REACTION BARRIER TRANSPARENCY FOR COLD FUSION WITH DEUTERIUM AND HYDROGEN

Transcription

1 REACTION BARRIER TRANSPARENCY FOR COLD FUSION WITH DEUTERIUM AND HYDROGEN Yeong E. Kim, Jin-Hee Yoon Department of Physics, Purdue University West Lafayette, IN 4797 Aexander L. Zubarev Racah Institute of Physics, Hebrew University Jerusaem 9194, Israe Mario Rabinowitz Eectric Power Research Institute Pao Ato, CA 9433 Abstract An improved parametric representation of Couomb barrier penetration is presented. These detaied cacuations are improvements upon the conventionay used Gamow tunneing coefficient. This anaysis yieds a reaction barrier transparency (RBT) which may have singuar ramifications for cod fusion, as we as significant consequences in a wide variety of fusion settings. 1. Introduction Recenty, Kim and Zubarev 1 deveoped a genera and reaistic barrier transmission mode which can accommodate simutaneousy both non-resonance and Couomb barrier transmission resonance contributions. The derivations for both cases wi be presented. The resonance anaysis cuminates in a reaction barrier transparency (RBT) which is due to the eraction of the transmitted and refected waves yieding constructive erference in a narrow energy regime. Athough RBT may have significant consequences for a wide variety of fusion probems, we wi expore cod fusion appications here.. Conventiona Parameterization The conventiona protoca for determining ow-energy (< kev) fusion cross-sections σ(e) is to extrapoate experimenta vaues of σ(e) measured at high energies using the parameterization where T G (E) = exp[ (E G /E) 1/ ], E G = (παz 1 Z ) μc / with the reduced mass μ = m 1 m /( m 1 + m ) and E is the center-of-mass (CM) kinetic energy. The transmission

2 coefficient ( Gamow factor) T G (E) resuts from the approximation E << B (Couomb barrier height). This technique is used for nucei in non-resonance reactions such as in standard soar mode, and magnetic and inertia confinement cacuations. In order to generaize the conventiona Gamow transmission coefficient, we roduce for the fusing system the foowing potentia which consists of an erior square-we nucear potentia and an exterior Couomb repusive potentia, For the potentia barrier given by eq. (), an approximate S-wave ( ) soution for T(E) can be cacuated in the Wentze-Kramers-Briouin (WKB) approximation as 3 where B is the Couomb barrier height, B Z Z e / R and r a is the cassica turning po, Z Z e / r 1 a E. Note that T WKB R (E) is defined ony for E B and that T WKB R 1. The traditiona Gamow transmission coefficient used in eq. (1) can be obtained from eq. (3) with R = (or equivaenty E << B): 1 3. Kim-Zubarev Parameterization T G (E), eq. (4), represents the probabiity of bringing two partices to zero separation distance. This impies that the Couomb barrier Z 1Z e / r aso exists inside the nucear surface of radius R, which is unphysica and unreaistic. In order to accommodate more reaistic transmission coefficients. Kim and Zubarev 1,4 have recenty roduced a more genera parameterization for σ(e) based on the P-matrix parameterization of the fusion reaction S-matrix. To obtain improved and more genera transmission coefficients, we use partia wave soutions of the Schrodinger equation. For the potentia described by eq. (), a genera soution of the radia Schrodinger equation for the exterior wave function in the exterior region (r R) is given by 5

3 where c () () is the Couomb phase shift and u is the compex conjugate of u. F and G are the reguar and irreguar Couomb wave functions normaized asymptoticay r as where γ is the Sommerfed parameter, Z Z E k /. 1 e / and k is reated to E by In terms of the partia wave S-matrix,, in eq. (5), the fusion reaction tota crosssection E is given by 5 r To accommodate the statistica factor and to compensate the two-body approximation invoved in deriving eq. (8), we roduce the partia wave S-factor, S E, which is expected to be neary energy-independent or weaky energy-dependent, and rewrite where S E is the -th partia wave S-factor and In order to determine the partia wave S-matrix as the ogarithmic derivative of the erior wave function in eq. (5), we roduce the P-matrix u r at r = R: where P and P are the rea and imaginary parts of exterior wave function, the P-matrix at r = R is defined as P, respectivey. For the

