o f the ordered f i e l d , and i t re a c h e s i t s maximum v a lu e (16.90) in a uniform f i e l d . Bremsstrahlung i s p o la r iz e d only when the e l e c t r o n d i s t r i b u t i o n f u n c tio n i s a n is o tr o p ic ( o r , s t r i c t l y sp e a k in g , th e d i s t r i b u t i o n f u n c t i o n f o r the r e l a t i v e v e lo c ity of the c o l l i d i n g p a r t i c l e s ) .

For in s t a n c e , p o l a r i z a t i o n of brems­

stra h lu n g a r i s e s when t h e r e i s a d i r e c t e d e l e c t r o n f lu x which i s s c a t t e r e d in a cold plasma.

Under cosmic c o n d itio n s ,

f o r g e t t i n g the Sun and a few non-

s t a t i o n a r y r e g i o n s , t h e r e a r e no s p e c i a l grounds to expect th e e x is te n c e of any strong a n is o tr o p y in th e e l e c t r o n v e l o c i t y d i s t r i b u t i o n (se e Chapter 15; moreover, when th e r e a r e c o l l i s i o n s , th e r e l a x a t i o n of an a n i s o t r o p i c e l e c t r o n v e l o c i t y d i s t r i b u t i o n , t h a t i s , i s o t r o p i z a t i o n , w i l l proceed even f a s t e r than in the c o l l i s i o n l e s s c a s e ) .

When e le c tro m a g n e tic r a d i a t i o n i s s c a t t e r e d by

p a r t i c l e s the deg re e of p o l a r i z a t i o n can i n p r i n c i p l e be l a r g e .

This i s w e ll

THEORETICAL PHYSICS AND ASTROPHY$2£ f

424

known, f o r i n s t a n c e , f o r the c ase of s c a t t e r i n g of l i g h t in g a se s o r i n a plasma (se e Chapter 14).

However, i n the case of s c a t t e r i n g of s o f t , unpola*

r iz e d photons by r e l a t i v i s t i c e l e c t r o n s , forming hard (X-ray and gamma-ray) photons the degree of p o l a r i z a t i o n of the l a t t e r i s of the o r d e r of (mc2 /E ) 2 t h a t i s , ve ry sm a ll. One should

expect

t h a t under

cosmic

c o n d itio n s

(e x c lu d in g

p o s s ib ly dense

magnetospheres of w hite dw a rfs, n e u tro n s t a r s , and so on) f o r a l l o th e r r a d i a ­ t i o n mechanisms, a p a r t from s y n c h r o tr o n and synchro-Compton mechanisms (see end of Chapter 15) no a p p r e c ia b le p o l a r i z a t i o n w i l l occur e i t h e r .

The observa­

t i o n of p o l a r i z a t i o n of cosmic X-ray or gamma-ray e m issio n , as i n the case of the cosmic r a d io - e m is s io n , allow s us t h e r e f o r e to assume t h a t the corresponding r a d i a t i o n i s syn c h ro tro n (or synchro-Compton) e m issio n .

I n p a r t i c u l a r , the

sync h ro tro n (and not brems) n a tu r e of th e X-ray e m issio n of the Crab Nebula was f i n a l l y recognized only as the r e s u l t of the o b s e r v a t i o n of th e p o l a r i z a ­ t i o n of The X-ray em ission. I f the o b s e r v a tio n of the p o l a r i z a t i o n of the X-ray e m issio n i n d i c a t e s i t s synchrotron n a t u r e , the o p p o s ite c o n c lu sio n i s , of c o u r s e , i n v a l i d — i t i s s u f f i c i e n t to say t h a t s y n c h ro tro n r a d i a t i o n i n a random m agnetic f i e l d i s not p o l a r i z e d .^

I t i s thus always d i f f i c u l t to e s t a b l i s h th e n a tu r e of the cosmic

X-ray em ission. the spectrum.

The main c r i t e r i o n ( a p a r t from p o l a r i z a t i o n ) i s the shape of Bremsstrahlung from a hot plasma has an e x p o n e n tia l spectrum

(s e e , f o r i n s t a n c e , (1 6 .2 4 )) and, moreover, i n the case of a h o t plasma one can o b se rv e , in p r i n c i p l e , c h a r a c t e r i s t i c X-ray l i n e s from heavy elem ents ( f i r s t of a l l i r o n ) .

Compton X-ray e m issio n i s produced by r e l a t i v i s t i c e le c ­

tr o n s which normally have a power-law spectrum (16.76) w ith an index y so t h a t f o r th e X-ray em ission — Ct 1 ly(Ey) “ Ey » & ■ ^ ( Y - l ) .

J (E ) * E, ® , 0 ■ |( Y + 0 (se e ( 1 6 .7 8 ) ) and T T Y ^ Moreover, fo r a g iv e n known o b j e c t the r e l a t i v i s ­

t i c e l e c t r o n s may a l s o produce sy n c h ro tro n r a d i a t i o n , the spectrum of which enables us to d e te rm in e th e index y (we remind o u r s e lv e s once a g a in t h a t fo r sy n c h r o tr o n r a d i a t i o n , as f o r Compton r a d i a t i o n , I ( v ) « V*01, a * y ( y - l ) ; see Chapter 5 ) .

We have a lr e a d y mentioned t h a t one can i n t h i s way f o r a known

r a d i a t i o n f i e l d i n which Compton s c a t t e r i n g f i e l d i n the same e m ittin g r e g io n .

t

o ccurs

fin d

a lso

the magnetic

U n f o r tu n a te ly , i n p r a c t i c e a l l t h i s i s not

One m ust, moreover, bear i n mind th a t p o l a r i m e t r i c measurements i n the X-ray r e g i o n , p a r t i c u l a r l y f o r low d e g re e s of p o l a r i z a t i o n and f o r r e l a t i v e l y weak cosmic X-ray e m issio n f l u x e s , a r e v e ry d i f f i c u l t .

X-RAY ASTRONOMY ao sim ple:

4*5

f i r s t of a l l we must n ote t h a t the s p e c tr a are not s t r i c t l y power

laws and the same r e l a t i v i s t i c e l e c t r o n s produce synchrotron a r t Compton radia­ t i o n w ith a g iv e n a on ly i n l im ite d and a p r i o r i unknown frequency i n t e r v a l s

Av ( in the r a d i o - and X-ray ba nds).

