Virtual photon cross sections for two-body electrodisintegration

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Nuclear Physics A163 (1971) 193-202; @ North-Holland Publishing Co., Amsterdam

2L Not to be reproduced by photoprint or microfilm without written permission from the publisher

VIRTUAL PHOTON CROSS SECTIONS

FOR TWO-BODY ELECTRODISINTEGRATION

B. F. GIBSON t and H. T. WILLIAMS tt

National Bureau of Standards, Washington, DC

Received 14 August 1970

Abstract: Two-body electrodisintegration cross sections (differential with respect to the angles of the ejected nuclear fragment) are developed in the long wavelength limit of virtual photon theory for the few-nucleon system. Only the non-spin-flip electric transitions are considered. Contributions from the multipole expansion through the El-E3 interference terms are retained. Comparable photodisintegration expressions are included to facilitate the extraction of photo cross sections from appropriate inelastic electron scattering data.

1. Introduction

The information to be gained from the study of photodisintegration and electrodisintegration of the nucleus has been discussed at length in the literature; see; for example, refs. ‘9 ‘) and th e work cited therein. In addition, the experimental difficulties associated with photon experiments, which have made useful the study of mathemati- cal procedures for extracting photo cross sections from inelastic electron scattering data, are well known. Furthermore, it is intended that this paper present for the experimentalist working in the field virtual photon formulae directly applicable to the determination of photo cross sections from analysed electrodisintegration data. Therefore, the repetition of much background information already quite familiar to this group of people is considered unnecessary.

Theoretical considerations leading to the differential cross sections for both photodisintegration and electrodisintegration l, 3, “) are briefly outlined in sect. 2 for the purpose of &fining certain concepts and notation. A direct application of the theory leads to the cross sections corresponding to the non-spin-flip electric transitions presented in sect. 3. We specifically consider photon absorption and inelastic electron scattering which lead to the two-body breakup of the nucleus where the initial state is comprised of two nuclear fragments having relative angular momentum I = 0. We discuss the cross sections differential with respect to the angles of the ejected nuclear fragment. This corresponds to the experimental situation in electrodisintegration in which the scattered electron is not detected. The results are directly appli-

t National Research Council Postdoctoral Research Associate. tt Present address: Institut fiir Theoretische Physik, der Universitat Erlangen-Niirnberg, Erlangen,

Germany.

193

194 B. F. GIBSON AND H. T. WILLIAMS

cable to experiments presently being carried out at the National Bureau of Stand- ards “) and at the University of Saskatchewan “).

We wish to emphasize that we treat only the non-spin-flip, electric transitions, since it is possible in this case under certain simplifying, physically reasonable assumptions to couch the problem in a calculable form. The lack of a simple and re- liable theory of the electromagnetic current including meson effects makes the calculation of the magnetic transitions unreliable. Thus the connection between photon and electron magnetic transitions, although theoretically straightforward to deduce, is in practice difficult to establish.

2. Formalism

For completeness, we reproduce here certain equations leading to the differentia1 cross sections for both photodisintegration and electrodisintegration of the nucleus. The interaction Hamiltonian, to lowest order in the electromagnetic coupling constant, is

H = <flJ,liP,, (1)

where (flJ,Ji) is the matrix element of the nuclear electromagnetic current between the initial and final states of the nuclear system. Here A,, is proportional to the photon polarization E,, in the case of the photon-induced reaction, and in the case of the electron-induced reaction A,, is given by

A = -eW)y,u(d P q2 ’

where e is the electron charge, p’ and p are the final and initial electron four momenta, and q = p -p’ is the four-momentum transfer +. The electron spinors are normalized such that u*u = 1. And in what follows u(p) and u(p’) are treated as free electron spinors (plane wave solutions to the Dirac equation).

In the case of the photon interaction the timelike component of the potential, A,,

vanishes because the real photon has polarization only in the spacelike directions. This is not true in the case of electrodisintegration, but A0 may be eliminated through use of the charge conservation relations for both electrons and nucleons:

q,,A,, = 0,

q,<flJ,li) = 0. (3)

One can verify that for A’ = A -h2/d)(4 * -44, (4)

the Hamiltonian can be expressed in terms of spacelike quantities:

H = <fIJIi) * A’. (5)

7 We use the metric in which q2 = 4*-q;; also we use the units in which fi = c = 1.

VIRTUAL PHOTON CROSS SECTIONS 195

Note that although e - k = 0 in the case of the photon, the “polarization” of the virtual photon in electron scattering is not transverse and hence A’ - q St 0.

