Measurement of the Curie
Temperature of Gadolinium
Department of Physics and Engineering Physics
University of Saskatchewan
PHYS 404
Laboratory #1
February 2, 2001
Christopher Foley (352253)
Devin Greene (314122)
Abstract
This experiment was designed to measure the Curie temperature of gadolinium.
A value of 287.12 +/- 0.62K was obtained, which does not fall within the range of
accepted values of 293K – 289K. The experiment was performed twice, once in a water
bath and once in an oil bath. Possible reasons for the low value of the Curie point include
impurities in the sample, a defective inductance meter, and improper heating and cooling
of the sample during the experiment. More research needs to be done on the improper
heating and cooling methods, as there may be a relaxation time for the alignment of and
creation of domains within a magnetic material.
1
Objective
The objective of this experiment was to determine the Currie temperature of
gadolinium.
Theory
Magnetic materials are magnetic because of the combined effects of individual
magnetic dipole moments of atoms in the material. In ferromagnetic materials, the atoms
tend to arrange themselves so that these dipoles are aligned. This only occurs on a small
scale, however, localized to regions called domains. Normally in a macroscopic sample,
these domains are randomly orientated, so that overall, the sample is non-magnetic
(Figure 1). If the sample is placed in an external magnetic field, a torque is exerted on
domains that are not aligned with the field, and the domains will rotate to align
themselves with the field. If the external field is then removed, most of the domains will
remain in an aligned state, so that overall the sample is now a permanent magnet (Figure
2).
Figure 1 – Random Domains
Figure 2 – Aligned Domains
2
At low temperatures, random thermal vibrations are not strong enough to dislodge this
dipole alignment. Above a certain temperature called the Curie temperature or Curie
point, however, the atoms now have enough thermal energy to overcome the binding
between neighboring dipoles to break up the domains. Once the order of the domains is
destroyed, the sample loses its magnetic properties: it is now paramagnetic. This is an
abrupt process and occurs at a well-defined temperature for a particular material. It is
called a phase transition, much like the transition from solid to liquid. Above the Curie
temperature, the sample is paramagnetic and below the Curie temperature, it is
ferromagnetic.
The magnetic susceptibility of a ferromagnetic material at temperatures above the
Curie temperature is given by the Curie-Weiss law:
   1 
C
T  Tc
(1)
where  is the magnetic susceptibility of the sample,  is the magnetic permeability of
the sample, C is the Curie constant which depends on the material, T is the temperature of
the sample, and Tc is the Curie temperature.
An easy way to measure the permeability of a sample is to measure the inductance
of an air-core solenoid, then place the sample in the solenoid, and measure how the
inductance changes. If the inductance of the air-core coil is Lo and the inductance of the
coil with the sample is L, then the permeability of the sample is given by:
 
L
Lo
(2)
where  is a geometrical factor that depends on the shape of the coil.
If equation (2) is now put into (1):
3
 L

