Transcript He3 Diffusion As a Probe of Delocalized Vacancies in HCP He4

Introduction
Our goal is to understand the properties of
delocalized vacancies in solid 4He.
Using a triplet of orthogonal diffusion measurements we plan to
locate the c-axis of a single crystal sample. This will allow us to
measure diffusion as a function of temperature along the c-axis
and along the basal plane to look for differences in the activation
energy.
Positive Ions
Diffusion of positive ions on the basal plane takes twice as much
energy as on the c-axis.
D+c ~ e-E/T; D+b ~ e-2E/T
Hypothesis: Two vacancies are needed to move along the basal
plane.
Why 3He?
3He
seems to diffuse by the
same mechanism as positive
ions. In the graph above, the
slope of the solid line shows
activation energies for both
3He and positive ion samples.
The tails on the right side of
the 3He lines are due to a
different diffusion mechanism.
Figure 1: Temperature
dependence of the diffusion
coefficient of impurities in a HCP
crystal of 4He. ●0.75% 3He,
○2.17% 3He, □positive ions
Figure 2: Log(D) vs. 1/T from the
literature. Suspected angle from caxis provided.
Previous measurements of
3He diffusion have assumed
isotropy, but the results vary.
Activation
energies
span
nearly a factor of 2, which
could be explained by the
anisotropy hypothesis.
Cryo-Cooling
A single-crystal helium sample
requires:
• temperatures under 2 K
• 25 atmospheres of pressure
1. Inner Vacuum Can (IVC)
isolates the experimental cell
2. IVC immersed in a bath of
liquid
helium,
4.2
K:
hydrogen exchange gas
while precooling
3. Thin tube leading to main
bath allows liquid helium to
enter 1 K pot at about 1
cc/min. Pump attached to 1
K pot activated. Achieves 1.5
K. Thin copper band from
1K plate to 3He plate
provides minimal thermal
coupling.
4. Pump for 3He pot activated.
Supply of helium fixed;
limited duration run. 3He
plate thermally coupled to
sample cell by a copper rod.
Cell approximately same
temperature as 3He pot.
If 3He, with lower vapor
pressure, were used, the 3He
pot could reach temperatures
as low as 0.3 K. For the 1 K
lower limit required, 4He is
sufficient.
Figure 3: Phase Diagram for 4He
Figure 4: Schematic of
cooling apparatus
Resonant Circuit
The resonant circuit for NMR is embedded in the cell before
crystal formation begins. The crystal grows around the circuit,
filling the inductor. As nuclear spins within the solenoid couple to
the RF potential across the circuit, the impedance of the inductor
changes, producing a signal.
Coupling occurs when the
precession rate of spins matches the RF frequency. Precession
rate is related to the applied magnetic field by B = γω, where
γ = 2.04x108 / Ts.
Figure 5: Resonant Circuit with Adjustable Coupling
An inductor and a capacitor in parallel form a primitive resonant
circuit with frequency 1/2π(LC)½. A large resistor (~1 MΩ) in
parallel with these and the small coupling capacitor in series with
the rest of the circuit match impedance of the circuit with the 50 Ω
line impedance at the resonant frequency. Two adjustments are
needed to match the real and imaginary components of
impedance. R is calculated once and soldered into the cell, while
CT is left exposed for easy adjustment.
NMR: Finding a Line
Figure 6: Electrical arrangement for detection of NMR resonance line.
RF power is fed into the resonant circuit, producing a reflection.
For circuit impedance Z, the reflection is given by:
Vr
Vi

Z  50
Z  50
With proper coupling, Z = 50Ω+r, where r is a perturbation caused
by aligned spins within the solenoid. Thus,
Vr
Vi

r
100
This reflection is amplified and mixed with the original signal to
produce a DC signal proportional to the power reflected. Care is
taken to match electrical path lengths of the reflected and
synchronization signal so that any phase difference is due to a
complex impedance in the circuit. The magnetic field in the cell is
modulated in the audio range so that any change in reflection is
picked up as an audio signal on top of the DC output of the mixer.
The signal is then sent through a transformer to strip out the DC
component. The remaining audio signal is fed into a lockin
detector, where any change in reflection will appear as a signal
synchronized with the original audio input. The NMR line is found
by slowly sweeping the static field in the cell while watching the
lockin detector. Because of the extreme precision required
(greater than 1 part in 1000), a smaller coil within the main
solenoid is swept through lower currents to pinpoint the exact field
of resonance.
Measuring Diffusion
NMR requires producing a net dipole moment in the atoms of the
sample with a strong magnetic field. In addition, we apply a static
field gradient to the sample in the direction to be measured.
Three pairs of magnet coils produce gradients in three orthogonal
directions. Any gradient desired can be created by linear
superposition of these. Only a thin plane normal to the gradient is
then in resonance. The width of the slice (W) is inversely
proportional to the strength of the gradient. An RF pulse at the
Larmour frequency tilts the spins of 3He atoms in this slice only.
A sufficiently strong pulse will randomize the spins in the resonant
slice, reducing the reflected signal from the circuit. As the
randomized spins diffuse out of the slice and are replaced by
preferentially oriented spins, the reflected signal will recover with
some characteristic time constant τ. The diffusion coefficient
along the gradient is then D = W2/τ. Because of the gradient,
spins will precess at different rates throughout the slice,
necessitating additional compensating techniques.
Locating the C-axis
Diffusion along the c-axis and diffusion on the basal plane
potentially have different diffusion constants and activation
energies. The diffusion constant for a gradient at an angle θ from
the c-axis must be given for an HCP lattice by:
D  Dc e Ec / T cos2    Dbe Eb / T sin 2  
When measurements are made in three orthogonal directions,
cos2(θ1)+ cos2(θ2)+ cos2(θ3)=1. With Dc, Db, Ec, and Eb as fitting
parameters, measurements at three different temperatures are
sufficient to locate the c-axis.
Figure 8: x, y, and z measurements used to locate c-axis.
The field gradient can then be oriented along the c-axis and along
at least one vector on the basal plane. The diffusion constant in
each direction can be assessed at various temperatures.
Data Analysis
To determine the activation energy needed for diffusion in a
particular direction, the diffusion coefficient must be measured at
various temperatures covering the range of interest (in this case
from 1 K to nearly 2 K). The energy is extracted using the
following prescription.
D  D0 e
E
T
 D
1
ln      E 
T
 D0 
A plot of ln(D) vs. 1/T is fit to a straight line. The slope of the line
will be –E.
References:
S.C. Lau and A.J. Dahm, J. Low Temp Phys. 112, 47 (1998).
K.O. Keshishev, JETP 45, 273 (1977).