Classical photon production in neutron-proton collisions

NUCLEAR PHYSICS A

Nuclear Physics A 575 ( 1994) 449-459

Classical photon production in neutron-proton collisions

Giuseppe Russo a,b a Dipartimento di Fisica dell’ Universitd di Catania, Corso Italia. 57-95129 Catania, Italy

b Istituto Nazionale di Fisica Nucleare. Laboratorio Nazionale de1 Sud, Catania, Italy

Received 15 June 1993; revised 23 February 1994

Abstract

The classical theory of bremsstrahlung is applied to neutron-proton scattering in order to obtain the elementary photon-production cross section. The so-called “internal radiation” from the Feynman diagrams is classically evaluated in the charged one-pion exchange hypothesis. The introduction of such a contribution produces a substantial enhancement of the pny yield for photon energies higher than 30 MeV. The results are compared with quantum calculations, which use a parametrization of the nucleon-nucleon T-matrix in terms of OBE amplitudes and with few available experimental bremsstrahlung data.

Keywords: Bremsstrahlung, pionic-exchange currents.

1. Introduction

The elementary photon-production cross section in nucleon-nucleon (NN) collisions

is a necessary input for photon-production calculations in heavy-ion reactions within

the theoretical frameworks based on the Boltzmann-Nordheim-Vlasov (BNV) [ 1] and

Quantum Molecular Dynamics (QMD) [ 21 approaches. At present, this essential ingre-

dient cannot be extracted from the experimental data over sufficiently wide enough kine-

matical range. There are various evidences that hard photons in the nucleus-nucleus col-

lision are mainly produced as a result of incoherent superposition of individual neutron-

proton bremsstrahlung contributions. Several attempts to reproduce such y-spectra in a

wide range of beam energies by means of the classical soft-photon expression [3] for

the elementary npy process, while seem to give a reasonable estimation of the overall

y-yields, they provide an underestimation of the associated inverse slope parameters [ 11.

On the other hand, various quantum-mechanical calculations [4-61 of the elementary

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450 G. Russo/Nuclear Physics A 575 (1994) 449-459

process including convection, magnetic and meson-exchange current contributions were

carried out. Some of these [5] have been applied successfully to explain experimental

y-data from nucleus-nucleus collisions especially close to the NN kinematic threshold.

It was emphasized that this better agreement is essentially due to the contribution of

radiation from virtual-pion exchange which is not included in the classical approxima-

tion [3]. It allows neutrons and protons to exchange their identity during the collision

leading to an increase of the pn scattering angular distribution at backward angles and

to an enhancement of the high-energy y-ray production. However, at variance with the

classical case, parametrizations of the elementary double-differential quantum y cross

sections are generally not available in the literature, while, for practical applications in

nucleus-nucleus dynamical calculations, an analytical representation of the elementary

npy process is required. For this reason and also in order to achieve a simple y cross

section, we apply the principles of the semiclassical theory of bremsstrahlung [ 31 to

the proton-neutron scattering assuming it proceeds via a virtual-pion exchange.

2. Internal radiation from charged one-pion exchange

We assume that the internucleon separation is large enough to describe their interaction

by the one-pion exchange contribution. The creation or annihilation of charged pions is

a process in which radiation is emitted. Such processes are purely quantum mechanical

in origin. There can be no attempt at a classical explanation of the basic phenomena.

Given that the process occurs, we may legitimately ask about the classical spectrum

and intensity of electromagnetic radiation accompanying it. Our starting point is the

semiclassical double-differential photon cross section

d2c, #, d21

dE, dL+ = tiEy dw, d0,

where d21/ dw, dL$ represents the energy radiated per unit solid angle per unit fre-

quency interval, a is the fine structure constant and the vector A(o), defined by [ 31

+CQ

A(w) = I nx [(n-P> XPI (1 -P.n)2

exp[iw(t--.r(t)/c)] dt,

--m

(2)

originates from the Fourier transform of the product RE, at the retarded time, between

the so-called acceleration electric field E in the observation point and the distance R

from this point to the point charge position.