4 We roduce the P-matrix for () u as where and s are the rea and imaginary parts of ogarithmic derivatives at r = R, i.e., P ext P () p, respectivey. By matching the, we obtain 1,4 In the Kim-Zubarev parameterization of T E, P and P are to be parameterized directy or in terms of a potentia mode wave function for the erior region (r < R). 4. Reaction Barrier Transparency We note that the reaction barrier transparency (RBT), E 1 T, can occur when P and s P. For simpicity, our discussion in this paper wi be imited to the S-wave case, =, in the foowing. Generaization to the cases is straight forward. 4 For the potentia given by eq. (), a genera soution for the erior (r R) wave function is where K / V E with E K /. We roduce two rea parameters and i and write e 1 c. If the owest partia wave ( = ) contribution is expected to be dominant for ow energies ( kev), then the tota cross-section E is given by

5 and T KZ E) 1 ( is given by T KZ, eq. (), is described by four parameters, V, R, and. T KZ contains both non-resonance and resonance contributions, and aso the erference term between them. The four parameters can be determined from the cross-section containing both a resonance part (resonance energy and width) and a non-resonance background. We note that T KZ 1 when RBT condition, K R and s K1R, is satisfied in eq. (). The resonance energy E (for T KZ ( E ) 1) and width are determined by r r the parameters and for fixed vaues of V and R. The resonance behavior of T KZ, generated from fitting E with particuar vaues of parameters, is a reaction barrier transparency (RBT) due to an erpay of Couomb barrier and nucear eraction, and is to be distinguished from conventiona resonances such as narrow neutron ( ) capture resonances, which are primariy due to the nucear eraction. The resonances present in E, which are shown by some reated experiments to be of a non-rbt type, are to be treated by conventiona methods. Very broad resonance behaviors for cross-sections observed in many nucear reactions 6 such as for reactions H(D,p) 3 He, H(D,n) 3 He, 3 He(D,p) 4 He, and 3 H(D,n) 4 He may correspond to RBT resonances and may yied different ow-energy extrapoations from those obtained by the use of the conventiona transmission coefficient, T G E, since the ow-energy tai of the RBT resonance is expected to be different from that of the conventiona case. For the case of a non-resonance cross-section,, and T KZ ( ) E, eq. (), reduces to the resut given by Batt and Weisskopf 5,

6 It shoud be noted that does. T KZ T BW E, eq. (5), does not have a resonance structure whie the resonance part of In the previous parameterizations of E E is parameterized with the Breit-Wigner resonance formua to be subtracted from the experimenta data,3 or incuded in S(E) in eq. (1). 6 The non-resonance formua, eq. (1), is then used to fit the resutant data. Our more genera formua for T KZ, eq. (), with eq. (19), wi aow us to parameterize the experimenta data exhibiting the RBT resonance behavior by the same formua, eq. (19), thus avoiding separate use of the Breit-Wigner formua for subtracting the resonance contribution from E. Furthermore, the erference term between the resonance and non-resonance contributions is automaticay incuded in eqs. (19) and (). The formuation described by eqs. (9), (15), (19), and () is a generaization of eq. (1) and thus can provide a more reaistic and genera parameterization method for ow-energy nucear fusion cross-sections needed for the soar neutrino and astrophysica cacuations, magnetic and inertia confinement fusion cacuations, and ow-energy (cod) fusion rate cacuations. 5. Fusion Rate Estimates with Narrow RBT Since coskr (in eqs. (3) and (4)) and sinkr sin KR 1 (in eq. (3)) satisfies cos KR, can be expressed in terms of K, K1 and K as For the case of K R and K1R Ns (RBT condition) where N > is a rea constant, we obtain using eq. (6) and After determining from eq. (7), can be determined from From eq. (), the maximum vaue of t KZ E is then given by which yieds T max 1 for N = 1 and T max 1otherwise.

7 Assuming that T KZ, eq. (), has a Breit-Wigner resonance form with a width at a resonance energy E, the width at ow energies can be written as 4 N >> 1 Er 7 s Er K1R1 ev. Since ser. 3T G Er and K R NsE r 1, we obtain for using T KZ 4 / N from eq. (3). Since T G E r is very sma near ambient temperature, kt E. 5eV, is aso very narrow; = 1-1 ev and = 1-1 ev r KZ 17 KZ 17 for T ( Er ) 1 TG ( Er ) and T ( Er ) 1 TG ( Er ), respectivey. Precise vaues of for different fusion reactions can ony be determined by experiments at present. KZ For the fusion cross-section E ST E/ E SE S D(D,p)T and D(D,n) 3 He, the fusion rate can be estimated as = 53 kev-barns for Since the conventiona estimate is given by we can concude for the equiibrium Maxwe-Botzmann distribution new conv f( ) at ambient temperature of kt.5 ev. However, non-equiibrium energy sweeping through the narrow RBT may resut in a greaty enhanced fusion rate 7 as in cod fusion experiments. Recent observations 8-11 of anomaous neutron bursts during therma cycing with deuterated high T C superconducting materias may be attributabe to energy sweeping invoving a non-equiibrium state during the superconducting phase transition.

