Resonance in crystal beams of sodium-ammonium...

414 E. C. Stoner

McDougall, J . and femner, E. C. 1938 Philos. Trans. A, 237, 67.M ott, N. F . 1935 Proc. Phys. Soc. 47, 571.M ott, N. F. and J ones, H. 1936 “ The Theory of the Properties of Metals and Alloys.”

Oxford.Slater, J . C. 1936 a Phys. Rev. 49, 537.— 1936 b Phys. Rev. 49, 931.

Stoner, E . C. 1931 Phil. Mag. 12, 737.— 1933 Phil.Mag. 15, 1018.— 1935 Proc. Roy. Soc. A , 152, 672.— 1936 a Proc. Roy. Soc. A, 154, 656.— 1936 b Proc. Leeds Phil. Soc. 3, 191.— 1936 c Philos. Trans. A, 235, 165.— 1936 d Phil. Mag. 22, 81.

Tyler, F . 1930 Phil. Mag. 9, 1026.— 1931 Phil.Mag. 11, 596.

Resonance in crystal beam s of sodium -am m oniumseignette salt

B y W. Mandell, Ph .D.Chelsea Polytechnic, London

(Communicated by O. W. Richardson, —Received 13 November1937)

Introduction

I t has been noticed by several observers that, with longitudinal oscillations in certain crystal beams, on keeping the length constant and gradually decreasing either the breadth or] thickness, a sudden change of considerable magnitude may occur in the wave-length. This has been attributed to some resonance effect. Lack (1929) describes interesting results obtained with rectangular quartz beams of 30° cut. The X axis, as with Voigt, coincided with an electric axis and constituted the length direction of the beam. The breadth was parallel to the optic axis. With such a beam there is a low vibration frequency depending upon the length, and a high vibration frequency varying inversely as the thickness, the field being applied in the direction of the thickness. A square plate was used, and on gradually reducing the thickness, several discontinuities occurred on plotting the wave-length against the thickness. According to Lack these

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Resonance in crystal beams 415

discontinuities occurred at frequencies that could be identified with harmonics of the frequency which the crystal would have if it were vibrating in the direction of the length. That is, the longitudinal oscillation along the length could affect the frequency which was supposed to depend only on the thickness, so that the system acted as two coupled electrical circuits, although the two sets of oscillations were in planes at right angles.

Somewhat similar results have been obtained for quartz beams by Hitchcock (1930). He used the Curie (or perpendicular) cut, where the length is perpendicular to an electric axis and the thickness parallel to it, whilst the breadth is parallel to the optic axis. Hitchcock showed that the frequency of oscillations along the length suddenly changed in value on decreasing the breadth.

I t will be shown that the beams used in this paper are suitable for the study of this phenomenon, owing to their unique elastic and piezo-electric properties.

The piezo-electric properties op the crystals op seignette salt

The crystals used were of sodium-ammonium seignette salt

(NaNH4C4H40 64H20).

The theory of piezo-electricity applicable to all crystals has been completely solved by Voigt (1910), and a short account of the modification of this general theory applicable to these crystals has been mentioned by the writer in a former paper (Mandell 1928 a). There are three piezo-electric moduli only, namely

du = + 56-0 x 10-8, d25 = - 149-5 x 10-8, and d36 = + 28-3 x

Fig. 1 shows the orientation of the beams cut from the crystal and used in the present experiments. X, Y and Z represent the a, b and c crystallographic axes and are all at right angles. Three planes are now drawn each containing a pair of axes so that the planes also are mutually perpendicular. Three rectangular beams may now be cut as shown. Consider any one of the planes shown, say XO YL. The dotted line OL bisects the angle . In the case of the beam shown, (de) parallel to represents its length, (fe) the breadth, and ( fh)the thickness. This will be the convention used in the present paper, namely, for all of the three planes the length of the beam will always be along the direction of the bisector of the angle between the two axes considered. The breadth will always lie in the plane and will be at right angles to the length, while the thickness will be measured in the



416 W. Mandell

direction at right angles to the plane, that is, it will always be perpendicular to the plane and parallel to a crystallographic axis.

Since there are only three piezo-electric moduli, the general equations of Voigt (1910, p. 902) reduce to

Vz = dl iE1 , zx = d25E2, and xy = d36 Es.