For a specified motion, r(t) is known, /3(t) and j?(t) can be computed and the

integral can be evaluated as a function of w and the direction of sight it (unity vector),

which is generally assumed constant in time, since the observation point is imagined to

be far away from the region of space where the acceleration occurs. The sudden creation

of a rapidly moving charged pion will be accompanied by the emission of radiation.

Either we can think of the pion initially at rest in the nucleon frame and being accelerated

G. Russo/Nuclear Physics A 575 (1994) 449-459 451

violently during a short time interval to its final velocity, or we can imagine that its

charge is suddenly turned on in the same short time interval. Because of its larger mass,

the involved nucleon receives a smaller acceleration so producing a negligible radiation

contribution. A similar consideration holds during the pion disappearance phase.

Then, we assume that the virtual charged pion suffers constant accelerations during

short and equal time intervals At = 7 corresponding to its emission and capture. The

acceleration phases are supposed to begin at the times --7 and T, respectively, the latter

being the time of flight of the emitted pion. During the time interval [0, T] the pion is

supposed moving with a constant velocity c&. Thus, by indicating with pi and & the

velocity (in c units) of the initial and final involved nucleons, the integration over the

time in Eq. (2) implies a coherent superposition of contributions to the electromagnetic

radiation due to the sudden emission and absorption of the virtual charged pion, namely

A(U) =Ai(m> + Af(w). (3)

We shall restrict ourselves to the non-relativistic limit. Then, we can simplify the first

amplitude as

0

Ai( J exp(iwt)[nx {(n-p) xP}](1+2j?.n) dt --7 = [n X (n X APi) - n X (Pi X &I 1 x

{

being in the [ --7,O] time interval,

p=pi +p(t+r)

with

(4a)

and

Analogously, for the [T, T + 71 time interval, we obtain

T+T

Mm) = J exp(iwt)[nx {(n-p) xp}](l+2j?+n)dt

T = nx(nxAPf)-nx(p,xPf>lexp[iw(T+7)l

(4b)

being in this case,


P=P,+PO-T)

with

and

By neglecting second- and higher-order terms, the total amplitude Eq. (3) is written as

sin UT/~ A(w) N exp( --iw7/2) -

W/2

X[n X (n X A&) +eXp[iw(T+r)]{n X (U X A&)}] (5)

Thus, one finally obtains

IAWl*= [y]*,/n x (n x Afli)12 + In x (n x Aflf)l*

+2[n x (n x A&>l[n x (n x A/?f)]cos[w(T+r)]]. (6)

By assuming momentum conservation during the pion emission and capture phases, in

the neutron-proton center of mass, we have

A& = (m-m,)Pi +mPf

m,

and

Eq. (6) can be made explicit by using, for any vectors Q and b, the general identity

[nx (nxa)].[nx (nxb)]=a.b-(n.a)(n.b)

= ab[ COS oab - COS ana COS e,&]

= ab sin e,, sin &b cos ( Qna - @&) ,

where O,,, Qna and &b, @nb are the polar and azimuthal angles formed from the vectors

a and b with the direction II of observation, respectively. Then, after averaging over the

direction (Qr,, Qnpt) of the final nucleon, we finally get

]A(u)]*II [~]*[&sin*&r8, + $&I

x{l+2~(~-1)(1-Cos,W(T+7)])}. (7)

Actually, we have to remark that to obtain Eq. (7) from Eq. (6) we supposed, in

the averaging procedure, both an isotropic NN scattering and a constant pion time of


flight T. Moreover, in the classical approach here studied, the energy and the momentum

of the emitted photon cannot be considered. A posteriori such a neglect was justified

because we used the soft-photon limit. Although, when the photon is included in the

balance, the radiative cross section obtained does not automatically conserve energy and

momentum, we can impose their validity by treating the kinematics correctly. Finally,

to account for the virtual-pion wave function, we should make an average of Eq. (7)

over its probability density. Thus, using the asymptotic form of the one-pion exchange

wave function [ 71, we have to evaluate

(lA(w)12) = f rIACw)[2exp [-2r/pl dr,

0

where p = fic/m,c2 N 1.4 1 fm is the range of the NN potential. It is possible to obtain a

reasonable estimate of the final result by the assumption that the virtual pion propagates

with the speed of light (Wick’s argument). Thus, setting T = Y/C and by inserting

