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Heat capacity of UP-USe solid solutions
Journal of Magnetism and Magnetic North-Holland Publishing Company
HEAT CAPACITY A. BLAISE,
Materials
OF UP-Use
J.E. GORDON
Centre d’Etudes Nucleaires
109
38 (1983) 109-122
SOLID SOLUTIONS
*, R. LAGNIER
de Grenoble, 85x, 38041 Grenoble
- cedex, France
M. MORTIMER Chemistry
Division, AERE
Hat-well Didcot, Oxon, OXI I ORA,
UK
R. TROC Institute for Low Temperature Received
19 April
and Structure Research, Polish Academy
of Sciences, 50 - 950 Wroclaw, POB 937, Poland
1983; in revised form 1 June 1983
The heat capacities of UP,_,Se, for x = 0.05, 0.1, 0.18, 0.27 and 0.3 have been measured from 5-300 K. The first three of these solid solutions show pronounced specific heat anomalies at the para-antiferromagnetic phase transition, whereas the comparable anomalies for the x = 0.27 and x = 0.3 are small, a fact which probably reflects the competing tendencies toward antiferromagnetic and ferromagnetic ordering in these latter two materials. All five compounds have a linear specific heat coefficient at high temperatures, y(300), which is much smaller than at low temperatures, y(0). A possible explanation for this difference is that well below the ordering temperature there is a magnetic contribution to y(O), which arises from magnetic moments which are only partially aligned in magnetic fields far smaller than those which characterize the ordered phase. The magnetic entropies of the five compounds are estimated to be about 10 J/mol K at room temperature.
1.Introduction UP is antiferromagnetic, whereas USe is a ferromagnet. Both compounds have the simple NaCl crystal structure, and when mixed. they form a complete range of solid solutions. The transition from one terminal composition to the other gives rise to several intermediate phases with long-period magnetic structures. A recent comprehensive set of magnetic and neutron diffraction measurements [l] leads to a magnetic phase diagram which is reproduced in fig. 1. Other magnetic [2-41 and neutron [5] studies on this same system have been reported. However, no prior heat capacity measurements have been made. Such thermal measurements are useful for checking, and supplementing, 0
* Permanent
address: Department lege, Amherst, MA 01002, USA.
of Physics,
Amherst
0.1
0.2
0.3 composltlon
I )I
4
Col-
0304-8853/83/0000-0000/$03.00 0 1983North-Holland
Fig. 1. Magnetic
phase diagram
for UPC, _*,Se,
- from ref. [I].
110
A. B&se et al. / Heat capacity of UP-U&
the magnetic phase diagram proposed in ref. [ 11. The present work reports studies of five different nominal compositions of UP,_,Se,: x = 0.05, 0.1, 0.18, 0.27, 0.30. Measurements were made in the temperature range 5 to 300 K, except for the x = 0.1 composition, where the measurements were extended down to 1.6 K.
solid solutions
were made on each sample at two different heating rates. The results were smoothed by computer processing to give single curves for the thermodynamical data. The x = 0.1 sample was also measured between 1.6 and 10.3 K at Harwell, using a transient method. In the overlap region of the Grenoble and Harwell results the data agreed to better than 3%. Full details on apparatus and errors of measurement are given in refs. [6,7].
2. Sample preparation and experimental methods The methods for preparing the samples used in these experiments are described by Trot et al. [ 11. The heat capacity samples are the same as were used in that work. They were prepared in Wroclaw and were sent to Grenoble in evacuated glass ampules for the specific heat measurements. The solid solutions were analyzed by X-ray diffraction. All diagrams proved the samples to be single-phase except for the presence of small quantities of UO, (also visible in some of the heat capacity curves). It is estimated in ref. [l] that for a nominal comis a position UP, _x,Se,O the true distribution Gaussian centred on x0 with 95% of the sample being at x0 + 0.05~~. This question of sample inhomogeneity will be referred to in the discussion of the experimental results on the samples with x0 = 0.05 and x,, = 0.1. For the specific heat measurements on four of the samples, about three grams of powdered compound were mixed with an equal amount of silver powder (5 N purity). The mixture was compressed under a pressure of = lo5 N/cm* in order to obtain a specific heat sample with good thermal conductivity. A compressed silver pellet with mass equal to the mass of silver in the sample was inserted in the reference chamber of the dynamic differential calorimeter described in ref. [6]. These operations were carried out in a helium atmosphere to prevent oxidation. In the case of the only about 1.5 g of compound were UPO.,SeO., available. The heat capacity sample in this case contained approximately 1.5 g of compound and 4.5 g of silver powder. Measurements on this solid solution are therefore somewhat more uncertain than those on the other solid solutions. The heat capacities were measured at Grenoble in the 5 to 300 K temperature region. Several runs
3. Experimental results The specific heat results, interpolated to integer values of temperature are given in tables 1 through 5 together with the values of the thermodynamic functions derived from these results. The specific heat data are also plotted in figs. 2-6. In fig. 3 the heat capacity curve for the terminal compositions UP and USe have been added to the curve for The room temperature data compare UP,.,Se,.,. reasonable well with the data observed on UP by Counsel1 et al. [8] and on USe by Takahashi and Westrum [9] (see table 7). The overall behaviour of all five C,(T) curves displays the usual sigmoid curve with several peaks superimposed. In table 6 we list the temperatures and CD values of these peaks.
60.
Y
50.
Y LOE 7
30
o"ZO.
10.
*I
/ 50
150
200
TEMPERATURE
(K )
100
Fig. 2. Specific heat of UP, g5Se,,,.
250
7
A. Blaise et al. / Heat capacity of UP-(/Se
Table 1 Thermodynamic T W) 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0
180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
3.1. Specific diagram
functions cp (J/mol 0.22 0.61
1.56 3.49 5.84 9.60 13.30 17.90 16.30 17.75 21.89 25.45 28.85 33.13 40.42 54.75 40.40 40.53 41.09 42.20 43.18 43.93 44.66 45.47 46.16 46.83 47.47 48.08 48.67 49.25 49.76 50.26 50.73 51.19 51.62
K)
111
solid solutions
for UP,,,$e,,, S (J/mol
(J/m4
(H - H(O))/T (J/mol K)
--Cc - H(O))/T (J/mol K)
0.5 2.4 7.5 19.7 47.0 84.4 139.8 223.7 307.3 392.1 589.6 826.8 1098.0 1406.6 1771.0 226 1.O 2710.5 3114.9 3522.5 3938.8 4366.0 4801.6 5244.6 5695.2 6153.2 6618.1 7089.7 7567.5 8051.2 8540.8 9035.9 9536.0 10040.9 10550.6 11064.6
0.11 0.24 0.50 0.98 1.88 2.8 1 4.00 5.59 6.83 7.84 9.83 11.81 13.72 15.63 17.71 20.55 22.59 23.96 25.16 26.26 27.29 28.24 29.14 29.97 30.77 31.51 32.23 32.90 33.55 34.16 34.75 35.32 35.86 36.38 36.88
0.00 0.11 0.25 0.46 0.77 1.19 1.71 2.34 3.08 3.85 5.45 7.12 8.82 10.55 12.30 14.11 16.00 17.86 19.68 21.45 23.18 24.87 26.5 1 28.10 29.66 31.18 32.66 34.11 35.52 36.91 38.26 39.58 40.87 42.14 43.38
(H - H(O)) K)
0.11 0.35 0.76 1.44 2.65 4.00 5.70 7.94 9.91 11.69 15.28 18.93 22.55 26.18 30.01 34.66 38.58 41.82 44.84 47.7 1 50.47 53.11 55.64 58.08 60.43 62.70 64.89 67.01 69.07 71.07 73.01 74.90 76.73 78.52 80.27
heat results and the magnetic
phase
In the following we shall discuss the specific heat results with reference to the magnetic phase diagram given by ref. [l] and reproduced in fig. 1. 3.1.1. UP,~,,Se,,,, The principal peak corresponds unambiguously to the NCel temperature, where the antiferromagnetic order (type I) sets in. Fig. 1 locates it at 115
K, but our value is in better agreement with the value obtained by interpolation on the straight line graph of T vs. x in ref. [ 11. The peak at 38.5 K corresponds fairly well to the onset of type IA ordering as seen by neutron diffraction. The situation at 30.7 K is less clear. [l] sees this temperature as the disappearance of type I order. However, it should be remembered that a small amount of UO, may be present in the sample and that 30.8 K is the NCel temperature for this substance. Moreover, values of C, and of the
112
Table 2 Thermodynamic T W) 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0
100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
A. Blake et al. / Heat capacity of UP - USe solid solutions
functions cp (J/mol 0.23 0.68
1.67 3.49 6.09 10.41 11.77 14.45 17.25 19.79 30.20 25.68 30.25 36.11 52.00 38.14 38.54 40.36 41.78 42.79 43.76 44.66 45.44 46.17 46.75 47.29 47.91 48.46 49.0 1 49.50 49.97 50.4 1 50.84 51.25 51.65
for UP,,,Seo,, (H - H(O))
K)
c/mol 0.20 0.48 0.92 1.62 2.68 4.08 5.76 7.50 9.37 11.31 15.76 19.73 23.43 27.32 31.95 35.95 39.26 42.42 45.46 48.38 51.17 53.86 56.43 58.91 61.29 63.59 65.80 67.94 70.0 1 72.03 73.98 75.87 77.71 79.50 81.25
K)
(J/m4 0.6 2.7 8.3 20.8 44.6 83.4 137.9 203.2 282.6 375.2 621.0 878.1 1155.9 1486.5 1928.4 2347.3 2727.3 3122.3 3533.3 3956.5 3289.1 4831.5 5282.0 5740.1 6204.8 6675.0 7150.9 7632.7 8120.0 8612.6 9109.9 9611.9 10118.2 10628.7
11143.2
magnetic entropy associated with the peak at 30.7 K seem very small for a bulk phase transition. At 22.85 K we see another somewhat higher peak, which is not evident in the neutron diffraction pattern, but which is seen in the susceptibility measurements. There are several possible explanations for this peak. One might be that this x = 0.05 sample is strongly inhomogeneous and the peak is evidence for the moment-jump transition characteristic of UP and of UP-rich samples. Another explanation would be that the disappearance of type I order is achieved only at 22.9 K, and still
(H - H(O))/T (J/mol K) 0.11 0.27 0.55 1.04 1.78 2.78 3.94 5.08 6.28 7.50 10.35 12.54 14.45 16.52 19.28 21.34 22.73 24.02 25.24 26.38 27.43 28.42 29.34 30.2 1 31.02 31.79 32.50 33.19 33.83 34.45 35.04 35.60 36.14 36.65 37.14
-(G - H(O))/‘T (J/mol K) 0.09 0.21 0.37 0.59 0.89 1.30
1.82 2.42 3.09 3.81 5.41 7.18 8.98 10.80 12.67 14.61 16.53 18.40 20.23 22.01 23.74 25.44 27.09 28.70 30.27 31.80 33.30 34.76 36.18 37.58 38.94 40.27 41.58 42.85 44.10
another that so-called “steplike” transition of the type IA phase [l] sets in at this temperature. Rossat-Mignod et al. [lo] have shown that in UP the moment-jump transition is first order, from a single-k type I with a collinear alignment of moments along (100) to a double-k type I stable at low temperature. The “step-like” transition in UP, _,Se, may have the same origin, being from a high temperature single k type IA to a low temperature double k type IA. For UAs, the transition is from single k type I to double k type IA [lo].
