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Vector-coupling coefficients for space groups based on simple cubic lattices
H. Kunert, M. Suffczynski
To cite this version:
H. Kunert, M. Suffczynski. Vector-coupling coefficients for space groups based on simple cubic lat- tices. Journal de Physique, 1980, 41 (11), pp.1361-1370. �10.1051/jphys:0198000410110136100�. �jpa- 00208963�
1361
Vector-coupling
coefficients for space groups based on simple cubic latticesH. Kunert
Institute of Physics, Poznan Technical University, Piotrowo 3, Poznan 60-965, Poland
and M. Suffczynski
Institute of Physics, Polish Academy of Sciences, Lotnikow 32/46, Warsaw 02-668, Poland
(Reçu le 12 février 1980, révisé le 1 er juillet, accepté le 9 juillet 1980)
Résumé. 2014 On présente les coefficients de Clebsch-Gordan pour les représentations irréductibles X x X et M x M des groupes du perovskite et cuprite.
Abstract. 2014 Clebsch-Gordan coefficients are calculated for the irreducible representations X x X and M x M
of space groups of perovskites and cuprite.
J. Physique 41 (1980) 1361-1370 NOVEMBRE 1980,
Classification
Physics Abstracts
61.50E
1. Introduction. - Space groups Oh-Oh based on simple cubic lattice are symmetry groups of important crystals. The symmorphic group Oh(Pm3m) is the
space group of the simple cubic, caesium chloride,
rhenium trioxide, perovskites and several other cubic
crystal types. Selection rules for this group have been
published [1]. Space group 0’ (Pn3n) is not represent- ed in nature. For this space group the Clebsch- Gordan coefficients have been considered [2]. For the
space group Oh (Pm3n), symmetry group of the
important A-15 or beta-tungsten structure, selection rules and Clebsch-Gordan coefficients have been
published [3-7]. Space group 04 (Pn3m) is the symme- try group of cuprite, Cu20, of Ag20, and of the salts
of certain heteropolyacids like the phosphotungstic
acid H3p’(W3010)4.5 H20 [8] and caesium phospho-
tungstate CS3P.(W3010)4.2 H20 and
The unit cell of cuprite is shown in figure 1, and the Brillouin zone for the simple cubic lattice in figure 2.
The representations for the group 0’ at the zone
centre and at the cube corner R are identical except for nonzero primitive translations. For the octahedral point group Oh the Clebsch-Gordan (CG) coefficients
are given in the tables of Koster et al. [11]. For the
space group 0’ the selection rules have been pu- blished [12-17].
In the present paper, starting from the published
selection rules, we compute the CG coefficients for the irreducible representations at the points M and X
Fig. 1. - Unit cell of cuprite, with origin taken at the copper and
at the oxygen ion.
in the space groups Oh and 0’. These CG coefficients have to be published for the first time to complete the
list of CG coefficients for space groups based on the
simple cubic lattice.
In section 2 we describe in detail the notation of the CG coefficients for space groups, in particular the
wave vector selection rules and the corresponding
block structure of the unitary matrix of CG coeffi- cients. In section 3 we describe the table of the wave vector selection rules and the tables of calculated CG coefficients. In section 4 we recapitulate the multipole expansion in crystals of class Oh to point out the representations which transform as the electric dipole,
electric quadrupole and the magnetic dipole. In
section 5 we use the calculated CG coefficients to
LE JOURNAL DE PHYSIQUE. - T. 41, N° 11, NOVEMBRE 1980
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198000410110136100
Fig. 2. - The first Brillouin zone with the representation domain
for the space groups based on the simple cubic lattice.
compute the symmetrized wave functions at points M
and X in cuprite which transform as the electric
dipole and the electric quadrupole. In section 6 we
quote the arguments of Brahms, Nikitine and Dahl for an assignment to the ultraviolet reflectivity
spectra in cuprite of the transitions at the symmetry
points M and X. These points at the Brillouin zone
boundary become important in cuprite spectroscopy and have to receive further investigation.
