Experimental and Theoretical
Optimization of the Nuclear
Demagnetization Stage in the Yeti
cryostat
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OFSCIENCE
in PHYSICS
Author : V.A. Wagemans
Student ID : s1532413
Supervisor : Prof.dr.ir. T.H. Oosterkamp
2ndcorrector : Dr. W. L ¨offler
Experimental and Theoretical
Optimization of the Nuclear
Demagnetization Stage in the Yeti
cryostat
V.A. Wagemans
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
July 1, 2020
Abstract
Testing the mass-proportional CSL model, that describes quantum-mechanical wave function collapse, by measuring a small offset in energy due to that same collapse, requires ultra-low temperatures. These temperatures can be reached,
using adiabatic nuclear demagnetization as a refrigeration method. To obtain the lowest temperature possible and to do so for a long time, dissipation has to
be minimized. Theoretical work in this thesis provides a way to decrease dissipation through an optimized demagnetization ramp, resulting in a final
magnetic field of 5 mT and a field ramp rate of 0.5 mT/s. Experimentally, a decrease in dissipation is found by comparing demagnetization ramps with and without an LCR circuit. The ramp with such a circuit has approximately 2.5 times less dissipation. Also discussed in this thesis is SQUID thermometry,
a reliable way of measuring the temperature at ultra-low temperatures. An analysis method is presented to reduce the influence of mechanical
Contents
1 Introduction 1
2 Theoretical results 7
2.1 Magnetization process 7
2.2 Demagnetization process 8
2.3 Cooling power of a Cu refrigerant 10
2.4 External heating and optimization of Bf and ˙B for a Cu refrigerant 10 2.5 Optimization of Bf and ˙B for the Van Vleck paramagnetic
rare-earth compound PrNi5 13
2.6 Thermal contact with Ag sample 19
3 Experimental results & discussion 23
3.1 Set-up 23
3.1.1 Cryostat & Nuclear Demagnetization Stage (NDS) 23
3.1.2 SQUID 25
3.2 SQUID thermometry 25
3.2.1 Spectrum to temperature conversion 26
3.2.2 Analysis 27
3.3 Nuclear Demagnetization Stage 33
3.3.1 LCR circuit 33
3.3.2 Heat leak 39
3.3.3 Nuclear demagnetization experiment 40
4 Conclusion & Outlook 43
4.1 Conclusion 43
A 49
A.1 Code for SQUID data analysis 49
A.2 Demagnetization ramp overview 51
Acknowledgements 53
Chapter
1
Introduction
For quite a while now, people are exploring the region of physics where quan-tum mechanics meets classical mechanics. Interesting questions arise; ’How big of an object can one place in a superposition?’ is one of them. Another question frequently asked is about the collapse of the wave function upon measuring a quantum state. Is it some real physical interaction/event or is it just an empty explanation of why we measure only one measurement outcome at a time?
One approach to the last question is by studying the properties of wave func-tion collapse models, such as the mass-proporfunc-tional version of the Continuous Spontaneous Localization (CSL) model [1]. These types of models predict an offset in the measured temperature-dependent thermal energy (E ∝ kBT), be-cause a little bit of energy is associated with every collapse of the wave func-tion. Therefore collapse will change the thermal energy, inducing a change in the temperature, as is shown in figure 1.1. This particular model has two free parameters: a collapse rate λ and a characteristic length rC. These phenomeno-logical constants are then bounded by experiments, like for example measur-ing the energy of a(n) (ultra-cold) nanocantilever. A so-called exclusion plot is made for this parameter space in figure 1.2. The goal of the research done in this thesis is developing the measurement technology necessary for improving the bounds (red line) on the parameters for this model. Though in this thesis it is still quite far away from doing the actual CSL experiments, we propose some improvements that can be already made. Two of them are the main focus of this research; some additional improvements are treated in the last chapter for future developments.
The first focal point is the optimization of the nuclear demagnetization pro-cess in one set-up. Nuclear demagnetization is a technique used to cool samples to mK or even lower temperatures. Low temperature means low thermal energy and this is needed to be able to detect the very small offset due to wave
func-Figure 1.1: This figure shows a plot of the cantilever temperature as a function of the bath temperature. Normally it would be a straight line through the origin. However, CSL models predict a slight offset in this relation, due to the extra energy from wave function collapse; the line does not go through the origin, but intersects slightly above it. Adapted from [1].
Figure 1.2: This figure shows a plot of the parameter space of the mass-proportional CSL model. The red curve indicates the experimental upper limit on the collapse rate λ, using an ultra-cold nanocantilever. The black dotted line is a suggested upper limit reached by a proposed upgraded cantilever set-up. Adapted from [1].
Figure 1.3: This figure shows the entropy-temperature curves for various values of the magnetic field for a nuclear refrigerant. Different stages in the process are shown, in-dicated by AB (magnetization), BC (demagnetization) and CA (warming up). Adapted from [2]
tion collapse. An explanatory figure of the (de)magnetization process is seen in figure 1.3. Starting at some initial temperature, where the randomly oriented nuclear spins are fluctuating due to thermal energy, a magnetic field is applied to a refrigerant, while keeping the temperature constant by coupling it to a ther-mal bath, e.g. the coldest stage of a dilution refrigerator (AB). The spins start to align with the field and therefore the entropy decreases, giving their energy to the bath. After that, the refrigerant is thermally decoupled from its envi-ronment (mixing chamber of a dilution refrigerator). Then the field is slowly ramped down to some final magnetic field, which could be chosen to be zero, as shown in the figure. This results in a decrease in temperature as well (BC). After a while, the nuclear spins in the refrigerant reorient themselves and the entropy and temperature of the system increase back to their initial values (CA). The second focal point in this thesis is SQUID thermometry. A SQUID (Su-perconducting QUantum Interference Device) can be used as an absolute ther-mometer, needing only one calibration point. A schematic drawing of a (dc) SQUID is shown in figure 1.4. It consists of a superconducting ring with two Josephson junctions (1 and 2 in figure). A bias current I is applied and splits into the two paths of the ring. A magnetic field, with the flux fully enclosed by the SQUID, produces a current J. Then the currents in the junctions are I1= I/2 + J and I2= I/2 - J. The flux modulates the current in and the voltage over the junctions, according to the first and second Josephson relation respectively.
Because the period of the modulation is φ0 = 2·10−15 Tm2, a SQUID is very sensitive to the change in flux and thus can be used as a flux meter.
Figure 1.4: Schematic of a SQUID. You can clearly see the bias current (I), the induced current (J) and the two Josephson junctions (1,2). Adapted from [3].
Figure 1.5: Power density spectrum, SI, of current fluctuations for different
tempera-tures. Dots are the acquired data and the solid lines are the fits to a derived expression. Adapted from [4].
From measuring the SQUID voltage one can calculate the flux directly, but usually the power density spectrum (PDS), SV, of the voltage is evaluated. This is obtained by taking the absolute value squared of V(ω), the Fourier transform
of the measured signal V(t). The shape of this spectrum can be seen in figure
1.5. A calculation for the spectral density for magnetic field fluctuations can be found in [5]. Because the PDS depends at low frequencies (linearly) on the temperature, a SQUID can be used as a thermometer. In this thesis, the SQUID is used as a magnetic flux/field fluctuation thermometer (MFFT), measuring the thermal noise of the magnetic field from a separate noise source. Fitting the spectrum and calculating the temperature can then be done using the ex-pressions found in [6]. In the simplest version of the thermometer (being a sec-ondary thermometer), one reference temperature must be known from already (reliably) calibrated thermometers. Details are given in chapter3.
The next chapter consists of theoretical details of the demagnetization process, combined with results from calculations and numerical simulations. In chap-ter3, experimental results from SQUID measurements are presented. Heat leak and other dissipation related measurements are also included. Finally, some changes for future developments are discussed.
