162 9 Irreducible Tensors
H 4 ≡Hμνλκ(^4 ) r̂μr̂νr̂λr̂κ =x^4 +y^4 +z^4 −3
5
. (9.32)
Herex,y,zstand for the components of the unit vector̂rwith respect to the cubic
axese(i),e.g.x=r̂μeμ(^1 ). The functionH 4 is proportional to the fourth order cubic
harmonicK 4 with the full cubic symmetry. Cubic harmonics were introduced in [26],
and also used in [27, 28]. These real functions can be expressed in terms of linear
combinations of spherical harmonics, e.g.
K 4 ≡
5
4
√
21 H 4 =
√
4 π[√
7
12
Y 4 (^0 )+
√
5
6
1
2
(
Y 4 (^4 )+Y 4 (−^4 )
)
]
. (9.33)
The other 8 cubic harmonics of order 4 can be found in [26–28]. Similarly, of the 13
and 17 cubic harmonics of order 6 and 8, only those are presented here, which are
obtained by analogy to (9.32), from the tensorsH...(^6 )andH...(^8 ):
K 6 ≡
231
8
√
26 H 6 , H 6 =x^2 y^2 z^2 −1
105
+
1
22
H 4 , (9.34)
K 8 ≡
65
16
√
561 H 8 , H 8 =x^8 +y^8 +z^8 −1
3
−
28
5
H 6 −
210
143
H 4. (9.35)
The numerical factors occurring in the relations between theKandH,for=
4 , 6 ,8, are chosen such that the orientational averages ofK^2 are equal to 1. The
functionK 6 is proportional to a linear combination ofY 6 (^0 )andY 6 (^4 )+Y 6 (−^4 ).For
K 8 ,itisY 8 (^0 ),Y 8 (^4 )+Y 8 (−^4 ), andY 8 (^8 )+Y 8 (−^8 ), cf. [28].
The values of the functionsH 4 ,H 6 , andH 8 , taken at the positions of the first
and second coordination shell ofsimple cubic(sc),body centered cubic(bcc), and
face centered cubic(fcc) crystals, are characteristic for these different types of cubic
crystals.Thecoordinates{x,y,z}ofarepresentativenearestneighborare{ 1 , 0 , 0 }for
the sc,{ 1 /
√
3 , 1 /
√
3 , 1 /
√
3 }for the bcc, and{ 1 /√
2 , 1 /
√
2 , 0 }for the fcc crystal.
ThenH 4 ,H 6 ,H 8 are equal to 2/5, 2/231, 2/65, respectively, for sc, the simple
cubic crystal. The corresponding values for bcc, the body centered cubic crystal, are
− 4 /15, 32/2079, 16/1755. For fcc, the face centered cubic structure, these values
are− 1 /10,− 13 /924, 9/520.