ULTRASONIC ABSORPTION MEASUREMENTS IN SINGLE MOLECULAR CRYSTALS

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ULTRASONIC ABSORPTION MEASUREMENTS IN SINGLE MOLECULAR CRYSTALS

A. Victor, H. Altman, R. Beyer

To cite this version:

A. Victor, H. Altman, R. Beyer. ULTRASONIC ABSORPTION MEASUREMENTS IN SIN- GLE MOLECULAR CRYSTALS. Journal de Physique Colloques, 1972, 33 (C6), pp.C6-166-C6-169.

�10.1051/jphyscol:1972638�. �jpa-00215156�

JOURNAL DE PHYSIQUE

Colloque C6, supplPment au

1 1-1 2, Tome 33, Novembre- Dtcembre 1972, page 166

ULTRASONIC ABSORPTION MEASUREMENTS IN SINGLE MOLECULAR CRYSTALS

A. E. VICTOR, H. E. ALTMAN Jr. a n d K . T. BEYER

Department of Physics, Brown University Providence R h o d e Island 02912 USA

RbumB. -

Les mesures d'absorption ultrasonore ont Cte realisees par une methode d'impulsion l e long des axes principaux d'un monocristal de benzene. Les mesures ont &te effectuees au moyen d'ondes longitudinales dans la gamme de frequence dc

MHz et dans le domaine de tern+- rature de 170-250

Suivant les theories actuelles de I'absorption, le coefficient d'absorption

doit Ctre proportionnel au carre de la frequence dans ces domaines de temperature et de frequence

cependant, on observe pour

un ecart A cette loi lorsque la temperature diminue I'Ccart etant different pour chaque axe.

Dans le cas de I'axe a, un pic de relaxation est observe (pour la courbe

au voisinage de 15 MHz et pour I'axe b legerement sugrieur a 20 MHz. Le comportement le long de I'axe c est plus complexe, ce qui laisse supposer la presence d'effets multiples.

L'hypothbe selon laquelle l'absorption resulte d'un echange d'tnergie entre les vibrations du reseau et les oscillations molCculaires de basse frequence a etC avancee

mais les valeurs theoriques (selon les theories de Liebermann et Danielmeyer des temps de relaxation sont de 10 ii 100 fois trop petites. Une modification de la theorie de Danielmeyer, pour tenir approximativement compte de I'anisotropie conduit

une amelioration, mais I'accord jusqu'ici n'est que dans les ordres de grandeur.

Des monocristaux de nitrobenzbne ont ete Cgalement synthetists et des mesures effectuees par deux techniques differentes. Les rcsultats de ces mesures seront egalement exposb.

Abstract.

- Ultrasonic absorption measurements have been made by a pulse method along the principal axes of single crystalline benzene. The measurements were made with longitudinal waves in the frequency range

MHz, and over a temperature range 170-250

Present theories of such absorption predict the absorption coefficient

to be proportional to the square of the frequency in the temperature and frequency range of this experiment

however,

was observed to depart from the dependence as the temperature was lowered, the change being different for each axis. In the case of the a axis, a relaxation peak was observed (for the

curve) in the neighborhood of 15 MHz, and for the b- axis, just above 20 MHz. The behavior along the c- axis was more complex, suggesting the presence of multiple effects.

It has been hypothesized that the absorption results from an exchange of energy between the lattice vibrations and the low frequency molecular oscillations, but the theoretically derived (from the theories of Liebermann and Danielmeyer) relaxation times are 10 to 100 times too small.

Modification of the Danielmeyer theory to take rough account of anisotropy leads to some impro- vement, but agreement thus far is order-of-magnitude.

Single crystals of nitrobenzene have also been grown, and measurements made by two different laboratory techniques. The results of these measurements will also be reported.

1.

Ultrasonic absorption in mole- cular crystals has been studied by many observers since Liebermann's pioneering work of 1959. T h e experimental work has h a d three principal results

(1) the absorption coefficient a has been found t o be roughly proportional t o the square of the frequency in the range 10-100 MHz

(2)

is orders of magnitude greater than that found in ordinary

hard

dielectric solids

(3) a is almost independent of temperature.