Chapter
2
Theoretical results
This chapter consists of calculations and simulations that describe the demag-netization process, starting from simple heat calculations to the more compli-cated optimization of the final magnetic field and the magnetic field ramp rate. Finally, the topic of heat leaks is touched upon.
2.1
Magnetization process
Calculating the entropy S and heat capacity CB of a perfect paramagnet, i.e. a paramagnet that has no interaction between its nuclear magnetic moments, re-sults in the two expressions below [7]. Here, I have already taken into account the fact that nuclear magnetic moments are three orders of magnitude smaller than electronic magnetic moments, therefore allowing a high temperature ap-proximation of the original expressions for both quantities [7].
S=nR log(2I+1) − nλB 2 2µ0T2 (2.1) CB = nλB 2 µ0T2 (2.2)
In these expressions, n is the number of moles of the paramagnet, R is the ideal gas constant, I is the nuclear angular momentum, B is the magnetic field, T is the temperature, µ0is the vacuum permeability and λ is the nuclear molar Curie constant.
From these two equations you can determine the following differential equa-tion, which takes the form:
TdS=CBdT−nλB
µ0TdB (2.3)
We are looking for an equation that says something about the (de)magnetiza-tion process. A good quantity with which you can describe and characterize these processes is the heat, i.e. the amount of energy you can put in or is needed to cool the system. A general expression for the heat is:
Q= Z
T dS (2.4)
Applying equation2.3and equation2.4on the magnetization process, where we change the magnetic field from zero to some initial field Bi, but keep the temperature constant, we get:
Qmagnetization,ideal = − Z Bi 0 nλB µ0T dB = − 1 2 nλB2i µ0Ti (2.5) Ti denotes the initial temperature at which you start the demagnetization process. This equation describes the ideal system, in which isothermal magne-tization can be obtained. Normally this is not viable; the temperature increases due to external heating (Eddy currents). For the non-ideal case one needs to use the full equation2.3.
Qmagnetization = Z Tmag Ti CBdT− Z Bi 0 nλB µ0T dB (2.6)
Tmag is the temperature that one gets through unwanted heating in the sys-tem. Qmagnetization is the amount of heat that is released by ramping up the superconducting magnet, where the heat that is produced by Eddy currents is implicitly accounted for in Tmag.
2.2
Demagnetization process
The demagnetization process is very similar to the magnetization process. The only assumption I make now is that the precooling bath takes in the excess heat produced by ramping the magnetic field up, such that we start this process again at an initial temperature of Ti.
Qdemagnetization = Z Tf Ti CBdT− Z Bf Bi nλB µ0T dB (2.7)
2.2 Demagnetization process
Table 2.1:Values of useful physical quantities for some refrigerants. Adapted from [7]. In the case of PrNi5, values for the Curie constant and Korringa constant are for141Pr
atoms only.
Refrigerant Internal field b (mT) Curie constant (10−12Km3/mole) Korringa constant (sK)
Cu 0.36 4.05 1.1
Ag 0.035 0.0205 10
In 250 17.4 0.086
PrNi5 65±5 [8] 1.7819·103[2]∗ <0.001 [2]
*Value for the nucleus with hyperfine enhancement
Here Tf and Bf are the final temperature and final magnetic field, respec-tively, of the demagnetization stage. Eddy currents warm up the demagnetiza-tion stage, so they effectively raise the final temperature Tf that can be reached. An important note is that at low enough B fields, the internal field b, arising from interactions between the nuclear magnetic moments, can start to play a role. This depends heavily on the material one wishes to use. For a few refrig-erants, the values of the internal field are listed in table2.1. If b is larger than, or of equal order of magnitude as, the magnetic field Bf, it influences the tem-perature in such a way that the final temtem-perature depends on initial conditions and internal field [2] (page 220).
Tf = Ti Bi q B2 f +b2 (2.8)
We get from this a hypothetical minimum temperature that can be reached.
Tf ,min = bTi
Bi (2.9)
Filling in the value of the internal field for PrNi5and an initial external field of 2T, we should get at least in the low mK range. Taking an initial temperature of 10mK, which should not be an insurmountable problem, we even get a final temperature that is between 300 and 350 µK, depending on which value you take for the internal field b. Note that PrNi5is not a normal metal, and therefore the expression is slightly modified (see section2.5).
2.3
Cooling power of a Cu refrigerant
When the demagnetization stage has reached its final temperature, it is used to cool the sample. To efficiently cool to ultra-low temperatures, it needs to have a high cooling power. It needs to be able to absorb a lot of heat from the sample. An expression for this amount of heat is the following:
Qcooling = Z ∞ Tf CBdT = nλB2f µ0Tf (2.10) If the final field and final temperature are known, also the amount of heat that you can absorb is known. The top bound going to infinity is allowed, because the entropy doesn’t change the amount of heat at some temperature T > Ti [2]. The ”cooling power” is plotted for various values of the final mag-netic field as a function of the final temperature in figure2.1below.
Figure 2.1: This figure shows a plot of the heat that can be absorbed by a Cu refrigerant of 0.75 moles used in a demagnetization stage. For realistically attainable values of the final magnetic field Bf and final temperature Tf, it spans the nJ range.
2.4
External heating and optimization of B
fand ˙
B
for a Cu refrigerant
This section discusses how the external heating influences the final magnetic field Bf, and therefore also the final temperature Tf, defined by equation 2.8. The magnetic field, Bf and the ramp rate, ˙B, are optimized accordingly.
As occasionally mentioned in previous sections, there is a heat flow due to ex-ternal sources that warms up the entire set-up. A big part of the heat flow is due
2.4 External heating and optimization of Bf and ˙B for a Cu refrigerant
to Eddy currents, an important phenomenon that gives rise to heating. These currents arise from a change in magnetic field and warm up every metallic part that is exposed to such changing magnetic fields in the cryostat. An expression for the heat rate is found in [7].
˙ Qeddy = σ AV 8π ∂B ∂t 2 (2.11) This equation describes the Eddy currents in a metal cylinder where the magnetic field is aligned along the cylindrical axis. Here, σ is the electrical conductivity, A is the cross-sectional area perpendicular to the magnetic field B, and V is the volume of the cylinder.
The total heat flow produced in the system, ˙Qtotal, is the sum of ˙Qeddy, ˙Qvibration and ˙Q0, which accounts for all the imperfections of the set-up and radiation from the magnet and outside. The last term can be taken constant, because it depends only on the specific set-up, which is unique; every cryostat therefore has a different ˙Q0. In equilibrium, the cooling power of the nuclei due to de-magnetization, ˙Qdemag, cancels out the heating due to Eddy currents and other sources of heating. Then ˙Qtotal is also the same as ˙Qdemag.
An important note before the derivation of the optimum magnetic field, is that there is a clear difference between the temperature of the nuclear spins, Tn and the electronic part of the sample, Te. Nuclear demagnetization works directly on the nuclear spins, and in their turn the nuclear spins need to cool down the electrons. Therefore, the nuclear spins are colder than the electrons, and a heat flow exists from the electrons to the nuclear spins. From [2] (pages 227, 228), we have: ˙Tn = (Te−Tn)Tn κ (2.12) ˙ Qtotal =CB˙Tn (2.13)
In this expression, κ is the Korringa constant, which is independent of tem-perature. From now on the subscript total in ˙Qtotal is omitted for brevity.
Combining equation2.12with equation2.13and rewriting, we get: Te = " µ0κ ˙Q nλB2f +1 # Tn (2.14)
(a) (b)
Figure 2.2: Plot of the electron temperature according to equation2.15for copper. We have (a) ˙Q = 10−7W, Bf ,opt ≈0.21 T, Te ≈2.13 mK and (b) ˙Q = 10−8 W, Bf ,opt ≈67.5
mT, Te ≈0.67 mK. The temperatures are the minimum electron temperatures as seen in
the graphs.
