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Crystal field and magnetic heat capacity in PrIn3 and CeIn3
J. Phys. Chem. Solids" Pergamon Press 1971. Vol. 32, pp. 1867-1872.
Printed in Great Britain.
CRYSTAL FIELD AND MAGNETIC HEAT CAPACITY IN Prln3 A N D Celn3* A. M. VAN DIEPEN,I" R. S. CRAIG and W. E. WALLACE
Department of Chemistry, University of Pittsburgh, Pittsburgh, Penn. 15213, U.S.A.
(Received 3 September 1970) Abstract- Heat capacities of Lalna, Celn3 and Prln3 have been measured between 7 and 300~
The Lain3 results fit a Debye curve with 0 = 170~ Prln3 has a Schottky-type anomaly around 36~ from which an overall crystal field splitting of about 170~ is derived, with the singlet FI as the lowest state. The heat capacity and magnetic behavior of Prlna indicate a value of the crystal field parameter x in the range --0-8 to --0-6. Celn3 shows a Schottky anomaly around 60~ which yields a crystal field spirting of 155~ (doublet lowest). A Iambda-type anomaly is observed at 10-4-+0.5~ the Nrel point. The magnetic entropy indicates that the doublet F7 is the lowest level in CeIn3, The sign of the crystal field splitting is in both compounds in agreement with point charge predictions. 1. INTRODUCTION T H E MAGNETIC behavior of the various RIn3
(R = a rare earth) compounds has recently been reported[l]. PrIn3 was observed to exhibit Van Vleck paramagnetism at the lowest temperatures studied. Efforts to deduce the overall crystal field splitting (Ec) for this compound from its susceptibility-temperature behavior were only partially successful due to uncertainty as to the relative importance of the fourth and sixth order contributions to the crystal field potential; Ec could not be established to better than a factor of two. It should be possible to refine this estimate, as is pointed out below, by appropriate heat capacity measurements. This provided the incentive for undertaking the present study. In due course the investigation was broadened to include CeIn3. The magnetic measurements on Celn3 had suggested[l] an appreciable crystal field interaction but the results could not be unambiguously analysed. Again it appeared that heat capacity studies could provide information from which the overall splitting and degeneracy of the ground state could be deduced. Measurements on non-magnetic Lain3 were
included to facilitate the resolution of the magnetic and crystal field contribution from the total heat capacity. It (i.e. Lain3) was used to approximate the lattice vibrational contribution to the heat capacity of CeIn3 and PrIn3. 2. EXPERIMENTAL
The samples used in the present investigation, about 85 grams per compound, were prepared by levitation melting. Pieces of about 10 grams were made at a time. Since the compounds form congruently only a short stress annealing was needed. The indium metal used was 99.999 per cent, the rare earths were 99.9 per cent pure. X-ray diffraction showed that Lain3, CeIn3 and PrIn3 have the cubic Cu3Au structure with lattice parameters close to those given in Ref. [1]. The heat capacities were studied between 7 and 300~ in a calorimeter of the adiabatic type[2], described in Ref. [3]. The experimental heat capacities are given in Table 1. The values at room temperature slightly exceed 99-8 Joule/~ mole, the value for C~ expected for a solid of four atoms per formula unit. The heat capacity of PrIn3 exceeds that of LaIn~ up to 125~ the difference between the heat capacity of the two compounds is given in Fig. 1. The anomaly is of the Schottky type and attributed t o the crystal field acting on the PP~+ ion. CeIn3 has