One may n e v e r th e le s s expect in the f u tu r e

a g r e a t d e a l of p r o g r e s s along the l i n e of combined (complex) s tu d ie s of the r a d i a t i o n i n a ve ry wide range of ( r a d i o - , o p t i c a l , and X -ray-) fre q u e n c ies in c o n ju n c tio n w ith p o l a r i m e t r i c measurements and a l s o w ith measurements w ith a high a n g u la r r e s o l u t i o n *

This l a s t requirem ent i s connected with the f a c t

th a t f o r a low a n g u la r r e s o l u t i o n (so f a r c h a r a c t e r i s t i c fo r the m a jo r ity of o b s e r v a tio n s in X-ray astronomy) one cannot only not d i s t i n g u i s h the s t r u c t u r e of the d i s c r e t e

X-ray

s o u rc e s

but

the

n a tu re of

the X-ray background a ls o

remains u n c le a r — i t may p a r t i a l l y (and in p r i n c i p l e even almost completely) be determined by a s e t of u n re so lv e d d i s c r e t e so u rc e s. The p re se n t s t a t e of X-ray astronomy which has a lr e a d y produced b r i l l i a n t suc­ cesses (th e o b s e r v a t i o n of pow erful X-ray sources — ’X-ray s t a r s ’ , in c lu d in g p u ls a r s , and so on) i s on the whole in a s ta g e of being e s ta b lis h e d and of f a s t development and i t seems p a r t i c u l a r l y i n a p p r o p r i a t e to d is c u s s h e re the a v a i l ­ able o b s e r v a tio n a l d a ta (G r a tto n , 1970; Schnopper and D e l v a i l l e , 1972; Friedman, 1973; Stecker and Trombka, 1973; Culhane, 1978; Cooke, Lawrence, and P e r o la , 1978).

We conclude t h e r e f o r e the p r e s e n t c h a p te r only f o r a c e r t a i n o r i e n t a ­

tion with a few remarks on t h i s s u b j e c t . The observed d i s c r e t e

X-ray sou rc e s

(X-ray

stars)

belong

Within the Galaxy m ainly two k in d s of so u rc e s a r e o bserved:

to a few ty p e s. supernova s h e l l s

(these a re in f a c t extended so u rc e s) and X-ray s t a r s i n the s t r i c t sense of the word — p r a c t i c a l l y p o in t so u rc e s which a r e b r i g h t in th e X-ray band.

In

the m a jo rity of c a se s and, p o s s i b l y , alm ost always the l a t t e r a re p a r t of binary system;

the X-ray p u l s a r s and 'f l u c t u a r s *

h o l e s ') belong to t h i s c l a s s ; the p u ls a r

(which a r e , p o s s i b l y , ’b lack

PSR 0532 in the Crab Nebula i s a w e ll

known exception — i t i s an X-ray ' p o i n t ' source (neutron s t a r w ith i t s magneto­ sphere) which i s not p a r t of a b in a r y system.

The X-ray em ission from s u p e r ­

nova s h e l l s has e i t h e r a s y n c h ro tro n c h a r a c t e r (Crab Nebula) or i t i s b a s ic a lly bremsstrahlung from a hot plasma ( c h a r a c t e r i s t i c te m pe ra ture corresponding to an average p a r t i c l e energy of

102 to

106

lO ^eV ).

to

1 0 B °K

The app e a r­

ance of powerful X-ray sources in t i g h t b in a r y systems i s v e ry u n d e rs ta n d a b le — in such a s i t u a t i o n th e r e occurs str o n g a c c r e t i o n , t h a t i s , t r a n s f e r of plasma from the l i g h t e r to th e h e a v ie r s t a r .

In t h a t c a s e , e s p e c i a l l y f o r a compact

s t a r (white dwarf, neutron s t a r ) th e plasma f l u x e s when they approach the

THEORETICAL PHYSICS AMD

426

s t e l l a r photosphere r e a c h h ig h v e l o c i t i e s and when th e y a r e d e c e l e r a t e d ( ' f a l l * on the s t a r ) th e plasma i s g r e a t l y h e a te d up (T ~ 10 7 to has a s u f f i c i e n t l y s tr o n g m agnetic f i e l d * one may e x pect

10 9

I f th e

t h a t n o t only th e

brems- but a l s o the magneto-brems s t r a t i lung w i l l t u r n o u t to be s i g n i f i c a n t * The power (X-ray lu m in o s ity ) of g a l a c t i c X -ray so u rc e e rg /s^

r e a c h e s 10 37 to

10®

which exceeds th e s o l a r lu m in o s ity

L = 3.86 x 10 33 e r g / s by 4 to 5 G For Lx ~ 10 38 e r g / s and an i s o t r o p i c e m issio n the X—ray

o r d e r s of m agnitude.

f lu x a t the E a rth e q u a ls $ where R(pc)

V 4 tt R2

10a

erg

[R (p c ) ] 2

[R(pc)

]2

(16.91)

cm . s

i s th e d i s t a n c e t o th e so u rc e in p a r s e c .

According to t h i s e s tim a te we ge t fo r th e Crab Nebula (R « 2000 pc) whole of the X-ray band 2 to lOkeV i s

^x w ^ x

F^ « 2 photons/cm 2 . s ) .

body of tem p e ra tu re

T (in

°K)

7T2 k 2 B

*o = OT , 0 =

distance

i n the

( th e f l u x of photons of energy

For a com parison we n o te t h a t a black

e m its from u n i t s u r f a c e a f l u x

5.67 x i o - 5 -------^ ------- , $(R)=$„ — , •> ,O-.x b 0 u 2 cm . s . ( K)

60 n 3 c 2 where $(R)

c rg /c m 2 . s

i s the f l u x from a b la c k body sp h e re of r a d i u s

r

(16.92)

observed a t a

R.

