JOU RN AL OF RESEARC H of the Notional Bureau of Standa rds-A. Physics and Chemistry Vol. BOA, Na. 1, January-February 1976 Optical Properties of Nuclear Matter J. S. O'Connell Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 (September 18, 1975) The index of refraction and a bsorptive properties are estimated in nuclear matter consistin g of protons and neutrons and in nuclear matter charge neutra li zed by electrons. Key words: Absorption cross secti on: index of refraction; nuclear matter: opti cal properties; photon; plas ma frequency. 1. Introduction neutrons). The average binding energy per nucleon in nuclear matter (i.e., the coefficient of the term lin ear Nuclear matter is a hypothe ti c substance consisting in number of nucleons) is of interacting neutrons and protons in large enough numbers that the system can be considered infinite. B = -15.68 Me V / nucleon. The properties of this medium are determined by the nucleon·nucleon potential (usually with th e Coulomb It is the goal of nuclear matter theory to re produce forces between the protons omitted to avoid the in· these "observed" properties, p and B, starting from a finite Coulomb potential). Two properties, the equi­ potenti al V(r) betw een two nucleons. A re vi ew of these librium density and binding energy per nucleon, can calculations is given by Bethe [1] I. The Reid soft­ be de rived from measurements on finite nuclei be­ core potential, one of the better phenomenological cause the conditions at the center of a very large nucleon-nucleon potentials, yields PHSC = 0.20 fm - 3 nucleus approximate those of nuclear matter. and B Hsc =-1l.25 MeV in a calculation of the two­ The purpose of this investi gation is to study the body contributions. Estimates of hi gher order contri­ important interaction mechanisms of photons in butions seem to account for discrepancies between nuclear matter as a function of frequency in order to the Reid potential predicti ons and the observables. estimate the photon propagation characteristics in Nuclear matter is usually discussed within the frame­ the medium. In the next section we review some of work of the Fermi gas model. For no interactions a mong the basic facts of nuclear matter. In section 3 the nucleons (V( r) = 0) the individual nucleon wave fun c­ photon-nucleon cross sections for various processes tion is a plane wave of momentum k are computed. These processes are then reexamined in section 4 for the case where electrons are added to l/J(r)=eikr • the medium causing net charge neutralization. Finally, a discussion of the results and conclusions are pre­ These functions are normalized in a unit volume and sented in section 5. are subject to antisymmetrization for identical nucle­ ons. Our units are chosen to have 1i = c = 1. 2. Nuclear Matter The distribution of nucleon momentum in the Fermi gas model is uniform from k = 0 to k = kF (the Fermi The charge density di stribution of finite nuclei as momentum). The maximum momentum kF is related determined by elastic electron scattering indicates to the nucleon density by the nucleon density of heavy nuclei is fairly constant in the interior of the nucleus with a value p=O.170 nucleons/fm 3 . which gives kF = 1.36 fm - I . This value of the Fermi We shall adopt this value for infinite nuclear matter. momentum is also in agreement with inelastic electron 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 I Fi gu res in brackels indica te the lil eralure references at the end of this paper. 9 scattering in the quasi elastic region. This value cor­ because only a fraction R of the nucleons will have an responds to a maximum kinetic energy per particle of initial momentum ksuch that the final momentum will exceed the Fermi momentum Ik + q I > k p . The k2F expression is derived by calculating the excluded 2M=37 MeV volume of the initial and final Fermi momentum spheres shown in figure 1. 