In virtually all these measurements, however, the orientation of the crystal has n o t been known. Mea- surement of the sound velocity were made o n a n oriented crystal of benzene by Heseltine, Elliot and Wilson a n d a few absorption measurements were reported by them a t one frequency and temperature.

T h e present paper reports systematic measurements of

such absorption coefficients o n an oriented single benzene crystal.

2.

- Before performing the absorption measurements a number of auxiliary experimental techniques had t o be developed o r applied. First, a vacuum sublimation technique was used t o purify liquid benzene. Large single crystals were then grown from the melt utilizing a method similar t o that developed by Liebermann [3] a n d Rasmussen

The average size of the crystals that were grown was about 5 cm o n a side, the largest being 10 x 7 x 7 cm.

Single crystalline benzene is optically transparent a n d highly birefringent (orthorhombic structure) ; hence, it is amenable t o optical orientation with pola-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972638

ULTRASONlC ABSORPTION MEASUREMENTS I N SINGLE MOLECULAR CRYSTALS C6- 1 67

rized light. A polarizing microscope was used in conjunction with a universal stage to orient the crys- tals (orientation accuracy was + ^2O).

A special mechanical jig was used to cut and shape the oriented sample to a form suitable for ultrasonic measurements. The final sample was a cylinder that was 1.9 cm in diameter and 1.54 cm in height. It was sealed in an annular brass ring. Thin strips of mylar were used to seal each face, which prevented the sample from sublimating. An x-cut quartz transducer (2 MHz fundamental frequency) was bonded onto one of the mylar strips. The transducer was coaxially plated and had an active area of 318-in. (112-in. over-all diameter).

The sample was enclosed in a helium dewar that was modified to enable the sample temperature to be varied from 250 OK down to 77 OK. The ultrasonic measurements were made using the standard pulse- echo technique.

3. Experimental results.

The sound absorption measurements obtained were recorded at higher fre- quencies and lower temperatures than previous mea- surements in solid benzene, and, more significantly, were made along the crystallographic axes a, b, c.

Measurements of a were obtained for all three axes between 6 and 26 MHz and up to 34 and 38 MHz for the a and c axes, respectively. The characteristics of the bond limited the temperature range of the experiment to 170-250 OK for the a and b axes and to 180-2500 for the c-axis. The latter limit is attributed to the higher value of the expansion coefficient along the c-axis which caused the transducer bond to break at a higher temperature.

Relaxation effects were observed for all three axes.

The data are shown for the a and c axes in figures 1 and 2 respectively. Since accurate velocity measure- ments were not obtained for all frequencies, the results were plotted in crlv

vs.v. Since velocity dispersion is generally rather small, the a/v plot should show the same characteristics as the more conventional c r i plot.

For the a-axis date (Fig. 1) standard relaxation curves [12] have been drawn through the experimental

v s 1 R E O U E N C Y BENZENI-

a a x i s

Fro. 1.

10

:I

7 . 0 2 10.K s axis

6 . r x 1 9 0 . ~ I

. ,

A..

,

3 '4 5 14 18 2 2 2 6 3 0 1) 33 do I 2 0

FREQUENCY (MI-lz)

points for 170, 190, 230, and 240 OK. The curves indicate that the relaxation frequency v, changes from 17 MHz to 12 MHz and the maximum value ofp', i. e. p:, decreases by almost 60 % between 240 OK and 170OK. The shift of v, t o a lower value at the lower temperature is characteristic of vibrational relaxations. All the points for the a-axis data do not fall on the curves as drawn, however. A better fit would be given by a curve that peaks more sharply than the standard relaxation curve

the dashed curve drawn through the points for 190 OK represents such a behavior. The latter curve was obtained by adding the contribution of three standard relaxation curves with different relaxation strengths and frequencies.

This suggests the possibility of multiple relaxations occurring in the same frequency range. Of course, multiple relaxation processes are quite complicated and the addition procedure only gives an indication that such a process may exist. It does not yield any details of the interaction, i. e., whether the processes are interdependent or independent of each other.