Standard procedures of quantum mechanics now lead to the differential cross sections of interest in terms of the matrix elements of the electromagnetic current J and the potential A’ defined above. In the case of the photodisintegration process leading to a two-body final state, the cross section for emission of a nuclear fragment into the solid angle dQ is given by

da - = 27CCPNJjj’EjE~, dC& e

(6)

- where xe signifies an average over the photon polarization, pnds2, is the nuclear density of final states, Ej are components of photon polarization, and

Jj~ = C (f lJjli)*(f [JYli). (7)

Here c represents a sum over the final nuclear states and an average over the initial nuclear states, and

J = s

j(r)eiq”dr. (8)

The double differential cross section for the electron-induced two-body (nuclear) breakup is given by

d% 27r ___ = -pNPeJjfNjr, dG.&A P

(9)

where /I is the velocity of the initial electron andp,d& is the electron density of final states. The quantity Njf is defined by

N,. = c A;*& (10)

where the summation represents a proper sum and average over the electron spins; for free electron spinors the result is

The cross section (differential with respect to the angles of the ejected nuclear fragment) is obtained fromeq. (9) by an integration over da,. Details of the treatment of the nuclear matrix element are discussed in the appendix.

3. Cross-section formulae

Following the procedure outlined in the previous section we have generated both the differential cross section for photodisintegration and the double differential cross


section for electrodisintegration. We have retained only the leading non-vanishing terms of those multipoles that contribute up to order k2(and q2) +; we quote results for pure transitions and interference terms. And in the case of electrodisintegration we have integrated over the scattered electron anglesin order to obtain a single ~fferential cross section comparable to the results in the case of photodisintegration,

For the photodisintegration process, we have the following differential cross sections (averaged over polarization):

$- (El) = 2nczpn k[Mr I2 sin2 8, N

$- (El -E2) = 2xz,oN k2{2 Re [M, M,]} Sin2 0 cos 8, N

g (E2) = 27tUpN k3/h&(2 Sin’ 8 COS2 0, N

$ (El -E3) = 27C%pNk3(2 Re [M, M3]) Sin’ @OS2 @-+], N

(12)

where a = ez = &, k is the photon momentum, and 6 is the angle made by the ejected nucIear fragment with respect to the photon momentum. For the particular case of the deuteron, where the initial state is considered to be pure 1 = 0, the nuclear matrix elements in the long wavelength limit are given by:

where r is the neutron-proton separation and fi is the radial function of the Ith partial wave of the final state. (In the case of the plane wave final state,fr is proportional to the spherical Bessel function il.)

The fact that a particular photon cross section is completely determined by the corresponding electron cross section has been discussed previously 3? “) We indicate here the form of the E2 double differential cross section to illustrate this point:

Here q is the momentum transfer, B is the velocity of the incident electron, and P’ is the relative momentum of the nuclear fragments in the final state. It is clear that the same matrix elements enter both the photon and electron cross sections within

t For those readers interested in the detailed development of the long wavelength approximation, we refer them to ref. “f.


the approximation of this work. However, such a cross section requires a coincidence measurement. And coincidence experiments are complex; the cross sections are small. Because most of the electrons scatter in the nearly forward directions, rp *) we wish to consider the simpler process in which the scattered electron is not detected but is assumed to scatter predominantly forward. This involves an integration over the scattered electron angles. After tedious algebraic rn~~~u~ation one obtains to first order in m/E and m/E (where m is the electron mass and q. = E-E’):

In these expressions we have defined J = 2EE’/mqo. (The El result was first reported by Dodge and Barber for the more general case of an arbitrary initial state “).I


In addition to the matrix elements defined in eq. (13) for the photodisintegration cross sections, we have

Me = (2~)~l;<f0iW)“li>. (16)

This is actually the second-order contribution to the EO matrix element, because in the long wavelength limit the EO matrix element vanishes due to the orthogonality condition “) (f 11 Ii) z 0. Therefore, the q-dependence of the EO cross sections is the same as that of the E2 cross sections.

An interesting check on the EI and E2 resuits is obtained by performing the dQN integration. This reduces the pure El and E2 cross sections in eq. (15) to

(17)

These expressions can be seen to agree with the form of the total cross sections as previously reported by Thie et al. lo)

4. Conclusions

The combined angular distribution of the photon multipole cross section described in eq. (12) is of the form

[A sin2 8 +B sin2 e cos 8+ C sin2 6 cos2 S]. (18)

As can be seen from identification of the angular coefBcients above with the terms of eq. (12), only the El-E2 interference term contributes to B. But the El-E3 interference term contributes to both A, as does the El term, and C, as does the E2 term. The El-E3 contribution to A is of course a factor of k2 smaller than that of the El term (as noted previously we have retained only the leading term of the El cross section), but in the case of C the El-E3 contribution is of the same order in k as the E2 contribution.