y (T )   
 1
 Lo

1

T  Tc
C
(3)
it can now be seen that y(T) is a linear function of T with slope 1/C. This function
crosses the temperature axis when T=Tc. Thus by changing the temperature of the
sample and recording the inductance of the coil (at temperatures above the Curie point),
the Curie temperature can be determined. Note that the values of C or  need not be
known. For plotting results, the assumption was made that =1. This is valid only for
determining the Curie temperature, however, and not for measurement of the
permeability.
Apparatus
This lab was performed by measuring the inductance of a sample of gadolinium
as its temperature changed. The range over which the temperature was varied was
designed to include the theoretical Curie point, around 18°C. The following components
made up the test bench on which the experiment was performed:
6)
7)
8)
8)
1)
3)
5)
2)
4)
Figure 3 Apparatus
4
1) Gadolinium sample (~.5cm x .5cm x1.5cm)
2) Water tank (half filled with water)
3) BX Precision 858 Universal LCR meter
4) 2.119 mH Inductive Coil
5) 250ml beaker (half filled light oil)
6) B. Braun Thermomix M D-3508 water circulator/heater
7) Barnant 90 600-7840 (type ‘K’ thermocouple) Digital Thermometer
8) Stand
Temperature control is essential in this experiment. The temperature must change
at a relatively slow rate to allow for thermal conduction lag in the water, the oil, and the
sample. The heater was used to heat the sample, while snow was added to the water to
cool the sample. Both temperature controls were used in discrete amounts to allow time
for thermal conduction. The thermocouple provides a constant reading of the
temperature around the sample. The LCR meter was used in inductance mode at a setting
of 1kHz. Note that the oil bath is necessary to prevent oxidation of the gadolinium
sample.
Procedure
Before the system above was constructed, the inductance of the empty coil was
measured, as it is needed for analysis purposes. The system shown in Figure 3 was
initially set to just above room temperature (as verified by the thermocouple) by using the
Thermomix heater. Snow was then added to the water in small amounts to decrease the
temperature from twenty-five degrees Celsius to eleven and a half degrees Celsius over a
period of twenty-seven minutes. The Thermomix was running with its heater turned off
to help mix the water as it cooled. The readings from the thermocouple and LCR meter
were recorded every minute until room temperature was reached and all subsequent
measurements were taken every thirty seconds. As the LCR meter and thermocouple
5
have auto-off features, the respective data holds were pressed before data was recorded
(effectively avoiding shutdown) to ensure continuity in the data.
Once eleven and a half degrees Celsius was reached, the Thermomix heater was
turned on. Measurements were taken every thirty seconds as the water heated. At
approximately eighteen degrees Celsius, measurements were taken every minute until a
final temperature of twenty-three and a half degrees Celsius.
Observations
See tables on the following pages.
6
Trial 1 - Water Bath Cold -> Warm -> Cold
Temperature Inductance
Time of measurement
(+/- 0.1C)
(+/- 0.002 mH)
(h:mm:ss) +/- 5s
10.9
5.701
0:00:00
11.5
5.699
0:01:00
12.2
5.696
0:02:00
12.9
5.691
0:03:00
13.6
5.684
0:04:00
14.0
5.674
0:05:00
14.4
5.663
0:06:00
14.9
5.651
0:07:00
15.3
5.637
0:08:00
15.5
5.630
0:08:30
15.6
5.623
0:09:00
15.9
5.616
0:09:30
16.0
5.609
0:10:00
16.2
5.601
0:10:30
16.2
5.594
0:11:00
16.6
5.586
0:11:30
16.8
5.577
0:12:00
17.0
5.568
0:12:30
17.2
5.559
0:13:00
17.4
5.550
0:13:30
17.6
5.540
0:14:00
17.6
5.530
0:14:30
18.0
5.520
0:15:00
18.3
5.509
0:15:30
18.4
5.498
0:16:00
18.8
5.486
0:16:30
19.0
5.473
0:17:00
19.2
5.459
0:17:30
19.4
5.444
0:18:00
19.6
5.429
0:18:30
19.8
5.412
0:19:00
