Selection rules for the space group of cuprite, the
symmetry points and lines
K. Olbrychski, R. Kolodziejski, M. Suffczyński, H. Kunert
To cite this version:
K. Olbrychski, R. Kolodziejski, M. Suffczyński, H. Kunert. Selection rules for the space group
of cuprite, the symmetry points and lines. Journal de Physique, 1976, 37 (12), pp.1483-1491.
<10.1051/jphys:0197600370120148300>. <jpa-00208551>
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LE JOURNAL DE
PHYSIQUE
TOME
37, DTCEMBRE 1976,
1483
Classification
Physics Abstracts
7.152
-
8.800
SELECTION RULES FOR THE SPACE GROUP OF CUPRITE,
THE SYMMETRY POINTS AND LINES
K. OLBRYCHSKI
(*),
R. KOLODZIEJSKI
(*),
M.
SUFFCZY0143SKI (*)
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
and H. KUNERT
Institute of Physics, Technical
(**)
University, Pozna0144, Poland
(Reçu le 2 juillet 1976, accepté le 23 aout 1976)
Résumé.
2014
Les
produits directs de représentations irréductibles du groupe double O4h de cuprite
correspond au point de symétrie, l’autre à la ligne de symétrie dans la zone
sont réduits. Un facteur
de Brillouin.
Abstract.
2014
The paper presents selection rules for the double space group
decompositions of direct products of the irreducible representations in which
high symmetry point and the other to a symmetry line of the Brillouin zone.
The present paper is a conti1. Introduction.
nuation of the paper [1], it lists further selection rules
for the space group of cuprite. Selection rules are useful
in the investigation of the electron band symmetries,
optical transitions, infrared lattice absorption, electron scattering and tunnelling, neutron scattering,
magnon sidebands, etc. [2]. Analysis of scattering
processes involving photons, phonons or magnons in
crystalline solids generally requires the appropriate
selection rules to be worked out.
Recently much attention has been directed to the
calculation of the Clebsch-Gordan coefficients of the
space group representations [3-9]. In particular,
Birman and coworkers [8, 9, 10] have shown that the
elements of the first-order, one-excitation, scattering
tensor are precisely certain Clebsch-Gordan coefficients or prescribed linear combinations; the elements
of the second-order, two-excitation, process are a
particular sum of products of Clebsch-Gordan coefficients.
The factorization of a matrix element or a scattering
tensor element into a Clebsch-Gordan coefficient and
a reduced matrix element yields a maximum realization of the simplifications due to the symmetry of a
.
of cuprite, i.e.
factor refers to a
The Clebsch-Gordan coefficients for the irreducible
-
problem.
O4h
one
representations of the crystal space groups or the
crystal point groups are useful for an analysis of the
Brillouin scattering tensor, scattering tensors for
morphic effects, two-photon absorption matrix elements, scattering tensors for multipole-dipole-resonance Raman scattering, higher-order moment expansions in infrared absorption and diagonalization of
the dynamical matrix of crystal vibrations.
For a calculation of the Clebsch-Gordan coefficients
tensors an elaboration of the selection
first necessary step. In fact, a reduction of
products of the irreducible representations of the
relevant crystal group gives the frequency of occurrence of each irreducible representation in a product
and thus a survey of the matrix elements which vanish
by symmetry alone and of those remaining for which
the calculation of the Clebsch-Gordan coefficients is
or
scattering
rules is
a
required.
A listing of the selection rules for the irreducible
representations of the space group of cuprite, Cu20,
seems of particular interest in view of wide scope of
experimental and theoretical investigations performed
on this crystal [11-28]. Results of extensive optical
experiments in CU20, including the two-photon
absorption [19] and the Raman scattering [20, 21, 24]
available.
It is established that the extrema of the electronic
bands in cuprite are at the center of the Brillouin zone.
In cuprite the similarity of the calculated density of
are
(*) Address : Instytut Fizyki PAN, Al. Lotnik6w 32/46, 02-668
Warszawa, Poland.
(**) Address : Instytut Fizyki Politechniki Poznanskiej, ul.
Piotrowo 3, 60-965 Poznan, Poland.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370120148300
1484
states [17] and the experimental absorption and
reflection spectra in the ultraviolet suggests [18] that
the maxima on the lower energy side could correspond to transitions to the symmetry points M and X,
and that the peaks in the violet can originate from
transitions to the conduction bands MFX. For the
phonon spectra in cuprite less information is available [25-28], the occurrence of the extrema of the phonon branches away from the zone center cannot be
excluded. Therefore a list of selection rules including
all symmetry points and symmetry lines seems
appropriate.
The transition ampli2. Decomposition formula.
tude of an electron from the state §§l to the state §f
due to an interaction described by the operator t/J is
proportional to the integral
-
The integral vanishes unless the representation Dh is
contained in the product D’ x D’. Thus the selection
rules are obtained from decomposition of the Kronecker product of two irreducible representations into
irreducible ones. The irreducible representations are
labelled by the wave vectors k, m, h and the indices
be coefficients expressing
p, q, r, respectively.
the frequency of occurrence of the representation
C;::;h will
Dh Dk
x
Dm
q
For the space group G with the elements { aI v } and
with the characters xp,
of the irreducible representations Dp, Dq , D’, respectively, the frequencies
of occurrence are given by
Xr;, i
FIG. 1.