*This work was supported by a grant from the U.S. Army Research Office, Durham. tPresent Address: Philips Research Laboratories, Eindhoven, The Netherlands. 1867
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A. M. V A N D I E P E N et aL
Table 1. Experimental heat capacities o f Lalna, Celn3 and Prln3 as depending on the temperature T (*K)
Cuol (Joule/~ mole)
T (~
Lain3 41.47 46.61 52.45 68-73 74.78 80-89 87-22 93.81 100-55 107" 11 114"02 122.33 130"52 138-27 9"30 10-67 12.99 15-97 18.78 21"41 23"90 26.43 29-10 31"94 35" 19 37"28 38"58 44.40 48.05 52"39 56-78 61-51 66-71 81-89 89- 73 97" 82 106"30 115"26 124-53 134"78 145"45 154"73 164"21
174.53 184"66 194"23 225.03
48.26 54.97 61.77 74.52 77.86 81-63 84-75 86"89 88" 14 90"30 91 "88 93-68 95-24 96-72 "84 1 "90 3"89 7" 18 11.06 15"37 20" 32 25-14 28"72 33"55 38-91 41.82 44.16 53-01 56.66 61.94 66.22 69-62 73-33 .81-81 85"83 87- 57 90"09 92"22 94.22 96"35 98" 17 99-90 100-45 100.23 101"51 100.63 102"44
Cuot (Joule/OK mole) CeIn3
88.09 93-03 98-18 103.67 110.04 152-40 160.35 169"09 192-90 202.00 218"54 226-95 37-66 39.72 42.39 45.30 48.10 61" 10 66.29 71-24 110.36 118.41 127.29 136.35 145" 18 154.10 163.33 119-44 125.43 130-98 136.75 149-01 156.21 163"44 170.32 177.43 185.35 193.84 202.50 210.74 218.92 227.03 6-26 7-78 8.44 9.08 9"56
89.17 89-02 90.68 92.24 93.54 101-79 103-15 102-28 104"87 102"41 102"26 102-76 45-05 48-11 56' 18 61 "87 64"38 76" 16 80"22 82"03 94-82 96"44 98"48 98"80 100-56 101"79 101 "19 97"28 98"90 100.17 101.59 102-66 102-97 102-89 104-73 105.04 103"86 104.45 105-90 104"28 105"44 105-86 1-94 5"73 7.08 10" 12 9"75
a Schottky anomaly around 60~ (Fig. 2) and, in addition, a lambda-type anomaly at 10-4~ (Fig. 3), the latter being attributed to the antiferromagnetic transition at the N6el point.
T (~
CMol (Joule/OK mole)
9.96 10.43 11.14 11.97 12-84 14.50 8" 19 9.40 10"02 10"41 10"73 I 1"10 11-41 I 1-71 12"07 12"48 13" 18 14"92 17-23 19"47 22"05 24"55 26-91 29"64 32"72 35"87 38-97 41 "91 44-90 48"37 52"32 56"77 61-13 65"73 70.86 62"68 65-61 69"23 77-09 81 "65 94"44 99"32 104" 10 108'7t 113"25 117"84
15.39 8.41 3.83 4.27 4.75 6.50 7-30 12"56 16'03 15"45 4"46 4"47 4"47 5.02 4.01 4" 10 4"98 6"91 9"75 13.03 17"07 21 "64 26"41 31 "45 37-31 42"40 47"50 52" 13 56-82 62"05 66-87 73"07 75.00 77-24 80"87 75" 42 76" 55 78" 10 83"62 85"89 90.64 91 '45 93'32 94-43 96-11 97.02
T (~
CMol (Joule/OK mole) Prln3
12-55 15.23 17.83 20.87 23.57 26-11 28"86 31 "48 37"82 41 "70 45"87 50" 14 58-90 62"80 66"80 71-58 77-46 83.70 95-81 102"01 108"01 113"79 119'63 125-48 131"55 138'26 145"23 151 "68 157"20 162"51 167"77 173"34 179-19 189"87 196" 12 20 I"93 207-66 213"41 219-27 226-09
3.08 6.81 11.94 19.04 25-77 32-07 38"22 43"57 53"56 58.44 62"99 67"21 73-57 75"42 78" 19 79-82 82-49 85"53 88'86 89.92 9 I" I0 93"03 93 '60 94.24 95-33 96.52 97"35 97-04 97-65 97.58 98"67 98"72 99-33 99"86 99- 87 100.52 100.72 100"92 101' 18 101 "56
3. DISCUSSION
(a) Lattice heat capacity The experimentally measured heat capacity is the sum of different contributions. The main
CRYSTAL FIELD AND MAGNETIC HEAT CAPACITY 20
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1869
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Temperature (~
Fig. 1. Crystal field heat capacity of Prln.~
I
0
OOo
5
I0
15
Temperature (*K)
"N
Fig. 3. Heat capacity of Celn3 and Lain3 (lower curve) around the N~el point of Celn3. Both curves are extrapolated to zero temperature on the basis of the results of Ref. [10].
2
o ox~
i t
0
0
50
I O0
150
Temperature (*K)
Fig. 2. Crystal field heat capacity of Celn3 (circles, dashed line), compared with theory for doublet lowest (curve D) and quartet lowest (curve Q).
