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Quantum tunneling through a fluctuating barrier: Enhancement of cold-fusion rate
Physica A 164 (1990) North-Holland
QUANTUM
147-168
TUNNELING
ENHANCEMENT
THROUGH
OF COLD-FUSION
A FLUCTUATING
BARRIER:
RATE
E. SIMANEK Department of Physics, University of California. Riverside, CA 92-521, USA
Received
25 September
1989
Using the method of functional integration we calculate the rate of tunneling of a quantum particle weakly coupled to a single mode of the heat reservoir. The extremal paths of the steepest descent method are obtained in the adiabatic limit, valid when the reservoir mode frequency is well below the attempt frequency of the tunneling particle. In this limit, the tunneling rate is shown to exhibit a considerable enhancement due to the barrier fluctuation induced by the thermally excited reservoir mode. The results suggest that the rate of cold fusion in deuterated metals can be enhanced by fluctuations of the deuterium-pair distance, induced by thermal phonons of the host metal.
1. Introduction The problem of the quantum decay of a metastable state is of interest in many areas of physics, including low temperature physics, nuclear and chemical physics. The progress in micro-fabrication has facilitated the observation of macroscopic
quantum
first pointed
out by Caldeira
state
corresponding
tunneling
in ultrasmall and Leggett
to a nonzero
strongly affected by the coupling quantum system, which interacts reservoir, they have demonstrated
Josephson
[3] the decay
supercurrent
junctions
[ 1,2].
As
rate of a metastable
in a Josephson
junction
is
to the heat bath. By studying the model of a linearly with the normal modes of the thermal that the decay rate at T = 0 is exponentially
suppressed by the dissipation. The problem of dissipative tunneling has been given a great deal of interest in recent years [4-71 and some theoretical predictions have indeed been confirmed experimentally [S, 91. At sufficiently high temperatures the decay of a metastable state occurs via thermal activation over the intervening potential barrier. The classical thermodynamic theory of thermally induced decay in Josephson junctions is analogous to that for the Brownian motion of a particle in a periodic potential, which has been worked out in a seminal paper by Kramers [lo]. For an overdamped Josephson junction the problem of thermally activated 0378-4371/90/$03.50 (North-Holland)
0
Elsevier
Science
Publishers
B.V.
decay has been treated.
I‘ = lo,, exp( ~ b’,,‘k,, T) the
following
the Kratner~
[ 111. The decay t-ate /‘for
Halpcrin
barrier.
where
proposed.
in which
extension
the correlation
of
time-dependent
time
time
of the velocity
by Ambeg~tokat-
hy the Arrhcniu~
frcquenq
the
of
of the
of’ the barrier
of the “jumping
and
formula
and C;, i4 the height
thuory
tluctuationa
[ 13. 131. It’ the correlation
account
apprcah.
case is given
(I),, is the attempt
an
Recently,
this
ref.
01
[ 111 ha5 hee~i arc token
into
noiac is tnuch longer
barrier
than
particle”.
the calculation\
[ 17, 131SADfASFc predict an enhancement of the tlccay rate over the standard [ 10, I I]. The physics of thi4 cnhancemcnt can hc understood as follow\. result For
barrier.
the
seizes for
where
I‘,, i5 3 cro44ovct-
the decay rate is clctcrmincd
potential In
7‘ ’ I‘,,.
temperaturcs
[I-l].
The
moment this
when
the
opportunity
the correlation shorter
harrier
than
an
instantaneous
Mot-cover,
the inverse
average
barrier
fluctuations.
Bccausc
height.
this
yields
average
of
to a time-independent theory
tnctastable
state
Since
to
the
weakly It
turns
small
by the
out
decay
of
.loscphson that
rate
through
ct’fcct
their
mutual
Coulomb
an effective of /‘on
i4
rate. ot
the
the harrier
that cot-t-e\ponding
T,,.
\,crsioti the
01 the
decav
of
the
Lvith ;I dccav rate which
the
harrier
height
t))
pi-c~cnc‘c ol
find
the
useful
I\
ot- width. the
applications
l‘ot-
at IOM tempcrturcs.
strong
in lvhich
of
should
lat-gc average
in a nonfluctuating
7’
diminished
with
is the cold fusion.
trathct. time
distribution
;I quantum
tunneling
particularly a very
the
dependence
tcmpcrature4
‘/‘,, is
junctions
condition
potential.
fluctuations
this
‘-particle”
cot-relation
of the rate over
by quantum
temperature
for systems
category tunnel
he tnodificd
noise.
the instantaneous
over
i\ to in\,cstigate
[ 12. l.31. For
is dominated
damped
cxpcctcd
work
the ahov~
‘, one observes rate
of the cxponcntial
a\:eragc harrier
[ 1-I]. ;I theory
dissipation
I‘
an cnhunccmcnt
of rd.
crossover
rate
to feel
The
in t.cC.
o~c‘r the
current
is small
if the velocity
instantaneous
of the prcscnt
deca!~ rate
expcctcd
decay
the
height
prohahility.
potential.
detincd
:tcti\,ation
of the critical
hart-icr
a higher
the particle
is
The purpose
with
cnahlcs
which
thermal
as ;I result
fluctuutcs
to jump times
than the average much
latter
tctiiperature
1~ the classical
Kramers
ctltlatlcctlletlts
01
harrier
potential).
An
two nuclei
with
harrict-
the
height
cicca\
intcrcsting
before
very
to
example
mall
fusion
t-ate
(leading
relati\,e reaction
;I
;tt-c
\‘cr-h
in this cnerg
can t:tkc
[ 151. The fusion rate is proportional to the barrier penetration fac‘tot B). For the D, molcculc in fret space the exponent LI = 175 making the high rate of nuclcatbarrier penetration extremely small [ 161. An unexpectedly
place
cxp(
fusion
(at room
cmbcdded that
temperature)
in palladium
the solid-state
barrier. Kecently,
leading
[ 17. IX] and titanium
environment
to a substantinl
Lcggctt
has been reported
and Barym
produces reduction
rcccntly
[IX]. These
;I large
auf
that
hydrogen
isotopes
experiments
suppression
of the exponent
[ tc)] h ;IVC pointed
for
of the
suggest <‘oulotnh
H to it value :I scvcrc
H --. SO.
constraint
is
E. Simgmek / Quantum tunneling through fluctuating barrier
149
imposed on all possible screening mechanisms by the observable binding affinity of 4He atoms to the metal in question. The fact that 3He desorbs from Ti at room temperature indicates that the affinity is well below 1 eV. On the other hand, the required suppression of B to a value below 80 suggests an affinity of over 100 eV. If the observed high fusion rates are caused by barrier fluctuations the later discrepancy may be removed. For example, due to its coupling to the host lattice, the D atom executes vibrations with lattice frequencies. The resulting "slow" fluctuations of the D - D distance produce the needed modulation of the Coulomb barrier. Though the average value of the exponent B is still large (as imposed by the low value of the aHe affinity), the average fusion rate is enhanced due to the exponential dependence of the instantaneous rate on B. In the present work we study the quantum decay rate using the simplest possible model for the reservoir-induced barrier fluctuations. We disregard the dissipation and represent the environment by a single harmonic oscillator which interacts with the tunneling particle through an anharmonic interaction. Our method of analysis is based on the Feynman functional integral formulation of quantum mechanics originally used by Langer [20] and developed further by Coleman [121]. The tunneling particle is assumed to move in a one-dimensional potential which has a single stable metastable minimum. Specifically, we consider the cubic potential which has been found relevant for tunneling in superconducting quantum interference devices [4]. The displacement of the bath oscillator acts to modulate the frequency of the oscillations in the metastable well. To make sure that the modulation does not drive the squared frequency into an unphysical region of negative values, we introduce a limiting factor which makes the interaction term anharmonic in the reservoir coordinate as well. This prevents us using the effective-action method in which the reservoir mode is integrated out from the outset [3]. Instead, we evaluate the path integrals for the partition function by applying the steepest descent method [20] directly to the two-variable problem of the coupled system of the tunneling particle and the reservoir oscillator. Though the nonlinear differential equations for the extremal paths cannot be solved exactly, an approximate solution is possible in the "adiabatic limit", when the frequency of the reservoir oscillator is much smaller than the frequency of the metastable well. In this limit the tunneling particle is able to see the instantaneous rather than the average barrier potential and an enhancement of the decay rate by barrier fluctuations is expected from the analogy with classical decay theory [12, 13]. The paper is organized as follows. In section 2 we introduce the model and present a general path integral formulation for the quantum decay rate. The approximations for the extremal paths are developed in section 3. Sections 4 and 5 are devoted to the evaluation of the imaginary part of the partition
ti. ,{'imhnek
150
Quantum tunneling through [lu('tuating harri('r
function. The real part of the latter is considered in section 6 which also contains the derivation of the f o r m u l a of the e n h a n c e m e n t of the decay rate. The results are discussed and s u m m a r i z e d in section 7.
2. Quantum tunneling rate T h e system u n d e r investigation is a particle moving in an o n e - d i m c n s i o n a l potential of the form
V(q, x)= Ml!~o~,(x) q-~
'~,,q~l,
(l)
where q is the position of the particle of mass M and x is the displacement of the bath oscillator responsible for the m o d u l a t i o n of (o.. Wc assume <(x)
=
+
, (0 n
where w, is the frequency of oscillations of thc particle about q absence of modulation) and
{
.~.
t(x) = x,, Nil
for X, ~< x ~ X,, , for .r -2" X,, , for x ~ X,, .
