Fortschritte der Physik / Progress of Physics: Band 24, Heft 1 1976 [Reprint 2021 ed.] 9783112559284, 9783112559277

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

24. BAND 1976

A K A D E M I E

- V E R L A G

•

B E R L I N

Inhalt des 24. Bandes Heft 1 WEBER,

HAASE,

H. J., Medium Energy Inelastic Hadron Scattering and Nuclear Structure D., Die Diracsche Störangstheorie in Operatorform

1 37

Heft 2 S. I . , Properties of Equilibrium States on Quasilocal 0+-algebras U. E., Structure Effects of Hadrons in Hadronic Atoms

ANDERSSON,

55

SCHRÖDER,

85

Heft 3 GRUPEN, C.,

Electromagnetic Interactions of High Energy Cosmic Ray Muons

127

Heft 4 Galilei-Invarianz W., Models of Elastic pp and pp Scattering with Spin Included

STEINWEDEL, H . ,

211

MAJEROTTO,

237

Heft 5 P. MINMAERT, and L. M I C H E L , Amplitude Reconstruction for Usual Quasi,Two Body Reactions with Unpolarized or Polarized Target 259

DONCEL, M . G.,

Heft 6 MANOUKIAN,

E. B., Vacuum Energy-Density in Quantum Electrodynamics 325 Urbaryon Model Approach to Unified Description of Large and Small PT spectra 3 4 1

KINOSHITA, K . ,

Heft 7 MATSUMOTO, H L ,

tivity SZASZ, GY.

and H .

UMEZAWA, A

Rigorous Formulation of Boson Method in Superconduc-.

I., Zerfallende Zustände als physikalisch nichtisolierbare Teilsysteme

357 405

Heft 8 H., Bewegung einer Punktladung in elektromagnetischen Feldern unter Berücksichtigung der Strahlungsdämpfung 417

STÖCKEL,

Heft 9 KASCHLUHN, F . ,

Waves

ACTOE, A . ,

and M.

MÜLLER-PREUSSKER,

Regge Models and Dispersion Relations for Partial

Weak Neutrino-Lepton Annihilation into Hadrons

477 507

Heft 10 Hadron Production by a .Thermodynamical Quark Bootstrap Pseudoscalar Mesons as Goldstone Bosons in Chiral 8U(i) x SU(é) Symmetry . KROLL, P., Nucleon-Nucleon Scattering at High Energies

WOLF, R . , BOSE, K . ,

529 .

.

555

565

Heft 11 T., Quantum Theory of Path-Dependent Operator Based on Characteristics of Displacement Operators 619 633 KASCHLUHN, P., S-Matrix and Yang-Feldman Relations JERSÂK, J . , Duality Properties of Virtual Compton Amplitudes 657 OKOBAYASHI,

Heft 12 SPATSCHEK, K .

H., Parametrische Instabilitäten in Plasmen

687

H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DER DEUTSCHEN DEMOKRATISCHEN

REPUBLIK

VON F. KASCHLUHN, A. LÖSCHE, R. R I T S C H L UND R. R O M P E

H E F T 1 . 1 9 7 6 • B A N D 24

A K A D E M I E - V E R L A G

EVP 1 0 , - M 31728

•

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3 - 4 — im sozialistischen Ausland an eine Buchhandlung f ü r fremdsprachige Literatur oder an den zuständigen Postzcitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die AuslicfcrungsslcUc K U N S T U N D WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhclmstraße 4 — 6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Ilöchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- uud Zeilschriftcnliandcl: den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 7 0 1 Leipzig, Postfach 160, oder an den Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3 - 4

Zeitschrift „Fortachritte der P h y s i k " Herausgebet: Prof. Dr. Frank Kaecliluhn, Prof. Dr. Arlur Lasche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe. im Auftrag der Physikalisohen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2 200441; Telex-Nr. 114420; Postscheckkonto: Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universitit zu Berlin, D D R - 1 0 4 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizcnznummcr 1324 des Presseamtes beim Vorsitzenden de« Ministerrates der Deutschen Demokratischem Republik. Geaamtheretellung: V E B Druokhaus „Maxim Gorki", D D R - 7 4 Altenburg, Carl-von-Ossiettky-StraBe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " ersoheint monatlich. Die 12 Hefte eines Jahres bilden einen B a n d . Bezugspreis je B a n d : 1 8 0 , - M zuzüglich Versandspesen (Preis für die D D R : 1 2 0 , - M). Preis je H e f t IS,— M (Preis Tür die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/24/1. © 1976 b y Akademie-Verlag Berlin. Printed in the German Demooratic Republic.