113
A. Blake et al. / Heat capacity of UP - USe solid solutions
Table 3 Thermodynamic T
functions
6)
5 (J/mol
5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
0.00 0.62 1.41 2.89 4.7 1 6.61 8.96 12.00 15.55 19.20 26.53 26.52 30.50 44.40 54.05 40.55 39.67 40.89 42.03 42.95 43.90 44.7 1 45.49 46.23 46.92 47.56 48.16 48.72 49.26 49.78 50.27 50.74 51.19 51.61 52.00
K)
for UP,,,,Se,,,, S (J/mol 0.00
0.23 0.61 1.20 2.04 3.06 4.25 5.64 7.25 9.08 13.23 17.37 21.11 25.39 31.03 35.45 38.90 42.12 45.19 48.13 50.93 53.61 56.19 58.67 61.06 63.37 65.59 67.75 69.83 71.85 73.8 1 75.72 77.57 79.38 81.13
K)
( H - H(O)) (J/m4
(H - HW)/T (J/mol K)
-(C - HWVT (J/mol K)
0.0 1.8 6.6 17.1 36.1 64.3 103.0 155.1 223.9 310.8 539.4 808.0 1088.6 1453.3 1990.0 2452.1 2848.2 3251.3 3666.0 4090.8 4525.1 4968.2 5419.3 5877.9 6343.7 6816.2 7294.8 7779.2 8269.2 8764.4 9264.7 9769.8 10279.5 10793.5 11311.5
0.00 0.18 0.44 0.86 1.44 2.14 2.94 3.88 4.97 6.22 8.99 11.54 13.61 16.15 19.90 22.29 23.74 25.01 26.19 27.27 28.28 29.22 30.11 30.94 31.72 32.46 33.16 33.82 34.45 35.06 35.63 36.18 36.71 37.22 37.71
0.00 0.05 0.17 0.34 0.60 0.92 1.31 1.76 2.28 2.87 4.24 5.83 7.5 1 9.25 11.13 13.16 15.16 17.11 19.01 20.85 22.65 24.39 26.09 27.74 29.34 30.9 1 32.43 33.92 35.38 36.79 38.18 39.54 40.86 42.16 43.43
3.1.2. lJP,,,Se,,, Here again the high temperature peak at 98.3 K coincides perfectly with the NCel temperature, which is also a triple point in the phase diagram. At 58.6 K the specific heat peak is also in good agreement with fig. 1. The transition is a “step-like” one of the antiferromagnetic type IA phase [I 11. Once again a small peak at 30.2 K is probably due to the presence of small amounts of UO,, which order antiferromagnetically. The size of this peak is comparable to that observed in UP,,,,Se,,,,,,
although the transition appears to take place at a slightly lower temperature. Finally, the very small peak at 22.65 K may be due to the presence of small traces of UP-rich clusters, although such a surmise would not be consistent with that in ref. [l] regarding sample inhomogeneity. 3.1.3. UP,.,,Se,,,, The NCel temperature (97 K) inferred from the specific heat measurements is higher than the 93 K obtained from fig. 1. Similarly the “step-like”
A. Blake et al. / Heat capacity of UP - USe solid solutions
114
Table 4 Thermodynamic T
W) 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
functions cp (J/mol 0.44
1.oo 2.32 4.35 7.25 10.66 13.42 15.40 17.33 19.99 25.16 29.23 32.76 36.56 39.11 39.44 40.73 41.62 42.44 43.44 44.37 45.19 45.93 46.6 1 47.24 47.72 48.22 48.72 49.19 49.64 50.07 50.47 50.86 51.22 51.56
K)
for UP,,,Se,
27
s (J/mol 0.22 0.65
1.28 2.20 3.47 5.09 6.96 8.88 10.80 12.76 16.95 21.13 25.26 29.34 33.34 37.08 40.57 43.87 46.98 49.94 52.78 55.49 58.10 60.60 63.00 65.32 61.55 69.71 71.79 73.8 I 75.76 77.66 79.50 81.29 83.03
K)
( H - H(O)) (J/mol K)
(H - HK’))/T (J/mol K)
-(G - HW)/‘T (J/mol K)
I.1 4.3 12.4 28.7 57.2 102.1 162.8 235.0 316.7 409.5 640.1 911.7 1221.3 1568.7 1948.0 2341.2 2741.7 3154.1 3514.1 4003.4 4442.6 4890.5 5346.2 5809.0 6278.2 6753.0 7232.6 7717.4 8206.9 8701.1 9 199.6 9702.3 10209.0 10719.4 11233.2
0.22 0.43 0.82 I .43 2.29 3.40 4.65 5.87 7.04 8.19 10.67 13.02 15.27 17.43 19.48 21.28 22.85 24.26 25.53 26.69 27.77 28.77 29.70 30.57 31.39 32.16 32.88 33.55 34.20 34.80 35.38 35.93 36.46 36.96 37.44
0.00 0.2 1 0.45 0.77 1.18 I .69 2.31 3.01 3.71 4.57 6.28 8.10 9.99 11.91 13.86 15.80 17.72 19.61 21.45 23.25 25.01 26.72 28.39 30.02 31.61 33.16 34.68 36.15 37.59 39.00 40.38 41.72 43.04 44.33 45.59
transition as reflected by the heat capacity anomaly appears at 62 K rather than the 57 K temperature quoted by ref. [l]. For this x = 0.18 solid solution, as well as for the x = 0.27 and x = 0.3 samples, there is no evidence in the specific heat results for traces of UO, and UP-rich solid solutions. 3.1.4. lJP,,,,Se,,, and UP,,,Se,,, It is evident from figs. 5 and 6 that these two solid solutions lack the pronounced specific heat
anomalies present in the data for the other solid solutions and in UP and Use. Such a situation may reflect inhomogeneities in the composition of these samples or, as seems more likely to us, may reflect that the x = 0.27 and x = 0.3 compositions lie in that region of the phase diagram, where the low-temperature phase changes from antiferro- to ferromagnetic. Similar decreases in the size of the specific heat anomalies for the UP, _xSx system were observed [ 121 for the x = 0.4 and x = 0.6 compositions.
A. Blaise et al. / Heat capaciiy of UP - USe solid solutions
Table 5 Thermodynamic T
functions cP (J/mol
W 5.0
for UP,,,,Se,,,, S (J/mol
K)
110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
K)
0.20 0.67 1.32 2.22 3.45 5.02 6.75 8.45 10.14 11.93 15.78 19.59 23.41 27.26 31.10 34.92 38.52 41.78 44.86 47.80 50.6 1 53.29 55.86 58.32 60.69 62.96 65.15 67.26 69.30 71.28 73.19 75.05 76.85 78.61 80.31
0.39 1.10 2.28 4.24 7.05 10.13 12.12 13.36 15.56 18.66 22.91 26.69 30.70 34.68 38.23 41.86 40.31 41.22 42.12 43.05 43.89 44.6 1 45.26 45.84 46.40 46.85 47.27 47.72 48.16 48.59 49.0 1 48.41 49.78 50.11 50.35
10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
Table 6 Temperatures and specific heat of the heat capacity for the five solid solutions, UP, _,Se, x T peak (K)
cp (&,k) (J/mol
K)
115
0.05
0.10
0.18
22.85 30.7 38.5 108.5
22.65 30.2 58.6 98.35
62.0 97.0
6.95 11.0 18.85 60.75
5.15 11.7 34.25 60.0
28.0 60.8
0.27
103.5
39.35
(H - H@))/T
( H - H&9)/T
(J/mot)
(J/mol
1.0 4.6 12.8 28.8 56.6 99.8 156.0 219.7 291.4 376.9 588.7 835.9 1122.8 1450.0 1814.9 2215.7 2629.0 2036.5 3453.1 3879.0 4313.9 4756.5 5205.8 566 1.4 6122.6 6589.0 7059.5 7534.5 8013.9 8497.6 8985.6 9477.6 9973.6 10473.1 10975.5
0.20 0.46 0.85 1.44 2.27 3.33 4.46 5.49 6.48 7.54 9.81 11.94 14.04 16.11 18.15 20.14 21.91 23.36 24.67 25.86 26.96 27.98 28.92 29.80 30.6 1 31.38 32.09 32.76 33.39 33.99 34.65 35.10 35.62 36.11 36.58
anomalies
0.30
112.0
42.0
K)
-CC - HWVT (J/mol K)(J/mol
K)
0.00 0.21 0.47 0.78 1.19 1.69 2.29 2.96 3.66 4.39 5.97 7.64 9.38 11.15 12.95 14.78 16.61 18.42 20.20 21.94 23.65 25.31 26.94 28.53 30.08 31.59 33.06 34.5 1 35.91 37.29 38.63 39.95 41.23 42.49 43.72
In the case of the x = 0.27 sample, the phase diagram would predict phase changes at T = 103, 75, 42 and 39 K. There is no pronounced specific heat anomaly at any of these temperatures, but we do observe a small “peak” at 103 K, and “humps” at 40 and 60 K - results which are in reasonable agreement with the phase diagram. In the case of the x = 0.3 sample, we see a rounded peak at = 112 K, consistent with the para-ferromagnetic transition temperature shown in fig. 1. Once again there are small humps in the Cp-T curve at = 35 and 60 K, but whether these
A. Blaise et al. / Heat capacity of UP - USe solid solutions
116
I
TEMPERATURE
I
I
[Kl
Fig. 3. Specific heat of UP0,90Se0 1O
C,/T vs. T2 graph is used. In table 7 we list the low-temperature values for the electronic specific heat coefficient, y(O), and Debye temperature, 8,, which are obtained from the intercepts and slopes of these straight line graphs. The Bb values are calculated under the assumption that 6N Debye modes exist which contribute to the lattice specific heat. If we were to assume that the lattice specific
humps are evidence for discrepancies in the magnetic phase diagram or simply reflect a systematic error in the specific heat data is uncertain. 4. Data analysis 4.1. Low temperature data The very low temperature data (between 5 and 15 K) are plotted in fig. 7. Here the traditional 70
I
60. ^ 50. Y w -I 40. P 7 30. o"zo. 10,
‘I
50
100 150 TEMPERATURE
Fig. 4. Specific heat of UP0,82Se0.,8.
