Dispersion of Velocity and Anomalous Absorption of Ultrasonics in Nitrogen

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 23, NUMBER 4 JULY, 1951

Dispersion of Velocity and Anomalous Absorption of Ultrasonics in Nitrogen* A•.r•.•) Jos•.•a Z•m•)A$

Department of Physics, The Catholic University of America, Washington, D.C. (Received March 1, 1951)

By means of the interferometer, the velocity and absorption of ultrasonic waves in nitrogen were measured at the frequency of 2.992 mc/sec, in the pressure range of 2.09 to 76 cm of Hg at a temperature of 29øC. A dispersion of velocity was found ranging from 354.3 meters/sec to 364.4 meters/sec. The ratio aexp/aela• dropped from 1.40 to 1.32, and the corresponding value of Cv/R dropped from 2.50 to 2.08.

Theoretical values for the change in velocity and in the absorption ratio, calculated by applying the equations for the exchange of energy between translational and vibrational degrees of freedom, show good agreement with the observed values. The increase in velocity and the decrease in aexp/a•ass is interpreted as due to the slow exchange of energy between the translational and rotational degrees of freedom. The relaxation time for rotation was found to be 1.2X 10 -9 second.

INTRODUCTION

N gases with internal degrees of freedom, ultrasonic waves show dispersion and absorption in excess of the classical value. Herzfeld and Rice • have ascribed

this to the slowness of the exchange of energy between internal and external degrees of freedom, and Kneser •' has given the detailed formulas for this process.

In diatomic molecules the internal degrees of freedom consist of one degree of vibration and two of rotation. The effect of vibrational degrees of freedom on sound absorption has been investigated in many cases, but the exchange between translational and rotational energy is so effective--i.e., the dispersion frequency

FIo. 1. Ultrasonic in-

terferometer for gases and vapors. Modified from form used by Stew- art. Note outside sepa- rator bars giving greater rigidity. C = interferometer head, G-- guide- bar framework, H= exhausting tube, L-lead through, P--piston, Q -- quartz crystal, S = syl- phon bellows.

* Work supported by ONR. $ A dissertation presented to the Faculty of the Graduate

School of the Catholic University of America in partial fulfillment

of •ti•e requirements for the degree of Doctor of Philosophy. ß F. Herzfeld and F. O. Rice, Phys. Rev. $1, 691 (1928). • H. O. Kneser, Ann. Physik 11, 761 (1931).

lies so high•that up to now only hydrogen has been investigated completely.

In the case of nitrogen, the vibrational specific heat, and (in dry nitrogen) the vibrational dispersion frequency is so low, that at frequencies above 1 mc/atmos the effect of the vibrational energy on sound velocity and absorption is negligible, and only the rotational degrees of freedom matter. For gases, in which energy transfer occurs by collision, the dispersion and the absorption per wavelength are (with possibly some exception for polyatomic molecules) functions of the ratio of frequency to pressure only.

Keller a studied sound velocity and absorption in N• at low f/p regions using the optical diffraction method. He secured experimental values of absorption which were about' 1.3 times as large as the classical absorption. Griffith 4 employed the steady flow technique devised by Kantrovitz * and showed that this relaxation time should be less than 2X 10 -"second.

The purpose of the present work was to undertake a systematic study of the velocity and absorption of sound waves in nitrogen with the ultrasonic interferometer over the frequency range of 3 to 109 mc/ atmos. The interferometer was chosen to study this problem because with it velocity and absorption measurements can be secured directly. Following the work of Hubbard, Curtis, Alleman, the Stewarts, Cheng, and others, the crystal vibrations are kept small and measurements are taken with the reflector close to the

crystal, thereby allowing a reduction of power below a value which could measurably affect temperature equi-

' librium between the crystal and its surroundings, and, what is more important, making possible measurements of some precision even at very high absorptions.

DESCRIPTION OF THE APPARATUS

The interferometer, Figs. 1 and 2, is of the variable path type developed by Hubbard • for ultrasonic work

a H. K. Keller, Physik. Z. 41, 390 (1940). 4 W. C. Griffith, J. Appl. Phys. 21, 1319 (1950). • J. Kantrovitz, J. Chem. Phys. 14, 150 (1946). • J. C. Hubbard, Phys. Rev. 35, 1442 (1930); 36, 1668 (1930);

38, 1011 (1931); 41, 523 (1932). 472

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23:53:22

DISPERSION AND ANOMALOUS ABSORPTION IN NITROGEN 473

with gases. To the ratchet stop of the head was fastened a hand wheel which had a diameter of 2-• inches and had 250 accurately spaced divisions. This wheel was divided on a circular dividing engine of the Societ• Genevois and the divisions were read by a microscope. This arrangement permitted accurate readings to at least 1/2000 mm.

