To cool down further than about 2 mK made possible by the state of the art dilution refrigerators, adiabatic nuclear demagnetisation is often used. In our case, as well as in many others, copper is used as the refrigerant to be demagnetised. Many other materials with nuclear magnetic moment could be used, for example PrNi5 [52, 53]. Nevertheless, copper is readily
available, pure, easy to work with and relatively inexpensive. Moreover, in the solid form copper has low internal magnetic field. In case of a fine copper powder this field increases significantly, though this can be beneficial in certain applications as discussed in Section6.1.
Similarly to Section 2.3 we start with the partition function Z. For a nucleus with magnetic moment I there are 2I +1 sub-states labelled m, wherem= I,I−1, . . .−I. The partition function is then
Z=
I
∑
m=−I
The entropy S of such system can be again derived using (2.26) and we obtain a similar result as for the two level system in (2.29)
S=n kBln(2I+1)− λn B2 2µ0T2 , (3.2)
wherenis number of moles andλn is molar nuclear Curie constant
λn =
NAI(I+1)µ0µ2Ng2
3kB
. (3.3)
For copper the situation is slightly more complicated since there are two stable isotopes 63Cu and 65Cu with relative abundances of 69 % and 31 % respectively. Both have I = 3/2 and thus 4 non-equally populated energy levels. However, the principle outlined here stays the same. We can plot the dependence of entropy on temperature in high and low magnetic field using (3.2) as in Figure 3.1. The total entropy needs to be adequately aver- aged with respect to the population of both isotopes and all energy levels. The line from point A to B represents precooling in high magnetic field, in
S
Ti TA
B
C
Tf Bi Bf dilution refrigeration demagnetisationC'
Tf'Figure3.1: A schematic representation of a path followed during precool-
ing and demagnetisation. Copper refrigerant is precooled with the dilution refrigerator in high magnetic field Bi (path A→B) until the base tempera- ture Ti is reached. Subsequently, the thermal connection with the dilution unit is broken and the field is reduced from Bi to Bf, following the path B→C and reaching Tf. However, the process of demagnetisation is not ideal so realistically the sample reaches the point C’ at a slightly elevated
temperature T0f.
f
From (3.2) it is evident that if the system is isolated and entropy is kept constant, the temperature must decrease proportionally with decreasing magnetic field and we come to a simple equation
Tf =Ti
Bf
Bi
. (3.4)
This suggests that if we demagnetise to zero final field, the final temper- ature would also be zero, thus violating the second law of thermodynamics. The field is actually a superposition of the external magnetic field B and internal interaction magnetic field from the neighbouring nuclei B0 (around 0.3 mT for copper). Tf =Ti· q B2f +B02 q B2i +B02 . (3.5)
The situation is similar to the effective field in the Curie-Weiss model de- scribed in Section2.3.1.
The initial field in our case is 8 T and we demagnetise down to about 30 mT, which is the magnetic field required for NMR measurements at our designed frequency of 1 MHz. This should result in the reduction of temper- ature by a factor of 260, but in reality this is lower. Equation (3.5) assumes the refrigerant is always in thermal equilibrium with3He and that there is no external heat leak into the system, which is naturally not the case. To get at least close to the internal thermal equilibrium the rate of change of magnetic field should be kept low and thus the demagnetisation is usually carried out on a timescale of several hours overnight. Moreover, one can not neglect the entropy of the3He itself, with SHe ∝ T. The proportionality
in (3.2) becomes
S ∝ B
2
T2 +T , (3.6)
agreeing with the line B→C’ in Figure 3.1. The low rate of change also reduces losses by eddy currents induced in copper which manifest as a heat leak with ˙Q∝ B˙2.