The third equation, for instance, states that if an electric field Ez be applied along the direction of the Z axis, then in the perpendicular plane 1 7 , a shear strain is produced. If therefore the length ( of the beam bisects the angle XOY, longitudinal displacements take place along the length and along the breadth.

z

F ig . 1

M

Y

On referring again to fig. 1, if light metal plates be placed on the upper and lower surfaces ( gdef)and ( kmnh) and a steady potential difference be appliedto the plates, the crystal will either extend along (de) and contract along (/e), or contract along (de) and extend along (fe ), according to the sign of the electric field. If an alternating field be now substituted then the crystal may oscillate longitudinally in one of two directions, either along the length, or along the breadth, on condition that the frequency of the applied field is exactly equal to either of these natural frequencies which are determined only by the elastic constants and dimensions of the beam. All three beams act in a similar manner and if electric fields of the required frequency be applied in the direction of a crystallographic axis, that is, perpendicular to a plane, longitudinal oscillations will occur either along the length or along the breadth.




The lateral contractions accompanying an increase in length

FOR BEAMS FROM CRYSTALS OF THE RHOMBIC SYSTEM

At a later stage reference will be made to the elastic properties of the beams so that their values are determined here. The geometrical nature of elastic moduli is clearly set out by Voigt who shows that the elastic properties of a crystal are completely characterized by the combination of a bitensor and a tensor surface. For crystals of the most complex type, namely, those belonging to the triclinic system, there are twenty-one elastic moduli. The bitensor system is built up by combining the components of four vectors and supplies fifteen terms, each term being a usual component of a bitensor surface while the tensor system supplies a further six terms. The expression for the thermodynamic potential is then transformed into an algebraic expression of twenty-one terms representing these two surfaces, the coefficient of each term being one of the elastic moduli or a combination of two of them, and at the same time a component of the two surfaces. The bitensor surface is therefore a fourth power expression in x, y and z and the tensor surface, a second power expression. With the aid of these two surfaces any elastic modulus for a beam cut in any direction can easily be determined. Suppose that a new set of rectangular axes Z', arerelated to the original axes X, Y, Z, by the scheme of direction cosines shown:

X ' r Z'

X <*1 fix 7iY a 2 /?2 72

Z a3 /*3 7s

Then Voigt (1910, p. 590) shows that

sn = sn af + s22oc I + s33cc% + (2s23 + s44) af a§ + (2s 31 + afaf + (2s12 + s66) af

+ 2(s14 + s56) ocf a2 oc3 + 2(s25 + s64) <x\ ot3 + 2(s36 + s45) a§ a2

+ 2<2i(Si5a3 + Si6 0c2) + 2a |(s26a 1 + 524a 3) + 2al(534a 2 + s35a 1). (1)

This expression gives the extension modulus along the length when tensile stresses are applied in this direction, which coincides with the new X ' axis. This complex expression simplifies considerably when it is remembered that due to the symmetry of the crystal considered, twelve of the elastic moduli vanish, whilst the remaining nine have already been determined by a



418 W. Mandell

statical method (Mandell 19286, p. 122). Thus for beam C, the expression reduces to

4 i = ^iiai + ̂ 22̂ 2 (2<s12 + s66) a |a |= l/4(sn + s22 + 2*12 + *66) = 47-8 x 10-10, (2)

since the angles measured in degrees are = 45, a2 = 45, and a3 = 90.Suppose now, in beam C of fig. 1, the axis now along (OL), be rotated

through a further 90 degrees in the plane (OXLY), so that it is parallel to the breadth, then on substituting the appropriate values for the direction cosines in this equation we obtain the same numerical value for the extension for an equal tensile stress, one gram weight per square centimetre, along the direction of the breadth.

In like manner for beam B,

4> = s22A! + *33A4 + (2s23 + M /?!/?! = 37-5 x 10-10, (3)gives the extension modulus for stresses either along the length or the breadth, and for beam A,

4* = %$yi+ sn 7 i +(2*31+s5s) = 94*2 x io - 10, (4)

for the two directions.Again Voigt (1910, p. 592) shows that the general expression for the lateral

contraction for a crystal beam referred to the new set of rectangular axes is given by the expression

sh = siiPly\ + S22Plyl + sZzPlyl + SM{Plyl+Plyl) + szl{Plyl+l]\yl)+sAPiyl+Ply\)+suP2y2Pzyz+$ttPzyzPiyi+sMpiyiPzy2+ 556 A r d A r s + ^ 3 7 2 ) + 564/?2y2(/?3yi+ A y 3) + y 2 + A27 i)■f 5 i4( A y 2y 3 "f y f A A ) + * 25( A 7 i 7 3 + y l A A ) ~ f s36(Ay2yi_f y |A A )