Eq. (7) into Eq. (8)) we get

x4[ 1 -coswr] + (Ey/m,c2)2 + (E,/m,c2> sin@7

4 + (Er/m,,c212 I- (9)

The spectrum of radiation at finite frequencies will depend on the details of the

collision, but its form at low frequencies depends only on the initial and final velocities.

In the very low-frequency limit, that is, for emitted photons whose energies are small

compared to the total energy available, being wr << 1, Eq. (9) goes to

((A(w) 12) 2 [j3’sin28npi + $;I

(10)

where the first factor in the right-hand side (r.h.s.) is just what one obtains in a classical

calculation [ 31 of the so-called external radiation, namely radiation from the proton only

in which the charged-pion exchange mechanism is not accounted for.

3. Bremsstrahlung cross section from np scattering

In order to calculate the differential cross section for neutron-proton bremsstrah-

lung, we should incoherently sum over the three-pion state of charge contributions with

appropriate weights.


In case of ?ro exchange, there is no classical radiation emitted from the pion but only

one contribution from the proton bremsstrahlung in the initial or final phase depending

on whether the neutral pion is emitted from proton or neutron. The associated radiation

can be evaluated starting from Eq. (2) and using a similar procedure it provides the

well-known classical expression [ 31, namely

IA( N [@sin28,pi + $@I. (11)

After that, for our purpose, we use the isospin decomposition of reaction cross section

for single real-pion production as parametrized in Ref. [ 81. Then, we will assume that

Cry = Waa,(?P) + w+(T#r+) + W_C~(Z=) ,

where

and

u”P mel =&&+alo+2U,, +3ao11.

Again from Ref. [ 81, in the Tlab < 500 MeV range, we deduce approximatively w. N $

and W+ = w_ N i. Thus, we finally obtain from Eqs, (10) and (11)

d2a,

dE, dL$ =

d2u,

[ 1 dEY’ d% Jackson

(12)

We would like to remark that the second factor in the r.h.s. of Eq. (12) has some

similarity with that suggested in Ref. [ 91 where a rough estimation of the bremsstrahlung

contribution from virtual pions is made.

In Fig. 1 we display the comparison between double-differential cross sections for

proton-neutron bremsstrahlung at &b = 200 MeV in the soft-photon approximation, with

and without internal contribution (solid and dashed lines, respectively), and the quantum

calculation [ 41 (dash-dotted line) with the parametrization of the NN T-matrix in terms

of OBE amplitudes. We find that the charged meson exchange is essential because it

raises the production yield for photon energies higher than 30 MeV. The result of our

non-relativistic approach generally underestimates the y-yield at forward and backward

angles with respect to the covariant quantal calculation carried out without contribution

from the magnetic moment.

For the first time, Malek et al. [ lo] measured the photon spectrum produced at 90” in

the npy process at 170 MeV neutron laboratory energy with an experimental uncertainty

in the neutron flux of about 30%. The spectrum observed after background correction

is shown in Fig. 2 together with our semiclassical calculation. Since experimentally the


30. (x4) -

-1 -_

- - _, - _

Fig. 1. Comparison between double-differential cross sections for proton-neutron bremsstrahlung at

&, = 200 MeV for various photon angles. Dash-dotted line: quantum calculations of Ref. [4] ; Our classical

calculation and that without internal radiation [ 31, both in the soft-photon approximation, are given by the

solid and the dashed lines, respectively.

‘io 40 60 60 100

R, [MeVI

Fig. 2. Comparison of the photon spectrum measured at 90°, in the np collision at 170 MeV of neutron energy [ 10) with our theoretical calculation.


