Utilization of a Low Temperature Pressure Cell for the Study of Heat Capacity in Thoriated UBe13 by REBECCA ANNE DUKE

Utilization of a Low Temperature Pressure Cell for the Study of Heat Capacity in Thoriated UBe13 by REBECCA ANNE DUKE BS (California State University, Stanislaus) 1998 THESIS Submitted in partial satisfaction of the requirements for the degree of MASTER OF SCIENCE in PHYSICS in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, DAVIS 2004
Contents Introduction Background The Dilution Refrigerator Modifications to the Refrigerator Heat Exchanger Construction Acquisition of Data Uranium Heating Results and Conclusions References	Figure 1 Critical Temperature vs. Thorium Concentration for Thoriated UBe13
Introduction The unconventional nature of the superconductivity in the heavy fermion superconductors is well known. One of the first indications of this was the discovery that the heat capacity of UBe13 had a power- law temperature dependence below Tc, rather than going exponentially to zero as expected [1]. Later, it was found that the heavy fermion superconductor UPt3 had a double transition in the heat capacity, as well as a large linear term in the heat capacity below Tc [2]. Doping UBe13 with a small amount of thorium to make U1-xThxBe13 (0.019 < x < 0.043) also produced two transitions in the heat capacity. Furthermore, it was found that increasing pressure on thoriated UBe13 mimics a decrease in the thorium concentration. This analogy between pressure and thorium concentration is very useful to the study of the material. In 1994, Zieve et al. did a study of the effect of uniaxial pressure on the double transition in the heat capacity of UBe13 [3]. The results clearly showed the two transitions merging into one at the phase boundary. However, the phase boundary wasn't crossed directly, because the pressure could only be changed at room temperature. In the experiment described here, a method was employed that enabled the pressure to be changed without warming the sample above 300 mK. The sample is mounted in a pressure cell that is in turn mounted on a dilution refrigerator. The pressure on the sample is varied by the introduction and removal of liquid 4He into the pressure system, which includes a bellows that puts the pressure on the sample, as well as an extensive system of heat sinking to keep the sample at low temperature while the pressure is changed. With this method, we were able to drive the sample directly across the phase boundary, rather than around it, and were able to study the phase transition directly. The results indicate that the phase transition is of second order, as no hysteresis or latent heat was found when crossing the transition. The linear term in the specific heat of thoriated UBe13 was also observed, and a marked increase was found with the addition of pressures up to 7.8 kbar. Background Steglich first observed heavy fermion superconductivity in 1979 while investigating the low temperature behavior of LaCu2Si2 and CeCu2Si2. Though LaCu2Si2 behaved like a normal metal, CeCu2Si2 became superconducting below 0.5K. The superconductivity was found to be type II [4]. There are also several other notable differences between heavy fermion superconductors and other metallic superconductors. In the normal state, we find " small resistivity, a large effective mass, strong antiferromagnetic spin correlations, and highly reduced ordered moments." [5] Also in the normal state, heavy fermion materials have a much higher specific heat than would be expected, and in the superconducting state the specific heat has a power-law dependence in temperature, rather than an exponential dependence. Other power-law dependencies are found in the spin lattice relaxation, ultrasound attenuation, and thermal conductivity. There is much evidence that the superconductivity displayed by heavy fermion superconductors is unconventional when compared with other metallic superconductors. The elemental superconductors have the simplest physics, and are said to be conventional. In a conventional s-wave superconductor, the addition of a magnetic impurity breaks pairs, while the addition of a non-magnetic impurity leaves the pairs unaffected. In a non-s-wave superconductor, the addition of any type of impurity can break pairs. One of the most easily observable qualities of heavy fermion superconductors is the unexpectedly large specific heat in the normal state. At low temperatures, the specific heat of a heavy fermion superconductor can be two or three orders of magnitude above the expected value for a metal [6]. This is due to extraordinarily strong electron-electron interactions. Because