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The temperature dependence of the thermodynamic properties of uranium compounds
nil
i
Journal of Magnetism and Magnetic Materials 108 (1992)87-88 North-Holland
The temperature dependence of the thermodynamic properties of uranium compounds G.A. Gehring and L.E. Major Physics Department, Sheffield Unirersity, Sheffidd S3 7RH, UK The mean field equations have been solved for an Anderson lattice corresponding to a uranium heavy fermion compound as a function of temperature, both numerically and using an analytic approximation for the Fermi function. We find that the analytic results are a good approximation to the numerical results and also give a good physical insight into the nature of the solution. We calculate the magnetic sucel:tibility as a function of temperature and compare the results with UPt 3. The application of the slave boson technique to a fo_f~ Anderson lattice, in the Kondo limit, has yielded a succesful microscopic picture of cerium heavy fermion compounds [1,2]. They are characterised by two, distinct :emperature regimes. At low temperatures, the behaviour is Fermi-liquid like, exhibiting a greatly enhanced quasipartiele mass. At high temperatures, the f-level hybridisation with the conduction band is relatively weak, and local moment behaviour prevails. In this paper, we consider the mean field equations that correspond to a uranium heavy fermion system and investigate the temperature dependence of the solutions. For the f2_f3 Anderson lattice, the approach becomes more complex because the valence, n t has to be restricted between two magnetic Kondo limits i.e. 2 < nf < 3. This necessitates two boson fields within the slave boson method [3], and leads to four equations at mca:: field level. Wc obtain self-consistent equations for the chcraical potential, #(T), thc rcnormaliscd f-level energy e t, and the valence, nf, each as a function of temperature. These have been solved both analytically, using a linear approximation for the Fermi-function, and numerically. We find the temperature range in which meanfield theory is valid far exceeds the effective Kondo temperature. This is thought to derive from the negative feedback effect, first considered by Evans et al. [4,5], and renders the valence, nf, almost temperature independent. We have also calculated the specific heat and magnetic susceptibility as a function of temperature. The result for the susceptibility is compared with experiment. The mean field equations for an N-fold degenerate Anderson lattice, as shown by Rasul and Dcsgrangcs [2], are given below:
,,, = - N p o f ' u - ' " * ' { A + ( k ) f ( E_ ( k ) - t~) -
dek,
x ., +
(4)
l)w
f ( E _ ( k ) - IS) + f ( E+(k ) - I . t ) d%, E+(k)-E_(k) = Npof
N-' a
_
(2)
wI( E _ ( k ) -
W
+f(E+(k)-ll)
(3)
d,k = 4 ,
where E±(k)=~
,( % + ~ f - t - ¢ ' ( ~ - ~ f ) ' + 4, 1 2 2
A ~( k ) - - - ( ~1 1 + (e k __ e f ) / / ~ / ( e / ,
),
(4)
__e r )-~ + 41]'2
)'
(5) where E. is the bare f-level cnergy, W is the c reduction clcctron bandwidth, I~ is thc hybridisation width and P0 is the conduction electron density of states (both assumed to be constant), and 1,7,z= I / ~ ( 2 - nfXn t -3). On integration, employing a linear approximation to the Fermi-function [6], we find in the low temperature regime:
e,.(T) - . ( T )
,,f(T) = Np,,g--~- in
(
+a
V2 <.(T) = G + ( 2 " , ( r ) - 5) --if-
"/~ +
El(T)-,(T)
t
,
(6)
,
(7)
+ A)
{ > ( T ) +,.1 - e l ( T ) ) 2A
xln
W
+A_(kIf(E+(k)-l,Z)}
~t = Eo + Np.M",f(N_w
e,.(T)-,(T)-A
~t(T) = W(3 - , , f ( T ) ) ,
0312-8853/92/$05.00 © i992 - Elsevier Science Publishers B.V. All rights reserved
+1
(8)
G.A. Gehring. L.E. Major / Thermodynamic properties of uranium compoun&
88
(where A is defined in the approximation to the Fermi-function as 2T). In the low temperature limit. these expressions become equivalent to those derived at zero temperature [3]. The integration that yields the chemical potential can bc performed exactly [6], and the above expression is found to be a good approximation for all temperatures. We took W to be 10 eV, V as I eV, and E 0 as - 0 . 5 eV. Apart from these constant terms, it is apparent that the chemical potential is, by far, the largest energy. We define an effective f-level energy relative to the chemical potential, ~ ' ( T ) = ~ f ( T ) - / z ( T ) , and a corresponding bare f-level, E~*. We found, even in the Kondo limit, an extremely slow variation of valence, nr(T), with temperature. The effective f-level energy, E~'(T), however, is very strongly temperature dependent, and is shown to be equal to e~*(0)+A in the Kondo limit, where El*(0) corresponds to the Kondo energy. The evidence suggests that the same negative feedback effect that was found observed in the cerium case when the chemical potential was allowed to vary [4,5], is occurring for the uranium system as well. As the temperature is increased, the valence rises, causing the chemical potential to decrease; this serves to increase the effective f-level, which in turn, lowers the valence, n r, thus countering the initial effect. In order to calculate the magnetic susceptibility, a "t • term hE~,,,mf~,,f~,,, is introduced into the Hamiltonian, the effect of which is to transform Etm ~ E~.+ hm. X., is then found by differentiating twice with respect to h, and taking the limit h ~ 0. We obtain the following expression:
X
~ 0
0.~
1"3
"
-
"
t'5
"
-
"
~'0
T/Tk
Fig. !. The variation of magnetic susceptibility with temperature, relative to the Kondo temperature, Tk. The degeneracy, N, is taken to be 6.