The f lu x of the s o l a r r a d i a t i o n i s a t the E a r th equal t o $

®

= ---- — = 1.4 x i o 6 - ^ f S 4 7VR2 cm2. s

(R « 1.5 * 10 13 cm , L = 4 t t t 2 $ , T « 5700 °K) , b u t th e r a d i a t i o n i s concen© ® © © 17 t r a t e d in the o p t i c a l p a r t of the spectrum . The X-ray f l u x from the q u ie t Sun equals

^ ~

i t re a c h v a lu e s

10"

$

1* to © ,A

10~5

erg/cm 2 . s

~ 1 erg /cm2 , s .

and only d u r in g p o w e rfu l f l a r e s does Hence i t f o llo w s t h a t th e X-ray em ission

frcmi the c l o s e s t s t a r s (R ~ 4 x i o ie cm) , i f th e y emit l i k e t h e Sun, w i l l be very weak:

®

X

10 "

$‘e ,x < 10

11

e rg /c m 2 . s .

T his was j u s t th e r e a s o n why

t h e o b s e r v a tio n in 1962 of a b r i g h t X-ray ’ s t a r ’ in S c o rp io (Sco X -l) f o r which

$x ~ 10“ 6 erg/cm2. s

was u n f o r e s e e n .

As f o r

Sco X-l

th e

d istan ce

R ^ 500 pc , the t o t a l X-ray lu m in o s ity of t h i s source d o e s , a p p a r e n t l y , not exceed

th at

of

the

Crab Nebula and

is,

po ssib ly ,

weaker by an o r d e r

U ndoubtedly, our Galaxy i s in t h a t r e s p e c t not a t a l l e x c e p t i o n a l . S im ila r so u rc e s have a lr e a d y been observed in the M a g e lla n ic Clouds and c l e a r l y e x i s t a ls o in o th e r g a l a x i e s .

of

427

X-RAY ASTRONOMY magnitude.

Of c o u r s e , a la r g e X-ray lum inosity i s connected w ith a r e l a t i v e l y

la rg e e l e c tr o n energy and, to be p r e c i s e , f o r thermal sources w ith a high e le c tr o n tem perature. face tem perature

For i n s t a n c e , a s t a r of the s iz e of the Sun and a s u r ­

T~

6

* 106 °K would have the enormous lum inosity

~

3 x l 0 **5 e r g / s , j u s t in the X-ray r e g io n (the maximum of the i n t e n s i t y i n the black-body spectrum occurs a t a wavelength ^ = hc/4.965 kgT » 3 * 107 /T (°K ) A). An ord in a ry s t a r cannot have such a high tem perature (we a r e ta l k i n g h ere about the photosophere which em its approximately as a b lack body) j u s t because of the la r g e , p r a c t i c a l l y u n s u p p lia b le energy lo s s e s through r a d i a t i o n . neutron s t a r of r a d i u s

r ~

* 10 5 cm ~ 10“ 5 r

7

and

T~

For a

x 10 6 °K we have

6

©

alre a d y

3 x 1G35 e r g s /s

which i s allow able for a c e r t a i n time.

As re g a rd s

thermal sources which are not so compact, they are ’thin* sources (or, as one says, o p t i c a l l y t h i n ;

we avoid usin g th a t term as in the X-ray band i t may

lead to c o n f u s io n s ) . e a s ily th a t for

I f we, t h e r e f o r e , use E q n s .( l6 .3 6 ) and (16.37) we see

T~

6

x | 0 6 °K a lum inosity

reached by a plasma cloud of volume small mass

M ~ I 0 22g

e x trag alactic

X-ray

s o la r mass

sources

(1 6 .9 1 )).

i s , fo r in s ta n c e , the average

N ~ V(N2) ~ 10 16 cm” 3 . (in p a r t i c u l a r ,

rad io -

The X-ray lum inosity of a normal

galaxy (such as our Galaxy) does not exceed 10 39 Virgo A = NGC 4486 = M87)

erg /s

Mq = 2x io33g)

are g a la x ie s

g a l a x i e s ) , q u a s a r s , and c l u s t e r s of g a l a x i e s . such a galaxy a t a d i s ta n c e of

1 0 3B

V ~ lO 30 cm3 and with a comparatively

~ I0"11 M@ (the

e le c tr o n d e n s ity in the cloud is in th a t case D isc re te

Lx ~

to

10**° e r g / s .

The flu x of

R ~ 107 pc (the d is ta n c e from the ra d io -g a la x y

would be

X ~ 10“ 12 to

10 ” 1 3 erg/cm2.s

(see

However, the r a d io - g a la x y M87 emits an a p p re c ia b ly more powerful

X-ray r a d i a t i o n so t h a t f o r i t

Lx ~ 10**3 to 101*1* e r g / s .

Both in t h i s case

and in the case of o th e r powerful X-ray sources — g a l a x i e s , q u a sa rs, and c l u s ­ t e r s of g a la x ie s — the

em ission does

clearly

not reduce

to a system of

r a d i a t i o n from ’ X-ray s ta r s * and the X-ray source is provided by the r e l a t i v i s ­ t i c e le c tr o n s (sy n c h ro tro n and Compton mechanisms) or the hot plasma (brems mechanism) which f i l l the g a la x y , the c l u s t e r , or the ’c o ro n a 1 of the quasar. In p r i n c i p l e the s i t u a t i o n is p a r t i c u l a r l y ’simple* in the case of the hot plasma in an extended source such as a c l u s t e r of g a l a x i e s . a temperature a lum inosity

T~

6

* 10 6 °K erg /s

M ~ 2 x i 0 ” 2** NV ~ 10 13 Mq ; g a la x ie s .

a plasma of volume for

For in s ta n c e , a t

V ~ 3 * 1073 cm3

w ill

have

N ~ 1 0 ” 3 an -3 which corresponds to a mass

such a mass i s f u l l y a c c e p tab le fo r a c l u s t e r of

However, even t h i s example shows th a t i t i s not easy to e x p la in the

enormous lu m in o s itie s of powerful e x t r a - g a l a c t i c sources or i t i s more c o r r e c t

428

THEORETICAL PHYSICS AMD AOTTMCTTCVy

t o Bay t h a t such an e x p l a n a t i o n i s accompanied by f a r - r e a c h i n g assumptions which m is t be checked (and i n p r i n c i p l e can f u l l y be checked) i n a number of ways. From t h i s i t i s a l s o c l e a r how e x c e p t i o n a l l y u s e f u l X -ray astronomy can be, for

in stan ce,

for

a study

of

the

hot

plasma

in

the U niverse

(see a ls o

Z e l ’d o v ic h , 1975). Apart from the d i s c r e t e s o u r c e s , one o b se rv e s an X-ray background, t h a t i s , r a d i a t i o n coming from a l l d i r e c t i o n s w ith o u t on the c e l e s t i a l sphere having any pronounced ’g ra n u la r* s t r u c t u r e .