the electrons are so highly correlated, the system is modeled well by a system of quasiparticles with a very high effective mass. The quasiparticles are called "heavy fermions", and their mass can be up to 200 times larger than that of an electron [6]. Below the superconducting transition temperature, several measurable quantities in the heavy fermion superconductors have power law dependencies, rather than exponentials as would be expected. Among these are the specific heat, sound attenuation, and relaxation rate [6]. The existence of a superconducting energy gap in a conventional superconductor implies that there are an exponentially small number of electrons and holes in the excited state at low temperatures. Because only the excited charge carriers can contribute to quantities such as the specific heat, we would expect the specific heat to fall off exponentially as the excited carriers do. However, if features exist such as nodes in the gap structure, various measurements may be dominated by these features, which could cause the measurements to reveal power laws, rather than the expected exponentials [5]. The heavy fermion superconductor UBe13 is the one that will be discussed in this paper. UBe13 is interesting because it's particularly susceptible to doping by thorium. Doping UBe13 with lanthanum or gadolinium only serves to depress the superconducting critical temperature, but with the addition of a certain amount of thorium, creating the alloy U1- xThxBe13, the critical temperature becomes a function of the thorium concentration. If the thorium concentration is 0.019 < x < 0.043, there are two transitions for critical temperature. Below is a graph of how the critical temperature of thoriated UBe13 varies with the percentage of thorium. The two superconducting phases can be clearly seen.

One very useful property of thoriated UBe13 is that changes in pressure mimic changes in thorium concentration. As pressure is increased on a sample, the substance behaves as if its thorium concentration were decreasing. The reason for this mimicry is that replacing the thorium atoms with larger uranium atoms (decreasing the thorium concentration) is analogous to crowding more matter into the structure of the material, which is rather like increasing the "microscopic pressure." This is useful because studying what happens as the thorium concentration is varied can be achieved by using only one sample. This eliminates the need for many samples, as well as eliminating differences in sample quality occurring between samples. The Dilution Refrigerator The Kelvinox 100 dilution refrigerator should be able to attain temperatures of 15mK. These temperatures are realized by dragging 3He across a 3He 4He phase boundary. This is analogous to a volume expansion cooling process, although neither 3He nor 4He is a gas when this occurs. Perhaps the best way to describe how the dilution refrigerator works is to examine the schematic diagram below while reading the description that follows of the process in which the refrigerator is cooled down.

Figure 2 Schematic of Dilution Refrigerator
When starting the cool down process, the refrigerator is put into a dewar and all systems are pumped out thoroughly with a vacuum pump. This is done to avoid contamination of the various He systems by air and water, and in the case of the copper-nickel tube that goes from the top of the refrigerator through the copper heat exchangers and down to the bellows (shown on the far left of the schematic diagram), to keep water vapor from freezing in it and causing it to become plugged. The copper-nickel tube is also flushed by filling it with helium and then pumping it out several times to try to decrease the chance of it becoming plugged. Liquid nitrogen is then transferred into the dewar to pre-cool the refrigerator to 77K, and later the liquid nitrogen in the inner part of the dewar is replaced with liquid 4He so that the entire refrigerator is submerged in a 4He bath. The 4He bath cools the refrigerator to 4K. After the refrigerator reaches 4K, the actual refrigeration process can be started. Liquid 4He is pumped from the bath continuously with a siphon tube into the 1K pot. Pumping on the 4He causes an evaporative cooling process, which enables the 1K pot to get down to about 1.3K (calling it the 1K pot is actually fudging a bit, but that is the common name for it). 