International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 2.1, pp. 311-312

Section 2.1.5. Glossary

G. Eckolda*

aInstitut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany
Correspondence e-mail: geckold@gwdg.de

2.1.5. Glossary

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[{\bf a}^*], [{\bf b}^*], [{\bf c}^*] reciprocal-lattice vectors
A Helmholtz free energy
[{\bf A}] element of the coset [{\bf S}_ - \circ G_o ({\bf q})]
[{\bf C}({\bf q})= (C_{\kappa \kappa '}^{\alpha \beta }({\bf q}) )] modified dynamical matrix
[c_{ij}] elastic stiffness in Voigt notation
[(c_{jklm})] tensor of elastic stiffnesses
[c_p] lattice heat capacity at constant pressure
[\tilde c_{{\bf q},j} ] contribution of phonon state ([{\bf q},j]) to the heat capacity at constant volume
[c_s] multiplicity of irreducible representation s
[c_V] lattice heat capacity at constant volume
[c_V^{\rm Debye} ] lattice heat capacity at constant volume according to the Debye model
[c_V^{\rm Einstein} ] lattice heat capacity at constant volume according to the Einstein model
[{\bf D}({\bf q}) = ({D_{\kappa \kappa '}^{\alpha \beta }({\bf q})}) ] dynamical matrix
[\overline {\bf D}^{(s)}({\bf q}) ] submatrix of the block-diagonalized dynamical matrix corresponding to irreducible multiplier representation σ
[{\bf D}_{\kappa\kappa'}({\bf q})] [3\times 3] submatrix of the dynamical matrix
[{\bf D}_{\alpha}^i ] matrix of rotation about axis i by the angle α
[{\bf e}_{\kappa}({\bf q},j)] polarization vector of atom κ corresponding to the phonon [({\bf q},j) ]
[{\bf e}({\bf q},j)] eigenvector of the dynamical matrix corresponding to the phonon [({\bf q},j) ]
[{E}] identity
[{\bf E}({\bf q},sa\lambda)][\quad=(E_\kappa ^\alpha ({\bf q},sa\lambda) )] matrix of symmetry coordinates
[E_o] zero-point energy
[E_{\rm ph}] lattice energy
[E_{{\bf q},j}] contribution of the phonon [({\bf q},j)] to the energy of the lattice
[f_o(\kappa, S)] atom transformation table
[f_{\sigma}] degeneracy of the eigenfrequency [\omega_{{\bf q}, \sigma}]
[{\bf F}({\bf q})=({\bf F}_{\kappa,\kappa'}({\bf q}))] Fourier-transformed force-constant matrix
g reciprocal-lattice vector
[G({\bf q})] space group of the wavevector q
[G_o({\bf q})] point group of the wavevector q
[G_o({\bf q},-{\bf q})] augmented point group of the wavevector q
[|G|] order of group G
[G(\omega)] density of phonon states
[G^{\rm Debye}(\omega)] density of phonon states according to the Debye model
[G^{\rm Einstein}(\omega)] density of phonon states according to the Einstein model
H Hamiltonian
[\hbar ] Planck constant (1.0546 × 10−34 J s)
I inversion
k Boltzmann constant (1.381 × 10−23 J K−1)
[{\bf K}_o] anti-unitary operator
M mass tensor
[m_i] mirror plane perpendicular to axis i
[m_{\kappa}] mass of atom κ
[n_{{\bf q},j}] Bose factor corresponding to the phonon state ([{\bf q},j])
N number of atoms within the primitive cell
[N_Z] number of primitive cells
p pressure
[{\bf p}_{\kappa l}] momentum of atom [(\kappa l)]
[p_n] occupation probability of quantum state n
[{\bf P}^{(s)}({\bf q}) = (P_{\lambda\lambda'}^{(s)}({\bf q}) )] projection operator
q phonon wavevector
qBZ wavevector on the Brillouin-zone boundary
[Q_{{\bf q},j}] normal coordinate corresponding to the phonon [({\bf q},j)]
[{\bf r}_l] vector to the origin of the lth primitive cell
[{\bf r}_{\kappa l}(t)] time-dependent position vector of atom [(\kappa l)]
[{\bf r}_\kappa ^o ] equilibrium position of atom [\kappa] with respect to the origin of the primitive cell