In the lapping of the piston into the cylinder, it is extremely important that the fit of the piston be suffi- ciently accurate to prevent any "wobbling" of the piston as it is displaced from the crystal. Any "wobble" can be detected by comparing gas reaction readings taken as the piston is moved away from the crystal with results secured moving toward the crystal.

The face of the piston and the plane on which the crystal rests were polished until they were plane parallel

TOP VIEW

Brass R•ng

Phosphor • •uartz Bronze Sprir• . Tubes

SIDE VIEW

Tc•rce• rnal

To Vacuum

System.

Fro. 2. Detailed drawing of the crystal mounting and leads to the external circuit.

to within two wavelengths of light. This eliminated multiple peaks and increased the acoustic reaction of the gas column on the driving crystal. Figure 3 shows the reaction peaks before and after processing.

Electrical connection was made to the upper electrode of the crystal by means of a phosphor bronze spring which was soldered to a small brass ring, and then to a wire connection to the external circuit through the exhausting tube in the interferometer head. This wire connection was insulated from the exhausting tube by small l•yrex beads. The crystal was X-cut with a frequency of 2.992 mc/sec, and plane parallel to three wavelengths of the green mercury line.

The electrical circuit, Fig. 4, consisted of an electron- coupled oscillator, two-stage amplifier, and an LC circuit.

The oscillator was a BC 221-N Army surplus fre-

•11111111 IIIIIIIII IIIIIIIII Iiiiiii

25

5 05 I0 .15 .20

Reflector D•splacement mm

E25

.-

*- 20

&

• 15

.05 .10 .15 .ZO

Reflector Displacement mm

Fro. 3. The structure of the reaction peaks depends very greatly upon the alignment of the crystal and the reflector piston. After the piston and crystal support are polished plane parallel, the double peaks disappear and the peaks are increased in magnitude and become much sharper.

quency meter which covered the range of 0.125 to 20 mc/sec and was used at 2.992 mc/sec. The tank circuit was modified by the addition of a cylindrical condenser which had a micrometer screw as an inner

grounded conductor. The requirements of frequency and amplitude sta-

bility are exceedingly demanding in interferometer work. In the work on N•. it was noted that at the frequency used a change in frequency of 60 cycles was

o

!

Oscdlator I Ampl•fer I L C. Cwcult ,

1%o. 4. Tn the electrical circuit the output of the amplifier was loosely coupled to the LC circuit to isolate the oscillator from the electrical changes occurring in the LC circuit, as the reflector is moved.'awa¾ from the crystal.


23:53:22

474 ALFRED JOSEPH ZMUDA

sufficient to move the galvanometer deflection out of the bottom of the crevasse to a point off the scale. This corresponded to a current change from 12.3X 10 -4 to 24X10 -4 ampere. Therefore, an oscillation drift of 2 cycles/sec at a frequency of 2.992 mc/sec is sufficient to nullify a set of readings.

The amplifier was a conventional two-stage resistance- capacitance amplifier and was link coupled to the LC circuit.

The LC circuit is a wave meter that is tuned to the

oscillator frequency and serves as the link between the amplifier and the crystal. It also serves to furnish power to the crystal as well as to register the reaction on the crystal.

The current was measured by means of a No. 15 vacuum thermocouple (American Thermo-Electric Com- pany) connected to a moving coil galvanometer (Leeds and Northrop). Shunts permit a variation of sensitivity.

The interferometer was connected to the nitrogen tank by the gas handling system shown in Fig. 4. The nitrogen was passed through a trap which was im- mersed in a mixture of dry ice and alcohol. A trap for the condensation of water vapor and a bulb for ad- mitting water vapor were also available, but were not used. A Cenco Hyvac vacuum pump was used to evacu- ate the system.

EXPERIMENTAL PROCEDURE

The interferometer and the gas handling system were thoroughly evacuated and then filled with nitrogen at a pressure of about an atmosphere. This nitrogen, in turn,

TAB•.V.I. Absorption coefficients.