contribution is from the lattice vibrations; its temperature dependence is according to a D e b y e function [4]. F o r RIn3: Cv(lattice) = 36 R
(T):~ f l
ett4dt 1)z'
( e t-
(1)
where R is the molecular gas constant, 0 the D e b y e temperature determined by the maximum vibration frequency to,, through 0 = hto,,/k. The actual integration is over all vibration frequencies up to the maximum, with t -hto/k. Because of the small expansion coefficients of metals the experimental heat capacity
can be considered as measured under constant volume. The Lain3 data were fitted to equation (1). Numerical values for the integral were taken from Ref. [5]. Up to about 100~ a good fit was obtained for 0 = 170~ At the higher temperatures, where the lattice heat capacity increases very slowly, other effects such as conduction electron heat capacity and impurities become more pronounced, so that deviations from equation (1) occur. It is also because of these effects that the value 0--170OK should be seen as a lower limit, rather than as an average. (b) Crystal field heat capacity The (2J + 1)-fold degeneracy of the ground state of the rare-earth ion is removed by the crystal field. The cubic crystal field splits the J----~ level in Celn3 into a doublet F7 and a quartet I"8. In Prln3 the J = 4 level is split into a singlet F1, a nonmagnetic doublet F3, and two triplets 1-'4 and F~. A redistribution over the different crystal field levels with a variation of temperature involves energy varia-
1870
A.M.
V A N D I E P E N et al.
tions, and hence a contribution to the heat capacity. This effect is much smaller than that from the lattice vibrations. To determine the crystal field heat capacity the other contribuions (lattice and electronic) must be subtracted from the experimentally measured total. It is for this purpose that the heat capacity of Lain3 was measured. A smooth curve was drawn through the measured Lalnz points. The circles in Figs. 1 and 2 represent experimental Prlnz and Celna data minus the corresponding points on the LaIna curve. The excess heat capacity in Prlna becomes negative for T > 125~ In Celn3 it is almost a constant for T > 100~ A possible explanation of these differences can be found in the presence of impurities such as rare-earth hydrides, nitrides, oxides and so on, which have at the high temperatures bigger gram heat capacities than the rare-earth metals and the RIn3 compounds [6]. These impurities change the absolute value of the observed crystalfield contribution by almost a constant in the high temperature range, but do not change its shape appreciably. Entropies under these Schottky anomalies are: Prlna (up to 130~ AS = 11"8 Joule/mole ( R l n 9 = 18.27), Celna (up to 100~ AS = 5.6 Joule/mole (R In 6/2 = 9.13). The failure of ACp to approach zero as temperature increases complicates attempts to assess AS associated with excitation within the crystal field spectrum. The upper temperatures cited are arbitrary and do not correspond to complete excitation. The experimental entropies are hence underestimated. The heat capacity at a temperature T of a system involving n energy levels is given by
~=1 -~ exp ( - ~ J r )
Z ,
(2)
where A~ is the energy difference between the i-th level and the lowest level and Z is the partition function. Theoretical calculations
have been made for Ce a+ (see Fig. 2) and Pr 3§ (see Fig. 4) in a cubic environment. For the Ce compound only two level systems are possible, viz., doublet lowest or quartet lowest. For Pr 3+ many different combinations x of fourth and sixth order potentials are possible, each involving a different arrangement of the energy levels [7]. The calculated heat capacities were compared with experiment by adjusting the overall aplitting so that the experimental and theoretical temperatures of maximum heat capacity, 36~ for Prlna and 60~ for Celn3, coincide. By comparing Figs. 1 and 4 it is immediately seen that in Prlnz, in agreement with the point charge model, W must be positive and the singlet F1 is the lowest state. Reasonable fits are obtained for x > I - 0 - 8 , from which an overall splitting of about 170~ follows. The separation between the singlet ground level and the next triplet F4 or F5 is around 100~ A pure fourth order crystal field potential ( x = - - l . 0 ) definitely cannot describe the experimental heat capacity, since for T > 45 ~ large discrepancies occur. The actual level scheme should also give a good fit to the susceptibility/l], so that x must be in the range - 0 . 6 to--0.8. The corresponding energy levels are given in Table 2. From heat capacity and magnetic susceptibility data alone it is not possible to distinguish between the given (and intermediate) sets. A method which measures the energy separations in a direct way, e.g. inelastic neutron scattering/8], will probably be more decisive. The situation for Celn3 is less clear, since the experimental errors in the crystal field heat capacity are more nearly comparable with the crystal field contribution. The result (Fig. 2) is a broad peak around 60~ If the quartet is lowest, the splitting is 135 • 30~ in the other case it is 155 • 30~ The data in Fig. 2, while not decisive, suggest that the ground state is the doublet. Firmer evidence that the doublet is lowest results from analysis of the magnetic transition, presented in the next section.
CRYSTAL FIELD AND MAGNETIC HEAT CAPACITY
20
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~
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1871
=-0.8
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\
t
x
W< 0 x =-0.6
,
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~
0
,
,
I
I
50
I00
,
I
~
,
I
150 200 Temperature (*K)
I
I
250
300
Fig. 4. Some calculated crystal field heat capacity curves for different proportions of fourth and sixth order potentials acting on the Pta+ ion in Prlna.
