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Phonon symmetry selection rules for coherent inelastic neutron scattering: application to BCCD
Physica B 276}278 (2000) 305}307
Phonon symmetry selection rules for coherent inelastic neutron scattering: application to BCCD J. Hlinka * Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Praha 8, Czech Republic
Abstract Recently demonstrated theorem for searching of the symmetry selection rules for coherent inelastic neutron scattering by phonons, independent of particular structural features, is applied to betaine calcium chloride dihydrate (BCCD). Selection rules for the zone center phonon scattering in crystalline phases of BCCD are given explicitly. 2000 Elsevier Science B.V. All rights reserved. Keywords: Phonons; Lattice dynamics; Structural transitions
1. Introduction Polarized Raman or IR spectroscopic data on phonons in crystals are currently presented with their symmetry assignment } each phonon frequency is related to the small irreducible representation (irrep) describing the symmetry properties of the observed mode. Such assignment is directly derived from the crystal symmetry and the geometry of the experiment. Contrary to the widespread belief, the coherent inelastic neutron scattering (INS) phonon structure factor also obeys systematic selection rules, which follows merely from the space group symmetry of the crystal, irrep of the phonon mode and the scattering geometry (total transferred momentum). These systematic rules can be used for planning neutron scattering experiments (for example, to prevent overlapping of spectral responses from modes of di!erent irrep's) or for symmetry assignment of the detected phonon resonances. The systematic approach to the selection rules presented in Ref. [1] purposely do not take into account actual atomic positions. This makes the method particularly simple and general. Its convenience would be most ap-
preciated when dealing with crystals with many atoms in unit cell or when structure model is not available. On the other hand, some additional (&accidental') selection rules induced by special positions of the atoms may exist in some simple highly symmetrical structures [1]. For such crystals, the approaches based on both the symmetry and structural details (e.g. Ref. [2]) may provide additional conditions. In this paper, we derive the systematic selection rules for the case of zone center phonon modes in the crystal of betaine calcium chloride dihydrate (BCCD) [3]. It undertakes a number of structural phase transitions under lowering temperature. Among them the high-temperature Pnma and the low-temperature Pn2 a ferroelectric phases have 4 BCCD molecules in the unit cell; the intermediate phases are incommensurate (IC) and commensurate (C) modulated phases with wave vector q ""cH. Majority of the experiments con"rms that the space groups of the observed C phases are given as follows [4]: q /cH"odd/oddNP2 2 2 , q /cH"even/oddNPn2 a and q /cH"odd/evenNP2 ca. 2. Selection rules for BCCD
* Corresponding author. Fax: #420-2-821227. E-mail address: [email protected] (J. Hlinka)
In order to obtain the desired selection rules, it is su$cient to know the crystal space group, set of symmetry operations +R"t, generating the equivalent
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 4 9 5 - 7
306
J. Hlinka / Physica B 276}278 (2000) 305}307
Table 1 Selection rules for Pn2 a structure. Note: First six columns show characters of irreps and selection rules for Raman and IR spectroscopy, last three columns provide the special Brillouin zone centers for which the phonon INS scattering is allowed by the systematic selection rules of Eq. (1)
A A B B
E
2
n
a
1 1 1 1
1 1 !1 !1
1 !1 1 !1
1 !1 !1 1
OKL D, y xz, } yz, z xy, x
K#L K#L K#L K#L
even odd even odd
HKO
OKO
H H H H
K even never never K odd
even odd odd even
Table 2 Selection rules for P2 ca structure
A A B B
E
2
c
a
1 1 1 1
1 1 !1 !1
1 !1 1 !1
1 !1 !1 1
D, x yz, } xz, y xy, z
HOL
HKO
HOO
L L L L
H H H H
H even never H odd never
even odd even odd
even odd odd even
Table 3 Selection rules for P2 2 2 structure
A B B B
E
2 X
2 W
2 V
1 1 1 1
1 1 !1 !1
1 !1 1 !1
1 !1 !1 1
D, } xy, z xz, y yz, x