+ Ayi[*i5(Ay8+ yx A )+ *i6(Ay2+ 7i A)]+ A y2l>26( A 7 i+ y2 A ) + A y3 + y2 A)]+ A y3I>34(Ay2+ y3 A )+ «35(Ayi+ y3 A)]-

For crystals of the rhombic system twelve of the elastic moduli again vanish. Let us apply this equation to beam B, where the Y' axis is along the direction of length ( OM), the Z' axis being at right angles to this and in the same plane, and the X ' and X axes coinciding. Then the angles measured in degrees have the values olx = 0, a2 = 90, a3 = 90, = 90, /?2 = 45, /?3 = 45,y x = 90, y 2 = 135, y3 = 45, and

*23 = *22A y !+ s33Ay3+523(Ay!+Ay2)+M Ay2Ay3)= -5-3 x 10-10. (6)




This gives the lateral contraction along the direction of breadth for beam B considered as a unit cube, for unit tensile stress along the direction of length. In exactly the same way as before, by considering the rotation of the Y' axis through a further 90° so that it coincides with the direction of breadth it can be shown that for unit tensile stress along the breadth the lateral contraction along the length is 5-3 x 10~10 for the unit cube.

For the contraction in the direction of the thickness the equation (5) is again used except that the terms are changed in a cyclical manner. Thus the equations reduce to

«L = si2 0ciftt + s3i aiflt = -21-05 x 10-10, (7)

*81 = *317i <*! +*12 7 la i = - 21-05 x 10~10. (8)

The first equation states that for beam B, unit tensile stress along the length for a unit cube gives a contraction along the thickness direction of 21-05 x 10-10. The second equation states that for a unit tensile stress along the breadth the contraction along the thickness is the same, namely 21-05 x 10-10.

In a similar way for beam A ,

*31 = *117 l a ! + *33 y l a I + *3 l(73 a l + 7 l a l ) + *55 73 a 3 7 l a l

= — 82-2 x 10~10 (9)

is the contraction along the breadth for the extension in length,

*|3 = *23/?!y!+*i2/?!yi = - e - 7 x i o - 10 (io)

the contraction in thickness for extension in length and

*12 = *i2<22i/?i + *23ai/?i = - 6 - 7 x 10-10 (11)

is the contraction in thickness for extension in breadth.Corresponding equations for beam C are

*12 = * lla i/?l + *22a iAi + *12(a iAi + a iAl) + *66a l A a 2/?2 = ~ 10*2 X 10"10( 12)

*31 = *3i7ia i + s2373a i = - 19-25 x 10-10, (13)

*23 = *23^171 + *3i/?!y! = - 19-25 x 10-10. (14)

These equations show, therefore, that all three beams have similar elastic properties.



420 W. Mandell

The electrical circuit

The electrical circuit used in the experiments was an improved form of that described by Pierce (1923). The crystal was placed across the plate and grid, but an oscillatory circuit replaced the plate resistance. It was found that the grid-leak was not essential and oscillations were much more vigorous without it. On varying the capacity of the condenser in the circuit the crystal started to oscillate strongly and continued to do so with constant frequency through a wide range of the condenser setting. A point of vital importance connected with this arrangement was that the frequency was solely determined by the crystal for there was no measurable change in wave-length on the substitution of other inductances and capacity in the tuned circuit.

The alternating field was applied to the crystal beam by means of two light vertical condenser plates which were made to press lightly against the faces of the beam by means of two fine metal springs. The dimensions of the plates were somewhat larger than the crystal faces to ensure a uniform field. The nature of all oscillations was checked by means of dust-figures with lycopodium powder.

Since the frequency of a crystal is dependent on the temperature, it was necessary to enclose the beam in a thermostat, “which was regulated by an electrical device, when measuring some of the small changes in wave-length on changing the lateral dimensions.

The theory of electrically coupled circuits applicable to the

SPECIAL CRYSTAL BEAMS UNDER TEST

Suppose for simplicity that square beams are used, and that the equivalent electrically coupled circuits are as shown in the diagram. The

e , = E s i n cot

Fig. 2

alternating potentials, assumed of sine form, impressed on the two circuits will differ in phase by 7r radians, for if the applied field causes an extension in one direction, an equal contraction occurs in the direction at right angles.