I t i s not excluded t h a t such a back­

ground i s n o n e th e le s s connected ( p a r t i a l l y or even co m p le te ly ) w ith a system of d i s c r e t e sources which our in s tr u m e n ts do not r e s o l v e .

I t i s a t the same time

f u l l y p o s s i b l e and even p ro b a b le t h a t t h e r e e x i s t s some ( t r u e ) X-ray background which i s produced in i n t e r s t e l l a r and p a r t i c u l a r l y i n i n t e r g a l a c t i c space. The background a r i s e s as the r e s u l t of the b re m sstra h lu n g of th e hot i n t e r g a l a c t i c gas a n d /o r of the Compton s c a t t e r i n g of r e l a t i v i s t i c e l e c t r o n s by the r e l i c t thermal r a d i a t i o n ( in i n t e r g a l a c t i c space) and in th e G alaxy, moreover, by o th e r kinds of r a d i a t i o n (we have in mind, in p a r t i c u l a r , Compton s c a t t e r i n g by i n f r a - r e d and o p t i c a l p h o to n s ) . The spectrum of the X-ray background ( in the energy range

> 1 keV, A 0 Ey) , averaged over the g a l a c t i c cosmic r a y s , as f u n c t i o n of the energy of the gamna-rays produced.

E

MeV

THEORETICAL PHYSICS AND ASTROPHYSICS

434

taken from S te c k e r ’ s book (1971) f o r th e spectrum of the cosmic r a y 6 a t the E arth ( i n t e n s i t y

J c .r .,o ( E ) = J 0 (E )).

Belov we s h a l l

use th e value

CJJ0 (Ey > I Ob MeV ) * 1CT26 s™1 . s t e r a d " 1 , and, hence, 10~ 26 NVw /w > lOOMeV) - --------------------= Y' Y R2 5 x 10~3 M w £ /w # photons

F (E

cm2 . s

R2 where wc

(17.3)

*

i s the cosmic ra y energy d e n s i t y in the so u rc e , and where we have

assumed th a t the shape of t h e i r spectrum i s the same as a t the E a rth (so th a t wc . r > 0 - J c . r . / J c . r .,0 » where wc . r . , 0 = wo ~ ra y energy d e n s i t y a t the E a rth ; see ( 1 5 . 9 ) ) .

e r 8 /cm 3 i s the cosmic W ithin the l i m i t s of the

1 0 " 12

approxim ations made f o r sources such as the Galaxy where the n e u t r a l hydrogen atoms dominate, M w 1.2 MflI , where

i s the mass of n e u t r a l hydrogen; the

accuracy of the c a l c u l a t i o n s can be in c r e a s e d as one can i n n e d i a t e l y fin d the ra tio

M^VR2

from the hydrogen li n e (X = 21 cm) d a t a .

We s h a l l i n what

follow s not aim a t such a r e fin e m e n t, which i s as y e t not p r e s s i n g . We n o te , f i n a l l y , t h a t the gamma-ray spectrum of n u c le a r provenance about which we a r e t a lk in g i s f o r obvious reasons c o n c e n tra te d in the energy range Ey ^ 50 to

lOOMeV (the cosmological red s h i f t i s h e r e , of c o u r s e , not taken

in to account and we a re t h e r e f o r e c o n s id e r in g sources which a r e not too fa r away).

What we have said i s c l e a r from Fig. 17.1 and more p r e c i s e l y from the

follow ing example ( F i c h t e l , Hartman, K niffen, and Sommer, 1972); f o r gammarays from

tt°

decay we have th e r a t i o F (E > 50 MeV) - FV(E > 100 MeV) £ = - J — *---------; F (E > I 00 MeV)

Moreover, fo r r e l a t i v i s t i c e le c tr o n s w ith a spectrum case of brems gamma-rays the ganma-ray flu x

0.12. —2 6

J e (E) = Ke E

in the

£ = 2 . 0 3 and fo r gamma ra y s of sy n c h ro tro n provenance

or emitted in the in v e rse Compton e f f e c t of

=

thus

e a s i l y t h e i r ’n u c le a r ’ n a tu r e .

allow us

£ = 0.7 4 .

in p r i n c i p l e

The s p e c t r a l measurements to

estab lish

rela tiv e ly

I f t h i s i s done, we g e t a t once from measuring

the f l u x Fy(Ey > 100 MeV) or the c orresponding i n t e n s i t y

th e r a t i o

in the source, t h a t i s , the b a sic parameter which i s now la c k in g .

wc ^ /w0

I n th a t

case we make, of co u rse , an assumption about the s i m i l a r i t y of the cosmic ray s p e c tr a in the source and a t the E a rth .

There a r e , however, good grounds for

such an assumption and, moreover, under r e a l i s t i c c o n d itio n s i t can, apparently* lead only to th e appearance o f a numerical c o e f f i c i e n t of the o rd e r of u n ity .

gamma ASTRONOMY

43S

At any r a t e , even such a d e te rm in a tio n of the energy d e n s ity or the t p t a l d e n s ity Wc#r# = wc r

V of the cosmic rays in the sources would be an impor­

t a n t ste p forward. We can make what we have said more concrete using the Magellanic Clouds and the c e n t r a l r e g io n of the Galaxy as examples. I t i s of i n t e r e s t by i t s e l f of course, to c onsider the Magellanic Clouds. However, t h i s example i s even more im portant in connection w ith a tte m p ts to answer the q u e s tio n :

how to e lu c id a te most convincingly the f a t e of the m eta-

g a l a c t i c model of the o r i g i n of the cosmic r a y s . sa id in Chapter 15 t h a t fo r

th is

it

is

I t i s c le a r from what we

su fficien t

d e n s ity w^g in the re gion surrounding the Galaxy.

to determine

the energy

I f i t tu rn s out th a t

w.. Mg

wc *00 MeV) » 1 * 10

photons/cm2. s , 7

photons/cm2. s .

