Proc. JCF4 (Oct. 17 – 18, 2002, Morioka, Japan)

 

Neutron Drops and Production of the Larger Mass-Number Nuclides in CFP


 

Hideo KOZIMA1)

Physics Department, Portland State University

Portland, OR 97207-0751

1) On leave from Cold Fusion Research Laboratory, Yatsu 597-16, Shizuoka, Shizuoka 421-1202, Japan. E-mail: cf-lab.kozima@nifty.ne.jp


 


Key words; nuclear excited level, neutron-proton interaction, neutron energy band

 

Abstract

Formation of the neutron valence bands (NVB) below zero in transition-metal hydrides is verified by quantum mechanical calculation of interaction between lattice nuclei and occluded protons or deuterons. The local coherence of neutron Bloch waves in the NVB results in formation of high-density neutron liquid (NL) and neutron drops (ND) in boundary regions. The NL and ND interact with lattice nuclei, protons (or deuterons) and minor nuclei in boundary regions to produce cold fusion phenomenon (CFP) in which large change of nucleon and proton numbers of nuclei occur with dissipating channels of liberated energy rather than gamma emission


.

 


1. Introduction

The nuclear structure of isolated nuclei AZX has been thoroughly investigated in about sixty years since the discovery of the atomic nucleus in 1911 in order to achieve fundamental understanding in the energy region up to several hundred MeV1,2). The global features of the exited levels of nucleons and their energy distribution seem to be fairly well described by the Fermi gas mode, while the results have had been mainly confined to light nuclei and a quantitative analysis is plagued with difficulties in the description of the reaction mechanism.1) This is true even now especially for excited levels with energies very close to the zero level; which corresponds to the neutron level with a binding energy of zero in the nucleus AZX, or to the state where a neutron and the separated nucleus A-1ZX remain still. (We use this energy standard in this paper unless otherwise stated.)

Therefore, it is interesting to investigate some phenomena that are directly related with the excited levels of nucleons at around zero energy in medium and heavy nuclei.

In this paper, these features of excited states of nuclei in solids are semi-quantitatively investigated on the knowledge of nuclear structures established in nuclear physics and apply them to cold fusion phenomenon (CFP). We use the Fermi gas model for nucleons in a nucleus throughout this work.

 

2. Excited States of Neutrons and its Density of States in Medium and Heavy Nuclei

It is a common knowledge in nuclear physics that average properties of the excitation spectrum are given by the Fermi gas model as a result of dominance of the particle degrees of freedom over the number of collective modes.1)

In the Fermi gas model, nucleons in a lattice nucleus at ai is treated as independent particles and their quantum states ƒÕ{n}(x, ai) are specified by quantum numbers {n} ß (n, l, m, s);

ƒÕ{n}(x, ai)  =ƒÕ{nlms}(xai, ƒÐ).            (1)

The wave function of a neutron in a nucleus AZX, however, extends far away from the nucleus when the energy E of the state is less than but close to zero and then the wave function outsides the nucleus is approximated by

ƒÕ{nlms}(xai, ƒÐ) = ci e ƒÅ|r – ai| Yl,m(ƒÆi,ƒÓi)ƒÔs(ƒÐ), (2)

whereƒÅ߃Å(|E|) is a damping factor of the radial wave function depending on the energy assumed for simplicity to be independent of quantum numbers, and (ƒÆi,ƒÓi) are angles measured from the lattice point ai. In the following treatment, we use the wave function (1) until we need the wave function (2).

The result of the calculation of the total level density for the Fermi gas in a nucleus AZX is given as:1)

ƒÏ(N,Z,ƒÃ)= (61/4g0/12(g0ƒÃ)5/4)exp((4ƒÎ2/6)g0ƒÃ)1/2

(N Z)        (3)

where ƒÃ is the excitation energy measured from the ground state level and g0 is the one-particle level density at the Fermi energy gF, representing the sum of the proton and neutron level densities

g0 ß g(ƒÃF) = (3/2)(A/ƒÃF),               (4)

for a case Z = N = A/2. These levels seem very sharp and have fairly long lifetime, which we take as an infinite in the following treatment.

