CALCULATION OF THE RAMAN FREQUENCIES AS

FUNCTIONS OF TEMPERATURE AND PRESSURE

IN PHASE I, II AND III(III’) OF BENZENE

H. Yurtseven* and Ö. Tarı

Benzene exhibits various solid phases, liquid, decomposed compounds and polymer phases, as given in the P-T phase diagrams [1-7].

Solid phase I is transformed into the solid phase II at 270oC.

As the pressure is increased further, solid benzene III occurs [2].

Among the phases of liquid, solid I and solid II triple point was observed [2] and a second triple point was also obtained at the temperature of 590oC 15oC and the pressure of 40kbar [3] , as given in the T-P phase diagram [5].

INTRODUCTION

Pressure-volume-temperature relations for reorientational crystalline benzene have been obtained experimentally from -20 to 51oC for pressures to 5kbar [9].

Isothermal variation of molar volume of benzene has been determined experimentally at various pressures [10].

Molar volume has been measured as a function of temperature [11, 12] and pressure and the Grüneisen parameters have been inferred in the different phases of benzene [12].

Thermodynamic properties of crystalline benzene have been investigated using various experimental techniques as reported in the literature.

Using the experimental phase diagram [5], we have constructed the phase line equations for all the phases of benzene by the mean field theory [7, 8].

The thermal expansivity has been measured at various pressures along isotherms in solid benzene and at melting [13].

Using the experimental data for the molar volume near the melting lines [9, 13], we have analyzed the molar volume as a function of temperature [14] and pressure [15] in our earlier studies.

The spectroscopic studies on the solid and liquid phases of benzene have also been reported in the literature.

Far infrared spectroscopy [16], X-ray [17], infrared [18] and Raman spectroscopy studies [5, 6, 12, 19-21] have been given previously.

Using the volume data [12] and the Raman frequency data [6], we have calculated tha Raman frequencies at various pressures through the mode Grüneisen parameter in phase II of solid benzene in our recent study [22].

1. Calculation of the Raman frequencies as a function

of temperature in phase I of benzene

This study gives our calculation of the Raman frequencies of the lattice modes A(A1g), B(AgB2g) and C(B1gB3g) as a function of temperature at constant pressures of 0, 1.4 and 3.05GPa for the solid phase I of benzene. The Raman frequencies of those lattice modes are calculated using the volume data from the literature through the mode Grüneisen parameters for the solid phase I of this molecular crystal.

Calculations and Results

Temperature (or pressure) dependence of the volume V and frequency can be correlated by the mode Grüneisen parameter defined as

If one takes the temperature dependence of V and at a constant pressure, the isobaric mode Grüneisen parameter P can be defined as

From this definition, the temperature dependence of the frequency can be obtained as follows:

ln

ln

d V

d

P P P

VT V T

1 1( ) ( ) exp ln ( ) /p P PT A P V T V

(1)

(2)

(3)

where the pressure-dependent term can be expressed as

with the constants a0, a1 and a2. In Eq. (3) 1 and V1 are the extrapolated values of the frequency and volume at a constant temperature at zero pressure ( P=0), respectively.

20 1 2( )A P a a P a P

We calculated here the Raman frequencies of the three lattice modes, namely, the A(A1g), B(AgB2g) and C(B1gB3g) as a function of temperature for constant pressures of 0, 1.4 and 3.05GPa in the solid phase I of benzene using the volume data [12] according to Eq. (3).

(4)

For this calculation, we first analyzed the temperature dependence of the observed volume at zero pressure (P=0 GPa) for the solid phase I of benzene according to the quadratic relation:

2V a bT cT

where a, b and c are constants. Values of those coefficients from our analysis are given in Table 1. The extrapolated values of the volume at various temperatures and pressures (Eq. 5) are given in Table 2. We also needed to analyze the pressure dependence of some observed [12] values of the Raman frequencies for the lattice modes (A, B and C), which were used as the initial data according to the relation

20 1 2b b P b P

where b0, b1 and b2 are constants.

(5)

(6)

Table 1. Values of the coefficients according to Eq. (4) for the phase I of benzene.