4He is always being lost from the system in this manner. Once the 1K pot has cooled, the second helium system in the refrigerator can be started. A 3He-4He mix is put into the refrigerator. It is very important that the mix have precisely the right ratio of 3He to 4He so that when the two isotopes separate, the boundary will occur in the mixing chamber. Due to the importance of the ratio, and because 3He is very costly, the mix is contained in its own isolated system. The mix is continuously circulated through the refrigerator by use of another pump. The mix is condensed at the 1K pot and goes into the mixing chamber by way of heat exchangers. The mix is then pumped up into the still by way of the same heat exchangers, but the exchangers are compartmentalized so the up going mix is physically separate from the down going mix. In the heat exchangers, the up going mix cools the down going mix to the temperature of the mixing chamber. At the still, pumping on the mix causes it to undergo an evaporative cooling process that gets the mix below 860mK. The 860mK temperature is very important, because this is the temperature at which the mix separates into two distinct phases. If the mix has the right ratio, in the bottom of the mixing chamber there will now be mostly 4He with about 6% 3He, and floating on top of this mixture will be the rest of the 3He. If the ratio is wrong, the boundary between the two phases will be outside the mixing chamber, and optimum cooling will not occur. Now the actual dilution part of the dilution refrigerator can begin to work, and as it does, a rather dramatic drop in temperature is seen at the mixing chamber. The pumping tube in the mixing chamber goes nearly all the way to the bottom, so the 4He rich phase is what is pumped up into the still. The vapor pressures in the still are such that almost all of the liquid that evaporates is 3He, and very little 4He evaporates. This causes the 4He rich phase in the mixing chamber to have less than the quantum mechanically ideal 6% 3He. Some of the 3He phase in the top of the mixing chamber is then free to expand (as a gas would) into the 4He rich phase. This expansion of 3He cools the mixing chamber to about 15mK. Much care must be taken to allow as little 4He to evaporate in the still as possible, because it does not contribute to the cooling process, but it must be re-cooled itself if it circulates. Modifications to the Refrigerator The dilution refrigerator has been modified so that uniaxial pressures of 7.8 kilobars can be applied to a sample, but more importantly, the pressure can be added and subtracted while the sample is at low temperature. This is accomplished with another 4He system. 4He from a gas cylinder is introduced into a system of tubes, valves, and gauges known as the panel. In the panel, the 4He goes through a cold trap held at liquid nitrogen temperatures to remove impurities. A long charcoal filled tube is attached to the panel. This tube is affectionately referred to as "the bomb". When the bomb is dipped into a liquid helium dewar, a huge amount of helium from the panel is trapped in the charcoal. After the bomb is removed from the dewar, the helium in it becomes a high-pressure gas. The gas can then be introduced to the refrigerator through another tube leading from the panel. Leading down from the top of the refrigerator is a long, thin (about 0.036" at the largest inner diameter) copper nickel tube that is connected to a series of heat exchangers. The helium will liquefy due to the low temperatures in the refrigerator and the heat exchangers cool the helium to the temperature of whatever part of the refrigerator the tube is passing through, so as not to put a heat load on the refrigerator while the helium is being introduced. At the bottom of the tube is a bellows, which expands to put pressure on the sample. Because of the cross sectional area difference between the sample and the bellows, a pressure of 25 bar (the pressure at which the liquid helium solidifies) in the bellows corresponds to a pressure of 7.8 kbar on the sample. See figure 2 for a schematic diagram of the pressure cell. Between the bellows and sample is a quartz piezocrystal that acts as a force transducer. This serves to measure the amount of pressure applied to the sample. The force transducer is connected to a superconducting coaxial cable that goes back up to the top of the refrigerator. The superconducting cable is used because superconductors make poor thermal conductors, and it's important that the cold lower portions of the refrigerator do not have a direct thermal link with the warmer upper portions. The signal from the force transducer is an electric charge that is produced by the piezocrystal during pressure changes. This signal is then sent to a Kistler model 5010 dual mode amplifier, which utilizes preprogrammed information about the piezocrystal to convert the charge into units of pressure.
