Page 8 - ggggg

53 COOPERATIVE JAHN-TELLER EFFECT AND ELECTRON-... 8441
Z 5E p du 2Acos6u/T2bcos2u/T 4ck 6ck sin2usin2c bG ~E2!
b
b
0 p e F 12 T cos2ucos2c 2 T
and, using also Eq. ~22! and the results of Appendix C,
3k @126x#2 ( E p du 4ck 6ck bG
1
1
15 e Acos6u/T2bcos2u/T F 12 cos2ucos2c 2 sinusin2c 3cos~2u22c !. ~E3!
b
b
T c b Z 0 2p T T
s b s s
The integrals in Eqs. ~E2!, ~E3! may be expressed in terms of products of Bessel functions I (A/T)I (b/T). The expres-
n m
sions become simple when A→0or ` or b→0or `, and lead after straightforward calculations to Eqs. ~22!, ~23!.

´
1 E. Wollan and W. C. Koehler, Phys. Rev. 100, 545 ~1955!. 9 K. I. Kugel and D. I. Khomskii, Zh. Eksp. Teor. Fiz. 64, 1429
2
J. B. A. A. Ellemans, B. van Laar, K. R. van derVeer, and B. O. ~1973!@Sov. Phys. JETP 37, 725 ~1973!#.
Loopstra, J. Solid State Chem 3, 328 ~1971!. 10 S. Ishihara, J. Inoue, and S. Maekawa ~unpublished!.
3 11
Y. Tokura, A. Urushibara, Y. Moritomo, T. Arima, A. Asamitsu, G. J. Inoue and S. Maekawa, Phys. Rev. Lett. 74, 3407 ~1995!.
Kido, and N. Furukawa, J. Phys. Soc. Jpn. 63, 3931 ~1994!. 12 Y. Okimoto, T. Katsufuji, T. Ishikawa, A. Urushibara, T. Arima,
4
A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett. and Y. Tokura, Phys. Rev. Lett. 75, 109 ~1995!.
74, 5144 ~1995!. 13 M. D. Sturge, in Solid State Physics, edited by F. Seitz, H. Turn-
5
A. J. Millis, B. I. Shraiman, and R. Mueller ~unpublished!. bull, and H. Ehrenreich ~Academic Press, New York, 1967!, Vol.
6
J. Goodenough, Phys. Rev. 100, 530 ~1955!. 20.
7 14
J. Kanamori, J. Appl. Phys. 31, 145 ~1961!. J. M. Ziman, Models of Disorder ~Cambridge University Press,
8
For a review see, e.g., K. I. Kugel and D. I. Khomskii, Usp. Fiz. Cambridge, England, 1989!; see especially Chap. 12.
15
Nauk 136, 621 ~1982!@Sov. Phys. Usp. 25, 231 ~1982!#. G. Matsumoto, J. Phys. Soc. Jpn. 29, 606 ~1970!.