with experimental data [7], especially with that tor UPt 3. In conclusion we have solved the mean field equations that correspond to a uranium heavy fermion system, at finite temperature, within the slave boson approach. We find that the negative feedback effect obsewed in the cerium case is even more pronounced for uranium, and lelds to an effectively temperature independent valence. Thc magnetic susceptibility has been calculated as a function of temperature and shows good qualitative agr":',nent with experiment. References
x = 2p,,~.,m21f"';"'
¢16 k
:2 ×f(6)(Et.(T)_/z(T)_%)
} 3 ,
(9)
where y,,,, + = E,,,[( N - 1)] - /x and y,~,, = E,,,(W) - #. The relative susceptibility is plotted against temperature in fig. 1. The same form of scaling relations for the relevant temperatures found for cerium type compounds [5] is observed. A good agreement is found
[I] P. Coleman. Phys. Rev. 28 (1983) 5255. [2] J.W. Rasul and I|.U. Desgranges, J. Phys. C 19 (1986) L671. [3] J.W. Rasul and A.P. Harrington, J. Phys. C 20 4783. [4] S.M.M. Evans et ai., J. Phys. Condens. Matter I (1989) 10473. [5] S.M.M. Evans Thesis, 1989, University of Oxford, unpublished. [6] J. Choi, G.A. Gehring and R. Wojciechowski to be published. [7] G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755.
i
Journal of Magnetism and Magnetic Materials 108 (1992)87-88 North-Holland
The temperature dependence of the thermodynamic properties of uranium compounds G.A. Gehring and L.E. Major Physics Department, Sheffield Unirersity, Sheffidd S3 7RH, UK The mean field equations have been solved for an Anderson lattice corresponding to a uranium heavy fermion compound as a function of temperature, both numerically and using an analytic approximation for the Fermi function. We find that the analytic results are a good approximation to the numerical results and also give a good physical insight into the nature of the solution. We calculate the magnetic sucel:tibility as a function of temperature and compare the results with UPt 3. The application of the slave boson technique to a fo_f~ Anderson lattice, in the Kondo limit, has yielded a succesful microscopic picture of cerium heavy fermion compounds [1,2]. They are characterised by two, distinct :emperature regimes. At low temperatures, the behaviour is Fermi-liquid like, exhibiting a greatly enhanced quasipartiele mass. At high temperatures, the f-level hybridisation with the conduction band is relatively weak, and local moment behaviour prevails. In this paper, we consider the mean field equations that correspond to a uranium heavy fermion system and investigate the temperature dependence of the solutions. For the f2_f3 Anderson lattice, the approach becomes more complex because the valence, n t has to be restricted between two magnetic Kondo limits i.e. 2 < nf < 3. This necessitates two boson fields within the slave boson method [3], and leads to four equations at mca:: field level. Wc obtain self-consistent equations for the chcraical potential, #(T), thc rcnormaliscd f-level energy e t, and the valence, nf, each as a function of temperature. These have been solved both analytically, using a linear approximation for the Fermi-function, and numerically. We find the temperature range in which meanfield theory is valid far exceeds the effective Kondo temperature. This is thought to derive from the negative feedback effect, first considered by Evans et al. [4,5], and renders the valence, nf, almost temperature independent. We have also calculated the specific heat and magnetic susceptibility as a function of temperature. The result for the susceptibility is compared with experiment. The mean field equations for an N-fold degenerate Anderson lattice, as shown by Rasul and Dcsgrangcs [2], are given below:
,,, = - N p o f ' u - ' " * ' { A + ( k ) f ( E_ ( k ) - t~) -
dek,
x ., +
(4)
l)w
f ( E _ ( k ) - IS) + f ( E+(k ) - I . t ) d%, E+(k)-E_(k) = Npof
N-' a
_
(2)
wI( E _ ( k ) -
W
+f(E+(k)-ll)
(3)
d,k = 4 ,
where E±(k)=~