200 (K )
TEMPERATURE
Fig. 5. Specific heat of UP,,,Se,,,.
(K 1
A. Blab
UPO.?O
;!/*
50
100
et al. / Heat capacity of UP-l&
SE0.30
150
200
TEflPERATURE
250
300
(K )
Fig. 6. Specific heat of UPo.,,Seo,,o.
heat were characterized by contributions from 3N Debye modes and from 3N relatively-high frequency Einstein oscillators [ 131, then only the former would contribute to the low-temperature lattice specific heat. In this case all B,, values in table 7 would have to be reduced by a factor of 2- 113. It is evident from fig. 7 and from table 7 that the y(O) values are all intermediate between those reported for UP and USe [14], while the 8,‘s are
117
solid solutions
essentially the same (except for x = 0.18). The assumption that y(O) and B,, can be obtained from the above graphs should be treated with some caution. Given both the magnetic and thermal evidence for magnetic transitions at temperatures of 100 K and below, the assumption that the low-temperature specific heat is a simple sum of lattice and conduction electron contributions is likely to be naive. Further, as has often been pointed out, it is really only at helium tempertures or lower that we can interpret C,/T vs.T2 graphs with confidence. For this reason the specific heat of one of the samples (x = 0.1) was measured down to 1.6 K at Harwell. These results, as well as the Grenoble data are plotted in fig. 8. The two sets of data agree reasonably well in the overlap region (5-10 K), but the Hat-well data appear to show a flattening of the straight line at temperatures below about 6 K. It is possible that these lowest temperature results indicate poor thermal contact within the powdered sample, or it may be that the flattening is evidence of many body effects [ 15,161. Certainly the very large y(0) values, when combined with the much smaller high temperature y’s (see section 4.2) lend weight to this latter possibility. Just as there is ambiguity about the interpretation of the linear contribution to the low-tempera10 ,_,’ ,_,’ ,’
90
,,*
+,*’
*/ i
80
301 0 ( TEMPERATURE
)’
( K2 )
Fig. 7. C,/T versus T2 graphs between 5 and 14 K for x = 0.05, 0.18, 0.27 and 0.3. UP,_,Se,:
,’
,’
I-’
25
50
100 125 150 75 (TEMPERATURE l2 [Hz1
175
Fig. 8. C’/Tvs. T2 graph between 1.5 and 14 K for UP,,Se,,,; 0 - data from Hanvell; X - data from Grenoble.
2(
118
A. Blake et al. / Heat capacity of UP-C/Se
solid solutions
Table 7 Characteristic values for the heat capacity and entropy of UP, _,Se,. Results for UP are from ref. [8], those for USe are from ref. [9]. Least squares fitting of the experimental data yields uncertainties of approximately 5% for 8, and y(O) and of approximately 10% for a and ~(300). Uncertainties in S,,,,,(300) are estimated to be of the order of 20% (see section 4.4 for a discussion) 0.0 a
x
0.05
0.10
0.18
0.27
0.30
1.0 b
C,(298.15) (J/mol K)
50.3
51.54
51.58
51.93
51.5
50.3
78.28
S(298.15) (J/mol K)
54.8
79.95
80.93
80.8
82.72
80.0
96.52
8,
225
233
227
254
222
224
226
(K) Y(0) (mJ/mol
K’)
Y(300) (mJ/mol
K’)
ax10-4 (J.K/mol) %&300) (J/mol K)
9.6
32.0
35.0
37.8
67.5
74.2
86.8
3.8
10.5
10.6
11.8
10.6
7.3
23.6
8.6
8.2
8.8
8.6
10.0
10.2
40.0
10.3
10.3
11.1
11.0
13.1
11.6
19.4
a See ref. [8]. h See ref. [9].
ture specific heat of these solid solutions, so too there is uncertainty as to whether or not the T 3 term contains a magnetic contribution. In principle, we could estimate the size of a spinwave contribution by subtracting the lattice part of the specific heat of ThP [ 17,181 from the measured T’ term. In practice, the term which remains after the subtraction is carried out is smaller then uncertainties in the data. 4.2. High temperature
data
expansion for the A general high temperature specific heat of magnetically ordering compounds is:
with Cdi, = C, - C,, a term accounting for the thermal expansion coefficient, Clat, = the phonon due to the conterm, Ccond = yT, the contribution duction elections, and Cm_ = the contribution due to the magnetic ions in the paramagnetic phase. Several approximations can be made in eq. (1).
Without sume
losing
C,;, = AC,, ‘T,
too much
generality,
one can as-
A = constant,
C ,att = Debye function, C mag = a/T2
(Heisenberg
model).
With these approximations it is then possible to make computer fits to the specific heat for temperatures well above the ordering temperature, and to seek those values of 8,, y, and (Ywhich, for a given A, minimize the differences between the computer fit and the experimental results. Such a procedure leads, for the five solid solutions, to values for 8, which vary from 330 to 350 K (considerably higher than B. as determined at low temperatures), to values of y which vary from 0 to about 10 mJ/mol K2, depending upon the assumption made concerning the size of A, and to values of cr which are about 5 X lo4 JK/mol. Unfortunately C,,,, Cmag are comparable in size (and much smaller and Ccond than C,att) for T > Tordering. Therefore, the above fitting procedure does not provide an unambiguously “best” computer fit to the data.
119
A. Blaise et al. / Heat capacity of UP - USe solid solutions
An alternative, and method of analyzing the one similar to that used hashi [12]. Let us assume ACp = C,(UP,_,Se,)
somewhat more direct, high-temperature data is by Yokohawa and Takathat for T > Torderingr
- C,(ThP)
= y’T+ a/T2, (2)
where C,(ThP) is the measured specific heat of ThP. A least squares fit of AC,/T vs. T-3 then yields values for y’ and (Y. Such an analysis assumes that the lattice and dilation terms are the same for UP,_,Se, and for ThP, an assumption which probably overestimates somewhat C,,,, for these solid solutions [ 191. However, once again, any more sophisticated assumption would not seem merited by the experimental data. We have carried out an analysis of the hightemperature specific heat data for UP [8], USe [9] and the UP, _XSeX solid solutions using eq. (2) and the ThP specific heat data of Gordon and Blaise [17]. In table 7 the results for (Y and y (300) are listed. Here y (300) = y’ + yThP, where y’ is the intercept obtained from the least squares fit and K2 [18]. Y ThP = 2.9 mJ/mol The general trend in the variation of y (300) with x is the same as that for y(0): an increase with x. This is consistent with the variation with x of the temperature-independent susceptibility [ 11. An exception is the value observed for x = 0.30. But, as the specific-heat data are very sensitive to the base-line corrections used [6], and as the x = 0.30 solid solution was one for which only 1.5g of the compound was available, this low value may be in error. The feature of table 7, which is most striking, is the fact that the ~(300) values for the solid solutions (7-12 mJ/mol K2) were far lower than the y(O) values. This situation appears to be general in the cubic uranium compounds. It is possible that the apparent change with temperature in y arises from a narrow peak in the electronic density of states curve, which is located close to the Fermi energy. However, it can be easily demonstrated that a decrease of a factor of four or more in the y value between helium and room temperatures would require that such a peak has a width considerably less than 0.1 eV. At least at the present time
there appears to be no other evidence, either experimental or theoretical, for such a narrow peak in the electronic density of states. It seems more likely that apparent differences in the high- and low-temperature values of y arise either because of low-temperature electron-phonon enhancement effects [16], or because there is a significant magnetic contribution to the experimental y(O) values listed in table 7 (see section 4.4). The values of the electronic density of states at the Fermi energy inferred from these ~(300) values are consistent with those calculated by Davis [20]. Therefore, in our analysis of the magnetic contributions to the heat capacity and entropy of these solid solutions (section 4.4) we shall assume conduction electron contributions to the measured heat capacity which are calculated from the ~(300) values. 4.3. High temperature
magnetic
contributions
to C,
If we associate the cr/T’ term in eq. (2) with the high-temperature “tail” of the magnetic ordering transition, then we can, in principle, calculate the exchange constant (J) by using the first order approximation [ 121:
(3) where R = gas constant, g, = Lande factor and pefr/pa = paramagnetic moment in Bohr magnetons. In practice, because we do not know the state of the uranium ion, we cannot obtain a unique value for (J) from eq. (3). For example, if we assume Russell-Saunders coupling and if the uranium ions are in the 4 + state, then g, = 0.8, whereas for the trivalent state, g, = 8/l 1. This seemingly small difference in g, produces an uncertainty in ((g, - l)/g,)2, and therefore in (J), of a factor of 5: 2.25. Since, however, pen/p8 is known approximately (it varies from 3.3 to 2.5 over the series x = 0 to x = 1 [l]), we can at least check that eq. (3) yields a value for (J)/kTordering, which is of order 1 to 5. Were this not the case, the identification of the high-temperature TP2 “tail” with the magnetic interaction would not be justified. Values of (Yare listed in table 7. If g, is assumed
120
A. Blarse et al. / Heat capaciiy
to be 0.8, we obtain values of ( J)/kTordering of 4 to 4.8 for x=0 to 0.18, and of = 6 for x=0.27 and x = 0.3. If g, = 8/l 1 these values are reduced to 1.8 to 2.2 for x = 0 to 0.18 and of - 2.7 for x = 0.27 and x = 0.3. As is evident from table 7, the value of the magnetic coefficient (11for USe is far higher than that for any of the solid solutions measured.
of UP - USe solid solutions to assume that the lattice specific heat of UP, _,Se, at temperature T were not equal to that of ThP at the same temperature but rather at temperature 0.9T [ 191.