f mc a•exp P in cm

of Hg p atmos •exp cm -1 c•class ½m -1 •class

75.11 3.04 1.79 1.22 1.47 74.03 3.08 1.80 1.24 1.45 61.41 3.71 2.14 1.49 1.44 58.88 3.87 2.23 1.56 1.43 47.08 4.84 2.9 1.95 1.49 35.06 6.50 3.57 2.62 1.36 30.80 7.41 4.32 2.99 1.44 25.79 8.8 5.25 3.57 1.47 25.79 8.8 5.25 3.57 1.50 23.18 9.84 5.27 3.96 1.33 20.56 11.09 5.97 4.46 1.34 14.52 15.70 9.01 6.32 1.43 13.64 16.7 9.00 6.73 1.34 11.14 20.46 11.63 8.25 1.41 9.36 24.3 13.7 9.82 1.40 7.06 32.2 17.4 13.00 1.34 6.87 38.18 18.13 13.3 1.36 6.69 34.08 18.27 13.70 1.33 5.44 41.9 23.88 16.9 1.42 5.19 43.9 23.70 17.6 1.35 4.82 47.31 27.6 19.07 1.45 4.35 52.41 27.85 21.1 1.32 4.04 55.47 27.54 22.76 1.32 3.76 60.6 34.16 24.4 1.40 3.59 63.5 33.3 25.6 1.30 3.07 74.0 39.2 29.95 1.31 3.07 74.0 41.33 29.95 1.38 2.65 86.0 45.79 34.69 1.32 2.09 10.90 58.5 44.00 1.33

was pumped out and new nitrogen was admitted to the system. After a delay of about 10 minutes the pre- liminary electrical adjustments were begun.

Following Hubbard, 6 the reflecting plate is placed at a distance r--(2n-1)X/4 away from the crystal; and then the crevasse is made symmetrical by the adjust- ment of the oscillator frequency and the C of the LC circuit. After the crevasse is made symmetrical, the reflector is moved away from the crystal and the crevasse bottom goes through cyclical changes lifting at resonance to sharply increased galvanometer current readings. The resonance positions gave the values of X/2 which were used in the velocity calculations. In determi- nation X/2 in the range between 38 and 76 cm of Hg, the first and last five peaks of about 40 peak readings were taken and the method of transits was applied. This method was used whenever the number of peaks permitted even a limited statistical analysis (down to •-6 cm of Hg). In the range between 2.09 and 6 cm of Hg, the difference between the first and the last peak reading was taken and this number was divided by n-1. The number of peaks was limited by the amount of reaction in each region and dwindled to 3 peaks at 2.09 cm of pressure.

Absorption measurements require a set of minimum and peak thermocouple currents as well as a detailed peak structure of one of the peaks; these measurements were taken as the reflector was moved away from the crystal. The form of the P and Q functions makes it possible to derive absorption and reflection coefficients from peak decrement and peak structure. Two methods were used to calculate the absorption' the peak width method 6.7 and a method developed by Hubbard for Swomley 8 for use in high absorption regions.

DISCUSSION OF ABSORPTION MEASUREMENTS ø

The classical absorption per cm, a, is given by

O•class-- •(oo2/•ogo3)(•+IK(•y -- 1)/½p). (1)

In view of the well-known relation between heat con-

ductivity for diatomic gases and their viscosity, this may be rewritten as

O/class = • ((.02/p go 3) 1.407•. = (2/3Vo)(1.407/1.40)•wVp, (2)

where the relation has been used

pVo •'= p•,RT/M= •,p= 1.40p. (3)

At 1 atmos and 29øC, one has for nitrogen at f--2.992 mc/sec,

? R. S. Alleman, Phys. Rev. 53, 898 (1938). 8 E. E. Swomley, Phys. Rev. 69, 632 (1946). 0 The author is indebted to Professor K. F. Herzfeld for the

theoretical developments in this and the succeeding sections.


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o c H TO k•erferom H pump

Air

Fro. 5. The vacuum system and its component parts. A="T" joint, B--hoke valve, C-metal-glass seal, D--mercury vapor trap, E--open end manometer, F--glass bulb, G--water vapor trap containing a mixture of dry ice and alcohol, H = stopcocks.

KohleH ø has shown that

7= 1.27 lprc, (4)

where rc is the time between collisions which for Ns at 1 atmos and 300øK equals 2X 10 -•ø second.

Since no detailed theory exists at present for the energy exchange between translation and rotation, the expressions for the exchange between translation and vibration will be used. The formula for the absorption excess in this case has the form

(5)

Here C' is the rotational specific heat, i.e., R, C•=(5/2)R, Cp=(7/2)R; r is a characteristic time which is connected with the relaxation time of rotation, rr, by

r= (C,-C')rr/C,= (3/5)rr. (6)

Equation (5) may be written

(a'V/o•)(V/Vo) •= (2/21)o•r/(l+o•r •) = (2/21)o•r(1-o•r•). (7)

At low frequencies, c0•r•<<l and V= V0 and using Eqs. (2), (4), and (7) we have

a'/acla,= (2/21)(3/2)(1.40/1.407 X 1.271)r/r, =O.067r•/r,. (8)

At a frequency so high that c0•r • is not negligible compared to 1, but still small, one has

aexp/ae lass = lq-O.O67(r•/r,)(1--co2r•). (9)

Therefore, one can from the ratio Otexp/Otelass at low frequencies find rr/r•, the number of collisions necessary for energy transfer between rotation and translation. The ratio should be constant until c0•r • is not any more negligible compared to unity. This is the region in which velocity dispersion also becomes noticeable.

