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On the specific heat of Pr amorphous alloys
Physica 107B {1981/83-84 North-Holland Publishing Company
BF 16
ON THE SPECIFIC HEAT OF Pr AMORPHOUS ALLOYS
E. Borchi + and S. De Gennaro ++
+ Istituto di Fisica and GNSM-35100 Padova, Italy ++ Istituto di Fisica and GNSM-50125
Firenze, Italy
Starting from the Fert and Campbell non-axial crystal field model and using a suitable distribution of the asymmetry parameter n we evaluate the magnetization and the specific heat for Pr amorphous compounds.
Experimental results on the specific heat, electrical resistivity and magnetic properties of light rare earth (RE) amorphous alloys show evidence of substantial deviations from the predictions of the random uniaxial model by Harris et el. (I). The departure from uniaxiality has been considered in the non-axial crystal field model proposed by Fert and Campbell (FC)(2).This model, however, does not give information about the possible distribution of the asymmetry parameter n from site to site in amorphous RE alloys. In a recent work of the present authors (3) substantial agreement with the experimental data on magnetic properties of amorphous Pr21Ag79 (4) was obtained assuming a ~-like ~ distribution, i.e. assuming that a most probable n-value can represent the physical situation "on the average". On the other hand both superconductivity and calorimetric measurements on Pr21Ag79 amorphous alloys (5) and specific heat measurements on Pr La^_ Au^_ amorphous alloys (6) by Garoche .x ~o-x z~ et el. suggest the presence of two low-lying non degenerate levels and the existence of very low crystal-field (CF) excitation energies distributed from site to site around some non-zero average value. This is in qualitative agreement with a quasi-gaussian distribution of CF levels obtained by Cochra~eet el. (7) using computer generated clusters of dense random packing of hard spheres. On this ground Bhattacharjee and Coqblin (BC) (8) recently proposed a simple phenomenological singlet-singlet model with an energy splitring varying with gaussian distribution over RE sites. In the BC approach the singlet-singlet matrix element of J. (the z-component of the angular momentum of t ~ i-th RE ion) is replaced by an average value. The resulting theory extends the Pr two-singlets theory of Bleaney and Cooper (9) by taking into account a weigh~ddistrlbution of the singlet-slnglet energy splitting. The existence of low-lying non degenerate singlets in Pr amorphous alloys is easily explained by the FC Hamiltonian (see (3) for further details); moreover the very low excitation energies
03784363/8|/0000-(KK~/$02.50
©No.h-Ho~andPub~shMgCompany
can be accounted for by allowing for a suitable distribution of the asymmetry parameter ~ from site to site. On the other hand Fert and Spanjaard calculations (lO),assuming values of n equally distributed between the two extreme cases of uniaxial and planar local symmetries (n-±l), do not consider the fact that the majority of Pr ions appear to be in a singlet ground state. This fact suggests a narrower n-distribution. A gaussian distribution of n is unsustainable due to the fact that l~I~l. In this work we assume that Pr ions are subject to a nonaxial local crystal-fleld (2) ~ef" 6C [3(l_~)j2._xz 3(l+n)J~z]" and suggest a cosine-type distribution law forn p(n)-~/4a c°s[J-(D-no)]za ' p(n)=o for ID-D l~a
I
o
The width of this distribution depends on the parameter a. This distribution is symmetrical around the average value n , tends to a 6-distri hution for a~o and, at the°lowest order in (n-q),has the same behaviour as proposed by Cjze~ for amorphous materials (see (5)). Our approach, which puts on a more fundamental basis the BC suggestion, takes into account variations in the CF intensity as well as in the directions of local CF axes, In the presence of an exchange interaction and of an external magnetic field H the total Hamiltonian is i
j
[iJiz
13 Within this model we evaluate both the magnetization and the specific heat of Pr amorphous alloys, Taking into account only the low-lying singlet states ~i and ~2 (3) and using the molecular field approximation we obtain for the magnetization in the ferromagnetic case (see (3) for the meaning of symbols) 83
84
Jz = J(BiB2+2"flY2)x[l-(2~)2]~
tanh
(TF--),
x--cos 8
F " A 2+ g~B
2Yo =L(~) {-~BJ(~I82+2YIY2)x(H+ gPB 7 )}2] ~ z
with
where <~llJzil~2> = J(BiB2+2YiY 2) and A=E2-E I are functions of the parameter q. Here Y E~Y... o
~1
zj
The specific heat C v per mole of Pr is analogously obtained as CV R
--
2 =
~
e (l+e~) ~
~
dF dT
+ --
l-e2~-2~e ~ __ . (l+ee) z
•
+
2y
~
o z dT
2F T Both for ~ and C V the given expressions are to z be considered as averaged over the q-distribution as well as over the random directions e of the local z-axis (o~e~/2). In Fig. i the behaviour of the specific heat Cv/R is reported versus T in the absence of any induced magnetism for values of the parameters C=lOO K, q =-0.6. When the width "a" of the q di. . O. strlbutlon increases the low temperature behaviour of C becomes almost linear. This result V is in qualltative agreement with the experiment al data on PrxLa80_x Au20 and confirms the predictions of the phenomenological singlet-singlet model (8). 0.5
~o-- -0.6
a=O,O
C =1oo I
0.4
0.3
0.2
/J
i¢I II 0.1
0 o
4
e
,2 T (K) ,6
Fig.l : Specific heat Cv/R versus T for several values of the ~-distribution width "a".The dashed curve gives Cv/R for a=O.4 and with an applied magnetic field of 7 Tesla.
