10. Appendix

668 Appendix B
Now V and Y form two linear homogeneous equations while U and X form
another set of two linear homogeneous equations. The fact that the modes
represented by U and X are decoupled from those represented by V and
Y means that the atoms of both kinds are moving either in phase or out of
phase.
For
the X and U modes we obtain the determinant:
mBˆ2 f f
f mAˆ2
f
By setting this determinant to be zero we can obtain the equation:
ˆ 4 ˆ 2
1
mA
1
f 0.
mB
By solving this equation we obtain the eigenvalues:
ˆ 0 and ˆ 2 f [(1/mA) (1/mB)]. (see(9.23))
By calculating the eigenvectors one can show that: for the ˆ 0 acoustic
mode X U i.e. the displacement of the A and B layers are in the same direction
as one may expect for an acoustic mode. Furthermore, this displacement
pattern is odd under reflection. Similarly, one can show that the eigenvector
for the other mode is given by: mAU mBX or mA(u v) mB(x y) 0.
In this mode the A and B layers vibrate against each other but their center of
gravity remains constant. Their displacement patterns are shown in Fig. 9.14 as
that of the 348 cm 1 mode. Again, one can show that this mode is odd under
reflection (since only modes of same parity can be coupled with each other in
the equations of motion).
The determinant of the remaining (even parity) Y and V modes is given
by: mBˆ 2 f f
f mAˆ 2 f
The eigenvalues are obtained by the solving the following equation:
ˆ 4 ˆ 2
1
mA
1
2 8f
f 0
mB mAmB
or
ˆ 2 f 3(mA mB) ± 9(mA mB) 2 4mAmB
2mAmB
as given in (9.24). The corresponding eigenvectors will give the displacement
patterns of the remaining two modes in Fig. 9.14.
When k apple/d, the phase factors exp[±ikd] exp[±iapple] (1).
Thus, the equations of motion become:
mAˆ 2 v f [x 2v u];
mBˆ 2 x f [y 2x v];
mBˆ 2 y f [x 2y u]; and
mAˆ 2 u f [v 2u y].