Fortschritte der Physik 24, 1 - 3 6 (1976)

Medium Energy Inelastic Hadron Scattering and Nuclear Structure*) H . J. WEBEB University

of Virginia,

Charlottesville,

Virginia

22901,

USA

Abstract Some aspects of medium energy ( ~ 1 5 0 M e V to 1 GeV) proton and pion scattering theory from complex nuclei are reviewed with emphasis on inelastic scattering as a tool to extract nuclear structure information. Contents Abstract

1

1. Introduction

1

2. a. Plane and Distorted Wave Impulse Approximations b. Tibell Plots

3 7

3. Optical Model Approach a. Watson Multiple Scattering Series b. Pair Correlations c. Collective Variables in the Optical Potential d. Wave Equations e. Discussion of Results

12 12 14 15 16 17

4. Glauber Theory a. Ray Optics and Eikonalization b. The Glauber Formalism c. Glauber Theory for Inelastic Excitations

20 20 22 24

5. Applications a. Neutron Distribution in Nuclei from Pion Reaction Cross Sections b. Local Versus Nonlocal Optical Potential for Pions c. 7rA Scattering in the (3,3) Resonance and Magnetic Transitions

29 29 31 33

References

35

1. Introduction Hadron scattering from nuclei has long been used as a tool to extract nuclear structure information, mostly a t low energy, though [J], One might expect t h a t a t the higher energy of several hundred MeV to ~ 1 GeV hadron scattering should be even more use* Supported in part by the National Science Foundation. 1

Zeitschrift „ F o r t s c h r i t t e der P h y s i k " , Heft 1

2

H. J . WEBER

ful. For one, the projectile's wave-length (i>d

da^^Ej) ^ (thJ

^

(g)

The general spin-isospin decomposition of the hA7 amplitude requires introduction in Eqs. (4, 5) of additional transition densities =

/

ff

*ip -

Q)

°

M p )

with Oa = 1, k • k', a • (k X k') for (n, n') and 0 a = = a[w) + b(w) q2 + ic{w) o (¿'I tnon(w) |A) = a(w) + b{w) k'

-{tc'xic),

k + ic{w) a • (k'

(15)

xk),

which are identical, for elastic on-shell scattering, .provided a(w) = a(w) + b(w) k2, b(w) = — b(w)i2, c(w) = k2c{w) since q2 = 2fc2 — 2k • k' in this case. Except near threshold the s-wave part a(w) is small compared with the p-wave term which, on-shell, may be written as a k2 exp (—q 2 /2fe 2 ). This is similar to the high energy parametrization, Eq. (13), of hadron nucleon cross sections. Thus pion scattering for Tn in the 3,3 resonance region looks diffractive. However, at higher energies more partial waves and (izN) resonances come into the picture beside the zl (1236) yielding energy-dependent average cross section a„±N> whose Lippmann-Schwinger equation Te=Ve+

VCG0TC

(27)

yields Tc by definition, with the expression Ve=it

7= 1

i e

.

(28)

Medium Energy Inelastic Hadron Scattering

13

The full multiple scattering series for the optical potential including excited intermediate states in second and higher orders is given by vc = E tic + 27 E ) = B

0

L

\ l + £ °clmYl*M(Q,

M ,

(40)

J

LM

the equidensity surfaces may be deformed according to L

LM

J

Ra is a length of the order of the nuclear radius, e.g. the mean square radius. Expanding o(r') to first order in the collective variables = 0,

(46)

Ei being the total energy of the particle in the laboratory system and m its rest mass. The Klein-Gordon equation [(p 2 + m2) - (.EL -

VF] IP = 0

(47)

is known to yield identical solutions when | F| m and V changes slowly within one de Broglie wave length. Usually the KG Eq. (47) is written in the form [A2 + fc£a - v(r)] $ = 0 , v{r) = 2ELV-

V

2

~ 2

(48)

ELV,

assuming | F | \ E L 1. When the finite target mass M is taken into account in Eq. (46), this leads to a wave equation of the form of Eq. (48) with v(r) =

b

™ "

FE2 =

(^h + Ea-M)

E a

V(r)

^

( 4 9 )

[{EH

EH + EA

+

E A

M ?

~

~

M2]

in the hA c.m.s. while for the KG Eqs. (47) no corresponding procedure seems to exist. Alternatively one may write the usual nonrelativistic Schrodinger equation in the hA c.m.s. expressing the reduced mass in terms of the total c.m. energies, i.e. 2 EHEA

c.m.

ET

I

-E'h +

IT

& A

V

*V —V

(50)

Comparison with the KG Eq. (48) in the hA c.m.s. yields Y

+ EA EA

V

/

\

V \ _ E

H

2 EB)

+ EÄ EA

when VjE h 1 at high energy. Numerically the results obtained from these forms of wave equations differ, at high energies of ~ 1 GeV, by only a few percent. e. Discussion of Results Elastic and inelastic proton scattering (p, p') at 1 GeV from 40Ca and 208 Pb has recently been analyzed by BORIDY and FESHBACH [27] using Hartree-Fock nucleon densities and transition densities from the Tassie model, and integrating a modified (see Sect. 3d) Schrodinger equation. As Coulomb effects are important at small angles they have been included. H . MCMANUS et al. [28] use essentially the same wave equation including R0, r 0. Eq. (53) makes contact with the J W K B and Sopkovich approximations which were used earlier. In momentum space the linearization or eikonalization of the propagator is even more transparent transforming the unperturbed Hamiltonian

upon neglecting the last term, into fl0 = - ( p - k ) - k + ^ - f c 2 TO

(56)