The combined angular distribution of the electrodisintegration cross section is quite similar to that for photodisintegration:

[A sin’ B+B sin’ t? cos @+ c sin2 8 cos2 ef D-J. (19)

But now, in addition to mixing the matrix elements MI., M2 and MS, the angular coefficients contain the matrix element MO which cannot be easily separated from M, . It is because of the presence of Me that some model dependence may be intro- duced into any analysis of electrodisintegration data to obtain photodisintegration cross sections. However, as is discussed below, the EO multipole, which introduces the M, dependence, gives only corrections to the dominant photon-like parts of the El,


E2, and E3 multipole terms at low-momentum transfer where these cross section expressions are applicable.

It should be clear from an examination of the E and E’ dependence of the electron cross sections that the dominant term is the In 1 which corresponds to the predomi- nant photon-like interaction of the forward scattered electrons. In fact, for E >> E’ the electron cross sections have the same angular distributions as the corresponding photon cross sections to order ln(E’/m). The corrections to this are of order (E’/ E)“; they tend to make the electron cross sections more isotropic than the corresponding photon cross sections.

The EO terms give rise to the 0+--O+ transitions that occur in electron reactions but which are absent in photon reactions. In the treatment of inelastic electron scattering outlined in sect. 2, they are the result of the longitudinal polarization of the electric field of the virtual photon defined by eq. (4); in reality they result from the Coulomb interaction of the electron. The fact that there are no EO photon cross sections is made apparent in the virtual photon cross-section expressions by the absence of the In I term that comes from the photon-like part of the electron interaction. Thus the EO cross-section contributions are (E/E)” corrections in the electrodisintegration process. In a particular example such as the two deuteron disintegration of the tl- particle where one may have (El/E) x A, the corrections to the dominant In L term of the E2 transition coming from the EO and EO-E2 terms are of the order of (A)’ or about 1%.

In conclusion we wish to reiterate that within the framework of the approximations made, the real photon process can be completely determined from a knowledge of the corresponding virtual photon reaction. The fact that the virtual photon transitions may contain contributions of the 0+-O+ transitions not present in the real photon reaction does mean that there may be some model dependence in the extraction of the desired photon cross sections. However, of more probable importance in a practical application of the results quoted in this work is the neglect of higher-order Coulomb effects in the electron interaction, since it is quite possible that the scattered electron will have little energy. No attempt has been made to estimate corrections due to such Coulomb corrections.

We gratefully acknowledge helpful discussions with Dr. R. W. Hayward and we wish to thank Dr. W. R. Dodge for introducing us to the problem.

Appendix

A comparison of Cartesian and multipole expansions of the electromagnetic transition matrix elements.

In the case of electrodisintegration the physics of the nuclear states is contained in the nuclear matrix element

Mi, = (fl s

drj(r)*A’ei4”li). (AJ)


(Photodisintegration matrix elements are obtained by making the substitution A’ + a, 4 -+ k). The unambiguous separation of this matrix element into transverse and longitudinal electric as well as magnetic multipoles is accomplished by expanding in terms of the irreducible tensors “)7”:, i.e.

A;e’4” = (47r)+ z i’(21f 1)3.1(qr)Cb’,“TZ(P).

The index m = 0, f 1 describes the spherical components of the vector A’, where we have in general A’ * q # 0. (In the special case of photodisintegration we have E * k = 0, and the longitudinal terms vanish.) We assume q to define the z-direction. Thus the electric 2L pole is the contribution from 1 = L+ 1 with parity (-)“, m = &- 1; the magnetic 2= pole is the contribution from A = L with parity (-)Lfl, m = + 1; the longitudinal multipoles are those due to m = 0, that is when we have A’llq.

In general one must specify the form of the nuclear electromagnetic current in order to compute the matrix elements defined in eq. (A.l). However, the proper inclusion of meson exchange currents is a major difficulty in carrying out such a program. Siegert I*) pointed out some years ago that in the long wavelength limit (i.e., to lowest order in qr for a given multipole) the expression in eq. (A.2) can be written as the gradient of a scalar, so that after an integration by parts the matrix element can be written in terms of V *j, which can be transformed into an integral involving only the nuclear charge density by means of the charge continuity equation.