On employing the two circuit equations of standard form it may be shown that

— o)L + (oM j 4-

A = V **>+(V ■4 - ^ ] 2+■gives the amplitude of the current in either circuit.

In order to find the optimum values of co, for maximum current amplitude, it is convenient to square the expression for the amplitude, differentiate it with respect to oj, and equate to zero. In crystal oscillations the effects of resistance on the period can quite safely be neglected so that the expression obtained then reduces to:

[oj6C3(L +M) (L — M)2 + co*G2(3M + L ) ( M - L ) + + 1][(o*C2{M2- L 2) + 2 - 1] [oj2CM - oj2CL + 1] = 0.

On solving this equation, there are two different values for (o2, namely,

0)2 = C(L + M) and " 2= C ( L - M ) '

Hence it is to be expected fhat before exact adjustment is made the crystal will oscillate with one frequency only, but when the length is exactly equal to the breadth, two new fundamental frequencies will appear.


E xperimental results for resonance with the electrically

COUPLED CIRCUITS

A perusal of the elastic properties will show that the beams are ideal for testing the resonance theory. The piezo-electric theory requires that they shall be cut with the orientations shown in fig. 1. Then on applying the elastic theory it is seen, for this special orientation only, that the elastic properties are exactly identical for the length and for the breadth, which are the directions along which the crystal may oscillate. An experimental test was made with a square beam of type C (fig. 1), of side 19-329 mm. This beam was chosen because a fairly large specimen was available. The difficulty of the test will be realized when it is pointed out that the frequency of the oscillation along the length or along the breadth, for a square beam of this size is about 86,000 c./sec. Before resonance can occur, it is probable that the frequency for the two directions must not differ by more than, say ten cycles or even a much smaller number. The method of procedure was to



422 W. Mandell

start with a square beam and reduce the length in steps of about one-ten- thousandth of an inch. After every adjustment the crystal was made to oscillate and one frequency only was found. Then the breadth was reduced in a similar manner for there was no method of determining whether one was approaching or receding from the required condition. A few score of adjustments were thus made with the length and breadth. When resonance occurred the crystal oscillated in a more vigorous manner causing the plates which were pressing lightly on the crystal and supplying the alternating field (henceforth called the terminals), to emit a high pitched audible note. Two new oscillations appeared of wave-lengths 3835 and 3190 m., together with the fundamental of wave-length 3487 m. The nature of the readings on the wavemeter, showed without any doubt, that they were new fundamental frequencies, and not overtones supplied by the wavemeter. The two new frequencies disappeared at the same time after a few minutes. It had previously been found that the wave-length increases with rise of temperature, both along the length and along the breadth, for beams of type C, and from the symmetry of the crystal the coefficients of expansion with temperature should be of equal amount in the two directions. The temperature of the crystal when oscillating rises a small fraction of a degree above the temperature of the surroundings. The disappearance of the oscillations was attributed to a slight change due to unequal heating. On breaking the circuit and allowing the crystal to cool for a short time the oscillations reappeared. The method was also tried of moving the crystal a short distance between the terminals in order to change the frequencies by a cycle or two, due to exceedingly small changes in the air-gap. This method also proved successful.

In some of the earlier tests it was found that the wave-lengths of the new frequencies drifted slightly. This may have been due to the damping caused by the vibrating terminals, which would have the same effect as the addition of a resistance to the coupled electrical circuits. This can also be explained by the fact that the natural frequencies of the two modes of vibration may not have been absolutely equal. A difference of a few cycles would alter the values of the new frequencies somewhat, but no means were available of determining at what instant the two natural frequencies were identical.

The experiment was repeated some months later, with the same crystal slightly reduced in size. For a square beam of side 19*218 mm. the fundamental wave-length was 3472 m. When resonance occurred, the longer wave-length was 3818 m., so that both values decreased by about 16 m. Both resonant frequencies disappeared before the shorter wave-length could be measured. This illustrates the extremely critical nature of the




adjustment required for resonance. In this case the resonant condition was probably obtained by a slight temporary change of temperature due to the handling of the crystal specimen. I t may perhaps, also be stated that more than two hundred adjustments were required to obtain the above results.

From the numerical results recorded, it may be calculated that the Coefficient of Coupling for the two equivalent electrically coupled circuits is 18%.

