R. M. Bowley

Quartz Tuning Fork Viscometers for Helium Liquids

Mechanical resonators, in the form of vibrating wires or torsional oscillators, have long been employed as sensors in liquid 3He and 3He–4He mixtures. The damping of resonators is due to the viscosity of the surrounding liquid which is a strong, well-known function of temperature for bulk Fermi liquids. It is therefore possible to use the viscous damping for thermometry in the millikelvin regime. An alternative sensor is the small quartz tuning fork which is driven by the piezoelectric effect and requires no external magnetic field. In this paper, we present measurements of the viscous damping of such a tuning fork when immersed in a 6.2% 3He–4He mixture, between 3 and 100 mK, and at zero and high (10 T) magnetic field. The measurements indicate that damping of the tuning fork resonance is dominated by the liquid helium properties and is insensitive to the applied magnetic field. The response of the tuning fork to the saturated helium mixture demonstrates that it could potentially be used for thermometry in any magnetic field. There is evidence of slip at the interface between the fork and the helium suggesting specular scattering from the smooth surface of the quartz. The fork is also able to detect the superfluid transition in pure liquid 3He.

Analytic calculations of the E-fields induced by time-varying magnetic fields generated by cylindrical gradient coils.

Analytic expressions which allow the direct calculation of the electric field generated inside an infinite conducting cylinder by varying the current through the wires of any cylindrical coil are presented. These expressions provide some general insight into the spatial characteristics of the electric field generated inside the body by switched gradients and can be used to evaluate the locations where nerve stimulation by rapid gradient switching is likely to occur. They may also be employed at the design stage to produce gradient coils which can provide higher gradient switching rates without causing nerve stimulation. Using these expressions the electric field patterns produced by transverse and longitudinal, whole‐body gradient coils were calculated. Example data are presented along with the associated magnetic field patterns. The effect on the induced electric field pattern of varying the body size and the size of the region of gradient linearity was explored. Magn Reson Med 44:782–790, 2000.

Electric fields induced in the human body by time-varying magnetic field gradients in MRI : numerical calculations and correlation analysis

The spatial distributions of the electric fields induced in the human body by switched magnetic field gradients in MRI have been calculated numerically using the commercial software package, MAFIA, and the three-dimensional, HUGO body model that comprises 31 different tissue types. The variation of |J|, |E| and |B| resulting from exposure of the body model to magnetic fields generated by typical whole-body x-, y- and z-gradient coils has been analysed for three different body positions (head-, heart- and hips-centred). The magnetic field varied at 1 kHz, so as to produce a rate of change of gradient of 100 T m(-1) s(-1) at the centre of each coil. A highly heterogeneous pattern of induced electric field and current density was found to result from the smoothly varying magnetic field in all cases, with the largest induced electric fields resulting from application of the y-gradient, in agreement with previous studies. By applying simple statistical analysis to electromagnetic quantities within axial planes of the body model, it is shown that the induced electric field is strongly correlated to the local value of resistivity, and the induced current density exhibits even stronger correlation with the local conductivity. The local values of the switched magnetic field are however shown to bear little relation to the local values of the induced electric field or current density.

Electric fields induced in a spherical volume conductor by temporally varying magnetic field gradients

A homogeneous spherical volume conductor is used as a model system for the purpose of calculating electric fields induced in the human head by externally applied time-varying magnetic fields. We present results for the case where magnetic field gradient coils, used in magnetic resonance imaging (MRI), form the magnetic field, and we use these data to put limits on the rates of gradient change with time needed to produce nerve stimulation. The electric field is calculated analytically for the case of ideal longitudinal and transverse linear field gradients. We also show results from computer calculations yielding the electric field maps in a sphere when the field gradients are generated by a real MRI gradient coil set. In addition, the effect of shifting the sphere within each gradient coil volume is investigated. Numerical analysis shows similar results when applied to a model human head.