The results for the b-axis are very similar therefore to the a-axis, whereas the c-axis curves (Fig. 2) appear to be quite different, especially at the lower tempe- ratures. The presence of two widely separated relaxa- tion processes is indicated. A standard relaxation curve roughly fits the data at 240

however, as the tempe- rature is lowered, there is evidence of a second process with a higher relaxation frequency. The dashed stan- dard relaxation curves shown in figure 2 add to give a close fit to the experimental data at 180 OK. The dashed curves suggest that at 180 OK the two relaxation frequencies are 4.5 M H z and 120 MHz. It would appear that both relaxation peaks shift t o a lower frequency as the temperature is lowered. The exact behavior of the upper relaxation cannot be deter- mined because of the limited experimental informa- tion. However, the low frequency relaxation appears to be centered at 14 MHz and 240 OK and at 4.5 MHz at 1800K

pL changes by more than 30 % over the same temperature range.

All the data shown were corrected for diffraction

effects. The errors associated with the measuremeqts

C6-168 A. E. VICTOR, H. E. ALTMAN JR. AND R . T. BEYER

is estimated to be 7 % at 26 MHz and below and 10 %

at the higher frequencies. The additional error at the higher frequencies is due to the weaker signals obtained in that region.

Preliminary measurements have also been made by the same technique along the b crystallographic axis in single crystalline nitrobenzene, over a frequency range of 2.5 to 62.5 MHz, and a temperature range of 175-250OK. The data recorded at low temperatures indicate the presence of a relaxation frequency of about 10 MHz

the behavior of the absorption coefficient is more complicated at higher tempe- ratures, somewhat similar to the c-axis behavior of benzene. The absorption coefficients in nitrobenzene are about 70 % of those in benzene along the b axis.

4. Comparison with theory.

The fact that a different relaxation was observed for each axis of benzene indicates that

depends on the molecular orientation within the unit cell. The lack of relaxation effects in the same frequency range in the liquid state further indicates that the sound absorption mechanism must be related to the structure of the solid, i. e.,

is due to interaction between lattice (acoustic) phonons only or is a result of acoustic-optic phonon interac- tion. The former process has been theoretically treated for dielectric solids by Woodruff and Ehrenreich [5].

When applied to benzene the theory predicts a sound absorption that is less than 1 O/, of the observed values, and relaxation frequencies that are much higher than those observed. Hence, Liebermann's original postu- late appears to be appropriate. However, since his theory is based on sound absorption in fluids, it cannot be readily adapted to account for the anisotropic properties of a solid. In contrast, Danielmeyer's theory was derived from a solid state viewpoint and, even though it was originally based on an isotropic solid, it can be modified to account for anisotropy.

The procedure found to be most useful in the analysis was to compare the relative change of the maximum value of p' as a function of crystal orientation and temperature.

One of the chief results of Danielmeyer's theory is an expression for absorption per wavelength p in terms of specific heats and elastic constants. When written in terms of p6, it becomes

Here

is the longitudinal sound velocity for the axis under consideration, C , and Cv are the molar heat capacities at constant pressure and constant volume, respectively, and C: is the molar heat capacity asso- ciated with the internal degrees of freedom. The quantity [: is a structure factor that combines the elastic constants associated with a given principal axis,

I.

e.,

where c:,,

and cl are the adiabatic elastic constants associated with the transverse and longitudinal sound velocities for a particular principal axis. By substituting the experimental values of the sound velocities and heat' capacities for benzene into eq. (I), was calcu- lated for each axis at 180 OK and 240 OK. The calculat- ed change in pk for the a and c axes, respectively, was 72 % and 68 %, whereas the corresponding expe- rimental changes were 44 % and 32 %.