2. Vector-coupling coefficients for space groups. - For the irreducible space group representation kl
contained m",,,,, i times in the direct product of the
irreducible representations k’ l’and k" 1" with wave vectors satisfying the wave vector selection rule
the basis functions IF"’I are linear combinations ôf
the basis function products IF"’ a with the vector-
coupling or Clebsch-Gordan coefficients :
To compute the Clebsch-Gordan (CG) coefficients [18-22]
we decompose the space group G into cosets with respect to the wave vector group G(k)
We find ck coset representatives { ({Ja 1 ta } and thugs ce
arms of the k wave vector star. The integer
i.e. the orders 1 G | of the point group of the space
group G divided by the order
1 G(k)
1 of the pointgroup of G(k). We find c,,, and ck" arms of the k’ and k"
wave vector stars respectively.
From the star arms we compose all wave vector selec- tion rules of equation (1) or
We choose one leading wave vector selection rule (LWVSR)
with two space group symmetry operations { qJ;.’ T }
and { qJ;." 1 t;." }. The principal, or a’ = À’, a" = À",
6 = 1, block of CG coefficients is computed from the
small representations dk"’, dk"""’ and dkl of dimen- sions dim (1 ’); dim (/") and dim (/) respectively
by performing summations over the space group elements S = { (ps 1 -rus ) belonging to the intersection of the three wave vector groups
The small representations in equation (7) have to be transformed to the tabulated representations of the first
arm of the wave vector star
1363
The indices b’ b" b in equation (7) have to be chosen such that the sum with diagonal indices has a nonvanishing
value. For each wave vector selection rule of equation (5) we compute one symmetry operation p 1 r_, )
which rotates the principal block into the (1’ (1" 6 block,
The (1’ Q" 6 block is computed from the principal block by matrix multiplication
In case of multiplicity m",,,,, > 1, i.e. multiplicity
index y >, 1, we have to find CG coefficients for each y and ensure orthonormalization of columns and
rows of the CG square matrix of dimension
ck, dim (l’) Ck,, dim (1 Ml’l",l ck dim (1) - (13) l
3. Description of tables. - Table 1 lists the wave vector stars at the symmetry points. Canonical wave
vectors in the first column are as given in the tables of Miller and Love (M-L) [23]. We use M-L number- ing of symmetry operations listed for cubic groups in their table 1 on page 123. We use M-L labels and
generators of the irreducible representations. The representation matrices and the principal blocks of
the CG coefficients have been computed with help
of the computer programs [24].
Table II lists the coset representatives, the wave
vector stars and wave vector selection rules for the direct products X Q X and M p M in groups Oh to 0’. For each wave vector selection rule the indices of the corresponding block of CG coefficients are
given and the symmetry operation which rotates the
111 block into the (1’ (1" (1 block. Finally the wave
vector group intersections for X (8) X and M (8) M
are indicated. The principal block corresponding to
Table 1. - Coordinates of the wave vector stars at the symmetry points of the cubic Brillouin zone. aL is the cubic lattice constant.
Table II. - Coset representatives { qJu 1 T« } and stars of the wave vectors
Wave vector selection
rules,
blocks and symmetry operations { ({JI: 1 T_, 1the leading wave vector selection rule (LWVSR) in
channel r and the principal block corresponding to
LWVSR in channel M have the indices 111. The blocks of CG coefficients have been computed with respect to the wave vector selection rules specified in
table II.
Tables 1II-V give the CG coefficients for X p X and M (D M in Oh and tables VI-VIII in 0’. In the
tables of CG coefficients we use the symbols
while the entries not written explicitly are zero. For
the representations listed below tables III, IV and VI
the CG coefficients will be obtained by multiplying
the column by the factor w* written under the column,
above the column index. In table III the left margin
Table III. - Clebsch-Gordan coefficients for Xi Q Xi, Xj Q X J, Mi Q Mi and Mj Q Mj(i = 1 +, 2 +) and ( j = 3 :1:, 4 +) of space group 0’
Table IV. - Clebsch-Gordan coefficients for X5 ± @ X5 ± and M.5 ± (D MS:f: of space group 01. The channel r.
