Lattice dynamics of a monoclinic CsH 2 PO 4 crystal

Caesium dihydrogen phosphate CsH2PO4 (CDP) belongs to a group of monoclinic phosphates of the KH2PO4-type, like KD2PO4 (DKDP), RbD2PO4 (DRDP), TlH2PO4 (TDP), RbHPO4 (LHP). Owing to their interesting properties, the CDP crystal and its deuterated analogue CsD2PO4 (DCDP) have been intensively studied by X-ray diffraction [11, 21, 34], neutron scattering [9, 12, 22, 33], dielectric [5,7, 31] acoustic [1, 14, 24], Raman scattering [8, 13, 20, 38], hyper-Raman scattering [28], IR absorption [30] and optic [35] measurements. However, a lattice dynamics of the CDP crystal in the region of phase transitions (PTs) and a triple point have been investigated insufficiently. In this paper a theoretical treatment of the lattice dynamics of CDP within the framework of a rigid molecular-ion model in the quasi-harmonic approximation

In this paper a theoretical treatment of the lattice dynamics of CDP within the framework of a rigid molecular-ion model in the quasi-harmonic approximation in paraelectric (PE) phase under the influence of hydrostatic pressure and temperature is presented.We restrict ourselves to the consideration of low-frequency external phonon modes which gives the most complete information about the nature of PTs.Moreover, in monoclinic phosphates of the CDP-type the PTs are often accompanied by unit cell multiplication (e.g., in DRDP [10,29] and TDP [2,26] crystals).Therefore, the use of a CDP crystal as a model object for the investigation of structural PTs seems to the quite reasonable.

Structure and symmetry
In a high-temperature PE phase the CDP crystal has a space group P 2 1 /m with two molecules per unit cell (a = 7.906A, b = 6.372A, c = 4.883A, β = 107.73o [21] ).The CDP crystal structure is depicted in figure 1 (according to [9]).The peculiarity of this structure is the presence of two types of hydrogen bonds O-H. ..O.The shorter bonds (R = 2.47 Å) link the PO 4 groups into zig-zag-like chains running along the b axis.In the PE phase the protons are disordered on those bonds.The longer bonds (R = 2.54 Å), which are directed approximately along the c axis, cross-link the chains to form (b, c) layers.The protons on the longer bonds are ordered on one of the two possible equilibrium sites at all temperatures.By means of the dielectric [32] and neutron [25] investigation, it was shown that at some values of hydrostatic pressure and temperature, the PT into the antifer-  [25] allowed the authors to identify the symmetry of the AFE phase both with the P 2 1 space group and with the P 2 1 /a one.Due to the Raman spectra studies of the DCDP crystal at T = 83 K and P = 800 MPa [20] a conclusion was drawn that the symmetry of the AFE phase is P 2 1 /a.Besides, as it follows from the general symmetry consideration (Curie principle [3]), if there is an inversion centre in the PE phase, it must persist in the AFE phase, too.We shall assume that in the AFE phase the P 2 1 /a symmetry is realized.For the lattice dynamics consideration, an orthonormal set of axes X, Y and Z was used.In the PE phase, the principal parameters of the monoclinic lattice mP with respect to the orthonormal axes are the following: where τ = c; τ y , τ z are the corresponding principal parameters along the Y and Z axes; α = 17.73 o .At the same time, the reciprocal lattice parameters can be written as b 3 = (2π/(τ cos α), 0, 0).

