Effect Of Pressure On The Structure And Intermolecular Phonons In Solid

The low-temperature orientationaly ordered crystalline phase of fullerene C 60 was investigated in dependence on the external pressure. The vibrational spectrum of C 60 crystal was calculated using the group theory and atom-atom potentials methods. The frequencies of inter-molecular modes as the functions of external pressure were studied. An assumption was made that the energy of the lattice includes two contributions: a Lennard-Jones (12-6) potential and electrostatic interaction of charges located on the single and double bonds of C 60 molecule. The results obtained are in good agreement with the available experimental data.


Introduction
After the discovery of fullerene C 60 molecules 1] a simple method was developed to condense them to a solid phase 2].The appearance of superconductivity in alcali-doped fullerides 3] led to considerable e orts to understand the di erent physical properties of these compounds as well as pristine C 60 .
The investigations of fullerides are of great importance for extensive use of multi-atom carbon clusters in technics 4].
In the present paper the low-temperature orientationaly ordered crystalline phase of fullerene C 60 was studied using the atom-atom potentials method 5].By means of the group theory methods 6] a calculation of the vibrational spectrum of fullerene C 60 crystal was carried out in the approximation of an intermolecular potential which includes two contributions: a Lennard-Jones  potential and electrostatic interaction.The dependencies of C 60 crystal lattice parameter and its intermolecular frequencies on the external pressure were investigated.The study of molecular dynamics under the high pressure give us the essential information about the intermolecular interaction, which is necessary to improve the phonon spectrum calculation for C 60 crystal.
It is to be pointed out the great interest to the external vibrational spectra of solid fullerides is due to the following reasons.First of all, they are necessary for the construction of intermolecular potentials which deverens the molecular dynamics, elastic constants and thermodynamic properties of these systems [7][8][9][10][11][12][13][14][15][16][17][18].Second, the superconductivity models remain actual, in which the low-frequency phonons are considered [19][20][21].Besides, the 48 Yu.I.Prilutskii, V.A.Andreev, G.G.Shapovalov intermolecular vibrations are to be considered in the investigation of the dynamics of the triplet excited states in organic molecular crystals [22][23].
The sc structure of fullerene C 60 crystal in dependence on the external pressure was investigated in experiments 32,39-42].It was found that the solid C 60 is stable when the external pressure is below 100 kbar.In the region above about 100 kbar, the C 60 crystal transforms from a sc structure phase to a low-symmetry insulator phase 32, 39].
Let's write out the free energy of C 60 crystal which is under the action of uniform external pressure P at the temperature T F = 3DU(a; ') + E ?TS + PV; ( where U is the lattice energy (the potential energy of interaction between the molecules), E is the vibrational part of energy, S is an entropy, V = 3D a 3 4 is a volume of the unit cell of cubic crystal per a molecule.At low temperatures the vibrations and entropy factor have not much signi cance and the equilibrium crystal structure is determined by the minimum of free energy F : F = 3DU(a; ') + Pa 3 4 : (2) Within the framework of atom-atom potential method 5] the energy of interaction between the molecules is determined as a sum of pair interactions of atoms i and j which belong to the di erent molecules and 0 : U = 3D 1 2 X 0 U 0 (jr i ?r 0 j j): (3) Concerning the choice of the function U 0 the following should be men- tioned.An attempt to use the intermolecular Lennard-Jones (12-6) potential which is known for graphite has shown this potential describes the molecular dynamics of C 60 crystal insatisfactory 8].Therefore, a discrimination between single and double bonds was taken into account by locating both additional interaction sites at the centers of the double bonds and charges at these centers and carbon atoms.For the further improvement of this model the authors 16] have proposed to consider Coulomb potential with accounting of the error function.The authors 15] besides the Lennard-Jones potential have taken into consideration the noncentral interaction which can appear due to overlapping the orbitals of neighbouring molecules.To solve the dynamic problem for C 60 crystal the atom-atom Buckingham (6-exp) potential has been used too 14, 17 -18].
In the present paper the function U 0 includes two contributions 12]: an atom-atom Lennard-Jones (12-6) potential and electrostatic potential of charges located on the single bonds and double ones U 0 = 3D4" 60 X i;j=3D1 " jr i ?r 0 j j 12 ?jr i ?r 0 j j 6 # + 90 X m;n=3D1 qmqn jr m ?r 0 n j ; (4)   where r i and r m are the coordinates of ith carbon atom and mth bond center in molecule respectively, q m is an e ective charge of mth bond: it equals to q in the case of single bond and ?2q for the double one.
Minimizing the free energy (2) over parameters a and ' at a constant pressure was carried out in approximation of rigid molecules under the condition of remaining symmetry of the crystal.It was taken into account only twelve nearest neighbours of each C 60 molecule located within the radius sphere a= p 2. The contribution of more remote molecules is insigni cant.Our calculations showed that the following values of intermolecular potential parameters "=3D2.935meV, =3D3.47A and q = 3D0:27 e (e is an elementary charge) give an opportunity to obtain the solid C 60 structure close to experimental one in the absence of external pressure (P = 3D0) [24][25].
The gure 1 exhibits the dependence of the lattice energy U on the setting angle of molecules ' at P = 3D0.As we can see the minimum of the lattice energy equals to U min = 3D ?1911 meV.It is reached at ' = 3D23:8 o .This minimum is due to the contribution of a Lennard-Jones (12-6) potential on about 90 per cents (the curve 1 on gure 1).The result obtained is in good agreement with the available experimental data: ' = 3D26 o 24] and ' = 3D22 o 25] at the temperature T = 3D4:2 ? 5 K.
The gure 2 represents the dependence of the crystal lattice parameter on the external pressure.When the pressure is increased from 0 to 100 kbar the lattice constant decreases with a rate of 0.004-0.014A/kbar.It must be pointed out that at pressure P = 3D100 kbar the volume of C 60 crystal decreases on about 15 per cents in comparison with its initial one.

