Letters to the Editor
909
A Note on Exciton Multiplicities*
R. S. Knox**
Solid State Science Division
Argonne National Laboratory,
Argonne Illinois, U. S. A.
July 20, 1960
In a recent publication in a Supplement to this Journal, Takeuti1) has commented on the relative positions of
singlet and triplet exciton states. He
maintains that it is impossible to find
a case in which a singlet state will lie
below a triplet (a violation of an ex"
citon "Hund's Rule").
Since the
author has found such a case in solia.
argon,2) and has discussed it recently
* Based on work performed under the auspices
of the U. S. Atomic Energy Commission.
** Address after August, 1960: University of
Rochester, Rochester 20, New York, U. S. A.
910
Letters to the Editor
in another connection,S) some doubt is
cast upon his calculation and it appears
worthwhile to examine Takeuti's comment carefully_
Let us rewrite Eq. (4 ·14) of reference
1), the energy of an exciton of multi·
plicity M, as follows:
MVZ/I'(O, 0) =a~I'(MVa~+aM2Vix
+aM2V.D~+MVlL)'
(4·14')
Here al = 1, as = 0, K is the wave vector
of the exciton, and A and p. label variou~
states of polarization. This equation
differs from (4 ·14) in that Va and VJi
are allowed to depend on multiplicity.
The remaining notation follows reference
1). Now
IV~~(O,
0)
_3V;~(0,
0)
=2 (Vi'x+ VE) +.dV\
(4·20")
where
.dV~=lV8-3V~+lViL-3ViL­
Takeuti has shown that Vix+ V.; is
always positive /) but the term .dV~ and
others which appear in the complete
formalism make it possible for (4·20")
to be negative. .d V~ does not appear
In
reference 1) because the same
Wannier functions were used in the
triplet and singlet states and VOL was
neglected. .dV~ does appear in the
tight-binding model when
proper
Hartree-Fock atomic functions are used
for states of different multiplicity, and
it will arise in the effective mass
model when different Wannier functions
are assumed for the two branches (j =
1/2 and 3/2) of the valence band. In
the author's previous work2),3) 2Vfi7x
and .dV~ were essentially considered as
a single term and 2 V 17 was discussed
separately. The possibility of a Hund's
rule violation dces not (lepend, as implied in reference 1) on the use of the
dipole wave sum approximation to 2 V.; ;
neither is it caused solely by 2 V.; itself,
as implied in reference 2).
It is to be emphasized that the author
agrees with Takeuti's result within the
framework of the standard approximations, but wishes to make clear that
inequality (4· 20) does not imply the
general inviolability of the exciton
Hund rule.
1)
2)
3)
Y. Takeuti, Prog. Theor. Phys. SuppJ. No.
12 (1959), 75.
R S. Knox, J. Phys. Chern. Solids 9 (1959),
265.
R S. Knox, Phys. Rev. 116 (1959), 1093.