In our experiment the copper demagnetisation is used to precool the aerogel sample with a layer of solid 3He with local initial magnetic field of 150 mT. This field will be reduced to 30 mT cooling the sample further. There is no need for a heat switch at this stage, because quasiparticle con- ductance rapidly drops below 140 µK and thus the heat contact with copper quickly becomes negligible. The aerogel demagnetisation stage is discussed in more detail in Chapter6.
only to measure the temperature of the superfluid. Conventional resistance thermometers are inaccurate at microkelvin temperatures due to the large thermal boundary resistance between the fluid and the thermometer in ad- dition to significant self-heating. The method of probing the superfluid directly by measuring the damping of a mechanical oscillator has proven to be a superb tool at ultralow temperatures thanks to the elimination of thermalisation problems as well as having a relatively low heat load.
The standard vibrating devices present in the experiments described in this thesis are Vibrating Wire Resonators (VWR) and Quartz Tuning Forks (QTF). This chapter describes the basic principles of their operation and the underlying mechanism of the fluid damping.
4.1
Oscillator fundamentals
An oscillating device can be modelled as a simple mass m0 on a spring
with spring constant kdriven at angular velocity ω by a periodic force F =
F0eiωt. The mass also experiences a restoring force −ω02x coming from the
spring and a damping force −λx˙ from the fluid. Note that we assume a
damping force that is linearly dependent on the velocity, which is true for
low velocities. For such a system we can write an equation of motion
m0x¨+λx˙ +m0ω20x =F0eiωt , (4.1)
whereω0 is the natural frequency of the oscillator
ω0= s
k m0
. (4.2)
In the case of an oscillator submersed in a fluid, the damping term λ is
not sufficient to describe all fluid effects. Extra massm00is added to the mass of the oscillator m0 to take into account the fluid dragged by the oscillator
as well as the effect of the backflow around it
m=m0+m00 . (4.3)
Naturally, we expect harmonic motion for the velocity
˙
x=x˙0eiωt . (4.4)
We can substitute this particular solution into the equation of motion (4.1), solve for ˙x0 and obtain a complex Lorentzian function for the velocity am-
plitude
˙
x0=
F0iω
−ω2m+iωλ+k , (4.5)
which can be split into real and imaginary components with Re(x˙0) being
the in-phase velocity of the object
Re(x˙0) = F0ω2λ ω2λ2+ (ω2m−k)2 , (4.6a) Im(x˙0) = F0ω k−ω2m ω2λ2+ (ω2m−k)2 . (4.6b)
The frequency dependence of both components is plotted in Figure 4.1. At resonance, the velocity of the oscillator is maximum and Re(x˙0) will
Figure4.1: Frequency dependence of real (red) and imaginary (blue) com-
ponents of velocity around resonance.
reach its maximum value. The condition
ω2m−k=0 , (4.7)
must apply, leading to a resonance frequency
ω0 = r k m = s k m0+m00 . (4.8)
Moreover, at maximum when ω2m−k=0 we get
Re(x˙0) =
F0
λ . (4.9)
The shift in frequency from an oscillator in vacuum to the one damped by liquid is then ∆f1 = ω −ω0 2π = 1 2π s k m0 − s k m0+m00 ! . (4.10)
The imaginary component of velocity reaches a maximum when Re(x˙0)
is at half of its maximum value. This occurs at a frequencyω1/2, which can
be found by solving ω21/2λ2+ ω21/2m−k 2 =0 , (4.11)
The width of resonance at half height is then the difference between the two roots of the quadratic equation (4.11),
∆f2 =
ω1/2+ −ω−1/2
2π = λ
2πm . (4.12)
Substituting into (4.9) and realizing that max[Re(x˙0)]is the velocity at max-
imum resonancev0, we get
∆f2 = F0 2πv0m = F0 2πv0 m0+m00 . (4.13)
For the steadily driven oscillator such as our case, the rate of energy loss relative to the stored energy in the oscillator is described by the Q factor, often called the ‘quality’ of the resonance
Q= f0
∆f2 . (4.14)
Lower the energy dissipation, higher theQfactor and longer longer the ring down time of the oscillator.
Lastly, we introduce a useful fundamental property of the resonance called ‘Height times Width over Drive’ (HWD). The value is independent of temperature and represents the area under the resonance curve. It is often used to describe the ‘strength’ of the resonance
HWD= v0∆f2 F0
= 1
2πm . (4.15)