Optical Properties of Nuclear Matter

The index of refraction and absorptive properties are estimated in nuclear matter consisting of protons and neutrons and in nuclear matter charge neutralized by electrons.


Introduction
Nuclear matter is a hypoth e ti c sub s tan ce co nsis tin g of interacting ne utron s and protons in large e nough numbers that th e syste m can be co nside red infinite. The properties of thi s me dium are de termined by the nucleon·nucleon pote ntial (us ually with th e Co ulomb forces be twee n th e proton s omitted to avo id the in· finite Coulomb potential). Two properties, the eq uilibrium density and bindin g e nergy per nu cleo n, can be de rived from meas ure me nts on finite nu clei beca use th e co nditio ns at th e ce nter of a very large nucleus approximate those of nuclear matter.
The purpose of thi s inves ti gation is to study the important inte raction mechani s ms of photon s in nuclear ma tter as a fun ction of frequency in order to estimate the photon propagation c haracteristic s in the medium. In the next section we re view so me of the basic fac ts of nuclear matter. In section 3 the photon-nucleon cross section s for various processes are computed. These processes are the n reexamined in sec tion 4 for the case where electrons are added to the medium causing net charge neutralization. Finally, a discussion of the results and conclusions are prese nted in section 5.

Nuclear Matter
The charge density di stribution of finit e nuclei as determined by elastic electron scattering indicates the nucleon de nsity of heavy nuclei is fairly constant in the interior of the nucleus with a value p=O.170 nucleo ns/fm 3 . We s hall adopt this value for infinite nuclear matter. The total binding energy of finite nuclei as expressed in a semiempirical mass formula can be extrapolated to an infinite number of nucleons (half protons, half 9 ne utron s). The ave rage binding e nergy pe r nuc leon in nuclear matter (i.e., the coefficie nt of the term lin ear in numb e r of nucleons) is It is the goal of nuclea r matter th eo ry to re produce th ese "o bserve d " prope rti es, p and B, startin g from a pote nti al V(r) be tw ee n two nucleo ns. A re vi e w of th ese calc ulati ons is give n by Be the [1] I. Th e Reid softcore potential, one of th e better ph e nome nological nucleo n-nu cleo n pote ntial s, yield s PHSC = 0.20 fm -3 and B Hsc =-1l.25 MeV in a calc ulation of th e twobody co ntributi ons. Estimates of hi ghe r ord e r co ntributions see m to acco unt for discrepancies between the Reid poten tial predicti ons and th e observables.
Nuclear matter is us ually discussed within the fra mework of the Fermi gas model. For no interaction s a mong nucleons (V( r) = 0) th e individual nucleo n wave fun ction is a plan e wave of mom e ntum k l/J(r)=e ikr • These functions are normalized in a unit volume and are subject to antisymmetrization for identical nucleons. Our units are chosen to have 1i = c = 1.
The distribution of nucleon momentum in the F ermi gas model is uniform from k = 0 to k = kF (the F ermi momentum). The maximum momentum kF is related to the nucleon density by which gives kF = 1.36 fm -I . This value of the F ermi mom e ntum is also in agree me nt with inelastic electron I Fi gu res in brackels indica te the lil eralure references at th e e nd of thi s paper. scattering in the quasi elastic region. This value corresponds to a maximum kinetic energy per particle of k2 F 2M=37 MeV and an average kinetic energy of 22 Me V. These plane waves form the basis set used in the solution of the problem of interacting nucleons.
In nuclear matter the Schrodinger equation for the wave function t/J (r) of two nucleons with relative momentum k = kJ -I\: 2 interacting through a potential V (r) is modified by a term which takes into account the fact that nucleons cannot scatter into momentum states already occupied by other nucleons. The modification as derived by Bethe and Goldstone for a s-wave potential is where u(r) = kr t/J(r) and The solution of the Bethe-Goldstone equation can be written as a free wave minus a defect part u(r) =sin kr-~u(r). Figure 2 shows the defect function [2] for the Reid soft-core potential. The function ~u/ k is not sensitive to either the relative momentum, k, or the total momentum kJ + k 2• An important feature of ~u(r) is that it is small by the time r approaches the average internucleon separation distance, p -I /3 = 1.8 fm. This means that for subsequent collisions nucleons approach each other in relatively pure plane-wave states. The defect function normalization is such that is the probability that a s-wave pair of particles is outside the Fermi sea. For the Reid soft-core potential K~ 0.14 summed over all partial waves. Th~ Pauli exclusion principle which prevented nucleon-nucleon scattering to occupied momentum states will also modify photon-nucleon interactions. If in free space a photon process imparts a momentum transfer q ~o a nucleon, the cross section in nuclear matter will be r~duced by a factor because only a fraction R of the nucleons will have an initial momentum ksuch that the final momentum will exceed the Fermi momentum I k + q I > kp . The expression is derived by calculating the excluded volume of the initial and final Fermi momentum spheres shown in figure 1.