5. Linear magnetic contribution to the low temperature specific heat
4.4. Magnetic entropy If we are to estimate the entropy associated with magnetic ordering we must make some assumptions about how much of the measured specific heat is of magnetic origin. Where a specific heat anomaly associated with magnetic ordering is large, it is possible to obtain a rough estimate of the magnetic entropy by measuring the area which lies between the anomaly and the “background” specific heat. However, for situations in which the anomaly is small, such a procedure will yield unreasonably small values for Smagnetic. It is our view that the most consistent and straightforward method for calculating the magnetic entropy in these solid solutions is from the approximation: S,,,(T)
=l’(l/T)(c,
- C-~,,P - C,> dT,
(4)
where C, = measured specific heat at temperature TT CThP = specific heat of ThP at this temperature, and C, = y’T (see section 4.2 for the definition of y’). Values for the magnetic entropy at 300 K have been calculated from eq. (4) and are listed in table 7. The values listed there are larger than those usually reported for the uranium rock salt compounds because in our calculation we have included as part of the magnetic entropy an excess “electronic” contribution AS = /Oq,rd[y(0) - y(300)] dT. This procedure assumes that the conduction electron contribution to the specific heat is given at all temperatures by y(3OO)T, and that the lowtemperature linear term, y(O)T, is a sum of conduction electron and magnetic terms. Such a seemingly arbitrary procedure has been followed because it leads to values of S,,,,,(300) which are roughly constant across the series (except for x = 1). The values in table 7 are about R In 4. They would increase to approximately R In 5 if we were
While we have found it convenient for calculational purposes to attribute the term [y(O) y(300)]T to magnetic effects, it is reasonable to ask whether there is any magnetic interaction which could make a linear contribution to the specific heat at temperatures well below the ordering temperature. One possible mechanism is that postulated by Marshall [21] and Klein and Brout [22] to explain the fact that small amounts of Mn in copper produce anomalously large, and concentration-independent, y’s. In this picture the Mn spins interact via an RKKY interaction. At low temperatures most are rigidly aligned in an effective field far larger than kT/p (p is the magnet moment associated with a spin). Some fraction of the spins, however, see a magnetic field comparable in size to kT/p. As is shown in refs. [21,22] these spins have a heat capacity which varies linearly with T at low temperatures. The UP,_,Se, solid solutions are materials in which the ordering mechanism may well come about through an RKKY interaction. Because of their complex magnetic structures at low temperatures it is also reasonable to assume that there exists some variation in the magnetic fields seen by the uranium ions. Perhaps, then, Marshall’s model can be applied to the present problem. Let us assume that at a temperature well below the ordering temperature, T,, the majority of moments see an effective magnetic field of size = kT,/p. The contribution of these moments to the specific heat would, presumably, be given by spin-wave theory. However if some small fraction of the moments see considerably smaller fields, then their contribution to the specific heat can be calculated as follows: let p(H) be the distribution function for fields seen by these moments. If, for simplicity we then the average spin-4 particles, assume
A. Blaise et al. / Heat capacity of UP-We
energy/moment c=
/
is
H’ p(H)pH
tanh(pHH/kT)
dH,
(5)
-H’
where H’ z==kT/p. The heat capacity of n such moments is n d
cm= -
El
x’ p(x)x2 J --x’
sech2x dx.
The term (x2 sech’x) in eq. (6) is negligible when x differs significantly from 1. Hence the limits on the integral can be replaced by f 00. Further, if p(H) does not vary rapidly with H, then we can rewrite eq. (6) as
= CITI
f!$p(~)j:~x~sech2x
dx,
The value of the integral in eq. (7) is 7r2/3. Thus we can write C,,, = y,,,T, where y, = r2k2np(0)/3p.
(8)
A comparison of these arguments with those used for determining the y for conduction electrons shows that here rip(O)))) plays a role something akin to that of the density of states at the Fermi energy for the free electron gas [23]. Two remarks about the arguments which lead to eq. (8) are in order. First, our expressions for c and C, are not valid for those spins in the problem, presumably the large majority, whose behavior at low temperatures is properly described by spin-wave theory. The size of n is thus uncertain, but doubtless it is far less than the total number of uranium ions. Because of this uncertainty, we have not bothered to include, as is done in refs. [21,22], a factor $ in the expressions for Z and C,,, to account for the fact that H arises from spin-spin interactions. Second, any careful estimate of y, would require a detailed calculation of p(O) comparable to that carried out for the Cu-Mn problem in [22]. We can, however, make a rough estimate by inverting the argument used to estimate the Fermi temperture, TF, for conduction electrons. There we obtain TF by setting the measured y equal to R/T,. Here we can estimate y,,, by setting y, - R/T,, where T, is the ordering tem-
solid solutions
121
perature for these solid solutions. Since T, = 100 K*. Such an estiK, we obtain y,,, = 80 mJ/mol mate of y,, admittedly crude, is larger than, but in considerably better than in order-of-magnitude the measured values of agreement with, y(O)-~(300). Klein and Brout [22] have pointed out that correlation effects would reduce the theoretical estimate of y,. Here, as in their calculations, such an effect might bring the predicted values closer to the measured ones. If the large y(O) values in these solid solutions are, in fact, evidence for a spread in magnetic field values, then we might expect the effect to be most pronounced in these compositions which lie close to the ferromagnetic-antiferromagnetic region of the phase diagram. Such is the case of the x = 0.27 and x = 0.3 solid solutions, solutions which have a y(O) almost double those for the lower x values. Since the effect might also be sample dependent, it could explain the difference in y(O) values reported for USe [9,14]. There are, of course, other possible explanations for the high y(O) values reported here. Electron-phonon many body effects [16] and spin fluctuation effects [15] are but two. Careful lowtemperature measurements will, perhaps, provide some clarification. It is also possible that specific heat measurements in large magnetic fields would be helpful although Scarborough et al. [24] measured the low-temperature specific heat of UN in magnetic fields as large as 35 kG and found no field dependence.
6. Conclusion The main features of the magnetic phase diagram of the phosphorus-rich UP, _XSeX solid solutions proposed by [l] have been checked and found to be consistent with the specific heat results, although some small corrections probably are in order. For x = 0.05 the type I phase has probably disappeared, on cooling down, well above 30 K. For x = 0.18 the para-antiferromagnetic transition and the “step-like” transition occur at temperatures higher than those quoted by ref. [ 11. For x = 0.27 and x = 0.3 the specific heat anomaly associated with the ordering of the paramagnetic
122
A. Blaise et al. / Heat capacity of UP - We solid solutions
phase is small, a fact which probably reflects the competing tendencies toward antiferromagnetic and ferromagnetic ordering. Finally, analysis of the high and low-temperature specific heats yields high-temperature y values in the range 7-12 mJ/mol K2, while the low-temperature values range from 32 to 74 mJ/mol K2. If one assumes that the change in y takes place with the onset of magnetic order, and if one classifies the entropy associated with this change in y as magnetic entropy, then one obtains values for the magnetic entropy at 300 K, which are consistent with the ordering of localized magnetic moments.
Acknowledgements JEG wishes to thank E. Bonjour of the Service des Basses Temperatures, Laboratoire de Cryophysique and A. Blaise of the Laboratoire de Physique des Solides, Departement de Recherche Fondamentale, Centre d’Etudes Nucleaires de Grenoble for the hospitality extended to him. He also wishes to thank E. Cervos and P. Bertrand for technical assistance.
References [I] R. Trot, J. Leciejewicz and G.H. Lander, J. Magn. Magn. Mat. 21 (1980) 173. [2] W. Trzebiatowski and T. Palewski, Bull. Acad. Pol. Sci., Ser. Sci. Chim. 19 (1971) 83. (31 W. Suski, T. Palewski and T. Mydlarz, Proc. COB. CNRS Physics in High Magnetic Fields, Grenoble (1974) 57. [4] W. Suski, A. Wojakowski and T. Palewski, Phys. Stat. Sol. (a) 27 (1975) K27.
