Our goal is to understand the properties of delocalized vacancies in solid 4 He.

1. 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.

2. 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. Why 3He? Figure 1: Temperature dependence of the diffusion coefficient of impurities in a HCP crystal of 4He. ●0.75% 3He, ○2.17% 3He, □positive ions 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. Figure 2: Log(D) vs. 1/T from the literature. Suspected angle from c-axis provided.

3. Cryo-Cooling • A single-crystal helium sample requires: • temperatures under 2 K • 25 atmospheres of pressure • Inner Vacuum Can (IVC) isolates the experimental cell • IVC immersed in a bath of liquid helium, 4.2 K: hydrogen exchange gas while precooling • 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. • 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

4. 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.

5. 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: With proper coupling, Z = 50Ω+r, where r is a perturbation caused by aligned spins within the solenoid. Thus,

6. 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.

7. 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.

8. 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: 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.

9. 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. 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).