LT 21 Procecdings of the 21st International Conference on Low Temperature Physics Prague, August 8-14, 1996

P a r t $ 5 - Techniques and applications: Instrumentation

Thermal links in high fields and low temperatures

Ewout ter llaar a and Clemens M.C.M. van Woerkens b

a Instituto de Ffsica da Universidade de S.Paulo, C.P. 66318, CEP 05389-970, S. Paulo, SP, Brazil b Kamerlingh Onnes Lab., Leiden University, Postbus 9506, 2300 P~A Leiden, The Netherlands

The highest B/T environments are achieved by connecting a sample in a high field with a nuclear demag- netization stage [1]. The design of the thermal link should be taken into consideration at high B/T's. For example, for fixed material properties, there is an optimalization with respect to the diameter of the thermal link, depending on the desired experimental conditions. Eventually, other materials than the canonical silver may have to be used. We discuss some simple design criteria and suggest new materials. Tin in particular seems a promising candidate.

1. Genera l Cons idera t ions

Clearly, a thermal link should fulfill at least the following requirements: 1) the entropy which must be removed to cool it must be small and 2), its ther- mal conductivity should be high. A third property which may be looked at, is its thermal relaxation time to the stage. Thermal relaxation times in the system are useful design criteria in cases where the demagnetization time is limited, for example when relatively large heat leaks need to be overcome, as expected in high fields.

It is of no use to demagnetize faster than some ap- propriate (see below) thermal relaxation time in the system, because this will lead to temperature gradi- ents and entropy production. First we consider what relaxation times are important. The relaxation time of the sample should always be taken into account. But also heat capacities which are "in parallel" are sometimes important. A large heat capacity in the system with a very large relaxation time (example: protons in the walls of a plastic cell), can be treated as a small heat leak. However, a large heat capacity with a small relaxation time must be cooled (exam- ple: the nuclei of a metal with a small Korringa con- stant). Therefore, though the nuclear heat capacity of the thermal link is in parallel with the sample, it is necessary to cool the nuclei, since their entropy will be dumped in the nuclear stage before the experi- ment is over. This means that one of the relaxation times to consider is the nuclear spin-lattice relaxation time of the thermal link.

We now consider somewhat more quantitatively the effect of a single relaxation time r of the sample

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0 D

i

lx10 4 2x10 4 3x10 4 4x10 4 5x10 4

BIT / Tesla K -I

Figure 1: Entropy for various materiMs which is to be removed if the material is to cool from B / T = 1000 T / K to B/T.

to the stage. If we want to keep a constant relative temperature difference between the stage (Tn) and the sample (Ts), AT/T , = constant, then we have

dTn Tn dT, Tn dB

(It Ts (It B dt

and, with dT,/dt = AT~r, the field should be changed according to

dB A T B

dt T, r

A longer ~- means longer demag times and smaller tolerable heat leaks. Note that, unlike its contri- bution to the entropy dumped in the nuclear stage, the contribution of the thermal link to the relaxation time can not be made smaller by using a larger stage.

Czechoslovak Joumal of Physics, Vol. 46 (1996), Suppl. $5 2731

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area / n ~

Figure 2: Results of numerical simulations: the low- est temperature of the aHe in 9 Tesla, after the de- magnetization, as a function of the area of the silver thermal link.

2. Mate r ia l s

Going over a table of the elements with nuclear moments and abundancies (e.g. [2]), reveals three promising metals: tin, zinc and maybe tungsten. The entropy which must be removed to cool them from a precool temperature and field of 10 T/10 mK to B/T, is shown in Fig. 1, for B/T's which are hoped to be achieved in the near future. For B/T's higher than 104 T/K, silver is at a disadvantage, not only because of its high entropy, but also because of its long spin-lattice relaxation time. Tin and zinc are available in high purities and are expected to have high (thermal) conductivities [3]. The Korringa con- stant of zinc is estimated to be 0:5 sK [4], and that of tin is 0.05 sK [5], both to be compared to 12 sK [5] for silver. A disadvantage is their superconductivity, with critical fields of 5.5 and 30 mT for resp. zinc and tin. At the lowest temperatures, the nuclear quadrupole splitting (~.. 100 #K) may compromise the utility of zinc.

3. S imula t ions

In situations where B/T 's of ~, 104 T /K are reached for, silver has the advantage of being thor- oughly tested and having well known thermal prop- erties. Numerical simulations ofan actual setup were made, to optimize a silver thermal link, connecting a 3He cell (0.3 mol) in 9 T with a copper stage (32 mol copper). The precool B /T was 6 T / 7 mK, the heat leak on the cell 25 nW, and the RRR of the link 100. The simulated demagnetizations were always carried out linearly to 100 mT, at various rates (indicated as the total demagnetization time in the legends of

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o 2 0 0 0 0 min

u 1 0 0 0 0

�9 1 0 0 0

o 100 o

. . . . . . . . i j r . . . . . . i i i i f - - - - I . . . . . . .

l x l O - S l x 1 0 - 4 l x 1 0 - 3 lX10-2

a r e a / m 2

Figure 3: The time the 3He stays under 2 inK.

the figures). From Fig. 2 it can be seen that there is an optimal diameter for the link. For smaller links, the heat leak and the thermal conductivity of the link keep the temperature high, and for larger links, the entropy flow from the link to the stage limits the temperature.

With a one-shot cooling method, the experimen- tal time available under a certain temperature is an issue. The time the 3He stayed under 2 mK is given in Fig. 3, for the same simulations as described above. There is now an optimum for larger link di- ameters. The effect of stopping the demag at higher fields was not investigated.

In conclusion, as the state of the art in both high fields and low temperatures progresses, the design of the thermal link should be looked at, and maybe other materials than silver should be considered.

R E F E R E N C E S

[1] F. Pobell, 'Matter and Methods at Low Tem- peratures', Springer-Verlag, Berlin Heidelberg (1992).

[2] CRC Handbook of Chemistry and Physics.

[3] Cahn, ttaasen and Kramer, eds., 'Material Sci- ence and Technology', Vol 3A, VCH Weinheim (1991).

[4] 3. Goyette and R. V. Pound, Phys. Rev. B, 35 391 (1987).

[5] Carter, Bennet and Kahan, Prog. Mater. Sci. 20 (1977).

2732 Czech. J. Phys. 46 (1996), Suppl. $5