positions in the unit cell, correspondent (factor) point group and the table of characters of the operations +R"t, for the irrep's of the point group. The irrep's of all zone center phonon modes in all space groups of BCCD are one-dimensional. The condition for systematic absence of the one-phonon scattering at the Brillouin zone center Q is thus given by a single (scalar) equation [1] s(+R"t,) exp iQ ) t"0,
(1)
+R t,
where, s(+R"t,) is the character for the chosen irrep and the sum goes over all symmetry operations of the factor group such that RQ"R. As an illustration, let us take a B mode in the crystal of space group Pn2 a. Providing that the scattering vec tor is (for example) in the plane p , the summation in Eq. V (1) should be taken over two operations: the identity operation and the p (n) glide plane. Eq. (1) then reduces V to 1#exp(ipK#ip¸))"0, i.e. B is extinct at O K ¸ zone centers with K#¸ odd. This procedure can be
HOO
OKO
OOL
H H H H
K K K K
L L L L
even odd even odd
even odd odd even
even even odd odd
repeated for all possible special scattering geometries and for all irrep's. Selection rules for the space groups of C phases of BCCD are reviewed in Tables 1}3, those for the Pnma structure was given in Ref. [1]. Results provided here show that the selection rules for neutron scattering may be in some cases as selective as those for light spectroscopy. For example, any phonon mode observed in 0 3 0 Brillouin zone of Pn2 a crystal can be directly assigned as B mode. On the other hand, there is no Brillouin zone where one observes the B modes exclusively (for this space group) and compari son with results of di!erent Brillouin zones is necessary.
Acknowledgements Presentation of this work was supported by the ECNS '99 organizers and the work itself by the Grant Agency of the Academy of Science of the Czech Republic (Project No A1010828) and by the Grant Agency of the Czech Republic (Associated project 202/99/D066).
J. Hlinka / Physica B 276}278 (2000) 305}307
References [1] J.M. Perez-Mato, M. Aroyo, J. Hlinka, M. Quilichini, R. Currat, Phys. Rev. Lett. 81 (1998) 2462.
307
[2] S. Devine, G. Peckham, J. Phys. C 4 (1971) 1091. [3] R.J. Elliot, M.F. Thorpe, Proc. Phys. Soc. London 91 (1967) 903. [4] G. Shaack, M. Le Maire, Ferroelectrics 208 (1998) 1.
Phonon symmetry selection rules for coherent inelastic neutron scattering: application to BCCD J. Hlinka * Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Praha 8, Czech Republic
Abstract Recently demonstrated theorem for searching of the symmetry selection rules for coherent inelastic neutron scattering by phonons, independent of particular structural features, is applied to betaine calcium chloride dihydrate (BCCD). Selection rules for the zone center phonon scattering in crystalline phases of BCCD are given explicitly. 2000 Elsevier Science B.V. All rights reserved. Keywords: Phonons; Lattice dynamics; Structural transitions
1. Introduction Polarized Raman or IR spectroscopic data on phonons in crystals are currently presented with their symmetry assignment } each phonon frequency is related to the small irreducible representation (irrep) describing the symmetry properties of the observed mode. Such assignment is directly derived from the crystal symmetry and the geometry of the experiment. Contrary to the widespread belief, the coherent inelastic neutron scattering (INS) phonon structure factor also obeys systematic selection rules, which follows merely from the space group symmetry of the crystal, irrep of the phonon mode and the scattering geometry (total transferred momentum). These systematic rules can be used for planning neutron scattering experiments (for example, to prevent overlapping of spectral responses from modes of di!erent irrep's) or for symmetry assignment of the detected phonon resonances. The systematic approach to the selection rules presented in Ref. [1] purposely do not take into account actual atomic positions. This makes the method particularly simple and general. Its convenience would be most ap-