[51 J. Leciejewicz, R. Trot and T. Palewski, Phys. Stat. Sol. (b) 65 (1974) K57. 17 161 R. Lagnier, J. Pierre and M.J. Mortimer, Cryogenics (1977) 369. [71 J.E. Gordon, R.O.A. Hall, J.A. Lee and M.J. Mortimer, Proc. Roy. Sot. London A351 (1976) 179. PI J.F. Counsell, R.M. Dell, A.R. Junkinson and J.F. Martin, Trans. Faraday Sot. 63 (1967) 72. and E.F. Westrum Jr., J. Phys. Chem. 69 [91 Y. Takahashi (1976) 3618. P. Burlet, S. Quezel, J.M. Effantin, 0. 1101 J. Rossat-Mignod, Vogt and H. Bartholin, Proc. VII Intern. Conf. on Solid Compounds of Transition Elements, Grenoble, 1982 (invited paper). and R. Trot, Solid State [Ill P. Burlet, J. Rossat-Mignod Commun. 43 (1982) 429. 1121 H. Tokohawa and Y. Takahashi, J. Phys. Chem. Solids 40 (1979) 603. Solid State [I31 J. Danan, C.H. deNovion and H. Dallaporta, Commun. 10 (1972) 775. J. Danan, C.H. deNovion, Y. Guerin, F.A. Wedgwood and M. Kuznietz, J. de Phys. 37 (1976) 1169. for USe gives y(0) = 17.3 mJ/mol [I41 A recent measurement K’. H.R. Ott, private communication. [I51 W.F. Brinkman and S. Englesberg, Phys. Rev. 169 (1968) 417. [I61 G. Grimvall, J. Phys. Chem. Solids 29 (1968) 1221. [I71 J.E. Gordon and A. Blaise, unpublished. [I81 V. Maurice, J.L. Boutard and D. Abbe, J. de Phys. 40-C4 (1979) 141. [I91 H.E. Flotow, D.W. Osborne and R.R. Walter, J. Chem. Phys. 35 (1979) 880. Electronic Structure and WI H.L. Davis, in: The Actinides: Related Properties, vol. II, eds. A.J. Freeman and J.B. Darby Jr. (Academic Press, New York, 1974) p. 1. PII W. Marshall, Phys. Rev. 118 (1960) 1519. WI M.W. Klein and R. Brout, Phys. Rev. 132 (1963) 2412. to Solid Physics, 4th ed. (John ~231 C. Kittel, Introduction Wiley, New York, 1971) p. 252. H.L. Davis, W. Fulkerson and J.O. v41 J.O. Scarborough, Betterton, Jr., Phys. Rev. 176 (1968) 666.
HEAT CAPACITY A. BLAISE,
Materials
OF UP-Use
J.E. GORDON
Centre d’Etudes Nucleaires
109
38 (1983) 109-122
SOLID SOLUTIONS
*, R. LAGNIER
de Grenoble, 85x, 38041 Grenoble
- cedex, France
M. MORTIMER Chemistry
Division, AERE
Hat-well Didcot, Oxon, OXI I ORA,
UK
R. TROC Institute for Low Temperature Received
19 April
and Structure Research, Polish Academy
of Sciences, 50 - 950 Wroclaw, POB 937, Poland
1983; in revised form 1 June 1983
The heat capacities of UP,_,Se, for x = 0.05, 0.1, 0.18, 0.27 and 0.3 have been measured from 5-300 K. The first three of these solid solutions show pronounced specific heat anomalies at the para-antiferromagnetic phase transition, whereas the comparable anomalies for the x = 0.27 and x = 0.3 are small, a fact which probably reflects the competing tendencies toward antiferromagnetic and ferromagnetic ordering in these latter two materials. All five compounds have a linear specific heat coefficient at high temperatures, y(300), which is much smaller than at low temperatures, y(0). A possible explanation for this difference is that well below the ordering temperature there is a magnetic contribution to y(O), which arises from magnetic moments which are only partially aligned in magnetic fields far smaller than those which characterize the ordered phase. The magnetic entropies of the five compounds are estimated to be about 10 J/mol K at room temperature.
1.Introduction UP is antiferromagnetic, whereas USe is a ferromagnet. Both compounds have the simple NaCl crystal structure, and when mixed. they form a complete range of solid solutions. The transition from one terminal composition to the other gives rise to several intermediate phases with long-period magnetic structures. A recent comprehensive set of magnetic and neutron diffraction measurements [l] leads to a magnetic phase diagram which is reproduced in fig. 1. Other magnetic [2-41 and neutron [5] studies on this same system have been reported. However, no prior heat capacity measurements have been made. Such thermal measurements are useful for checking, and supplementing, 0
* Permanent
address: Department lege, Amherst, MA 01002, USA.
of Physics,
Amherst
0.1
0.2
0.3 composltlon
I )I
4
Col-
0304-8853/83/0000-0000/$03.00 0 1983North-Holland
Fig. 1. Magnetic
phase diagram
for UPC, _*,Se,
- from ref. [I].
110
A. B&se et al. / Heat capacity of UP-U&
the magnetic phase diagram proposed in ref. [ 11. The present work reports studies of five different nominal compositions of UP,_,Se,: x = 0.05, 0.1, 0.18, 0.27, 0.30. Measurements were made in the temperature range 5 to 300 K, except for the x = 0.1 composition, where the measurements were extended down to 1.6 K.
solid solutions
were made on each sample at two different heating rates. The results were smoothed by computer processing to give single curves for the thermodynamical data. The x = 0.1 sample was also measured between 1.6 and 10.3 K at Harwell, using a transient method. In the overlap region of the Grenoble and Harwell results the data agreed to better than 3%. Full details on apparatus and errors of measurement are given in refs. [6,7].
2. Sample preparation and experimental methods The methods for preparing the samples used in these experiments are described by Trot et al. [ 11. The heat capacity samples are the same as were used in that work. They were prepared in Wroclaw and were sent to Grenoble in evacuated glass ampules for the specific heat measurements. The solid solutions were analyzed by X-ray diffraction. All diagrams proved the samples to be single-phase except for the presence of small quantities of UO, (also visible in some of the heat capacity curves). It is estimated in ref. [l] that for a nominal comis a position UP, _x,Se,O the true distribution Gaussian centred on x0 with 95% of the sample being at x0 + 0.05~~. This question of sample inhomogeneity will be referred to in the discussion of the experimental results on the samples with x0 = 0.05 and x,, = 0.1. For the specific heat measurements on four of the samples, about three grams of powdered compound were mixed with an equal amount of silver powder (5 N purity). The mixture was compressed under a pressure of = lo5 N/cm* in order to obtain a specific heat sample with good thermal conductivity. A compressed silver pellet with mass equal to the mass of silver in the sample was inserted in the reference chamber of the dynamic differential calorimeter described in ref. [6]. These operations were carried out in a helium atmosphere to prevent oxidation. In the case of the only about 1.5 g of compound were UPO.,SeO., available. The heat capacity sample in this case contained approximately 1.5 g of compound and 4.5 g of silver powder. Measurements on this solid solution are therefore somewhat more uncertain than those on the other solid solutions. The heat capacities were measured at Grenoble in the 5 to 300 K temperature region. Several runs
3. Experimental results The specific heat results, interpolated to integer values of temperature are given in tables 1 through 5 together with the values of the thermodynamic functions derived from these results. The specific heat data are also plotted in figs. 2-6. In fig. 3 the heat capacity curve for the terminal compositions UP and USe have been added to the curve for The room temperature data compare UP,.,Se,.,. reasonable well with the data observed on UP by Counsel1 et al. [8] and on USe by Takahashi and Westrum [9] (see table 7). The overall behaviour of all five C,(T) curves displays the usual sigmoid curve with several peaks superimposed. In table 6 we list the temperatures and CD values of these peaks.
60.
Y
50.
Y LOE 7
30
o"ZO.
10.
*I
/ 50
150
200
TEMPERATURE
(K )
100
Fig. 2. Specific heat of UP, g5Se,,,.
250
7
A. Blaise et al. / Heat capacity of UP-(/Se
Table 1 Thermodynamic T W) 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0
180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
3.1. Specific diagram
functions cp (J/mol 0.22 0.61
1.56 3.49 5.84 9.60 13.30 17.90 16.30 17.75 21.89 25.45 28.85 33.13 40.42 54.75 40.40 40.53 41.09 42.20 43.18 43.93 44.66 45.47 46.16 46.83 47.47 48.08 48.67 49.25 49.76 50.26 50.73 51.19 51.62
K)
111
solid solutions
for UP,,,$e,,, S (J/mol
(J/m4
(H - H(O))/T (J/mol K)
--Cc - H(O))/T (J/mol K)
0.5 2.4 7.5 19.7 47.0 84.4 139.8 223.7 307.3 392.1 589.6 826.8 1098.0 1406.6 1771.0 226 1.O 2710.5 3114.9 3522.5 3938.8 4366.0 4801.6 5244.6 5695.2 6153.2 6618.1 7089.7 7567.5 8051.2 8540.8 9035.9 9536.0 10040.9 10550.6 11064.6
0.11 0.24 0.50 0.98 1.88 2.8 1 4.00 5.59 6.83 7.84 9.83 11.81 13.72 15.63 17.71 20.55 22.59 23.96 25.16 26.26 27.29 28.24 29.14 29.97 30.77 31.51 32.23 32.90 33.55 34.16 34.75 35.32 35.86 36.38 36.88
0.00 0.11 0.25 0.46 0.77 1.19 1.71 2.34 3.08 3.85 5.45 7.12 8.82 10.55 12.30 14.11 16.00 17.86 19.68 21.45 23.18 24.87 26.5 1 28.10 29.66 31.18 32.66 34.11 35.52 36.91 38.26 39.58 40.87 42.14 43.38
(H - H(O)) K)
0.11 0.35 0.76 1.44 2.65 4.00 5.70 7.94 9.91 11.69 15.28 18.93 22.55 26.18 30.01 34.66 38.58 41.82 44.84 47.7 1 50.47 53.11 55.64 58.08 60.43 62.70 64.89 67.01 69.07 71.07 73.01 74.90 76.73 78.52 80.27
heat results and the magnetic
phase
In the following we shall discuss the specific heat results with reference to the magnetic phase diagram given by ref. [l] and reproduced in fig. 1. 3.1.1. UP,~,,Se,,,, The principal peak corresponds unambiguously to the NCel temperature, where the antiferromagnetic order (type I) sets in. Fig. 1 locates it at 115
K, but our value is in better agreement with the value obtained by interpolation on the straight line graph of T vs. x in ref. [ 11. The peak at 38.5 K corresponds fairly well to the onset of type IA ordering as seen by neutron diffraction. The situation at 30.7 K is less clear. [l] sees this temperature as the disappearance of type I order. However, it should be remembered that a small amount of UO, may be present in the sample and that 30.8 K is the NCel temperature for this substance. Moreover, values of C, and of the
112
Table 2 Thermodynamic T W) 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0
100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
A. Blake et al. / Heat capacity of UP - USe solid solutions
functions cp (J/mol 0.23 0.68
1.67 3.49 6.09 10.41 11.77 14.45 17.25 19.79 30.20 25.68 30.25 36.11 52.00 38.14 38.54 40.36 41.78 42.79 43.76 44.66 45.44 46.17 46.75 47.29 47.91 48.46 49.0 1 49.50 49.97 50.4 1 50.84 51.25 51.65
for UP,,,Seo,, (H - H(O))
K)
c/mol 0.20 0.48 0.92 1.62 2.68 4.08 5.76 7.50 9.37 11.31 15.76 19.73 23.43 27.32 31.95 35.95 39.26 42.42 45.46 48.38 51.17 53.86 56.43 58.91 61.29 63.59 65.80 67.94 70.0 1 72.03 73.98 75.87 77.71 79.50 81.25
K)
(J/m4 0.6 2.7 8.3 20.8 44.6 83.4 137.9 203.2 282.6 375.2 621.0 878.1 1155.9 1486.5 1928.4 2347.3 2727.3 3122.3 3533.3 3956.5 3289.1 4831.5 5282.0 5740.1 6204.8 6675.0 7150.9 7632.7 8120.0 8612.6 9109.9 9611.9 10118.2 10628.7
11143.2
magnetic entropy associated with the peak at 30.7 K seem very small for a bulk phase transition. At 22.85 K we see another somewhat higher peak, which is not evident in the neutron diffraction pattern, but which is seen in the susceptibility measurements. There are several possible explanations for this peak. One might be that this x = 0.05 sample is strongly inhomogeneous and the peak is evidence for the moment-jump transition characteristic of UP and of UP-rich samples. Another explanation would be that the disappearance of type I order is achieved only at 22.9 K, and still
(H - H(O))/T (J/mol K) 0.11 0.27 0.55 1.04 1.78 2.78 3.94 5.08 6.28 7.50 10.35 12.54 14.45 16.52 19.28 21.34 22.73 24.02 25.24 26.38 27.43 28.42 29.34 30.2 1 31.02 31.79 32.50 33.19 33.83 34.45 35.04 35.60 36.14 36.65 37.14
-(G - H(O))/‘T (J/mol K) 0.09 0.21 0.37 0.59 0.89 1.30
1.82 2.42 3.09 3.81 5.41 7.18 8.98 10.80 12.67 14.61 16.53 18.40 20.23 22.01 23.74 25.44 27.09 28.70 30.27 31.80 33.30 34.76 36.18 37.58 38.94 40.27 41.58 42.85 44.10
another that so-called “steplike” transition of the type IA phase [l] sets in at this temperature. Rossat-Mignod et al. [lo] have shown that in UP the moment-jump transition is first order, from a single-k type I with a collinear alignment of moments along (100) to a double-k type I stable at low temperature. The “step-like” transition in UP, _,Se, may have the same origin, being from a high temperature single k type IA to a low temperature double k type IA. For UAs, the transition is from single k type I to double k type IA [lo].