I) (in the
I3)
The cutoff amplitude X. is chosen so thal the "mlodulation index" A dclined as A
cX( }
~
(4)
(O 0
is smaller than 1. This ensures that the quantity w0(x) stays always positivc, while x is allowed to assume values in the interwd ( ~'-. z). The Lagrangian of the system is L-
I Mgt2
V( q, x) + }rn(,("-
.(J~x'~) ,
(5)
where m and ~Q,, are the mass and resonance frequency of the bath oscillator. T h e potential V(q, x) describes (for x fixed) a metastable potential well. T h e r m a l and q u a n t u m fluctuations allows the q-particle to escape this well. At low t e m p e r a t u r e s ( T < T,), the decay is d o m i n a t e d by q u a n t u m mechanical tunneling [14]. T h e crossover t e m p e r a t u r e T o is given, in the absence of dissipation, by (/i = 1)
151
E. Simdnek / Quantum tunneling through fluctuating barrier
k BT(, = WB/2V ,
(6)
where wB is the barrier frequency, given by the curvature of the potential at the top of the barrier [4]. In what follows, we will consider the decay rate in the quantum regime by assuming that T < T 0. We start from the imaginary time Green's function [3]
q,,x, /3),
/~(qi, qf, / 3 ) = f d x K ( q , ,
(7)
where x(~3 ) = x
K ( q i, qf, x, / 3 ) =
f
q(~)=ql
Dx(r)
.~(()) = x
/3
f Dq(r)exp(-fL[q(r),x(r)ldr),
q(0)=qi
{I
(8)
where L[q(r), x(r)] is the Euclidean version of the Lagrangian (5) (involving an inverted potential) and /3 = I/T. For large/3 the spectral expansion of the space-diagonal Green's function /~'(q, q,/3) is dominated by the lowest energy eigenvalue, which develops a small imaginary part due to the quantum decay. Following refs. [20-22] the quantum decay rate is calculated from eqs. (7) and (8) using the formula
F
2TImlnZ=2TImln(Z where Z = qlim /~(q, q , / 3 ) . ~(}
o +iZ~)-
2TZ I
Zo
(9)
3. Extremal paths The path integral (8) is evaluated using the steepest descent method, adapted to the two-variable {q, x} problem. The Euler-Lagrange equations for the extremal trajectories follow from the Euclidean version of the Lagrangian (5),
Mc df(x)
d2x dr 2
~Q°x- 2m
d2q d,r2
OJo(X) q + ttq 2 = 0.
~
2 d~-- q '
(10)
2
(11)
152
L. .q'i,ul,ek
C)uanlum tu,neling dmmgh /luctualit G, I~artivt
The b o u n d a r y conditions to be satisfied by tile solutions of these equations arc x(it)
-
.v(/3
) -
x,
(121
q((I)
q(/J)
q.
An a p p r o x i m a t e solution of the system of equations ( I ( I ) ;rod ( 1 1) is possible when the frequency m o d u l a t i o n it, slow and weak. Thus wc assume (o,, <~ _¢2,, .
A ~
1
(
.
13
In this case the variable x(r) changes \ c r y slowly in c o m p a r i s o n x~ilh q(r) which (for fixed ~o~,) is described by a bounce solution of ~l narrow width 1/(%. C o n s e q u e n t l y , ltlc quantity ~oi~(v) in cq. ( 11 ) can bc replaced b \ ~o;i(x )
where x i s the valuc o f . v ( r ) at .v
.v(r
r~
],/3 (lhc position of the bounce).
!/~).
(t4)
With this rcplacernenl, lhc standard nomrivial solution ol cq. (11
q~(r)
3,o,](.;:)scch:l!~o,,(.~)ir~/3)1. 2u
is
(151
Eq. (15) c o r r e s p o n d s to a single bOtlllCC situalcd at r ~./3. For A <- 1. it is sufficicnt to calculate A: to first o r d c r ill A. Wrilmg ~(r) v~(r) 4 .v~(r) where .v,,(r) satisfies the c q u a t i o n
dT with the boundary condition x.((I) d:.,-, d;'
d./] G]
_
....
.~,,(/3), wc ha',c from cq. ( I l l )
qi: lTi
"
(171
To solve cq. (17). we take advantage of the sh~w-nlodulation condition (13). which allows us to replacc the strongly p e a k e d function q : ( r ) by a 5-function situated at r - ! / 3 . Thus cq. 17) is reduced t o a single G r e e n ' s funclion equation
de.v, ctr e
*2~,.v~ Na(r
~/3
(IX)
E. ~q'im~nek / Quantum tunneling through fluctuating barrier
153
to be solved with the boundary condition xl(0 ) = x~(/3) = 0. The normalization constant N is according to eqs. (15) and (17) given to first order in h by Mc N = 2 - m F ( x ° ) ./ qB(r) d c = NoF(2o) ,
(19)
where 3 M coo l~, No ---
roll
2
1 f o r - X . ~ < 2 . < ~ X o, 0 otherwise,
F(2")=
and where 2,, = x,,(r = X x,,(r) - sinh(•o/3)
½/3). The
(2O)
solution of eq. (16) is
{sinh(-Oor) + sinh[.Q,,(/3 - r)]} ,
(21)
which implies 2,,
(22)
x sech(½~'2,,fi).
Eq. (18) has the solution
(
| - ~ XI(3" )
N
1 sech(2-Q./3) sinh(.Q,r)
|v~-N~, sech(½n,/3)sinhlg2,(r-/3) L~az,,
forO
!fi,
(23) 1
for ~ fi < r < [3 ,
from which x, : x , ( r : ½fi)
N tanh(!fi~2o). 2~'2,,
(24)
From eqs. (14). (22) and (24) we have N x = x 0 + 2, = x sech( ~$2,,/3) - 777- tanh( ~/352,,).
(25)
4. Fluctuations about extrema
According to eqs. (7-9). the functional integral for Z t is. in the one-bounce approximation,
l:'. ~'im~'mek
154
,'
Quantum tunm'lin~ through ,fluctuatin~ harriet
~(/J)
,/t M I
q
Dq(r) exp.{ Slq. xl}) ..... ., ........ ~ .
Dx(r)
Z I
ff(()l
/(11)
(26)
'/
where j~
] d~- L[q(~).x(T)l.
s[0. xl
(27)
H
The functions x(r) and q(r) in cq. (26) are given by the extremal solution of eq. (10) (to o r d e r a) and the fluctuation about the extremal solution q , ( r ) , respectively. H e n c e
.v(r) = . v ( r ) q(r)
x,,(r) + x , ( r ) ,
q , ( r ) + ((r) ,
where x,,(r) aml x~ (r) arc given in eqs. (21) and (23). respectively. Introducing expressions (28) into cq. (27) wc obtain with use of cq. (5)
Slq, ~1- _Jt d~-l~' M d ~ + V ( q . . . v , ) ] H
t
+
H
I
dr
;I M~ ~ + ~l
a ~V( q. ~,. ) ] ! E ¢tq q qu
+ ] dT ( ~,,,_i-~ + O(,.,-:. t ii
= ,"'u + 5'~ + 3; .
( 29 )
The first term on tile right-hand side of eq. (29) c o r r e s p o n d s - b o u n c e - a c t i o n ' " ,S'~. Using eqs. (11) and (15) we have
to the
.
,
6M
s, = M. d~ O;~ = 5,~ 1"~,(-,)1 ~
.,<,( _
~:O H
1+
2
,
(3())
(O o
where V,, is the barrier height (for c
0).
E. Sim~nek / Quantum tunneling through fluctuating barrier
155
The fluctuation contribution to the path integral (26) is calculated using the expansion [20] :
,7,,&(r),
(31)
n
where the eigenfunctions s%(r) satisfy the equation
and the normalization condition
f ~ , ( r ~ , v ( r ) dr = ¢l~nn.
(33)
0
Introducing the expansion (31) into the Se term of eq. (29) we have S~ = ~M/3 2 (w~ + %)'q~.
(34)
n
The slow-modulation assumption (13) allows us to simplify eq. (32) by replacing x~ by £ = x~.(½/3). Subsequent change of variables y = ½ooo(£)(r - ½/3),
~,,(r) = s%(Y)
(35)
and the use of eqs. (1) and (15) changes eq. (32) into the form
d2
dY2
12sech2y) s~ = o5,,~,, ,
(36)
where 2
o,,, :
"~
-
4[e,, + w , , - w0(x)]
(37)
5. Bound state contribution to Z~ Eq. (36) can be reduced by a further substitution to a hypergeometric equation [23]. The eigenvalues o3n follow from the condition o3 = - ( s - n )
2,
n=0,1,2,...,
(38)
t:...¢~'im~)#u'k
15(~
(),a/ttu,z
:,#,u'/i*~<¢ t/mJu.c.tl /It. t.alilLt~ harpi{'~
where the parameter .v inusl satisfy the condition .s'(s + 1) 12. from which .v 3. For #t O. 1,2, cq. (38) yields ~o,, ¢1, 4, 1, respectively, which {tl-t2 the discrete cigcnvalucs of the thi-cc bound s,l:.llCS t)f cq. (3f~). Introducing these cigenvalucs inlo cct. (37) wc obtain
0 1
for #z = (). for H I . for
11 = ~ --
(3 tj }
"
The eigcnvalucs with #z , 3 belong to the conthluous spcclruln oI oq. {3()). The action 3; . ~i,<'cn b \ the last iorm of cq. (2c)). is c v a l u a l c d tish~17 integration b \ parl,,; as follows:
#1l
]
2 . dr (22 ~ S.~/>.v ) it
7;
m.~ [.C<(/3 )
[Jsing cots. (1(~) and
Ill
.f- (0) I 4 ~
]
cir'(
v
t -~2,i.t<.).t<
.
(41)t
IS) the inlcgral on the right-hand side of cq. (411) can bc
,
7;
.\' ] ctr D(r
.] d r (
\'~,l
',/J) =
17).t ( r \'t
.
Substituting eelS. (21). (23) and 125) l o t t< ~¢ ccllculatu from CCl~,. 41il and (41)
#11,"%';
(42)
The path integral for + Z I Call ilo'¢, bc forllltll;.liCd with ti~,c ()1 ccly,. ( ,~~ ) . (2~). (2()) and (31) as
z,
lm(]
t.V,, S
tl I1 ,'
]
xpt S.. t).
where J is the Jacobian of the transformation (31). l h c Gaussian integration
E. Simgmek / Quantum tunnelingthroughfluctuating barrier
157
over the amplitude of the goldstone mode rh (corresponding to the n = 1 zero eigenvalue) leads to the factor [20, 21] f3
1
I,'~
\1/2
J f dn,=~(~ f o~dr) (~SB) -.
(44)
0
The integration over r/0 [corresponding to the unstable mode with negative "~ 2 eigenvalue ~w0(x)] is analytically continued [20, 21] and produces an imaginary factor,
) 1."-
i
2"rr
2
M/31E,,+ w ,l
(45)
From eqs. (43)-(45) the net fluctuation contribution to Z~ is
i( M~le,,
dT?,, exp(--S~) = ~
2"nS B
~
+ w~l(e 2 + ,og) ,
2"rr
)1/2
M/J(e,, + w~) (46)
6. Enhancement of decay rate
Expression (9) for the decay rate F involves, besides the imaginary part Z t , the real part of the partition function Z., which is given by ~(f3 I=t
Zl,= f dx
f Dx(~')fDq(-r)exp{-S(l')[q,x]}, I:(O)
(47)
where q(r) is represented by the fluctuation about the trivial extremal path q 0 ( r ) = 0 , and x(~-) is the solution of eq. (16). Using the eigenfunction expansion for the fluctuation ~:l°)(r) we have for the paths in (47) x(~) =
x,O), (48)
=
=
~,,
¢,,
t~';,
it
where scl,°)(r) are normalized according to eq. (33) and satisfy the equation (
62 ) + oJ~ /:(c~ = (w~ + ~lo)) ,~lo) dr 2 ~,, ,, / oil
- --
(49)
K. Si#mbwk I Quantwn tunneling through ltuctuatittg barrier
158
~(11)
.