Also in the presence of an external magnetic field the low-temperature behaviour of C v appears to be in qualitative agreement with experiment. In fact C V is not sensibly affected by an external magnetic field up to =i Tesla but it is strongly depressed by a field of 7 Tesla (see the dashed line in Fig.l). Numerical calculations which take into account a possible contribution to C V both of ferromagne~c and antiferromagnetic origin are presently in progress. REFERENCES (i)
Harris, R., Plischke, M. and Zuckermann,M. J., Phys.Rev.Lett. 31 (1973) 160. (2) Fert, A. and Campbell, I.A., J.Phys.F: Metal Phys. 8 (1978) L57. (3) Borchi, E. and De Gennaro, S., J.Phys.F: Metal Phys. ii (1981) L47. (4) Pappa, C. and Boucher, B., J.Magn.Magn.Mater. 15-18 (1980) 97. (5) Garoche, P., Fert., A., Veyssi~, J.J. and Boucher,B, J.Magn.Magn.Mater. 15-18 (1980) 1397. (6) Garoche, P., Veyssi~,J.J. and Durand, J., J. Phys.Lett. 41 (1980) 357. (7) Cochrane, R.W., Harris, R., Plischke, M., Zobin, D and Zuckermann, M.J., J.Phys.F: Metal Phys. 5 (1975) 763. (8) Bhattacharjee, A.K. and Coqblin, B., J.Phys. Coll. 41 (1980) C8-626. (9) Cooper, B.R. in "Magnetic Properties of Rare-Earth Metals", ed. Elliot R.J. (Plenum Press, London and New York, 1972). (IO) Fert, A. and Spanjaard, D., J. Phys. Coll. 40 (1979) C5-248.
BF 16
ON THE SPECIFIC HEAT OF Pr AMORPHOUS ALLOYS
E. Borchi + and S. De Gennaro ++
+ Istituto di Fisica and GNSM-35100 Padova, Italy ++ Istituto di Fisica and GNSM-50125
Firenze, Italy
Starting from the Fert and Campbell non-axial crystal field model and using a suitable distribution of the asymmetry parameter n we evaluate the magnetization and the specific heat for Pr amorphous compounds.
Experimental results on the specific heat, electrical resistivity and magnetic properties of light rare earth (RE) amorphous alloys show evidence of substantial deviations from the predictions of the random uniaxial model by Harris et el. (I). The departure from uniaxiality has been considered in the non-axial crystal field model proposed by Fert and Campbell (FC)(2).This model, however, does not give information about the possible distribution of the asymmetry parameter n from site to site in amorphous RE alloys. In a recent work of the present authors (3) substantial agreement with the experimental data on magnetic properties of amorphous Pr21Ag79 (4) was obtained assuming a ~-like ~ distribution, i.e. assuming that a most probable n-value can represent the physical situation "on the average". On the other hand both superconductivity and calorimetric measurements on Pr21Ag79 amorphous alloys (5) and specific heat measurements on Pr La^_ Au^_ amorphous alloys (6) by Garoche .x ~o-x z~ et el. suggest the presence of two low-lying non degenerate levels and the existence of very low crystal-field (CF) excitation energies distributed from site to site around some non-zero average value. This is in qualitative agreement with a quasi-gaussian distribution of CF levels obtained by Cochra~eet el. (7) using computer generated clusters of dense random packing of hard spheres. On this ground Bhattacharjee and Coqblin (BC) (8) recently proposed a simple phenomenological singlet-singlet model with an energy splitring varying with gaussian distribution over RE sites. In the BC approach the singlet-singlet matrix element of J. (the z-component of the angular momentum of t ~ i-th RE ion) is replaced by an average value. The resulting theory extends the Pr two-singlets theory of Bleaney and Cooper (9) by taking into account a weigh~ddistrlbution of the singlet-slnglet energy splitting. The existence of low-lying non degenerate singlets in Pr amorphous alloys is easily explained by the FC Hamiltonian (see (3) for further details); moreover the very low excitation energies
03784363/8|/0000-(KK~/$02.50
©No.h-Ho~andPub~shMgCompany