20.0
5.395
0:19:30
20.2
5.376
0:20:00
20.4
5.336
0:21:00
20.8
5.294
0:22:00
21.2
5.249
0:23:00
21.2
5.202
0:24:00
21.7
5.155
0:25:00
22.1
5.106
0:26:00
22.5
5.053
0:27:00
22.8
5.001
0:28:00
23.2
4.948
0:29:00
23.4
4.897
0:30:00
23.8
4.847
0:31:00
24.3
4.796
0:32:00
24.8
4.747
0:33:00
25.2
4.698
0:34:00
25.5
4.655
0:35:00
Begin cooling here
Temperature Inductance
Time of measurement
(+/- 0.1C)
(+/- 0.002 mH)
(h:mm:ss) +/- 5s
25.3
4.623
0:36:00
24.9
4.551
0:37:00
24.9
4.505
0:38:00
24.7
4.468
0:39:00
24.5
4.435
0:40:00
24.5
4.402
0:41:00
24.0
4.371
0:42:00
missed
0:43:00
23.9
4.316
0:44:00
23.5
4.290
0:45:00
missed
0:46:00
23.3
4.247
0:47:00
missed
0:48:00
23.1
4.201
0:49:00
22.5
4.183
0:50:00
22.2
4.164
0:51:00
22.2
4.152
0:52:00
21.9
4.139
0:53:00
20.8
4.127
0:54:00
20.8
4.126
0:55:00
20.1
4.127
0:56:00
20.2
4.131
0:57:00
19.7
4.137
0:58:00
19.4
4.147
0:59:00
19.3
4.154
1:00:00
19.1
4.164
1:01:00
18.7
4.176
1:02:00
18.5
4.188
1:03:00
18.1
4.200
1:04:00
17.7
4.213
1:05:00
17.5
4.227
1:06:00
17.6
4.241
1:07:00
17.3
4.254
1:08:00
17.1
4.269
1:09:00
17.0
4.282
1:10:00
16.6
4.296
1:11:00
16.5
4.308
1:12:00
16.4
4.323
1:13:00
15.9
4.336
1:14:00
15.6
4.350
1:15:00
15.3
4.364
1:16:00
15.2
4.377
1:17:00
14.7
4.390
1:18:00
14.0
4.407
1:19:00
13.9
4.417
1:20:00
13.7
4.430
1:21:00
13.3
4.443
1:22:00
13.1
4.455
1:23:00
12.3
4.481
1:24:00
7
Trial 2 - Oil Bath Warm -> Cold -> Warm
Temperature
Inductance
Time of measurement Temperature
Inductance
Time of measurement
(+/- 0.1C)
(+/- 0.002 mH)
(m:ss)
(+/- 0.1C)
(+/- 0.002 mH)
(m:ss)
25.0
2.878
0:00:00
11.5
4.432
0:26:30
25.0
2.863
0:02:00
11.5
4.454
0:27:00
25.2
2.858
0:03:00
11.3
4.496
0:27:30
25.1
2.855
0:04:00
11.7
4.536
0:28:00
24.0
2.859
0:05:00
12.2
4.573
0:28:30
22.3
2.872
0:06:00
12.8
4.609
0:29:00
21.5
2.885
0:07:00
13.3
4.642
0:29:30
21.0
2.903
0:07:30
13.8
4.672
0:30:00
20.7
2.926
0:08:00
14.2
4.700
0:30:30
20.4
2.955
0:08:30
14.8
4.727
0:31:00
20.0
2.990
0:09:00
15.1
4.751
0:31:30
19.6
3.033
0:09:30
15.5
4.774
0:32:00
19.4
3.081
0:10:00
15.8
4.795
0:32:30
19.0
3.133
0:10:30
16.1
4.813
0:33:00
18.7
3.189
0:11:00
16.3
4.831
0:33:30
18.5
3.247
0:11:30
16.4
4.846
0:34:00
18.3
3.305
0:12:00
16.5
4.861
0:34:30
18.1
3.363
0:12:30
16.7
4.874
0:35:00
18.0
3.420
0:13:00
16.8
4.886
0:35:30
17.7
3.477
0:13:30
17.0
4.897
0:36:00
17.6
3.531
0:14:00
17.3
4.907
0:36:30
17.5
3.582
0:14:30
17.5
4.916
0:37:00
17.3
3.678
0:15:00
17.5
4.925
0:37:30
17.1
3.726
0:15:30
17.6
4.932
0:38:00
16.9
3.771
0:16:00
17.6
4.939
0:38:30
16.6
3.814
0:16:30
17.7
4.946
0:39:00
16.6
3.855
0:17:00
18.0
4.951
0:39:30
16.3
3.895
0:17:30
18.3
4.957
0:40:00
16.2
3.934
0:18:00
18.4
4.961
0:40:30
16.1
3.971
0:18:30
18.6
4.965
0:41:30
15.9
4.007
0:19:00
18.8
4.968
0:42:30
15.4
4.060
0:19:30
18.9
4.970
0:43:30
15.3
4.075
0:20:00
19.0
4.972
0:44:30
15.3
4.109
0:20:30
19.5
4.973
0:45:30
15.1
4.140
0:21:00
20.0
4.973
0:46:30
14.9
4.171
0:21:30
20.5
4.971
0:47:30
14.6
4.206
0:22:00
21.0
4.968
0:48:30
13.8
4.230
0:22:30
21.5
4.961
0:49:30
13.2
4.257
0:23:00
21.7
4.951
0:50:30
12.9
4.285
0:23:30
22.0
4.938
0:51:30
12.5
4.311
0:24:00
22.3
4.923
0:52:30
12.4
4.337
0:24:30
22.5
4.904
0:53:30
12.0
4.364
0:25:00
22.7
4.887
0:54:30
11.8
4.385
0:25:30
22.7
4.867
0:55:30
11.7
4.410
0:26:00
23.0
4.847
0:56:30
Begin warming here
23.3
4.827
0:57:30
23.5
4.806
0:58:30
8
Analysis
Because of the large deviation from the accepted value and the large number of
errors associated with it, the water trial data was not used in the analysis. Only the oil
bath data is discussed here.
To derive the Curie temperature for the sample of gadolinium the acquired data
had to be massaged so it could be applied to the Curie-Weiss law. This involved
choosing which data points to use, and thus how to fit the best curve to our data. Given
the formula, as in (3) assuming a gamma factor of 1:
 L

