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Two-phonon absorption processes in liquidoctane, decane, hexadecane and nujolC. Hall a b , J.W. Fleming a c , G.W. Chantry a c & J.A.D. Matthew da National Physical Laboratory , Teddington, Middlesexb Division of Numerical Analysis and Computingc Division of Materials Applicationsd Department of Physics , University of York , YorksPublished online: 11 Aug 2006.

To cite this article: C. Hall , J.W. Fleming , G.W. Chantry & J.A.D. Matthew (1971) Two-phononabsorption processes in liquid octane, decane, hexadecane and nujol, Molecular Physics: An InternationalJournal at the Interface Between Chemistry and Physics, 22:2, 325-333, DOI: 10.1080/00268977100102591

To link to this article: http://dx.doi.org/10.1080/00268977100102591

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MOLECULAR PHYSICS, 1971, VOL. 22, NO. 2, 325-333

Two-phonon absorption processes i n l i q u i d octane, decane, hexadecane and nujol

by C. HALL t, J. W. FLEMINGJ;, G. W. CHANTRY~:

National Physical Laboratory, Teddington, Middlesex

and J. A. D. M A T T H E W

Department of Physics, University of York, Yorks

(Received 28 June 1971)

Similarities in the measured 100-300 cm -I absorption spectra of liquid paraffins over a wide range of chain lengths suggest that an identical absorption mechanism is dominant in each case. The intensity and temperature dependence of the 250 cm -1 band in nujol, in particular, are indicative of a multiphon0n process. The present paper offers a theoretical explanation for the spectra of octane, decane, hexadecane and nujol, in terms of two-phonon absorptions. Calculations based on the Lax and Burstein approach (suitably modified to take account of finite chain length) reproduce the general features of the spectrum fairly well in each case, and provide an adequate account of the observed temperature dependence of the band in nujol.

1. TNTRODUCTION

The infra-red absorption spectra of liquid paraffins in the 100-300 cm-1 region show remarkable similarities over a wide range of chain lengths. Typically for the simple linear paraffins octane, decane and hexadecane, one finds a fairly distinct peak in the 300 cm -1 region flanked on the low frequency side by a wide flat shoulder decaying steadily down to the low frequency limit (cf. figure 1). Nujolw too has this characteristic form of spectrum, with the main absorption peak in this case lying near 250 cm -1 (cf. figure 2) (although observations by Wyss [1] tend to place the peak nearer to 300 cm -1, more in agreement with that found for the simple linear paraffin chains). Detailed calculations on the normal modes of the n-paraffins in the solid state have previously been reported by Schachtschneider and Snyder [2], but their results on the paraffins considered in this paper show no active modes close to 300 cm -1. The effects of finite chain length and disorder in the liquid state could give rise to some activity at the observed frequency, but a more likely explanation for the dominant absorption mechanism is suggested by the marked temperature dependence of the 250 cm -1 band of nujol (cs figure 2). This temperature dependence points strongly to a muldphonon process. Moreover, the band has an intensity which is comparable with that found for two-phonon absorptions in silicon and germanium [3]. In the present paper we offer an explanation of the absorption spectra of these paraffins in the 100-300 cm -1 region, in terms of two-phonon processes (sum and difference bands are both considered). Our theoretical analysis reproduces the general features of the spectrum fairly well in each case, and also provides an adequate account of the observed temperature

t Division of Numerical Analysis and Computing. :~ Division of Materials Applications. w Nujol consists of hydrocarbons of different lengths, possibly branched.

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326 C. Hall et al.

I 4 i OCTANE 3

_

�9 < : f loo 15.o '

T 4 i"" DECANE

u

< 0 lO0

I I I 200 250 300

Frequency co (cm 1}

(a)

350

- - I I I I 150 200 250 300 350

F r e q u e n c y c~ (era -1}

(b)

i4i �84 .u_ 2 .u_

81 ~ - -

8 .Jo

100

HEXADECANE

i I I I 150 200 250 300

Frequency~ (cm -1}

(~)

350

Figure 1. Experimentally determined far infra-red spectra of: (a) octane; (b) decane; (c) hexadecane. The spectra shown in figures I and 2 were obtained by the technique of Fourier transform spectroscopy [8] using the N P L - G r u b b Parsons modular cube interferometer.

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Two-phonon absorption processes in liquids 327

NUJOL 5 T -

4 -

3

2

1

Figure 2.

I I I 100 150 ZOO z50 300 350

Frequency ~(cm 1)

Experimentally determined far infra-red spectra of Nujol at 300 K and 120 K.

dependence of the band in nujol. We take as the basis for our treatment the theory of multiphonon absorption in crystals, adapted for the finite chains known to be present in the liquid.