Using equation2.8without b2, we find: Te = µ0κ ˙ Q nλBf Ti Bi +Bf Ti Bi (2.15) From the above expression the derivative with respect to the magnetic field is taken and set to zero for optimization purposes. We find for the final result:
Bf ,opt = s
µ0κ ˙Q
nλ (2.16)
Using equation2.8in the derivation is not quite correct. It assumes that there is no heat load at all. However, this approximation gives us just a small error, and therefore, the optimum final magnetic field is still a good estimate of what you should aim for in practice. Equation2.15 is also plotted for two different values of the heat leak, shown in figure2.2. Appropriate values for our set-up are n = 0.75 moles, Ti = 10mT and Bi = 2T. The magnet in our set-up ranges from 0 to 2T, which is shown in the figure as the maximum value on the x axis.
Besides optimizing the final magnetic field, the rate at which to ramp from Bito Bf is important. This is done by calculating the total heat from all the heat leaks, and minimize for ˙B.
Qtotal = Z τ
0 ˙
Qeddy+Q˙vibration+Q˙0 dt (2.17) Here ˙Qeddy =γ ˙B2, ˙Qvibration =δ|B|and ˙Q0is a constant [9], [10]. Comparing
2.5 Optimization of Bf and ˙B for the Van Vleck paramagnetic rare-earth compound PrNi5
to2.11, γ is the (mostly) geometrical factor in front of ˙B2. The ramp takes a time τ = |Bf−˙BBi|. Taking the derivative with respect to ˙B and putting it to zero yields the following solution:
˙Bopt = v u u t ˙ Q0 γ + 1 2 δ γ |B2f −B2i| |Bf −Bi| (2.18) Here we can take Bf ,opt from equation2.16 as our final magnetic field and put that into the equation above. In this way a prediction can be made for the ramp rate, using the already optimized final magnetic field.
2.5
Optimization of B
fand ˙
B for the Van Vleck
para-magnetic rare-earth compound PrNi
5An important refrigerant is the rare-earth compound PrNi5, which has the prop-erty that you do not need as big of an external magnetic field B as, for example, copper. This is much more practical and PrNi5 is then preferred above copper when only refrigerating to the mK range. For temperatures much below this, µK or nK range, there is just one option: copper.
The difference between copper and PrNi5 is the response to an externally ap-plied magnetic field. In the rare-earth compound this field changes the elec-tronic structure and a hyperfine field Bh f is generated, due to an induced elec-tronic magnetic moment. This field then modifies the derived expressions in this chapter with a factor (1 + K)n, where K is the Knight shift Bh f/B, and n is an integer. The magnetic field gets modified by a factor (1 + K) and the nuclear molar Curie constant by (1 + K)2. For PrNi5the hyperfine enhancement factor is K = 11.2±0.05 [8]. Going back to equation2.9, we get now a modified version:
Tf ,min= bTi (1+K)Bi
(2.19) As a reminder, b is the internal magnetic field, which is especially large in PrNi5. Therefore it is a necessity to take this field into account when dis-cussing properties of this material. Filling in the same values as in section 2.2
we now have a new estimate of the hypothetical minimum temperature, namely between 25 and 30 µK. However, the lowest temperature reached with the com-pound is 0.19 mK [11]. This is mostly due to the fact that PrNi5has a magnetic ordering at a temperature around 0.4 mK, increasing the minimal obtainable temperature [12].
Also taking a look at the cooling power in the case of PrNi5, putting a factor of(1+K)4 and the internal magnetic field into equation2.10 and filling in the appropriate values from table2.1, we see a vast difference between figure2.1
and figure2.3below. It is clear that PrNi5 can cool very sufficiently compared to a Cu refrigerant.
Figure 2.3: This figure shows a plot of the heat that can be absorbed by a PrNi5
refrig-erant of 0.75 moles used in a demagnetization stage. For realistically attainable values of the final magnetic field Bf and final temperature Tf, it spans the mJ range.
Qcooling = Z ∞ Tf CBdT = nλPrNi5(1+K) 2 µ0Tf h (1+K)2B2f +b2i (2.20) Here, (1+K)2λPrNi5 is the value found in 2.1. The fact that we take only
the Pr nuclei into account is because the Ni ions behave as Cu [2] (section 10.7). Therefore their Curie constant, not enhanced, is negligible compared to that of the Pr nuclei.
Now coming to the main subject of this section: the optimization of the final magnetic field and the ramp rate. Following the same procedure as in the pre-vious section, we have:
Te = " µ0κ ˙Q nλBf 1 (1+K)4 +Bf # Ti Bi (2.21) It has a similar solution to equation2.16.
Bf ,opt = 1 (1+K)2
s µ0κ ˙Q
2.5 Optimization of Bf and ˙B for the Van Vleck paramagnetic rare-earth compound PrNi5
Remembering that the internal magnetic field, b, is very large for PrNi5, this needs to be taken into account as well. Then equation2.21becomes:
Te = " µ0κ ˙Q nλBe f f 1 (1+K)2 +Be f f # Ti q (1+K)2B2 i +b2 (2.23) In this equation, Be f f = q (1+K)2B2
f +b2. Putting the derivative to zero for optimization purposes then gives the following expression:
Bf ,opt = 1 (1+K)
s
µ0κ ˙Q
nλ(1+K)2 −b2 (2.24) One can immediately see that the expression under the square root is neg-ative for some value of the heat leak ˙Q. What this means is that equation2.24
is purely a theoretical predicted optimum that can not always be achieved in practice. If the heat leak is small enough, such that the theoretical optimum is imaginary, the optimum in practice will be 5mT, because the magnetic field should not go under the critical field of the cadmium solder, which is about 3mT, because when the solder becomes superconducting, its heat conductivity becomes very poor. For two different heat leaks the electron temperature as a function of magnetic field according to equation2.23is plotted in figure2.4. One value of the heat leak that gives a real optimum field and one value that gives an imaginary theoretical optimum. Also the internal magnetic field is varied, according to its approximate value from table2.1: b = 60mT (2.4b,2.4d) and b = 70mT (2.4a, 2.4c). For the more appropriate heat leak of 5.5·10−8W, obtained by optimizing the magnetic field ramp rate, the electron temperature is as low as 35-38µK, which is five times colder than the current record with PrNi5 [11]. The model then seems to be incapable of predicting the exact temperature of the electrons. However, it predicts a decrease of temperature when decreasing the final magnetic field, up until the lowest field in practice. Even if the temper-ature is not correct, demagnetization until you reach a field of 5mT still gives you the lowest temperature.
If one wants to be precise, there is a small correction to be made. The Kor-ringa constant depends on the magnetic field if the field becomes comparable to the internal magnetic field b.
κ(B) = κ(∞) B 2+b2 B2+
αb2 (2.25)
Here α, with a value between two and three [2], is determined by the internal interactions, and κ(∞) is the Korringa constant in the regime Bf b, given in
(a) (b)
(c) (d)
Figure 2.4: Plot of the electron temperature according to equation 2.23 for the Van Vleck paramagnet PrNi5for two different values of the heat leak ˙Q and internal field b.
Looking at the lowest obtainable electron temperature, we have (a) ˙Q = 5.5·10−8 W, Te= 38.1 µK; (b) ˙Q= 5.5·10−8W, Te= 35.1 µK; (c) ˙Q=1.6·10−2W, Te= 100.5 µK; (d)
˙
Q=1.6·10−2W, Te= 100.5 µK.
table2.1. Calculating an optimum magnetic field gets a bit harder. Using equa-tion2.23with the Korringa constant as in2.25will yield the following result:
Bf ,opt = s − w2 2w1 ± 1 2w1 q w22−4w1w3 (2.26a) w1= nλ(1+K)2 µ0κ(∞)Q˙ (1+K) 4 (2.26b) w2 = 2αb 2w 1 (1+K)2 − (1+K) 2 (2.26c) w3 = α2b4w1 (1+K)4 + (2−α)b 2 (2.26d)
2.5 Optimization of Bf and ˙B for the Van Vleck paramagnetic rare-earth compound PrNi5
(a) (b)
(c) (d)
Figure 2.5: Plot of the electron temperature according to equation 2.23 for the Van Vleck paramagnet PrNi5with a small correction from the Korringa constant. Evaluated
at ˙Q = 5.5·10−8W. Looking at the lowest obtainable applied magnetic field of 5 mT, we have (a) α=2, b = 70 mT, Te= 38.1 µK; (b) α=3, b = 70 mT, Te= 38.1 µK; (c) α=2,
b = 60 mT, Te= 35.1 µK; (d) α=3, b = 60 mT, Te= 35.1 µK.