The energy range, where the above formula is applicable, is determined by a relation

ƒÃF /A á E áƒÃFA1/3,                   (5)

whereƒÃF 37 MeV for heavy nuclei.1) This relation gives an energy range 0.4 – 170 MeV of applicability of the relation (3) for nuclei with mass numbers A 100.

High density of nuclear levels at high excitation energies, amounts of the order 106 times higher than that corresponding to single-particle motion, has been revealed by densely spaced, sharp resonances in the slow neutron capture reactions and results in formation of the compound nucleus in a nucleus with A 100.1,3) The figure 106 will be increased further by several orders when the energy of the slow neutron capture reactions goes down to  1 eV. In the following discussion, we will take this factor as 109 at its maximum suggested by experimental data for Ag in the range of 2 to 8 MeV3) considering later application to Pd isotopes in the energy range up to 10 MeV.

 

3. Effective Potential for the Super-nuclear Interaction between Neutrons in Adjacent Lattice Nuclei of Metal Hydrides and Deuterides

In the transition-metal hydrides MeHx, on the other hand, the crystal structure is dependent on the concentration x of hydrogen isotopes which can be introduced into the crystal lattice of the metal Me continuously until a definite limit and kept stably there (occluded).4,5) We confine our investigation to crystals of stoichiometric compounds PdH for our object in the following treatment. In this compound, hydrogen atoms occluded in the crystal are ionized and occupy octahedral interstices having six Pd atoms each as nearest neighbors on the crystallographic axes half way of the lattice constant a. The lattice constant a of the compound PdHx depends on the composition and that of PdH is a little larger than that of Pd crystal 3.89 ð. In the following treatment, however, we ignore the dependence of a on the composition x and use the value for Pd crystal as for the compound PdH.

Dynamical behavior of the proton occluded in transition-metal hydrides is described as a harmonic oscillator in its ground and lower excited states. The wave function, ƒÓp(Rbj,ƒÐ), of a proton in a state specified by quantum numbers p ß (np,l,m,sp) at an interstice bj can have finite probability density at nearby lattice point at  ai, a nearest neighbor of bj, especially when the proton is in its excited states. If we ignore mutual interaction of Z protons on different interstices, the total proton wave function may be expressed as a product of wave functions on the interstices (neglecting anti-symmetrization),

ƒ³{pƒ¿}(X1, X2,   ,Xz,) = ƒ®jƒÓ{pj}(Rjbj,ƒÐj),   (6)

where {pƒ¿} ß{p1, p2,      , pz}.

The overlapping of the proton wave functionƒÓ{pj}(Rjbj) on the interstice bj with a nucleon (neutron) wave functionƒÕ{n}(rai), Eq. (1), of an adjacent lattice nucleus at ai results in the proton-neutron interaction through the nuclear force. The nuclear interaction is expressed by a potential whose form is taken, for example, as the square-well type;

Vs(rR) = – V(s)0,  (|rR| < b)              (7)

= 0,       (|rR  > b) 

where V(s)0 3.5 MeV and  b 2.2 ~10–13 cm.6) The choice of this potential out of several possible types does not make a large difference to the result for low energy phenomena we are considering in this paper.

This interaction pulls two neutron states in different lattice nuclei into coupling as shown below that we will call the "super-nuclear interaction." In the following investigation, we concentrate on excited neutrons in lattice nuclei than protons, which needs more energy to be raised to the excited levels with the same energy than neutrons due to the fact Z á N.  (In Pd, Z = 46 and N = 56 – 64.)