It would appear that this resonance effect, requiring such critical adjustment, differs entirely from any previously shown.

Other interesting results were obtained with beams of type C. Broad beams of small thickness were employed and the wave-length was measured, while the length and thickness were kept constant and the breadth was gradually decreased. Since the beams used were of different dimensions a perusal of the table of results does not readily furnish much information nor does a graphical representation showing the relation between the wavelength and breadth. I t is much more instructive to plot the actual wavelength in metres divided by the lengths of the beams in millimetres, as ordinates, and the breadths as abscissae, the length and thickness being kept constant for each beam.

The results obtained for two beams only, of type C, are shown in graph 1, although the same effect was obtained with six different beams. On decreasing the breadth, it is seen that the wave-length decreases and a close examination shows that the curvature changes sign. When the breadth was reduced to about 0-60 of the length the wave-length suddenly increased by about 160 in 3000 m., and on further decreasing the breadth, a new curve was obtained. Experiments with dust-figures seemed to prove without doubt that the beam was oscillating in a longitudinal manner along the length for all breadths, although, for the very broad beams, the dust figures were not very sharply defined. In the table 3) readings are given for one beam where both wave-lengths were obtained separately for the critical breadth of the beam, the breadth being 10-965 mm. and the length 18-314mm., so that the ratio of breadth to length was 0-60. These discontinuities in the curves seem to be similar to those obtained by Lack and Hitchcock, when using quartz. If the results shown in the present paper can be attributed to resonance effects, then they seem to differ considerably from the results obtained with the square beams. With the latter two new frequencies appear at resonance. For the beams of decreasing breadth, single fundamental frequencies occur on both sides of the discontinuity. Still, it will be seen in the later part of the paper, that it is possible for oscillations along the length of a beam of length (l) and breadth (0-60Z), to



424 W. Mandell

have the same frequency as oscillations along the length of a beam of length (0-60Z) and breadth ( l).

Experiments with changes in the lateral dimensions of the beamsExperimental results were obtained with the three beams shown in

fig. 1. In the first part of the experiments the length and breadth were kept constant and the wave-lengths were measured on gradually reducing the thickness of the beam. In the second part of the experiment these thin broad beams were used and, keeping the length and thickness constant, the breadth was gradually cut down and the wave-length measured. For a long beam of very small lateral dimensions, the fundamental frequency of the longitudinal oscillations by this dynamical method should be determined by the simple equation:

^ 2 l j ~ p ’°r (15)

where l is the length of the beam, p the density of its material, Y the Young’s modulus and E the ‘ ‘ Dehnungsmodul ’ ’, a term used in crystal physics. I t should be pointed out, however, that this formula does not apply for beams of large lateral dimensions.

Rayleigh (1894) has estimated theoretically the error involved in neglecting the inertia of the parts of a vibrating rod of isotropic material which are not situated on the axis of the rod. Chree (1889) considers isotropic beams of rectangular cross-section and shows theoretically that any increase in the moment of inertia due to the lateral dimensions of the beam causes an increase of wave-length, whilst Davies (1933) has obtained a formula of corrections for crystals of Rochelle salt. I t was found by experiment that Chree’s formula subject to certain modifications, agreed best with the obtained results, if modified to the general form:

+ K bf ) , <16>

where A is the observed wave-length for a beam of length l, breadth b and thickness t, A0 is the wave-length when b and t are very small, and kx and are constants varying as the square of the appropriate Poisson ratio, namely for the directions of length-thickness and of length-breadth respectively. These latter values can be obtained from the calculations already made for the lateral contractions accompanying an increase in length for the beams under test. I t should be pointed out, however, that all the above correction formulae are based on the fact, that for an extension in