2. THEORY OF TWO-PHONON ABSORPTION IN CRYSTALS

The problem of two-phonon interactions in crystals has been investigated by Lax and Burstein [3] (hereafter referred to as LB). For a lattice having two identical atoms per unit cell and with a centre of inversion symmetry, i.e. a homopolar crystal, there is no linear electric moment and hence there will be no fundamental or ' reststrahl ' absorption. In the absorption mechanism for homopolar crystals suggested by LB, one vibration mode induces charges on the atoms while a second mode simultaneously causes a movement of these charges. This produces an electric moment (of second order in the atomic displacements) which can couple to the electric field of incident radiation. The second-order moment fluctuates at the sum or difference frequency of the two phonons involved, giving rise to absorption at these frequencies.

The coupling of two phonons implies that anharmonic forces are present between the atoms. The existence of anharmonicity is not sufficient of itself to produce fundamental absorption in homopolar crystals (because of their inversion symmetry), but it enables absorption to occur via the second-order moment.

The coupling between two modes from branches t and t' can be characterized by a matrix element H(k, tt'). If the temperature dependence is included specifically [4] the two-phonon summation bands have an absorption coefficient

As(w) oc ~o • E ] H(k, tt')12[(n(k, t) + 1)(n(- k, t') + 1 ) - n(k, t )n(- k, t')] t')

• 3(oJ--co(k, t ) - r t')), (1 a)

where ~o(k, t) is the frequency of the (k, t) mode with wave number k and n(k, t) the

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328 C. Hall et aL

occupation number of the phonon states in thermal equilibrium, is given by Bose- Einstein statistics as

1] -1

The absorption of a photon at a summation frequency oJ creates two phonons at frequencies ~o(-k, t') and oJ(k, t) where oJ=~o(k, t ) + ~ ( - k , t'). In order to conserve momentum, these phonons must possess equal and opposite wave numbers k and - k , the photon momentum being effectively zero. For difference bands, the absorption coefficient

Ao(o~)ocoj • ~ [H(k, tt')]2[n(k, t)(n(k, t)+ 1)-n(k, t)(n(k, t ')+ 1)] o,(k, t)o4k, r)

x 8(oJ- oJ(k, t) + co(k, t')) (1 b)

has the same matrix elements H(k, tt') as before, differing only in its temperature dependence.

We shall discuss the nature of the absorption matrix elements H(k, tt') later, but we note here that their accurate evaluation requires a detailed knowledge of the normal modes and their associated frequencies.

Major contributions to difference bands occur when at least one of the two phonons has a low frequency, and for a summation band at 300 cm -1 two low frequency modes must be involved. For this reason we have restricted our investigation to combination bands arising from the skeletal modes of the n- paraffins.

3. CALCULATION OF NORMAL MODES FOR FINITE CHAINS

Our model for the paraffins was the simple planar zig-zag chain with point masses representing each CHz group. For in-plane vibrations a harmonic force field was used with separate bond-bending and bond-stretching terms. The bond- stretching and bond-bending force constants were taken to be 4.53 x 10 -8 and 0.44x 10 -3 Nm -1 respectively. These are effectively the values obtained by Snyder [5] who used bond-bending and stretching forces as part of a more sophisticated force field. We have neglected the out-of-plane skeletal vibrations in our present calculation--it can in fact be shown that the charge deformation caused by these out-of-plane motions must be relatively small in comparison with that due to the in-plane vibrations.

The time independent equations of motion for in-plane vibrations at a frequency o~ may be written in matrix notation as

[ A - m I]V = o, (3 where A is the dynamical matrix of order 4n x 4n, which depends upon force constants and details of chain geometry, and V is a vector containing atomic amplitudes; m is the mass of the CH2 group. A is symmetric and if we impose cyclic boundary conditions it can be written in the n • n block circulant form

I -A1 A2 ". A2 T ] A8 T A1 A2 O

A = "., A8 T AI Ae ".. , 0 . . . . . . . . . . . . . . . . A2

_A2 "-., A2 ~ A1

(4)

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Two-phonon absorption processes in liquids 329

where A1, A2 are submatrices of order 4 x 4 (there are 4 degrees of freedom/unit cell) and A2 T is the transpose of A2. The eigenvalues of A are the eigenvalues of the n matrices [6]

A'(O) = A1 + A2 exp (iO) + A2 T exp ( - iO)

where

O= 2~rj, j = l , 2 , . . . , n n

and n is the number of unit cells. The corresponding eigenvectors of A have the form

b(0, t) b(0, t) exp (iO) b(0, t) exp (i20)

v(0, t)=

(5)

(6)