Let us consider a neutron in an excited state {n} of one of lattice nuclei. The regularity of the crystal lattice determines the coefficients of the linear combination as required by the Bloch's theorem.7) Then in a periodic potential of lattice nuclei, a neutron in an excited state {n} of a lattice nucleus at ai should be expressed by a Bloch function (omitting the spin part)

ƒÕk(r)  = ƒ°i ei(kai)ƒÕ{n}(rai).              (8)

Therefore, the total wave function of the system composed of a neutron Bloch waveƒÕk(r) and z occluded protons in the state {pƒ¿} = {p1, p2,     , pz,} at interstices is expressed as (omitting spin parts)

ƒµk,{p}(r;R1, R2    , Rz) 

=ƒÕk(r) ƒ³{pƒ¿}(X1, X2,   ,Xz,) .           (9)

The total energy Ek,{pƒ¿} of this system in the second-order perturbation approximation is expressed as follows taking the square well potential for the nuclear interaction:

Ek,{pƒ¿} = E{n,pƒ¿}

+ ƒ°kf,i,if,jexp(–i(kaikfaif))vnp(iifj),  (10)

vnp(ii'j)

=ƒ°p'(<np;ij|V|n'p';ij><n'p';i'j|V|np;i'j>)/(E{n',p'}E{n,p}),

=ƒ°{pf}{p}PçdEƒÏ~

(<np;ij|V|n'p';ij><n'p';i'j|V|np;i'j>)/(E+ƒÃp'p),    (11)

E{n,pƒ¿}  = E{n}(p) + ƒ°jƒÃpj, V(r) = Vs(r),        (12)

<np;ij|V|n'p';ij> = èdrdRjƒÕ*{n}(rai)ƒÓ*{p}(Rjbj,)

~Vs(rR) ƒÕ{nf}(rai)ƒÓpf(Rjbj,),        (13)

where summations over i and i' in (10) are only over the nearest neighbor lattice points ai and aif of an interstice bj, ƒÏn(E) is a density of states for neutron quantum states, ƒÃp'p ߃ÃpfƒÃp, and E ßE{n'} E{n}.  Further, the summation over {pf} reduces to a factor, (np+1)(np+2), the degeneracy of the energyƒÃnp. E{n}(p) is an energy of a neutron in an excited stateƒÕ{nf}(rai) in a lattice nucleus at ai when occluded protons are in states {pƒ¿}, and ƒÃpj in (12) is an energy of a proton in a stateƒÓpj(Rjbj,) at an interstice bj. We ignore, however, p-dependence of E{n}(p) hereafter in this work.

For the neutron wave function (1) in the Fermi gas model, we can describe wave functions ƒÕ{nf}(rai) by those determined in the nuclear harmonic oscillator potential in a nucleus to calculate matrix elements (13) in the above equation (11):

ƒÕnlms(r,ƒÆ,ƒÓ,ƒÐ)

= Rnl(r)Ylm(ƒÆ,ƒÓ)ƒÔs(ƒÐ),  (|m| l)    (14)

Enlms = (n + 3/2)(h/2ƒÎ)ƒÖn + ƒ¢ƒÃlms          (15)

whereƒ¢ƒÃlms expresses the l·s and other coupling energies taken symbolically into consideration to distinguish energies of the states with the same n and different l, m, and s, ƒÖn is the circular frequency of the harmonic oscillator and Ylm(ƒÆ,ƒÓ) are the spherical harmonics.

In nuclei of palladium isotopes, we can use an excited neutron state 2f7/2 as shown by shell model calculation with a Woods-Saxon potential1) for the order of magnitude estimation of (14):

ƒÕ2f 7/2,s (r,ƒÆ,ƒÓ,ƒÐ)

= R53(z)Y3m(ƒÆ,ƒÓ)ƒÔs(ƒÐ), (|m| 3)    (16)

R53(z) = Cn(32/210)1/2z3/2(1 – (2/9)z)e–z/2,      (17)

Cn = 2(8ƒ¿n3/ƒÎ)1/4,  z = 2ƒ¿n r2, ƒ¿n = ƒÎmnƒÖn/h,

where mn is the mass of the neutron and ƒÖn = 41/A1/3 MeV.8)

For the interstitial proton wave functionsƒÓpf(Rjbj,) in PdH, on the other hand, we can use a wave functionƒÓ1d(R,ƒ¦,ƒ³) in a lattice harmonic oscillator potential centered at an interstice determined by diffusion data;9)

ƒÓpf(Rj) =ƒÓnplmsf(R, ƒ¦,ƒ³,ƒÐp)

= ƒÌnpl(R)Ylm (ƒ¦,ƒ³)ƒÔs(ƒÐp),  (|m|l)  (18)

