4 X-ray Dichroisms in Spherical Tensor and Green’s Function Formalism
93
parameters: with adjustable parameters, one runs the risk of producing some spectra
in good agreement with experiments not for a theoretically justifiable reason but
rather due to some lucky cancellation between inappropriate choices of the parameters and inaccurate theoretical approximations employed in solving the model. It
is therefore important that one uses some sensible limits for these parameters and
critically examines the values used in the model.
Equation 4.21 is a useful expression because the matrix elements are integrals
again over three spherical harmonics and are given by the Gaunt coefficients. Furthermore, the series can be truncated according to the triangular condition. For 3d
orbitals (like in the case of an Fe ion), 1 = 2 = 2 so the maximum value of m
possible is m = 4 with k ≤ 4. The actual form of the matrix elements depends on
the symmetry of the CF potential.
We will present the example of an octahedral (O h ) cluster to illustrate the procedure of calculating the CF matrix elements. In the case of an O h cluster, six neighbours
are positioned at equal distances from the central ion as shown in Fig. 4.1. We will
first calculate the multipole terms possible for this configuration according to (4.20).
This can be easily evaluated and many softwares are available such as the “multipoles” Python package [8]. One finds that the multipole expansion of the octahedral
potential reduces to the following terms with the coefficients listed below C 0,0 → 6,
C 4,0 →
49
4
, C 4,4 →
35
8
, C 4,−4 →
35
8
.
Now we can evaluate the matrix elements of (4.21) using the Gaunt coefficients.
One finds the following matrix for O h crystal field
k,m
Y 2,m 1 |C k,m |Y 2,m 2 ∝
⎡
⎢
⎢
⎢
⎢
⎣
1 0 0 0 5
0 −4 0 0 0
0 0 6 0 0
0 0 0 −4 0
5 0 0 0 1
⎤
⎥
⎥
⎥
⎥
⎦
.
(4.22)
Diagonalizing this matrix gives the following eigenvalues E = 6, −4, −4, 6, −4 for
the eigenvectors
⎡
⎢
⎢
⎢
⎢
⎣
1
√
2
0
0
0
1
√
2
⎤
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎣
−
1
√
2
0
0
0
1
√
2
⎤
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎣
0
1
0
0
0
⎤
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎣
0
0
1
0
0
⎤
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎣
0
0
0
1
0
⎤
⎥
⎥
⎥
⎥
⎦
.
(4.23)
This illustrates the splitting of the 3d one electron orbitals in an O h crystal field as
shown in Fig. 4.1. The energy difference between the t 2g and e g orbitals is referred
to as 10Dq and its magnitude depends on the radial part A k,m . The five degenerate
orbitals split into two types of orbital:
1. Three orbitals of energies −4Dq referred to as the t 2g orbitals and which are
1
√
2
(Y 2,−2 − Y 2,2 ), Y 2,1 , and Y 2,−1 .
93
parameters: with adjustable parameters, one runs the risk of producing some spectra
in good agreement with experiments not for a theoretically justifiable reason but
rather due to some lucky cancellation between inappropriate choices of the parameters and inaccurate theoretical approximations employed in solving the model. It
is therefore important that one uses some sensible limits for these parameters and
critically examines the values used in the model.
Equation 4.21 is a useful expression because the matrix elements are integrals
again over three spherical harmonics and are given by the Gaunt coefficients. Furthermore, the series can be truncated according to the triangular condition. For 3d
orbitals (like in the case of an Fe ion), 1 = 2 = 2 so the maximum value of m
possible is m = 4 with k ≤ 4. The actual form of the matrix elements depends on
the symmetry of the CF potential.
We will present the example of an octahedral (O h ) cluster to illustrate the procedure of calculating the CF matrix elements. In the case of an O h cluster, six neighbours
are positioned at equal distances from the central ion as shown in Fig. 4.1. We will
first calculate the multipole terms possible for this configuration according to (4.20).
This can be easily evaluated and many softwares are available such as the “multipoles” Python package [8]. One finds that the multipole expansion of the octahedral
potential reduces to the following terms with the coefficients listed below C 0,0 → 6,
C 4,0 →
49
4
, C 4,4 →
35
8
, C 4,−4 →
35
8
.
Now we can evaluate the matrix elements of (4.21) using the Gaunt coefficients.
One finds the following matrix for O h crystal field
k,m
Y 2,m 1 |C k,m |Y 2,m 2 ∝
⎡
⎢
⎢
⎢
⎢
⎣
1 0 0 0 5
0 −4 0 0 0
0 0 6 0 0
0 0 0 −4 0
5 0 0 0 1
⎤
⎥
⎥
⎥
⎥
⎦
.
(4.22)
Diagonalizing this matrix gives the following eigenvalues E = 6, −4, −4, 6, −4 for
the eigenvectors
⎡
⎢
⎢
⎢
⎢
⎣
1
√
2
0
0
0
1
√
2
⎤
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎣
−
1
√
2
0
0
0
1
√
2
⎤
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎣
0
1
0
0
0
⎤
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎣
0
0
1
0
0
⎤
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎣
0
0
0
1
0
⎤
⎥
⎥
⎥
⎥
⎦
.
(4.23)
This illustrates the splitting of the 3d one electron orbitals in an O h crystal field as
shown in Fig. 4.1. The energy difference between the t 2g and e g orbitals is referred
to as 10Dq and its magnitude depends on the radial part A k,m . The five degenerate
orbitals split into two types of orbital:
1. Three orbitals of energies −4Dq referred to as the t 2g orbitals and which are
1
√
2
(Y 2,−2 − Y 2,2 ), Y 2,1 , and Y 2,−1 .