(l)
with the boundary condition a,, (I)) ~;, ( / 3 ) = 1 ) . Using cqs. (48) and ( 4 9 ) the action S("~[q, x] is calcuhtted from eqs. (5). (27) and (42) with the result
s<<"lq,-,1- s<,'''+ ',,w/3
,•
(0)
tli)
it ii
I,o,,~ + ~,, (,7,,)~1,
51)
where
S"~'''
x:m`(2,, tanh( !/3.(2,,)
51 (111
Performing the Gaussian integrations over rl,, . ;
2Tr
=
'
)'~/
,
----~
~
M/3(~,,
....
+,o~)'
)
-
d_v cxp(
S< <'') .
52
where the thrcc lowest eigenwllucs e~<'~, arc a p p r o x i m a t e d by zcro, whictl introduces a negligible error for /3--,~-. In this way the product of the tt " 3 terms can be matched with the continuum product in eq. (46). Using cqs. (43), (46) and (52) wc obtain Z1 Z,,
(2/15"rr)l e/3°)" t d_vcxp( S"'''~
l d.v ''~ ~;' ::R(x)exp[-(Sl~ + S )] ( 53 ) ] 1 t !]1(2_)
where eq. (39) has bccn used for the discretc cigcnvalucs. The function R(x) is the continuum product-ratio, dclincd as
R(x)=
!]
, ,
I
--
( l ~ e
(1~)
,
2
..~o,,
l
"
1+<,,'(o~ )
(54)
In the appendix we derive, following the m e t h o d of k a n g c r [2()1, 15
c]'(_v )
--
(l)
ll
~ -" '
For vanishing coupling constant c. we have, using eqs. (30). (42) and (51). S~(c = 0) -
36Vo 5tO o
(5~)
E. L{'imhnek / Quantum tunneling through fluctuating barrier Sx(c = 0) = S'°).~ •
159 (57)
Hence the integrals in the denominator and n u m e r a t o r of eq. (53) mutually cancel and we are left, as expected, with the standard result for the quantum decay rate of an "uncoupled particle" [22],
r(c =0) =2
T(ZI] / 3Vo\ '/2 exp( \eo/c=o=lZwo~T~w)
36V{, )
5wo/"
(58)
Using eqs. (53)-(58), we obtain for the relative e n h a n c e m e n t of the rate E = - -l'(cI"= 0)
mN2(x))tanh(!¢'(l") I
f dx (1 + (f(£))? 4 exp 36V{,(1 (J~x))~ : _ (m~,ox2 me - 5w,~ + wo , + ~ ~
j dxexp[
mg&x:tanh(~[3.(lo)]
(59)
where the function N(x) is, according to eqs. (19) and (20), given by
N(x)=
N,, 0
for - X o cosh( ½/312o) ~< x ~< X o cosh( ½/3g2,,), otherwise.
(60)
The functional dependence o f f ( Y ) upon x is determined by eqs. (3), (19), (20) and (25). When the magnitude of the second term in eq. (25) exceeds X 0 the function f(x) goes over to a simple form,
f ( x ) = X o sgn[x - X o cosh( ½~o13)]
for a > 1 ,
(61)
where N(} c~ - 2~oX(~} tanh( ½/3ao).
(62)
For c~ < 1, f ( Y ) assumes a somewhat more complicated form. However, we confine ourselves to the case a > 1, which is pertinent to the physical applications of interest. Further simplification of the integrand of eq. (59) can be made by noting that by virtue of eqs. (3), (4) and (13) we have 2
(63)
This allows us to use in eq. (59) a binomial expansion to order cf(Y)l~o~.
I0()
I:..i4mmm'k
. Omtnlum throttling tl to ~gl tluctuamtg harrier
P e r f o r m i n g the i n t e g r a t i o n s o v e r the three s e p a r a t e regions ol the variable .~. we o b t a i n from cq. (50) with the use of cqs. (601 and (~1) t:"
E I + I:~ + E ~ .
(~4)
where
r{x,,V 2~
a)cxpITa~S,<""' )11
lc~
(I
:
I:,
(1
:~ A) cxp( ~ A,S'~,'~''
l:~
(1t
:~ a ) c x p (
cosh(e ~1312,,))1 .
~3 )12f'(.v,,V2y cosh( :'/3.~2,, ))
"A,Su ..... )ll
/'(.~,,\..'2y cosh(i:/3-(2,,)) I .
w h e r e l'(.v) is the n o r m a l p r o b a b i l i t y function d c l i n c d as
l'(.v)
c
~
\, 2"rr -I
it,~
I I.
¢,(,
67
124]
dt .
T h c p a r a m c t c r s 6 and y arc givcn b\'
'~
I t l ,'\' (,
=
4-Q~7tanh( ~/3.Q, ) .
-y = m_C~, tanh( ~/3~2,, ) .
and S'i'~'~ is thc bounce action for c qH)
7.
69 )
O. given by cq. (30). as
36t],
(7())
Discussion
T h e q u a n t i t y t:. givcn by cqs. ( {~4 )-( 07 ). cxhibils ;.ill interesting bchax.ior a s a function of t e m p e r a t u r e . For T : ( ) . t h c a r g u m e n t of the n o r m a l probability function gocs to inlinity, so that 1241 f'(x,,V'2~ cosh( ~,/3.(2,, )) -
/
,l)
, I .
(71}
This implies lhat E,(T
and
0)
E~(T
O)
o
(72)
E. ,~imhnek / Quantum tunneling through fluctuating barrier
E= E2(T=O ) = ( 1 - 7 a ) e x p
(5AS~ '1 2
161
mN~) 40,,
"
(73)
We see from eq. (73) that the e n h a n c e m e n t at T = 0 depends on the sign of the exponent which is determined by the ratio 10AS~" g20
r-
mN~
( ~ ) X ~
-
(x2--~o '
(74)
where (x2)o = (m~20) ~ is the mean-square amplitude of the bath oscillator at T = 0. If r > 1 the exponent is positive and an enhancement E ~> 1 is possible. According to eq. (74) this requires a weak frequency modulation, satisfying the condition
8 h ~< -5- ~(~1 B (x25o
(75)
A physically interesting case corresponds to very large values of 2 (°~ u - For example, the barrier factor for the cold fusion rate in isotopic hydrogen molecules [15] involves an exponent of order S~"= 102. Assuming X~/(x~) = 10, eq. (75) yields for a the condition a<~0.16. Taking a = 0.1 , uvB (°~ = 1 0 2 , we have from eq. (74) r = s_ and eq. (73) then yields an enhancement factor E = 0.82 exp(9.37) = 9.6 × 103. This rather large zero-point enhancement can, however, turn into a suppression of the decay rate if the ratio X~/(x2), becomes smaller than 6.2 (for AS~') = 10). For example, taking X~/(x2)o = 1, a = 0.1, S~ ~) = 102, eq. (74) yields r = 0.16 and eq. (73) predicts a "suppression factor" E = 0.82 e x p ( - 5 . 2 5 ) . This suppression is presumably due to the small overlap of the x-oscillator wave functions, caused by the displacement due to the " b o u n c e " of the q-particle. The size of the cutoff length x 0 is at present uncertain, but one expects it to be (for fusion in condensed matter) of the order of the interatomic distance or less; hence the interval 1 ~< X~/(x 2). ~ 10 is not an unreasonable choice. It should be also pointed out that the condition for the validity of eqs. (64)-(67) is, according to eqs. (61) and (62), S(O)
12
X[~
15 ( x 2 ) 0
(76)
, 2)o and a lower bound o n -v(o) This puts an upper limit on Xo/(x u • For example, when xa/(x2),, 10, eq. (76) restricts u u to values larger than 8. At finite temperatures, all the terms, given by eqs. (65)-(67), contribute to 2 2 the enhancement. Taking Xo/(x )0 = 10, /3,Qo = 1, a = 0.1 and S~ ~t 10 2, we obtain E. = exp(16.8), E 2 = e x p ( - 4 7 . 7 ) and E 3 = e x p ( - 2 6 . 8 ) . In the latter
162
1:. /;imanek
Quantum tunneling through [hwtuating barrier
case. the overlap reduction term , 5 - 72.2 plays an overriding role in the suppression of E,. However, the overall enhancement --cxp(22.8) is vet\ strong, owing to thc thermal excitation of the x-oscillator. The temperature dependence of this enhancement shows a rapid onsct at a tempcraturc 7~=~1o/k~ For T ~ 7 , the enhancement is practically zero. while for T > T it exhibits an approximately linear 7"-dependencc. It should be pointed out, however, that if the tunneling particle interacts simuhancously with a whole set of bath oscillators and the latter exhibit a distribution of rcsonancc frequencies ,Q,,, the resulting temperature depcndencc of the decay rate is expected to deviate from the above rapid onset and generally a smoother increase of the enhancement with thc temperature is expected. The Coulomb barrier intervening in the cold fusion process is obviously very different from our model potential, cq. (1). Nevertheless, the barricr penetration factors share common features which allow us to appply the results of the present calculations, at least qualitatively, to the problem of cold fusion in metals [17, 18]. In fact, the well known WKB expression for the Coulomb barrier factor shows that the exponent B is proportional to (MU) ~ed wherc U and d is the average height and width of the barrier [15]. Similarly, the "'bounce action" S B ( c - 0 ) is, according to eqs. (I) and (5(~), proportional to (MV.)~ ~-qowhere V~,is the barrier height and % - c0~/u is the tunnel distance in the barrier. The sensitivity of the barrier factor to the distancc d is an indication of the difficulty of the tunneling particles to pass through the rcgion of large internuclear distances. Recent estimates show [16] that the reaction rates observed in refs. [17, 18] could be explained by reducing thc equilibrium D - D distance in the metal well below the free-space 0.74 A value. The ab initio density functional calculations of the D - D distance in a hypothetical PdH: crystal show an increase by about 0.2 A over the value of 0.74 A [25]. This large internuclear distance suggest that it cold-fusion reaction of deuterium, via an ordinary tunneling process, is very improbable. Enhancement of cold fusion by screening of thc deuteron charges has been recently considered by Burrows 126]. Such a mechanism is made effective by the fact that the screening length is considerably shorter than the equilibrium D - D distance in the metal. Howevcr, there is a problem with this and all other screening mechanisms, which has been pointed out recently by Leggett and Baym [ 191, As mentioned in our introduction an efficient screening mechanism (needed to suppress the exponent B to B < 80) implies an unrealistic enhancemerit of the binding affinity of a 4He atom to the metal [19]. The barrier-fluctuation enhancement of the quantum tunneling provides an alternative avenue for the theory of cold fusion in metals. A possible origin of the barrier fluctuations is suggested by a previous study on the diffusion of hydrogen dissolved in metals [271. In particular, it has been proposed that the
E. Simanek / Quantum tunneling through fluctuating barrier
163
diffusion process involves quantum mechanical transitions between the lattice sites assisted by host-lattice phonons. The results of these studies show a considerable increase of the tunneling rate above its value in the rigid lattice. Though the barrier height in the diffusion process is much smaller than that for fusion, there is no apparent reason why a host-lattice phonon should not play a role in the modulation of the fusion barrier. The dissolved hydrogen at sites of low symmetry tends to form a covalent bond with the nearest metal atom [25]. The vibrations of the latter can thus be directly transmitted along this rigid bond to the hydrogen atom. In certain special steric situations the vibration amplitude of the H atom may well exceed that of the host metal atom. Such is the case in the bcc Nb crystal containing hydrogen atoms at the interstitial sites of tetrahedral symmetry [28]. Thus we may anticipate that the host-lattice vibrations in the deuterated metals will produce fluctuations in the D - D distance, which can be modeled (recalling that qo = w ~ / u ) as the fluctuations of the w~2, parameter in eq. (1). Large vibrational amplitudes may be necessary to obtain significant enhancements of the tunneling rate. The authors of ref. [18] have not seen any evidence of cold fusion in equilibrated deuterated metals. A possible explanation may be that unusual steric constraints allowing for large D - D vibrational amplitudes are met only during the process of electrolysis. Another possibility is the enhancement by nonequilibrium fluctuations, which is, however, outside the scope of the present work. In summary, we have calculated the rate of the quantum decay of a metastable state in the presence of a fluctuating barrier. The path integral formulation of quantum mechanics of Feynman is applied to the model of a tunneling particle anharmonically coupled to a single harmonic oscillator of the heat bath. Applying the steepest descent method, an approximate solution for the extremal paths is obtained in the limit of a slowly and weakly fluctuating potential. In this approximation, an analytic expression is derived for the enhancement of the decay rate over that of the uncoupled tunneling particle. The temperature dependence of the enhancement factor shows a rapid onset at a temperature of the order of the energy level spacing of the bath oscillator. Above this temperature significant enhancements of the decay rate are possible. The application of the studied model to the case of cold fusion in deuterated metals is discussed. It is suggested that the fluctuations of the Coulomb barrier are induced by the "slow" modulations of the d e u t e r i u m deuterium distance caused by host-lattice phonons.