Table 2. Extrapolated values of the volume data [12] at various temperatures and pressures for the phase I of benzene (Eq. (5)).

a (cm3) b (cm3/K) c (cm3/K2)

69.38 -1.47x10-2 1.42x10-4

T(K) V(cm3) P(GPa) V(cm3)

274 76 0 77.3

294 77.3 1.4 66.0

300 77.7 3.05 61.8

Table 3. Values of the coefficients according to Eq. (5) for the Raman bands indicated. Values of the mode Grüneisen parameter (T, P) and the values of the Raman frequency 1 at P=0 (T=294K) for the lattice modes are also given here for the phase I of benzene.

Table 4. Values of the Raman frequencies of the lattice modes indicated for three pressures according to Eq. (6) for the phase I of benzene.

Raman bands

b0 (cm-1) b1 (cm-1/GPa) b2 (cm-1/GPa2) T 1 (cm-1)

A (A1g) 44.94 20.54 -2.48 3.0 44.94

B (AgB2g) 64.35 27.19 -3.50 2.5 64.35

C(B1gB3g) 105.45 49.49 -6.33 3.0 105.45

P(GPa)P(cm-1)

Mode A Mode B ModeC

0 44.94 64.35 105.45

1.4 68.84 95.56 162.33

3.05 84.52 114.72 197.51

Table 5. Values of the coefficients for the lattice modes indicated according to Eq. (4) for the phase I of benzene.

Raman bands a0 (cm-1) -a1 (cm-1/GPa) a2 (cm-1/GPa2)

A(A1g) -0.70 3.87 0.83

B(AgB2g) -0.84 0.92 0.46

C(B1gB3g) 1.65 11.21 2.20

Figure 1 The Raman frequency calculated as a function of temperature for the lattice mode A(A1g) for constant pressures indicated for the phase I of benzene according to Eq. (3).

Mode A

Temperature (K)

50 100 150 200 250 300

40

45

50

55

60

65

P=0GPa P=1.4GPaP=3.05GPa

(cm

-1)

Figure 2 The Raman frequency calculated as a function of temperature for the lattice mode B(AgB2g) for constant pressures indicated for the phase I of benzene according to Eq. (3).

Mode B

Temperature (K)

50 100 150 200 250 300

(cm

-1)

60

65

70

75

80

85

90

P=0GPaP=1.4GPa P=3.05GPa

Figure 3 The Raman frequency calculated as a function of temperature for the lattice mode C(B1gB3g) for constant pressures indicated for the phase I of benzene according to Eq. (3).

Mode C

Temperature (K)

50 100 150 200 250 300

(cm

-1)

90

100

110

120

130

140

150

160

P=0 GPaP=1.4GPa P=3.05GPa

Conclusions Raman frequencies of the lattice modes (A, B and C) were

calculated as a function of temperature at constant pressures for the solid I of benzene.

Volume data was used through the mode Grüneisen parameters for this calculation of the Raman frequencies.

It was found that the Raman frequencies of the three lattice modes decrease with increasing temperature, as expected.

The pressure dependence of the Raman frequencies of the lattice mode C at various temperature is somehow different from the other lattice modes B and C in the phase I of benzene.

Our calculated Raman frequencies can be compared with the experimental data.

2. Raman frequencies calculated at various pressures

in phase I of benzene

We calculate in this study the pressure dependence of the frequencies for the Raman modes of A(Ag), B(AgB2g) and C(B1gB3g) at constant temperatures of 274 and 294K (room temperature) for the solid phase I of benzene. Using the mode Grüneisen parameter of each lattice mode, which correlates the pressure dependence of the crystal volume and the frequency, the Raman frequencies of those modes are computed for phase I of benzene.

Volume dependence of the frequency can be determined in a crystalline system and from this dependence the frequency can be calculated as functions of temperature and pressure. At constant temperatures, from the pressure dependence of both the crystal volume and the frequency, the isothermal Grüneisen parameter can be defined as

In Eq. (1), the pressure dependence of the crystal volume and the frequency can be considered as quadratic relations given by

T T T

VP V P

20 1 2V a a P a P

20 1 2b b P b P

respectively. In Eqs. (2) and (3), a0, a1 and a2, b0, b1 and b2 are constants.

