%Neutron Bands Made of Excited Neutron States of Nuclei on the Lattice Mediated by Protons in Transition-Metal Hydrides (to be

1. Introduction

The nuclear structure of isolated nuclei ^A_ZX 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 MeV^1,2). 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 ^A_ZX, or to the state where a neutron and the separated nucleus ^A-1_ZX 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 a_i is treated as independent particles and their quantum states ψ_{n}(x, a_i) are specified by quantum numbers {n} ≡ (n, l, m, s);

ψ_{n}(x, a_i) =ψ_{nlms}(x – a_i,σ). (1)

The wave function of a neutron in a nucleus ^A_ZX, 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

ψ_{nlms}(x – a_i,σ) = c_i e ^–^η^{|r
– ai|} Y_l,m(θ_i,φ_i)χ_s(σ), (2)

whereη≡η(|E|) is a damping factor of the radial wave function depending on the energy assumed for simplicity to be independent of quantum numbers, and (θ_i,φ_i) are angles measured from the lattice point a_i. 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 ^A_ZX is given as:¹⁾

ρ(N,Z,ε)= (6^1/4g₀/12(g₀ε)^5/4)exp((4π²/6)g₀ε)^1/2

(N ≃ Z) (3)

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

g₀ ≡ g(ε_F) = (3/2)(A/ε_F), (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

ε_F /A ≪ E ≪ε_FA^1/3, (5)

whereε_F ≈ 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 MeH_x, 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 PdH_x depends on the composition and that of PdH is a little larger than that of Pd crystal 3.89 Å. 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, φ_p(R–b_j,σ), of a proton in a state specified by quantum numbers p ≡ (n_p,l,m,s_p) at an interstice b_j can have finite probability density at nearby lattice point at a_i, a nearest neighbor of b_j, 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),

Φ_{p_α_}(X₁, X₂, ,Xz,) = Π_jφ{p_j}(R_j –b_j,σ_j), (6)

where {p_α} ≡{p₁, p₂, , p_z}.

The overlapping of the proton wave functionφ{p_j}(R_j – b_j) on the interstice b_j with a nucleon (neutron) wave functionψ{n}(r – a_i), Eq. (1), of an adjacent lattice nucleus at a_i 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;

V_s(r – R) = – V^(s)₀, (|r – R| < b) (7)

= 0, (|r – R > 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 ≪ 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 a_i should be expressed by a Bloch function (omitting the spin part)

ψ_k(r) = Σ_ie^i(kai)ψ_{n}(r – a_i). (8)

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

Ψ_k,{p}(r;R₁, R_{2 ,}R_z)

=ψ_k(r) Φ_{p_α_}(X₁, X₂, ,Xz,) . (9)

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

E_k,{p_α_} = E_{n,p_α_}

+Σ_k_’,i,i’,jexp(–i(ka_i – k’a_i’))v_np(ii’j), (10)

v_np(ii'j)

=Σ_p'(<np;ij|V|n'p';ij><n'p';i'j|V|np;i'j>)/(E_{n',p'} – E_{n,p}),

=Σ_{p’}_≠_{p}P∫dEρ×

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

E_{n,p_α_}= E_{n}^(p)₊Σ_jε_pj, V(r) = V_s(r), (12)

<np;ij|V|n'p';ij> = ∬drdR_jψ*_{n}(r – a_i)φ*{p}(R_j –b_j,)

×V_s(r – R) ψ_{n’}(r – a_i)φ_p’(R_j –b_j,), (13)

where summations over i and i' in (10) are only over the nearest neighbor lattice points a_i and a_i’ of an interstice b_j, ρ_n(E) is a density of states for neutron quantum states, ε_p'p ≡ε_p’ –ε_p, and E ≡E_{n'} – E_{n}. Further, the summation over {p’} reduces to a factor, (n_p+1)(n_p+2), the degeneracy of the energyε_np. E_{n}^(p) is an energy of a neutron in an excited stateψ_{n’}(r – a_i) in a lattice nucleus at a_i when occluded protons are in states {p_α}, and ε_pj in (12) is an energy of a proton in a stateφ_pj(R_j –b_j,) at an interstice b_j. 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 ψ_{n’}(r – a_i) by those determined in the nuclear harmonic oscillator potential in a nucleus to calculate matrix elements (13) in the above equation (11):

ψ_nlms(r,θ,φ,σ)

= R_nl(r)Y_l^m(θ,φ)χ_s(σ), (|m| ≦ l) (14)

E_nlms = (n + 3/2)(h/2π)ω_n+ Δε_lms (15)

whereΔε_lms 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, ω_n is the circular frequency of the harmonic oscillator and Y_l^m(θ,φ) are the spherical harmonics.