103

102

101

100 25 50 75 100 125 150

E, [MeVl Fig. 3. Photon spectra observed at three laboratory angles in the pdy reaction at 200 MeV of proton en-

ergy [ 111. Also shown in the figure are the results (solid lines) of our theoretical calculation for the pny

reaction at 200 MeV, in absence of reliable calculations for the pdy process.

energy profile of the neutron beam had a FWHM of 70 MeV, we folded our calculated

cross section with a gaussian in energy having the same width. We found our results in

fair agreement with the first direct measurements of y-spectrum in np collisions.

It is experimentally confirmed, by comparison between the ppy and pdy reactions [ 111,

that the ppy radiation is negligible with respect to the npy one. Thus, it is also possible

to extract information on the elementary npy process from radiation emitted in pdy

reactions. Pinston et al. [ 121 measured at several angles the pdy reaction at 200 MeV

of protons. In Fig. 3 such data are compared with our semiclassical calculation for the

elementary pny process. Our result generally underestimates the data as it also hap-

pens for full quanta1 calculations [ 121, probably depending on the Fermi motion of the

neutron inside the deuteron which is not taken into account.

Edgington and Rose [ 111 studied the pdy reaction at 140 MeV. The measured angle-

integrated y cross section, reported in Fig. 4, where it is compared with our pny

calculation (full line). Although they agree in the absolute values, a different trend is

observed.

By integrating, in the lab system, the double-differential photon cross section Eq. ( 12))

both over the whole angular range and energies above E, = 40 MeV we obtained the

full curve reported in Fig. 5 which is also compared with the Jackson’s prediction and

the few available experimental data [ 10,11,13].

G. Russff/~~cle~r Physics A 57.5 (1994) 449-459

! I 0.01 g

20 40 60 80 100

R, [MeVl

457

Fig. 4. Comparison between total-energy y-spectrum for pd bremsstrahlung at 140 MeV incident proton energy [ 121 with the results of our calculation for the pny reaction.

II r I j I I I, ,-I 100 200 300 400

Ttab [MeVl

Fig. 5. The total bremsstrahlung cross section (for E, 2 40 MeV) versus the bombarding energy of the proton-neutron system. The full and dashed lines are the classical soft-photon limit with and without infernal radiurio~, respectively. They are compared with the available data from Refs. [IO] (diamond), [ 1 l] (square) and 1131 (cross).


4. Summary and conclusions

We have applied the classical theory of bremsstrahlung to the neutron-proton scatter-

ing. The so-called internal radiation from the Feynman diagrams, in the single virtual-

pion exchange hypothesis, is evaluated as a coherent superposition of electromagnetic

fields produced during the pion emission and its capture. We find that the charged-meson

exchange mechanism is essential because, allowing neutrons and protons to exchange

their identity during the collision, it produces an enhancement for high-energy y-ray

emission with respect to the predictions of Jackson’s expression.

The comparison of our non-relativistic and semiclassical calculation, in the soft-

photon limit, with quanta1 results seems to evidence a slight difference in the angular

distribution of emitted photons which has in our case a more marked dipolar component.

We observe that, although the pion emission process is purely quantum mechanical, the

classical approach has the advantage that working directly with electromagnetic fields

does not present the usual ambiguities noticed in quantum-mechanical approaches when

one uses different gauges in the one-pion exchange diagram [ 141.

The comparison with the npy and pdy available experimental data has shown that our

calculation in general gives the right order of magnitude for hard-photon yields.

All that suggests the use of such an elementary double-differential photon cross

section as a convenient input for hard-photon-production calculations in nucleus-nucleus

collisions within the frameworks of BNV and QMD simulations. Recently, perturbative

BNV calculations of hard-y production in nucleus-nucleus collisions, using Eq. (12) as

elementary cross section, have been able to reproduce [ 151, as functions of the impact

parameter, both the multiplicity and slope of the y-yield obtained in measurements

performed with an almost 47r multidetector array.

Finally, we would like to emphasize that, following the procedure described in Sect. 2,

a fully relativistic expression also removing the soft-photon approximation could be

derived in a straightforward way.

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Classical photon production in neutron-proton collisions