Table 3. Values o f the energy distance between the lowest levels in CeIn3 and PrIn3 as derived fi'om heat capacity data
Table 2. Energy levels f o r the Pr~+ ion in Prln3 after removal o f the degeneration o f the J = 4 state by crystalline fields for different values o f the parameter x Level x = - 0 . 8 Fl F~ Fa F5
- 120~ -33 +30 +53
x=-0-7
x=-0-6
--114~ --23 +43 +32
-- 117~ -16 +55 + 18
Celn~ Prln.~
At low temperatures (T ~ A2) equation (2) can be approximated by c<, = R g~ \ T ] exp (--A2/T),
(3)
where gl and g2 are the degeneracies of the first and the second level [9]. A plot of In CelT 2 vs. 1/T should give a straight line, the slope of which is --A 2. Az can also be determined from the intersection with the In CctT 2 axis. F o r
A2/~ ai (~
A2/k tbl
Az/ktc]
(~
(~
167 112
145 120
155 -96
[aJObtained from the slope of the plot ofln CctT-' vs. 1/T. t~}Obtained from the intersection with the In CctT"- axis of the same plot. t~lObtained from the position of the peak and the overall shape of the Schottky anomaly.
Prln3 data between 16 and 32~ lay on a straight line; for Celn3 data between 30 and 50~ are used. The results are given in Table 3. It should be noted that, if in Prln3 the energy separation between the second and the third level is about kT, to first approximation still a straight line is obtained. However,
A . M . VAN DIEPEN et al.
1872
the derived 'A2' will be bigger than the actual value. (c) Magnetic heat capacity A lambda-type anomaly was observed in CeIn3 at 10-4_+0.5~ (Fig. 3). This agrees well with the N6el point, found in the susceptibility at ll~ An estimated peak value of the heat capacity of 25 Joule/~ mole was used to evaluate the entropy change associated with the anomaly. The latter was found to be 6-0Joule/mole, very near to R l n 2 = 5 . 7 6 . The experimental AS was obtained from the difference of Cp for CeIn3 and Lain3. Preliminary heat capacity data in the range 1-5 to 4.2~ of both compounds indicate, however, that the low temperature heat capacity of CeIns is far in excess of that of LaIns[10], So that the actual entropy associated with the antiferromagnetic transition is less than 6.0Joule/mole. Since the quartet would require twice as much entropy, which by no means can be found, this result unambiguously shows that the doublet is the lowest level in CeIn3. 4. CONCLUSION
Heat capacity data together with susceptibilities have lead to a satisfactory description
of the crystal fields in Prln3 and CeIn3. The heat capacity data indicate the lowest level to be a singlet in PrIn3 and a doublet in CeIn3. These results are in qualitative agreement with the point charge model.
REFERENCES 1. BUSCHOW K. H. J., deWlJN H. W. and VAN D I E P E N A. M.,J. chem. Phys. 50, 137 (1969). 2. WESTRUM E. F., Jr., H A T C H E R J. B. and OSBORNE, J. chem. Phys. 21, 419 (1953); RUEHRWEIN R. A. and H O F F M A N M., J. Am. Chem. Soc. 65, 1620 (1943); STERRETT K. F., BLACKBURN D. H., BESTUL A. B., C H A N G S. S. and H O R M A N J., J. Res. Nat'l. Bur. Std. 69C, 19 0965). 3. W A LLA CE W. E., D E E N A D A S C., THOMPSON A. W. and CRA I G R. S., J. Phys. Chem. Solids, 32, 805(1971). 4. DEBYE P.,Ann. Phys. 39, 789 0912). 5. BEATI'IE J. A.,J. Math. Phys. 6, 1 (1926). 6. GERSTEIN B. C., T A Y L O R W. A., SHICKELL W. D. and S P E D D I N G F. H., 8th Rare Earth Conference, Reno (1970). 7. LEA K. R., LEASK M. J. M. and WOLF W. P., J. Phys. Chem. Solids 23, 1381 (1962). See this reference for definitions of W and x. 8. B I R G E N E A U R. J., BUCHER E., PASSELL L., PRICE D. L., T U R B E R F I E L D K. C., J. appl. Phys. 41,900 (1970). 9. BUCHER E., GOSSARD A. C., ANDRES K., MAITA J. P. and COOPER A. S., 8th Rare Earth Conference, Reno (1970). 10. VAN DIEPEN A. M., NASU S., N E U M A N N H. H. and CRAIG R. S., to be published.
Printed in Great Britain.