ƒÃnplm = 2ƒÎ(np + 3/2)hƒÖp,                 (19)

ƒÓ1df(R, ƒ¦,ƒ³) =ƒÌ1d(Z)Y20(ƒ¦,ƒ³), (n = 2)    (20)

ƒÌ1d(Z) = Cp(4/15)1/2Zexp(–Z/2),            (21)

Cp = 2(8ƒ¿p3/ƒÎ)1/4, Z = 2ƒ¿pR2, ƒ¿p = {mpƒÎƒÖp/h},  ƒÖp = (K/mp)1/2,

or by Hermite polynomials Hn(ƒÌ);10)

ƒÓpf(Rjbj, ƒÐp) = unx(x) uny(y) unz(z) ƒÔs(ƒÐp),  (22)

unx(x) = NnHn(ƒ¿x)exp(–(1/2)ƒ¿2x2),           (23)

ƒ¿4 = 4ƒÎ2mpK/h2, Nn = (ƒ¿/ƒÎ 1/22nn!))1/2.

where R = (R, ƒ¦,ƒ³), np is an integer, l np and |m| l, ƒÃnlm is the proton energy of the stateƒÓnlmf(R), ƒÖp = (K/mp)1/2, mp is the mass of the proton, K is the force constant, and ni (i = x, y, z) are integers.

The proton wave functions thus determined include already effects of screening by itinerant electrons and electrons bound in atoms, and also the effect of Coulomb repulsion by lattice nuclei.

The analysis based on the diffusion data9) showed that appropriate wave functions for a proton in the NbH is that with n = 2 in the above equation and the corresponding force constant K is given as

 KH = 1.44 ~1019 eV/m2  (NbH)         .  (24)

We use this value for PdH to make an order of magnitude estimation in this paper.

A concrete expression of the matrix element (14) for PdH is expressed as follows using wave functions (15), (19), and others:

<2f{7/2}1d;ij|V|2p{3/2}2s;ij>

= – èdrdRjR53(zi)Y3,0(ƒÆi,ƒÓi)ƒÔ1d(Zj)Y2,0(ƒ¦j,ƒ³j)

~Vs(r  Rj)R51(zi)Y1,0(ƒÆi,ƒÓi )ƒÔ2s(Zj) Y0,0(ƒ¦j,ƒ³j), (25)

zi =  2ƒ¿n|rai|2,  Zj = 2ƒ¿p|Rj bj|2,

where ai is a nearest neighbor lattice site of an interstice bj, K = KH inƒ¿p in Eq.(22), and (ƒÆi,ƒÓi) and (ƒ¦j,ƒ³j) are angles measured from origins at ai and bj, respectively.

To estimate an order of magnitude of the effective potential vnp(ii'j) (11), we utilize the property of the densely spaced excited states explained before and ignore selection rules associated with single configurations. Furthermore, we put the numerator of (11) as a constant and take it as the value of the matrix element (25) for PdH.

Then, the order of magnitude of the effective potential vnp(ii'j) given in Eq.(11) is estimated as follows: the proton wave functionƒÓpf(R) is slowly varying in the range of the nuclear force, and the nuclear wave functionƒÕn(r) is approximated by a delta-function. Then, an order of magnitude of the matrix elements <np;ij|V|n'p';ij> is given as

|<np;ij|V|n'p';ij>|

çƒÕn(r)* ƒÕn(r)dr<V>ƒÓpf(R) *ƒÓpf(R)ƒ¶  (26)

1~{4/3}ƒÎr03~|u2(xN)|2|u0(0)|2|u0(0)|2

= 3.2 ~10–14  eV,                       (27)

where ƒ¶is the volume of the Pd nucleus, <V> = |V0(s)|= 3.5 MeV (Eq.(7)), ƒÓpf(R) is taken as u2(x)u0(y)u0(z) and xN = 1.95ð is the position of the lattice nucleus measured from the interstice.

Putting this value (30) into Eq.(11), we can estimate the effective potential vnp(ii'j) as a function of the principal value of the integration appeared in that equation, assuming the insensitiveness of the matrix elements to the energy:

vnp