Acknowledgement The author acknowledges stimulating discussions with R. Gaupsas.
A. ,c;imam'k
1~4
()uanlum :umu'ling :hrotLR,h llm'luatin,~ harric~
Appendix
(Tdcu/ation <~I d i e / , ' o d m ' l R(.v) Following Langcr [20] wc start from the identity • which alloxvs us to express the product R(x) defined in cq. (54) as 2
![
R(x): ,
I ?
+ ~<,<>>.'~o,,
I
cxp(()
(,,\,1)
1 4 E ,".~<~
~
where
(
7. I t d~'lf,'"'(~')
(,,\.2 1
p(~')lh~(l~'.,><,
ii
The unperturhcd and pcrturbed densities of states ~ll-C dclinecl, respectively.'(E r ) = , _"N>, "~, $ ( E '
~,( ( I ) ).
#l
(,,X.3) ps
Introducin~ the dimensionless cncrov variable ¢o [see cc 1. (37)]. L"~ _
41~' + ,o:~-,o,',(x)l
O)
-
(A.4)
•
.J~,(.v )
expression (A.2) can be written, with use of cq. (2).
(
~ .-
d~olp
'(~)
f,(~)l in(i+ '~,~)*h~(l+ 4(x))
.
(A.5)
(Oip
Expressing thc density of states in tcrms of the (;rccn's operator of cq. hlivC
<,,
,,-,
(
'
36)x~,c
)
whcTc
H,
d-','dv:
v:
12 scch:v
(A.7)
Introducing cq. (A.6) into eq. (A.5) and integrating by parts, ~+rC obtain
E. Sirnanek / Quantum tunneling through fluctuating barrier
C=~o
'f
I (
'
d05Im T r l n 1 + H 0 _ 0 5 _ i 0
v
)l'
~ + ~
165
to~i
' (A.8)
where the factor 3 in the second term on the right-hand side of (A.8) comes from the use of the Levinson theorem for the three bound states of eq. (36). The integral of eq. (A.8) can be done by contour integration in the complex 05 plane. We introduce the constant 4M
~Trln
1+ H~,L05 v ,
(A.9)
I"
where F ' is the contour which surrounds the negative real axis from 05 = - 9 to 05 = 0. We note that the integrand of eq. (A.9) has a branch cut along the real axis, extending from 05 = - 9 to to = 2. The trace in eq. (A.9) can be evaluated in terms of the eigenvalues of the equation 1 + H , , ~1
v) 0,,
A,,(05) 0,,,
(A.10)
from which follows:
i f d05 4~ri
05 + 4 ,, =0 I"
=d0+dl+ &+d~,
(A.11)
where
lid
4wi
05 + 4 ,
In A,,(05)
(A.12)
/"
and the constants (~0, C~ and C2 correspond to the three bound states of eq. (36). Eq. (A.10) can be reduced to the hypergeometric equation with eigenvalues 05, satisfying [25] o3=-(s,,-n)
2,
n
0,1,2 ....
(A.13)
The parameter s,,, however, satisfies [in distinction from eq. (36)] the condition 12 1 - A,,(05)
- s,,(s,, + 1).
(A.14)
166
t'~'. .q'imanek . Quantum tunnelittg tt rou
T o e v a l u a t e the c o n s t a n t (',,. wc s u b s t i t u t e . :=0 into eq. ( A . 1 3 ) . which yields s% = V' ~o. I n t r o d u c i n g this result into eq. ( A . 1 4 ) we o b t a i n 12
.l,,(uJ)
1 + o)
(A.I'~)
V'- ~o
F r o m eqs. (A. I I ) a n d ( A . 5 ) we have
1
(-"'
1 d,o
5Wi .
I"
in(l+
-
,o
~
12 ~_). ' --; \,
(A.I(~)
,o
T h e i n t e g r a n d has a p o l e at to -- 4 and a b r a n c h cut on the negative rcal axis with a b r a n c h p o i n t at tD - 9 . F v a l u a t i n g these c o n t r i b u t i o n s to (A.I(~) wc obtain ''/I
2
In ~1
(A.17)
'
Substituting n 1 into eq. ( A . 1 3 ) a c c o r d i n g to eq. ( A . 1 4 ) ,
wc o b t a i n s'~
12
(2+\ /
o~)(1+\ /
co)
1
\,-
¢o, which implies.
(A. 1~4)
F r o m eq. (A. IN) we have :11(¢0 4) --l) a n d h e n c e In l~(oD) has a b r a n c h cut on the n e g a t i v e real axis with a b r a n c h p o i n t at ~o - 4 . This b r a n c h p o i n t h a p p e n s to c o i n c i d e with the p o l e of the i n t e g r a n d of U~. T h e b r a n c h cut c o n t r i b u t i o n to C I is { In 4, while the p o l e c o n t r i b u t e s { In ~ . T h u s we o b t a i n ("l
For n _ ,~(,~)
/ h'l i7 .
( A . 19)
2 eqs. ( A . 1 3 ) and ( A . 1 4 ) give 12
l
(3+,/
~)(2+x/=-,;)
(A.20)
Since ,12(o3 -1) 0, the function In ~12(oJ) has a b r a n c h on the n e g a t i v e real axis e x t e n d i n g from w 1 to zero. This cut c o n t r i b u t e s to (-'~ a value of { In {. T h e r e is also a c o n t r i b u t i o n from the p o l e at ~a - 4 , which a m o u n t s to - ( In 2q , T h u s the net v a l u e o f (-', is ('~ = ~ l n !~ .
(A.21)
E. Simdmek / Quantum tunneling through fluctuating barrier
167
For n ~> 3, all the branch points of In An(o3 ) are on the positive real axis, so that the only contribution to C~ comes from the pole at o3 = - 4 . Using eq. (A.12) we have C~ -
1 2 ,~3 In A,,(o3 = - 4 ) .
(A.22)
For 03 = - 4 , eq. (A.13) gives s n = n + 2. With this value of s,,, eq. (A.14) yields A,,(-4) = (n - 1)(n + 6) (n+Z)(n+3) "
(A.23)
Inserting eq. (A.23) into eq. (A.22) we obtain C~=-
1 (i~i ( n - 1 ) ( n + 6 ) ) ~ l n ,,=3 ( n 7 2 ) ( n + 3 )
1 = 21n14"
(A.24)
Collecting the results (A.17), (A.19), (A.21) and (A.24) we obtain from eq. (A.11) (~ = In ~ .
(A.25)
Introducing eq. (A.25) into eq. (A.8) we finally obtain
el(2) )3,,2, w~ /
C=ln~+ln(l+
(A.26)
which implies, according to eq. (A.1),
,5(,+
)32
References [1] R.F. Voss and R.A. Webb, Phys. Rev. Lett. 47 (1981) 265. [21 L.D. Jackel, J.P. Gordon, E.L. Hu, R.E. Howard, L.A. Fetter, D.M. Tennants, R.W. Epworth and J. Kurkijarui, Phys. Rev. Lett. 47 (1981) 697. [3] A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46 (1981) 211. [4] A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y.) 149 (1983) 374. [5] A.J. Bray and M.A. Moore, Phys. Rev. Lett. 49 (1982) 1546. [6] S. Chakravarty, Phys. Rev. Lett. 49 (1982) 681. [7] H. Grabert and V. Weiss, Phys. Rev. Lett. 53 (1984) 1787.