(3)

(1)

(2)

Calculations and Results

In this study, we calculated the Raman frequencies of the lattice modes of Ag (A), AgB2g (B) and B1gB3g (C) as a function of pressure at constant temperatures (T=294K, and 274K) in the solid phase I of benzene. The volume data was used and through the values of the mode Grüneisen parameters [12] the Raman frequencies of those lattice modes were calculated.

We first analyzed the pressure dependence of the observed volume [12] in phase I of benzene according to Eq. (2) and the coefficients a0, a1 and a2 were determined, as given in Table 1. We then determined the functional form of the pressure dependence of the Raman frequency for the lattice modes of A, B and C using the observed Raman frequency data [12] at T=294K according to Eq. (3). The coefficients (b0, b1 and b2) determined are given in Table 2. In this table, we also give the experimental values [12] of the isothermal mode Grüneisen parameter T for the lattice modes studied here.

In order to calculate the pressure dependence of the lattice modes of A, B and C, we also needed the extrapolated values of the volume at the temperatures of T=274K and 300K for zero pressure (P=0) in the solid phase I of benzene. These volume values were extracted from the volume data [12] at various temperatures (P=0) using the quadratic relation

2V a bT cT

where a, b and c are constants. The extracted values of the volume and the coefficients a, b and c are tabulated in Table 3. Correspondingly, we used some observed values [12] for the Raman frequencies of the lattice modes (A, B and C) as the initial data (Table 4) to predict the pressure dependence of the Raman frequencies for the lattice modes studied here.

(4)

Using the observed values of the volume at various pressures at room temperature [12] and the values of the mode Grüneisen parameter T for the lattice modes of A, B and C (Table 2), the Raman frequencies of those modes were predicted according to the relation

1 1( ) ( ) exp ln ( ) /T T TP A T V P V

where the temperature-dependent term is given by

0 1 1( ) ( )A T c c T T

with the constants c0 and c1. In Eq. (5), 1 is the value of the lattice modes (A, B and C) at T=294K (P=0) and V1 represents the volume value at T1=300K (P=0), as given in Tables 2 and 3, respectively.

(5)

(6)

Table 1 Values of the coefficients according to Eq. (2) for the phase I of benzene.

Table 2 Values of the coefficients according to Eq. (3) for the Raman bands indicated. Values of the isothermal mode Grüneisen parameter γT and the values of the Raman frequency 1 at P=0 (T=294K) for the lattice modes are also given here for the phase I of benzene.

a0 (cm3) a1 (cm3/GPa) a2 (cm3/GPa2)

75.63 -9.19 1.56

Raman bands

b0 (cm-1) b1 (cm-1/GPa) -b2 (cm-1/GPa2) T 1 (cm-1)

A (Ag) 44.94 20.54 2.48 3.0 44.94

B (AgB2g) 64.35 27.19 3.50 2.5 64.35

C(B1gB3g) 105.45 49.49 6.33 3.0 105.45

Table 3 Values of the coefficients according to Eq. (4) for the phase I of benzene. The values of the molar volume extrapolated (Eq. 4) are also given here.

Table 4 Observed values of the Raman frequencies [12] for the lattice modes in the phase I of benzene. Values of the coefficients c0 and c1 according to Eq. (6) are also given here.

a(cm3)

-b (cm3/K)

cx10-4

(cm3/K2)V274K (P=0)

(cm3)

V294K (P=0)

(cm3)

V1 (cm3)

T=300K P=0

69.38 0.015 1.42 76.0 77.3 77.7

Lattice Modes

T=274K (P=0)

T=294K (P=1.4GPa)

-c0

(cm-1)

c1x10-2

(cm-1/K)

νA (cm-1) 40 68 5.39 -10.13

νB (cm-1) 62 96 1.96 7.49

νC (cm-1) 100 162 11.32 -5.27

Figure 1 The Raman frequencies calculated as a function of pressure at constant temperatures of 274 and 294K for the lattice mode A(Ag) according to Eq. (5) for the phase I of benzene.