FIGURE 1. Initial (left sphere) and final (right sphere) momentum distributions in a Fermi gas model in which a momentum q is to be given each nucleon.
Only those nuc leons for whi ch I k + q I > k f ' are allowe d to make tht: tran sition. Tht: fraction R(q) is the ratio of excluded volu me (s haded) to the total (4/31T kF 3 ).

Photon Interactions
Consider a photon of frequency w incident on a semi-infinite slab of nuclear matter. The index of refraction (ratio of wavelength in vacuum to that in the material) is related to the forward elastic scattering amplitude f( w) by n2(w) = 1 + 47rp~(w).

W
In general, both f and n are complex numbers_ The imaginary part of f is related to the absorption cross section through the optical theorem The real part of f can be calculated from the imaginary part by the use of a dispersion relation, but we will not make use of this approach in our estimates. We now examine the photon interaction mechanisms [3].

Thomson Scattering
Electromagnetic waves scatter from the free proton (mass M, charge e) with an amplitude e 2 f= -M = -1.54 X 10-3 fm, which is independent of frequency. The forward amplitude is the same in nuclear matter because no momentum is transferred to the proton. It is conveni ent to introduce a quantity called the plasma frequ ency which for pp = 0.085 protons/fm 3 has the value Wo = 8.0 MeV. The index of refraction due to Thomson scattering can be writte n as Note that for w < 8 Me V the ind ex n is imagin ar y which mean s an in cide nt p hoton will be re fl ec ted as it tries to e nter th e medium. This phe nomenon is analogous to th e re fl ec tion of li ght waves from the surface of a metal due to Thom son scatterin g by the conducti on electrons.
Within the medium the ph oton inte nsity falls off exponentially and the 1/ e pe ne tration de pth is give n by For w < < Wo thi s distan ce is 25 fm .
Above the plasma frequen cy n{w) is real and the photons propagate freely with the index approac hin g unity from below.

Proton Compton Effect
Photon scatte rin g by the proton's c harge is inelas ti c at angles other than 0° a nd therefore a bsorptive . Th e diffe re ntial c ro ss sec tion for scatterin g by a free proton is giv e n by the Klein-Ni shina formula

M 2
In nuclear matter thi s cross section must be multiplied by the Pauli reduction factor. The mome ntum transfer to the proton is 1 q= 2w sin "2 O.

Pair Production
An important absorpti on mec hanis m is th e conv ersion of th e photon to a n elec tron -positron pair in th e Coulomb field of th e proton. The free proton cro ss sec tion for w > > m e is give n b y where me is the mass of the elec tron. The proton is required to take up an average mome ntum transfer of q == me. Thus, the nuclear matter cro ss sec tion per proton is  The imaginary part of the forward sc atter ing amplitude exceeds the real Thomson aplitude above 50 MeV.

Absorption on Correlated Neutron-Proton Pairs
Single particle ejection , which is an important photoabsorption mechanism in finit e nuclei, is not possible in a noninteracting system in which the nucleon has definite linear momentum because e nergy and momentum cannot be simultaneously conserved. However, in an interacting Fermi gas the defect wave function has a spread of mome ntum components that allows two nucleons to absor b a photon and be ejected from the F ermi sea.
An es timate of the absorption cross section of this process can be made by considering the electric dipole transition of a correlated ne utron-proton pair using the s-wave defect wave fun c tion as the initial state and a free p-wave for the final s tate The density of final two-nucleon states is a function 1.0, ,-a' of the final relative momentum p The El matrix element is calculated from th e momentum space wave functions as where E is the photon polarization vector.
A properly normalized initial state is where for convenience we use the Fourier transform of the defect wave function The final state with both particles outside the Fermi momentum sphere is The matrix element is simply evaluated as The total c ross section averaged over photon polarization is then We take the threshold for the reaction to be twice the a verage single particle binding energy, 16 Me V. Thus, photon energy and final relative momentum are related by The resulting cross section is plotted in figure 2. At the peak w = 100 MeV the nuclear matter cross section is about 13 percent of the free deuteron cross section.