Group-theoretical classification of lattice vibrations
A general group theory analysis of the normal modes of a CDP crystal in the PE phase for 14 wavevectors q which correspond to all the symmetrical points and lines of the Brillouin zone (BZ) of the monoclinic lattice mP was made.A symmetrized form of the dynamical matrix D(q), the matrices of irreducible multiplier representations (IMRs) T (q, h) (here h is an element of the wavevector point group G 0 (q)) and the symmetry vectors E(q) for wavevectors q i (i = 1, 14) were obtained.However, in this work we present the results for the most interesting from the physical point of view cases (q 7 , q 1 , q 12 , q 13 , q 3 , q 11 ) which will be used in numerical calculations.Kovalev's notation [15] was used for the identification of wavevectors, elements of the wavevector point groups G 0 (q), symmetrical points and lines of the BZ.
We carried out the lattice dynamics study of a CDP crystal within the rigid molecular-ion approximation [36] where the H 2 PO 4 groups are treated as rigid and nondeformable.Hence, there are two Cs + ions (k = 1, 2) and two (H 2 PO 4 ) − ions (k = 3, 4) in the unit cell of a CDP crystal.In this case, there are 18 external branches in the phonon spectrum of a CDP crystal.
The symmetry vectors in the centre of the BZ (q 7 = 0) corresponding to the translational displacements and librations of the structural units of the crystal are given in table 1.They are used for diagonalization of the dynamical matrix and phonon frequencies classification according to IMRs.The point groups of the wavevectors (q 7 , q 1 , q 12 , q 13 , q 3 , q 11 ) and normal modes classification by the IMRs of these groups are shown in table 2. The IMRs of the above mentioned groups are given in table 3.

Model
The crystal potential energy Φ in the rigid molecular-ion approximation is written as where a = 1822 eV, b = 12.364; l, l ′ are unit cell indices; K, K ′ are the indices of atomic or molecular vibrating units in the unit cell; k, k ′ are the indices of atoms in molecular units; k ⊂ K means that atom k belongs to molecule K; e is an electron charge; Z Kk and R Kk are effective charge and radii parameters of Kk atom, respectively; r is a distance between Kk and K ′ k ′ atoms.The first and second terms of expression (3) represent the long-range Coulomb and the shortrange Born-Mayer-type repulsive energy, respectively.Moreover, we assumed that the total potential energy is a sum of the pair interactions of the atoms of different molecules.Interactions between the atoms within a molecule have been neglected.The unknown parameters Z Kk and R Kk are determined from the lattice equilibrium conditions with respect to any macroscopic internal strains [4] and also Table 1.Symmetry vectors for CsH 2 PO 4 in the PE phase in the centre of the BZ (q 7 = 0).Designations x, y and z refer to the translations along the X, Y and Z axes; x, ỹ and z refer to the librations about X, Y and Z axes.

Representation
Symmetry vectors Point group Elements of Wavevector G 0 (q) of G 0 (q) Classification wavevector Table 3. Irreducible multiplier representations of some point groups G 0 (q) of wavevectors for the P 2 1 /m space group.
G 0 (q 7 ), G 0 (q 12 ), G 0 (q 13 ) with respect to the condition of molecule electroneutrality.Thus, the constraints for the crystal potential are where U i α (lK) is a displacement of molecule K in the unit cell l; α is the Cartesian components X, Y and Z; i refers to the translation (t) or rotation (r); S αβ is a macroscopic strain.
At the same time, we suppose that a priori the quantum particle proton located on the hydrogen bond O-H. ..O cannot be taken into account directly in the quasi-harmonic approximation.Therefore, the influence of the protons is taken into consideration indirectly by means of unequal values of the effective charges Z Kk and radii R Kk of oxygen ions in H 2 PO 4 groups.With the help of ab initio calculations within the extended Huckel method it was shown [27] that the charges and radii of oxygen ions in the H 2 PO 4 group depend on proton localization on the O-H. ..O bond in one of the minima of a double-well type potential in which the proton moves.
Proceeding from the aforesaid, we have used the following values of the model parameters Therefore, in reality the lattice dynamics of a crystal which consists of the Cs + and (PO 4 ) − ions with indirect consideration of the proton influence is studied.