The first Brillouin zone and the representation domain
for the space group 0: of cuprite. b1, b2, b3 are the basic vectors
of the reciprocal lattice.
-
leading wave vector selection rules (LWVSRs),
Lewis [30], i.e. over the elements determined
the expansion of the point group G into double
vant
see
by
cosets
([29], p. 208),
where G’ is the point group of the wave vector
group Gh of h and G’ is the point group of Gk. The
index P(a) means that fl is dependent on a ; it is an
arbitrary element of G satisfying
means equality modulo a vector of the recilattice of the group G. The symbol If is to
remind us that, if for given k, m, h and a no element fl
of G satisfying eq. (5) exists, then we have zero
instead of the sum over S. La is the point group of
La G A Gb, the intersection of the group Gak of
the vector ak and the group Gh of h. The is, ta and rp
are fixed, e.g. the simplest fractional translations
associated with S, a and p, respectively, in the group’G.
and t/J’q are given by the relations of the type
where
=
procal
=
m, h belong to the suitably choof
Brillouin zone, the so-called
the
1/1 UlI part
representation domain 0, containing one arm from
each star. The integerI UI is the order of the single
point group G of the group G. An example of the
representation domain of the space group Oh is shown
in figure 1, by heavy lines. Eq. (3) can be expressed in
terms of the characters ql’ of the small representations d’ which induce the representations D k of
G [29, 30] :
Here
the vectors k,
sen
Here the
sum
indexed
by
a
is taken
over
the rele-
t/J:p
For the small representation d’ of the unbarred
primitive translation { Et} we assume the convention
we choose the + sign in the exponent on the
right-hand side. 1 is the unit matrix of dimension of
the representation d’. In the above considerations G
can be a single or a double space group. Correspondingly all the groups considered are single
groups or double groups, respectively. If G is a
double space group, we use the same symbols for the
point operations of a double group as for correspond-
i.e.
1485
ing operations of the single group. For given k, m from
the representation domain 0 all the vectors h for
which the coefficient (4) may be different from zero
can be found as follows [1] : we consider the vectors ki
from the star of k and mj from the star of m. We
construct the vectors ki + mj with one of the vectors
ki, mj fixed and the second varied. In this way, on
account of eq. (5), we obtain representants h
ki + mj
of the star of the vector h for which the coefficient (4)
may be nonvanishing.
=
,
TABLE I
Table I list coordinates
3. Description of tables.
of the symmetry points, lines and planes of the representation domain 0 for cuprite. In table II we give
the characters of the small representations for all the
symmetry planes of the representation domain of the
group 01 : the labels I of the operations h, are those
of Miller and Love (M-L) ([31], p. 123).
Tables III-XII present the decompositions of the
Kronecker products of the irreducible representations
of the space group Oh into irreducible representations, eq. (2), where k runs over the four symmetry
points and m-over the six symmetry axes of the
Brillouin zone. The irreducible representations of the
group 0’ are labelled by labels of the corresponding
small representations of Miller and Love [31]. Powers
mean frequency of occurrence c; of the given irreducible representation [32].
In some cases there is a need to reduce the wave
vectors to the representation domain 0 : if in the
-
TABLE II
Symbol.d in the left side margin refers only to the selection rules r x
Symbol.d in the right side margin refers only to the selection rules R
A
x
TABLE IV
=
A
Zci A j.
Eci Ti’.
=
1486
TABLE V
Symbol I in the left side margin refers only to the selection rules for F
Symbol 2; in the right side margin refers only to the selection rules for
x E = 2;ci fi.
R x 2; = Eci S’*
TABLE VI
TABLE VII
TABLE VIII
1487
TABLE IX
TABLE X
selection rules, eq. (2), a vector h goes out of the
representation domain 0 for certain values of k
and m, then a vector h’ can be found in 0 such that
for 0 , ii
The vector
-1
but
for 2
q 5 1 it goes
out of 0.
for ’ 2 I I lies just in 0.
where a
E.g. for
E
G and K is
a
vector of the
reciprocal lattice.
In general, the relation between representation dh
and a representation d"’ which induces the same full
group representation as dh is
,
and
we
have
and
The latter vector lies in the
representation
domain 0
where I#lvlc-G h, I LXI r..}is an element of G with
the rotational part a, qlh and §ti are characters of the
representation dh and d"’, respectively.
A closer examination of the tables reveals that a
reduction of the wave vectors h to the representation
domain 0 is required only in tables VIII, IX and XI
1488
TABLE XI
TABLE XII
where the relations (8) are written out for each case.
