132
6 Interactions
Fig. 6.1 The Mexican hat potential-energy surface of the E × e linear JT problem. The nuclear
displacement coordinates are the tetragonal elongation, Q θ , and the orthorhombic in-plane distortion, Q ǫ
Then the secular equation of the force element matrix in Eq. (6.69) becomes:
ε
2
k − F
2
E ρ
2 cos
2 ϕ − F
2
E ρ
2 sin
2 ϕ = 0
(6.71)
Two roots are found, which are independent of the angular coordinate. The corresponding eigenfunctions are:
ǫ 1 = F E ρ −→ | ψ 1 =cos
ϕ
2
|Eθ+sin
ϕ
2
|Eǫ
ǫ 2 =−F E ρ −→ | ψ 2 =−sin
ϕ
2
|Eθ+cos
ϕ
2
|Eǫ
(6.72)
The surface consists of two sheets and exhibits rotational symmetry.
E ± = E 0 +
1
2
K E ρ
2 ± F E ρ
= E 0 +
1
2
K E
Q
2
θ + Q
2
ǫ
± F E
Q 2
θ + Q 2
ǫ
(6.73)
A cross section of this surface looks like a two-well potential, with two displaced
parabolæ. The depth of the well is called the JT stabilization energy:
E JT =−
F 2
E
2K E
(6.74)
In the 2D space of the active modes these parabolæ revolve around the centre, giving rise to the Mexican hat appearance. At the origin this surface has the shape
of a conical intersection, indicating that the high-symmetry point is unstable, and
will spontaneously relax to the circular trough surrounding the degeneracy [8]. The