Phonon dispersion relations in CDP
The lattice dynamics calculation of the CDP crystal is carried out by means of the program DISPR [6] which has been modified by us in order to use the grouptheory information to a greater extent.Figure 3 shows the calculated phonon dispersion relations along the directions b 1 , b 3 (q . The compatibility relations between the IMRs along these directions are presented in table 4. The accuracy of the calculation was controlled at different stages by the correspondence of the calculated dynamical matrix D(q) to the one determined theoretically by means of general symmetry requirements [18].As one can see from figure 3, at the boundary of the BZ at point q 11 = 1 2 b 2 , the phonon modes become two-fold degenerate.This is a consequence of two-dimensionallity of the irreducible multiplier representation E of the group G 0 (q 11 ).Note, that the time-reverse symmetry does not require any additional degeneracy for all the considered cases (q 7 , q 1 , q 12 , q 13 , q 3 , q 11 ).
The disagreement in frequencies while approaching the centre of the BZ from different directions can be explained by LO -TO splitting.Only polar modes which transform according to the IMRs A u and B u possess LO -TO splitting.It follows from the eigenvectors analysis that the vibrations propagating along the b 1 and b 3 directions are purely transverse (A u symmetry) or quasi-transverse (B u symmetry).At the same time, the modes propagating along the b 2 direction are purely longitudinal (A u symmetry) or purely transverse (B u symmetry).Therefore, at b 2 → 0, the modes of A u symmetry have larger frequencies as compared with those when b 1 → 0 or b 3 → 0 (ω LO > ω TO ).On the contrary, the frequencies of the B u symmetry at b 2 → 0 are smaller than the frequencies at b 1 → 0 and b 3 → 0, because the modes along the b 1 and b 3 directions are quasi-transverse and have some longitudinal components.acoustic acoustic acoustic acoustic 97, QTO * , q = (0.001, 0, 0) 76 96, TO, q = (0, 0.001, 0) 106, QTO, q = (0, 0, 0.001) 122, QTO, q = (0.001, 0, 0) 106 108, TO, q = (0, 0.001, 0) 112, QLO, q = (0, 0, 0.001) 149, QTO, q = (0.001, 0, 0) 146 123, TO, q = (0, 0.001, 0) 150, QLO, q = (0, 0, 0.001) * QTO, quasi-transverse optical mode OLO, quasi-longitudinal optical mode Comparison of the calculated phonon frequencies in the centre of the BZ (q 7 = 0) with those obtained from Raman and IR investigations [19] is presented in table 5.For most of the calculated frequencies, a good agreement with the corresponding experimental results is obtained.The calculation of the external phonon dispersion relation along the b 1 direction attracts special interest since the PT into the AFE phase occurs in a CDP crystal at some values of temperature and hydrostatic pressure.Usually, such a PT is related to the external mode condensation at the BZ boundary at point q 13 = 1 2 b 1 .With the help of group theory consideration [17], one can show that the IMR A u is responsible for the AFE PT in a CDP crystal with the symmetry change P 2 1 /m → P 2 1 /α.The same IMR A u is responsible for the PT into the FE phase (space group P 2 1 ).This PT is caused by the external mode condensation in the centre of the BZ.The phonon spectra calculation of a CDP crystal at different values of temperature and hydrostatic pressure was carried out.Moreover, we remained within the framework of the quasi-harmonic approximation at specific values of T and P .The influence of interatomic anharmonicity on lattice dynamics is taken into consideration indirectly through the change of the lattice parameters a, b and c, which were determined from the experimental data of thermal expansion and ultrasonic measurements.We assumed that the temperature and hydrostatic pressure influence the lattice parameters only (without a change of fractional atomic coordinates in a unit cell) according to the next linear laws: where K a , K b and K c are linear compressibility components along the a, b and c axes, respectively, at the applied hydrostatic pressure P ; a T , b T and c T are the lattice parameters at temperature T .The linear compressibility K lmn of a monoclinic crystal with the P 2 1 /m symmetry along the given direction [l, m, n] in the Cartesian coordinate system X, Y and Z is written as [23] K lmn = (S 11 + S 12 + S 13 )l 2 + (S 12 + S 22 + S 23 )m 2 (6) + (S 13 + S 23 + S 33 ) n 2 + (S 16 + S 26 + S 36 )lm, here S ij are components of the elastic compliance matrix which is inverse to the elastic constant matrix C ij .The values of S ij components for the CDP crystal, determined at room temperature by the ultrasonic waves velocities measurements [24], are presented in table 6.
Thus, the components of the CDP crystal linear compressibility along the crystallographic a, b and c axes have the following values: K a = 0.022, K b = −0.260and K c = 0.390.At the same time, we assume that the linear compressibility is independent of both the temperature and hydrostatic pressure.
To determine the thermal dependence of the lattice parameters a, b and c, the dilatometric investigations of monodomain specimen have been carried out.The linear thermal expansion (∆1/1) observed along the a * , b and c directions as a function of temperature, where a * ⊥ (b, c), is shown in figure 4. As one can see, ∆1/1 along the b and c axes are essentially larger than ∆1/1 along a * direction.This can indicate a quasi-layer nature of the CDP crystal along the (b, c) plane.In other words, interactions between the ions within the same layer are to a great extent larger than interactions between the ions from different layers.The perfect cleavage that occurs along the (b, c) plane confirms a relative weakness of interlayer forces.Therewith, the linear thermal expansion along the c axis increases with the temperature decrease, i.e. the CDP crystal expands along this axis at cooling.The thermal expansion coefficient a along c axis becomes negative.This unusual behaviour of the coefficient a is considered in detail in [37].Using the data of thermal expansion (figure 4) and linear compressibility (table 6) of a CDP crystal, with the help of expression (5) it is easy to obtain the lattice parameters at various values of T and P .The calculated dispersion relations for external phonon modes of the B symmetry (G 0 q 1 ) along the b 1 direction at different values of T and P are presented in figure 5.As one can see, at increasing the hydrostatic pressure and decreasing the temperature, a lowering of most of the phonon branches is observed.At T = 130 K and P = 241 MPa, the lower optic phonon branch of the A u symmetry falls to zero in the centre of the BZ (at room temperature and atmospheric pressure the value of this optic mode is 39 cm −1 .This implies that in a CDP crystal the PT into the PE phase must occur.