CRYSTAL FIELD AND MAGNETIC HEAT CAPACITY IN Prln3 A N D Celn3* A. M. VAN DIEPEN,I" R. S. CRAIG and W. E. WALLACE
Department of Chemistry, University of Pittsburgh, Pittsburgh, Penn. 15213, U.S.A.
(Received 3 September 1970) Abstract- Heat capacities of Lalna, Celn3 and Prln3 have been measured between 7 and 300~
The Lain3 results fit a Debye curve with 0 = 170~ Prln3 has a Schottky-type anomaly around 36~ from which an overall crystal field splitting of about 170~ is derived, with the singlet FI as the lowest state. The heat capacity and magnetic behavior of Prlna indicate a value of the crystal field parameter x in the range --0-8 to --0-6. Celn3 shows a Schottky anomaly around 60~ which yields a crystal field spirting of 155~ (doublet lowest). A Iambda-type anomaly is observed at 10-4-+0.5~ the Nrel point. The magnetic entropy indicates that the doublet F7 is the lowest level in CeIn3, The sign of the crystal field splitting is in both compounds in agreement with point charge predictions. 1. INTRODUCTION T H E MAGNETIC behavior of the various RIn3
(R = a rare earth) compounds has recently been reported[l]. PrIn3 was observed to exhibit Van Vleck paramagnetism at the lowest temperatures studied. Efforts to deduce the overall crystal field splitting (Ec) for this compound from its susceptibility-temperature behavior were only partially successful due to uncertainty as to the relative importance of the fourth and sixth order contributions to the crystal field potential; Ec could not be established to better than a factor of two. It should be possible to refine this estimate, as is pointed out below, by appropriate heat capacity measurements. This provided the incentive for undertaking the present study. In due course the investigation was broadened to include CeIn3. The magnetic measurements on Celn3 had suggested[l] an appreciable crystal field interaction but the results could not be unambiguously analysed. Again it appeared that heat capacity studies could provide information from which the overall splitting and degeneracy of the ground state could be deduced. Measurements on non-magnetic Lain3 were
included to facilitate the resolution of the magnetic and crystal field contribution from the total heat capacity. It (i.e. Lain3) was used to approximate the lattice vibrational contribution to the heat capacity of CeIn3 and PrIn3. 2. EXPERIMENTAL
The samples used in the present investigation, about 85 grams per compound, were prepared by levitation melting. Pieces of about 10 grams were made at a time. Since the compounds form congruently only a short stress annealing was needed. The indium metal used was 99.999 per cent, the rare earths were 99.9 per cent pure. X-ray diffraction showed that Lain3, CeIn3 and PrIn3 have the cubic Cu3Au structure with lattice parameters close to those given in Ref. [1]. The heat capacities were studied between 7 and 300~ in a calorimeter of the adiabatic type[2], described in Ref. [3]. The experimental heat capacities are given in Table 1. The values at room temperature slightly exceed 99-8 Joule/~ mole, the value for C~ expected for a solid of four atoms per formula unit. The heat capacity of PrIn3 exceeds that of LaIn~ up to 125~ the difference between the heat capacity of the two compounds is given in Fig. 1. The anomaly is of the Schottky type and attributed t o the crystal field acting on the PP~+ ion. CeIn3 has
*This work was supported by a grant from the U.S. Army Research Office, Durham. tPresent Address: Philips Research Laboratories, Eindhoven, The Netherlands. 1867
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A. M. V A N D I E P E N et aL
Table 1. Experimental heat capacities o f Lalna, Celn3 and Prln3 as depending on the temperature T (*K)
Cuol (Joule/~ mole)
T (~
Lain3 41.47 46.61 52.45 68-73 74.78 80-89 87-22 93.81 100-55 107" 11 114"02 122.33 130"52 138-27 9"30 10-67 12.99 15-97 18.78 21"41 23"90 26.43 29-10 31"94 35" 19 37"28 38"58 44.40 48.05 52"39 56-78 61-51 66-71 81-89 89- 73 97" 82 106"30 115"26 124-53 134"78 145"45 154"73 164"21
174.53 184"66 194"23 225.03
48.26 54.97 61.77 74.52 77.86 81-63 84-75 86"89 88" 14 90"30 91 "88 93-68 95-24 96-72 "84 1 "90 3"89 7" 18 11.06 15"37 20" 32 25-14 28"72 33"55 38-91 41.82 44.16 53-01 56.66 61.94 66.22 69-62 73-33 .81-81 85"83 87- 57 90"09 92"22 94.22 96"35 98" 17 99-90 100-45 100.23 101"51 100.63 102"44
Cuot (Joule/OK mole) CeIn3
88.09 93-03 98-18 103.67 110.04 152-40 160.35 169"09 192-90 202.00 218"54 226-95 37-66 39.72 42.39 45.30 48.10 61" 10 66.29 71-24 110.36 118.41 127.29 136.35 145" 18 154.10 163.33 119-44 125.43 130-98 136.75 149-01 156.21 163"44 170.32 177.43 185.35 193.84 202.50 210.74 218.92 227.03 6-26 7-78 8.44 9.08 9"56
89.17 89-02 90.68 92.24 93.54 101-79 103-15 102-28 104"87 102"41 102"26 102-76 45-05 48-11 56' 18 61 "87 64"38 76" 16 80"22 82"03 94-82 96"44 98"48 98"80 100-56 101"79 101 "19 97"28 98"90 100.17 101.59 102-66 102-97 102-89 104-73 105.04 103"86 104.45 105-90 104"28 105"44 105-86 1-94 5"73 7.08 10" 12 9"75
a Schottky anomaly around 60~ (Fig. 2) and, in addition, a lambda-type anomaly at 10-4~ (Fig. 3), the latter being attributed to the antiferromagnetic transition at the N6el point.
