Nuclear Physics AZ87 (1977) 501-505 ; © Nortk-BoUand P~lifhüty Co., Muterdant Not to be reprodaoed by photopslnt or mioro®lm wIthont wrütm parmi~lon from the pablisher
A STATISTICAL THEORY OF NUCLEAR NEUTRINO CAPTURE NAOKI ITOH, YASUHARU KOHYAMA and AKIHIKO FUJII Department of Physics, Sophia University, 7 Kioicho, Chiyoda-ku, Tokyo 102, Japan Received 31 January 1977 (Revised 15 March 1977) Ahstract : A statistical theory is proposed for the calwlation of the total capture crosssection of a neutrino by a nucleus. Close references are made to the statistical theory of decay and electron capture developed by Yamada and his collaborators . As a test of the theory, we calculate the total cross section for the capture of a solar neutrino by a "CI nucleus.
1. Introduction Processes of neutrino capture by nuclei play important roles in astrophysics. A well-known example is the capture of a solar neutrino by a 3 'Cl nucleus : v~+ 3'Cl -~ 3'Ar+e -. In this case, the total rapture cross section has been obtained "almost experimentally" by ßahcall') who noticed the use of the experimental ft values of the mirror ß+ decays 3'Ca -" 3 'K+e + +v~. Another important example is a neutrino capture process in a dense stellar core which undergoes a supernova explosion z )" Although its total cross section is small compared with the neutrino scattering cross sections, the neutrino capture process plays an important role in establishing the chemical equilibrium via weak interaction in the dense stellar matter, as noted by Sato 3). Even apart from the astrophysical applications, the study of neutrino capture processes is an interesting theoretical subject by itself However, rather few works have so far been done in this field One of the most frequently referred to, among others, is the work by l3ahcall and Frautschi ~). In their paper, however, their interest was mainly in neutrinos with rather high energy (E z 15 MeV~ For the nuclear capture of low energy neutrinos, the statistical theory of ß-decay developed by Yamada and his collaborators' -8) is applicable. In this paper, we shall extend the statistical method to the nuclear neutrino capture. In setrt. 2 we formulate the theory. In sect. 3 we calculate the total capture cross section ofa solar neutrino by a "Cl nucleus and make a comparison with Bahcall's almost experimental value. 2. Formulation The total cross section for the capture of a neutrino by a nucleus v~+(A, Z) -. 501
N . ITOH et al.
Z+ 1)+e - is given by 9) C\s
Iz + CCv/Z I where G is the Ferrai coupling constant, ~
G = (3.001 f0.002) x 10,1 a,
m is the electron mass, f 1~ and fa~ are the Fenmi and Gamow-Teller nuclear matrix elements corresponding to the transition to the jth nuclear state, p1 and E~ are the momentum and energy of the emitted electron, respectively, and F(Z, E~) is the Ferrai function
with a being the fine structure constant . The first point of the statistical theory lies in the approximation that the summation over many final states in eq. (2.1) is replaced by an integral . For this purpose let us label the final states by their excitation energies E, and introduce a strength function defined by IMd~71 2 = I
~~ I2 ~~,
~ = F, GT,
where cè denotes either a Ferml (1) or a Gamow-Teller (Q) operator, Iflih is the average ofthe nuclear matrix element at the excitation energy E, and n(E) the density of the final states at the energy E. By using the above strength function, we write the expression for the total cross section as
where E~ is the neutrino energy, and Q the Q-value of the capture reaction. Here we have assumed that the parent nucleus is unpolarized, so that Q 3 f~~ (2-~ J I2 I I2~ The second point of the statistical theory is to factori~e the strength function IMn(E)IZ into the single-nucleon strength function Dr,(E, s) :
, ~~ IMn(E7I2 .= Dn(E, e)W(E, s) ~ de,
where E is the energy of the single-nucleon state. The function Dr,(E, E) describes the square of the transition matrix element by the operator Q of a single neutron with energy E imbedded in the initial nucleus into a single proton in the final nucleus with excitation energy E, irrespective of the Pauli exclusion principle. In the Fermi gas model, the exclusion principle is manifested by the constraint factor W(E, E), where
~ being the Fertni level of the initial neutron aggregate. Here we remark that Q+~ corresponds to the Fermi level of the final proton aggregate. We denote by dN/ds the density of the single-nucleon states in the initial nucleus. Furthermore, we assume that the function Dr,(E, E) does not depend on E in the spirit of the impulse approximation, and use the Fermi gas model for the nucleon aggregate. Then we obtain immediately forE
where d~ and Qn represent the peak position and the width ofthe strength function, respectively . These parameters are the expectation values of certain operators in the initial nuclear state. These operators depend also on the characteristics of the nuclear Hamiltonian. In the simplest model where the nuclear force is assumed to be isospin independent (except the Coulomb force and the neutron-proton mass di6erence) these expectation values can be evaluated with the aid of a uniformly charged sphere motel with radius R = 1.3 A} fm, dF = (1.33 ZA -~ -0 .7825) MeV,
oF = 0.145 ZA'~ MeV.