Figure 3 The Pressure Cell
Even though the pressure system is in very good thermal contact with the rest of the dilution refrigerator, care must be taken when changing the pressure at low temperatures. Initially the thin copper nickel tube is flushed with helium several times and then pumped out very thoroughly to avoid it becoming obstructed by solidified air when cooled down. During pressure changes, if helium is allowed to rush into the system, the heat load from its introduction can be so large that it can cause the refrigerator to warm up so that the circulation is upset and must be restarted. If the goal is to change the pressure on a sample while it remains below its critical temperature, the helium must be introduced very slowly. This is done by means of the panel. The panel has several valves and regions to allow a controlled introduction of helium. After sitting in the cold trap to remove impurities, the helium is let into a region with a pressure gauge. Another valve is used to let this helium into the refrigerator a little at a time. The gauge on the panel serves as an indicator of how much helium is being put into the system. At the same time, a labview program records a pressure versus time graph from a Keithley multimeter that is constantly displaying the pressure signal it receives from the Kistler dual mode amplifier. Heat Exchanger Construction The biggest challenge in modifying the dilution refrigerator was the creation and mounting of the four copper heat exchangers that serve to keep the refrigerator at a constant temperature as pressure is added or removed. Each heat exchanger consists of a copper body and flanged lid. The copper body has been machined to include a cylindrical space for the containment of sintered material, and a threaded hole at the bottom to provide a means of attaching the heat exchanger to the dilution refrigerator. A small hole has been drilled through the side of the copper body into the bottom of the sinter space to facilitate the copper nickel tube that serves as the helium outlet for the heat exchanger. The flanged lid is designed to fit snugly on top of the copper body and is soldered into place to form a leak tight seal. The lid also has a hole drilled into its center to facilitate the copper nickel tube that serves at the helium inlet for the heat exchanger.
Figure 4 Heat exchanger Schematic
The heat exchangers are packed with sinter. This provides a very large surface area (over 3000cm2) that is in good thermal contact with itself and the heat exchanger body. Because of the large surface area and the good thermal contact, helium passing through the sinter should cool to the temperature of the exchanger before leaving it for the next exchanger. The first sinter that was used was a silver powder packed to a high density. It was found that simply packing the powder tightly was not sufficient to make a sinter that was effective in cooling the helium. This was blatantly obvious when comparing the thermometer calibration curve before helium was introduced into the pressure system to the calibration curve after the helium was introduced. Calibration data is taken each time the temperature is changed, and serves to match the uncalibrated semiconductor thermometer attached to the sample with the nicely calibrated one on the fridge. It is also useful for making sure that the sample is getting as cold as the fridge. One would expect the calibration curve to be a nicely shaped smoothly decreasing curve because as the temperature goes down on a semiconductor thermometer the resistance goes up, and at lower temperatures the resistance rises more steeply. In figure 3, it can be seen that without helium in the system the calibration curve exhibits smoothly decreasing behavior, but after the helium is introduced into the bellows system the calibration curve flattens. This indicates that even though the fridge is cooling, the sample is at a much higher temperature. It was also extremely hard to get the fridge much below 300 mK when helium was in the pressure system. These two findings indicated that the pressure system was putting too great a heat load on the fridge.
Figure 5 Silver Heat Exchanger Calibration
Above is a sample of a graph obtained by changing the pressure. The actual pressure changes are the steep lines, and the rest of it is showing what is happening when we are not actively changing the pressure. Notice the long slope at the left, which describes what the piezo is doing before pressure is added. This is not indicating a change of pressure on the sample, but is a bleed-off of charge from the piezo. Sometimes this bleed off is prominent, as in this graph, and at other times it is nonexistent. It is unknown why the nature of the bleed-off changes. In any case, the calculated pressures from different runs with and without bleed-off have been compared, and found to be the same, so it is certain that it is not a pressure change. Care must be taken to account for the bleed-off though when calculating the pressure, because the bleed off tends to make the pressure look smaller than it is. In the above graph, it appears that about 0.20 kbar of pressure has been added to the sample, but when the bleed-off is taken into account by measuring each steep line separately, we see that 0.34 kbar was added. If this mistake were made each time the pressure was changed, there would be a huge error at the higher pressures. The heat