y (T )  
 1
 Lo

1

T  Tc
C
(4)
One could graph the dependence of 1/(L/Lo –1) against temperature. This provides a
means to calculate the experimental Curie point through extrapolation. The method
employed was to select a point that was central in the region that behaves according to
the Curie-Weiss law. Then the first point at a higher temperature was used with the
selected point to make a linear least squares fitted line. Where the line crossed the x-axis
revealed the Curie temperature predicted by those two points. Then the same process
was done with the first point below the selected point. Many subsequent fits were
performed using the first n points above with the first n-1 points below, and then with the
first n-1 points above and the first n points below. Making a histogram of the Curie
temperature predicted by certain point combinations aided in locating the region that best
seemed to obey the Curie-Weiss law. Once a point that was included above (or below)
caused a significant change in the predicted value, the point previous was considered the
cutoff point. The histogram has been included in the Appendix (Figure A1). The points
that were include were four points below the median and three points above the median,
9
Plot 1: Curie Temperature
2.500
2.000
1/(L/Lo -1)
1.500
Curie Trend
Linear (Curie Trend)
y = 0.411x - 5.723
1.000
0.500
0.000
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
21.0
-0.500
Temperature [degrees C]
10
22.0
where the median was defined as the data point at 18.7 degrees Celsius. This point was
chosen as it consists of a significant number of points (eight) and the inclusion of more
points decreases the slope of the line.
Error values for plotting the graphs were calculated using the derivative method.
For a y-axis value given by:
 L

y   
 1
 Lo

1
(5)
the error in the calculated value is:
y  y 2
L  L

  1 (6)
Lo  Lo

Rearrange (5) to solve for  and substitute back into (6) to get:
y  y y  1
L  L

  1
L  Lo

(7)
Here L and Lo are taken to be the same value, 0.002mH.
The error in the x-axis is that associated with the Digital Thermometer, thus it has been
taken as the accuracy of the instrument: +/- 1 oC.
This least squares fit line crosses the x-axis at 13.959 oC (287.119 K), which is
the resulting experimental Curie temperature for gadolinium, as shown in Plot 1. The
graph shows only the Currie-Weiss region; the full data graphs are presented in the
Appendix (Figures A2 – A4)
The confidence interval for this value can be derived as followed:
(   t / 2 S
1
x