Table I shows this ratio over the entire f/p range covered. Since no dispersion was found in the region from 3 to 50 mc/atmos, the ratio should be a constant or, at the most, should have changed only a small amount. Taking the mean in this range gives aexp/aelass

10 M. Kohler, Z. Physik $, 715 (1949).

= 1.40, from which it follows that

or

r•/rc=0.40/O.067 = 6 collisions

r•=6X2X 10-•ø= 1.2X 10 -ø second.

(lO)

(11)

It further follows that the frequency of the midpoint of the dispersion curve is

f•= 1/2,rr= (1/2,r)(5/3)l/r•

= 220 mc/atmos. (12)

The change in the ratio of ae,•p/a•]•, that one should expect at 100 mc/atmos is, according to Eqs. (8) and (9),

(alfaclass)- a0fa½lass = (1-- 1.40)20.25X 10 -• =--0.08, (13)

where a•=ae,•p at 100 mc/atmos and a0=a•xp at 3 mc/atmos.

OtexpfOtelaas = 1.40-0.08= 1.32.

Looking at the graph Fig. 6, it is shown that there is a definite trend toward this value.

Analyzing the graph, one sees that the ratios are much higher in the f/p region extending from 3 to 10 mc/atmos than they are in the region from 10 to 50 mc/atmos. The former measurements were taken at the beginning of the experiment before the handwheel and microscope were used. The reaction peaks are exceedingly narrow in this region, making it very difficult to measure the /Xr used in the peak width method of calculation in Hubbard's theory. In higher f/p regions this did not exist, because the peak structure started to broaden out. Under these circumstances, it was concluded that the decrease in the ratio between 3 and 50

mc/atmos did not have any physical significance. Measurements are being made in this laboratory in the low f/p range to improve these values.

OCcloss

_: % o o 140 -- '"

• o e c,

• e • o o

• oo o c I .•0

• 20

I10

I001 5 I0 • m C, 50 i00 e-T"•.

Fro. 6. Plot of Otexp/Otclass V$ logliP mc/atmos showing the decrease in the ratio in the dispersion region.


23:53:22

476 ALFRED JOSEPH ZMUDA

TABLE II. Velocity of sound in Ns at 29øC, 2.992 mc/sec. Vc•ass= 354.3 meters/sec.

vmeters sec

v2ometers V2oe (meters • • non- C_2 Vexp 2 Vtheør2 f/p sec sec / ideal R Vo 2 Vo •

3 354.4 12.56 X104 354.45 2.50 1.00 1.000 6.05 354.4 12.56 354.38 2.50 1.00 1.000

13.3 354.4 12.56 2.50 1.00 1.000 27.3 354.7 12.58 2.50 1.00 1.000 31 354.3 12.56 2.50 1.00 1.000 42.9 354.7 12.58 2.50 1.00 1.000 49.2 354.2 12.55 2.50 1.00 1.000 55 354.4 12.56 2.50 1.00 1.000 60 356.8 12.73 2.38 1.014 1.013 74.4 359.0 12.89 2.29 1.027 1.019 86.1 360.2 12.97 2.24 1.033 1.025

105 362.0 13.10 2.17 1.044 1.035 109.2 364.4 13.28 2.08 1.058 1.037

The peak width method of calculating absorption was used from 3 to 50 mc/atmos. Throughout this range the 1/q•,• vs m plot gave a straight line; and when a peak structure of the first or second peak was taken, Hubbard's approximation of tanh«•km=«%•, where •b,• = 2raq-• was valid.

Hubbard's method developed for Swornely s was also applied in the range 30 to 50 mc/atmos and gave results compatible with values secured by the peak width method; it was used exclusively for absorption calculations from 50 to 109 mc/atmos.

The value o•xp/O•lass= 1.40 was secured by taking the arithmetical mean of the ratios secured in the range 3 to 50 mc/atmos. The error in this value was secured by taking the deviation of each value from the mean, squaring the deviation, dividing by n-1 and then extracting the square root. This procedure gave a final value of O•xp/O•lass in this range of

aexp/ae lass = 1.404- 0.06, (14)

which is in satisfactory agreement with Keller.