Using the vector potential in evaluating the likelihood of peripheral nerve stimulation due to switched magnetic field gradients

The time‐varying magnetic field gradients used in MRI can cause peripheral nerve stimulation (PNS) in human subjects, as a result of the electric fields induced in tissue. The local electric field, E, is given by E = −∂A/∂t − ∇ϕ where A, is the vector potential and ϕ is the scalar electric potential generated by charges accumulated at boundaries between regions of different conductivity. Difficulties in calculating ϕ have led some investigators to use −∂A/∂t alone as a predictor of the induced field. Here the spatial variation of −∂A/∂t and E is investigated for the case of a simple spherical conductor exposed to time‐varying gradients produced by two different gradient coils that generate identical internal magnetic fields, but very different vector potentials. The results indicate that the temporal derivative of A bears little relation to the induced electric field, and that consequently neglecting the effect of the scalar potential introduces significant errors in estimating the likelihood of PNS. Magn Reson Med 50:405–410, 2003.

Calculations of static properties of spin-polarized3He-4He mixtures

The theory of dilute mixtures of3He in4He that have been polarized by a strong magnetic field is developed. The interaction between the quasiparticles is taken to be constant, an approximation valid at low temperatures. The polarization of the mixture depends on the strength of the interaction. The internal energy, the specific heat, the osmotic pressure, and the velocity of second sound are also calculated. The specific heat is relatively insensitive to the interaction, but it does change significantly with magnetic field. The osmotic pressure is more sensitive to the effects of the interaction for some3He concentrations and temperatures. The velocity of second sound behaves qualitatively like the osmotic pressure. The measurement of these quantities as a function of temperature and magnetic field is discussed with a view to obtaining the strength of the interaction between quasiparticles.

Analysis of spin wave resonances in3He-4He mixtures

Spin waves have been reported previously in3He-4He mixtures using cw-NMR. The spin waves form standing waves in the rf coil used to detect the NMR signal. Only one resonance was observed, corresponding to ak vector of 1 mm−1. This is in sharp contrast to similar experiments performed on pure3He, where several resonances were seen. An explanation is given of why this occurs and the resonant mode is identified. Furthermore, it is explained why there is an extra width to the resonance for temperatures below 2 mK. This is caused by radiation damping of the resonant mode. The theory is found to be in excellent agreement with the data. Further experiments are suggested, including details of an improved rf coil design, which should lead to information about the quasiparticle interaction.

Heat capacity of3He/4He mixtures

We measured the specific heat capacity of three dilute mixtures (2.6, 1.07, and 0.44%) of3He in4He and of pure4He. The4He contribution to the specific heat of the mixture is subtracted, leaving only the3He part. This is fitted to a theoretical expression over the whole temperature range from 10 to 700 mK. Assuming a dispersion relation of the form ε=ħ2k2(1+γk2)/2m*, the fits yield the value of γ and the effective mass m* of each mixture. The average value of γ is −0.076±0.01 A2 and the effective mass in the limit of zero concentration is (2.23±0.02)m3. These are compared to values deduced from other measurements.

Transport properties of spin polarized3He-4He mixtures

We have calculated the viscosity, thermal conductivity, and longitudinal spin diffusion coefficient in3He-4He mixtures which are spin polarized. The calculation applies to all temperature regimes. We have also calculated the Onsager cross coefficient which arises because of the coupling between heat and longitudinal spin currents. The interaction between3He quasiparticles is taken to be a constant as a first approximation. We have also investigated the changes brought about by allowing the interaction to vary with the momentum of the quasiparticle.

Second Sound Measurements in 3He–4He Mixtures

We report measurements of the damping of second sound in a 0.1% 3He–4He mixture in the temperature range 100 mK to 1K and for frequencies from the fundamental cavity resonance of -700 Hz to 12 KHz. The measurements enable us to examine the phonon contribution to the damping over a much wider frequency range than hitherto. The data are compared to a theory based on ideas which are developed in detail elsewhere.