In figure2.5the results for this small correction are shown, using a heat leak of ˙Q = 5.5·10−8 W. The plots were made for two values of the internal mag-netic field and two values of the material dependent constant α. The differences in temperature are less then 1 nK between the two models. It means that the correction is so small we can neglect the fact that the Korringa constant is de-pendent on the magnetic field. For a heat leak ranging from 10−8W to 10−7W, the difference (∼0.4 nK), is also negligible.
Optimization of the magnetic field ramp rate in the case of PrNi5uses the same integral as in equation2.17. As heating through Eddy currents and vibrations is only due to conduction electrons, there is no hyperfine enhancement or internal field that needs to be accounted for. The expression is thus the same for PrNi5 as it is for copper.
˙Bopt= v u u t ˙ Q0 γ + 1 2 δ γ |B2f −B2i| |Bf −Bi| (2.27) From [9] and [10] we get approximate values for γ, δ and ˙Q0. This is allowed, because our cryostat and the one from [10] have both a demagnetization stage made from PrNi5. However, considering [10] uses a wet cryostat, the parameter δshould be larger for us [9]. For the Eddy currents, there are of course some dif-ferences as well. We have three times more PrNi5, resulting in a 6-fold increase in Eddy currents within the PrNi5 itself. Looking at the contributions to these currents [10], this increase does not matter for the total amount of Eddy current heating. The copper in the set-up is much more important. Within the magnet coil the effect is the strongest; here is no copper in our set-up. Compare this with 2.8 moles of copper wire in [10]. We do have a lot of copper outside the mag-net, but we did not analyze how much Eddy current heating may be expected from the magnetic field outside the magnet. Thus possibly, the parameter γ is smaller than in [10]. Concluding, the values from [10] are sufficient to optimize the magnetic field ramp rate, but they are most likely to be slightly different, as discussed above. Using γ ≈ 0.15±0.05, δ ≈ 10−8 and ˙Q0 ≈ 22 nW, a value for the optimum ramp rate is calculated. The demagnetization ramp starts at 2 T and ends at 5 mT; for any appropriate heat leaks the final magnetic field is limited to 5 mT. We get ˙Bopt = (4.88±0.88) ·10−4T/s. According to [2] (page 235), in practice, a ramp rate equal to or less than 1 T/h should be appropriate. The here calculated optimum is between 1.44 and 2.04 T/h, which makes our estimate to be within a factor of two of the above inequality. Then again, the parameters were taken from measurements in a different cryostat and set-up, but even in [9], [10] it was not a completely made out case; the constant heat leak ˙Q0 was varied as 2nW plus 5, 10 or 20 nW. At the lowest value, the ramp rate would be 1.05 T/h, which is almost within the bounds of what seems to be appropriate. A list of values of the optimized ramp rates is shown in table
2.2. To be able to make equation2.27truly work, an analysis has to be made of temperature data from demagnetization runs in our own cryostat. From that, the parameters can be estimated properly, and optimization for our set-up can be done with greater accuracy.
Using this optimized ramp rate ˙Bopt ( ˙Q0 = 22 nW), one can also calculate the heat leak ˙Qtotal according to equation2.17. An estimate of this quantity is
˙
Qtotal = (7.32±2.52) ·10−8 W. A rate that is used often in our experiments is ˙B = 56 ·10−4 T/s (it is equivalent to 0.1A/min). Accounting for the different values of ˙Q0(top three values in table2.2), we get ˙Qtotal = (2.55±0.75) ·10−8 W. We now have a feeling for what the value of the heat leak should be. To
2.6 Thermal contact with Ag sample
Table 2.2: Calculated values of the optimized ramp rate for different given values of the constant heat leak ˙Q0.
˙ Q0(nW) ˙Bopt (10−4T/s) 22 4.88±0.88 12 4.005±0.685 7 3.52±0.60 2 2.955±0.505
summarize, the heat leak is in between 10−8W and 10−7W.
Optimization of the magnetic field ramp rate results in a larger ˙Qtotal than using a ˙B from practice. This might feel counter intuitive. However, we do not want to minimize the heat flow per second, but the total amount of heat that is dissipated. This is calculated as ˙Q·τ, where τ =
|Bf−Bi|
˙B . This leads for ˙Bopt and ˙Q0 = 22 nW to a total amount of heat Qopt = (3.3±1.6) ·10−4 J. For a common field rate in practice, as stated above, the total heat will be Qnorm = (6.1±1.8) ·10−4 J. Again, the lowest value indicates ˙Q0 = 7 nW and the highest value indicates ˙Q0= 22 nW. The optimized case has about 0.62 times the dissipated heat of the not optimized case. Indeed, optimization is useful in lowering the temperature as much as possible.
2.6
Thermal contact with Ag sample
This section gives a simple example of a demagnetization stage in thermal con-tact with some given sample, as seen in figure2.6. We take a block of PrNi5for our demagnetization stage with a volume and temperature of V1and T1 respec-tively. The silver wire and block are taken into account separately, getting the subscript 2 and 3 only for their volume. Using conservation of energy combined with equation2.2, we get the following equation:
n1λPrNi5B2(1+K)4 µ0 Z 1 T2dT+ (n2+n3)λAgB2 µ0 Z 1 T2dT = [n1λPrNi5(1+K)4+n2λAg+n3λAg]B2
µ0
Z 1
T2dT (2.28)
Where ni is the volume in number moles of the demagnetization block and the sample, and the factor (1+K)4 comes from the hyperfine enhancement.
The first indefinite integral on the LHS is evaluated at a temperature T1and the other one is evaluated at T2. On the RHS the integral should be evaluated at the equilibrium temperature of the system Teq. You can easily see that the magnetic field and the permeability constant drop out.
Figure 2.6: Simple configuration of a demagnetization stage consisting of a block of PrNi5 in thermal contact with a silver wire and block. A constant magnetic field is
applied over the whole system. (It is in fact the final field after the demagnetization process Bf.)
Finally, this equation results in the following expression: Teq= T1T2
n1λPrNi5(1+K)4+ (n2+n3)λAg n1λPrNi5T2(1+K)4+ (n2+n3)λAgT1
(2.29) The molar Curie constant of PrNi5 is three orders of magnitude larger than that of silver. As seen in the expression above, n2and n3can be made very large without changing Teqmuch, i.e. silver looses its heat easily and therefore a con-siderable amount of it can be cooled down efficiently.
As a last item in this section, let us calculate the amount of energy that is needed to cool down the sample from an initial temperature to a final tempera-ture, considering of course that the demagnetization stage is at its coldest. The magnetic field then is Bf.
Qsilver = Z Tf Ti CBdT = − (n2+n3)λAgB2f µ0 " 1 Tf − 1 Ti # (2.30)
2.6 Thermal contact with Ag sample
To get a feel for how much (or little) energy is needed to cool down silver, we put in some numbers. Ramping down to 5 mT, as usual, from a temperature of 10 mK to 5 mK is a realistic scenario. A silver wire, 1 mm diameter and 20 cm long, with a block of 1 cm3 silver at the end is about 0.11 moles (n2 + n3). Using the value for the Curie constant from table2.1, we get the low number of 4.5 pJ. This is easily achieved when using a PrNi5 refrigerant, due to its enor-mous cooling power. The rest of the cooling power may be used for the power dissipated in the experiment.