capacity data- gathering program is a bit more complicated than the pressure program. The program instructs the LR 700 to set the fridge temperature, and then has a hold time to allow all parts of the refrigerator to equilibrate. It then instructs an HP power supply to send the pulse to the sample heater and dictates the voltage of the pulse. It also dictates when and how often the sample thermometer is read, and concurrently stores readings of the fridge thermometer for the purpose of the calibration curve. The pulse is a voltage pulse, but there is a 1MW resistor in series with the sample thermometer, which ranges from 1kW to 10 kW, effectively turning the pulse into a current pulse (the 1MW resistor dominates the smaller, variable resistor, so that the current in the circuit due to the voltage change stays fairly constant and does not vary wildly when the thermometer resistance changes). The whole program is nicely automated, so that one can set the parameters in the beginning and then allow the experiment to run on its own. The parameters that are set are the starting temperature, the number of temperature steps and the distance between them, the time the program should wait for the temperature to equilibrate before sending a pulse, the number of heat pulses sent at each temperature, the pulse voltage, and the rate of acquisition of data points (frequency). These parameters can be set in several stages. For example, the wait time does not need to be nearly so long at higher temperatures, so the program can be set to have a wait time of 360s for temperatures ranging from 100mK to 540mK, and a wait time of 150s from 550mK to 900 mK. The voltage of the pulses needs to be varied at different temperatures so that the temperature of the sample doesn't go up too high during a pulse (because the specific heat at a given temperature is actually a composite of the specific heats of all the temperatures the sample achieves during a pulse), or not change enough to produce a good signal. The frequency of data acquisition also often optimized for different temperatures. The program samples a certain number of points per pulse. If the frequency is too high, the entire relaxation of the sample will not be included in the data, and if it's too low, there will be a lot of extraneous, uninteresting data taken. The frequency needs to be set higher at higher temperatures because the relaxation happens faster. The data analysis is not entirely straightforward and has several steps, so yet another labview program is employed so that the steps can be worked through quickly. The aim is to get a graph showing the way specific heat changes with temperature like the one below.

Figure 9 A sample specific heat graph
The first step in finding how the specific heat changes with temperature is to sort out the calibration between the fridge thermometer and the sample thermometer. To do this the program first asks the user to look at the curve and throw out any clearly bad points. If the run went to very low temperature, the first couple of points tend to be bad because the pressure cell is out of thermal equilibrium with the fridge, and this is evidenced by the first part of the graph being less steeply sloped than expected. Once this is accomplished, the program seeks to fit the calibration curve to a polynomial, so it asks the user to pick what order polynomial to use. Conveniently, one can test different fit orders by looking at a graph of the calibration curve against a graph of the fit curve, and seeing how well they agree. In addition, for each order tested one can see a graph of the error between the curve and the fit at each temperature. The purpose of this graph is to provide a more detailed view of the errors, to see if they are correlated. Correlated errors indicate that there is some sort of regular deviation of the polynomial curve from the fit curve, and that perhaps a better fit would be achieved with a higher order polynomial. After this step, the rest of the program is automated. The program first takes the original pulse data, which shows how the sample thermometer resistance changes with time, and applies the calibration polynomial obtained earlier to get the data to show how the sample temperature changes with time. In the next step, the program determines the part of the data that is just the relaxation of the sample temperature, rather than the relaxation of the surroundings, and chops the rest of the data off. The data should now be a nice exponential curve, so the next step is to take the natural log of the data to get a straight line. This line is used by the program to make a reasonable guess at the temperature change of the sample, while trying to exclude information like the temperature change of the sample surroundings. The program then uses this information as the DT in the equation C=DQ/DT. For the DQ, the program uses DQ=(V/Rheater + leads)2 Rheater Dt, which comes simply from P=DQ/Dt, and P=I2R. The program does this automated bit over and over for each temperature, and for each temperature gets one point on the goal graph of Cv/T vs. T, showing how the specific heat changes with temperature for a given pressure. Plotting Cv/T instead of just Cv for our purposes is really just conforming to convention. In normal state metals, C/T is constant as the temperature goes to zero, and in superconductors that are more traditional C/T goes as T2, and goes to zero