Study of the vibrational spectrum of fullerene C 60 crystal
Let's calculate the vibrational spectrum of solid C 60 using the obtained structure.In the harmonic approximation the frequencies and eigenvectors ! 2 e s = 3D X 0 s 0 D s 0 s 0 e 0 s 0 ; ( where e s is the sth component of displacement vector of the th molecule, D s 0 s 0 is a symmetric dynamic matrix which describes the elastic inter- action at translational and librational motions of the molecules and their coupling: Here the sum is carried out over all units of the C 60 crystal.M s means either mass of the C 60 molecule (M = 3D720 a.u.m) if index s corresponds to the translational motions or the inertia moments of this molecule (I x = 3DI y = 3D6:2 10 3 A 2 a.u.m., I z = 3D5:9 10 3 A 2 a.u.m., z k C 2 ) if index s corresponds to the librational motions.The values of the force constants b s 0 s 0 are calculated in the equilibrium state according to the  (7)  For the pure translational vibrations of the lattice of C 60 crystal its symmetry dynamics matrix of order 12 12 includes only ve independent components (the additional conditions on the dynamic matrix elements appear owing to the existence of threefold degenerated acoustic mode) 11]: d 1 + d 3 + d 6 + d 8 = 3D0; d 7 = 3D ?d 2 ; d 5 = 3Dd 4 : (8) The classi cation of intermolecular modes at k=3D0 can be found by a positional symmetry method 6] = 83 = 3DA g + E g + 3F g + A u + E u + 3F u ; (9) where the even (g) representations are related to the librational modes and odd (u) ones to the translational modes of C 60 molecule in a crystal.All even modes are Raman active, one of the F u modes is acoustic and only two modes F u are infrared active.
Finally, for undegenerated modes A g and A u the frequencies are equal to 11]: For threefold degenerated modes F g and F u the frequencies can be found from the following characteristic equation: where for the librational vibrations The obtained expressions ( 13) and ( 14) di er from the result 11] owing to multiple choice of the normal vibrations forms for the C 60 crysral 44].
They have more simple form in our case.The obtained above parameters for the potential (4) turned out to be insatisfactory for the calculation of intermolecular frequencies being compared with the experimental data 27].By means of potential simulation the following values were received: " = 3D0:059 meV and q = 3D0:03 e (the value of was remained constant).
The calculated intermolecular frequencies of C 60 crystal in the Brillouin zone center at T = 3D4:2 ?5 K and P = 3D0 are represented in table 1.It is seen that all translational modes are located higher then the librational ones.The values obtained for the frequencies are distinguished from the experimental data 26-29] up to 7 percents.
Figures 3 and 4 depict the dependencies of frequencies for translational modes and librational ones on the external pressure correspondingly.When the pressure is increased from 0 to 100 kbar the librational modes and translational ones shift toward higher frequencies with a rate of 0.35-0.65 cm ?1 /kbar and 0.77-1.38cm ?1 /kbar respectively.This results is in good agreement with the experimentally observed values for the librational modes 30]: the strongest libron mode A g shifts at a rate of 0.37 cm ?1 /kbar, while a weaker, higher energy libron mode F g shifts at a rate of 0.52 cm ?1 /kbar.

Figure 1 .
Figure 1.Dependence of the lattice energy on the setting angle of molecules calculated by the atom-atom potentials method at the temperature T = 3D4:2 ? 5 K and external pressure P = 3D0. of crystal vibrations can be found from the equation 5]

Figure 2 .
Figure 2. Calculated dependence of the lattice constant on the external pressure at the temperature T = 3D4:2?5 K. formulas 43].At the value of wave vector k = 3D0 the dynamic problem (5) is solved for translational and librational motions of the molecules in a crystal separately.The symmetry analysis of the dynamic matrix of solid C 60 11] has shown that for the pure librational vibrations of the lattice of C 60 crystal its symmetry dynamic matrix of order 12 12 consists of eight independent components only: d 1 = 3DD 1x1x ; d 2 = 3DD 1x1y ; d 3 = 3DD 1x2x ; d 4 = 3DD 1x2y ; d 5 = 3DD 1x2z ; d 6 = 3DD 1x3x ; d 7 = 3DD 1x3z ; d 8 = 3DD 1x4x :(7)  For the pure translational vibrations of the lattice of C 60 crystal its

Figure 3 .Figure 4 .
Figure 3. Calculated dependence of the translational frequencies on the external pressure at the temperature T = 3D4:2 ? 5 K.The numerical calculations carried out for the pure translational vibrations have shown that d 6 d 8 .Then we can found from equations (8) and

Table 1 .
Frequencies (in cm ?1 ) of intermolecular modes k = 3D0 calculated by the atom-atom potentials method at the temperature T = 3D4:2 ? 5 K and external pressure P = 3D0.