Chapter
3
Experimental results & discussion
This chapter provides the used set-up for the nuclear demagnetization stage (NDS) and for the SQUID data acquisition. The data together with its analysis and discussion is then presented, both for SQUID thermometry and the NDS.
3.1
Set-up
This section provides the used set-up for the different experiments.
3.1.1
Cryostat & Nuclear Demagnetization Stage (NDS)
All the experiments are done in a dilution refrigerator with a single pulse tube, model CF-1400 made by Leiden Cryogenics (LC), which has a base temperature of 10 to 12 mK. A picture of the cryostat is shown in figure 3.1. It consists of multiple plates with a different temperature. Thermometers are placed on each one of them and are calibrated correctly. Between the 50mK and 1K (Still) plate the NDS is placed, which consists of rods of the material PrNi5. It is thermally coupled to the mixing chamber (MC) with a conducting/superconducting alu-minium heat switch. To measure the temperature as close to the NDS as possi-ble, a copper block with two thermometers and a heater is attached to the PrNi5 through silver wires and to the mixing chamber through the heat switch. The copper block is annealed in vacuum at 950◦C for 72 hours for a higher thermal conduction. For the demagnetization experiments, a 2T superconducting mag-net is installed, with a PS 120-10 Oxford Instruments power supply that sends 40A to achieve its maximum field. The heat switch is turned on/off by sending a current between 150 and 200 mA using a Yokogawa GS200 current source. By
sending a current, the superconducting (closed) state of the aluminium changes to normal conducting, and thus also thermally conducting (open).
Figure 3.1: Picture of the cryostat. Every plate has a different temperature and needs to be thermalized. In the middle is the PrNi5nuclear demagnetization stage (NDS). It
is thermally coupled to the mixing chamber through an aluminium heat switch.
(a) (b)
Figure 3.2:Pictures of (a) the NDS before mounting onto the cryostat, where the PrNi5
is seen at the top and the aluminium heat switch at the bottom, and (b) the annealed (and gilded) Cu block.
3.2 SQUID thermometry
The cryostat is challenged by the heat load on the 4K plate. There are so many wires going down from room temperature that the pulse tube has trouble keep-ing it sufficiently cold, especially when the magnet is at maximum current and the heat switch is powered. The thermometers are read out with Labview and the data is analyzed with Origin.
3.1.2
SQUID
SQUID experiments were done in a different, but similar cryostat. The SQUID was attached to a transformer that was inside an aluminium housing to shield it from external magnetic sources. A golden strip attached to the box was used as a source for thermal noise, to be read out by the SQUID. The aluminium box and SQUID were then placed on the 50 mK plate. Figure3.3 shows the set-up. The SQUID was tuned with SQUIDviewer and read out with Labview. Data was analyzed using Python.
Figure 3.3: Picture of the SQUID set-up, taken when it was placed on the 10 mK plate for a second run. To the left one can see the aluminium box. In the middle is the SQUID (silver), obscuring the golden strip behind it.
3.2
SQUID thermometry
In this section we discuss how to analyze the data obtained by the SQUID, using a theoretical model to extract a temperature from it.
3.2.1
Spectrum to temperature conversion
As mentioned in the introduction, a conversion needs to be made from the power density spectrum to a temperature. From [6] we have the following ex-pression for the flux PDS:
SΦ(f , T) = S0(T) [1+ (ff
c)
2a]b (3.1)
Here S0(T)is the zero-frequency value of the spectrum, fc, a and b are con-stants. The zero-frequency value can be obtained from the data and figure1.5
looking at the horizontal part of the spectrum at very low frequencies. It is a function of temperature, directly and through the electrical conductivity. As it is based on thermal noise (Johnson noise) we have a direct influence which goes linear with temperature.
SV(T) =4kBTR (3.2)
The electrical conductivity at low temperatures is dominated by a residual resistivity ρ0and can be taken to be independent of temperature. The full curve is seen in figure3.4, using ρ =ρ0+ρi(T), where ρi(T)depends on temperature according to the Bloch-Gr ¨uneisen equation.
3.2 SQUID thermometry
At low temperatures, ρi behaves as T5 and at high temperatures as T. One can see that in the mK range the resistivity is constant, gaining a linear relation for the power density spectrum. Thus, at low temperatures, the temperature dependency of S0 is solely due to Johnson noise, and is the only parameter in equation 3.1 that is dependent on temperature. Calculating the temperature from this is then done by the simple equation:
T =Tre f
S0(T) S0(Tre f)
(3.3) Using a single reference temperature with its data is enough to calculate every other temperature after gaining the spectrum. Fits of the spectra are made using equation3.1, with S0already evaluated from the low-frequency part, but fc, a and b free parameters. Note that SΦ is related to SV by some factor, which can be calculated. However, the form of equation 3.1 is maintained; only the zero-frequency value is changed by that factor. The formula thus works for any power density spectrum of a SQUID measuring thermal noise.
3.2.2
Analysis
As the data obtained in this experiment was not, unfortunately, particularly good, we present here only the analysis method. In five steps, using the data at a temperature of around 1 K, a simple guide has been made with accompany-ing figures, in order to obtain the parameter S0. This parameter tells us what the temperature is, calculated from one reference point. To get the right value of S0 we have to smooth out the noise peaks first before calculating the fit curve that gives us a value for S0. At the end of the last step, a little discussion explains what possible effects caused the data to be this way.
Step 1: Importing and plotting data
Reading data with Python can be done using an inbuilt function. There are multiple of these (very similar) functions; we chose here to read from an Excel file.
N = 100000
#Load file paths
x = pd.read excel(”P:\SQUID processed data\SQUID data test.xlsx”,nrows=N, usecols=[0], header=None, dtype=np.float64, squeeze=True)
y = pd.read excel(”P:\SQUID processed data\SQUID data test.xlsx”,nrows=N, usecols=[2], header=None, dtype=np.float64, squeeze=True)
#Plotting the data plt.plot(x,y, label=”Data”) plt.xlabel(”Frequency (Hz)”) plt.ylabel(”SV (Vrms2 /Hz)”) #Define axes plt.xscale(’log’) plt.yscale(’log’)
This piece of code will result in the following figure:
Figure 3.5: Data from SQUID measurement at a temperature of 1 K. The PDS is plotted against frequency.
Immediately apparent in this figure are the many interference peaks, most of which are mechanical in origin. This is a common problem for SQUID ther-mometry. Nonetheless, the data also shows the baseline noise in between the interference peaks.
Step 2: Cutting the data
This is the most important step. Here, we will get rid of the noise peaks seen in the data. We already know that it follows equation3.1, so we make the ansatz: ymax = A· f−α+B, where A and B are constants to be determined from two ap-propriate points obtained from the data. Note that this only works on the part that is frequency dependent. A list of maxima for all frequencies is made. If a data point, y, is higher than this value, ymax, it will automatically be set to this maximum. If the value is less, it keeps that value. The data here also shows a dependency at low frequencies, which we will talk about later. The list is sliced to account for the part that we want to change, i.e. the part with a lot of high
3.2 SQUID thermometry
peaks. Adding to the previous code, we get: alpha = 2.2
#Set cutting parameters
X = get function(x[1000], y[1000], x[30000], y[30000])
#Calculate list of maxima
ymax = get list(X,x) y max = ymax[400:]
#Cutting the data
y new = np.zeros(N) for i in range(len(y new)):
if i>400: if y[i]>y max[i-401]: y new[i] = y max[i-401] else: y new[i] = y[i] else: y new[i] = y[i]
plt.plot(x,y new, label=”VW smooth”,color=”g”)
The parameter α can be changed, but should be around two as it is the same as 2·a·b in equation3.1, where a and b are both close to one [6]. This extra code gives us figure3.6below.
