ANÁLISIS SOCIO-ECONÓMICO A LA POBLACIÓN DE LA

One o f the most popular paramagnetic refrigerants for the high temperature region in recent years has been gadolinium gallium garnet (GGG). This material can be used to cool from temperatures as high as 15K in an ADR, and so has been used in a variety o f applications, including commercial helium liquefaction (Barclay, Jaeger and Prenger, 1990; Hashimoto et al., 1988; Prenger et al., 1990). For space applications DGG is more useful, since it can produce roughly the same cooling effects using only one third o f the magnetic field. Nevertheless, a series o f tests were performed on GGG as a precursor to the DGG test programme, in order to highlight any problems that might occur.

A new salt pill was constructed for the M SSL laboratory ADR described in Chapter 2, containing 0.180 magnetic moles o f GGG powder in a stainless steel case 19mm in diameter and approximately 10.5cm long with a wall thickness o f 0.3mm. The distinction between chemical and magnetic moles is important in materials, such as GGG, which contain more than one paramagnetic ion per chemical formula unit. The pill contained a thermal bus composed o f 100 gold-plated copper wires 0.25mm in diameter and 95mm long. The top rod, base disk, and support structure were all similar to those o f the CPA salt pill described in Chapter 2. A series o f three demagnetisations were carried out at demagnetisation rates o f lT m in '1 from 6T to IT, 0.2Tm in'' from IT to 0.5T, and O .lTm in'1 from IT to zero field, from initial fields o f 6, 2.5 and IT and an initial temperature o f 4.2K. The field was completely removed in each case and the lowest temperature noted. The thermometry system described in previous Chapter 2 was used. The results are shown in Table 4.1, and graphically in Fig. 4.1.

The CPA pill described in the last chapter typically reached its lowest temperature within 180s o f the end o f a demagnetisation. It was clear from the time taken to reach the lowest temperature in the GGG

1 2 3 4 5 6

Field (T)

Fig. 4.1. Experimental results (points) and model predictions from the GGG tests. The lower curve shows the model predictions for the nominal GGG pill parameters, and the upper curve shows the effect o f reducing the amount o f salt and the area o f the thermal bus to 6% o f their

nominal values.

Field (T) Initial pill T (K) Lowest pill T (K) Time to lowest T (s) Model lowest T (K) Difference (%) 1 4.2086 2.6074 3120 2.6306 0.9 2.5 4.1384 1.2721 2100 1.1609 8.7 6 4.1384 0.6892 9900 0.7281 5.6

Table 4.1. Numerical results from the GGG experiments and model.

pill that the thermal contact inside the pill was poor. There was no discernible relationship between the time and either the starting field or the temperatures, indicating that conditions inside the pill changed between demagnetisations. The problems arising from compressing garnet powders were discussed in Chapter 2. The changes between demagnetisations indicated that the GGG powder was moving inside the pill due to the force applied by the magnetic field gradient, supporting the idea that these crystals are too hard to be compressed to the required extent by the hand compression jig used at MSSL. When the pill was cut open after the experiment, much o f the powder was found to be loose

inside it. Barclay and Steyert (1982) have published some measurements o f the thermal conductivities o f various garnets, and report low thermal conductivities even in well-compressed garnet powders, attributing it to boundary scattering o f phonons at the interfaces between crystal grains. The thermal conductivity o f GGG in single crystal form is relatively high, 40W m‘IK '1 at 4K (Barclay and Steyert, 1982) or around three orders o f magnitude higher than that o f CPA (Hagmann, Benford and Richards, 1994), but falls dramatically in the powder form.

The GGG pill could be improved in two ways. A hydraulic press could be used to increase the pressure applied, but a much simpler alternative would be to mix the powder with vacuum grease to fill the spaces between the grains and so increase the thermal conductivity. It was clear from the results that this will be required for future powdered garnet pills. No calculation o f the thermal conductivity in the MSSL GGG pill could be made, due to the complexity o f the pill design and the changes between demagnetisations.

The thermal model was applied to the GGG results. The lower curve in Fig. 4.1 shows the results o f the model, using the values for pill dimensions given earlier and the data on GGG given in Chapter 2. No data were available for the boundary resistance between GGG and gold or copper and so the values for CPA were used. Suomi, Anderson and Holstrom (1968) have stated that, at low temperatures, the m agnitude o f the boundary resistance does not vary by more than a factor o f 10 between different materials. In most cases, the boundary resistance does not become significant until the temperature falls below lOOmK, and so any difference between GGG and CPA should have little effect on the model. The predicted temperatures were an average o f 332mK lower than the experimental results. The difference between the model and the results was constant to within -1 0 % over the field range covered, indicating that the model was accounting for most o f the processes occurring in the system. The difference between the model and the results again indicated poor thermal contact between the cold stage and the salt, since the model made no attempt to account for thermal gradients within the pill, and this was the only area in which it was obviously lacking.

An attempt was made to correct the model by entering a lower value for the molar quantity o f salt in the pill. This approximated a low thermal conductivity in the GGG powder through simulating a long

thermal time constant, by assuming that little o f the salt was in therm al contact with the top rod during the relatively short time-span o f the demagnetisation. The num ber o f moles o f salt in the model was reduced to 6% o f its measured value, a value determined by fitting the model to the experimental results. The thermal bus area was reduced by same fraction, to ensure that the hypothetical sm aller pill did not benefit from a relative increase in thermal bus area, and so the effective boundary resistance remained the same in both cases. The results from this adapted model are shown as the upper curve in Fig. 4.1. The altered model provided a good fit to the experim ental results, being accurate to within 10% over the range o f magnetic fields and temperatures covered, indicating that modelling a reduced amount o f the salt was an effective way o f accounting for poor therm al conductivity in the pill.

The main conclusion from the GGG experiment was that com pressed garnet powder pills require vacuum grease or some other thermally conducting matrix between the grains o f powder to improve the thermal conductivity within the powder. The salt pill thermal model discussed in Chapter 3 indicated that the temperature difference between the cold stage and the coldest point in the salt should be around 6mK in the CPA salt pill for a cold stage temperature o f 1 OOmK. The two pills used similar thermal buses, but the operating temperature of the GGG pill was much higher, and the thermal conductivity of GGG is around two orders o f magnitude higher than that o f CPA. It should therefore be possible to achieve negligible temperature differences between the cold stage and the coldest point in the pill in a well designed GGG pill. The unaltered thermal model, which ignored the temperature profiles inside the pill, would then apply. A well designed GGG pill would therefore cool to temperatures around 330mK lower than those seen in the experiment.

It was also shown that the thermal model could be adapted to account for poor thermal contact within the pill by reducing the molar quantity o f the paramagnetic m aterial in the pill and the contact area between the material and the thermal bus by a factor determined from experimental results. It would be impossible to draw any quantitative conclusions about conditions within the pill from the reduction factor, since a variety o f processes might contribute to the difference between model and the results. However, it was important to demonstrate that the altered model could accurately predict base temperatures over a wide range o f magnetic fields and initial temperatures. It could then be used as a component o f a larger model to predict the performance o f multi-stage ADRs using the same salt pill

construction technique, estimating both the theoretical best performance and the likely real-world performance o f the multi-stage system.