(17.4) j

As we have a lre a d y mentioned these f lu x e s can a lso be c a lc u la te d more e x a c tly . Here something e l s e i s im portant — the f lu x e s (17.4) are d i r e c t l y obtained fo r any known m e ta g a la c tic models as in those models the r o l e of the i n t r i n s i c cosmic ray sources in the Magellanic Clouds, as in the Galaxy, i s small so t h a t w.. » w _ « w____ » wci,_. On the other hand, th e r e i s no ground fo r Mg C. r . , G LMC bMC expecting the o b s e rv a tio n of such an e q u a lity in g a l a c t i c models. Even for equal cosmic ray source a c t i v i t y i t i s very probable th a t

w >w_,_ C. r . G LMC SMC because of the sm a lle r s i z e of the Clouds and the corresponding f a s t e r depar­ tu r e from them of the cosmic r a y s . Thus, for a convincing r e f u t a t i o n of the m e ta g a la c tic models of the o r i g i n of cosmic r a y s ^

i t would be s u f f i c i e n t , fo r i n s t a n c e , to e s t a b l i s h th a t from the t see footnote on next page

436

THEORETICAL PHYSICS AND ASTROPHYSICS

two clouds taken to g e th e r

F r,(Ev > lOOMeV) < k 3 * 1 0 ~ 7 p h o to n s /c n 2 • ■ or th a t T >ML T cur 2 F , u«' Y, aHU Y,LWLWhen the c a l c u l a t i o n s have been made more a c c u r a te one can in p r i n c i p l e r e p l a c e the - s i g n by a < - s i g n . I t i s in t h i s case im portant t h a t any c o n t r i b u t i o n to the gamma-ray f lu x connected w ith r e l a t i v i s t i c e l e c ­ tr o n s le a d s only to an in c r e a s e in the f lu x e s

Fy and, hence, in no way a f f e c t s

the i n t e r p r e t a t i o n g iv e n , say, of the r e s u l t

Fy MC(Ey > lOOMeV)

photons/cm 2 • s .

3*10

7

Such a r e s u l t we c a l l a n e g a tiv e one and we note t h a t th e r e is

a c e r t a i n asymmetry, as o f te n i s the c a s e , when i n t e r p r e t i n g p o s i t i v e and nega­ t i v e r e s u l t s of an e x p e rim e n t•

For i n s t a n c e , i f the measurement of the gamna-

ra y

Clouds would

f l u x from

the M agellanic

a p p r e c ia b le f l u x

F

in d ic a te

^ 3 * 1 0 ” 7 photons/cm2. s

(such

the

p resence

a re su lt

of

an

we would

c u sto m a rily c a l l p o s i t i v e ) t h i s would not y e t prove the v a l i d i t y of m eta g a la ct i c models, as in p r i n c i p l e such a f lu x could a ls o be g e n e ra te d by the cosmic ra y s (and a ls o by the r e l a t i v i s t i c e l e c t r o n s ) which a re a c c e l e r a t e d in the Clouds them selves.

U n f o r tu n a te ly , the measurement of the gamm-ray f l u x from

the M agellanic Clouds i s a d i f f i c u l t problem and i t s s o l u t i o n l i e s in the fu tu re.

I t i s a p p r e c ia b ly s im p le r , but a l s o very i n t e r e s t i n g , to study the

gamoa—emi s s i o n from the c e n t r a l r e g io n of the Galaxy.

Such r a d i a t i o n (with

Ey > lOOMeV) has a lre a d y been observed and the i n t e n s i t y of the corresponding ( 'e q u i v a l e n t ' ) l i n e a r source i s 1 to 2 . 5 * 1 0 ** photons/cm 2 . s . r a d i a n (see F i c h t e l , Hartman, K n iffe n , and Somner, 1972; K ra u sh a a r, C la r k , Garmire, Borken, Higbie, Leong, and Thorson, 1972; Ginzburg, 1973a). by the a ngular r e s o l u t i o n (about

I f we m u l t i p l y t h i s v a lu e

t t / 6 ) we ge t fo r th e f l u x from the c e n t r a l

g a l a c t i c source Fy(E^ > lOOMeV) = 3 to

10* 10” 5 photons/cm 2. s .

(17.5)

Doubts have been expressed about the r e a l i t y of the o b ta in e d r e s u l t and the v a l i d i t y of i t s i n t e r p r e t a t i o n as evidence fo r the p re se n c e o f an extended ganma-ray

source

in

the d i r e c t i o n

of

i s now c l e a r (Stecker and Trombka, 1973;

the

cen tre

of

the Galaxy,

Puget and S te c k e r , 1974)

but

it

t h a t the

I t i s im portant th a t ve a r e t a l k i n g h e re about a l l known m e ta g a l a c t i c models, whereas a measurement of th e i s o t r o p i c background of gammara y s g e n e ra te d in the i n t e r g a l a c t i c space ( se e Ginzburg and S y r o v a t s k i i , 1964a, 1965; Ginzburg, 1969; S te c k e r , 1971; G a l 'p e r , K irillov-U gryum ov, Luchkov, and P r i l u t s k i i , 1972; S te c k e r and Trombka, 1973; G a l 'p e r , K irillov-U gryum ov, and Luchkov, 1974) can se rv e as a r e f u t a t i o n only of those models in which the cosmic r a y s f i l l a ve ry la r g e volume, in p a r ­ t i c u l a r , the whole of i n t e r g a l a c t i c space (moreover, th e d e n s i t y of the m e ta g a la c t ic gas has not y e t been e s t a b l i s h e d ) .

437

GAMMA ASTRONOMY

corresponding gamna-emission e x i s t s although i t s source i s not c o n c e n tra te d itt the g a l a c t i c c e n t r e i t s e l f ( t h i s was assumed e a r l i e r ; 1972; Kraushaar e t a l . , 1972; Ginzburg, 1973a). the va lu e (17.5) i s r e a l .

see F ic h te l e t a l . ,

We s h a l l assume below t h a t

In t h a t c a se , both on the b a s is of s p e c t r a l measure­

ments and a l s o from a s e r i e s of i n d i r e c t c o n s id e r a tio n s i t i s probable t h a t we a re d e a lin g w ith gamma-rays generated by cosmic rays ( t h a t i s , mainly w ith ir° decay p r o d u c ts ) .