[{\bf r}_{\kappa l}^o ] equilibrium position of atom [\kappa] within the lth primitive cell
R element of the point group of the wavevector [G_o({\bf q})]
[\bar{\bf R}] element of [G_o({\bf q}, -{\bf q})]
[\{{{\bf S}| {{\bf v}({\bf S}) + {{\bf x}(m)}\}}} ] symmetry operation (Seitz notation)
[{\bf S}=(S_{\alpha\beta})] matrix of rotation
[{\bf S}_-] space-group element that inverts the wavevector
t time
T temperature
[{\bf T}({\bf q},{\bf R})][\quad= (T_{{\rm K}\kappa }^{\alpha \mu }({\bf q},{\bf R})) ] matrix operator associated with a symmetry operation r of the point group of the wavevector q
[{\bf u}^o] polarization vector for elastic waves
[{\bf u}_{\kappa l}(t)] displacement vector of atom [(\kappa l)]
V potential energy
V volume
[{\bf V}(\kappa l,\kappa' l')][\quad=(V_{\alpha\beta}(\kappa l,\kappa' l'))] matrix of force constants acting between atoms [(\kappa l)] and [(\kappa' l')]
[{v}_s] sound velocity
[{\bf v}({\bf S})] fractional translation associated with symmetry operation [{\bf S}]
[{\bf x}(m)] lattice translation
Z partition function
[{\boldalpha}=(\alpha_{kl})] tensor of thermal expansion
β coefficient of volume expansion
γ mean Grüneisen parameter
[\gamma_{{\bf q},j}] averaged-mode Grüneisen parameter
[\gamma_{{\bf q}j,kl}] generalized-mode Grüneisen parameters
[\boldGamma=(\Gamma_{jl})] propagation tensor
[\boldGamma =(\Gamma_{{\rm K}\kappa }^{\alpha \mu }({\bf q},\{{\bf S}| {{\bf v}({\bf S})}][\quad +\, {\bf x}(m )\})) ] transformation matrix
[\delta_{kl}] Kronecker delta
[\delta(\omega)] Dirac delta function
[\boldDelta({\bf q},{\bf R})] block-diagonal matrix of irreducible representations
[\boldvarepsilon=(\varepsilon_{kl})] strain tensor
χ character of a representation
[\varphi({\bf q}, {\bf r}_i,{\bf r}_j)] multiplier associated with two symmetry operations [{\bf r}_i] and [{\bf r}_j] of the point group of the wavevector q
[\Phi] potential energy
[\kappa] isothermal compressibility
[\Theta_{D}] Debye temperature
[\Theta_{E}] Einstein temperature
[\rho] density
[\boldsigma=(\sigma_{kl})] stress tensor
[\boldtau ^{(s )}({{\bf q},{\bf R}})] irreducible representation
[\quad= ({\tau _{\lambda \lambda '}^{(s)}({{\bf q},{\bf R}})}) ]  
[\overline {\boldtau ^{(s)}}({{\bf q},{\bf R}}) ] conjugated representation
[\boldtau_v] vector representation
[\boldtau_T] tensor representation
[\omega_D] Debye frequency
[\omega_E] Einstein frequency
[\omega_{{\bf q},j}] frequency of phonon [({\bf q},j)]
[\boldPsi] arbitrary vector
* denotes the complex-conjugate quantity
+ denotes the Hermitian conjugate matrix
T denotes the transposed matrix

References

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Giannozzi, P., de Gironcoli, S., Pavone, P. & Baroni, S. (1991). Ab initio calculation of phonon dispersions in semiconductors. Phys. Rev. B, 43, 7231–7242.
Natkaniec, I., Bokhenkov, E. L., Dorner, B., Kalus, J., Mackenzie, G. A., Pawley, G. S., Schmelzer, U. & Sheka, E. F. (1980). Phonon dispersion in d8-naphthalene crystal at 6 K. J. Phys. C, 13, 4265–4283.
Pintschovius, L., Pyka, N., Reichardt, W., Rumiantsev, A. Yu., Mitrofanov, N. L., Ivanov, A. S., Collin, G. & Bourges, P. (1991). Lattice dynamical studies of HTSC materials. Physica C, 185–189, 156–161.
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Trampenau, J., Heiming, A., Petry, W., Alba, M., Herzig, C., Miekeley, W. & Schober, H. R. (1991). Phonon dispersion of the bcc phase of group-IV metals. III. Bcc hafnium. Phys. Rev. B, 43, 10963–10969.








































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