T (~
CMol (Joule/OK mole)
9.96 10.43 11.14 11.97 12-84 14.50 8" 19 9.40 10"02 10"41 10"73 I 1"10 11-41 I 1-71 12"07 12"48 13" 18 14"92 17-23 19"47 22"05 24"55 26-91 29"64 32"72 35"87 38-97 41 "91 44-90 48"37 52"32 56"77 61-13 65"73 70.86 62"68 65-61 69"23 77-09 81 "65 94"44 99"32 104" 10 108'7t 113"25 117"84
15.39 8.41 3.83 4.27 4.75 6.50 7-30 12"56 16'03 15"45 4"46 4"47 4"47 5.02 4.01 4" 10 4"98 6"91 9"75 13.03 17"07 21 "64 26"41 31 "45 37-31 42"40 47"50 52" 13 56-82 62"05 66-87 73"07 75.00 77-24 80"87 75" 42 76" 55 78" 10 83"62 85"89 90.64 91 '45 93'32 94-43 96-11 97.02
T (~
CMol (Joule/OK mole) Prln3
12-55 15.23 17.83 20.87 23.57 26-11 28"86 31 "48 37"82 41 "70 45"87 50" 14 58-90 62"80 66"80 71-58 77-46 83.70 95-81 102"01 108"01 113"79 119'63 125-48 131"55 138'26 145"23 151 "68 157"20 162"51 167"77 173"34 179-19 189"87 196" 12 20 I"93 207-66 213"41 219-27 226-09
3.08 6.81 11.94 19.04 25-77 32-07 38"22 43"57 53"56 58.44 62"99 67"21 73-57 75"42 78" 19 79-82 82-49 85"53 88'86 89.92 9 I" I0 93"03 93 '60 94.24 95-33 96.52 97"35 97-04 97-65 97.58 98"67 98"72 99-33 99"86 99- 87 100.52 100.72 100"92 101' 18 101 "56
3. DISCUSSION
(a) Lattice heat capacity The experimentally measured heat capacity is the sum of different contributions. The main
CRYSTAL FIELD AND MAGNETIC HEAT CAPACITY 20
12
10
Jl
! !
8
"6:=: o
6
'~
4
o
1869
_~ 15 o
./
g
3a o
o o
o o
-2 510
0
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150
200 o
Temperature (~
Fig. 1. Crystal field heat capacity of Prln.~
I
0
OOo
5
I0
15
Temperature (*K)
"N
Fig. 3. Heat capacity of Celn3 and Lain3 (lower curve) around the N~el point of Celn3. Both curves are extrapolated to zero temperature on the basis of the results of Ref. [10].
2
o ox~
i t
0
0
50
I O0
150
Temperature (*K)
Fig. 2. Crystal field heat capacity of Celn3 (circles, dashed line), compared with theory for doublet lowest (curve D) and quartet lowest (curve Q).