Accepting t h i s i n t e r p r e t a t i o n we draw a few c o n c lu s io n s .^

S u b s t i t u t i n g the v a lu e (17.5) i n t o Eqn.(17.3) we reach the co n c lu sio n th a t in the c e n t r a l g a l a c t i c source (we in d ic a te d the corresponding q u a n t i t i e s by an index GC) cosmic r a y s a r e c o n c e n tra te d w ith a t o t a l energy W =w v GC GC GC

3 to

10x10 6 6

o

N GC

~ 3*JO

in5 5*f

N,'GC

erg

(17.6)

as

w0 = w ft ~ 10" 12 erg/cm 3, w hile the d i s t a n c e from the Sun to the cenu c.r ., u t r a l source R * 1 0 k p c . I f we assume t h a t the c e n t r a l source i s not small ( c h a ra c te ris tic size

ngc cannot -------- ’be put volume ~ 10 63 GC f o r N ~ 10 cm* 3

bb

le s s than or of the order of

l' a r g e r than approxim ately u n ity ( fo r cm3

Lnr, ~ bb

1 0 21

cm, the

m^ V v ~ 10 6 N M : ~ 2 x 10"* HN GC GC GC e ’ M_n ~ 10 7 M which i s , p ro b a b ly , the

and the gas mass

we have a lr e a d y

300 pc) the gas d e n s ity

M

bb

bb

9

l i m i t f o r a r e g io n w ith the volume chosen). When

NGC ~ 1 cm- 3 we ge t from (17.6) the e s tim a te

WGC ~ 3 to

1 0 x l 0 51* erg

which i s only one o r d e r of magnitude sm a lle r th a n the t o t a l energy of a l l cosmic r a y s in the Galaxy (se e ( 1 5 .1 6 ) ) . o r d e r of

1 0 55

On th e o th e r hand, a v a lu e of the

erg i s j u s t o b ta in e d from an a n a l y s i s of astro n o m ic a l d a ta

which g iv e evidence

fo r

an o u tb u r s t

of

the g a l a c t i c

nucleus

about

107

y e a rs ago. I f the s i z e of the c e n t r a l gamma-ray source turned out to be le s s than 200 to 300 pc,

a v a lu e

N-_ ^ 10 cm” 3 bb

would not be excluded.

The energy W

bb

the n,

of c o u r s e , becomes s m a lle r (se e ( 1 7 . 6 ) ) , b u t the i n t e n s i t y of the cosmic r a y s Jc r

GC H J GC

*s » *n Ke n e r a l > not changed.

For i n s t a n c e , f o r

NGC~ I O c m “ 3

As the r e s u l t of more r e c e n t measurements (P lo v d iv , 1977; Ginzburg, 1978) i t h a s , a p p a r e n t l y , become c l e a r t h a t an a p p r e c ia b le p a r t of the ganma-emission from the r e g io n of the g a l a c t i c d is c i s connec­ ted not only w ith tt° de c a y, b u t a l s o w ith d i s c r e t e so u rc e s and, p e rh a p s, w ith b r e m sstra h lu n g from r e l a t i v e l y s o f t e l e c t r o n s (Ee £ lOOMeV). We must t h e r e f o r e t r e a t the a c t u a l e s tim a te s and v a lu e s g iven i n th e p r e s e n t c h a p te r f o r th e g a l a c t i c ganmaem ission as having m erely an i l l u s t r a t i v e v a lu e .

THEORETICAL PHYSICS AMD ASTROPHYSICS

438

we g e t from (1 7 .6 ) WGC~

3

t0

• J GC/ J c . r . fo “ WGC/w o ~

1 0 x , °53« 8

300

to

300°*

I t i s v e r y d i f f i c u l t to c o n t a i n cosmic r a y s f o r 10 7 y e a r s i n a s m a l l e r regiom ( f o r in s ta n c e * f o r D ~ 1 0 27 cm2/ s

TGC- 3 x i o 1- s

th e d i f f u s i o n p a t h i s V 2 DTqC ~

which c o r r e s p o n d s to a v e r y s m a ll v a l u e o f

1 0 21

L ~ 0 .0 3 p c

cm f o r

f o r the

e f f e c t i v e mean f r e e p a th

fl, ~ D/v v v ~ 10 1 0 cm/s) .

~ 3 x 10 s 3 vL erg t h e r e f o r e seems to us to be th e s m a l l e s t p o s s i b l e one and one should r a t h e r

have WGC ~

In t h a t c a s e t h e p r e s e n c e o f a c e n t r a l cosmic ra y

so u rc e

3

x 105** e r g .

would have

an

im p o r ta n t

v a lu e

for

The v a l u e

th e whole

e nergy b a la n c e

of

the

~ W--/T n > 10 1*0 bt (it GC which i s o f th e same o r d e r of m a g n itu d e as th e t o t a l power of i n j e c t i o n

cosmic r a y s i n the Galaxy ( a v e ra g e power o f i n j e c t i o n e rg /s

i n g a l a c t i c m odels;

see ( 1 5 . 2 0 ) ) ,

I f the v a lu e (17.5) i s v a l i d th e t o t a l f l u x from the c e n t r a l so u r c e e q u a ls (E > lOOMeV) ■ 4ttR 2 Fy ~ 3 to 10 x ]Qkl p h o t o n s / s , c o r r e s p o n d in g to a lumiY Y n o s i t y Ly ~ Ey3fy ~ 1O30 e r g / s . M oreover, th e whole G ala x y , i f i t were f i l l e d u n if o r m ly w ith cosmic r a y s would em it p h o to n s /s