DISCUSSION OF VELOCITY MEASUREMENTS

There were some velocity measurements, outside the dispersion region, taken at temperatures different from 29øC. These readings were reduced to the velocity at 29 ø by the classical temperature correction equation.

V2o= Vt[(273+29)/(273+ (15)

In the pressure region between 76 and 38 cm of Hg, it was possible to secure between 40 and 50 reaction peaks. Very precise measurements were taken in this region with the aid of the microscope and hand wheel which were previously described. These measurements were compared with theoretical calculations of velocity taking into account the deviations from ideality. This theoretical expression is given by

V2: V0a{l+2p[Bq-1---(dBq 1RT dabs]

where

v0= c0= C•ø= (7/2)R, and -•= 1.40.

(16)

B is the second virial coefficient as defined in the

equation of state

PV=RT[lq-B(T)i/V. . . ]. (17)

Values of B and .its derivatives were secured from

two sources for comparison' a report by Hirschfelder, Bird, and Spotz • and from values of Van der Waal's a and b taken from the International Critical Tables.

Both sets gave practically identical correction values. As had been done with the absorption measurements,

the experimentally secured velocities were compared with theoretical values secured from the equations of Kneser.

The dispersion formula is

(V•/Vo•) -- l=RC'co%•/Cv(Cv-C')(l+co%•). (18)

At the beginning of the dispersion region, this may be written

(V/Vo)- 1= (2/21)•0% •. (19)

Table II and Fig. 7 give the experimentally secured values of the velocity with a frequency of 2.992 mc/sec. The agreement of these values with the theoretical values calculated for a non-ideal gas at low f/p is very good. At pressures lower than 37.66 cm of Hg the deviations from ideality are negligible. No increase in velocity was found in the region between 3 and 50 mc/atmos; but the velocity started rising at about 50 mc/atmos and increased to a value about 2« percent larger than the classical value at 109 mc/atmos, which was the highest f/p value attained in this experiment. The table shows that the dispersion values lie between the two values of V•=12.553X104 meters/sec and V2= 14.888X104 meters/sec, which are the lower and

102

z

z

z

z

z

z

z

V 2,

106

104

I00

98

96

, 5 •o •p mc. 50 •)o •--f--•.

Fro. 7. Plot of V•/Vo • vs logf/p mc/atmos for nitrogen at 29øC.

n Hirschfelder, Bird, and Spotz, Contract N. Ord. 9938.


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upper velocity limits, respectively, in the dispersion region, calculated with -• = 7/5 = 1.40 and -• = 5/3 = 1.67, respectively.

The agreement of the measured velocities and the values calculated from the theoretical expression for the beginning of the dispersion region is good. However, it may be noted that the experimental values rise a little more rapidly than predicted by theory. A dis- agreement in the detailed shape of the curve is not surprising, since the theory is developed for vibrational degrees of freedom.

However, since only 6 collisions are necessary to transfer energy between translational and rotational degrees of freedom, the rotational dispersion region covered overlaps the region in which a dispersion due to translational effects should begin. Translational dispersion was found in helium by Greenspan ;•' and if the Stokes-Navier dispersion theory is applied to a diatomic gas, one finds that the change in velocity due to translational relaxation is, in first approximation

V/Vo= 1+4/3R •' or at 109 mc/atmos, 1+0.007, (20)

where R is the Reynold's number and equals It is concluded therefore that the velocity change

found is a combination of translational and rotational

dispersion. The changes in C•/R were calculated from

C• 1 --= . (21) R

• M. Greenspan, J. Acoust. Soc. 22, 568 (1950).

TABLE III.

f mc

p atmos •theoretical •exp

3.04 0.06 0.21 3.71 0.07 0.28 3.87 0.07 0.26 7.41 0.10 0.21 8.80 0.12 0.24

32.2 0.24 0.26 41.9 0.26 0.27 55.5 0.32 0.35 60.6 0.33 0.32-

THE REFLECTION COEFFICIENT

According to Alleman, 7 Herzfeld's intensity reflection coefficient is related to the /5 in •k = 2raq-lg, which is used in interferometric calculations, by -•=e-•.

Table III shows a comparison between some theoretical and indicative experimental values of/5. In the region between 3 and 30 mc/atmos, the experimental values are between 2 and 4 times as large as the theoretical value. Beyond 30 mc/atmos the agreement is very good.

ACKNOWLEDGMENT

The author wishes gratefully to acknowledge the aid and guidance given throughout the problem by Pro- fessor J. C. Hubbard. He also sincerely appreciates the theoretical discussions and interest of Professor K. F.

Herzfeld.


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Dispersion of Velocity and Anomalous Absorption of Ultrasonics in Nitrogen