(7)

b(0, t) exp [i(n- 1)6]

where b(O, t) is an eigenvector of A'(0). The subscript t labels differentbranches of the dispersion curve, there being one branch for each degree of freedom in the unit cell. In this way one can find approximate normal mode frequencies for finite chains by sampling the dispersion curve at different values of 0 appropriate to the chain length 2n. Sampling at these points (6) is equivalent to imposing the condition that the normal mode wave train possesses an integral number of wavelengths along the chain. However, Pitzer [7] pointed out that a more appropriate condition for finite chains with free ends would be to choose a wave train with an integral number of half wavelengths along the chain. This latter convention, which is equivalent to sampling the dispersion curve at

O= ~rJ, j = 0 , 1 . . . . , n - l , (6a) n

is the one we have adopted in our present calculation.

4 . EVALUATION OF ABSORPTION INTENSITY

The expression for the absorption coefficient expressed in equations (1 a) and (1 b) is derived from the basic formula for second-order absorption

A(o)) cc to x Average ~.] (Pl2)tf] ~ x 8(E~- E i - hoJ), (8) i f

where i and f represent initial and final vibrational states of the system with energies Ei and Ef respectively, and (M2)if is a matrix element connecting the two states. The second-order dipole moment 1~2 takes the form

M2(r) = ~r~ ~ . C~m ~ . rm p, (9) lm

where l, m denote unit cells while a and fl refer to different atoms within a unit cell; rl ~ is the displacement of the a type atom in the / th unit cell and the Clm ~ are

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330 C. Hall et al.

expansion coefficients. coordinates q(k, t):

where

Alternatively, H2 can be expanded in terms of normal

M2 = Z q(k, t')*q(k, t)H(k, tt'), (9 a) k, it"

H(k, tt')ocEb~(k , t')* exp [-ikld].Ctm~.b#(k, t) exp [ikmd]. (10) lra

b~(k, t) is the polarization vector for a type atoms in the (k, t) mode, d is the lattice constant and * denotes the complex conjugate. Evaluation of the matrix element in (8), with the expansion (9 a) leads to expressions of the form (1 a, b) for summation and difference absorption intensities.

We now make the assumption that motion in the x direction does not affect the y component M2y of the electric moment and vice versa (x is along the chain axis and y is in the plane of the chain perpendicular to the x axis). This leads to the simplification

I bt(k, tt')l z= ]Hx(k, t t ' ) [2+ [Hu(k, tt')12, (11)

where the matrix elements Hz(k, tt') and Hy(k, tt') depend respectively on the x and y components of the polarization vector b(O, t); specifically they take the form

H~(k, tt')=_ g~( o, tt')oc Y bx~(O, t')* exp [-ilO]Cz, Zm~#bxP( O, t) exp [ira0]

lm Hu(k, tt')-Hy(O, tt')oc 2by~(O , t')* exp [-ilO]Cy, zm~by~(O, t) exp qm0], (12)

lm

where 0 is related to the wave number k of the combining modes by 0 = kd. Our skeletal model for the paraffin chain possesses a centre of inversion symmetry

and we may apply the theory developed by LB to calculate the absorption of combination bands. Thus for a chain of two atoms per unit cell with inversion symmetry, it can be shown that general invariance requirements restrict the possible values of Cz, ~m ~ and Cy, tm~. The only non-zero terms are those with a =/3, and if we consider nearest cell interactions in the electric moment expansion the matrix elements Hz(O, tt') can be expressed in terms of a single expansion coefficient Cx, and Hu(O , tt') in terms of a coefficient Cy. Thus,

Hx(O, t t ' ) ~ 2 ( 1 - cos O)[bxl(O, t')*bxl(O, t)-bx-l(O, t')*bx-l(O, t)]Cx

Hu(O , t t ' )oc2(1- cos O)[byl(O , t')*byl(O, t)-by-l(O, t')*bu-l(O, t)]Cu, (13)

where the superscripts + 1, - 1 denote the two types of atom in the unit cell.

5. DISCUSSION OF RESULTS

We have used equations (1) and (13) to evaluate the two-phonon absorption intensities for chains of different lengths. This procedure gives the results in the form of discrete spectral lines at the various sum and difference frequencies. The spectra were then smoothed by replacing each line by a Lorentzian profile of halfwidth 60 cm -1. This relatively large halfwidth]" is meant to account for

]" The corresponding line shifts are already implicitly included in the calculation as a result of Snyder's [4] choice of effective force constants.

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Two-phonon absorption processes in liquids 331

'~ 1"0 l 0.75

i 0 " 5

o-zs

I t ! r

100 150 zoo 250 300 r-','equency ca (cm -11

(a)

o

OCTANE

I'0

350

~

o w- .~ 0.75

~ 0 .5

;o u

~ 0 . 2 5

Q. o .o < 0

100

DE N 1"0

0.75

0-5

0-25

0

Figure 3.