3

y ^G(Ey > 100 MeV) * 4 x 1 0

a s th e t o t a l mass of gas in the Galaxy i s

3

4ttMq ~ 10

Mq ~ 3 x 10 1*2 g

T h is i s n o t the end of th e s t o r y , as one can (and m ust) compare th e gamnaastronomy d a t a w ith r a d io - a s tr o n o m y r e s u l t s . extended so u r c e

of

U n f o r t u n a t e l y , i n the c a se o f an

n o n - th e r m a l r a d i o - e m i s s i o n i n th e c e n t r a l

reg io n

of

the

Galaxy we d i s p o s e o n ly of o b s o l e t e d a t a ( f o r r e f e r e n c e s see G in z b u rg , 1973a). A ccording to th o s e d a t a th e f l u x from an e x te n d e d s o u r c e o f s i z e ( p r o b a b le volume V ~ 10 6 3 cm3) i s V* 85.5MHz

= 3 x 10” 2° e r g /c m 2 .s .H z

w ith a s p e c t r a l in d e x a = 0 .7 (y = 2a + 1 = 2 . 4 ) ,

l ° x 3°

a t a frequency

S u b s t i t u t i n g th o se

p a r a n e t e r s i n t o E q n s .( 5 .7 0 ) and ( 5 .7 1 ) we a r r i v e f o r th e c e n t r a l r a d i o - s o u r c e a t th e v a l u e s ( t h e index

r

i n d i c a t e s t h a t we a r e h e r e u s in g r a d i o - d a t a ) 3 ** I WGC,r “ lce We t r ~ 9 * , o 5 1 V Kl e r g » H ~ 1 ° " 5 ( KH ? 0 e ° 7 ' 7)

where we have assumed t h a t th e r a d i o - e m i s s i o n has a s p e c t r a l in d e x th e r a n g e

107

to

109

Hz;

a * 0.7 in

th e e s t i m a t e s a r e n o t v e r y s e n s i t i v e t o th e c h o ic e

of t h i s r a n g e o r , w i t h i n c e r t a i n l i m i t s , t o t h a t of th e o t h e r p a r a m e t e r s . we now p u t , as a t th e E a r t h , tce ~ 100 and

GC.r

3 x 10 52 e rg , W

~ 1

we have ( f o r

3 x l o 50 e rg ,

If

V ~ 10 63 cm3)

439

gamma astronomy

w

GC,r

GC,r . -3xi0 V

e,r

e.r V

11 erg/cm3 *- 30 w

c.r

- 3 x 10” 13 erg/cm3 *- 30 w e G

Even with all inaccuracies of the initial data (and therefore also of the estimates to a certain extent) the value of smaller than the value

W

GC

~ 3 x io53

to

W given here is appreciably GC }r 10s5 erg determined above from the

ganma-dat a . One might find a rather natural explanation for such a discrepancy (Ginzburg, 1973a) but we have very strong arguments for not discussing this problem. The fact is that the estimates given above to some extent are based upon the assumption that the gamma-radiation sources with an enhanced intensity which are observed in the direction of the galactic centre are concentrated in some region close to the centre itself.

Just this interpretation (Fichtel et al.,

1972; Kraushaar et al.f 1972; Ginzburg, 1973a) was thought to be the most probable one until the appearance of measurements which became known in 1973/4 (Stecker and Trombka,

1973; Puget and Stecker, 1974).

According to these

later data the observed radiation does not come from the central galactic region itself, but from a very extended section situated between the Sun and the centre of the Galaxy.

One possible interpretation of the results consists

in assuming that the cosmic ray intensity (perhaps, only as far as particles with energies

E ~ 1 to

lOGeV

tance between 4 and 5 kpc

are concerned) in a toroidal region at a dis­

from the galactic centre is about an order of

magnitude higher than that near the Earth.

The increased cosmic ray intensity

might in turn be explained to be due to their additional acceleration and trapping in that region (Puget and Stecker,

1974).

It is, however, possible

that the cosmic ray intensity in that region is the same as at the Earth but that the gas density there is appreciably higher due to the presence of mole­ cular hydrogen (Fichtel et al., 1975).

However, as we are dealing with pre­

liminary data and hypotheses, it is inappropriate to go into details, the more so as we here have another aim.

Namely, we wanted not only to state, but also

to illustrate by an actual example how one can use data on cosmic gamma-rays in the energy range

Ey > 50MeV,

which are formed in the decay of

7T°-mesonB

and other particles which are produced in the gas by the cosmic rays (see also Ginzburg and Ptuskin, 1976a,b; Plovdiv,

1977; Ginzburg, 1978).

Above we concentrated our attention on gamma-rays of nuclear origin, that is, those which are formed in the gas during nuclear collisions of cosmic rays.

440

THEORETICAL PHYSICS AND ASTROPHYSICS

Moreover, in that case we were talking only about energies It is at the same time well known, of course, other possibilities* interest;

Ey ^ 50

to 100 MeV.

that there are a whole number of

For instance, the energy range

Ey ■ I

to

50 MeV

is of

this is the range of energies of gamna-rays emitted in the ir° decay

in regions with a large red-shift parameter

z

(to be precise, we are talking

about gamma-radiation in the Lemaitre cosmological model, about the observa­ tion of matter-antimatter annihilation with when

Ey | 9 and so on).

the competing processes, other than

important than for

Ey>50MeV.

Of course,

decay are much more

There are undoubtedly certain possibilities

to perform measurements in the energy range et al., 1972, 1974;

tt°

Ey ~ J to

Stecker and Trombka, J973).

observations of gamna-rays with energies

50MeV (see

Gal'per

One can say the same about

Ey > JO11 eV

which are performed by

the method of Cherenkov fluorescence in the atmosphere (Gal’per et al., 1972, 1974).

Particular mention deserve gamma-rays emitted by excited nuclei which

appear in nuclear reactions and the garana-radiation from positron-electron annihilation (if we neglect the red shift the energy of those gamma-rays must be concentrated near an energy of

0.51 MeV).

We have already mentioned these

gamma-radiation processes at the start of Chapter 16. (J965;

Ginzburg,

Ginzburg and Syrovatskii

1969b), for instance, have given a scheme for calculating the

corresponding intensities (see also the books by Stecker (1971; Trombka, ]973;

and the literature quoted by Ginzburg,

al., 1972, 1974).

It is very well possible

that

Stecker and

J973a and by Gal'per et

the various possibilities

for gamna-astroncany enumerated here, or most of them, will be developed in the future.