n S xx
,
  t / 2 S
1
x

)
n S xx
(8)
where = 287.11K (13.959 oC)
11
t/2 = 2.306 (for 95% Confidence Interval)
S= 0.024651 (square root of variance estimator)
n=8
x(avg) = 291.1 oK
Sxx = 2.44
This leads to a 95% confidence interval of (13.34, 14.58) oC, or (290.73, 289.49 K) for
the experimental Curie temperature.
Discussion
Accepted values for the Curie temperature of gadolinium vary somewhat. Values
range from 293.4K (CRC Handbook, 81st ed), 293K (Ashcroft and Mermin 1976, Wilkes
1973), 290K (von Ardenne 1973), 290.1K (Lewowski and Wozniak 1997) down to 289K
(Kittel 1956, Legwold et al 1953, Flippen 1963).
Trial 1 – Water Bath
From the water bath trial, a Curie temperature of 272.7K was determined. This is
much lower than the excepted value. There are a number of possible reasons for this.
The first is that the sample was placed in a water bath. This should not have been done.
Water contains dissolved oxygen, which will react with gadolinium. This oxidation
process reduced the purity of the sample, which altered the Curie point. It is not known
what magnitude or direction this impurity would change the Curie temperature. Because
this was the first trial, the second trial results in the oil bath would also be affected.
The second problem arose from the inductance meter. The meter had an
automatic shut off if no buttons were used after a few minutes. When the meter was
turned back on after one of these automatic shut offs, the inductance reading was
drastically different. The amount of difference varied between 0.02mH and 2.0mH, and
12
the direction of the change appeared to be random as well. Sometimes it was a higher
reading than before it shut off; sometimes it was lower. This same effect was not
apparent when the meter was shut off manually. To avoid this effect, whenever a
measurement was taken, the hold button on the meter was pushed so that it did not shut
off automatically. While this means there are no large ‘jumps’ in the readings as they
were taken, it does mean there could have been a large offset to the data. While the
experiment was being set up, the meter was turned on and it shut itself off several times.
An offset in the data would result in the curve being shifted up or down by a constant
amount. The overall shape of the curve would not be changed, but the x-intercept of the
best-fit line would, and hence the Curie temperature would change.
The biggest problem, however, was the way in which the sample was heated and
cooled. In the water bath, the first half of the cycle where the temperature started below
the Curie point then heated had a very low Curie temperature, but the shape of the curve
seemed reasonable. When the sample was cooled back down again, the curve was quite
different than expected. During the warming phase, the inductance decreased as
expected. Once the cooling cycle started, this trend in the inductance continued. It was
only after over 20 minutes and approximately five degrees that the inductance started to
increase again. Two hypotheses were suggested to explain this. The first was that there
was a temperature gradient in the sample. Because there was no waiting time between
the warming and cooling cycles, the sample didn’t have time to ‘catch up’ with the
temperature of the surrounding water bath. The temperature reading did not correspond
to the actual temperature of the bulk sample. This explanation has two major flaws.
First, the thermocouple was in direct contact with the sample. Second, the sample was
13
relatively small, and should not take over 20 minutes to begin to come into equilibrium
with the surrounding water bath
The second hypothesis has to do with the individual domains in the sample, and
the rate at which they are created or destroyed as the Curie temperature is crossed. As the
sample was warmed above the Curie point, the domains in the sample were mostly
destroyed by the thermal motion of the atoms. There is a drop in the inductance of the
sample at the Curie point, but a general downward trend of the inductance continues as
the sample is heated. This is because domains still exist and are being destroyed by
random thermal motions. When the sample first cooled, the temperature was still above
the Curie temperature. There was still enough thermal energy in the sample to continue
to destroy domain arrangements. This is why the inductance reading still decreased
although the temperature was now being lowered. There seems to be some kind of
‘relaxation time’ for the destruction of the domains in a sample to reach equilibrium with
the sample temperature. If the sample had been left at the high temperature for some
time and allowed to reach a steady inductance reading before it was cooled, better results
would have been obtained. This same effect was also taking place before the experiment
even started. The sample was being stored at room temperature before the experiment
took place. It was then placed in the chilled water bath, and the experiment was started.
It was roughly 10 minutes between the time the sample was in the water and the time the
experiment was started. This would have a strong effect on the data because the sample
was quickly brought across the Curie point, and then slowly warmed back up to cross it
