Fortschritte der Physik / Progress of Physics: Band 16, Heft 11/12 1968 [Reprint 2021 ed.] 9783112500606, 9783112500590

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE D E R PHYSIKALISCHEN GESELLSCHAFT IN D E R DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R . R O M P E

B A N D 16 • H E F T 11/12 • 1968

A K A D E M I E

-

V E R L A G

•

B E R L I N

A. I. A N S E L M

Einführung in die Halbleitertheorie Übersetzung aus dem Russischen In deutscher Sprache herausgegeben von HANS NEUMANN , 1964. X I I I , 405 Seiten — 90 Abbildungen — 4 Tabellen — gr. 8° — Leinen 4 4 , - M Der Verfasser behandelt in diesem Buch vorwiegend Transportprozesse in der üblichen Näherung mit Hilfe der Boltzmann-Gleichung. E r untersucht ausführlich die Gitterschwingungen in Kristallen mit Basis und das Elektron im periodischen Potentialfeld des Kristalls. Unter den verschiedenen Streuprozessen, die in der Boltzmann-Gleichung berücksichtigt werden, richtet er das Hauptaugenmerk auf die Wechselwirkung der Elektronen mit den Phononen. Dabei geht er ausführlich auf einzelne Modelle für diese Wechselwirkung ein. Schließlich untersucht er Lösungen der Boltzmann-Gleichung bei Anwesenheit äußerer Magnetfelder, elektrischer Felder, Temperaturgradienten etc., berechnet also thermoelektrische, galvanomagnetische, thermomagnetische u. ä. Effekte. Der Autor geht abschließend auf Transporterscheinungen in Yieltalhalbleitern ünd den Phonondrang ein. Optische Phänomene und Oberflächeneigenschaften und daher alle mit Kontakten und dem Transistoreffekt zusammenhängenden Erscheinungen sind teilweise nur kurz erwähnt. Ausführliche Rechenbeispiele empfehlen das Buch als Hochschullehrbuch. Es bietet den Standard-Stoff der Halbleitertheorie, dringt aber an einzelnen Stellen bis zur modernen Entwicklung der Halbleitertheorie vor. Bestellungen

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Fortschritte der Physik 16, 595-634(1968)

Proton Compton Effect P . S. B a r a n o v , L. V. F i l ' k o v , G. A. S o k o l P. N. Lebedev Physical Institute, Moscow,

U.S.S.R

Abstract A method of calculating cross-sections of Compton scattering on protons based on dispersion relations at a fixed square of the momentum transferred with a subtraction for invariant amplitudes is thoroughly discussed in this review. Dispersion sum rules for the interaction constants of 71°, 7) and X° mesons are analysed. A comparison is made by means of j; 2 -test. The results obtained from the comparison of the experimental data in the range of energies up to 300 MeV with the theoretical cross-sections are presented. In the range of gamma-ray energies from 180 to 220 MeV a strong difference between theoretical calculations and experimental data is observed. In the other ranges of energy the experimental data available are in agreement with the theoretical calculations, the contribution from the annihilation channel being considered and the sign of the contribution from the 7r°-meson pole being chosen equal to the sign of the main amplitude. Table of Contents I. Introduction II. Compton effect theory 1. Introduction 2. Kinematics 3. Derivation of dispersion relations for the amplitudes Ti at a fixed t 4. Single integrals in the Mandelstam representation 5. Consideration of the annihilation channel states not described by the diagrams of the fourth order perturbation theory 6. Dispersion relations with subtraction in the point u 0 = m? 7. Sum rules for interaction constants of tt°, 7) and X°-mesons 8. Numerical analysis of dispersion relations I I I . Experimental investigations of gamma-ray elastic scattering on hydrogen . . . . 1. Identification of the process of gamma-ray elastic scattering on hydrogen . . . 2. Experimental data concerning Compton scattering on protons in the energy range up to 120 MeV 3. Experimental data concerning Compton scattering on protons in the energy range from 120 to 300 MeV 4. Experimental data concerning Compton scattering on protons in the energy range from 300 to 1450 MeV 44

Zeitschrift „Fortschritte der Physik", Heft 11/12

596 597 597 598 599 601 605 606 607 609 614 614 622 625 626

596

P . S. BABANOV, L . V. FIL'KOV, G. A. SOKOL

IV. Comparison of experimental data with theory 627 1. Method of quantitative analysis 627 2. Qualitative picture of the comparison with theory 628 3. Analysis of Compton scattering on protons in the gamma-ray energy ranges from 60 to 150 MeV and from 240 to 280 MeV 629 V. Conclusions

