Quantum tunneling of Hydrogen in Niobium
Schematic picture of a Hydrogen atom (white), in BCC Niobium, tunneling
between two energetically equivalent sites. These sites are low in
energy because of the presence of the impurity atom Oxygen (red).
Trapping of H in Nb
Niobium (Nb) is a metal which is superconducting below a temperature of
9.2K. Nb has a body-centered-cubic (BCC) structure.
Hydrogen (H), being a small atom, if introduced in a Nb sample, can
sit in the interstitial sites between the Nb atoms. At very low
temperatures H is seen to move freely through the Nb lattice of atoms.
However, if there are certain impurity atoms like Oxygen (O) or
Nitrogen (N), present in the Nb sample, they form trapping
centers for the otherwise freely moving hydrogen atoms. A typical
location of an Oxygen atom in the Nb lattic is shown by in red in the
In the presence of Oxygen impurities, the H atom likes to stay around
such impurity sites. It tries to sit at the site where its energy will
be the lowest. It turns out that the two sites shown in the figure are
the lowest energy sites for the H atom (shown in white). These sites
are physically quite close, and also energetically identical for the H
atom. The hydrogen atom near such sites "sees" a potential which is
like a double-well. However, in a given sample of Nb, there are many
such double-wells because there are many H atoms trapped around Oxygen
impurities. The presence of other impurities affect the symmetry of the
double-well, and it will in general be asymmetric, as shown in the
figure below. It can stay in either of the two wells, but not in
between. At high temperatures H atom has enough energy to jump the
barrier between the two wells. At low temperatures it can stay in
either of the two wells, but cannot jump from one to the other.
This is true for things which follow Newton's laws. But a hydrogen atom
is too small to be contrained by Newton's laws. It takes advantage of
the laws of the Qunatum theory, which is a more fundamental theory of
nature, and is seen only when one goes to atomic scales. Even if the H
atom doesn't have enough energy to jump over the barrier, it can grow
fuzzy and just "tunnel" to the other site! Quantum mechanic
allows this. Left to itself at a particular site, the H atom will
execute a coherent clock like motion between the two sites. This is
called "quantum tunneling".
This was observed in beautiful experiments by Helmut Wipf and
collaborators at Grenoble, France (H. Wipf et al, Europhys.
Lett. 4 (1987) 1379). In these experiments neutrons with
known energy are thrown on a sample of Niobium containing some Oxygen
impurities, in which Hydrogen has been introduced. After the neutrons
scatter off the sample atoms, their energy gain/loss is measured.
There are basically three kinds of scattering of neutrons:
- Elastic scattering: In this process there is no energy
gain or loss in the scattered neutron. If neutron bumps into a rigid
immobile atom, this is the kind of scattering that takes place.
- Inelastic scattering: In this kind of scattering there is
a gain or loss of discrete quanta of energy from the neutron. If the
neutron comes across an atom which is executing a periodic oscillation,
there is inelastic scattering.
- Quasi-elastic scattering: In this kind of scattering there
is loss or gain of energy in randomly varying amounts. If a neutron
collides with an atom which is randomly moving around, there is
The energy gain/loss measured by Wipf and collaborators, at a
temperature of 0.1K, is shown in the figure on the right. There is big
peak at the center, which represents elastic scattering from immobile
However, one can see a distinct inelastic peak at some
discrete energy. This means there is an atom inside, which is
executing a coherent clock-like motion! At this low temperature
the H atom cannot jump the barrier between the two wells, so the only
way it can move between the two sites is through tunneling. So, this
experiment shows evidence for tunneling of H in Nb.
The quasi-elastic peak (shown in green) narrows down as the
temperature is increased.
This, however, is just one part of the story. When the experiments
were repeated at higher temperatures, inelastic scattering disappeared
completely, and was replaced by a quasi-elastic peak. This, one would
recall, indicates that the scattering atom is not moving coherently but
incoherently. This means that the tunneling motion has been destroyed
by something which is causing randomness.
The width of the quasi-elastic peak is supposed to quantify the rate
at which the scattering atom is jumping randomly between two sites.
Normally one expects that as the heat is turned on, the scattering
atom is kicked around more and more and so the width of the
quasi-elastic peak will increase with temperature. But here it was
observed that the quasi-elastic peak actually narrows
with the increasing temperature! This is really strange, for it means
that as the temperature is increased, the atom tends to sit more on
one site rather than jump around!
This is really weird - so let us try to see how this can happen. Recall
that Nb is a metal. So the H atom which moves between the two sites,
not only sees the double-well potential but also feels the effect of
the sea of electrons formed by the outer electrons of Nb (see the
The electrons in Nb solid seem to have a drag on the moving H atom.
Each atom is surrounded by an electron "cloud", and in metals these
clouds overlap so that there is a sea of electrons present throughout.
Effect of conduction electrons on tunneling of H
Effect of conduction electrons is not easy to study theoretically, as
the problem involves studying so many particles all together. We
simplified the problem to make it tractable. We approximate the
tunneling H atom by a quantum two-level system. This two level
system interacts with electrons, which are also treated approximately.
If one grinds through the calculation (S. Dattagupta
and T. Qureshi, Physica B, 174 (1991) 262), one gets the
- At very low temperatures, the effect of electrons is not destructive,
and one observes coherent tunneling motion of H.
- This tunneling frequency is highly suppressed because the motion
of the H atom from one site to another requires the dragging of the
surrounding electron cloud, together with it.
- As the temperature is raised, the electrons destroy the coherent
motion of H. Tunneling is possible still, but now it is incoherent,
like random jumps.
- The destruction of coherence lead to the H atom tending to sit on
- The frequency of jumps between the two sites is proportional to
T2K-1, where T is the temperature and K=0.055. This agrees
with the experimental observation of decrease in the jump rate with
- This is an example of the "watched pot effect", which says
that a watched pot never boils! It is also known as the Quantum Zeno effect. If one observes a quantum
system, its time evolution is inhibited. In the limit of continuous
observation, the system completely stops evolving and freezes in its
initial state. Here the conduction electrons of the metal keep
"watching" the H atom, and so it is not able to tunnel to the other