contribution is from the lattice vibrations; its temperature dependence is according to a D e b y e function [4]. F o r RIn3: Cv(lattice) = 36 R
(T):~ f l
ett4dt 1)z'
( e t-
(1)
where R is the molecular gas constant, 0 the D e b y e temperature determined by the maximum vibration frequency to,, through 0 = hto,,/k. The actual integration is over all vibration frequencies up to the maximum, with t -hto/k. Because of the small expansion coefficients of metals the experimental heat capacity
can be considered as measured under constant volume. The Lain3 data were fitted to equation (1). Numerical values for the integral were taken from Ref. [5]. Up to about 100~ a good fit was obtained for 0 = 170~ At the higher temperatures, where the lattice heat capacity increases very slowly, other effects such as conduction electron heat capacity and impurities become more pronounced, so that deviations from equation (1) occur. It is also because of these effects that the value 0--170OK should be seen as a lower limit, rather than as an average. (b) Crystal field heat capacity The (2J + 1)-fold degeneracy of the ground state of the rare-earth ion is removed by the crystal field. The cubic crystal field splits the J----~ level in Celn3 into a doublet F7 and a quartet I"8. In Prln3 the J = 4 level is split into a singlet F1, a nonmagnetic doublet F3, and two triplets 1-'4 and F~. A redistribution over the different crystal field levels with a variation of temperature involves energy varia-
1870
A.M.
V A N D I E P E N et al.
tions, and hence a contribution to the heat capacity. This effect is much smaller than that from the lattice vibrations. To determine the crystal field heat capacity the other contribuions (lattice and electronic) must be subtracted from the experimentally measured total. It is for this purpose that the heat capacity of Lain3 was measured. A smooth curve was drawn through the measured Lalnz points. The circles in Figs. 1 and 2 represent experimental Prlnz and Celna data minus the corresponding points on the LaIna curve. The excess heat capacity in Prlna becomes negative for T > 125~ In Celn3 it is almost a constant for T > 100~ A possible explanation of these differences can be found in the presence of impurities such as rare-earth hydrides, nitrides, oxides and so on, which have at the high temperatures bigger gram heat capacities than the rare-earth metals and the RIn3 compounds [6]. These impurities change the absolute value of the observed crystalfield contribution by almost a constant in the high temperature range, but do not change its shape appreciably. Entropies under these Schottky anomalies are: Prlna (up to 130~ AS = 11"8 Joule/mole ( R l n 9 = 18.27), Celna (up to 100~ AS = 5.6 Joule/mole (R In 6/2 = 9.13). The failure of ACp to approach zero as temperature increases complicates attempts to assess AS associated with excitation within the crystal field spectrum. The upper temperatures cited are arbitrary and do not correspond to complete excitation. The experimental entropies are hence underestimated. The heat capacity at a temperature T of a system involving n energy levels is given by
~=1 -~ exp ( - ~ J r )
Z ,
(2)
where A~ is the energy difference between the i-th level and the lowest level and Z is the partition function. Theoretical calculations
have been made for Ce a+ (see Fig. 2) and Pr 3§ (see Fig. 4) in a cubic environment. For the Ce compound only two level systems are possible, viz., doublet lowest or quartet lowest. For Pr 3+ many different combinations x of fourth and sixth order potentials are possible, each involving a different arrangement of the energy levels [7]. The calculated heat capacities were compared with experiment by adjusting the overall aplitting so that the experimental and theoretical temperatures of maximum heat capacity, 36~ for Prlna and 60~ for Celn3, coincide. By comparing Figs. 1 and 4 it is immediately seen that in Prlnz, in agreement with the point charge model, W must be positive and the singlet F1 is the lowest state. Reasonable fits are obtained for x > I - 0 - 8 , from which an overall splitting of about 170~ follows. The separation between the singlet ground level and the next triplet F4 or F5 is around 100~ A pure fourth order crystal field potential ( x = - - l . 0 ) definitely cannot describe the experimental heat capacity, since for T > 45 ~ large discrepancies occur. The actual level scheme should also give a good fit to the susceptibility/l], so that x must be in the range - 0 . 6 to--0.8. The corresponding energy levels are given in Table 2. From heat capacity and magnetic susceptibility data alone it is not possible to distinguish between the given (and intermediate) sets. A method which measures the energy separations in a direct way, e.g. inelastic neutron scattering/8], will probably be more decisive. The situation for Celn3 is less clear, since the experimental errors in the crystal field heat capacity are more nearly comparable with the crystal field contribution. The result (Fig. 2) is a broad peak around 60~ If the quartet is lowest, the splitting is 135 • 30~ in the other case it is 155 • 30~ The data in Fig. 2, while not decisive, suggest that the ground state is the doublet. Firmer evidence that the doublet is lowest results from analysis of the magnetic transition, presented in the next section.
CRYSTAL FIELD AND MAGNETIC HEAT CAPACITY
20
rW>O
r
F
/X =-0"6/~
~
r
"~ .~ .~:
-o.,H \
r i:,:.o
15
1871
=-0.8
W>O x =- .~
\
t
x
W< 0 x =-0.6
,
I
7 !,~/ 0
~
0
,
,
I
I
50
I00
,
I
~
,
I
150 200 Temperature (*K)
I
I
250
300
Fig. 4. Some calculated crystal field heat capacity curves for different proportions of fourth and sixth order potentials acting on the Pta+ ion in Prlna.