I:. ,i,'imanek
Quantum tunm'ling through ttuctmttint: I~arrtcr
[~J S. W a s h b u r n and R . A . W c b b , A n n . NY Acad. Sc. 4~41) (1O:lii~. I~J;~£). c h 3. 1_~41 .~,. AbranlOWilz alld J.:\. Sic'gull. Jhlldb,.>ok o l l%,|clthclll~itlcLiI J:ullclJoll~, (I.:.S (lo\t_,ll/lil. Publ. ()flicc, W a s h i n g t o n , l)('. 1c)64l, oh. 27. J251 Z. Still alld l). T o m a n c k . Phys. Roy. [,oft. f~3 (]UNtJ) 5u 12{~i A. llurrm~s. Phys. l~.cv. I>> 40 (10
QUANTUM
147-168
TUNNELING
ENHANCEMENT
THROUGH
OF COLD-FUSION
A FLUCTUATING
BARRIER:
RATE
E. SIMANEK Department of Physics, University of California. Riverside, CA 92-521, USA
Received
25 September
1989
Using the method of functional integration we calculate the rate of tunneling of a quantum particle weakly coupled to a single mode of the heat reservoir. The extremal paths of the steepest descent method are obtained in the adiabatic limit, valid when the reservoir mode frequency is well below the attempt frequency of the tunneling particle. In this limit, the tunneling rate is shown to exhibit a considerable enhancement due to the barrier fluctuation induced by the thermally excited reservoir mode. The results suggest that the rate of cold fusion in deuterated metals can be enhanced by fluctuations of the deuterium-pair distance, induced by thermal phonons of the host metal.
1. Introduction The problem of the quantum decay of a metastable state is of interest in many areas of physics, including low temperature physics, nuclear and chemical physics. The progress in micro-fabrication has facilitated the observation of macroscopic
quantum
first pointed
out by Caldeira
state
corresponding
tunneling
in ultrasmall and Leggett
to a nonzero
strongly affected by the coupling quantum system, which interacts reservoir, they have demonstrated
Josephson
[3] the decay
supercurrent
junctions
[ 1,2].
As
rate of a metastable
in a Josephson
junction
is
to the heat bath. By studying the model of a linearly with the normal modes of the thermal that the decay rate at T = 0 is exponentially
suppressed by the dissipation. The problem of dissipative tunneling has been given a great deal of interest in recent years [4-71 and some theoretical predictions have indeed been confirmed experimentally [S, 91. At sufficiently high temperatures the decay of a metastable state occurs via thermal activation over the intervening potential barrier. The classical thermodynamic theory of thermally induced decay in Josephson junctions is analogous to that for the Brownian motion of a particle in a periodic potential, which has been worked out in a seminal paper by Kramers [lo]. For an overdamped Josephson junction the problem of thermally activated 0378-4371/90/$03.50 (North-Holland)
0
Elsevier
Science
Publishers
B.V.
decay has been treated.
I‘ = lo,, exp( ~ b’,,‘k,, T) the
following
the Kratner~
[ 111. The decay t-ate /‘for
Halpcrin
barrier.
where
proposed.
in which
extension
the correlation
of
time-dependent
time
time
of the velocity
by Ambeg~tokat-
hy the Arrhcniu~
frcquenq
the
of
of the
of’ the barrier
of the “jumping
and
formula
and C;, i4 the height
thuory
tluctuationa
[ 13. 131. It’ the correlation
account
apprcah.
case is given
(I),, is the attempt
an
Recently,
this
ref.
01
[ 111 ha5 hee~i arc token
into
noiac is tnuch longer
barrier
than
particle”.
the calculation\
[ 17, 131SADfASFc predict an enhancement of the tlccay rate over the standard [ 10, I I]. The physics of thi4 cnhancemcnt can hc understood as follow\. result For
barrier.
the
seizes for
where
I‘,, i5 3 cro44ovct-
the decay rate is clctcrmincd
potential In
7‘ ’ I‘,,.
temperaturcs
[I-l].
The
moment this
when
the
opportunity
the correlation shorter
harrier
than
an
instantaneous
Mot-cover,
the inverse
average
barrier
fluctuations.
Bccausc
height.
this
yields
average
of
to a time-independent theory
tnctastable
state
Since
to
the
weakly It
turns
small
by the
out
decay
of
.loscphson that
rate
through
ct’fcct
their
mutual
Coulomb
an effective of /‘on
i4
rate. ot
the
the harrier
that cot-t-e\ponding
T,,.
\,crsioti the
01 the
decav
of
the
Lvith ;I dccav rate which
the
harrier
height
t))
pi-c~cnc‘c ol
find
the
useful
I\
ot- width. the
applications
l‘ot-
at IOM tempcrturcs.
strong
in lvhich
of
should
lat-gc average
in a nonfluctuating
7’
diminished
with
is the cold fusion.
trathct. time
distribution
;I quantum
tunneling
particularly a very
the
dependence
tcmpcrature4
‘/‘,, is
junctions
condition
potential.
fluctuations
this
‘-particle”
cot-relation
of the rate over
by quantum
temperature
for systems
category tunnel
he tnodificd
noise.
the instantaneous
over
i\ to in\,cstigate
[ 12. l.31. For
is dominated
damped
cxpcctcd
work
the ahov~
‘, one observes rate
of the cxponcntial
a\:eragc harrier
[ 1-I]. ;I theory
dissipation
I‘
an cnhunccmcnt
of rd.
crossover
rate
to feel
The
in t.cC.
o~c‘r the
current
is small
if the velocity
instantaneous
of the prcscnt
deca!~ rate
expcctcd
decay
the
height
prohahility.
potential.
detincd
:tcti\,ation
of the critical
hart-icr
a higher
the particle
is
The purpose
with
cnahlcs
which
thermal
as ;I result
fluctuutcs
to jump times
than the average much
latter
tctiiperature
1~ the classical
Kramers
ctltlatlcctlletlts
01
harrier
potential).
An
two nuclei
with
harrict-
the
height
cicca\
intcrcsting
before
very
to
example
mall
fusion
t-ate
(leading
relati\,e reaction
;I
;tt-c
\‘cr-h
in this cnerg
can t:tkc
[ 151. The fusion rate is proportional to the barrier penetration fac‘tot B). For the D, molcculc in fret space the exponent LI = 175 making the high rate of nuclcatbarrier penetration extremely small [ 161. An unexpectedly
place
cxp(
fusion
(at room
cmbcdded that
temperature)
in palladium
the solid-state
barrier. Kecently,
leading
[ 17. IX] and titanium
environment
to a substantinl
Lcggctt
has been reported
and Barym
produces reduction
rcccntly
[IX]. These
;I large
auf
that
hydrogen
isotopes
experiments
suppression
of the exponent
[ tc)] h ;IVC pointed
for
of the
suggest <‘oulotnh
H to it value :I scvcrc
H --. SO.
constraint
is
E. Simgmek / Quantum tunneling through fluctuating barrier
149
imposed on all possible screening mechanisms by the observable binding affinity of 4He atoms to the metal in question. The fact that 3He desorbs from Ti at room temperature indicates that the affinity is well below 1 eV. On the other hand, the required suppression of B to a value below 80 suggests an affinity of over 100 eV. If the observed high fusion rates are caused by barrier fluctuations the later discrepancy may be removed. For example, due to its coupling to the host lattice, the D atom executes vibrations with lattice frequencies. The resulting "slow" fluctuations of the D - D distance produce the needed modulation of the Coulomb barrier. Though the average value of the exponent B is still large (as imposed by the low value of the aHe affinity), the average fusion rate is enhanced due to the exponential dependence of the instantaneous rate on B. In the present work we study the quantum decay rate using the simplest possible model for the reservoir-induced barrier fluctuations. We disregard the dissipation and represent the environment by a single harmonic oscillator which interacts with the tunneling particle through an anharmonic interaction. Our method of analysis is based on the Feynman functional integral formulation of quantum mechanics originally used by Langer [20] and developed further by Coleman [121]. The tunneling particle is assumed to move in a one-dimensional potential which has a single stable metastable minimum. Specifically, we consider the cubic potential which has been found relevant for tunneling in superconducting quantum interference devices [4]. The displacement of the bath oscillator acts to modulate the frequency of the oscillations in the metastable well. To make sure that the modulation does not drive the squared frequency into an unphysical region of negative values, we introduce a limiting factor which makes the interaction term anharmonic in the reservoir coordinate as well. This prevents us using the effective-action method in which the reservoir mode is integrated out from the outset [3]. Instead, we evaluate the path integrals for the partition function by applying the steepest descent method [20] directly to the two-variable problem of the coupled system of the tunneling particle and the reservoir oscillator. Though the nonlinear differential equations for the extremal paths cannot be solved exactly, an approximate solution is possible in the "adiabatic limit", when the frequency of the reservoir oscillator is much smaller than the frequency of the metastable well. In this limit the tunneling particle is able to see the instantaneous rather than the average barrier potential and an enhancement of the decay rate by barrier fluctuations is expected from the analogy with classical decay theory [12, 13]. The paper is organized as follows. In section 2 we introduce the model and present a general path integral formulation for the quantum decay rate. The approximations for the extremal paths are developed in section 3. Sections 4 and 5 are devoted to the evaluation of the imaginary part of the partition
ti. ,{'imhnek
150
Quantum tunneling through [lu('tuating harri('r
function. The real part of the latter is considered in section 6 which also contains the derivation of the f o r m u l a of the e n h a n c e m e n t of the decay rate. The results are discussed and s u m m a r i z e d in section 7.
2. Quantum tunneling rate T h e system u n d e r investigation is a particle moving in an o n e - d i m c n s i o n a l potential of the form
V(q, x)= Ml!~o~,(x) q-~
'~,,q~l,
(l)
where q is the position of the particle of mass M and x is the displacement of the bath oscillator responsible for the m o d u l a t i o n of (o.. Wc assume <(x)
=
+
, (0 n
where w, is the frequency of oscillations of thc particle about q absence of modulation) and
{
.~.
t(x) = x,, Nil
for X, ~< x ~ X,, , for .r -2" X,, , for x ~ X,, .