113
A. Blake et al. / Heat capacity of UP - USe solid solutions
Table 3 Thermodynamic T
functions
6)
5 (J/mol
5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
0.00 0.62 1.41 2.89 4.7 1 6.61 8.96 12.00 15.55 19.20 26.53 26.52 30.50 44.40 54.05 40.55 39.67 40.89 42.03 42.95 43.90 44.7 1 45.49 46.23 46.92 47.56 48.16 48.72 49.26 49.78 50.27 50.74 51.19 51.61 52.00
K)
for UP,,,,Se,,,, S (J/mol 0.00
0.23 0.61 1.20 2.04 3.06 4.25 5.64 7.25 9.08 13.23 17.37 21.11 25.39 31.03 35.45 38.90 42.12 45.19 48.13 50.93 53.61 56.19 58.67 61.06 63.37 65.59 67.75 69.83 71.85 73.8 1 75.72 77.57 79.38 81.13
K)
( H - H(O)) (J/m4
(H - HW)/T (J/mol K)
-(C - HWVT (J/mol K)
0.0 1.8 6.6 17.1 36.1 64.3 103.0 155.1 223.9 310.8 539.4 808.0 1088.6 1453.3 1990.0 2452.1 2848.2 3251.3 3666.0 4090.8 4525.1 4968.2 5419.3 5877.9 6343.7 6816.2 7294.8 7779.2 8269.2 8764.4 9264.7 9769.8 10279.5 10793.5 11311.5
0.00 0.18 0.44 0.86 1.44 2.14 2.94 3.88 4.97 6.22 8.99 11.54 13.61 16.15 19.90 22.29 23.74 25.01 26.19 27.27 28.28 29.22 30.11 30.94 31.72 32.46 33.16 33.82 34.45 35.06 35.63 36.18 36.71 37.22 37.71
0.00 0.05 0.17 0.34 0.60 0.92 1.31 1.76 2.28 2.87 4.24 5.83 7.5 1 9.25 11.13 13.16 15.16 17.11 19.01 20.85 22.65 24.39 26.09 27.74 29.34 30.9 1 32.43 33.92 35.38 36.79 38.18 39.54 40.86 42.16 43.43
3.1.2. lJP,,,Se,,, Here again the high temperature peak at 98.3 K coincides perfectly with the NCel temperature, which is also a triple point in the phase diagram. At 58.6 K the specific heat peak is also in good agreement with fig. 1. The transition is a “step-like” one of the antiferromagnetic type IA phase [I 11. Once again a small peak at 30.2 K is probably due to the presence of small amounts of UO,, which order antiferromagnetically. The size of this peak is comparable to that observed in UP,,,,Se,,,,,,
although the transition appears to take place at a slightly lower temperature. Finally, the very small peak at 22.65 K may be due to the presence of small traces of UP-rich clusters, although such a surmise would not be consistent with that in ref. [l] regarding sample inhomogeneity. 3.1.3. UP,.,,Se,,,, The NCel temperature (97 K) inferred from the specific heat measurements is higher than the 93 K obtained from fig. 1. Similarly the “step-like”
A. Blake et al. / Heat capacity of UP - USe solid solutions
114
Table 4 Thermodynamic T
W) 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
functions cp (J/mol 0.44
1.oo 2.32 4.35 7.25 10.66 13.42 15.40 17.33 19.99 25.16 29.23 32.76 36.56 39.11 39.44 40.73 41.62 42.44 43.44 44.37 45.19 45.93 46.6 1 47.24 47.72 48.22 48.72 49.19 49.64 50.07 50.47 50.86 51.22 51.56
K)
for UP,,,Se,
27
s (J/mol 0.22 0.65
1.28 2.20 3.47 5.09 6.96 8.88 10.80 12.76 16.95 21.13 25.26 29.34 33.34 37.08 40.57 43.87 46.98 49.94 52.78 55.49 58.10 60.60 63.00 65.32 61.55 69.71 71.79 73.8 I 75.76 77.66 79.50 81.29 83.03
K)
( H - H(O)) (J/mol K)
(H - HK’))/T (J/mol K)
-(G - HW)/‘T (J/mol K)
I.1 4.3 12.4 28.7 57.2 102.1 162.8 235.0 316.7 409.5 640.1 911.7 1221.3 1568.7 1948.0 2341.2 2741.7 3154.1 3514.1 4003.4 4442.6 4890.5 5346.2 5809.0 6278.2 6753.0 7232.6 7717.4 8206.9 8701.1 9 199.6 9702.3 10209.0 10719.4 11233.2
0.22 0.43 0.82 I .43 2.29 3.40 4.65 5.87 7.04 8.19 10.67 13.02 15.27 17.43 19.48 21.28 22.85 24.26 25.53 26.69 27.77 28.77 29.70 30.57 31.39 32.16 32.88 33.55 34.20 34.80 35.38 35.93 36.46 36.96 37.44
0.00 0.2 1 0.45 0.77 1.18 I .69 2.31 3.01 3.71 4.57 6.28 8.10 9.99 11.91 13.86 15.80 17.72 19.61 21.45 23.25 25.01 26.72 28.39 30.02 31.61 33.16 34.68 36.15 37.59 39.00 40.38 41.72 43.04 44.33 45.59
transition as reflected by the heat capacity anomaly appears at 62 K rather than the 57 K temperature quoted by ref. [l]. For this x = 0.18 solid solution, as well as for the x = 0.27 and x = 0.3 samples, there is no evidence in the specific heat results for traces of UO, and UP-rich solid solutions. 3.1.4. lJP,,,,Se,,, and UP,,,Se,,, It is evident from figs. 5 and 6 that these two solid solutions lack the pronounced specific heat
anomalies present in the data for the other solid solutions and in UP and Use. Such a situation may reflect inhomogeneities in the composition of these samples or, as seems more likely to us, may reflect that the x = 0.27 and x = 0.3 compositions lie in that region of the phase diagram, where the low-temperature phase changes from antiferro- to ferromagnetic. Similar decreases in the size of the specific heat anomalies for the UP, _xSx system were observed [ 121 for the x = 0.4 and x = 0.6 compositions.