Table 3. Values o f the energy distance between the lowest levels in CeIn3 and PrIn3 as derived fi'om heat capacity data
Table 2. Energy levels f o r the Pr~+ ion in Prln3 after removal o f the degeneration o f the J = 4 state by crystalline fields for different values o f the parameter x Level x = - 0 . 8 Fl F~ Fa F5
- 120~ -33 +30 +53
x=-0-7
x=-0-6
--114~ --23 +43 +32
-- 117~ -16 +55 + 18
Celn~ Prln.~
At low temperatures (T ~ A2) equation (2) can be approximated by c<, = R g~ \ T ] exp (--A2/T),
(3)
where gl and g2 are the degeneracies of the first and the second level [9]. A plot of In CelT 2 vs. 1/T should give a straight line, the slope of which is --A 2. Az can also be determined from the intersection with the In CctT 2 axis. F o r
A2/~ ai (~
A2/k tbl
Az/ktc]
(~
(~
167 112
145 120
155 -96
[aJObtained from the slope of the plot ofln CctT-' vs. 1/T. t~}Obtained from the intersection with the In CctT"- axis of the same plot. t~lObtained from the position of the peak and the overall shape of the Schottky anomaly.
Prln3 data between 16 and 32~ lay on a straight line; for Celn3 data between 30 and 50~ are used. The results are given in Table 3. It should be noted that, if in Prln3 the energy separation between the second and the third level is about kT, to first approximation still a straight line is obtained. However,
A . M . VAN DIEPEN et al.
1872
the derived 'A2' will be bigger than the actual value. (c) Magnetic heat capacity A lambda-type anomaly was observed in CeIn3 at 10-4_+0.5~ (Fig. 3). This agrees well with the N6el point, found in the susceptibility at ll~ An estimated peak value of the heat capacity of 25 Joule/~ mole was used to evaluate the entropy change associated with the anomaly. The latter was found to be 6-0Joule/mole, very near to R l n 2 = 5 . 7 6 . The experimental AS was obtained from the difference of Cp for CeIn3 and Lain3. Preliminary heat capacity data in the range 1-5 to 4.2~ of both compounds indicate, however, that the low temperature heat capacity of CeIns is far in excess of that of LaIns[10], So that the actual entropy associated with the antiferromagnetic transition is less than 6.0Joule/mole. Since the quartet would require twice as much entropy, which by no means can be found, this result unambiguously shows that the doublet is the lowest level in CeIn3. 4. CONCLUSION
Heat capacity data together with susceptibilities have lead to a satisfactory description
of the crystal fields in Prln3 and CeIn3. The heat capacity data indicate the lowest level to be a singlet in PrIn3 and a doublet in CeIn3. These results are in qualitative agreement with the point charge model.
REFERENCES 1. BUSCHOW K. H. J., deWlJN H. W. and VAN D I E P E N A. M.,J. chem. Phys. 50, 137 (1969). 2. WESTRUM E. F., Jr., H A T C H E R J. B. and OSBORNE, J. chem. Phys. 21, 419 (1953); RUEHRWEIN R. A. and H O F F M A N M., J. Am. Chem. Soc. 65, 1620 (1943); STERRETT K. F., BLACKBURN D. H., BESTUL A. B., C H A N G S. S. and H O R M A N J., J. Res. Nat'l. Bur. Std. 69C, 19 0965). 3. W A LLA CE W. E., D E E N A D A S C., THOMPSON A. W. and CRA I G R. S., J. Phys. Chem. Solids, 32, 805(1971). 4. DEBYE P.,Ann. Phys. 39, 789 0912). 5. BEATI'IE J. A.,J. Math. Phys. 6, 1 (1926). 6. GERSTEIN B. C., T A Y L O R W. A., SHICKELL W. D. and S P E D D I N G F. H., 8th Rare Earth Conference, Reno (1970). 7. LEA K. R., LEASK M. J. M. and WOLF W. P., J. Phys. Chem. Solids 23, 1381 (1962). See this reference for definitions of W and x. 8. B I R G E N E A U R. J., BUCHER E., PASSELL L., PRICE D. L., T U R B E R F I E L D K. C., J. appl. Phys. 41,900 (1970). 9. BUCHER E., GOSSARD A. C., ANDRES K., MAITA J. P. and COOPER A. S., 8th Rare Earth Conference, Reno (1970). 10. VAN DIEPEN A. M., NASU S., N E U M A N N H. H. and CRAIG R. S., to be published.