I) (in the
I3)
The cutoff amplitude X. is chosen so thal the "mlodulation index" A dclined as A
cX( }
~
(4)
(O 0
is smaller than 1. This ensures that the quantity w0(x) stays always positivc, while x is allowed to assume values in the interwd ( ~'-. z). The Lagrangian of the system is L-
I Mgt2
V( q, x) + }rn(,("-
.(J~x'~) ,
(5)
where m and ~Q,, are the mass and resonance frequency of the bath oscillator. T h e potential V(q, x) describes (for x fixed) a metastable potential well. T h e r m a l and q u a n t u m fluctuations allows the q-particle to escape this well. At low t e m p e r a t u r e s ( T < T,), the decay is d o m i n a t e d by q u a n t u m mechanical tunneling [14]. T h e crossover t e m p e r a t u r e T o is given, in the absence of dissipation, by (/i = 1)
151
E. Simdnek / Quantum tunneling through fluctuating barrier
k BT(, = WB/2V ,
(6)
where wB is the barrier frequency, given by the curvature of the potential at the top of the barrier [4]. In what follows, we will consider the decay rate in the quantum regime by assuming that T < T 0. We start from the imaginary time Green's function [3]
q,,x, /3),
/~(qi, qf, / 3 ) = f d x K ( q , ,
(7)
where x(~3 ) = x
K ( q i, qf, x, / 3 ) =
f
q(~)=ql
Dx(r)
.~(()) = x
/3
f Dq(r)exp(-fL[q(r),x(r)ldr),
q(0)=qi
{I
(8)
where L[q(r), x(r)] is the Euclidean version of the Lagrangian (5) (involving an inverted potential) and /3 = I/T. For large/3 the spectral expansion of the space-diagonal Green's function /~'(q, q,/3) is dominated by the lowest energy eigenvalue, which develops a small imaginary part due to the quantum decay. Following refs. [20-22] the quantum decay rate is calculated from eqs. (7) and (8) using the formula
F
2TImlnZ=2TImln(Z where Z = qlim /~(q, q , / 3 ) . ~(}
o +iZ~)-
2TZ I
Zo
(9)
3. Extremal paths The path integral (8) is evaluated using the steepest descent method, adapted to the two-variable {q, x} problem. The Euler-Lagrange equations for the extremal trajectories follow from the Euclidean version of the Lagrangian (5),
Mc df(x)
d2x dr 2
~Q°x- 2m
d2q d,r2
OJo(X) q + ttq 2 = 0.
~
2 d~-- q '
(10)
2
(11)
152
L. .q'i,ul,ek
C)uanlum tu,neling dmmgh /luctualit G, I~artivt
The b o u n d a r y conditions to be satisfied by tile solutions of these equations arc x(it)
-
.v(/3
) -
x,
(121
q((I)
q(/J)
q.
An a p p r o x i m a t e solution of the system of equations ( I ( I ) ;rod ( 1 1) is possible when the frequency m o d u l a t i o n it, slow and weak. Thus wc assume (o,, <~ _¢2,, .
A ~
1
(
.
13
In this case the variable x(r) changes \ c r y slowly in c o m p a r i s o n x~ilh q(r) which (for fixed ~o~,) is described by a bounce solution of ~l narrow width 1/(%. C o n s e q u e n t l y , ltlc quantity ~oi~(v) in cq. ( 11 ) can bc replaced b \ ~o;i(x )
where x i s the valuc o f . v ( r ) at .v
.v(r
r~
],/3 (lhc position of the bounce).
!/~).
(t4)
With this rcplacernenl, lhc standard nomrivial solution ol cq. (11
q~(r)
3,o,](.;:)scch:l!~o,,(.~)ir~/3)1. 2u
is
(151
Eq. (15) c o r r e s p o n d s to a single bOtlllCC situalcd at r ~./3. For A <- 1. it is sufficicnt to calculate A: to first o r d c r ill A. Wrilmg ~(r) v~(r) 4 .v~(r) where .v,,(r) satisfies the c q u a t i o n
dT with the boundary condition x.((I) d:.,-, d;'
d./] G]
_
....
.~,,(/3), wc ha',c from cq. ( I l l )
qi: lTi
"
(171
To solve cq. (17). we take advantage of the sh~w-nlodulation condition (13). which allows us to replacc the strongly p e a k e d function q : ( r ) by a 5-function situated at r - ! / 3 . Thus cq. 17) is reduced t o a single G r e e n ' s funclion equation
de.v, ctr e
*2~,.v~ Na(r
~/3
(IX)
E. ~q'im~nek / Quantum tunneling through fluctuating barrier
153
to be solved with the boundary condition xl(0 ) = x~(/3) = 0. The normalization constant N is according to eqs. (15) and (17) given to first order in h by Mc N = 2 - m F ( x ° ) ./ qB(r) d c = NoF(2o) ,
(19)
where 3 M coo l~, No ---
roll
2
1 f o r - X . ~ < 2 . < ~ X o, 0 otherwise,
F(2")=
and where 2,, = x,,(r = X x,,(r) - sinh(•o/3)
½/3). The
(2O)
solution of eq. (16) is
{sinh(-Oor) + sinh[.Q,,(/3 - r)]} ,
(21)
which implies 2,,
(22)
x sech(½~'2,,fi).
Eq. (18) has the solution
(
| - ~ XI(3" )
N
1 sech(2-Q./3) sinh(.Q,r)
|v~-N~, sech(½n,/3)sinhlg2,(r-/3) L~az,,
forO
!fi,
(23) 1
for ~ fi < r < [3 ,
from which x, : x , ( r : ½fi)
N tanh(!fi~2o). 2~'2,,
(24)
From eqs. (14). (22) and (24) we have N x = x 0 + 2, = x sech( ~$2,,/3) - 777- tanh( ~/352,,).
(25)
4. Fluctuations about extrema
According to eqs. (7-9). the functional integral for Z t is. in the one-bounce approximation,
l:'. ~'im~'mek
154
,'
Quantum tunm'lin~ through ,fluctuatin~ harriet
~(/J)
,/t M I
q
Dq(r) exp.{ Slq. xl}) ..... ., ........ ~ .
Dx(r)
Z I
ff(()l
/(11)
(26)
'/
where j~
] d~- L[q(~).x(T)l.
s[0. xl
(27)
H
The functions x(r) and q(r) in cq. (26) are given by the extremal solution of eq. (10) (to o r d e r a) and the fluctuation about the extremal solution q , ( r ) , respectively. H e n c e
.v(r) = . v ( r ) q(r)
x,,(r) + x , ( r ) ,
q , ( r ) + ((r) ,
where x,,(r) aml x~ (r) arc given in eqs. (21) and (23). respectively. Introducing expressions (28) into cq. (27) wc obtain with use of cq. (5)
Slq, ~1- _Jt d~-l~' M d ~ + V ( q . . . v , ) ] H
t
+
H
I
dr
;I M~ ~ + ~l
a ~V( q. ~,. ) ] ! E ¢tq q qu
+ ] dT ( ~,,,_i-~ + O(,.,-:. t ii
= ,"'u + 5'~ + 3; .
( 29 )
The first term on tile right-hand side of eq. (29) c o r r e s p o n d s - b o u n c e - a c t i o n ' " ,S'~. Using eqs. (11) and (15) we have
to the
.
,
6M
s, = M. d~ O;~ = 5,~ 1"~,(-,)1 ~
.,<,( _
~:O H
1+
2
,
(3())
(O o
where V,, is the barrier height (for c
0).
E. Sim~nek / Quantum tunneling through fluctuating barrier
155
The fluctuation contribution to the path integral (26) is calculated using the expansion [20] :
,7,,&(r),
(31)
n
where the eigenfunctions s%(r) satisfy the equation
and the normalization condition
f ~ , ( r ~ , v ( r ) dr = ¢l~nn.
(33)
0
Introducing the expansion (31) into the Se term of eq. (29) we have S~ = ~M/3 2 (w~ + %)'q~.
(34)
n
The slow-modulation assumption (13) allows us to simplify eq. (32) by replacing x~ by £ = x~.(½/3). Subsequent change of variables y = ½ooo(£)(r - ½/3),
~,,(r) = s%(Y)
(35)
and the use of eqs. (1) and (15) changes eq. (32) into the form
d2
dY2
12sech2y) s~ = o5,,~,, ,
(36)
where 2
o,,, :
"~
-
4[e,, + w , , - w0(x)]
(37)
5. Bound state contribution to Z~ Eq. (36) can be reduced by a further substitution to a hypergeometric equation [23]. The eigenvalues o3n follow from the condition o3 = - ( s - n )
2,
n=0,1,2,...,
(38)
t:...¢~'im~)#u'k
15(~
(),a/ttu,z
:,#,u'/i*~<¢ t/mJu.c.tl /It. t.alilLt~ harpi{'~
where the parameter .v inusl satisfy the condition .s'(s + 1) 12. from which .v 3. For #t O. 1,2, cq. (38) yields ~o,, ¢1, 4, 1, respectively, which {tl-t2 the discrete cigcnvalucs of the thi-cc bound s,l:.llCS t)f cq. (3f~). Introducing these cigenvalucs inlo cct. (37) wc obtain
0 1
for #z = (). for H I . for
11 = ~ --
(3 tj }
"
The eigcnvalucs with #z , 3 belong to the conthluous spcclruln oI oq. {3()). The action 3; . ~i,<'cn b \ the last iorm of cq. (2c)). is c v a l u a l c d tish~17 integration b \ parl,,; as follows:
#1l
]
2 . dr (22 ~ S.~/>.v ) it
7;
m.~ [.C<(/3 )
[Jsing cots. (1(~) and
Ill
.f- (0) I 4 ~
]
cir'(
v
t -~2,i.t<.).t<
.
(41)t
IS) the inlcgral on the right-hand side of cq. (411) can bc
,
7;
.\' ] ctr D(r
.] d r (
\'~,l
',/J) =
17).t ( r \'t
.
Substituting eelS. (21). (23) and 125) l o t t< ~¢ ccllculatu from CCl~,. 41il and (41)
#11,"%';
(42)
The path integral for + Z I Call ilo'¢, bc forllltll;.liCd with ti~,c ()1 ccly,. ( ,~~ ) . (2~). (2()) and (31) as
z,
lm(]
t.V,, S
tl I1 ,'
]
xpt S.. t).
where J is the Jacobian of the transformation (31). l h c Gaussian integration
E. Simgmek / Quantum tunnelingthroughfluctuating barrier
157
over the amplitude of the goldstone mode rh (corresponding to the n = 1 zero eigenvalue) leads to the factor [20, 21] f3
1
I,'~
\1/2
J f dn,=~(~ f o~dr) (~SB) -.
(44)
0
The integration over r/0 [corresponding to the unstable mode with negative "~ 2 eigenvalue ~w0(x)] is analytically continued [20, 21] and produces an imaginary factor,
) 1."-
i
2"rr
2
M/31E,,+ w ,l
(45)
From eqs. (43)-(45) the net fluctuation contribution to Z~ is
i( M~le,,
dT?,, exp(--S~) = ~
2"nS B
~
+ w~l(e 2 + ,og) ,
2"rr
)1/2
M/J(e,, + w~) (46)
6. Enhancement of decay rate
Expression (9) for the decay rate F involves, besides the imaginary part Z t , the real part of the partition function Z., which is given by ~(f3 I=t
Zl,= f dx
f Dx(~')fDq(-r)exp{-S(l')[q,x]}, I:(O)
(47)
where q(r) is represented by the fluctuation about the trivial extremal path q 0 ( r ) = 0 , and x(~-) is the solution of eq. (16). Using the eigenfunction expansion for the fluctuation ~:l°)(r) we have for the paths in (47) x(~) =
x,O), (48)
=
=
~,,
¢,,
t~';,
it
where scl,°)(r) are normalized according to eq. (33) and satisfy the equation (
62 ) + oJ~ /:(c~ = (w~ + ~lo)) ,~lo) dr 2 ~,, ,, / oil
- --
(49)
K. Si#mbwk I Quantwn tunneling through ltuctuatittg barrier
158
~(11)
.