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 modi­ fication as derived by Bethe and Goldstone for a s-wave potential is (::2 + k2 ) u (r) = MV(r) u(r) -~ 100 X (r,r') V(r') u(r') dr' 7r 0 where u(r) = kr t/J(r) and FIGURE 1. Initial (left sphere) and final (right sphere) momentum ') sin kdr-r') sin kp(r+r') distributions in a Fermi gas model in which a momentum q is to X ( r,r = ------- r- r' r+ r' . be given each nucleon. Only those nucleons for which I k + q I > k f ' are allowed to make tht: transition. Tht: The solution of the Bethe-Goldstone equation can be fraction R(q) is the ratio of excluded volu me (s haded) to the total (4/31T kF 3). written as a free wave minus a defect part 3. Photon Interactions u(r) =sin kr-~u(r). Consider a photon of frequency w incident on a semi-infinite slab of nuclear matter. The index of Figure 2 shows the defect function [2] for the Reid refraction (ratio of wavelength in vacuum to that in soft-core potential. The function ~u/ k is not sensitive the material) is related to the forward elastic scattering to either the relative momentum, k, or the total momen­ amplitude f( w) by tum kJ + k 2• An important feature of ~u(r) is that it is small by the time r approaches the average inter­ nucleon separation distance, p - I/3 = 1.8 fm. This n2(w) = 1 + 47rp~(w). means that for subsequent collisions nucleons approach W each other in relatively pure plane-wave states. In general, both f and n are complex numbers_ The The defect function normalization is such that imaginary part of f is related to the absorption cross section through the optical theorem 47r Imf(w) = W<T(w). is the probability that a s-wave pair of particles is The real part of f can be calculated from the imaginary outside the Fermi sea. For the Reid soft-core potential part by the use of a dispersion relation, but we will K~ 0.14 summed over all partial waves. not make use of this approach in our estimates. We Th ~ Pauli exclusion principle which prevented now examine the photon interaction mechanisms [3]. nucleon-nucleon scattering to occupied momentum states will also modify photon-nucleon interactions. 3.1. Thomson Scattering If in free space a photon process imparts a momentum transfer q ~o a nucleon, the cross section in nuclear Electromagnetic waves scatter from the free proton matter will be r~duced by a factor (mass M, charge e) with an amplitude 3 q [ 1 ( q ) 2] e 2 R(q)=-- 1-- - , f= = -1.54 X 10- 3 fm, 4 kp 12 kp -M 10 which is independent of frequency. The forward ampli­ The total Compton cross section for a noninteracting tude is the same in nuclear matter because no momen­ Fermi gas is then tum is transferred to the proton. It is convenient to introduce a quantity called the plasma frequency free 2 3 q (du ) 276 (e ) 2 w U NM (w) = J-- -- dfl =--7T - - Com pton 4 k p dfl 105 M kl" For example , u~~:nPto n ( 100 MeV) = 0_73 X 10- 31 cm 2• which for pp = 0.085 protons/fm 3 has the value Wo = 3.3. Pair Production 8.0 MeV. The index of refraction due to Thomson scattering can be written as An important absorption mechanis m is the conver­ sion of th e photon to an electron-positron pair in the Coulomb field of the proton. The free proton cross section for w > > m e is given by Note that for w < 8 MeV the index n is imaginary which means an incident photon will be re fl ected as it tries to enter the medium. This phenomenon is analo­ gous to the re fl ection of li ght waves from the surface of a metal due to Thomson scattering by the conducti on where me is the mass of the electron. The proton is electrons. required to take up an average momentum transfer of Within the medium the photon intensity falls off q == me. Thus, the nuclear matter cross section per exponentially and the 1/ e penetration depth is given by proton is NM 3 m e r a (w) = - - U ree( w) • I)air 4 kF pair For w < < Wo this distance is 25 fm . Above the plasma frequency n{w) is real and the A l/e absorption length can be computed using photons propagate freely with the index approaching unity from below. 3.2. Proton Compton Effect Typi cal values are: Photon scatterin g by the proton's charge is inelasti c (J" NM W dpa1r at angles other than 0° and therefore absorptive. The pair 1mf pa;r (MeV) (x IO - 21 em') (X lO - 3 fm) (fm) differential cross section for scattering by a free proton is giv en by the Klein-Nishina formula 10.