Green's function theory and high temperature series expansion for magnetic systems with crystal-fields

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Green’s function theory and high temperature series expansion for magnetic systems with crystal-fields

Y.-L. Wang

To cite this version:

Y.-L. Wang. Green’s function theory and high temperature series expansion for magnetic sys- tems with crystal-fields. Journal de Physique Colloques, 1979, 40 (C5), pp.C5-112-C5-113.

�10.1051/jphyscol:1979538�. �jpa-00218955�

JOURNAL DE PHYSIQUE Colloque C5, supplément au n° 5, Tome 40, Mai 1979, page C5-112

Green's function theory and high temperature series expansion for magnetic systems with crystal-fields

Y.-L. Wang

Department of Physics, Florida State University Tallahassee, Florida 32306, USA

Résumé. — La théorie des fonctions de Green et la technique des développements en série haute température sont exposées pour les systèmes magnétiques avec champ cristallin. Nous appliquons les deux méthodes à l'éva- luation de la température critique d'un ferromagnétique de moment angulaire / = 5/2 (Ce3+) avec champ cris- tallin cubique. La variation de la température critique avec l'intensité du champ cristallin est étudiée et comparée aux prédictions de l'approximation du champ moléculaire.

Abstract. — Green's function theory and high temperature series expansion technique have been developed for magnetic systems with crystal-field potentials. We apply the two methods to evaluate the critical temperature of a J = 5/2 (Ce3 +) ferromagnet in a cubic crystal-field. The dependence of the critical temperature on the crystal- field strength is shown and compared with the prediction in the molecular field approximation.

Crystal-field effects in magnetic systems have long been recognized. In rare earths, unlike the transition metals, the spin-orbit coupling is much stronger than the crystal-field and the total angular momentum J remains a good quantum number. As a consequence, the behavior of a rare earth system can be described within the / multiplet of the lowest energy. The Hamiltonian consisting of a crystal-field potential and a Heisenberg exchange interaction energy has been found fruitful in the qualitative discussions of the magnetic behavior of rare earth systems. (Magneto- elastic interactions and higher order pair exchange interactions may be important in certain circums- tances.) However, a quantitative comparison between the theoretical predictions and the experimental data is hampered by the crudeness of the molecular field approximation, which is almost exclusively used in the theoretical calculations for such a complicated Hamil- tonian.

Only very recently [1, 2] have the methods of the Green's function theory and the high temperature series expansion been reformulated to allow the inclusion of the crystal-field potential. In this paper, we apply both methods to obtain the critical tempe- rature of a C e3 + ( J = 5/2) ferromagnetic system in a cubic crystal field of an arbitrary strength.

We first apply the Green's function diagrammatic technique for complicated energy level systems intro- duced by Yang and Wang [1]. We shall only discuss the evaluation of the critical temperature to the order of 1/z, which consists of a summation of the single loop free energy diagrams (to infinite order in the exchange interaction).

We divide the Hamiltonian into two parts.

J6 = J€o + <3Cj,

J£⁰ contains the crystal-field potential and the mole- cular field part of the exchange Hamiltonian; X,t des- cribes the correlations of the fluctuations.

* i = - 1 w

j

r+(

J

i-<

Jz

» v;-<

Jz

»] •

u

(1) Diagonalizing 3£⁰ we find the molecular field eigen- states | s„ > with energy e„ and the zeroth order free energy F⁰ = — & r i n £ e x p ( — /feⁿ). The corrections

n

to F⁰ can be shown in a standard procedure to be - £ A F = £ i _ i - d z J d T2- x

« = i " J o Jo

x J dz„<r

^t

«je

¹

(Ti)3ei^2)--^A-.)>c (2)

Jo

where the angular brackets denote the canonical thermal averages over 3t0. The subscript c denotes the cumulant part of the t-ordered product, or, in the diagrammatic analysis the contribution of the con- nected diagrams. The diagrams can be classified according to the number of wave vector sums in each diagram: an «-th order diagram has n free wave- vector variables summed over in the calculation of its contribution. The correction to the order of \\z is,

- fi AF, = - 1 £ In [1 - 0 d{q) g⁺ "(to,)] -

i V q.l

- ^ 5 > [1 - 2 jB 5(g) »"(«»«)] (3)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979538

GREEN'S FUNCTION THEORY AND HIGH TEMPERATURE SERIES EXPANSION C5-113

where

Here D m = exp( -

PE,)IC

^{exp( -}

PC,)

^{and E,, = E,} - ^E,.

The susceptibility

x

can then be obtained from the free energy, and the critical temperature is located at which

x

diverges. We show the results of this calculation (dotted curve) in figure 1 where an fcc lattice has been taken and nearest-neighbor-only exchange interaction has been assumed. A in the figure is the splitting between the two crystal-field multiplets.

The HTSE can be constructed from Eqs. (2) directly.

We note that each interaction line in the diagrams carries a factor

pa.

An expansion in powers of

pa

is easily obtained. The basic procedures of the HTSE have been given by Wang and Lee 121. We have followed essentially their method with the aid of a computer. We have obtained four terms in the power series expansion of X. The standard ratio test for the convergence of power series has been used to estimate the value of T, and the extrapolated value of T, is shown in figure 1 (solid curve).

In conclusion, we emphasize that both the GFT and the HTSE methods used here calculate the critical

J = 5 / 2

Cubic Crystalline Field f c c Lattice

- HTSE

...

GFT - - M F T ---

Fig. 1. - Variation of the critical temperatures with the crystal- field strength. The results of GFT are shown in the dotted curve, the HTSE in solid curve, and the MFT in dashed curve. The almost exact values, in the limits of A = 0 and m, obtained in the long HTSE are shown in open circles.

temperature for a magnetic system with an arbitrary strength of the crystal-field. The results of the two calculations are amazingly close. Comparing with the almost exact values from the long series estimations which are available only in the two limits, A = 0 and co. We find our values 3

%

off at A = 0 and 4

%

off a t A = a. However, the molecular field values are off by 20

%

and 50

%

respectively in the two limits.

We are in the process of obtaining the fifth coefficient of the HTS. The work on HTSE in this paper was done in collaboration with S. Jafarey and K. Rauchwarger.

Support by NSF and the FSU Computing Center are also acknowledged.

References

[l] YANG, D. H. and WANG, Y. L., Phys. Rev. B 10 (1974) 4714;

12 (1975) 1057.

[2] WANG, Y. L. and LEE, F., Phys. Rev. Lett. 38 (1977) 913.