,( % + ~ f - t - ¢ ' ( ~ - ~ f ) ' + 4, 1 2 2
A ~( k ) - - - ( ~1 1 + (e k __ e f ) / / ~ / ( e / ,
),
(4)
__e r )-~ + 41]'2
)'
(5) where E. is the bare f-level cnergy, W is the c reduction clcctron bandwidth, I~ is thc hybridisation width and P0 is the conduction electron density of states (both assumed to be constant), and 1,7,z= I / ~ ( 2 - nfXn t -3). On integration, employing a linear approximation to the Fermi-function [6], we find in the low temperature regime:
e,.(T) - . ( T )
,,f(T) = Np,,g--~- in
(
+a
V2 <.(T) = G + ( 2 " , ( r ) - 5) --if-
"/~ +
El(T)-,(T)
t
,
(6)
,
(7)
+ A)
{ > ( T ) +,.1 - e l ( T ) ) 2A
xln
W
+A_(kIf(E+(k)-l,Z)}
~t = Eo + Np.M",f(N_w
e,.(T)-,(T)-A
~t(T) = W(3 - , , f ( T ) ) ,
0312-8853/92/$05.00 © i992 - Elsevier Science Publishers B.V. All rights reserved
+1
(8)
G.A. Gehring. L.E. Major / Thermodynamic properties of uranium compoun&
88
(where A is defined in the approximation to the Fermi-function as 2T). In the low temperature limit. these expressions become equivalent to those derived at zero temperature [3]. The integration that yields the chemical potential can bc performed exactly [6], and the above expression is found to be a good approximation for all temperatures. We took W to be 10 eV, V as I eV, and E 0 as - 0 . 5 eV. Apart from these constant terms, it is apparent that the chemical potential is, by far, the largest energy. We define an effective f-level energy relative to the chemical potential, ~ ' ( T ) = ~ f ( T ) - / z ( T ) , and a corresponding bare f-level, E~*. We found, even in the Kondo limit, an extremely slow variation of valence, nr(T), with temperature. The effective f-level energy, E~'(T), however, is very strongly temperature dependent, and is shown to be equal to e~*(0)+A in the Kondo limit, where El*(0) corresponds to the Kondo energy. The evidence suggests that the same negative feedback effect that was found observed in the cerium case when the chemical potential was allowed to vary [4,5], is occurring for the uranium system as well. As the temperature is increased, the valence rises, causing the chemical potential to decrease; this serves to increase the effective f-level, which in turn, lowers the valence, n r, thus countering the initial effect. In order to calculate the magnetic susceptibility, a "t • term hE~,,,mf~,,f~,,, is introduced into the Hamiltonian, the effect of which is to transform Etm ~ E~.+ hm. X., is then found by differentiating twice with respect to h, and taking the limit h ~ 0. We obtain the following expression:
X
~ 0
0.~
1"3
"
-
"
t'5
"
-
"
~'0
T/Tk
Fig. !. The variation of magnetic susceptibility with temperature, relative to the Kondo temperature, Tk. The degeneracy, N, is taken to be 6.
with experimental data [7], especially with that tor UPt 3. In conclusion we have solved the mean field equations that correspond to a uranium heavy fermion system, at finite temperature, within the slave boson approach. We find that the negative feedback effect obsewed in the cerium case is even more pronounced for uranium, and lelds to an effectively temperature independent valence. Thc magnetic susceptibility has been calculated as a function of temperature and shows good qualitative agr":',nent with experiment. References
x = 2p,,~.,m21f"';"'
¢16 k
:2 ×f(6)(Et.(T)_/z(T)_%)
} 3 ,
(9)
where y,,,, + = E,,,[( N - 1)] - /x and y,~,, = E,,,(W) - #. The relative susceptibility is plotted against temperature in fig. 1. The same form of scaling relations for the relevant temperatures found for cerium type compounds [5] is observed. A good agreement is found
[I] P. Coleman. Phys. Rev. 28 (1983) 5255. [2] J.W. Rasul and I|.U. Desgranges, J. Phys. C 19 (1986) L671. [3] J.W. Rasul and A.P. Harrington, J. Phys. C 20 4783. [4] S.M.M. Evans et ai., J. Phys. Condens. Matter I (1989) 10473. [5] S.M.M. Evans Thesis, 1989, University of Oxford, unpublished. [6] J. Choi, G.A. Gehring and R. Wojciechowski to be published. [7] G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755.