1365
Table V. - Clebsch-Gordan coefficients for X.5 i Q XS:f: and M.5 ± (& M 5 ± of space group 0’. The channel M.
Table VI. - The Clebsch-Gordan coefficients for Xi 0 Xi, Mi @ Mi (i = l, 2) andXj @ Xi’ Mj @ Mj ( j = 3, 4) of space group 0:. The channel r.
Table VII. - Clebsch-Gordan coefficients for Xl (D Xl and X 2 Qx X 2 of space group Oh. The channel M.
refers to the representations listed above, the right margin to the representations listed below the table.
In table VI the upper signs refer to the upper (i = 1, 2),
the lower signs refer to the lower ( j = 3,4) description
line above and below the table.
4. Multipoles in crystals of class Oh. - The electron- photon interaction is described by the vector potential
with the propagation vector k = k(12, m2, n2) and perpendicular unit polarization vector E = (li, ml, nl) projected on the electron momentum p = - iwmr,
In crystals of class Oh the multipole expansion starts
with
i.e. sum of the electric dipole (ED), electric quadru- poles (EQ) and magnetic dipole (MD). We write
first Elliott [25, 26] and then the M-L [23] or Kos-
ter [11] representation label.
Birman [30, 31] has shown that the elements of the
first order scattering tensor are CG coefficients mul-
tiplied by a constant and elements of the second order scattering tensor are bilinear sums of CG
coefficients. For crystals of class Oh Birman [31]
calculated the multipole-dipole scattering tensors
1367
Table VIII. - Clebsch-Gordan coefficients for Xj Q X ., M . 0 Mj ( j = 3, 4) and Mi 0 Mi (i = 1, 2) of space
group 0:. The channel M.
and applied them to the analysis of Raman scatter- ing by LO and TO phonons on the 1 S yellow exciton
line at the zone centre of cuprite.
5. Symmetrized products of wave functions at points X and M in Oh. - In view of the selection rule for the space group Oh [15]
the vertical transitions at the point X between the states of the Xl symmetry are possible via both the
electric dipole and the electric quadrupole radiation.
From table VI of CG coefficients we write down the
products of basis functions of the representation Xi
which transform as components of the electric dipole, F4-, representation,
and as components of the electric quadrupole, T3+
and 7"5+, representations,
The selection rule
restricts the vertical transitions at the M point bet-
ween the states of M3 symmetry to the electric qua-
drupole transitions only. From our table VI of CG
coefficients the products of the wave functions of M3
symmetry which transform as the components of the electric quadrupole of symmetry r 3 + and F5, are
6. Applications for cuprite. - Cuprous oxide, Cu20, cuprite, a semiconductor crystal with a direct band
gap of about 2.17 eV, is since many years an object
of intense investigations, both experimental and theo-
retical. In particular its exciton spectrum has attract- ed considerable interest. The exciton spectra investi- gated in cuprite consist of four series : yellow, green,
blue and indigo [32], and the main lines originate
from the transitions at the zone centre [33-35].
Raman scattering studies of the yellow exciton
series in cuprite with participation of one and two phonons have been performed [36-38]. Two-photon absorption in cuprite has been measured [39, 40]
and the analysis has contributed to the unambiguous assignment of the parity of the band edge states at the
zone centre. Electroabsorption studies further contri- buted to the interpretation of the absorption spectra in cuprite [41-43].
It is established that the extrema of the valence and conduction bands in cuprite are at the centre of
the Brillouin zone. The electron bands for cuprite
have been computed by the APW method by Dahl
and Switendick [44]. They used the notation of
Bouckaert, Smoluchowski, Wigner [27] and Elliott [25]
for the zone centre, and the notation of Kovalev [28]
for the points M and X at the zone boundary, see
table IX. Dahl and Switendick have found in parti-
cular that the Ml and X3 i.e. M3 and X, states of
M-L occur in the valence and in the conduction band close to the forbidden energy gap and thus may contribute to the optical absorption at sufficiently high photon energies.