Phonon dispersion relations in a CDP crystal at different values of temperature and hydrostatic pressure
At the same time there occur such displacements of structural units of the crystal to new equilibrium sites which define the structure of the PE phase.These displacements, determined from the analysis of the eigenvector corresponding to the soft phonon mode A u , are schematically presented in figure 6.As follows from this figure, the Cs + and (PO 4 ) − ions shift along the b axis in opposite directions with simultaneous PO 4 group rotation in the ac-plane.The calculated displacements of structural units both translational and rotational are in good qualitative agreement, with neutron scattering data [12].For other values of T , there is no good coincidence between the experimental and the calculated values of P at which the PT into the FE phase occurs.It should be noted that the calculated frequencies of external phonon modes are more sensitive to the influence of hydrostatic pressure on the lattice parameters than to the temperature influence.So, at T = 121 K and P = 286 MPa (an experimental value of hydrostatic pressure obtained from the P − T phase diagram for T = 121 K is P = 370 MPa), the lowering to zero of the same lower optic phonon branch of the A u symmetry, active at PT into the FE phase, takes place at the BZ boundary at point q 13 = 1 2 b 1 (figure 5).In other words, the PT into the AFE phase, accompanied by the unit cell doubling along the a axis, must occur in a CDP crystal at some values of T and P .Here the displacements of the crystal structural units to new equilibrium sites of the AFE phase are similar to those presented in figure 6, only they are opposite in the neighbouring cells.In figure 7