Figure 3.6: Data smooth from SQUID measurement at a temperature of 1 K. The PDS is plotted against frequency.
Step 3: Fitting the data
This step is pretty straightforward. We take the part that follows f−α the best and use an inbuilt function to plot a fitting curve.
#Plotting and fitting to last part of the smoothed data
x best = x[900:] y best = y new[900:]
popt3,pcov3 = curve fit(PDS,x best,y best,p0 = (1,1,70,10**-7),maxfev=10000) plt.plot(x best, PDS(x best, *popt3), label=”PDS partly fit to smooth”,color=”r”) print(popt3)
print(np.sqrt(np.diag(pcov3)))
Values of the fitting parameters are a = 4.40±1.81·10−1; b = 2.45·10−1±1.04·10−2; fc = 9.10·101 ±5.56·10−1; S0= 1.37·10−8±2.05·10−10. The fit curve is shown in figure3.7, and it follows the last part, that goes like f−α, really well.
Figure 3.7: Fitting curve of data from SQUID measurement at a temperature of 1 K. The PDS is plotted against frequency.
Step 4: Plotting function
This step uses the fitting parameters to plot the function from equation3.1.
#Plotting PDS
PDS = PDS(x,popt3[0],popt3[1],popt3[2],popt3[3]) plt.plot(x,PDS, label=”PDS”)
3.2 SQUID thermometry
The function used to plot this is: def PDS(x,a,b,f c,S 0):
term1 = 1 + (x/f c)**(2*a) S = S 0 / (term1)**b return S
The plotted curve is seen in figure3.8in orange.
Figure 3.8: Plotting equation3.1with parameters from the fitting function (S0 =1.37·
10−8). Using data from SQUID measurement at a temperature of 1 K. The PDS is plotted against frequency.
Other functions used in this analysis are listed below. The full code is found in appendixA.1.
#Obtaining constants A and B for ymax def get function(x 0, y 0, x 1, y 1):
m list = [[x 0**(-alpha), 1], [x 1**(-alpha), 1]] M = np.array(m list)
b = np.array([y 0, y 1]) X = np.linalg.inv(M).dot(b) return X
# Getting the list of all maxima
def calculate maximum(A,B,f): F = A*f**(-alpha) + B
def get list(X,x):
ymax = np.zeros(N) for i in range(len(ymax)):
ymax[i] = calculate maximum(X[0],X[1],x[i]) return ymax
Step 5: Compare to reference value
Now we have a value for S0 we can calculate the temperature using equation
3.3and the reference point S0(T =586mK) = 1.7·10−8. We get a temperature of T = 472 mK, which is much lower than expected. The problem might be that the orange fitting curve looks good in figure3.8, but should go up to lower fre-quencies. This can be achieved by slicing x and ynew differently.
x best = x[400:] y best = y new[400:]
Now using the same value as earlier in the calculation of the list of maxima.
Figure 3.9: Plotting equation3.1with parameters from the fitting function (S0 =4.10·
10−8). Using data from SQUID measurement at a temperature of 1 K. The PDS is plotted
against frequency.
Calculating the temperature now, we have T = 1413 mK. Concluding here, there are some parameters that you can tweak that alter the outcome significantly. Therefore, a good case needs to be made for each of the different values of the parameters. Then again, the reference value used here, was from a different SQUID (and different tuning). This would change the value of S0as well.
3.3 Nuclear Demagnetization Stage
As a last part of this section, some remarks about the data itself. Usually, the data exhibits the behaviour described by equation3.1: constant at low frequen-cies and roughly f−2 at higher frequencies. We will give a possible reason for the peculiar f−1/2behaviour in the data. The expression for the PDS is based on the thin plate approximation [5], which breaks down at d ≥λskin. Here, d is the thickness of the thermal noise source (golden strip), and λskin = (µ0σπ f)−1/2is the skin depth. With a measured thickness of 160 µm, the frequency at which d = λskin is 2176 Hz, well above the frequency fc. This points in the direction that the effects of the skin depth are negligible. However, approximations usu-ally break down only when, in this example, d λskin. When equating the thickness to only a tenth of the skin depth, the frequency at which that happens is only 21.76 Hz, below fc, in the low frequency range. From this we can see that it does influence the data, but to be able to answer the question with good confidence, it needs to be studied in a next project. Having said that, if we just take a thinner thermal noise source, we can somewhat or entirely negate this skin depth effect. From calculations, we can see that for gold a thickness of 60 µm has a frequency of 154.80 Hz at d = λskin/10, while silver for example has a frequency of 7.04 Hz. A gold strip of 60 µm would then be a great improvement for this experiment.
3.3
Nuclear Demagnetization Stage
In section 3.3.1 we discuss a way to decrease one source of dissipation, Eddy currents, in the cryostat while sweeping the magnetic field. In section3.3.2we discuss a heat leak measurement and calculation using results from the theory of chapter2.
3.3.1
LCR circuit
During a ramp up or down with the magnet power supply, a current source, Eddy currents warm up the system. The sweep rate can be varied, but because it is a digtal current source, the current is increased in small dicrete steps. A possible way to decrease this heating is by using a capacitor to act as a current buffer, smoothing out the voltage peaks, and thus also smoothing out dIdt. The electrical circuit that belongs to this set-up is shown in figure3.10.
The current source sends a step wise signal through the magnet coil, with inductance L. The sudden increase in current gives rise to a lot of heating due to Eddy currents. We want to smooth this step out, getting a more gradual in-crease in current. Then dIdt will be smaller over the whole period of heating. Although the time period will increase, the net heating will be less. This effect
Figure 3.10: Electrical circuit. I is the current from the magnet power supply. R1, R2are
the resistances of the wires, R3 is an extra resistor put in series with the capactor (C),
and L is the inductance of the magnet coil.
can be obtained by placing a capacitor (C) with resistor (R3) parallel to the coil, where R3is used to limit the Q factor of the capacitor combined with the induc-tance L of the magnet.
Recall from section 2.4 that the Eddy currents scale with ˙B2. As the magnetic field is produced by a current, thus B ∼ I, we need to look at the following quantity: QEddy ∼ Z ˙I2dt = 1 L2 Z VL2dt (3.4)
Here, VLis the voltage across the coil, which is the same as the voltage across the capacitor with resistor. The latter signal is measured with a capacitance of 16mF and a resistance of 2.43Ω. The voltage over the coil is also measured, without the capacitor in the system. This is shown in figure3.11.
As can be seen from the data, the signal is periodic. This periodicity comes from the way the current source increases its current input. Every 6-6.5 seconds it raises the current by 1mA, considering the set rate of 10mA/min. From equa-tion3.4 we need to calculate the time integral over the voltage signal squared.
3.3 Nuclear Demagnetization Stage
(a) (b)
Figure 3.11: Voltage signal with (a) and without capacitor and resistor (b). The peri-odicity is about 6s for (a) and 6.5s for (b). Note the difference in vertical scale and that the voltage with the capactior oscillates at a frequency 1/√LC and the voltage becomes negative.
(a) (b)
Figure 3.12: Voltage signal squared with (a) and without capacitor and resistor (b). Only one period is shown.
Because the signal is periodic and we only want to compare the two cases, tak-ing only one period is enough. Figures 3.12 and 3.13 are the voltage squared and the cumulative integral respectively. The integral gives almost a factor 3 in difference between using or not using a capacitor. There is less dissipation with the capacitor (integral gives 2.8 V2s) than without (integral gives 7.4 V2s).