(l)
with the boundary condition a,, (I)) ~;, ( / 3 ) = 1 ) . Using cqs. (48) and ( 4 9 ) the action S("~[q, x] is calcuhtted from eqs. (5). (27) and (42) with the result
s<<"lq,-,1- s<,'''+ ',,w/3
,•
(0)
tli)
it ii
I,o,,~ + ~,, (,7,,)~1,
51)
where
S"~'''
x:m`(2,, tanh( !/3.(2,,)
51 (111
Performing the Gaussian integrations over rl,, . ;
2Tr
=
'
)'~/
,
----~
~
M/3(~,,
....
+,o~)'
)
-
d_v cxp(
S< <'') .
52
where the thrcc lowest eigenwllucs e~<'~, arc a p p r o x i m a t e d by zcro, whictl introduces a negligible error for /3--,~-. In this way the product of the tt " 3 terms can be matched with the continuum product in eq. (46). Using cqs. (43), (46) and (52) wc obtain Z1 Z,,
(2/15"rr)l e/3°)" t d_vcxp( S"'''~
l d.v ''~ ~;' ::R(x)exp[-(Sl~ + S )] ( 53 ) ] 1 t !]1(2_)
where eq. (39) has bccn used for the discretc cigcnvalucs. The function R(x) is the continuum product-ratio, dclincd as
R(x)=
!]
, ,
I
--
( l ~ e
(1~)
,
2
..~o,,
l
"
1+<,,'(o~ )
(54)
In the appendix we derive, following the m e t h o d of k a n g c r [2()1, 15
c]'(_v )
--
(l)
ll
~ -" '
For vanishing coupling constant c. we have, using eqs. (30). (42) and (51). S~(c = 0) -
36Vo 5tO o
(5~)
E. L{'imhnek / Quantum tunneling through fluctuating barrier Sx(c = 0) = S'°).~ •
159 (57)
Hence the integrals in the denominator and n u m e r a t o r of eq. (53) mutually cancel and we are left, as expected, with the standard result for the quantum decay rate of an "uncoupled particle" [22],
r(c =0) =2
T(ZI] / 3Vo\ '/2 exp( \eo/c=o=lZwo~T~w)
36V{, )
5wo/"
(58)
Using eqs. (53)-(58), we obtain for the relative e n h a n c e m e n t of the rate E = - -l'(cI"= 0)
mN2(x))tanh(!¢'(l") I
f dx (1 + (f(£))? 4 exp 36V{,(1 (J~x))~ : _ (m~,ox2 me - 5w,~ + wo , + ~ ~
j dxexp[
mg&x:tanh(~[3.(lo)]
(59)
where the function N(x) is, according to eqs. (19) and (20), given by
N(x)=
N,, 0
for - X o cosh( ½/312o) ~< x ~< X o cosh( ½/3g2,,), otherwise.
(60)
The functional dependence o f f ( Y ) upon x is determined by eqs. (3), (19), (20) and (25). When the magnitude of the second term in eq. (25) exceeds X 0 the function f(x) goes over to a simple form,
f ( x ) = X o sgn[x - X o cosh( ½~o13)]
for a > 1 ,
(61)
where N(} c~ - 2~oX(~} tanh( ½/3ao).
(62)
For c~ < 1, f ( Y ) assumes a somewhat more complicated form. However, we confine ourselves to the case a > 1, which is pertinent to the physical applications of interest. Further simplification of the integrand of eq. (59) can be made by noting that by virtue of eqs. (3), (4) and (13) we have 2
(63)
This allows us to use in eq. (59) a binomial expansion to order cf(Y)l~o~.
I0()
I:..i4mmm'k
. Omtnlum throttling tl to ~gl tluctuamtg harrier
P e r f o r m i n g the i n t e g r a t i o n s o v e r the three s e p a r a t e regions ol the variable .~. we o b t a i n from cq. (50) with the use of cqs. (601 and (~1) t:"
E I + I:~ + E ~ .
(~4)
where
r{x,,V 2~
a)cxpITa~S,<""' )11
lc~
(I
:
I:,
(1
:~ A) cxp( ~ A,S'~,'~''
l:~
(1t
:~ a ) c x p (
cosh(e ~1312,,))1 .
~3 )12f'(.v,,V2y cosh( :'/3.~2,, ))
"A,Su ..... )ll
/'(.~,,\..'2y cosh(i:/3-(2,,)) I .
w h e r e l'(.v) is the n o r m a l p r o b a b i l i t y function d c l i n c d as
l'(.v)
c
~
\, 2"rr -I
it,~
I I.
¢,(,
67
124]
dt .
T h c p a r a m c t c r s 6 and y arc givcn b\'
'~
I t l ,'\' (,
=
4-Q~7tanh( ~/3.Q, ) .
-y = m_C~, tanh( ~/3~2,, ) .
and S'i'~'~ is thc bounce action for c qH)
7.
69 )
O. given by cq. (30). as
36t],
(7())
Discussion
T h e q u a n t i t y t:. givcn by cqs. ( {~4 )-( 07 ). cxhibils ;.ill interesting bchax.ior a s a function of t e m p e r a t u r e . For T : ( ) . t h c a r g u m e n t of the n o r m a l probability function gocs to inlinity, so that 1241 f'(x,,V'2~ cosh( ~,/3.(2,, )) -
/
,l)
, I .
(71}
This implies lhat E,(T
and
0)
E~(T
O)
o
(72)
E. ,~imhnek / Quantum tunneling through fluctuating barrier
E= E2(T=O ) = ( 1 - 7 a ) e x p
(5AS~ '1 2
161
mN~) 40,,
"
(73)
We see from eq. (73) that the e n h a n c e m e n t at T = 0 depends on the sign of the exponent which is determined by the ratio 10AS~" g20
r-
mN~
( ~ ) X ~
-
(x2--~o '
(74)
where (x2)o = (m~20) ~ is the mean-square amplitude of the bath oscillator at T = 0. If r > 1 the exponent is positive and an enhancement E ~> 1 is possible. According to eq. (74) this requires a weak frequency modulation, satisfying the condition
8 h ~< -5- ~(~1 B (x25o
(75)
A physically interesting case corresponds to very large values of 2 (°~ u - For example, the barrier factor for the cold fusion rate in isotopic hydrogen molecules [15] involves an exponent of order S~"= 102. Assuming X~/(x~) = 10, eq. (75) yields for a the condition a<~0.16. Taking a = 0.1 , uvB (°~ = 1 0 2 , we have from eq. (74) r = s_ and eq. (73) then yields an enhancement factor E = 0.82 exp(9.37) = 9.6 × 103. This rather large zero-point enhancement can, however, turn into a suppression of the decay rate if the ratio X~/(x2), becomes smaller than 6.2 (for AS~') = 10). For example, taking X~/(x2)o = 1, a = 0.1, S~ ~) = 102, eq. (74) yields r = 0.16 and eq. (73) predicts a "suppression factor" E = 0.82 e x p ( - 5 . 2 5 ) . This suppression is presumably due to the small overlap of the x-oscillator wave functions, caused by the displacement due to the " b o u n c e " of the q-particle. The size of the cutoff length x 0 is at present uncertain, but one expects it to be (for fusion in condensed matter) of the order of the interatomic distance or less; hence the interval 1 ~< X~/(x 2). ~ 10 is not an unreasonable choice. It should be also pointed out that the condition for the validity of eqs. (64)-(67) is, according to eqs. (61) and (62), S(O)
12
X[~
15 ( x 2 ) 0
(76)
, 2)o and a lower bound o n -v(o) This puts an upper limit on Xo/(x u • For example, when xa/(x2),, 10, eq. (76) restricts u u to values larger than 8. At finite temperatures, all the terms, given by eqs. (65)-(67), contribute to 2 2 the enhancement. Taking Xo/(x )0 = 10, /3,Qo = 1, a = 0.1 and S~ ~t 10 2, we obtain E. = exp(16.8), E 2 = e x p ( - 4 7 . 7 ) and E 3 = e x p ( - 2 6 . 8 ) . In the latter
162
1:. /;imanek
Quantum tunneling through [hwtuating barrier
case. the overlap reduction term , 5 - 72.2 plays an overriding role in the suppression of E,. However, the overall enhancement --cxp(22.8) is vet\ strong, owing to thc thermal excitation of the x-oscillator. The temperature dependence of this enhancement shows a rapid onsct at a tempcraturc 7~=~1o/k~ For T ~ 7 , the enhancement is practically zero. while for T > T it exhibits an approximately linear 7"-dependencc. It should be pointed out, however, that if the tunneling particle interacts simuhancously with a whole set of bath oscillators and the latter exhibit a distribution of rcsonancc frequencies ,Q,,, the resulting temperature depcndencc of the decay rate is expected to deviate from the above rapid onset and generally a smoother increase of the enhancement with thc temperature is expected. The Coulomb barrier intervening in the cold fusion process is obviously very different from our model potential, cq. (1). Nevertheless, the barricr penetration factors share common features which allow us to appply the results of the present calculations, at least qualitatively, to the problem of cold fusion in metals [17, 18]. In fact, the well known WKB expression for the Coulomb barrier factor shows that the exponent B is proportional to (MU) ~ed wherc U and d is the average height and width of the barrier [15]. Similarly, the "'bounce action" S B ( c - 0 ) is, according to eqs. (I) and (5(~), proportional to (MV.)~ ~-qowhere V~,is the barrier height and % - c0~/u is the tunnel distance in the barrier. The sensitivity of the barrier factor to the distancc d is an indication of the difficulty of the tunneling particles to pass through the rcgion of large internuclear distances. Recent estimates show [16] that the reaction rates observed in refs. [17, 18] could be explained by reducing thc equilibrium D - D distance in the metal well below the free-space 0.74 A value. The ab initio density functional calculations of the D - D distance in a hypothetical PdH: crystal show an increase by about 0.2 A over the value of 0.74 A [25]. This large internuclear distance suggest that it cold-fusion reaction of deuterium, via an ordinary tunneling process, is very improbable. Enhancement of cold fusion by screening of thc deuteron charges has been recently considered by Burrows 126]. Such a mechanism is made effective by the fact that the screening length is considerably shorter than the equilibrium D - D distance in the metal. Howevcr, there is a problem with this and all other screening mechanisms, which has been pointed out recently by Leggett and Baym [ 191, As mentioned in our introduction an efficient screening mechanism (needed to suppress the exponent B to B < 80) implies an unrealistic enhancemerit of the binding affinity of a 4He atom to the metal [19]. The barrier-fluctuation enhancement of the quantum tunneling provides an alternative avenue for the theory of cold fusion in metals. A possible origin of the barrier fluctuations is suggested by a previous study on the diffusion of hydrogen dissolved in metals [271. In particular, it has been proposed that the
E. Simanek / Quantum tunneling through fluctuating barrier
163
diffusion process involves quantum mechanical transitions between the lattice sites assisted by host-lattice phonons. The results of these studies show a considerable increase of the tunneling rate above its value in the rigid lattice. Though the barrier height in the diffusion process is much smaller than that for fusion, there is no apparent reason why a host-lattice phonon should not play a role in the modulation of the fusion barrier. The dissolved hydrogen at sites of low symmetry tends to form a covalent bond with the nearest metal atom [25]. The vibrations of the latter can thus be directly transmitted along this rigid bond to the hydrogen atom. In certain special steric situations the vibration amplitude of the H atom may well exceed that of the host metal atom. Such is the case in the bcc Nb crystal containing hydrogen atoms at the interstitial sites of tetrahedral symmetry [28]. Thus we may anticipate that the host-lattice vibrations in the deuterated metals will produce fluctuations in the D - D distance, which can be modeled (recalling that qo = w ~ / u ) as the fluctuations of the w~2, parameter in eq. (1). Large vibrational amplitudes may be necessary to obtain significant enhancements of the tunneling rate. The authors of ref. [18] have not seen any evidence of cold fusion in equilibrated deuterated metals. A possible explanation may be that unusual steric constraints allowing for large D - D vibrational amplitudes are met only during the process of electrolysis. Another possibility is the enhancement by nonequilibrium fluctuations, which is, however, outside the scope of the present work. In summary, we have calculated the rate of the quantum decay of a metastable state in the presence of a fluctuating barrier. The path integral formulation of quantum mechanics of Feynman is applied to the model of a tunneling particle anharmonically coupled to a single harmonic oscillator of the heat bath. Applying the steepest descent method, an approximate solution for the extremal paths is obtained in the limit of a slowly and weakly fluctuating potential. In this approximation, an analytic expression is derived for the enhancement of the decay rate over that of the uncoupled tunneling particle. The temperature dependence of the enhancement factor shows a rapid onset at a temperature of the order of the energy level spacing of the bath oscillator. Above this temperature significant enhancements of the decay rate are possible. The application of the studied model to the case of cold fusion in deuterated metals is discussed. It is suggested that the fluctuations of the Coulomb barrier are induced by the "slow" modulations of the d e u t e r i u m deuterium distance caused by host-lattice phonons.