Better agreement was obtained by considering the possibility of partial relaxation processes, i. e., the energy that is coupled into the internal modes does not relax with all the modes at on time. Such behavior has been observed for vibrational relaxations in mole- cular liquids. T o determine which optical modes are involved in the relaxation process, the internal specific heat contribution due to each mode 6Ci, which can be calculated by using the Einstein model together with spectroscopic data [7], [8] must be used in eq. (1) in place of c:. When this is done, it is seen that, in general, ApL(T) becomes greater as the internal vibrational frequency increases. For the lowest internal vibration, which corresponds to a skeletal-out-of- plane vibration, the relative change in pL between 180 and 240 OK for the a and c axes is 49 % and 42 %,

respectively. (The benzene molecule is planar and has the shape of a regular hexagon with a carbon atom a t each vertex. One hydrogen atom is bonded to each carbon atom and the C-H bonds extend radially outward from the center of the hexagon along lines joining the center and the vertices of the hexagon.) Although the correlation is not as good for the c- axis as for the a- axis, it is much better than when the total C: was used. For the tenth internal vibration (there are a total of thirty), the relative changes in

are 86 % and 82 %. Hence, these calculations suggest that the lower frequency internal modes are the ones which play the principal role in the observed relaxa- tions. Another fact that lends credence to this assertion is that the magnitude of pk calculated by using 6Ci is of the same order as the experimental values, whereas those obtained using C; in eq.

are an order of magnitude greater.

5. Relationship of a to crystalline structure.

An intuitive argument as to why one might expect the

low-frequency optical modes to participate in the

relaxation processes rather than the other modes can

be obtaining by trying to visualize the effect of the

sound waves on the molecules in the unit cell. In the

cell (Fig. 3) [9], the planar benzene molecules are

arranged in what appears to be a series of corrugated

sheets that are stacked upon one another in the direc-

tion of the c- axis. Closer inspection shows that the

C-H bonds of adjacent molecules intermesh in a bevel

ULTRASONIC ABSORPTION MEASUREMENTS IN

MOLECULAR CRYSTALS

a)

Diagram of the crystal structure

benzene viewed down

Diagram of

crystal structure of benzene viewed

the b-axis. the c-axis.

Frc. 3.

- Crystal structure of benzene (from Cox, Cruikshank, and Smith

gear type arrangement. Hence, when a longitudinal sound wave is launched down the c- axis, the ensuing -compressions and rarefactions will tend t o induce the molecular planes to oscillate about axes parallel to the b- axis. This motion would cause the benzene ring to distort out of its plane and displace nearest- neighbor H atoms from their equilibrium positions.

One would expect that the internal modes most affected would be the skeletal out-of-plane vibrations, the three lowest modes in the optical spectrum. Thus, there is a possibility that relaxations are taking place between an excited state of a rotational lattice mode and excited states of the lower frequency optical modes.

A similar effect would take place along the a- axis.

Here the molecular planes would tend t o move in an

N

accordion-like

fashion, which would also tend t o excite the lower optical modes.

The above discussion is, of course, oversimplified.

The actual interaction between the sound wave and the molecules is certainly much more complicated

however, if molecular resonance is the primary ound absorption mechanism in organic crystals, one can see that the molecular arrangement within the unit cell should be an important factor. More insight as t o the mechanisms involved would be gained by experi- ments on substances with less complicated molecular.

and crystalline structure. Also, better theoretical understanding of the problem would be provided by using a more realistic expression for the intermolecular.

potential than has been used to date.

References

(11 LIEBERMANN L.,

(1959) 1063.

also [5] WOODRUFF

and EHRENREICH

the review

LIEBERMANN,

Resonance Absorp- (1961) 1553.

tion

in Physical Acoustics (W. P. Mason, ed.) [6] DANIELMEYER H.

(1966) 102.

Vol. IV

A (Academic Press, New York, 1966), 183. [7] GEE A. G. and ROBINSON G. W., J. Chem. Phys. 46 (21 HE~ELTINE J. C., ELLIOT D. W. and WILSON Jr.

B., (1967) 4847.

J . Chem. Phys. 40

(1964) 2584. [8] HOLLENBERG J. L. and Dows D. A.,

(1962) 1300.

(31

L.,

[g] COX

CRUIKSHANK

and SMITH

A...

[4] RASMUSSEN R.

(1967) 211.

(London) 247A (1958) 1.