This is exactly what we expected, however, the data for the voltage has quite a lot of noise and it has a slight offset. This asks for a revision: subtract the DC
(a) (b)
Figure 3.13: Cumulative integral with (a) and without capacitor and resistor (b). The full period has a value for the integral of (a) 2.8·10−4V2s and (b) 7.4·10−4V2s. The integral becomes non-zero at a time (a) 1 s and (b) 4.5 s, as those are the moments a period starts. For every moment in time, the integral has been evaluated.
offset voltage and put a filter over the data before the signal is squared. A sec-ond important notion is the current. Before it was stated to be a step function, but looking at the integrated voltage in figure 3.14it clearly increases linearly, while exhibiting the aforementioned steps. This is due to an offset in the data. The current has a different rate for each of the cases. Looking back at3.10 we immediately see that there is a current split. Therefore we expect less current to flow into the coil if there is a second, parallel, path. Showing in figure3.15
is the integrated voltage when the offset is subtracted. It is now clear that the current is indeed a step wise function, still with a slightly different dI/dt. Us-ing the value of the inductance from the magnet specs, L=3H, we have rates of 15.36mA/min and 18.32mA/min with and without capacitor respectively. Comparing these rates to the expected rate of 10mA/min, there is a significant difference. The current without capacitor should be exactly the same as the set current from the source, while the current with capacitor should be smaller, which is still the case. Most probable is that the inductance is not what it says in the specs. Calculating the value from the data without capacitor, it gives L=5.5H.
In figure 3.16 the voltage is shown after the offset is subtracted, together with a smooth of the data. The filter or smooth over the data is done with adjacent averaging (AAv). The data with a capacitor held up its features of interest up until a 1000 point AAv with no further smoothing to be needed. The data without capacitor could only be smoothed to a 100 point AAv while
3.3 Nuclear Demagnetization Stage
(a) (b)
Figure 3.14: Cumulative integral of the voltage with (a) and without capacitor and re-sistor (b). This is the same as the current through the coil multiplied by the inductance. It increases linearly, while exhibiting step wise behaviour. For every moment in time, the integral has been evaluated.
(a) (b)
Figure 3.15: Cumulative integral of the voltage with (a) and without capacitor and re-sistor (b). This is the same as the current through the coil multiplied by the inductance. It increases step wise. For every moment in time, the integral has been evaluated.
maintaining its features.
Integration of smoothed and unsmoothed data is presented in 3.17. As ex-pected, after smoothing, the value of the integral is lower. Smoothed it gives us 7.5·10−5V2s (capacitor) and 2.2·10−4V2s (no capacitor), which is a bit less than what integration yield without smoothing (8.9·10−5V2s and 2.4·10−4V2s respectively). Using these two values we can give a lower bound on the relative
dissipation of 2.5, i.e. having the capacitor and resistor in this system decreases the dissipation with a factor of 2.5 compared to the same system without capac-itor. This could be improved by using a larger capacitance.
(a) (b)
Figure 3.16: Voltage signal with (a) and without capacitor and resistor (b). The offset and AAv smooth are (a) +5mV, 1000pts and (b) +8mV, 100pts.
(a) (b)
Figure 3.17: Cumulative integral with (a) and without capacitor and resistor (b). The full period has a value for the integral of (a) 8.9·10−5V2s unsmoothed and 7.5·10−5V2s
smoothed; (b) 2.4·10−4V2s unsmoothed and 2.2·10−4V2s smoothed, the last value in the graph. For every moment in time, the integral has been evaluated.
3.3 Nuclear Demagnetization Stage
3.3.2
Heat leak
Another way to see a decreased heating due to an added capacitor and resis-tance, is by looking at the temperature of the 50mK plate. The plate is close to the magnet and has been seen to increase in temperature while sweeping the magnet. Comparing two ramps, one with and one without LCR, and calculat-ing the heat ratio Q1/Q2, will show the influence of the LCR circuit.
Consider the expressions for the different contributions to the heat from sec-tion2.4: Qtotal = " γ ˙B2+1 2δ |B2f −Bi2| |Bf −Bi| +Q˙0 # τ (3.5)
Here, τ is the time it takes to ramp from Bi to Bf. As a note: rates and field values will be given in A/min and Ampere for practical reasons in stead of T/s and Tesla.
We compare the following two ramps: 1A→0A with LCR and 0.5A→1A without LCR. They have the same rate of 0.1A/min. We can now calculate the heat ratio. Looking at the extremes, we have a ratio Q1/Q2= 2 when Eddy cur-rents (first term in equation3.5) or the constant heat leak (last term) is dominant; only the ramp time influences the difference between Q1(ramp with LCR) and Q2(ramp without LCR). This is the upper bound on the ratio. When vibrational heating is dominant, we get a ratio of 1.33, the lower bound. We now know that the ratio has a value between 1.33 and 2. However, the calculation uses only the values for the different magnetic fields, magnetic field rate, ˙Q0 and τ. It does not take into account an LCR circuit. In that way, a lower ratio than calculated means that the LCR circuit lowers the heating.
To make a much more precise estimate of this ratio, we need to take a look at what percentage is approximately caused by vibrational heating. Filling in equation3.5with the lowest value of γ and the largest value of the vibrational heating term, we get an upper estimate of 12.23% heating caused by vibrations. This gives us the estimate of the heat ratio Q1/Q2≈1.914. A direct calculation from equation3.5gives us Q1/Q2≈1.92-1.99, depending on γ and ˙Q0. Figure
3.18 shows the plots of the two sweeps. Calculating the heat from this data by integrating T(t) (surface under the graph) in the range of the sweep, subtracting the heat from integrating over the area from zero to the temperature at which the rate starts, gives a heat ratio of Q1/Q2≈5.74. The calculated value does not take into account the effect of an LC resonator (LCR circuit). Having a lower heat ratio from the data would imply a decrease in heating when coupling the system with a LCR circuit, giving more evidence that a heat reduction can be
achieved by putting a capacitor and resistance into the system. Unfortunately, this is not the case here. In fact, it is higher than the upper limit. This should be impossible, but there are some reasons why this method fails to conclude anything about the influence of the LCR circuit. Most important is the fact that the current running through the conduction heat switch heats up the 50 mK plate, but those currents are not likely to be the same between different runs of the cryostat. In this case it is unknown if they are the same or not. Therefore, the two ramps above cannot be compared properly.
(a) (b)
Figure 3.18:Temperature versus time plots of the NDSprobe50mK thermometer. Sweep (a) is with LCR (17-03-2020) and sweep (b) is without (23-12-2019). Red arrows point to the start and end of the sweep.
For future experiments, the ramps should be exactly the same; meaning, having the same field rate, start and final field, starting temperature and the current through the heat conductance switch.
3.3.3
Nuclear demagnetization experiment
Another data set to look at is the temperature data obtained from a thermome-ter on an annealed copper block at the demagnetization stage. This was done in conjunction with Wim Bosch from Hightech Development Leiden (HDL) [14]. The data is shown in figure3.19together with data of the NDSprobe50mK ther-mometer. They show the temperatures from 09:00 (Cu block) and 09:22 (ND-Sprobe) to 00:00. The demagnetization data of the AR-100 thermometer (Cu block) gives a lowest temperature of 1.47 mK. However, there are some doubts about this value. Namely, the AR-100 thermometer was calibrated with a poly-nomial that is valid for a temperature range between 10 mK and 207 mK.
Be-3.3 Nuclear Demagnetization Stage
low the 10 mK, the thermometer is not reliable any more, because extrapolation becomes very dependent on the accuracy of the fit. As discussed in chapter
2, if the temperatures and magnetic fields are not too low, the temperature is proportional to the magnetic field. Using the plateau in figure 3.19a around 5000 seconds, we have 19.1 mK at an external field of 200 mT (using the ramp overview from appendixA.2to get a value of the magnetic field). Extrapolating to 50 mT, we expect a temperature of 4.8 mK. However, when the ramp stops at that field in the last part of the graph, a temperature of 16.54 mK is obtained. This is not what we expect, but the temperature is still dropping down, and fi-nally, just before the ramp from 50 mT to zero field begins, it is as low as 1.98 mK. This delay is due to cooling relaxation: it takes time to cool down until the equilibrium temperature. With this, we can conclude that the lowest tempera-ture reached is roughly 3.6 mK, even though the thermometer reading gives a lower temperature. It is useful still to use the thermometer to check whether the temperature has reached an equilibrium.