The results of the band calculations show that the excitons in cuprite are associated with the copper ions,
rather than with the oxygen ions. Exciton transitions may make their appearance not only near the funda-
mental edge but also at higher energies where higher
conduction or lower valence bands are involved.
In the ultraviolet strong absorption and reflection spectra are observed [45]. The similarity of the calculat- ed density of states and the experimental data, mainly
on reflection spectra, allowed Nikitine [32, 45] to
Table IX. - Labels of the irreducible representations for the points r, R, M and X in space group Oh (Pm3m)
and 0: (Pn3m).
1369
suppose that the maxima on the lower energy side of the band, i.e. A, B, El, E2 and E3 peaks should correspond to transitions from d copper states to the conduction band Ml Fl’2 X3 Rl_, i.e. M3 F3- Xi R4-
of M-L at the corresponding symmetry points of the
Brillouin zone. The E1, E2, E3 peaks could be due to
exciton formation. The intense El peak probably ori- ginates from a transition at the X point. The strongly temperature-dependent peak A could be associated with exciton formation at the point M, but its width is
unusually large, which is rather difficult to reconcile with an exciton assignment. The very intense absorp-
tion above 5 eV with C maximum corresponding to
a peak of reflectivity of more than 40 %, and the
maximum D at 6 eV, could be related to transitions from the d copper valence band states to a higher
excited state which should lie about 2 eV above the conduction band Ml F’12 X3 i.e. M3 F3- Xi of M-L.
In this case the transition should take place at the M
and X points and are forbidden at the r point having a representation T25 i.e. F5, of M-L.
The reflectivity maxima of about 10 % in the 9 eV region could be due to transitions from the oxygen
2p states to the conduction band M 1 -rl +X3 i. e.
M3 Fi 1 Xi of M-L. Dahl’s [44] calculations give
too large a separation between the 3d copper valence band and the 2p oxygen band [46].
In section 5 we have seen that in cuprite the vertical transitions between the states of Xi symmetry are
possible via electric dipole and electric quadrupole
transitions. Vertical transitions between the states of M3 symmetry are possible only via electric qua-
drupole, like at the zone centre where the electric
quadrupole and magnetic dipole transitions arc
allowed and the electric dipole transitions are forbid- den because of equal parity of conduction and valence band. Thus the vertical transitions between the Xi
and those between the M3 states can be definitely
distinguished. Experimental detection of this distinc- tion poses an interesting problem for optical spec-
troscopy.
We have no access to the wave functions of cuprite
and we are not in position to advance our present considerations nor compute intensity ratio of the
absorption peaks. Nevertheless the calculated CG coefficients can be of use for the future interpretation
of the optical experiments.
In cuprite the gradient with respect to the wave
vector of the electron and the phonon bands may vanish at the off-centre symmetry points in the Bril-
louin zone [47] and the density of states at such points
may contribute to the absorption and scattering spectra.
Future experiments with polarized synchrotron
radiation can add further detailed information on the
reflectivity and absorption spectra in the ultraviolet and thus help to understand the role of the off-centre symmetry points in the optical processes.
One of the authors (H. K.) was working under project No. I. 5.6.04 coordinated by the Institute of Experimental Phys. of Warsaw U.
APPENDIX. - We take this opportunity to correct
errors found in our previous publications on the
selection rules for cuprite. In ref. [15] in table III in the right-hand margin X3 has to be exchanged with X4,
in table IV in the right-hand margin M3 with M4
and in table VI in the left-hand margin M3 with M4.
In table V in the decomposition Ms x Ms = GM32
instead of GMj’. We thank Prof. A. P. Cracknell for pointing this. In ref. [16] in table XI the line
S’ = (1 - 1, 1, 1 - q) n/a. In Kovalev tables [28]
for the double-valued representations at the points M
and X in Oh, P147, one should take n3,4 = nI x îs,6
since n3 and n4 as given by Kovalev are equivalent to nI and ft2 respectively.
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