As an additional exercise on the calculation of double group character table, we shall consider the X point of the zincblende structure.

The first step is to decide what are the classes in the double group. In this case we need only to compare the single group and double group classes at the zone center since the classes of X form a subset of these classes. It should not be difficult to see that there are now 7 classes:

{E}, {}, {2}, {2S₄}, {2m_d}, {} and {S₄}.

Using the results of Prob. 2-10 one can show that the characters of these operations on the two spin wavefunctions are:

E			S4	md		S4
2	0	0		0	-2	-

Using this result we can show that the character table for the double group of the X point in the Zincblende crystal is given by:

	E			S4	md		S4
X1	1	1	1	1	1	1	1
X2	1	1	1	-1	-1	1	-1
X3	1	1	-1	-1	1	1	-1
X4	1	1	-1	1	-1	1	1
X5	2	-2	0	0	0	2	0
X6	2	0	0		0	-2	-
X7	2	0	0	-	0	-2

Using this character the reader should show that the X₁ X₃, and X₅ representations in the zincblende structure (see Problem 2.10) goes over to the X₆xX₁ =X₆, X₆xX₃=X₇ and X₆x X₅=X₆+X₇ representations in the double group (see, for example, the band structure of GaAs in Fig.2.14).