A. Blaise et al. / Heat capaciiy of UP - USe solid solutions
Table 5 Thermodynamic T
functions cP (J/mol
W 5.0
for UP,,,,Se,,,, S (J/mol
K)
110.0 120.0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0
K)
0.20 0.67 1.32 2.22 3.45 5.02 6.75 8.45 10.14 11.93 15.78 19.59 23.41 27.26 31.10 34.92 38.52 41.78 44.86 47.80 50.6 1 53.29 55.86 58.32 60.69 62.96 65.15 67.26 69.30 71.28 73.19 75.05 76.85 78.61 80.31
0.39 1.10 2.28 4.24 7.05 10.13 12.12 13.36 15.56 18.66 22.91 26.69 30.70 34.68 38.23 41.86 40.31 41.22 42.12 43.05 43.89 44.6 1 45.26 45.84 46.40 46.85 47.27 47.72 48.16 48.59 49.0 1 48.41 49.78 50.11 50.35
10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 60.0 70.0 80.0 90.0 100.0
Table 6 Temperatures and specific heat of the heat capacity for the five solid solutions, UP, _,Se, x T peak (K)
cp (&,k) (J/mol
K)
115
0.05
0.10
0.18
22.85 30.7 38.5 108.5
22.65 30.2 58.6 98.35
62.0 97.0
6.95 11.0 18.85 60.75
5.15 11.7 34.25 60.0
28.0 60.8
0.27
103.5
39.35
(H - H@))/T
( H - H&9)/T
(J/mot)
(J/mol
1.0 4.6 12.8 28.8 56.6 99.8 156.0 219.7 291.4 376.9 588.7 835.9 1122.8 1450.0 1814.9 2215.7 2629.0 2036.5 3453.1 3879.0 4313.9 4756.5 5205.8 566 1.4 6122.6 6589.0 7059.5 7534.5 8013.9 8497.6 8985.6 9477.6 9973.6 10473.1 10975.5
0.20 0.46 0.85 1.44 2.27 3.33 4.46 5.49 6.48 7.54 9.81 11.94 14.04 16.11 18.15 20.14 21.91 23.36 24.67 25.86 26.96 27.98 28.92 29.80 30.6 1 31.38 32.09 32.76 33.39 33.99 34.65 35.10 35.62 36.11 36.58
anomalies
0.30
112.0
42.0
K)
-CC - HWVT (J/mol K)(J/mol
K)
0.00 0.21 0.47 0.78 1.19 1.69 2.29 2.96 3.66 4.39 5.97 7.64 9.38 11.15 12.95 14.78 16.61 18.42 20.20 21.94 23.65 25.31 26.94 28.53 30.08 31.59 33.06 34.5 1 35.91 37.29 38.63 39.95 41.23 42.49 43.72
In the case of the x = 0.27 sample, the phase diagram would predict phase changes at T = 103, 75, 42 and 39 K. There is no pronounced specific heat anomaly at any of these temperatures, but we do observe a small “peak” at 103 K, and “humps” at 40 and 60 K - results which are in reasonable agreement with the phase diagram. In the case of the x = 0.3 sample, we see a rounded peak at = 112 K, consistent with the para-ferromagnetic transition temperature shown in fig. 1. Once again there are small humps in the Cp-T curve at = 35 and 60 K, but whether these
A. Blaise et al. / Heat capacity of UP - USe solid solutions
116
I
TEMPERATURE
I
I
[Kl
Fig. 3. Specific heat of UP0,90Se0 1O
C,/T vs. T2 graph is used. In table 7 we list the low-temperature values for the electronic specific heat coefficient, y(O), and Debye temperature, 8,, which are obtained from the intercepts and slopes of these straight line graphs. The Bb values are calculated under the assumption that 6N Debye modes exist which contribute to the lattice specific heat. If we were to assume that the lattice specific
humps are evidence for discrepancies in the magnetic phase diagram or simply reflect a systematic error in the specific heat data is uncertain. 4. Data analysis 4.1. Low temperature data The very low temperature data (between 5 and 15 K) are plotted in fig. 7. Here the traditional 70
I
60. ^ 50. Y w -I 40. P 7 30. o"zo. 10,
‘I
50
100 150 TEMPERATURE
Fig. 4. Specific heat of UP0,82Se0.,8.
200 (K )
TEMPERATURE
Fig. 5. Specific heat of UP,,,Se,,,.
(K 1
A. Blab
UPO.?O
;!/*
50
100
et al. / Heat capacity of UP-l&
SE0.30
150
200
TEflPERATURE
250
300
(K )
Fig. 6. Specific heat of UPo.,,Seo,,o.
heat were characterized by contributions from 3N Debye modes and from 3N relatively-high frequency Einstein oscillators [ 131, then only the former would contribute to the low-temperature lattice specific heat. In this case all B,, values in table 7 would have to be reduced by a factor of 2- 113. It is evident from fig. 7 and from table 7 that the y(O) values are all intermediate between those reported for UP and USe [14], while the 8,‘s are
117
solid solutions
essentially the same (except for x = 0.18). The assumption that y(O) and B,, can be obtained from the above graphs should be treated with some caution. Given both the magnetic and thermal evidence for magnetic transitions at temperatures of 100 K and below, the assumption that the low-temperature specific heat is a simple sum of lattice and conduction electron contributions is likely to be naive. Further, as has often been pointed out, it is really only at helium tempertures or lower that we can interpret C,/T vs.T2 graphs with confidence. For this reason the specific heat of one of the samples (x = 0.1) was measured down to 1.6 K at Harwell. These results, as well as the Grenoble data are plotted in fig. 8. The two sets of data agree reasonably well in the overlap region (5-10 K), but the Hat-well data appear to show a flattening of the straight line at temperatures below about 6 K. It is possible that these lowest temperature results indicate poor thermal contact within the powdered sample, or it may be that the flattening is evidence of many body effects [ 15,161. Certainly the very large y(0) values, when combined with the much smaller high temperature y’s (see section 4.2) lend weight to this latter possibility. Just as there is ambiguity about the interpretation of the linear contribution to the low-tempera10 ,_,’ ,_,’ ,’
90
,,*
+,*’
*/ i
80
301 0 ( TEMPERATURE
)’
( K2 )
Fig. 7. C,/T versus T2 graphs between 5 and 14 K for x = 0.05, 0.18, 0.27 and 0.3. UP,_,Se,:
,’
,’
I-’
25
50
100 125 150 75 (TEMPERATURE l2 [Hz1
175
Fig. 8. C’/Tvs. T2 graph between 1.5 and 14 K for UP,,Se,,,; 0 - data from Hanvell; X - data from Grenoble.
2(
118
A. Blake et al. / Heat capacity of UP-C/Se
solid solutions
Table 7 Characteristic values for the heat capacity and entropy of UP, _,Se,. Results for UP are from ref. [8], those for USe are from ref. [9]. Least squares fitting of the experimental data yields uncertainties of approximately 5% for 8, and y(O) and of approximately 10% for a and ~(300). Uncertainties in S,,,,,(300) are estimated to be of the order of 20% (see section 4.4 for a discussion) 0.0 a
x
0.05
0.10
0.18
0.27
0.30
1.0 b
C,(298.15) (J/mol K)
50.3
51.54
51.58
51.93
51.5
50.3
78.28
S(298.15) (J/mol K)
54.8
79.95
80.93
80.8
82.72
80.0
96.52
8,
225
233
227
254
222
224
226
(K) Y(0) (mJ/mol
K’)
Y(300) (mJ/mol
K’)
ax10-4 (J.K/mol) %&300) (J/mol K)
9.6
32.0
35.0
37.8
67.5
74.2
86.8
3.8
10.5
10.6
11.8
10.6
7.3
23.6
8.6
8.2
8.8
8.6
10.0
10.2
40.0
10.3
10.3
11.1
11.0
13.1
11.6
19.4
a See ref. [8]. h See ref. [9].
ture specific heat of these solid solutions, so too there is uncertainty as to whether or not the T 3 term contains a magnetic contribution. In principle, we could estimate the size of a spinwave contribution by subtracting the lattice part of the specific heat of ThP [ 17,181 from the measured T’ term. In practice, the term which remains after the subtraction is carried out is smaller then uncertainties in the data. 4.2. High temperature
data
expansion for the A general high temperature specific heat of magnetically ordering compounds is:
with Cdi, = C, - C,, a term accounting for the thermal expansion coefficient, Clat, = the phonon due to the conterm, Ccond = yT, the contribution duction elections, and Cm_ = the contribution due to the magnetic ions in the paramagnetic phase. Several approximations can be made in eq. (1).
Without sume
losing
C,;, = AC,, ‘T,
too much
generality,
one can as-
A = constant,
C ,att = Debye function, C mag = a/T2
(Heisenberg
model).
With these approximations it is then possible to make computer fits to the specific heat for temperatures well above the ordering temperature, and to seek those values of 8,, y, and (Ywhich, for a given A, minimize the differences between the computer fit and the experimental results. Such a procedure leads, for the five solid solutions, to values for 8, which vary from 330 to 350 K (considerably higher than B. as determined at low temperatures), to values of y which vary from 0 to about 10 mJ/mol K2, depending upon the assumption made concerning the size of A, and to values of cr which are about 5 X lo4 JK/mol. Unfortunately C,,,, Cmag are comparable in size (and much smaller and Ccond than C,att) for T > Tordering. Therefore, the above fitting procedure does not provide an unambiguously “best” computer fit to the data.