Nevertheless at the present time it seems that none of them has such

a potentially wide and universal character as the gamna-astronomical study of the proton-nuclear cosmic ray component. so much attention to it;

This was the reason why we paid here

ve shall not discuss in detail any other, particular

possibilities (see also Ginzburg and Ptuskin,

1976a,b; Plovdiv,

1977, Ginzburg,

1978). Concluding this chapter we shall touch upon the problem of the absorption of gamma-rays (and also partially of X-rays) as this problem is one of interest because of the principles involved.

For a source with a distance from us characterized by the parameter z the energy observed on the Earth is Ey « E y ^ / ( I + z), where Ey

q

is the photon energy in the source.

441

GAMMA ASTRONOMY

y for gamma- and X-rays it is usually

To evaluate the absorption coefficient important

to

take

into

account

how

the

primary

flux

of

the

radiation

attenuatedv that is 9 to pay attention to absorption and to scattering* # -|* definitionv the quantity y occurs in the equation

- where

a

MJ ,

is

By

y = CTN ,

(17.9)

is the total cross-section for absorption and scattering and

N

the

particle (atcmi, electron) density responsible for the absorption and scatter­ ing.

In the geometric optics approximation (it is always applicable in the

cases of interest to us), and this is just the case to which E q n . (17.9) refers, —T

we have for the intensity (or in a uniform m edium

and for the optical length

T * J o

ydR

T = yL).

We assumed in (17.9), moreover, line of sight.

«L

J = J 0e

that there is no emission of photons along the

If, on the other hand, emission takes place, the transfer equa­

tion becomes

* *S y ) where

q(v)

considered

is the emittance v = Ey/h

= q(v) — y(v) J(V)

,

(17.10)

(in the photon number scale) at the frequency

(see (16.7)).

There are, in principle, very many processes which contribute to

a , namely:

1.

Photoeffect

2.

Compton scattering.

3.

Transitions in the continuous spectrum (free-free absorption).

4.

Transitions between atomic levels (excitation of atoms).

5.

Formation of

e+ , e"

6.

Formation of photons (the

e+ , e” pairs on thermal and, in general ’soft1 y + Y* e+ + e~ process, where y' is a soft photon).

7.

Absorption by nuclei (nuclear photoeffect and excitation of nuclei. ± n Production of tt“ and ir mesons at protons and nuclei. Production of other particles.

8.

(ionization of atoms).

pairs in the medium.

We have already considered some of the processes listed here.

For instance,

the absorption coefficient for free-free transitions (process 3) in a hydrogen plasma is given by E q n . (16.33).

+

Absorption due to bound-bound transitions

.

t

#

We do not consider here the possibility of induced absorption or scattering; as far as one can judge this assumption is justified in the X-ray and gamma bands, but on the whole this problem requires a more detailed analysis.

442 (process elements

4)

in the X-ray region can play a role only for comparatively

for

the

simple

reason

that

the

ionization

potential

elements, even for the K-shell, is too small (for instance, for 2-13,

it equals approximately

for A1

baaey light

atoms with

1500 V, which for the edge of the K-band corres­

ponds to absorption at a wavelength of

6 1).

The cross-section for Compton

scattering (process 2) was given in Chapter 16. For low energies the main part in the absorption is played by the photoeffect, while

with

increasing

energy

Compton

consideration of the photoeffect

scattering

starts

to

dominate.

A

(process 1) requires, in general, taking into

account the actual chemical composition of the medium and its degree of ioniza­ tion.

We shall not dwell upon this process which determines the absorption of

not too hard X-rays (see Bell and Kingston, 1967;

Vainshtein, Kurt, and Sheffer,

1968; Brown and Gould, 1970; Fireman, 1974), but ve emphasize that a detailed study of the absorption of soft X-rays in the interstellar and intergalactic medium is of exceptional interest.

Possibly,

that is the way to obtain useful

information about the density, composition, and degree of ionization of the gas in regions about which we know very little at the moment (particularly in the intergalactic medium).

However, this is a special problem and it is not

possible to elucidate it here in the necessary manner. When the energy increases absorption due to the photoeffect decreases and in air the contributions from Compton scattering and from the photoeffect are equal at an energy

w 25keV.

At

Ey

= 50 keV photo-absorption is already

about five times smaller than Compton absorption. and up to energies

2 m c 2 « I MeV

when

e+ , e ”

For X-rays with

pairs start to be formed, one

therefore needs take into account only Compton scattering. Ey - h v «

Ey > 50 keV

For

m e 2 « 5 x 105 eV

the total cross-section for scattering

Oq which occurs in (17.9) is, when ve

neglect other processes, equal to the Thomson cross-section 0^ “ - TT (e2/mc2 )2 * 6.65 x 10~2S c m 2 . When the frequency increases the cross-section diminishes and for ve have already encountered energies

O c ■ 0.43 O p

Therefore with the usually

1astrophysical accuracy1 one can put

E y ^ 1 MeV.

instance, for

When

Ey » m e 2

*= I03 me* * 5 * 10* eV

one

must

we have

more detailed formulae and also tables with

oc

o^ ~ use

(but not alvaysl) for all gamma-ray

Eqn. (16.72)

a Q = 3 * I0-3 ctp in

5 36

hv * m e 2

and,

for

One can find

of Heitler's book

GAMMA

443

astronomy

(1947).

We note also that when Compton scattering is taken into account we

must in (17.9) understand by we put

o =

N

the total electron density in the medium.

O j , we have in the interstellar medium

(N

If

is the total density

of all electrons) Pc In the energy range

= 6 . 6 5 x l O " 2 5 N « 0.4 cm2/g .

Ey
10e eV pair production occurs to a first

approximation under conditions of complete screening.

The corresponding value

of the absorption coefficient in the interstellar medium equals

U

-

Hpair

= 1.2 x 10“ 2 cm2/g = 2 x 10’ 26 N °

a

we have used here the value of the t-unit length equal to 16) and

Na

is the density of atoms.

;

cm"1

(17.12)

66 g/cm2 (see Chapter

In a plasma (a completely ionized gas)

one can neglect screening in the cases of interest to us and 4Z(Z + l)e2 Rc

«

( £ ) * »•

= 3.6 x | 0 “ 27 (*n \ me

»

-

J

- 1.9 ) Na cm"1

1.9') cm2/g ,

= 2.1 x 10“ 3 (