preciated when dealing with crystals with many atoms in unit cell or when structure model is not available. On the other hand, some additional (&accidental') selection rules induced by special positions of the atoms may exist in some simple highly symmetrical structures [1]. For such crystals, the approaches based on both the symmetry and structural details (e.g. Ref. [2]) may provide additional conditions. In this paper, we derive the systematic selection rules for the case of zone center phonon modes in the crystal of betaine calcium chloride dihydrate (BCCD) [3]. It undertakes a number of structural phase transitions under lowering temperature. Among them the high-temperature Pnma and the low-temperature Pn2 a ferroelectric phases have 4 BCCD molecules in the unit cell; the intermediate phases are incommensurate (IC) and commensurate (C) modulated phases with wave vector q ""cH. Majority of the experiments con"rms that the space groups of the observed C phases are given as follows [4]: q /cH"odd/oddNP2 2 2 , q /cH"even/oddNPn2 a and q /cH"odd/evenNP2 ca. 2. Selection rules for BCCD
* Corresponding author. Fax: #420-2-821227. E-mail address: [email protected] (J. Hlinka)
In order to obtain the desired selection rules, it is su$cient to know the crystal space group, set of symmetry operations +R"t, generating the equivalent
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 4 9 5 - 7
306
J. Hlinka / Physica B 276}278 (2000) 305}307
Table 1 Selection rules for Pn2 a structure. Note: First six columns show characters of irreps and selection rules for Raman and IR spectroscopy, last three columns provide the special Brillouin zone centers for which the phonon INS scattering is allowed by the systematic selection rules of Eq. (1)
A A B B
E
2
n
a
1 1 1 1
1 1 !1 !1
1 !1 1 !1
1 !1 !1 1
OKL D, y xz, } yz, z xy, x
K#L K#L K#L K#L
even odd even odd
HKO
OKO
H H H H
K even never never K odd
even odd odd even
Table 2 Selection rules for P2 ca structure
A A B B
E
2
c
a
1 1 1 1
1 1 !1 !1
1 !1 1 !1
1 !1 !1 1
D, x yz, } xz, y xy, z
HOL
HKO
HOO
L L L L
H H H H
H even never H odd never
even odd even odd
even odd odd even
Table 3 Selection rules for P2 2 2 structure
A B B B
E
2 X
2 W
2 V
1 1 1 1
1 1 !1 !1
1 !1 1 !1
1 !1 !1 1
D, } xy, z xz, y yz, x
positions in the unit cell, correspondent (factor) point group and the table of characters of the operations +R"t, for the irrep's of the point group. The irrep's of all zone center phonon modes in all space groups of BCCD are one-dimensional. The condition for systematic absence of the one-phonon scattering at the Brillouin zone center Q is thus given by a single (scalar) equation [1] s(+R"t,) exp iQ ) t"0,
(1)
+R t,
where, s(+R"t,) is the character for the chosen irrep and the sum goes over all symmetry operations of the factor group such that RQ"R. As an illustration, let us take a B mode in the crystal of space group Pn2 a. Providing that the scattering vec tor is (for example) in the plane p , the summation in Eq. V (1) should be taken over two operations: the identity operation and the p (n) glide plane. Eq. (1) then reduces V to 1#exp(ipK#ip¸))"0, i.e. B is extinct at O K ¸ zone centers with K#¸ odd. This procedure can be
HOO
OKO
OOL
H H H H
K K K K
L L L L
even odd even odd
even odd odd even
even even odd odd
repeated for all possible special scattering geometries and for all irrep's. Selection rules for the space groups of C phases of BCCD are reviewed in Tables 1}3, those for the Pnma structure was given in Ref. [1]. Results provided here show that the selection rules for neutron scattering may be in some cases as selective as those for light spectroscopy. For example, any phonon mode observed in 0 3 0 Brillouin zone of Pn2 a crystal can be directly assigned as B mode. On the other hand, there is no Brillouin zone where one observes the B modes exclusively (for this space group) and compari son with results of di!erent Brillouin zones is necessary.
Acknowledgements Presentation of this work was supported by the ECNS '99 organizers and the work itself by the Grant Agency of the Academy of Science of the Czech Republic (Project No A1010828) and by the Grant Agency of the Czech Republic (Associated project 202/99/D066).
J. Hlinka / Physica B 276}278 (2000) 305}307
References [1] J.M. Perez-Mato, M. Aroyo, J. Hlinka, M. Quilichini, R. Currat, Phys. Rev. Lett. 81 (1998) 2462.
307
[2] S. Devine, G. Peckham, J. Phys. C 4 (1971) 1091. [3] R.J. Elliot, M.F. Thorpe, Proc. Phys. Soc. London 91 (1967) 903. [4] G. Shaack, M. Le Maire, Ferroelectrics 208 (1998) 1.