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
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. 0.38 0. 15 306. 30. 0.77 0.93 153. d~fr ee 100. 1.19 4.80 99. v Compton dfl The imaginary part of the forward scattering ampli where the scattered photon energy is tude exceeds the real Thomson aplitude above 50 MeV.
1 W i = W ( 1 -+ Z(1 - cos 0) r 3.4. Absorption on Correlated Neutron-Proton Pairs For incident photon energies small compared to the Single particle ejection, which is an important proton mass, W < < M = 938 Me V, the photon energy photoabsorption mechanism in finite nuclei, is not loss w - W I is small and the free cross section can be possible in a noninteracting system in which the approximated by nucleon has definite linear momentum because energy and momentum cannot be simultaneously conserved. dufree e 2 2 1 However, in an interacting Fermi gas the defect Compton = ( _ ) _ (1+ cos 2 0) . dfl M 2 wave function has a spread of momentum components that allows two nucleons to absorb a photon and be In nuclear matter this cross section must be multi ejected from the Fermi sea. plied by the Pauli reduction factor. The momentum An estimate of the absorption cross section of this transfer to the proton is process can be made by considering the electric 1 dipole transition of a correlated neutron-proton q= 2w sin "2 O. pair using the s-wave defect wave function as the initial state and a free p-wave for the final state 11 The density of final two-nucleon states is a function 1.0, ,-a' of the final relative momentum p
.8
N E .6 u
c The El matrix element is calculated from the momen b~ .4 tum space wave functions as
.2 .Jt(p) = f IjIJ~·kljlid3k where E is the photon polarization vector. °0~--~~~--~--~--~--~--~--~--~~200· w, MeV A properly normalized initial state is FIGURE 2 . Upper right insert shows s-wave defect wave function fo r a soft-core nucleon-nucleon potential. ljIi (k) = (3 p/8)1 /2 . tlljl(k) The curve IIp'l is the cross section per nucleon for the absorption of a photon by a where for convenience we use the Fourier transform correlated neutron-proton pair calculated llsing6.u. of the defect wave function occurs for both neutrons and protons. The cross section for free nucleons reaches a peak at w = 300 MeV of A (k) - _1_ f ikr tlu(r) d3k 0.5 X 10- 27 cm2 (the 3, 3 resonance). After a few uljl - (27T)3/2 e kr . smaller resonances the cross section is essentially constant between 2 Ge V and 20 Ge V at 0.1 X 10 - 27 The final state with both particles outside the Fermi cm 2• One might expect that the nuclear matter cross momentum sphere is section would be the free cross section modified by the Pauli reduction factor R (q) where the nucleon recoil momentum is
The matrix element is simply evaluated as 2m"M 11 /2 q= [ M+m,,(w -m")J . .Jt(p) = (3 p/8) 1/2 (27T)3/2 E· ptlljl(p). However, the A dependence of the high energy ab The total cross section averaged over photon sorption cross section for finite nuclei polarization is then
(J"YA = {AO.75, w= 0.14 to 2 GeV (J" y N A 0.91, w >2 GeV
indicates [4] a shadowing phenomenon which is more We take the threshold for the reaction to be twice the complicated than the incoherent sum of individual a verage single particle binding energy, 16 Me V. nucleon events in nuclear matter. Present theories Thus, photon energy and final relative momentum contend that a high energy photon spends a fraction are related by 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 The resulting cross section is plotted in figure 2. At photon energy. In this case, the high energy absorption the peak w = 100 MeV the nuclear matter cross section will be dominated by the ever-increasing e+e- pro is about 13 percent of the free deuteron cross section. duction process.
3.5. Pion Production 4. Photon Interactions in Charge Neutralized Nuclear Matter At photon energies greater than the mass of the pion w> mrr = 140 MeV the reaction 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
12 and Rafelski [6] argue that charge neutralization occurs nuclei occurs coherently in the intense Coulomb spontaneously when the Coulomb potentia] exceeds field at the edge of the nucleus rather than from [k F2 + me )1/2 = kF = 268 MeV. This occurs for finite individual p:-otons in the interior. nuclei which have more than 10 4 protons. Electron-proton-neutron matter has a much higher The plasma frequency of the electrons is much plasma frequency so that photons do not penetrate higher than that of the protons because of the smaller the medium until w > 342 MeV. Pion production from electron mass, the protons and neutrons dominates the absorption up to about 2 Ge V, then pair production becomes more w~lectron = (47T p e e2 /me ) 1/2 = 342 Me V, important; however, both these processes are difficult to calculate accurately. 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· I ACCESSIBLE infinite slab of this material. / / REGION FOR 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 e- which is produced must INACCESSIBLE have a momentum greater than kv. Thus, the threshold for pair production is raised from 2me to kv and the '"E u In 2w term in the high energy cross section formula me becomes In ~:. In the threshold region the e+ and e energies are not shared symmetrically and there will be mutual srreening of the electron and proton Coulomb fields. An estimate of the combined pair -29 10 production cross section can only be made for w > 2 ~~~~~~--~~~~~~~~~~'-~~~~'~04 GeV, viz., w, MeV
FIGURE 3. Cross sections for dominant photoabsorption processes in nuclear matter.
The vertical line al w = 8 Me V marks the rrC{IUcncy below which photons will not prop agate in matter composed of protons and ne utrons. The line at w = 342 MeV is the limiting The photo pion absorption cross section is unaf· energy for mailer composed of electrons. protons, and neutrons. The so li d curve la beled lT palr is the absorption cross section per proton for e ~ e - production in pn matter: th e dashed fected by the electron component of nuclear matter. curve for epn matter. The curve labeled U -YTr is the free nucleon photo pion absQ rption cross Above 2 Ge V the pair cross section is larger than section. This c ross section is probably an upper limit to the process in nu clear mail er Other photon interaction cross secti ons. Compton scatt ering, and nucleon pair absorption, (JY 7T, but the reverse is probably true between 342 are less that 1O - 29 ic m2 at all energies. MeV and 2 GeV. Compton scattering on the electrons is smaller than both these cross sections. 6. References
5. Discussion and Conclusions [1] Bethe, H. A., Theory of Nuclear Maller in Annual Review of Nuclear Science, Vol. 21, 1971. [2] Siemens, P. J., Nucl. Phys. A141, 225 (1970). The main results are displayed in figure 3. Proton [3] Roy, R. R., and Reed, R. 0., Interactions of Photons and Leptons neutron matter is dominated by reflective Thomson with Matter (Academic Press, 1968). scattering below 8 MeV and pair production above the [4] Weise, W., Hadronic Aspects of Photon-Nucleus Interactions, proton plasma frequency. All real nuclei are too small Physics Reports 136, No.2 (1974). [5] Johnson, M., LAMPF Summer School on Pion-Nucleus Scat to exhibit the low frequency imaginary index of re tering, 1973, LA-5443-C. fraction expected of np matter since the largest [6] Miiller, B., and Rafelski, J., Phys. Rev. Lett. 34,349 (1975). diameter (14 fm) is still small compared to the pene tration depth (25 fm). Also, pair production on finite (Paper 80Al-876)
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