Pion Production
At photon energies greater than the mass of the pion w> mrr = 140 MeV the reaction   [4] a shadowing phenomenon which is more complicated than the incoherent sum of individual nucleon events in nuclear matter. Present theories contend that a high energy photon spends a fraction of the time as a spin-one meson (vector dominance model). Mesons propagating in nuclear matter have a large and complex index of refraction [5]. For our purposes, however, it is probably safe to assume that the photo pion absorption cross section per nucleon is no larger than that of the free nucleon at the same photon energy. In this case, the high energy absorption will be dominated by the ever-increasing e+eproduction process.

4 . Photon Interactions in Charge Neutralized Nuclear Matter
If electrons are added to the proton-neutron matter to achieve charge neutrality (Pe= pp=O.085 fm -3) the optical properties are drastically changed. Miiller and Rafelski [6] argue that charge neutralization occurs spontaneously when the Coulomb potentia] exceeds [kF 2 + me )1 /2 = kF = 268 MeV. This occurs for finite nuclei which have more than 10 4 protons.
The plasma frequency of the electrons is much higher than that of the protons because of the smaller electron mass, w~lectron = (47T p e e2/m e ) 1/2 = 342 Me V, with an associated penetration depth less than a fm. As a consequence, any photons with energy less than 342 MeV will be reflected from the surface of a semi· infinite slab of this material.
Photon propagation in the medium becomes possible again for w > 342 MeV. Pair production in the Coulomb fields of the electrons and protons is influenced by the requirement that the ewhich is produced must have a momentum greater than kv. Thus, the threshold for pair production is raised from 2me to kv and the In 2w term in the high energy cross section formula me becomes In ~:. In the threshold region the e+ and eenergies are not shared symmetrically and there will be mutual srreening of the electron and proton Coulomb fields. An estimate of the combined pair production cross section can only be made for w > 2 GeV, viz., The photo pion absorption cross section is unaf· fected by the electron component of nuclear matter. Above 2 Ge V the pair cross section is larger than (JY 7T, but the reverse is probably true between 342 MeV and 2 GeV. Compton scattering on the electrons is smaller than both these cross sections.

Discussion and Conclusions
The main results are displayed in figure 3. Protonneutron matter is dominated by reflective Thomson scattering below 8 MeV and pair production above the proton plasma frequency. All real nuclei are too small to exhibit the low frequency imaginary index of refraction expected of np matter since the largest diameter (14 fm) is still small compared to the penetration depth (25 fm). Also, pair production on finite nuclei occurs coherently in the intense Coulomb field at the edge of the nucleus rather than from individual p:-otons in the interior.
Electron-proton-neutron matter has a much higher plasma frequency so that photons do not penetrate the medium until w > 342 MeV. Pion production from the protons and neutrons dominates the absorption up to about 2 Ge V, then pair production becomes more important; however, both these processes are difficult to calculate accurately.  The ve rti cal lin e al w = 8 Me V marks th e rrC{IUc nc y belo w whi ch ph otons will no t pro paga te in matt er co mposed of protons and ne utrons . The lin e at w = 342 Me V is th e limiting energy for mail er co mposed of elec trons. prot ons, and neutron s. The so li d curve la be led lTpalr is the absorption cross sec tion per proton for e~eproduction in pn matt e r : th e das hed curve fo r epn matter. The curve labeled U -YTr is th e free nucleon photo pion absQ rption cross section. This c ross section is probabl y an upper limit to th e process in nu c lea r mail er Other ph oton int eraction cross secti ons. Compton scatt ering, and nucl eo n pair absorpti on, are less that 1O -29 i c m2 at all energies.