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For the Gamow-Teller matrix element, the assumption of the spin independence of the nuclear force is certainly far from reality. For the sake ofsimplicity we assume that the spin-dependent force simply broadens the strength function but does not shift the peak position, where by experience in the ß~ecay case we may take QN = 6 MeV. The moments do not uniquely determine the functional form . For practical calculation we adopt the explicit form as (i) Gaussian ;
and (ü) exponential :
The integration in (2.6) has to be performed numerically to give the total capture cross section. 3. A test of the theory and disassion In the preceding section, we have given the general method of the calculation of the total cross section in the framework of the stâtistical model. In this section, we shall calculate the total cross section for the capture of a solar neutrino emitted by the process BB -" 8Be*+e + +ve by a 3'Cl nucleus. The rapture process of the neutrinos ofthis origin plays a vital role in the Brookhaven solar neutrino experiment led by Davis lo). Bahcall') has calculated the total capture cross section averaged over the neutrino spectrum by using the experimentally known ft values for the decays 3'Ca -" 3 'K+e+ +ve, which are symmetric to the processes ve + 3 'Cl ~ 3'Ar+e- . He has obtained an almost experimental value (Q)~ _ (1.35f 0.1) x 10 -~z cm2. (3.1) As this is "almost an experimental value, one cannot claim any theoretical estimate to be more accurate than this Rather, one may examine the accuracy of the theor:tical calculations by comparing with this almost experimental cross section. We have applied the statistical theory to this process using the nucleon effective mass parameter M*lM = 0.6 [refs. s" 6)], and obtained the results (6~G
= 1.50 x 10 -~s cm2 ,
for the Gaussian DA(E),
for the exponential Dn(~.
At first sight, the agreement with the almost experimental value (3.1) appears to be surprisingly good, if one takes due account of the statistical nature of the present theory . However, when one further analyses the relative Fermi and Gamow-Teller contributions to the total capture cross section, one finds that the Fermi contribution in the statistical model is about 30 %, while Bahcall's estimate'" t l) gives an approximately 60 ~ Fermi contribution. The reason for this discrepancy might be that the peak of the Gamow-Teller strength function is located at a higher energy than the Fermi peak in the actual nucleus 3'Ar, so that a small part of the Gamow-Teller strength function participates in this neutrino capture process. When one takes into account this situation, one concludes that the statistical theory gives a total capture cross section which is in agreement with the almost experimental value within a factor of 2. Part of the discrepancy might be due to the failure of the uniformly charged sphere model. Nevertheless, this agreement is much better than the ßt decay case s.6). The reason seems to be that in our case the neutrino energy is high (the maximum energy is 14.06 MeV for the neutrino emitted by the process 8B -" eBe*+e+ +ve) and the threshold energy is low (0.81 MeV), so that many excited states in the 3'Ar nucleus participate in the transition, while in the usual ßf decays participating final states are much less. Thus, we have shown that the present statistical theory, though simple enough, can be a powerful method of calculating the total cross section for the capture of a neutrino by a nucleus, ifmany final states participate in the transition. An application of the present theory to the neutrino capture processes in dense stars is in progress. The authors thank H. Tansei and T. Taguchi for their help in numerical computation. References 1) 2) 3) 4) S) 6) 7) 8) 9) 10) 11)
J. N. >~hcall, Phys . Rev. Left. 17 (1966) 398 D. L. Tubbs and D. N. Schramm, Astrophys. J. 201 (1975) 467 K. Sato, Prog . Theor. Phys. 54 (1975) 1325 J. N. ~hcall and S. C. Frautschi, Phys . Rev. 136 (1964) B1S47 K. Takahashi and M. Yamada, Prog. Theor. Phys. 41 (1969) 1470 S. I. Koyama, K. Takahashi and M. Yamada, Prog . Theor. 1?hys. 44 (1970) 663 K. Takahashi, 1?rog. Theor. lfiys. 45 (1971) 1466 Y. Egawa, K. Yokoi and M. Yamada, Frog . Theor. Phys . 54 (1975) 1339 M. Monta, in Beta decay and moon capture (Benjamin, Reading, Massachusetts, 1973) p. 11S J. N. ~hcall and R Davis, Jr., Scicnce 191 (1976) 264 J. N. Bahcall, 1?hys. Rev. 135 (1964) B137