Acknowledgement The author acknowledges stimulating discussions with R. Gaupsas.
A. ,c;imam'k
1~4
()uanlum :umu'ling :hrotLR,h llm'luatin,~ harric~
Appendix
(Tdcu/ation <~I d i e / , ' o d m ' l R(.v) Following Langcr [20] wc start from the identity • which alloxvs us to express the product R(x) defined in cq. (54) as 2
![
R(x): ,
I ?
+ ~<,<>>.'~o,,
I
cxp(()
(,,\,1)
1 4 E ,".~<~
~
where
(
7. I t d~'lf,'"'(~')
(,,\.2 1
p(~')lh~(l~'.,><,
ii
The unperturhcd and pcrturbed densities of states ~ll-C dclinecl, respectively.
~,( ( I ) ).
#l
(,,X.3) ps
Introducin~ the dimensionless cncrov variable ¢o [see cc 1. (37)]. L"~ _
41~' + ,o:~-,o,',(x)l
O)
-
(A.4)
•
.J~,(.v )
expression (A.2) can be written, with use of cq. (2).
(
~ .-
d~olp
'(~)
f,(~)l in(i+ '~,~)*h~(l+ 4(x))
.
(A.5)
(Oip
Expressing thc density of states in tcrms of the (;rccn's operator of cq. hlivC
<,,
,,-,
(
'
36)x~,c
)
whcTc
H,
d-','dv:
v:
12 scch:v
(A.7)
Introducing cq. (A.6) into eq. (A.5) and integrating by parts, ~+rC obtain
E. Sirnanek / Quantum tunneling through fluctuating barrier
C=~o
'f
I (
'
d05Im T r l n 1 + H 0 _ 0 5 _ i 0
v
)l'
~ + ~
165
to~i
' (A.8)
where the factor 3 in the second term on the right-hand side of (A.8) comes from the use of the Levinson theorem for the three bound states of eq. (36). The integral of eq. (A.8) can be done by contour integration in the complex 05 plane. We introduce the constant 4M
~Trln
1+ H~,L05 v ,
(A.9)
I"
where F ' is the contour which surrounds the negative real axis from 05 = - 9 to 05 = 0. We note that the integrand of eq. (A.9) has a branch cut along the real axis, extending from 05 = - 9 to to = 2. The trace in eq. (A.9) can be evaluated in terms of the eigenvalues of the equation 1 + H , , ~1
v) 0,,
A,,(05) 0,,,
(A.10)
from which follows:
i f d05 4~ri
05 + 4 ,, =0 I"
=d0+dl+ &+d~,
(A.11)
where
lid
4wi
05 + 4 ,
In A,,(05)
(A.12)
/"
and the constants (~0, C~ and C2 correspond to the three bound states of eq. (36). Eq. (A.10) can be reduced to the hypergeometric equation with eigenvalues 05, satisfying [25] o3=-(s,,-n)
2,
n
0,1,2 ....
(A.13)
The parameter s,,, however, satisfies [in distinction from eq. (36)] the condition 12 1 - A,,(05)
- s,,(s,, + 1).
(A.14)
166
t'~'. .q'imanek . Quantum tunnelittg tt rou
T o e v a l u a t e the c o n s t a n t (',,. wc s u b s t i t u t e . :=0 into eq. ( A . 1 3 ) . which yields s% = V' ~o. I n t r o d u c i n g this result into eq. ( A . 1 4 ) we o b t a i n 12
.l,,(uJ)
1 + o)
(A.I'~)
V'- ~o
F r o m eqs. (A. I I ) a n d ( A . 5 ) we have
1
(-"'
1 d,o
5Wi .
I"
in(l+
-
,o
~
12 ~_). ' --; \,
(A.I(~)
,o
T h e i n t e g r a n d has a p o l e at to -- 4 and a b r a n c h cut on the negative rcal axis with a b r a n c h p o i n t at tD - 9 . F v a l u a t i n g these c o n t r i b u t i o n s to (A.I(~) wc obtain ''/I
2
In ~1
(A.17)
'
Substituting n 1 into eq. ( A . 1 3 ) a c c o r d i n g to eq. ( A . 1 4 ) ,
wc o b t a i n s'~
12
(2+\ /
o~)(1+\ /
co)
1
\,-
¢o, which implies.
(A. 1~4)
F r o m eq. (A. IN) we have :11(¢0 4) --l) a n d h e n c e In l~(oD) has a b r a n c h cut on the n e g a t i v e real axis with a b r a n c h p o i n t at ~o - 4 . This b r a n c h p o i n t h a p p e n s to c o i n c i d e with the p o l e of the i n t e g r a n d of U~. T h e b r a n c h cut c o n t r i b u t i o n to C I is { In 4, while the p o l e c o n t r i b u t e s { In ~ . T h u s we o b t a i n ("l
For n _ ,~(,~)
/ h'l i7 .
( A . 19)
2 eqs. ( A . 1 3 ) and ( A . 1 4 ) give 12
l
(3+,/
~)(2+x/=-,;)
(A.20)
Since ,12(o3 -1) 0, the function In ~12(oJ) has a b r a n c h on the n e g a t i v e real axis e x t e n d i n g from w 1 to zero. This cut c o n t r i b u t e s to (-'~ a value of { In {. T h e r e is also a c o n t r i b u t i o n from the p o l e at ~a - 4 , which a m o u n t s to - ( In 2q , T h u s the net v a l u e o f (-', is ('~ = ~ l n !~ .
(A.21)
E. Simdmek / Quantum tunneling through fluctuating barrier
167
For n ~> 3, all the branch points of In An(o3 ) are on the positive real axis, so that the only contribution to C~ comes from the pole at o3 = - 4 . Using eq. (A.12) we have C~ -
1 2 ,~3 In A,,(o3 = - 4 ) .
(A.22)
For 03 = - 4 , eq. (A.13) gives s n = n + 2. With this value of s,,, eq. (A.14) yields A,,(-4) = (n - 1)(n + 6) (n+Z)(n+3) "
(A.23)
Inserting eq. (A.23) into eq. (A.22) we obtain C~=-
1 (i~i ( n - 1 ) ( n + 6 ) ) ~ l n ,,=3 ( n 7 2 ) ( n + 3 )
1 = 21n14"
(A.24)
Collecting the results (A.17), (A.19), (A.21) and (A.24) we obtain from eq. (A.11) (~ = In ~ .
(A.25)
Introducing eq. (A.25) into eq. (A.8) we finally obtain
el(2) )3,,2, w~ /
C=ln~+ln(l+
(A.26)
which implies, according to eq. (A.1),
,5(,+
)32
References [1] R.F. Voss and R.A. Webb, Phys. Rev. Lett. 47 (1981) 265. [21 L.D. Jackel, J.P. Gordon, E.L. Hu, R.E. Howard, L.A. Fetter, D.M. Tennants, R.W. Epworth and J. Kurkijarui, Phys. Rev. Lett. 47 (1981) 697. [3] A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46 (1981) 211. [4] A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y.) 149 (1983) 374. [5] A.J. Bray and M.A. Moore, Phys. Rev. Lett. 49 (1982) 1546. [6] S. Chakravarty, Phys. Rev. Lett. 49 (1982) 681. [7] H. Grabert and V. Weiss, Phys. Rev. Lett. 53 (1984) 1787.
I:. ,i,'imanek
Quantum tunm'ling through ttuctmttint: I~arrtcr
[~J S. W a s h b u r n and R . A . W c b b , A n n . NY Acad. Sc. 4~41) (1O