as T goes to zero. In our material C/T neither goes to zero, nor goes as T2 as T goes to zero, but it is convenient to plot it this way so the data can be seen in the manner people are accustomed to. Uranium Heating There is some concern that at low temperatures, the heating of the sample due to the alpha decays of the uranium atoms can become significant. In some experiments, the rise in temperature associated with this heating has been sufficient to distort the data at low temperatures. For example, if the sample temperature were to change by 0.1 mK on the order of the time of a pulse (100 ms), then that would introduce a 1% error into the measurement of the heat capacity. The U-238 in our 5.9X10-5 mol sample decays at an approximate rate of 170 decays/second. The energy given off in one decay is 4.2 MeV. It was important to find the temperature increase of the sample due to the Uranium during the time data was being taken. The sample is heat linked very weakly to the fridge so that the amount of heat that leaks from the sample to the fridge is very small. The time constant for the heat leaving the sample can be given by the usual RC, with R being the thermal resistance of the link between the sample and the fridge, and C being the heat capacity of the sample. (Technically, C is the heat capacity of the sample and the link, but the heat capacity of the link is negligible.) The way the temperature of the sample relaxes after a heat pulse is given by dQ/dt=DT/R which becomes:

With this equation we can use the rate of heating due to the alpha decays, which for our sample was 1.14�10-10 J/s, and values from the data for heat capacity at various temperatures, along with the time constant determined from the measurement to find out how much the temperature changes due only to the uranium heating. It was found that even at the lowest of our temperatures, 125 mK, the heating due to the uranium only raised the temperature of the sample by about 0.2 mK. At higher temperatures the effect was less, and the temperature change became less than 0.1 mK at 150 mK. Results and Conclusions The experiment has thus far yielded several interesting results. The larger linear term in the heat capacity of thoriated UBe13 in the superconducting phase was found to increase by over a factor of 2 with the addition of pressure. This change in the linear term was found to be completely reversible, so there was no hysteresis. In addition, no latent heat was found. The reversibility and lack of latent heat imply that the transitions in the region studied are of second order. This large change in the linear term in the heat capacity was engineered simply by changing the pressure on the sample, which is an indication that the linear term is an intrinsic property of the material, rather than the result of impurities. For this experiment, a 0.021g (5.90X10-5) sample of U0.98Th0.02Be13 was used. The sample was made by J.L. Smith of Los Alamos National Laboratory. It was designed to have two faces parallel to one another for use in our uniaxial pressure system.
Figure 10 Specific Heat Under Pressure
Above is a graph of how the specific heat changes with temperature for various pressures. At low pressures, there is an obvious double transition between 400mK and 600mK. A "shoulder" can be seen between these two transitions that is centered at about 500mK. As the pressure is increased, the two transitions both decrease in temperature and appear to merge into one transition. This would make sense, as increasing pressure is analogous to decreasing thorium concentration. It appears that we are crossing the boundary (see phase diagram, figure 6) near 2% thorium concentration, as this is where a higher concentration/lower pressure yields two transitions, and a lower concentration/higher pressure yields only one. The addition of pressure also suppresses the peak specific heat, but enhances the specific heat at low temperatures. This is important to conserving the entropy, which is discussed below. Heavy fermion superconductors often exhibit the odd property of having a large linear term in specific heat, and our sample is no exception. The specific heat of a heavy fermion in the superconducting phase can be given by C(T) = gsT+ Cnon-linear(T). In our sample, the linear term was found to be 0.64 J/mol K2, at a pressure of 0.03 kbar. Under pressure, gs increases fairly steadily to 1.6 J/mol K2 at 7.8 kbar. In comparison, pure UBe13 has no linear term in the superconducting state, and a linear term of 1.1 J/mol K2 just above Tc. The linear term in specific heat is largely a normal state phenomenon. However, it persists into the superconducting state for several heavy fermion superconductors. These superconductors include the (UTh)Be13 studied here, as well as UPt3, CeCoIn5, URu2Si2, and UPd2Al3, among others. The underlying principle for this linear term is not completely understood. It would make sense to have a linear term if some portion of the material remained normal while the rest was superconducting, but there is no clear indication that this is the case. In fact, if gs was coming from a normal portion of the sample, it would be expected that there be a linear Korringa relaxation, as this is usual for the normal state. However, the Korringa relaxation has been found to be cubic in nature [7]. Another explanation for the linear term in the specific heat is that it is caused by impurities in the