188
7 Quadrupole Contributions from Interface and Bulk
Next we examine the diagonal element χ [pppp] (= χ [xxxx] = χ [yyyy] =
χ [zzzz]). As a simple example, let us rotate the coordinate system by 45 ◦ around
the z axis,
x →
x + y
√
2
, y →
−x + y
√
2
, z → z.
Then a tensor element χ [xxxx] is transformed as a direct product of the coordinates,
χ [xxxx] → “χ
x + y
√
2
x + y
√
2
x + y
√
2
x + y
√
2
”
=
1
4
{χ [xxxx] + χ [xxxy] + χ [xxyx] + χ [xxyy] + χ [xyxx] + χ [xyxy]
+ χ [xyyx] + χ [xyyy] + χ [yxxx] + χ [yxxy] + χ [yxyx] + χ [yxyy]
+χ [yyxx] + χ [yyxy] + χ [yyyx] + χ [yyyy]} .
Since the transformed element should be invariant to the original element in the
isotropic condition, the following relation should hold,
χ [xxxx] =
1
4
{χ [xxxx] + χ [xxxy] + χ [xxyx] + χ [xxyy] + χ [xyxx] + χ [xyxy]
+ χ [xyyx] + χ [xyyy] + χ [yxxx] + χ [yxxy] + χ [yxyx] + χ [yxyy]
+χ [yyxx] + χ [yyxy] + χ [yyyx] + χ [yyyy]}
=
1
4
{χ [xxxx] + χ [xxyy] + χ [xyxy] + χ [xyyx]
+χ [yxxy] + χ [yxyx] + χ [yyxx] + χ [yyyy]} ,
where the shaded elements are null. Therefore,
χ [pppp] =
1
4
{2χ [pppp] + 2χ 1 + 2χ 2 + 2χ 3 } ,
and hence
χ [pppp] = χ 1 + χ 2 + χ 3
(7.39)
is obtained.