A. Würger (ILL).
Interstitial hydrogen trapped by an oxygen or nitrogen impurity in Nb
is confined to two states and shows a rich dynamic behaviour due to
quantum tunnelling. At temperatures below 10 K, the hydrogen impurity
performs coherent oscillations about the trap atom. At higher T,
adiabatic interaction with conduction electrons drives a cross-over
to overdamped motion; the relaxation rate decreases with temperature
according to the law 
T2K-1, as derived first by Kondo. At about 60 K, however,
the rate shows a minimum as a function of T and increases on further
rising the temperature. We report strong evidence for the behaviour
above 60 K being due to coupling to lattice vibrations; the resulting
rate is governed by multiphonon processes.
Hydrogen in metals shows a variety of quantum
phenomena. In clean samples, it diffuses through the whole crystal
through jumps between well-defined interstitial sites. On the other
hand, oxygen or nitrogen impurities may capture the hydrogen atom,
thus forming a two-state tunnelling centre on equivalent tetrahedral
sites. The resulting tunnel matrix-element in niobium has been
measured by inelastic neutron-scattering; it takes a value of 
= 2.4 K for H trapped by an oxygen
impurity, and = 1.9 K for a
nitrogen trap.
Interaction with conduction electrons gives rise to dissipation and leads to overdamped, or incoherent, tunnelling above T = 10 K. Quasielastic neutron-scattering confirmed Kondo's prediction for the rate,
(1) el = (
/ 
K) [(2
kT/)2K-1]
that decreases with rising temperature as T 2K-1, with a Kondo parameter K ~ 0.055. This law describes quantitatively measurements on several impurity systems at various concentrations. In real metals, however, coupling to lattice vibrations provides a second damping mechanism that prevails at sufficiently high temperature.
Fig. 1 shows the potential energy landscape of
a two-state defect as a function of the defect position
q and one
phonon coordinate xk. Due to the linear coupling term 
qxk, the energy minima
occur no longer at xk = 0, but are shifted
to finite values of the vibrational coordinate. As a consequence, the
tunnelling motion involves both defect and lattice modes; when
tunnelling from one well to the other, the hydrogen impurity drags
its phonon cloud. This may be viewed as a two-state polaron, in
analogy to the diffusion of a light particle in a solid.
This polaron effect gives rise to an effective potential that changes shape for the particle dwelling in the left or right well, as shown in Fig. 2. At very low T, phonon dressing results in an apparent increase of the barrier height and hence slows down the tunnelling motion.
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Figure 1: Potential energy landscape as a function of the defect position q and the elastic mode xk . |
Recently, we have pointed out a few subtleties concerning the phonon damping rate. With rising temperature, the rigid phonon cloud 'melts', thus lowering the effective barrier and enhancing the jump rate.
When summing an infinite series of multiphonon contributions, we find a temperature-dependent correction factor to the electron-driven rate (1),
=
el [1 +
F(T)]
(2)
In the limit of low temperature,
T <<
T0, the additional term
in brackets vanishes, F(T) 
0,
whereas in the opposite case, T
>> T0, it exceeds unity and hence governs the temperature
dependence of the rate. The cross-over temperature is given by
kT0 = (3
v5 / 3
2)1/2 , with sound
velocity v, mass density , and
the elastic deformation potential . There are well-known analytic expressions for
F(T),
valid for T both well below and above the Debye temperature. For
hydrogen in Nb, none of these limits is satisfied, and the correction
factor F(T) has to be evaluated numerically.
In Fig. 3 we plot quasielastic
neutron-scattering data for Nb(OH)x and
Nb(NH)x, observed by Wipf and co-workers. Because of the
smaller tunnel energy, the relaxation is slower at nitrogen traps.
The dashed lines account for damping by conduction electrons only,
according to Eq. (1). The solid lines (2) take phonon coupling into
account. The parameters used for the fit are v = 2 260 m/s, = 8.4 g/cm3, and = 0.2 eV, resulting in
T0 = 23 K. The minimum in
the rate occurs at about 2 T0.
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Figure 2: Effective potential for the hydrogen dwelling in the left or in the right well.
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Figure 3: Jump rates for hydrogen impurities for N and O traps. Dashed lines give the rate (equation 1) driven by conduction electrons only; solid lines (equation 2) take phonon coupling into account. |
Below 20 K, the solid lines differ little from
the dashed ones, i.e., dissipation is driven by conduction electrons
only. Between 30 and 70 K, the polaron effect reduces the rate
slightly below the undressed value. Above 70 K, however, multiphonon
processes prevail, and the rate increases with T. This behaviour is
characteristic for phonon-assisted tunneling. At still higher
T, the
rate levels off towards the high-temperature result
T-1/2. In the whole
range, the data agree well with the theoretical curve. In particular,
the constant ratio of the rates for N and O traps assures that we are
in the quantum regime, where
2-2 K holds true.
In summary, Fig. 3 shows how the polaron effect
influences the damping rate. At low T, screening by conduction
electrons results in a decreasing with
rising temperature. Above T0 phonon coupling causes a strong increase of the rate.
Comparison with quasielastic neutron-scattering data proves the
relevance of phonon-assisted tunnelling for the dissipative dynamics
of trapped hydrogen impurities above about 60 K.