An alternative approach to the separation of eq. (A.l) into multipoles is taken by Bosco and Quarati “). They use the separation suggested by Foldy ‘)

I 1

Ateil” = ds(V[A’ * reiq’r]- isr x [q x A’]eis4”j. 0

(A-3)

Expressions resulting from the first term within the integral having A’l_q are considered to be the “electric” multipole terms, and those from the second term, A’ I q,

are considered to be the “magnetic” multipole terms. When A’l[q, the second term vanishes, and all the longitudinal contribution then comes from the first term and must therefore coincide with the longitudinal terms from eq. (A.2). The separation of eq. (A.3) into electric and magnetic multipoles is not identical to the separation of eq. (A.2), as was pointed out by Foldy. To what extent the two separations coincide can be seen by direct comparison. Consider the “electric” term

E = s

‘dsV[A; * r eisq “I, (A.4) 0

where q defines the z-direction and A ; * q = 0. The angular dependences can be expanded in terms of spherical harmonics:

e iSq ” = (4n)+ F i’(21+ l)*jr(sqr)Y,O(P), (A.5)

rm = r(+n)+Yr(P).


Coupling the two spherical harmonics in accordance with angular momentum conventions, one is led to

E, = s

rdsV[(4rc)* g i’(2rZ-k l)*jl(sqr)rCbb”Cfi Y?(P)]. (A-6) 0

Then using the gradient formula 11)

(A-7) with cp(r) = rj,(sqr), one obtains

Utilizing the differential and recurrence properties of the spherical Bessel functions one is led to

E,,, = s ‘ds[ -(2rr)i C i”+’ 0

W+l))*((J+ l)*j~+l(sqr)TtA+I +~+~~-l(w)~~~-I)l. 1.

(A-9)

Retaining results only to lowest order in qr for each Iz, one can reduce the above to

E, = -(4~)~~i’+‘[f(l+l)lfj~-,(qr)T;:,-,, (A.lO)

where we have defined

(4r)’ 1.3+*(21+1)’

This coincides with the electric multipoles from eq. (A.2) to lowest order in qr for each 1. To the next highest order in qr this correspondence between the terms in the Foldy expansion and the exact multipole expansion no longer exists.

Thus, the Foldy separation, eq. (A.3), has the same criteria of validity as the Siegert theorem, and since the “electric” terms are already written as gradients it is the convenient form for application of the Siegert theorem. Consider the electric transition matrix element

MFr = (fl i

“dsj(r) * V[A’ * reisq”]~i). (A.ll) 0

Performing an integration by parts, employing the current conservation relation


(assuming point charge nucleons) one obtains the expression

MTf = i(Er-EJ(f//O1dsA’ * reisq’rji)

= (E,-Ei) C P(f iD&), L

where

D, = (A’. r) (ti’-l. L!

(A.12)

(A.13)

The last step, expanding in terms of qr cos 8, is equivalent to lowest order in qr to the expansion in terms of spherical harmonics and spherical Bessel functions and thus places no further restrictions on the validity of the results.

The work of Bosco and Quarati, as well as the results presented in sect. 3 of this paper, were achieved using eq. (A.12) and eq. (A.13), and they are only valid to lowest order in qr for a given physical process.

A difficulty would seem to arise in the case of the EO transitions. To lowest order in qr the EO operator is proportional to 1, and since the initial and final states are orthogonal in the disintegration process, the expectation value of the operator vanishes. Hence the second-order term is dominant. All EO transitions, however, arise from that part of A’ satisfying the condition A’llq, and thus they are due to that part of the expansion correct to all orders. It is perhaps worth pointing out explicitly that the EO and E2 transition matrix elements are thus of the same order in qr; therefore their effects must be added to give the correct lowest order results.

References

1) W. C. Barber, Ann. Rev. Nucl. Sci., ed. E. Segre (Annual Reviews, Palo Alto, Calif., 1962) vol. 12, p. 1.

2) W. R. Dodge and W. C. Barber, Phys. Rev. 127 (1962) 1746 3) J. M. Eisenberg, Phys. Rev. 132 (1963) 2243 4) B. Bosco and P. Quarati, 33 (1964) 527 5) W. R. Dodge, private communication 6) Y. M. Shin, private communication 7) L. L. Foldy, Phys. Rev. 92 178 (1953);

R. W. Hayward, Handbook of physics, ed. E. Condon and H. Odishaw (McGraw-Hill, New York, 1967) pp. 9-172

8) R. H. Dalitz and D. R. Yennie, Phys. Rev. 105 (1957) 1958 9) L. I. Schiff, Phys. Rev. 96 (1954) 765

10) J. Thie, C. Mullin and E. Guth, Phys. 87 (1952) 962 11) M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) 12) A. J. F. Siegert, Phys. Rev. 52 (1937) 787

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Virtual photon cross sections for two-body electrodisintegration