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53 COOPERATIVE JAHN-TELLER EFFECT AND ELECTRON-... 8439

values are most representative of the undoped material, and 2sinF 2~n 2n !p 2G
1
2
that the LaMnO sample studied in Ref. 15 was inadvertently 3 12d 22d 5asin6d 1 ~C4!
1
3
doped.
The actual crystal structure of LaMnO is complicated; and
3
for example, every Mn-O-Mn bond is buckled. It is assumed
here that the important quantities are the Mn-O bond lengths, sin6d 52sin6d . ~C5!
2
1
and that the remaining distortions are subsidiary, being
Equation ~C5! implies either d 52d or
driven by rotations of the ~distorted! MnO octahedra re- 1 2
6
2
1
quired to fit the rigid MNO octahedra into a lattice with d 5d 1(2n11)p/6; the latter solution would imply that
6
lattice constants smaller than twice the Mn-O distances. the anharmonicity energy vanishes. Such an extremum can-
2
1
LaMnO was found to be orthorhombic, with three un- not produce an absolute energy minimum. If d 52d , then
3
equal Mn-O distances, which are Eq. ~C4! may be solved. Define
p 2 arcsinFA 11a 2 1
1
u 52.187 Å, d5 2 G . ~C6!
2
x
12 2 4a 2a
u 51.905 Å, ~B1! Then d 52d 5d if n 2n 51 or 22 and
y
2
1
1
2
d 52d 52d in n 2n 521 or 2. A different formula ap-
2
1
2
1
u 51.956 Å. plies if n 5n ; however, this case may be seen not to lead to
z
2
1
the global energy minimum because if n 5n , then the in-
Here u ,u are the distances most nearly parallel to the 1 2
x y tersite term is positive unless p/8<udu<p/6, in which case
nearest-neighbor Mn-Mn bonds in the basic ~orthorhombic
the anharmonicity term is positive. The mean-field energy is
a-c) plane. The x-y directions are about 45° rotated from the
orthorhombic a-c axes. u is the distance most nearly paral- thus minimized by any of the six configurations with n 1
z
2
2
1
lel to the orthorhombic b axis. Þn and appropriate d and d .
The z oxygen ion is equidistant from the Mn above and It is instructive to suppose that at all but one of the sites
the angles take values minimizing E and to study the en-
below it; the x and y oxygen ions are not equidistant from MF
ergy function E of the remaining angle. Assume the isolated
the in-plane Mn; indeed, if one moves from one Mn to its 0
site is on the ‘‘1’’ sublattice and n 50. Then
in-plane nearest neighbor, the roles are reversed. We there- 1
fore assume that the observed Mn-O bond lengths have been p a
obtained from an ideal peroviskite structure with Mn-O dis- E 52cosF 2u2 2dG 1 cos6u. ~C7!
0
1/3
0
tance of u 5(u u u ) >2.01 Å by composing a uniform 3 3
x y z
u
tetragonal distortion D 520.112 Å (1,21/221/2) and a For a,a*54/3, Eq. ~C7! has only one minimum, at the
s
staggered distortion D 50.15 Å (1,21,0). The mean lattice u which satisfies the mean-field equation. For a.a* there
u
constant is 4 Å; thus D corresponds to a uniform strain are three minima. For a@a* these occur at
xx
zz
yy
components e 5e 50.014, e 520.028, while the stag-
s
gered distortion is u 50.038(1,21,0). ~2n11!p 2sin2np/3 1
u 5 1 1O 2 ~C8!
n
6 3a a
APPENDIX C: SOLUTION OF THE MEAN-FIELD
EQUATIONS and correspond to energies
In this appendix details are given of the solution of the a 2np 2 2np
mean-field equations and of the energetics of small devia- E 52 12cos sin 3 . ~C9!
n
3
tions from the mean-field solution. 3 A3a
Solution: Assume a two sublattice solution with u5u on
1
one sublattice and u5u on the other. Take A.0 without This is the expected form of the energy of a three-state
2
loss of generality and choose units in which 3k51. Write Potts model with a first-neighbor ‘‘antiferromagnetic’’ and
second-neighbor ‘‘ferromagnetic’’ interactions. A precise
2n 11 mapping is discussed in Appendix D.
1,2
u 5 p1d 1,2 , ~C1! Small deviations: Assume that on every site the angle is
1,2
6
close to one of the two-sublattice solutions; thus, if a51,2
with
u 5u 1c , ~C10!
i
i
a
2p/6<d <p/6. ~C2! with u given by Eqs. ~C1!, ~C6! according to whether i is
1,2
a
From Eq. ~7! one has on sublattice 1 or 2 and c small. Substituting Eq. ~C10! into
5cosF 2~n 2n !p 2G a
Eq. ~7! and expanding yields
1
2
E MF 12d 22d 2 ~cos6d 1
1
3 6 2
1
E5E MF ( v c c . ~C11!
k 2k
k
1cos6d !, ~C3! k
2
with a5A/k.0. The energy v (n 1 ,n 1 ) depends on the quantities n ,n describ-
2
1
k
Minimizing yields ing the possible ordered states. There are three independent