2to

The corresponding Green's function Wr, r ' ) = J

sxp (ip • (r — —

(p — k)-k

^

_,

=

+ i0+

TO

V ) d[z

_ „

e x p ( oo, see Fig. 14,

*

(

r

)

r =

-

¿

/

t

^

'

'

^

2

6

'

-

( 5 8 )

(b, z)

Fig. 14. Geometry used for interpolation from the near-zone plane (z t ) to the asymptotic region (r), in Eqs. (60), (61)

22

H . J. WEBER

On this plane, then, ip consists of the plane wave eikz° and a remainder — -T(S') eikz° according to Eq. (54). Thus Eq. (58) interpolates from a plane in the near zone to the Fraunhofer diffraction pattern of the asymptotic region, the far zone, where .

^

eikz

_j_

r

f 0 ) _

(59)

As a consequence of Eqs. (54), (58) and (59), the scattered wave is z) =

eikz' [ — r(b') d*b'. ¿711 J Q

(60)

Since r = z0z + b + qt, i.e. r = z0z • f + b' • r + Q, where z • f ~ 1 and ic' = r, k' = k, we see that for large r ^ eikr giirpjQ ,—I g-ikza g-ib'-k' r

!

If we use this for the scattered wave tpsc and compare with Eq. (59) we obtain the basic result [40] 6) = — / r(b) e^ b d2b, "2n)

(61)

where q = k — k' and k • 6 = 0. Eq. (61) relates the diffraction pattern 3^(k, 6) to the Fourier transform of the "complementary screen" -T(S). A few examples for profile functions J1 are in order. At high energy, a purely absorptive system or a black sphere is a fairly realistic approximation. Then J1 is real, and r = 1 when b < R and r = 0 when b > R. This resembles the profile of a large nucleus as seen by hadrons, although the nuclear edge is diffuse and nuclei are not totally opaque. The corresponding Jr{k, d) = ikR2Jl(qR)lqR

(62)

describes the Fraunhofer diffraction pattern of a black disk. For a Gaussian profile r(b) = coßji (1 — ioc) exp (—c&2) ikcr 3r(k, 0) = — (1 - ioc) exp (-g 2 /4c), 4tt

(63)

while for a spherically symmetric system generally oo Jr(k, 0) = ik J r(b) J0{qb) bdb.

(64)

b. The Glauber Formalism Since at high energy the hadronic projectile travels through the nucleus in such a short time that the nucléons cannot rearrange themselves until the probe has left, the transition amplitude F(q ;r1; ..., rA) is calculated in the "frozen-nucleus" approximation. This basic assumption eliminates the complex nuclear spectrum from the outset. It is only

23

Medium Energy Inelastic Hadron Scattering

later in the averaging process Fn{q) = .

(68)

The expansion of the product expression in Eq. (68) shows that the characteristic series of diffraction minima and maxima of high energy hA scattering cross sections is described by the Glauber formalism in terms of interference between single, double and so forth scattering of the incident hadron with individual nucleons in the nucleus (see Fig. 16). Assuming independent target nucleons, i.e. a product wave-function for the nuclear ground state, for elastich A scattering r00(b)

= l -

[ i - •••,rA)~ £ e»,o(*,i) Pm >=i

• • •> ri-1> ri+i> • • •> *•.„. Instead of >p„*{r) f0(r) one usually writes the transition density g„{r) whose Fourier transform is Qn(q). For inelastic scattering Eq. (70) is replaced by rn(E, b) = = ¿ r J d*q?**Mq)

(74)

q),

where rn(E, $) = A f d3r Qn(r) rhN(E,

$-3).

Using Eqs. (61, 72 and 73), the scattering amplitude

rr

ik c

g) = 2 - J

r

b) J . . . J e^in,...,

¡J ( I - rf(l - ?,•)) dr. (75)

is obtained. From elastic scattering we recall that 1 - rfrHE,

b) ~ e^X)

= (ip0(r,..., rA^)\ [j\ 1 - rf{E, b - *,)) \Mr» ;= 1 = [1 - (r(b

so that

,..,rA-,)>

-

can be written as

ik • r S?nhA{E, q) ~ — / d% e^-brn(E,

b) e « ' » ) .

(76)

If the finite nucleon size is ignored in Eq. (74), we can use rhN(E, b) ~ 27iJFhN(E, 0) 8{t)jik for the elementary hTV profile function implying rn(E, b) =

2JI

o) A

r

dz en(b, z)

(77)

in Eq. (76). It is quite instructive to compare the basic Glauber result of Eqs. (76), (77) for inelastic scattering with the rather similar DWIA formula (see Sect. 2 a) Fn(E, q) = AtFhN(E, 0) f cPr ^7>*(r) e ,(r) **