again. Overall, this argument seems reasonable on a qualitative basis. No data was
found however to confirm or deny it.
14
Trial 2 –Oil Bath
From the oil bath, a Curie temperature of 287.11 +/- 0.62K was obtained. While
this value is closer to the accepted values than the result from the water bath, it is still not
in agreement. The possible reasons for this are similar to the water bath trial. First, the
purity of the sample was in question after the water bath trial. This would affect the
Curie temperature. Second, the inductance meter still had its problems with the
automatic shut off feature. Again, during the course of the experiment, the meter was not
allowed to shut off between readings, but there still could be a large shift of the data due
to the meter. Depending on the direction of the shift, the measured Curie temperature
would be lower or higher than the actual value.
Finally, the same problem with switching between cooling and heating phases
occurred. This time, the sample was first cooled, and then warmed. As the temperature
was decreased, the inductance measurement increased. Once the cooling stopped and the
warming began, the inductance continued to increase. Much like the water bath trial, it
was about 20 minutes and nine degrees before the inductance started to decrease. The
temperature gradient explanation can be ignored for the same reasons as before. The
‘relaxation time’ explanation can still be used in this case. As the sample is cooled below
the Curie point, domains begin to form between neighboring atoms. Most of this
formation occurs at the Curie temperature, but some will occur after the Curie point has
been reached. Once the sample started to be warmed up again, the temperature was still
below the Curie point, so domains were still being formed in the sample, and the
inductance measurement would still increase. If the sample had been allowed to come to
15
an equilibrium state where no new domains were being formed before being warmed up
again, the data would have given results that are more accurate.
Recommendations
A few things could be done differently when the experiment is repeated. First, the
sample should remain in oil whenever possible. This wasn’t obvious from the
documentation for the lab. Secondly, the reliability of the inductance meter was in
question. Perhaps a different meter could be used, or if there was a known sample to
calibrate the meter from, data could be adjusted for any kind of offset it may introduce.
Finally, further research into the ‘relaxation time’ for the domains to begin
forming/destroying should be done. Because we came up with the hypothesis late in the
lab, there was not enough time to do a thorough literary search. Until this hypothesis can
be confirmed or reputed, it would be a good idea to leave the sample at a fixed
temperature for a long time before beginning the experiment. If a cooling cycle is being
performed, then the sample should be stored at a warm constant temperature (~25°C) for
many hours, overnight if possible. Similarly, if a warming cycle is being performed, then
the sample should be kept at a constant cold temperature (~12°C) for many hours before
the experiment is started. Another idea that might make the transition at the Curie
temperature more drastic would be to put the sample in the coil and run a current through
the coil to saturate the susceptibility of the sample. Of course, this will only work if a
warming cycle is being performed. This is one way to ensure that the sample is in a
known state before the experiment begins.
16
References
Griffiths, D.J. Introduction to Electrodynamics. 3nd edition, Reed College, 1999.
Hook and Hall Solid State Physics. 2nd edition, John Wiley & Sons Ltd, England 1998.
Johnson and Bhattacharyya, Statistics Principles and Methods. 3rd edition, John Wiley &
Sons Inc, Toronto, 1996.
Kraftmakher, Yaakov, “Curie point of ferromagnets”. IOP Publishing LTD & The
European Physical Society, 1997.
Lewowski and Wozniak, “Measurement of Curie temperature for gadolinium: a
laboratory study for students”. IOP Publishing LTD & The European Physical Society,
1997.
17
Appendix
18
1u
1b
1b1u
2b1u
1b2u
2b2u
2b3u
3b2u
3b3u
4b3u
3b4u
4b4u
4b5u
5b4u
5b5u
5b6u
6b5u
6b6u
7b6u
6b7u
7b7u
7b8u
8b7u
8b8u
8b9u
9b8u
9b9u
10b8u
10b9u
11b8u
11b9u
12b8u
12b9u
13b8u
13b9u
14b8u
14b9u
15b8u
15b9u
16b8u
17b9u
Curie Temperature (degrees Celsius)
Figure A1 Finding the Curie-Weiss Range
15
14.5
14
13.5
13
12.5
12
11.5
Combination of points below and above the median
19
Figure A2
Oil Bath Warm to Cold
Data Used for Analysis
3.500
3.000
1/(L/Lo - 1)
2.500
2.000
1.500
1.000
0.500
282.0
284.0
286.0
288.0
290.0
292.0
294.0
296.0
Temperature (K)
20
298.0
Figure A3
Oil Bath Cold to Warm
Data Discarded
0.900
0.880
0.860
1/(L/Lo - 1)
0.840
0.820
0.800
0.780
0.760
0.740
0.720
284.0
286.0
288.0
290.0
292.0
294.0
296.0
Temperature (K)
21
Figure A4
Water Bath Cold to Warm
Data Used for Analysis
0.850
0.800
1/(L/Lo - 1)
0.750
0.700
0.650
0.600
0.550
282.0
284.0
286.0
288.0
290.0
292.0
294.0
296.0
Temperature (K)
22
298.0
Figure A5
Water Bath Warm to Cold
Data Discarded
1.100
1.050
1/(L/Lo - 1)
1.000
0.950
0.900
0.850
0.800
284.0
286.0
288.0
290.0
292.0
294.0
296.0
Temperature (K)
23
298.0