Conclusions
In this paper we have reported the results of lattice-dynamical calculations of a CDP crystal based on the rigid molecular-ion model which includes the Coulomb and short-range interactions.The protons on hydrogen bonds in the quasi-harmonic approximation have not been taken into account immediately but indirectly by means of choice of the unequal values of effective charges and radii of oxygen ions in H 2 PO 4 groups.This approach provides a reasonable explanation of the observed Raman and IR data for a CDP crystal in the external modes region where the effects of internal vibrations are not expected to be felt.In the case when anharmonic particles of the protons were taken into consideration immediately as atoms (e.g., Cs, P, or O) there was a significant aggravation of lattice equilibrium conditions and appearance of unphysical results (the phonon frequencies with imaginary values).
The lattice dynamics calculation of a CDP crystal at different values of temperature and hydrostatic pressure was carried out.Thereat, we supposed that in the quasi-harmonic approximation the influence of T and P would be displayed only through the change of the lattice parameters a, b and c.For this purpose, dilatometric investigations of a CDP crystal were performed.Using the ultrasonic waves velocities measurements [24], the values of linear compressibility along the a, b and c axes were obtained.This enabled us to calculate the phonon spectra of CDP at various values of T and P .At decreasing temperature and increasing hydrostatic pressure the lowering of most of the phonon branches was observed.At T = 130 K and P = 241 MPa, the lower optic phonon branch of the A u symmetry falls to zero in the centre of the BZ (q 7 = 0).At T = 121 K and P = 286 MPa, the same lower optic phonon branch tends to zero at the boundary of the BZ at point q 13 = 1 2 b 1 .In other words, the obtained results qualitatively show that the PT in a CDP crystal takes place either into the FE phase (condensation of the A u mode in the BZ centre) or into the AFE phase (condensation of the A u mode at the BZ boundary) depending on a correlation of interatomic forces which depend on the temperature and hydrostatic pressure.It confirms once more an extremely important role of a proton subsystem in phase transitions in the compounds of this type.

Figure 1 .
Figure 1.The P-T phase diagram of a CsH 2 PO 4 crystal in the PE phase (according to [20]).At T c = 156 K and atmospheric pressure, a CDP crystal undergoes the PT into the PE phase with the space group P 2 1 and the spontaneous polarization P s b (Z = 2, a = 7.87 Å, b = 6.32 Å, c = 4.89 Å, β = 108.3o at T = 83 K [12]).In the PE phase a proton ordering on the shorter hydrogen bonds occurs.By means of the dielectric[32] and neutron[25] investigation, it was shown that at some values of hydrostatic pressure and temperature, the PT into the antifer-

Figure 2 .
Figure 2. The structure of a CsH 2 PO 4 crystal in the PE phase (according to [9]).

Figure 3 .
Figure 3.The phonon dispersion relations in a CsH 2 PO 4 crystal in the PE phase at room temperature and atmospheric pressure along the following directions: a) b 1 , b) b 2 , c) b 3 .
because the protons on the bonds O 1 -H. ..O 2 are localized near the oxygen ions O 1 at all temperatures.The charges Z(O 3 ), Z(O 4 ) and radii R(O 3 ), R(O 4 ) are equal, respectively, because the protons on the bonds O 3 -H. ..O 4 tunnel between the two possible off-centre equivalent positions in the PE phase.

Table 4 .
Compatibility relation between the irreducible multiplier representations along the b 1 , b 2 and b 3 directions in CsH 2 PO 4 in the PE phase.

Figure 4 .
Figure 4.The linear thermal expansion along the orthonormal a * , b, and c axes of a CsH 2 PO 4 crystal.

Figure 5 .
Figure 5.The dispersion relations of phonon modes of the B symmetry along the b 1 direction in a CsH 2 PO 4 crystal at different values of temperature and hydrostatic pressure: a) T = 150 K, P = 200 MPa; b) T = 135 K, P = 230 MPa; c) T = 130 K, P = 241 MPa; d) T = 121 K, P = 286 MPa.

Figure 6 .
Figure 6.Schematic representation of ions displacements corresponding to the ferroactive A u soft phonon mode determined from the eigenvector analysis.

Figure 7 .
Figure 7.The dispersion relation of the A u soft phonon mode in the (b 1 , b 3 ) plane at a) T = 130 K and P = 241 MPa; b) T = 121 K and P = 286 MPa.

Table 2 .
Point groups of wavevectors and classification of normal modes for CsH 2

Table 5 .
Comparison of the experimental and theoretical values of external modes frequencies in the centre of the BZ in CsH 2 PO 4 .

Table 6 .
The components S ij of the elastic compliance matrix in CsH 2 PO 4 at T = 293 K (according to Prewer et al. (1985))