In nuclei of palladium isotopes, we can use an excited neutron state 2f_7/2 as shown by shell model calculation with a Woods-Saxon potential¹⁾ for the order of magnitude estimation of (14):

ψ_2f
7/2,_s (r,θ,φ,σ)

= R₅₃(z)Y₃^m(θ,φ)χ_s(σ), (|m|≦ 3) (16)

R₅₃(z) = C_n(32/210)^1/2z^3/2(1 – (2/9)z)e^–z/2, (17)

C_n = 2(8α_n³/π)^1/4, z = 2α_nr², α_n = πm_nω_n/h,

where m_n is the mass of the neutron and ω_n
=41/A^1/3 MeV.⁸⁾

For the interstitial proton wave functionsφ_p’(R_j –b_j,) in PdH, on the other hand, we can use a wave functionφ_1d(R,Θ,Φ) in a lattice harmonic oscillator potential centered at an interstice determined by diffusion data;⁹⁾

φ_p’(R_j) =φ_nplms’(R, Θ,Φ,σ_p)

= ξ_npl(R)Y_lm (Θ,Φ)χ_s(σ_p), (|m|≦l) (18)

ε_n_plm= 2π(n_p + 3/2)hω_p, (19)

φ_1d’(R, Θ,Φ) =ξ_1d(Z)Y₂₀(Θ,Φ), (n = 2) (20)

ξ_1d(Z) = C_p(4/15)^1/2Zexp(–Z/2), (21)

C_p = 2(8α_p³/π)^1/4, Z = 2α_pR², α_p = {m_pπω_p/h}, ω_p = (K/m_p)^1/2,

or by Hermite polynomials H_n(ξ);¹⁰⁾

φ_p’(R_j –b_j, σ_p) = u_nx(x) u_ny(y) u_nz(z) χ_s(σ_p), (22)

u_nx(x) = N_nH_n(αx)exp(–(1/2)α²x²), (23)

α⁴= 4π²m_pK/h², N_n = (α/π ^1/22ⁿn!))^1/2.

where R = (R, Θ,Φ), n_p is an integer, l ≦ n_p and |m| ≦l, ε_nlm is the proton energy of the stateφ_nlm’(R), ω_p = (K/m_p)^1/2, m_p is the mass of the proton, K is the force constant, and n_i(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

K_H = 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>

= – ∬drdR_jR₅₃(z_i)Y_3,0(θ_i,φ_i)χ_1d(Z_j)Y_2,0(Θ_j,Φ_j)

×V_s(r – R_j)R₅₁(z_i)Y_1,0(θ_i,φ_i )χ_2s(Z_j) Y_0,0(Θ_j,Φ_j), (25)

z_i = 2α_n|r – a_i|², Z_j = 2α_p|R_j– b_j|²,

where a_i is a nearest neighbor lattice site of an interstice b_j, K = K_H inα_p in Eq.(22), and (θ_i,φ_i) and (Θ_j,Φ_j) are angles measured from origins at a_i and b_j, respectively.

To estimate an order of magnitude of the effective potential v_np(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 v_np(ii'j) given in Eq.(11) is estimated as follows: the proton wave functionφ_p’(R) is slowly varying in the range of the nuclear force, and the nuclear wave functionψ_n(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>|

≅ ∫ψ_n(r)* ψ_n(r)dr<V>φ_p’(R) *φ_p’(R)Ω (26)

≅ 1×{4/3}πr₀³×|u₂(x_N)|²|u₀(0)|²|u₀(0)|²

= 3.2 ×10^–14eV, (27)

where Ωis the volume of the Pd nucleus, <V> = |V₀^(s)|= 3.5 MeV (Eq.(7)), φ_p’(R) is taken as u₂(x)u₀(y)u₀(z) and x_{N =}1.95Å is the position of the lattice nucleus measured from the interstice.

Putting this value (30) into Eq.(11), we can estimate the effective potential v_np(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:

v_np