(a) (b)
Figure 3.19: Temperature versus time plots of two different thermometers of the de-magnetization run on 17-03-2020. Graph (a) is data from the thermometer on the Cu block and graph (b) is data from the 50 mK plate. The red arrows point to the different temperatures mentioned in the text.
As a test for the paramagnetic model from chapter2, we can put in the num-bers for this ramp. As there are multiple ramps done in this run, we need to choose a ramp that should be most representative. The highest magnetic field achieved was 1 T, at 13:53 (17.552 in figure 3.19a). We thus take a ramp from 1 T to zero field, with a starting temperature obtained from the data of 23.7 mK. Then the model predicts a minimum temperature between 117 and 136 µK. As already stated in chapter2, the temperatures are rather low due to at least one
missing ingredient: magnetic ordering at around 0.4 mK. Comparing the exper-imental result and the theoretical result, there is a factor of 26 to 31 in differ-ence, depending on which value for the internal field one takes. Another factor to play a role is that there is a cooling relaxation. During ramping down, the temperature will firstly increase due to Eddy currents, but eventually the tem-perature decreases. However, equilibrium of the temtem-perature is reached only some time after the ramp is over. This relaxation alters the starting temperature in the model. The model assumes only one ramp down, so not accounting for the multitude of ramps before the final one.
Before we end this chapter, a final piece of data from the thermometer on the copper block, shown in figure3.20. The peaks are due to the heat switch coil being conducting. However, the feature of interest is the constant line at the end of figure3.20a(the same line pops up earlier as well).
(a) (b)
Figure 3.20:Temperature versus time plots of the thermometer on the Cu block of the demagnetization run on 16-03-2020, with zero magnetic field. A heat leak is seen at the end of the left graph. Graph (a) is data from whole day and graph (b) is a zoom-in of the heat leak at the end of3.20a.
In that moment the heat switch is superconducting, but there is a heat flow between the PrNi5 refrigerant and the annealed Cu block, ˙Qexchange. Also a constant heat leak is always present, but no Eddy currents or vibrational heat-ing, because there is zero magnetic field. Then we have a heat leak Qtotal = Qexchange +Q0. As for now, obtaining values for these (contributions to the) heat leaks from data or theory is a too difficult a task, and is left for future de-velopments.
Chapter
4
Conclusion & Outlook
This chapter gives an overview of all the sub-projects and their conclusions, as well as some future improvements on theory and experiments.
4.1
Conclusion
Chapter 2
In this chapter, we looked at the paramagnetic model for nuclear magnetic re-frigeration as a means to calculate certain quantities for the nuclear demagneti-zation process. Expressions for the heat of magnetidemagneti-zation and demagnetidemagneti-zation were derived, as well as the cooling power of a nuclear refrigerant, both for cop-per and the special material PrNi5. After this ”introduction”, the final magnetic field and the magnetic field rate in the demagnetization ramp were optimized to gain the lowest electronic temperature. Finally, a short introduction was made for a test case: contact between a PrNi5 nuclear demagnetization stage and a silver wire and block. A list of conclusions are presented below.
1. For realistic values of final magnetic field and final temperature after the demagnetization ramp, the ”cooling power” (heat that the refrigerant can absorb) spans the whole nJ range for copper and lower part of the mJ range for PrNi5. Sections2.3,2.5.
2. Optimization of the magnetic field gives a value of 5 mT, mainly because of practical limitations. Section2.5.
3. Optimization of the magnetic field ramp rate is given by equation 2.27. The values of the parameters ˙Q0, γ and δ are chosen from [9] and [10].
The values are sufficient for a first estimate of ˙Bopt, but not completely correct. The optimized field rate has a value of approximately 0.5 mT/s. Section2.5.
4. Optimization of the magnetic field rate is used to calculate the (optimized) heat leak ˙Qtotaland Qtotal, with values of (7.32±2.52)·10−8W and (3.3±1.6) ·10−4 J respectively. A constant heat leak ( ˙Q0) of 22 nW is used. Section
2.5.
5. Using common, not optimized, ramp rates to calculate heat leaks, it is found that the total heat leak, ˙Qtotal, is in the range of 10−7W and 10−8W. Section2.5.
6. The model does not change in electronic temperature when expanding to a magnetic-field-dependent Korringa constant. Section2.5.
7. The electronic temperatures predicted in the model for PrNi5 are much lower than can be reached with the most optimized set-up in the world. This indicates that the model in this thesis is insufficient in capturing all the effects that are playing an important role in the nuclear demagnetiza-tion stage. Secdemagnetiza-tion2.5.
8. Cooling a silver wire and block with a PrNi5 refrigerant is very efficient. The large cooling power makes sure large quantities of silver can be cooled down. It takes 4.5 pJ of energy to cool down 0.11 moles of silver from 10 mK to 5 mK. Section2.6.
Chapter 3
In this chapter, we showed the experimental part of this thesis. First, experi-ments were done to measure the temperature with a SQUID thermometer that is able to measure at ultra-low temperatures when calibrated the right way. Sec-ondly, experiments were done to decrease heating of the cryostat during de-magnetization, to allow for a lower electronic temperature. This was achieved by putting a capacitor and an extra resistance into the electrical circuit of the superconducting magnet coil. Also, using resistance thermometers, the temper-ature was measured and analyzed.
1. A method to analyze SQUID data (PDS vs frequency plots) is presented. We find that the noise of the thermometer does not follow the simple first order behaviour expected from a LCR circuit. If one tries to fit a model
4.1 Conclusion
to the data to determine the temperature, the calibration depends signif-icantly on the parameters chosen, which is not a good thing. The alter-native is to integrate the power spectrum density to determine the noise power. Section3.2.2.
2. The data that was obtained had a f−1/2 dependency at low frequencies. This could be due to a break down of the thin plate approximation. Section
3.2.2.
3. The data with an LCR circuit shows a smaller current rate (15.36mA/min) than the data without (18.32mA/min). Calculated with L=3H. However, the current input was 10mA, so in the system without LCR, it should have been 10mA. From this we conclude that the inductance L of the magnet coil is 5.5H. Section3.3.1.
4. The dissipation when sweeping the magnet may be reduced by as much as a factor of 2.5 when using the LCR circuit. Section3.3.1.
5. Comparing the temperature-vs-time plots from the NDSprobe50mK ther-mometer of ramps with and without LCR gives a heat leak ratio. This ratio can be calculated from theory, with the only difference that the the-ory predicts for the case that both ramps are without LCR. The supposed influence of the LCR circuit is shown as a lower ratio from the data than was calculated from theory. The ramps that were available did not show this behaviour. As the ramps were too different, the result stays inconclu-sive. Section3.3.2.
6. The temperature on a copper block near the demagnetization stage was measured. The lowest temperature reached was 1.47 mK, using a calibra-tion that is valid from 10 mK to 207 mK. However, the validity in the range below 10 mK is very questionable. With the fact that the temperature and magnetic field are linear at low temperatures, a different estimate has been made of the lowest temperature. It is roughly 3.6 mK now. Section3.3.2. 7. A calculation was made, using the model of chapter2 to predict the
elec-tronic temperature of the same copper block. Result was a temperature in between 117 and 136 µK, a factor 26 to 31 in difference of the measured value. Reasons for the big difference might be due to multiple consecu-tive runs, cooling relaxation and a magnetic ordering that are not present in the current model. Another reason for this difference is experimental of nature. Extra heating due to pulse tube vibrations and having more Eddy currents from the set-up than expected, could raise the temperature significantly. Section3.3.2.