Note that our representation domain 0 can be defined
by the coordinates of its point (kx, ky, k) 7r/a which
satisfy the inequalities
ei (i 1, 2, 3, 4) are full
corresponding to wave vector
2°
=
group
representation
and
In the tables the wave vectors h parametrized in a way
different from that of table I are distinguished by
primes and their coordinates are specified in the
appropriate table. The same symbols label the vectors h and their representations, e.g. in table VIII :
io eis the vector
and
In table XIII we summarize the notations of the
single-valued and the spinor, separated by space,
representations for symmetry points r, R, M and X,
according to Miller and Love [31], Zak et al. [33],
Kovalev [34] and Bradley and Cracknell [29]. Labels
of the irreducible representations for the symmetry
lines are given in table XIV.
establishing the correspondence between
representations one should notice different definitions
of the small representations dp of the primitive
While
1489
TABLE XIII
TABLE XIV
TABLE II A
1490
translations { EI R,, I where E is the rotation
0° : Kovalev and Bradley have
whereas Miller and Love and’Zak
through
assume
(1 is the unit matrix).
We follow Miller and Love and Zak and we compare
their representations with the complex conjugate representations of Kovalev and of Bradley and Cracknell.
In the tables II A-VI A we supplement the selection
rules of ref. [1] :we list the decompositions of Kronecker products of single-valued by the double-valued
representations at four symmetry points in the
Brillouin zone.
Numbers with the minus sign above correspond
to the odd representations, numbered by M-L with
sign; those without any sign correspond to even
representations, numbered by M-L with + sign.
-
TABLES III A
AND
IV A
TABLE V A
interaction [9]. In a cubic crystal
of the symmetry class Oh the term reduces to Electric-
electron-photon
Quadrupole
EQ(r 25 +) + Electric-Quadrupole
+
EQ(Fl2+) Magnetic-Dipole MD(Fl5+)- When the
incoming photon is in resonance with an EQ(F251)
transition and the outgoing photon corresponds to
an ED(F,,-) transition the phonon symmetries
which can arise are given by
For in photon in resonance with an EQ(Fl2l)
transition and out photon at ED transition the phonon
symmetries which can arise are given by
For in photon in resonance with a MD(FL 5 1)
transition and out photon at ED transition the phonon
symmetries which can arise are given by
All these selection rules can be read from table II,
with help of the table VII, in paper [1]. The selection
rules T x R, T x M, T x X, r x 11, FxA, Fx2:, etc.
have analogous meaning.
In our papers we derive and list the selection rules
at all symmetry points and lines in the Brillouin zone
of cuprite. While at present experimental data have
not been yet accumulated for detailed comparison,
these general selection rules are important, particularly
interesting is the isomorphism of the selection rules
at the r and the R point. The isomorphism is seen
in table II of paper [1], and examples of similarities
can be seen in tables III, IV, V, VI of the present paper.
Furthermore, in cuprite the gradient of the electron
and the phonon energy versus wave-vector dispersion
curves may vanish at the off-center symmetry
points [17, 18, 26, 28]. Ascertainment of their possible
contribution to the interband and the exciton phononassisted optical transitions requires further inves-
tigation.
Comparison of the irreducible representations of Miller and Love (M-L) [31], Zak et al.
[33], Bradley and Cracknell (B-C) [29] with those of
Kovalev [34] is facilitated by noticing the following
relations between their spinor matrices D1/2. We
denote the matrices of the two-dimensional, doublevalued representation of the full orthogonal group
by D112 with the symbol of the respective authors.
1° The relation between the matrices D112 of Miller
and Love and of Kovalev is ([31], p. 14, 123, [34],
Appendix.
TABLE VI A
Numerous examples of the appli4. Examples.
cation of the selection rules at the center of the
Brillouin zone of cuprite have been given for the multipole radiation transitions in ref. [13] and compared
with experiment in ref. [16], and for the Raman
scattering tensor in ref. [9] and compared with experiment in ref. [24].
In cuprite the Is exciton of the yellow series is
electric-dipole forbidden, and we have to consider
the next term beyond electric-dipole (ED) in the
-
p.
-
24-26)
sign - holds for the Miller and Love’s
rotations j 3, 6, 7, 8, 17, 24, 27, 30, 31, 32, 41, 48
(or Kovalev’s operations hi h3, h65 h7, hs, hl7g
h249 h27, h3o, h31, h32, h41, h4s);; for the remaining
where the
=
=
1491
rotations the matrices
K 2 (h j)
D2L(j)
are
the
same
as
where the
.
p.
angles of
the active rotation R are
([29],
52) :
and the matrix S is of the form
where R
{ 0, n } is the active rotation through the
about
the axis with the versor n and 6/2
angle 4?
is the Pauli spin vector. The angles of the successive
rotations are :
qf(O 4/
8(0 K 0
T(O T
after the two
2
7r) about the z axis,
n) about the new x axis,
2 7r) about the z axis which
previous rotations.
([29], p. 418) and of
Bradley
is obtained
20 The relation between the matrices of
and Cracknell
30 The relation between the matrices of
and Cracknell and of Zak ([33], p. 5) is
with
Bradley
Kovalev is
where the
angles
of the successive rotations
are :
4/(0 ik 2 7r) about the z axis,
e(o e , 7c) about the new y axis,
T(O , T 2 7r) about the z axis which is obtained
after the two previous rotations, and the matrix S is
with
of the form
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the
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