Heat Capacity Application Note 1085-151
Anomalously Low Heat Capacity Below 1 K
Problem
At temperatures below about 1 K, the measured heat capacity is observed to decrease
more rapidly than expected as the temperature is decreased. For example, Fig 1 shows
the measured specific heat of a 20 mg sample of copper compared to the expected value.
Note that this data is obtained using a measurement time of 1 time-constant.
Figure 2 shows another example of this effect for a 65 mg sample of gold. In this case
the measurements were obtained with measurement times of 1 time-constant and 2 timeconstants. Note that the measurement obtained at 2 time-constants does not show the
downturn.
Figure 3 shows the effect of varying the measurement time at a fixed temperature (0.350
K) for the gold sample. Measurements are shown with measurement times of 0.5, 1, 2,
and 3 time-constants, respectively. Note the systematic decrease in the reported specific
heat at measurement times less than 2 time-constants
0.0009
0.00085
PPMS
C p/T (J/mole K2)
CRC Handbook
0.0008
0.00075
0.0007
0.00065
0.0006
0
0.5
1
1.5
2
T (K)
Figure 1: 20 mg copper sample at low temperatures. Measurement obtained with a measurement
time of 1 time-constant.
Quantum Design
Heat Capacity Application Note 1085-151 A
05/03/2002
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0 .0 0 1 5
P P M S 2 tim e c o n s t
C R C H andbook
P P M S 1 tim e c o n s t
2
Cp/T (J/mole K )
0 .0 0 1 3
0 .0 0 1 1
0 .0 0 0 9
0 .0 0 0 7
0 .0 0 0 5
0
0 .5
1
T
2
1 .5
2
2
(K )
Figure 2: 65 mg gold sample obtained with a measurement time of 1 time-constant and 2 timeconstants.
Quantum Design
Heat Capacity Application Note 1085-151 A
05/03/2002
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0.00028
65 mg gold at 0.35 K
0.00027
0.00026
3 time constants
Cp (J/mole K)
0.00025
2 time constants
0.00024
0.00023
0.00022
1 time constant
0.00021
0.0002
0.00019
0.5 time constants
0.00018
0
2
4
6
8
10
12
Measurement Time (seconds)
Figure 3: Specific heat measurements of 65 mg of gold at 0.35 K at different measurement times (0.5,
1, 2, and 3 time constants).
Explanation
Ideally, the reported specific heat value will be independent of the specified measurement
time. However, this will be true only to the extent that the sample holder and sample
behave as idealized point-like specific heats, and the wires are mass-less thermal
resistors. In reality, there are multiple thermal relaxation times in the calorimeter and
wires that result in different reported specific heats for different measurement times. At
higher temperatures this effect is usually negligibly small, but it becomes problematic
below 1 K, where the thermal boundary resistance, or Kapitza resistance, becomes large
between the various materials comprising the calorimeter.
The addenda heat capacity was obtained with a measurement time of 2 time constants in
Figs. 1 thru 3. Because of the nature of the fitting model used in the Heat Capacity
software, a substantial improvement in the accuracy of the data is obtained if the sample
measurements are also obtained with the same measurement time (2 time constants) as
used for the addenda measurements.
Solution
To obtain the best accuracy below 1 K, it is recommended that specific heat
measurements be obtained using 2 time-constants for the measurement time to best match
the thermal conditions of the calorimeter when the addenda measurements were obtained.
Quantum Design
Heat Capacity Application Note 1085-151 A
05/03/2002
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