8 6. RBT Mechanism for Other Fusion Reactions In view of our new resut T KZ, eq. () or T (E), eq. (15), it is appropriate to ask whether some fusing systems can support an RBT at ow energies near the fusion threshod. This can ony be answered at present by experiments. It shoud be emphasized that RBT cod fusion is possibe not ony with deuterium but aso with hydrogen since T KZ, eq. (), is appicabe to both cases as ong as the RBT exist in fusing systems invoving deuterium or hydrogen, such as in nucear fusion reactions with the entrance channes, D + D, D + Li, D + Pd, H + D, H + K, etc. Given the RBT mechanism for cod fusion the question remains why fusion products are observed in cod fusion experiments at a much ower eve than commensurate with the observed excess heat. This question can ony be addressed separatey for each fusion reaction since the exit channes are different for each reaction. The anomaous excess heat and tritium production reported in many eectroysis or simiar experiments may not be due to D - D fusion, but may incude nucear fusion with hydrogen and/or impurity 6 7 nucei which are aways present. This scenario and others such as Lid, p Li, Lid, He He, etc. may expain the resuts of excess heat, tritium and neutron production observed in heavy water(with Li) eectroysis experiments. Scenarios for other cases invoving both deuterium and hydrogen may be possibe and need to be investigated. 7. Summary and Concusion Our progressivey more generaized parametric representation of Couomb barrier tunneing yieds significant improvements upon the conventionay used Gamow tunneing coefficient. This anaysis yieds RBT which is due to the eraction of the transmitted and refected quantum waves yieding constructive erference in a narrow energy regime. RBT appears to have important ramifications for cod fusion. References 1. Y. E. Kim and A. L. Zubarev, Couomb Barrier Transmission Resonance for Astrophysica Probems, Purdue Nucear Theory Group Report PNTG-93-8 (August 1993), to be pubished in Modern Physics Letters B.. W. A. Fower, G. R. Caughan, and B. A. Zimmermann, Thermonucear Reaction Rates, Annua Review of Astronomy and Astrophysics, 5, 55 (1967). 3. Y. E. Kim, M. Rabinowitz, J.-H. Yoon, and R. A. Rice, Internationa Journa of Theoretica Physics, 3, 31 (1993). 4. Y. E. Kim and A. L. Zubarev, P-Matrix Parameterization of S-Matrix for Fusion Reaction Cross-Section, Purdue Nucear Theory Group Report PNTG-94-8 (1994), to be pubished. 5. J. M. Batt and V. F. Weisskopf, Theoretica Nucear Physics, Wiey, New York (195), Chapter VIII. 6. G. S. Chuick, Y. E. Kim, R. A. Rice, and M. Rabinowitz, Extended Parameterization of Nucear-Reaction Cross Section for Few-Nuceon Nucei, Nucear Physics, A551, 55 (1993).

9 7. Y. E. Kim, Time-Deayed Apparent Excess Heat Generation in Eectroysis Fusion Experiments, Mod. Phys. Lett. 6, 153 (1991). 8. F. Ceani et a., Search for Neutron Emission from Deuterated High Temperature Superconductors in a Very Low Background Environment, Proc. of the Second Internationa Conference on Cod Fusion, Como, Itay, June 6-Juy 4, 1991 (Itaian Physica Society Conference Proceedings, Vo. 33), p F. Ceani et a., Measurements in the Gran Sasso Laboratory: Evidence for Nucear Effects in Eectroysis with Pd/Ti and in Different Tests with Deuterated High Temperature Superconductors, AIP Conf. Proc. No. 8 (1991), p G. Lipson et a., Possibe Cod Nucear Fusion in Deuterated Ceramic YBa Cu3O7x, Sov. Phys. Dok. 36, 849 (1991). 11. G. Lipson et a., Neutron Generation in YBaCu3O7 Dy High Temperature Superconductors Stimuated by a Superconducting Phase Transition, Sov. Phys. Dok. 38, 119 (1993).