DECANE

] I I !

150 Z00 250 300 350 Frequency ca (cm -1)

(b)

I I I I

100 150 2 0 0 2 5 0 3 0 0 3 5 0

Frcq.uency ca (r

(c)

Calculated two phonon absorption spectra, in the 100-300 cm ~1 region, of: (a) octane; (b) decane; (c) hexadecane.

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332 C. Hall et al.

statistical fluctuations in the liquid state and for the increased broadening which occurs when a line involves the action of two phonons rather than just one.

In each case, one finds the separate x and y contributions to the absorption spectrum to be fairly similar in character, so that the overall shape of the combined profile is not strongly sensitive to the relative magnitudes of Cx and C u. The results displayed in the present paper correspond to the choice C~ = Cy.

Figure 3 shows our smoothed combination band spectra, with intensities in arbitrary units, for the simple linear paraffins octane, decane and hexadecane at 300 K. Nujol, although a more complex substance, probably contains a significant proportion of chains with lengths in the region of Ce0H4~, so in figure 4 we give the calculated spectrum for a planar zig-zag chain of this length and composition at 300 K and 120 K.

1'2Sl

t ~

i 1.o L

0 -7S

u

O-S

u

g ~a. 0.25 o

<C

o 1oo

C20 H42

I I I I 150 200 250 300

Frequency ,-~ (cm -1 )

Figure 4.

350

Calculated two phonon absorption spectra of C~0H42 in the 100-300 cm -1 region at 300 K and 120 K.

The model calculations exhibit the same general features as the experimental results, namely an absorption peak whose frequency is relatively independent of chain length, with the intensity falling sharply on the high frequency side and with a shoulder at low frequencies. Although the calculated profiles show more sensitivity to chain length than do the experimental ones, nevertheless the positions of the theoretical and observed maxima correspond roughly in all cases.

As implied in w167 1 and 2, the important aspect of combination bands is their temperature dependence. It is impractical to measure the absorption spectra of the simple paraffins at low temperature, since they crystallize just below room temperature. However, in the case of nujol, for which extended low temperature measurements are possible, our calculated temperature dependence (cf. figure 4) is in substantial agreement with that observed experimentally (figure 2).

Considering the simplicity of our theoretical model, the similarities between observed and calculated spectra are very encouraging indeed. We would conclude that, although the spectra of the paraffins must contain contributions from

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Two-phonon absorption processes in liquids 333

fundamental modes (disorder-induced localized modes might be expected to be particularly active) and from higher multiphonon processes, it appears that the two-phonon process makes a significant contribution to the absorption intensity in the 100-300 cm -1 region.

The Pitzer sampling technique we have used in our calculation gives only an approximate solution to the vibrational problem, assuming, as it does, individual k-values and strict selection rules for the modes. More accurate calculations are now being performed in which the dynamical matrix of equation (3) is solved exactly with a free end boundary condition and where no rigorous selection rules are assumed. This will enable us to examine, more closely, the variation of the spectrum with chain length.

Although the results given in the present paper seem particularly encouraging, we should perhaps end on a note of caution. The linear zig-zag chain is a much over simplified model of liquid paraffins. Certainly non-symmetric configurations known to exist in the liquid state will give rise to fundamental absorption. This needs to be checked by specific calculations for non-symmetric chains, when sufficient information on electron density distributions for these cases becomes available. In particular, it would be desirable to investigate the temperature dependence of chain configuration and its effect upon the temperature dependence of the fundamental absorption spectrum.

We would like to thank Dr. R. J. Bell for many helpful discussions relating to the content and form of this paper.

REFERENCES

[I] WYSS, H. R., WERDER, R. m., and G~NTHARD, Hs. H., 1964, Spectrochim. Acta, 20, 573. [2] SCHACHTSCHNEIDER, J. H., and SNYDER, R. G., 1963, Spectrochim..4cta, 19, 117. [3] LAX, M., and BURSTEIN, E., 1955, Phys. Rev., 97, 39. [4] HOUGHTON, J., and SMITH, S. D., 1966, Infra-Red Physics (Oxford University Press),

p. 101. [5] SNYDER, R. G., 1967, ft. chem. Phys., 47, 1316. [6] DEAN, P., 1967, ft. Inst. Maths. Applics, 3, 98. [7] PITZER, K. S., 1940, J. chem. Phys., 8, 711. [8] GEBBIE, H. A., and TwIss, R. Q., 1966, Rep. Prog. Phys., 29, 729.

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