632

I. Introduction Gamma-ray elastic scattering on. nucleons (Compton scattering on nucleons) is one of the fundamental processes, the analysis of which can give important information about elementary particles. Contrary to the classical Compton scattering on electrons, with regard to gamma-ray scattering on nucleons a strong interaction in intermediate states plays an eminent role. That is why an investigation of Compton scattering on nucleons may be a source of information concerning the nature of both electromagnetic and strong interactions of particles involved in the intermediate states of the process under consideration. The analysis of Compton scattering at small gamma-ray energies (E y 100 MeV) made it possible to explore the spatial structure of nucleons. The problems of electrical and magnetic polarisation of protons are thoroughly considered in the papers [i—4], That is why we do not want to dwell at length upon this problem. The decisive role of strong interactions in Compton scattering on nucleons indicates that the method of dispersion relations is the only sensitive method of analysing the given process. On the other hand, the fact that photons take part in the process justifies a number of simplifying assumptions and the use of the perturbation theory with respect to the electromagnetic constant of interaction. Applying dispersion relations to the amplitudes of Compton scattering on nucleons, no integral equations are produced as for instance in the case of n-n scattering, but integral relations connecting the Compton effect amplitudes with quadratic functions of the photoproduction amplitudes. Thus, concerning the process under consideration we did not come across any specific difficulties arising from the solution of integral equations. Therefore, by using information about photoproduction amplitudes we are able to determine the unknown parameters introduced into the theory from a comparison of the theory with the experiment without any unnecessary assumptions. At present the range of greatest interest for studying Compton scattering is the range of incident gamma-ray energies E-i < 300 MeV. This limit is due to the fact that the information needed for the theoretical estimation in the case of photoproduction of two and more 7r-mesons as well as the information about photoproduction of heavier particles is extremely poor nowadays. Thus, when using dispersion relations in this range of energies one ought to introduce additional unknown parameters into the theory. One of the problems to be solved by studying Compton scattering is a definition of the 7t° 2y decay matrix element sign, which in its turn may help us to examine whether the 7i°, rj, X°-mesons are „elementary" particles or reggeons. In this paper an attempt is made at analysing the state of affairs in studying Compton scattering in the range of energies Ey < 300 MeV.

Proton Compton Effect

597

II. Compton Effect Theory 1. I n t r o d u c t i o n After the publication of the paper by GELL-MANN, GOLDBERGER and THIRRING [5] dedicated to the application of dispersion relations to the theory of elementary particles and after the publication of the paper by BOGOLYUBOV and SHIRKOV [6] where some dispersion relations for the Compton scattering amplitude on nucleons at an arbitrary angle were suggested, a lot of theoretical papers dealing with the analysis of dispersion relations for the amplitudes of Compton scattering on protons [7—29] were published. In most of these papers ordinary one-dimensional dispersion relations were used written in terms of the scattering amplitudes at 0°. This was due to the fact that ordinary one-dimensional dispersion relations at a fixed square of the momentum transfer have a i-dependent subtractional function which can be determined by the low-energy limit in the case of 0 = 0°, while for 0 =)= 0° the subtractional function is still unknown. The use óf dispersion relations for the amplitudes of scattering at an angle of 0° leads to the fact that only a small number of partial waves can be taken into account in calculating differential cross-sections for 0 0. In other papers where the one-dimensional dispersion relations were written for the scattering amplitudes at an arbitrary angle [21, 22] it was assumed that an asymptotic of the amplitudes of Compton scattering on nucleons and, consequently, a subtractional function is defined either by a low-energy limit [22] or by a low-energy limit and the 7r°-meson pole [21]. The equivoques mentioned may be removed by using the so-called rule of substitutions. In the case of Compton scattering on nucleons this rule leads to the statement that the amplitudes of the three processes Yi + -^í

Ta +

N

2

Y2 + N1 -> Yi + N2

N1 + N:2

Yi + y2

(channel I), (channel II), (channel I I I )

are the boundaiy values of the same analytical function. In particular, the equivoques mentioned above may be removed by starting from the double dispersion relations (Mandelstam representation [30]) when analysing Compton scattering. Thus, in the papers [24, 25] the Mandelstam representation is used to build dispersion relations for partial waves. In paper [23] one-dimensional dispersion relations on the basis of double dispersion relations at a fixed scattering angle were obtained. Note that a numerical estimate was fulfilled only in one of the three papers mentioned [25], I t was shown in this paper that the consideration of the fourth-order Feynman diagrams for channel I I I gives a considerable contribution to the differential cross-section of Compton scattering on protons. A characteristic feature of the approaches discussed in the papers [23—25] is the fact that the functions under investigation in these cases have complicated analytical properties in a complex region of a variable s (where s is the square of the full energy in the centre of mass system of channel I). Besides the application of 44*

598

P. S. BABANOV, L. V. FIL'KOV, G. A. SOKOL

dispersion relations to the partial amplitudes causes an inevitable restriction in the number of the partial amplitudes in the case of a numerical analysis. These drawbacks may be avoided by constructing one-dimensional dispersion relations for invariant amplitudes at a fixed t with one subtraction. To find a subtractional function one ought to use both one-dimensional dispersion relations with regard to t and the low-energy limit. The papers [26—29] are dedicated to the construction of this dispersion relation, the results of which will be given in Sect. 2—5 of this review. 2. K i n e m a t i c s Designate four-momenta of the input and output photons by k1 and k2 and fourmomenta of the input and output nucleons by px and p2. Starting from these vectors the following invariants can be derived: a = (Pi + ¿i)2 = (P* + u

(

=

i,)» = (p2

~

P l

= (Pl

t = (ia-

(i)

-

-

The invariants obtained are connected by the equation: s + u + f = 2m 2 ,

(2)

where m is the nucleón mass. We represent $-matrix as: Sfi = dfi +

im OO, and as COS 0 entering the angle operaS - - Diagram corresponding to interactors equals tions of gamma rays with mesons Fi

cos 0 = 1 +

—

2

St m*

the function Au (s, t) — Au (.s, 0) is diminishing as s -> oo supposing that Axi (s, 0) increases slower than s as s —> oo. The limitations pointed out above are the reason why the asymptotic of the function T( (s, u, t) — T^s, u, t = 0) is expressed by the poles in channel I I I and the integral from Xsi oo Tt (s, u, t) - Ti (s, u, 0) -> B? - B