can be accounted for by allowing for a suitable distribution of the asymmetry parameter ~ from site to site. On the other hand Fert and Spanjaard calculations (lO),assuming values of n equally distributed between the two extreme cases of uniaxial and planar local symmetries (n-±l), do not consider the fact that the majority of Pr ions appear to be in a singlet ground state. This fact suggests a narrower n-distribution. A gaussian distribution of n is unsustainable due to the fact that l~I~l. In this work we assume that Pr ions are subject to a nonaxial local crystal-fleld (2) ~ef" 6C [3(l_~)j2._xz 3(l+n)J~z]" and suggest a cosine-type distribution law forn p(n)-~/4a c°s[J-(D-no)]za ' p(n)=o for ID-D l~a
I
o
The width of this distribution depends on the parameter a. This distribution is symmetrical around the average value n , tends to a 6-distri hution for a~o and, at the°lowest order in (n-q),has the same behaviour as proposed by Cjze~ for amorphous materials (see (5)). Our approach, which puts on a more fundamental basis the BC suggestion, takes into account variations in the CF intensity as well as in the directions of local CF axes, In the presence of an exchange interaction and of an external magnetic field H the total Hamiltonian is i
j
[iJiz
13 Within this model we evaluate both the magnetization and the specific heat of Pr amorphous alloys, Taking into account only the low-lying singlet states ~i and ~2 (3) and using the molecular field approximation we obtain for the magnetization in the ferromagnetic case (see (3) for the meaning of symbols) 83
84
Jz = J(BiB2+2"flY2)x[l-(2~)2]~
tanh
(TF--),
x--cos 8
F " A 2+ g~B
2Yo =L(~) {-~BJ(~I82+2YIY2)x(H+ gPB 7 )}2] ~ z
with
where <~llJzil~2> = J(BiB2+2YiY 2) and A=E2-E I are functions of the parameter q. Here Y E~Y... o
~1
zj
The specific heat C v per mole of Pr is analogously obtained as CV R
--
2 =
~
e (l+e~) ~
~
dF dT
+ --
l-e2~-2~e ~ __ . (l+ee) z
•
+
2y
~
o z dT
2F T Both for ~ and C V the given expressions are to z be considered as averaged over the q-distribution as well as over the random directions e of the local z-axis (o~e~/2). In Fig. i the behaviour of the specific heat Cv/R is reported versus T in the absence of any induced magnetism for values of the parameters C=lOO K, q =-0.6. When the width "a" of the q di. . O. strlbutlon increases the low temperature behaviour of C becomes almost linear. This result V is in qualltative agreement with the experiment al data on PrxLa80_x Au20 and confirms the predictions of the phenomenological singlet-singlet model (8). 0.5
~o-- -0.6
a=O,O
C =1oo I
0.4
0.3
0.2
/J
i¢I II 0.1
0 o
4
e
,2 T (K) ,6
Fig.l : Specific heat Cv/R versus T for several values of the ~-distribution width "a".The dashed curve gives Cv/R for a=O.4 and with an applied magnetic field of 7 Tesla.
Also in the presence of an external magnetic field the low-temperature behaviour of C v appears to be in qualitative agreement with experiment. In fact C V is not sensibly affected by an external magnetic field up to =i Tesla but it is strongly depressed by a field of 7 Tesla (see the dashed line in Fig.l). Numerical calculations which take into account a possible contribution to C V both of ferromagne~c and antiferromagnetic origin are presently in progress. REFERENCES (i)
Harris, R., Plischke, M. and Zuckermann,M. J., Phys.Rev.Lett. 31 (1973) 160. (2) Fert, A. and Campbell, I.A., J.Phys.F: Metal Phys. 8 (1978) L57. (3) Borchi, E. and De Gennaro, S., J.Phys.F: Metal Phys. ii (1981) L47. (4) Pappa, C. and Boucher, B., J.Magn.Magn.Mater. 15-18 (1980) 97. (5) Garoche, P., Fert., A., Veyssi~, J.J. and Boucher,B, J.Magn.Magn.Mater. 15-18 (1980) 1397. (6) Garoche, P., Veyssi~,J.J. and Durand, J., J. Phys.Lett. 41 (1980) 357. (7) Cochrane, R.W., Harris, R., Plischke, M., Zobin, D and Zuckermann, M.J., J.Phys.F: Metal Phys. 5 (1975) 763. (8) Bhattacharjee, A.K. and Coqblin, B., J.Phys. Coll. 41 (1980) C8-626. (9) Cooper, B.R. in "Magnetic Properties of Rare-Earth Metals", ed. Elliot R.J. (Plenum Press, London and New York, 1972). (IO) Fert, A. and Spanjaard, D., J. Phys. Coll. 40 (1979) C5-248.