material. It appears that impurities play a role, as the magnitude of the linear term in UPt3 varies from sample to sample [2]. However, in our sample the magnitude of the linear term changed greatly with the addition and removal of pressure. Though it is possible that the addition of pressure could damage the crystal structure and create impurities that would raise gs, the fact that the change is nicely reversible indicates that this is not the case. These findings are an indication that impurities are not the sole cause of the linear heat capacity. For thoriated UBe13 Cnon- linear(T) can be expressed as ATn, so that the total specific heat becomes C(T) = gsT + ATn. The exponent n is very important in determining the superconducting gap structure of the material; more specifically it is used to determine the nature of the nodes. For our sample, n was found to be near 3 for all pressures, and a specific heat that goes as T3 indicates that there are point nodes in the superconducting gap. Having point nodes in the gap means that for some particular values of k in k-space, there is no energy gap, and that the energy gap depends on which direction in k space one looks in. The specific heat curves were extrapolated to T=0, and as a check of the validity of the curves the entropy was explored. The entropy is given by:

Because entropy is a state function, if you integrate from zero to a temperature where the sample is normal (presuming transitions in the specific heat are of second order, to avoid a jump in entropy), you should get the same answer as if the sample stayed normal all the way to T=0. Furthermore, it was found that when our sample is above Tc, the specific heat is pressure independent. From this we can presume that if the sample was normal at any temperature, the specific heat would continue to be independent of pressure. With these two facts, it is permissible to compare the entropy of the sample at different pressures, and expect them to be the same. It was found that in fact, when integrating from T=0 to 700 mK, the entropy of the sample at 0.03 kbar and 5.58 kbar differed by only a few percent. It appears that the lost area under the diminished peaks of a high-pressure specific heat curve is made up by extra area under the low temperature region. This result confirms that the specific heat curves achieved by the experiment, along with the extrapolations down to T=0 are plausible. It also indicates that the transitions in specific heat are of second order. Perhaps the largest indicator that the specific heat transitions are second order comes from the reversibility of the results. No hysteresis was found in either of the methods used to measure the specific heat under different pressures. In the first method, the pressure is changed while keeping the sample below 300 mK. Then, the specific heat is measured from about 100 mK to a temperature well above Tc. After this, the sample is re-cooled, and the specific heat measurement is repeated. The two measurements of specific heat were found to agree within 0.5%. In the second method, we attempted to hold the temperature constant and measure the specific heat while varying the pressure. This was hard to accomplish, as the fridge is not easily able to handle the heat load due to the changing of the pressure. Still, there was no evidence of hysteresis in the specific heat, whether increasing or decreasing the pressure. The most important advantage of the pressure system we designed for this experiment is that it has the ability to keep the sample cool while changing the pressure. Because of the analogy of changing pressure to changing thorium concentration, we picked a sample with a thorium concentration close to the transition just below x=0.02. With the addition of pressure, the transition is crossed. While changing pressure at low temperature, and crossing this transition, no latent heat was ever observed. This is an indication that the transition in pressure is of second order. In conclusion, the experiment that the modifications to the dilution refrigerator were designed for has been quite successful in exploring the specific heat of our U0.98Th0.02Be13 sample. We were able to vary the pressure on the sample from 0.03 kbar to 7.8 kbar while keeping the sample below 300 mK in temperature. This is a great improvement over previous experiments, during which the pressure could only be changed at room temperature. Looking at the specific heat at different pressures clearly shows the double transition of thoriated UBe13 merging into a single transition at higher pressures, and a study of the entropy confirmed that these results are valid. The linear term in the specific heat was found to increase markedly with the addition of pressure, and the change was found to be completely reversible. This lack of hysteresis suggests that the linear term in the specific heat is an intrinsic property of the material, rather than being simply due to impurities. It also suggests that the transitions are of second order. The non-linear portion of the specific heat was found to be cubic in order, which suggests point nodes in the superconducting gap. In addition, no latent heat was found upon changing pressure. This indicates that the transitions in pressure are of second order, and also suggests that the analogous transitions in thorium concentration might also be second order.
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