119
A. Blaise et al. / Heat capacity of UP - USe solid solutions
An alternative, and method of analyzing the one similar to that used hashi [12]. Let us assume ACp = C,(UP,_,Se,)
somewhat more direct, high-temperature data is by Yokohawa and Takathat for T > Torderingr
- C,(ThP)
= y’T+ a/T2, (2)
where C,(ThP) is the measured specific heat of ThP. A least squares fit of AC,/T vs. T-3 then yields values for y’ and (Y. Such an analysis assumes that the lattice and dilation terms are the same for UP,_,Se, and for ThP, an assumption which probably overestimates somewhat C,,,, for these solid solutions [ 191. However, once again, any more sophisticated assumption would not seem merited by the experimental data. We have carried out an analysis of the hightemperature specific heat data for UP [8], USe [9] and the UP, _XSeX solid solutions using eq. (2) and the ThP specific heat data of Gordon and Blaise [17]. In table 7 the results for (Y and y (300) are listed. Here y (300) = y’ + yThP, where y’ is the intercept obtained from the least squares fit and K2 [18]. Y ThP = 2.9 mJ/mol The general trend in the variation of y (300) with x is the same as that for y(0): an increase with x. This is consistent with the variation with x of the temperature-independent susceptibility [ 11. An exception is the value observed for x = 0.30. But, as the specific-heat data are very sensitive to the base-line corrections used [6], and as the x = 0.30 solid solution was one for which only 1.5g of the compound was available, this low value may be in error. The feature of table 7, which is most striking, is the fact that the ~(300) values for the solid solutions (7-12 mJ/mol K2) were far lower than the y(O) values. This situation appears to be general in the cubic uranium compounds. It is possible that the apparent change with temperature in y arises from a narrow peak in the electronic density of states curve, which is located close to the Fermi energy. However, it can be easily demonstrated that a decrease of a factor of four or more in the y value between helium and room temperatures would require that such a peak has a width considerably less than 0.1 eV. At least at the present time
there appears to be no other evidence, either experimental or theoretical, for such a narrow peak in the electronic density of states. It seems more likely that apparent differences in the high- and low-temperature values of y arise either because of low-temperature electron-phonon enhancement effects [16], or because there is a significant magnetic contribution to the experimental y(O) values listed in table 7 (see section 4.4). The values of the electronic density of states at the Fermi energy inferred from these ~(300) values are consistent with those calculated by Davis [20]. Therefore, in our analysis of the magnetic contributions to the heat capacity and entropy of these solid solutions (section 4.4) we shall assume conduction electron contributions to the measured heat capacity which are calculated from the ~(300) values. 4.3. High temperature
magnetic
contributions
to C,
If we associate the cr/T’ term in eq. (2) with the high-temperature “tail” of the magnetic ordering transition, then we can, in principle, calculate the exchange constant (J) by using the first order approximation [ 121:
(3) where R = gas constant, g, = Lande factor and pefr/pa = paramagnetic moment in Bohr magnetons. In practice, because we do not know the state of the uranium ion, we cannot obtain a unique value for (J) from eq. (3). For example, if we assume Russell-Saunders coupling and if the uranium ions are in the 4 + state, then g, = 0.8, whereas for the trivalent state, g, = 8/l 1. This seemingly small difference in g, produces an uncertainty in ((g, - l)/g,)2, and therefore in (J), of a factor of 5: 2.25. Since, however, pen/p8 is known approximately (it varies from 3.3 to 2.5 over the series x = 0 to x = 1 [l]), we can at least check that eq. (3) yields a value for (J)/kTordering, which is of order 1 to 5. Were this not the case, the identification of the high-temperature TP2 “tail” with the magnetic interaction would not be justified. Values of (Yare listed in table 7. If g, is assumed
120
A. Blarse et al. / Heat capaciiy
to be 0.8, we obtain values of ( J)/kTordering of 4 to 4.8 for x=0 to 0.18, and of = 6 for x=0.27 and x = 0.3. If g, = 8/l 1 these values are reduced to 1.8 to 2.2 for x = 0 to 0.18 and of - 2.7 for x = 0.27 and x = 0.3. As is evident from table 7, the value of the magnetic coefficient (11for USe is far higher than that for any of the solid solutions measured.
of UP - USe solid solutions to assume that the lattice specific heat of UP, _,Se, at temperature T were not equal to that of ThP at the same temperature but rather at temperature 0.9T [ 191.
5. Linear magnetic contribution to the low temperature specific heat
4.4. Magnetic entropy If we are to estimate the entropy associated with magnetic ordering we must make some assumptions about how much of the measured specific heat is of magnetic origin. Where a specific heat anomaly associated with magnetic ordering is large, it is possible to obtain a rough estimate of the magnetic entropy by measuring the area which lies between the anomaly and the “background” specific heat. However, for situations in which the anomaly is small, such a procedure will yield unreasonably small values for Smagnetic. It is our view that the most consistent and straightforward method for calculating the magnetic entropy in these solid solutions is from the approximation: S,,,(T)
=l’(l/T)(c,
- C-~,,P - C,> dT,
(4)
where C, = measured specific heat at temperature TT CThP = specific heat of ThP at this temperature, and C, = y’T (see section 4.2 for the definition of y’). Values for the magnetic entropy at 300 K have been calculated from eq. (4) and are listed in table 7. The values listed there are larger than those usually reported for the uranium rock salt compounds because in our calculation we have included as part of the magnetic entropy an excess “electronic” contribution AS = /Oq,rd[y(0) - y(300)] dT. This procedure assumes that the conduction electron contribution to the specific heat is given at all temperatures by y(3OO)T, and that the lowtemperature linear term, y(O)T, is a sum of conduction electron and magnetic terms. Such a seemingly arbitrary procedure has been followed because it leads to values of S,,,,,(300) which are roughly constant across the series (except for x = 1). The values in table 7 are about R In 4. They would increase to approximately R In 5 if we were
While we have found it convenient for calculational purposes to attribute the term [y(O) y(300)]T to magnetic effects, it is reasonable to ask whether there is any magnetic interaction which could make a linear contribution to the specific heat at temperatures well below the ordering temperature. One possible mechanism is that postulated by Marshall [21] and Klein and Brout [22] to explain the fact that small amounts of Mn in copper produce anomalously large, and concentration-independent, y’s. In this picture the Mn spins interact via an RKKY interaction. At low temperatures most are rigidly aligned in an effective field far larger than kT/p (p is the magnet moment associated with a spin). Some fraction of the spins, however, see a magnetic field comparable in size to kT/p. As is shown in refs. [21,22] these spins have a heat capacity which varies linearly with T at low temperatures. The UP,_,Se, solid solutions are materials in which the ordering mechanism may well come about through an RKKY interaction. Because of their complex magnetic structures at low temperatures it is also reasonable to assume that there exists some variation in the magnetic fields seen by the uranium ions. Perhaps, then, Marshall’s model can be applied to the present problem. Let us assume that at a temperature well below the ordering temperature, T,, the majority of moments see an effective magnetic field of size = kT,/p. The contribution of these moments to the specific heat would, presumably, be given by spin-wave theory. However if some small fraction of the moments see considerably smaller fields, then their contribution to the specific heat can be calculated as follows: let p(H) be the distribution function for fields seen by these moments. If, for simplicity we then the average spin-4 particles, assume
A. Blaise et al. / Heat capacity of UP-We
energy/moment c=
/
is
H’ p(H)pH
tanh(pHH/kT)
dH,
(5)
-H’
where H’ z==kT/p. The heat capacity of n such moments is n d
cm= -
El
x’ p(x)x2 J --x’
sech2x dx.
The term (x2 sech’x) in eq. (6) is negligible when x differs significantly from 1. Hence the limits on the integral can be replaced by f 00. Further, if p(H) does not vary rapidly with H, then we can rewrite eq. (6) as
= CITI
f!$p(~)j:~x~sech2x
dx,
The value of the integral in eq. (7) is 7r2/3. Thus we can write C,,, = y,,,T, where y, = r2k2np(0)/3p.
(8)
A comparison of these arguments with those used for determining the y for conduction electrons shows that here rip(O)))) plays a role something akin to that of the density of states at the Fermi energy for the free electron gas [23]. Two remarks about the arguments which lead to eq. (8) are in order. First, our expressions for c and C, are not valid for those spins in the problem, presumably the large majority, whose behavior at low temperatures is properly described by spin-wave theory. The size of n is thus uncertain, but doubtless it is far less than the total number of uranium ions. Because of this uncertainty, we have not bothered to include, as is done in refs. [21,22], a factor $ in the expressions for Z and C,,, to account for the fact that H arises from spin-spin interactions. Second, any careful estimate of y, would require a detailed calculation of p(O) comparable to that carried out for the Cu-Mn problem in [22]. We can, however, make a rough estimate by inverting the argument used to estimate the Fermi temperture, TF, for conduction electrons. There we obtain TF by setting the measured y equal to R/T,. Here we can estimate y,,, by setting y, - R/T,, where T, is the ordering tem-
solid solutions
121
perature for these solid solutions. Since T, = 100 K*. Such an estiK, we obtain y,,, = 80 mJ/mol mate of y,, admittedly crude, is larger than, but in considerably better than in order-of-magnitude the measured values of agreement with, y(O)-~(300). Klein and Brout [22] have pointed out that correlation effects would reduce the theoretical estimate of y,. Here, as in their calculations, such an effect might bring the predicted values closer to the measured ones. If the large y(O) values in these solid solutions are, in fact, evidence for a spread in magnetic field values, then we might expect the effect to be most pronounced in these compositions which lie close to the ferromagnetic-antiferromagnetic region of the phase diagram. Such is the case of the x = 0.27 and x = 0.3 solid solutions, solutions which have a y(O) almost double those for the lower x values. Since the effect might also be sample dependent, it could explain the difference in y(O) values reported for USe [9,14]. There are, of course, other possible explanations for the high y(O) values reported here. Electron-phonon many body effects [16] and spin fluctuation effects [15] are but two. Careful lowtemperature measurements will, perhaps, provide some clarification. It is also possible that specific heat measurements in large magnetic fields would be helpful although Scarborough et al. [24] measured the low-temperature specific heat of UN in magnetic fields as large as 35 kG and found no field dependence.
6. Conclusion The main features of the magnetic phase diagram of the phosphorus-rich UP, _XSeX solid solutions proposed by [l] have been checked and found to be consistent with the specific heat results, although some small corrections probably are in order. For x = 0.05 the type I phase has probably disappeared, on cooling down, well above 30 K. For x = 0.18 the para-antiferromagnetic transition and the “step-like” transition occur at temperatures higher than those quoted by ref. [ 11. For x = 0.27 and x = 0.3 the specific heat anomaly associated with the ordering of the paramagnetic
122
A. Blaise et al. / Heat capacity of UP - We solid solutions
phase is small, a fact which probably reflects the competing tendencies toward antiferromagnetic and ferromagnetic ordering. Finally, analysis of the high and low-temperature specific heats yields high-temperature y values in the range 7-12 mJ/mol K2, while the low-temperature values range from 32 to 74 mJ/mol K2. If one assumes that the change in y takes place with the onset of magnetic order, and if one classifies the entropy associated with this change in y as magnetic entropy, then one obtains values for the magnetic entropy at 300 K, which are consistent with the ordering of localized magnetic moments.
Acknowledgements JEG wishes to thank E. Bonjour of the Service des Basses Temperatures, Laboratoire de Cryophysique and A. Blaise of the Laboratoire de Physique des Solides, Departement de Recherche Fondamentale, Centre d’Etudes Nucleaires de Grenoble for the hospitality extended to him. He also wishes to thank E. Cervos and P. Bertrand for technical assistance.
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