Mode A

P(GPa)

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

(cm-1)

40

50

60

70

80

90

T=274KT=294K

Figure 2 The Raman frequencies calculated as a function of pressure at constant temperatures of 274 and 294K for the lattice mode B(AgB2g) according to Eq. (5) for the phase I of benzene.

Mode B

P(GPa)

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

(cm-1)

60

70

80

90

100

110

120

T=274KT=294K

Figure 3 The Raman frequencies calculated as a function of pressure at constant temperatures of 274 and 294K for the lattice mode C(B1gB3g) according to Eq. (5) for the phase I of benzene.

Mode C

P(GPa)

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

(cm-1)

100

120

140

160

180

200

T=274KT=294K

The Raman frequencies of the lattice modes (A, B and C) were calculated as a function of pressure for constant temperatures of 274 and 294K in the solid phase I of benzene.

It was obtained that the Raman frequencies increase with increasing pressure, as expected.

Those frequencies calculated for the lattice mode C(B1gB3g) are the same at the temperatures of 274 and 294K in phase I of benzene, whereas for the lattice modes of A(Ag) and B(AgA2g) the calculated Raman frequencies which follow the same trend, are the temperature dependent.

Our calculated frequencies can be compared with the experimental measurements for the lattice modes (A, B and C) in the solid phase I of benzene.

Conclusions

3. Calculation of the Raman frequencies as a function of

pressure in the solid phases II and II (III’) of benzene

We calculate here the Raman frequencies of the lattice modes A(Ag), B(B2g) and C(B1gB3g) as a function of pressure at room temperature for the solid phases (II, III and III’) of benzene. This calculation is performed using volume data through the mode Grüneisen parameter.

The Raman frequencies can be calculated as a function of pressure at a constant temperature for various solid phases of benzene. This calculation can be performed using the volume data through the mode Grüneisen parameter.

By defining the volume dependence of the Raman frequency as the mode Grüneisen parameter,

V d

dV

at various pressures, the isothermal mode Grüneisen parameter T can be defined as

T T T

VP V P

(1)

(2)

Calculations and Results

Thus, the volume dependence of the Raman frequency ( anharmonicity ) can be obtained from Eq. (2) as

1 1( ) ( ) exp ln ( ) /T T TP A T V P V

where the temperature-dependent term is assumed to be linear as

0 1( )A T c c T

with the constants c0 and c1. In Eq. (3) 1 and V1 denote the values of the Raman frequency and the crystal volume at a constant temperature. The temperature-dependent A(T) term is added to the solution for T(P) from Eq. (2) in order to predict the Raman frequencies as a function of pressure at some other constant temperature using the volume data.

(3)

(4)

To calculate the Raman frequencies T(P) for various solid phases of benzene, we first analyzed the pressure dependence of volume as measured experimentally [12] according to a quadratic function

where a0, a1 and a2 are constants. By analyzing the volume data for the solid phases of II, III and III’ at various pressures (T=294K) [12], we determined the coefficients, as given in Table 1. The V1 values obtained at T=294K for the solid phases of II, III and III’, are also given in Table 1.

20 1 2V a a P a P (5)

We predicted in this study the pressure dependence of the lattice modes of A(Ag mode), B(AgB2g mode) and C(B1gB3g mode) at T=294K using the volume data [12], as stated above. As the initial data, we used the observed Raman frequencies [12] of those lattice modes (A, B and C), which we analyzed according to the relation

where b0, b1 and b2 are constants.

20 1 2b b P b P (6)

Table 1 Values of the coefficients a0, a1 and a2 according to Eq. (5) using the experimental data [12] at T= 294K for the phases indicated of benzene. The V1 values were determined for pressures of 2.7GPa (solid II), 4.9 GPa (solid III) and 11 GPa (solid III’) using Eq. (5).

Table 2 Observed (first column) [12] and calculated frequencies (second column) using Eq. (6) for the lattice modes A, B and C at constant pressures indicated for the solid phases II, III and III’ of benzene.

