7.8 Problems
189
To conclude, we present the eigenenergies in the notation of Satten [14], who used
parameters g 1 and g 2 . The reduced matrix elements are expressed as
J p =
μ B
2
g 1 + 9g 2
10
J f =
μ B
2
3(g 1 − g 2 )
10
(7.80)
Furthermore, the magnetic field is represented by directional cosines as B z =
Bn z ,B x = Bn x ,B y = Bn y . The four eigenvalues then become
E =±
μ B
2
B
1
2
g
2
1 + 9g
2
2
±
1
4
√
2
(g 1 + 3g 2 )
9(g 1 − 9g 2 )(g 1 − g 2 )F
−
g
2
1 − 42g 1 g 2 + 9g
2
2
1/2
1/2
(7.81)
The three isotropic cases are reflected by the zeroes of the three factors preceding
the function F = (n 4
x + n 4
y + n 4
z ).
7.8 Problems
7.1 Find a relationship between the crystal-field potentials of an octahedron and a
cube.
7.2 The product of two rotations is a rotation. Obtain an expression for the Cayley–
Klein parameters of the product as a function of the parameters of its factors.
Is the product commutative? The SU(2) matrices may also be identified as normalized quaternions.
7.3 Work out the group multiplication table for the D ∗
3 double group and derive the
class structure.
7.4 Consider a set of three eigenlevels transforming as A 1 + E in D 3 symmetry.
A general matrix for the interaction between the states can be written as
H
|A 1 | E x | E y
A 1 |− 2
a + ib
c + id
E x |
a − ib
e − if
E y |
c − id
e + if
Introduce a fictitious spin operator ˜
S that recognizes these states as the components of a triplet spin, ˜
S = 1, and consider the spin Hamiltonian
H Ze =
μ B
g B z ˜
S z + g ⊥ (B x ˜
S x + B y ˜
S y )
(7.82)
Express the a,...,f parameters as functions of the two g-parameters.
189
To conclude, we present the eigenenergies in the notation of Satten [14], who used
parameters g 1 and g 2 . The reduced matrix elements are expressed as
J p =
μ B
2
g 1 + 9g 2
10
J f =
μ B
2
3(g 1 − g 2 )
10
(7.80)
Furthermore, the magnetic field is represented by directional cosines as B z =
Bn z ,B x = Bn x ,B y = Bn y . The four eigenvalues then become
E =±
μ B
2
B
1
2
g
2
1 + 9g
2
2
±
1
4
√
2
(g 1 + 3g 2 )
9(g 1 − 9g 2 )(g 1 − g 2 )F
−
g
2
1 − 42g 1 g 2 + 9g
2
2
1/2
1/2
(7.81)
The three isotropic cases are reflected by the zeroes of the three factors preceding
the function F = (n 4
x + n 4
y + n 4
z ).
7.8 Problems
7.1 Find a relationship between the crystal-field potentials of an octahedron and a
cube.
7.2 The product of two rotations is a rotation. Obtain an expression for the Cayley–
Klein parameters of the product as a function of the parameters of its factors.
Is the product commutative? The SU(2) matrices may also be identified as normalized quaternions.
7.3 Work out the group multiplication table for the D ∗
3 double group and derive the
class structure.
7.4 Consider a set of three eigenlevels transforming as A 1 + E in D 3 symmetry.
A general matrix for the interaction between the states can be written as
H
|A 1 | E x | E y
A 1 |− 2
a + ib
c + id
E x |
a − ib
e − if
E y |
c − id
e + if
Introduce a fictitious spin operator ˜
S that recognizes these states as the components of a triplet spin, ˜
S = 1, and consider the spin Hamiltonian
H Ze =
μ B
g B z ˜
S z + g ⊥ (B x ˜
S x + B y ˜
S y )
(7.82)
Express the a,...,f parameters as functions of the two g-parameters.