RAPID COMMUNICATIONS
PHYSICAL REVIEW C 79, 031602(R) (2009)
Lattice effect in solid state internal conversion
P´ eter K´ alm´ an and Tam´ as Keszthelyi
Budapest University of Technology and Economics, Department of Experimental Physics,
Budafoki ´ ut 8. F. I.I.10, H-1521 Budapest, Hungary
(Received 12 November 2008; revised manuscript received 5 March 2009; published 30 March 2009)
The effect of the crystal lattice on nuclear fusion reactions p + d →
3
He taking place in internal conversion
channels is studied. Fusionable particles solved in the investigated crystalline material form a sublattice. Fusion
reaction is generated by a flux of incoming fusionable particles. The calculated cross sections are compared with
those of an ordinary fusion reaction. The internal conversion coefficients are also calculated.
DOI: 10.1103/PhysRevC.79.031602 PACS number(s): 23.20.Nx, 25.60.Pj
The enhancement of deuteron fusion-reaction rates in
metallic environments at very low energies has been investi-
gated by various experimental groups [1–5], but the theoretical
explanation of the so-called anomalous screening effect seems
to be still missing [6]. A few years ago we suggested a
possible mechanism, called the solid state internal conversion
process [7], that may be partly responsible for the increase
of unknown origin in the observed fusion rates. In Ref. [7]
the solid was considered only as a dense material and its
crystal-like structure was not taken into account. Now we study
the effect of the crystal lattice in the problem.
The crystal is irradiated by a flux of fusionable particles
(of mass m
1
and of charge z
1
e, where e is the elementary
charge) that may be different from or similar to (e.g., deuteron)
those solved in the metal lattice forming a sublattice. It is
assumed that nuclear fusion reaction takes place between the
incoming particles and the particles of the sublattice. It is also
supposed that the incoming particles interact with the host
crystal and the sublattice via the Coulomb interaction that
assists the fusion reaction between a bound and an incoming
free particle. In what follows the cross section of the Coulomb-
assisted nuclear fusion reaction (solid state internal conversion
process) is calculated taking into account the effect of the
periodic structure of the material.
The Coulomb interaction between the incoming free parti-
cle and the crystal disturbs the stationary state of the crystal
causing competing processes such as ionization, lattice defect
formation, and phonon creation. These processes can be treated
as step by step processes, that follow each other [8]. In the
case investigated it is supposed that the energy and the flux
(the current of the beam) of the incoming particles are low
enough to consider the state of the crystal unchanged in one
step. The effects of screening on fusion reaction [9] and lattice
polarization, which are considered in pycnonuclear fusion rate
calculations [10], are neglected.
The model of the investigated process is the following.
Let us consider a crystalline metallic (e.g., Pd of f.c.c.
lattice) solid (of N lattice points) that already contains N
2
fusionable particles, e.g. deuterons, that form a sublattice (at
the octahedral sites in the case of Pd, N
2
= N in the case
of 1 : 1 deuteron number/metal atom number ratio). In the
calculation deuterated Pd is used as a model material, for this
is the case where one of the largest discrepancies between
theory and experiment was observed [6].
A state of a particle of rest mass m
3
of the host lattice, that
is localized around all of the lattice points, is described by a
Bloch function of the form [11]
ϕ
k
3,i
(r
3
) =
1
√
N
l
s
e
i k
3,i
·l
s
a
3
(r
3
− l
s
− u
3
(l
s
)), (1)
where r
3
is its coordinate, k
3,i
is a wave vector of the first
Brillouin zone (BZ) of the reciprocal lattice, and a
3
(r
3
− l
s
)
is a Wannier function, which is independent of k
3,i
within
the BZ and is well localized around lattice site l
s
. u
3
(l
s
) is
the displacement of the atom located at lattice site l
s
. [The
Coulomb interaction between the incoming free particles and
the crystal may change the energy distribution of the states
of form (1).] In the solid state internal conversion process Pd
ions, deuterons, and electrons can take part as particle 3.
The state of a particle of coordinate r
2
and of rest mass m
2
of the sublattice is also described by a Bloch function of the
form
ϕ
k
2,i
(r
2
) =
1
√
N
2
l
s
e
i k
2,i
·l
s
a
2
(r
2
− l
s
− u
2
(l
s
)), (2)
where u
2
(l
s
) is the displacement of the particle located at lattice
site l
s
. For the Wannier functions a
2
and a
3
a
j
(x) =
β
2
j
π
3/4
e
−
β
2
j
2
x
2
(3)
is used, that is, the wave function of the ground state of a
three-dimensional harmonic oscillator of angular frequency
ω
j
with β
j
=
m
j
ω
j
/¯ h [12]. (The suffixes j = 2, 3 refer
to the sublattice and the host lattice particles, respectively,
throughout the article. It will be seen later that the calculation
is not sensitive to the concrete value of parameter β
3
.)
The initial state
i
of the three particles that participate
in solid state assisted fusion reaction is the product of one-
particle Bloch states of initial wave vectors k
2,i
and k
3,i
and a
Coulomb wave function ϕ
1
(r
1
− r
2
) corresponding to the state
of a sublattice particle, one host particle and one incoming
particle, respectively,
i
= ϕ
k
2,i
(r
2
)ϕ
k
3,i
(r
3
)ϕ
1
(r
1
− r
2
), (4)
where
ϕ
1
(r
1
− r
2
) = e
i k
1
·(r
1
−r
2
)
f (k
1
, r
1
− r
2
)/
√
V. (5)
0556-2813/2009/79(3)/031602(4) 031602-1 ©2009 The American Physical Society