Vm(cm3) a0 (cm3) a1 (cm3/GPa) a2 (cm3/GPa2) V1 (cm3)

Phase II 70.34 -3.52 0.16 61.99

Phase III 65.97 -2.25 0.09 57.05

Phase III’ 56.73 -0.44 0.002 51.62

Lattice modes

(cm-1)Solid II (2.7GPa) Solid III (4.9GPa) Solid III’ (11GPa)

A(Ag) 98 96 117 117 158 157

B(B2g) 156 158 190 193 262 262

C(B1gB3g) 181 185 220 224 295 302

Table 3 Values of the coefficients b0, b1 and b2 for the lattice modes indicated according to Eq. (6) using the observed data [12] in solid II of benzene. Values of the mode Grüneisen parameter T [12] and the 1 values at T= 294K for the lattice modes are given. Values of the coefficients c0 and c1 according to Eq. (4) for each lattice mode at T=294K are also given here.

Raman bands (cm-1)

b0

(cm-1)

b1

(cm-1/GPa)

b2

(cm-1/GPa2)

T 1

(cm-1)

c0

(cm-1)

c1

(cm-1/K)

A(Ag) 67.81 9.20 0.39 2.9 96 -146.70 0.50

B(B2g) 77.45 37.28 -2.79 3.1 158 -147.64 0.49

C(B1gB3g) 113.74 30.77 -1.58 2.6 185 -146.90 0.50

Table 5 Values of the coefficients b0, b1 and b2 for the lattice modes indicated according to Eq. (6) using the observed data in solid III’ of benzene.

Raman bands (cm-1)

b0

(cm-1)

b1

(cm-1/GPa)

b2

(cm-1/GPa2)

T 1

(cm-1)

c0

(cm-1)

c1

(cm-1/K)

A(Ag) 76.46 9.42 -0.19 2.0 157 -144.95 0.50

B(B2g) 172.45 9.19 -0.09 2.0 262 -144.04 0.51

C(B1gB3g) 217.57 8.27 -0.06 1.8 302 -144.43 0.51

Raman bands (cm-1)

b0

(cm-1)

b1

(cm-1/GPa)

b2

(cm-1/GPa2)

T 1

(cm-1)

c0

(cm-1)

c1

(cm-1/K)

A(Ag) 66.91 12.10 -0.37 2.7 117 -145.83 0.50

B(B2g) 109.60 19.87 -0.55 2.9 193 -145.40 0.50

C(B1gB3g) 128.63 22.95 -0.71 2.8 224 -146.63 0.50

Table 4 Values of the coefficients b0, b1 and b2 for the lattice modes indicated according to Eq. (6) using the observed data in solid III of benzene.

Figure 1 Calculated Raman frequencies of the lattice modes A, B and C as a function of pressure for the solid phase II of benzene at T= 294K according to Eq. (3). The observed data [12] are also shown here.

Phase II

Pressure (GPa)

1.5 2.0 2.5 3.0 3.5 4.0

(c

m-1

)

60

80

100

120

140

160

180

200

220

A

B

C

Figure 2 Calculated Raman frequencies of the lattice modes A, B and C as a function of pressure for the solid phase III of benzene at T= 294K according to Eq. (3). The observed data [12] are also shown here.

Phase III

Pressure (GPa)

4 6 8 10 12

(c

m-1

)

50

100

150

200

250

300

350

A

B

C

Figure 3 Calculated Raman frequencies of the lattice modes A, B and C as a function of pressure for the solid phase III’ of benzene at T= 294K according to Eq. (3). The observed data [12] are also shown here.

Phase III'

Pressure (GPa)

10 12 14 16 18 20 22

(c

m-1

)

100

150

200

250

300

350

400

A

B

C

The Raman frequencies of the lattice modes (A, B and C) were calculated from the volume data through the mode Grüneisen parameter T at various pressures at room temperature for the solid phases of II, III and III’ of benzene.

We obtained that the Raman frequencies of those lattice modes increase as the pressure increases for all the solid phases studied, as expected.

Our calculated Raman frequencies are in good agreement with the experimental data. This shows that the method of calculating the Raman frequency from the volume data using the mode Grüneisen parameter is adequate, as applied to the solid phases of II, III and III’ of benzene.

This method can also be applied to the other phases of benzene and also to some other molecular crystals.

Conclusions

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