A2-Aj> = A2( kx



•049 6-287 5-605659 4632 4604659 4628 4603

0 - 4 - 1

6-9602654

(facing p. 424)




length there are accompanying contractions along the breadth and along the thickness due solely to that extension. However, owing to the special piezo-electric properties of the beams used here, none of these formulae would be expected to apply. Consider beam C for instance, although the same reasoning applies to all of them, and suppose that a steady potential be applied in the direction OZ. Then the beam extends say, along the length and contracts along the breadth. The extension along the length causes a contraction along the thickness, and the contraction along the breadth an extension of equal magnitude along the thickness, so that for static fields there should be no change in the thickness of the beam and the correction term for the length-thickness directions is zero. For alternating fields experiments showed that the change in wave-length due to change in thickness was exceedingly small. The readings for the three types of beams are tabulated. I t will be seen that, where measurements are made on reducing the thickness, there is a very small reduction of wave-length, in fact it is almost of negligible amount. For instance, with one of the beams ((73) the thickness was reduced in steps from 16-403 to 5-105 mm. and the reduction in wave-length was from 3085 to 3055 m., a difference of about 1 % for a change in thickness of 11-3 mm. Similar results were obtained for all the beams so that the correction term (Aq may be omitted for thin beams. With regard to the second term the application of the staticfield produces an extension along the length which causes a contraction along the breadth as the correction formulae require, but there is now an added complication due to the fact that a further contraction takes place along the breadth due to the electric field. Hence it would be expected that for thin beams, on keeping the length constant and gradually reducing the breadth, the formula

V -A $ = A ^ 5 )

should apply, but k2 for the different beams would not be expected to vary as the square of the Poisson’s ratio.

Graph 2 shows the results obtained for beams A of fig. 1, with their lengths and breadths in the ZOX plane. Comparatively broad thin beams were used. While the length and thickness were kept constant the breadth was gradually decreased and the wave-length measured. From the theoretical calculations for a centimetre cube, for an extension along the length of 94-2 x 10~10 cm. the lateral contraction along the direction of thickness is 6-7 x 10-10cm. and along the direction of breadth 82-2 x 10~10 cm. Poisson’s ratio for the length-breadth direction (0-87) is therefore

Vol. CLXV. A. 28



426 W. Mandell

• l =19-857

Graph 2

l = 16-561l =20-253

Graph 1




exceedingly large. The readings for beams 4 are tabulated and the results for three of them are shown in Graph 2. Since the three curves do not coincide it follows that the wave-length per millimetre length is a function of both the length and the breadth. Furthermore, for the beams of lengths 16-573, 18-026 and 19-857 mm., on reducing the breadths to a very small value the wave-lengths per millimetre length taken from the graph are 245-0, 244-7 and 245-0 m. respectively, while for beam {A 5), which is not shown on the graph, the value is 244-7. It is of interest to compare these values with those obtained on substituting the values found by the statical experiment equation (4) in (15). This calculated value is 234-1 m./mm. length, as compared with 245-0 m. by the present dynamical method, a difference of rather less than 5 %.

Graph 4 is drawn with (A2 —Aq)/A2 as ordinates and as abscissae. For the three beams shown in graph 2 it is found that all the points lie closely to a straight line of slope 0-430, which therefore gives the value of k2. I t is of interest to note that the formula applies over a very wide range, for in some of the broader beams the ratio of breadth to length is between 0-8 and 0-9. The results obtained with beam (.4 5) are not plotted on the graph. Instead, two more rows are added to Table (A 5), one giving the wavelength as calculated by the formula

A2- (4510) /A2I 0-430- (17)

where 4510 is obtained by multiplying the length of the beam in millimetres by 244-7. The lower row gives the difference between the measured and calculated wave-length. With this particular beam many readings were made to discover any small discontinuities due to resonance, and great care was taken that the temperature remained constant throughout the experiment. I t will be seen that the calculated and experimental values agree very closely. The wave-length increases by nearly 22 % for the broadest beam of (4 3) as compared with the infinitely narrow beam, and it may be noticed that this exceedingly large increase is associated with the abnormally large value of the Poisson’s ratio for the length-breadth direction of the beam.

The experimental results obtained with the beams of fig. 1 are tabulated and are shown in graph 3. A unit tensile stress, 1 g. weight/sq. cm., along the length of a centimetre cube, gives an extension along the length of 37-5 x 10~10cm., a contraction along the breadth of 5-3 x 10-10cm., and a contraction along the thickness of 21-05 x 10-10 cm. I t will be seen from the tabulated results that the change of wave-length on decreasing the thickness

28-2



428 W. Mandell

is again small in spite of the large value of the lateral contraction in this direction. The Poisson’s ratio for the length-breadth direction is small,

l =18-354l =23-210l =26-975

Graph 3

namely, 0-14, or about one-sixth of the corresponding value for the A beams. In graph 3 three thin beams are used of lengths 18-354, 23-210 and 26-975 mm. and the results show that the percentage increase in wave-




l =16-573l =18-026l =19-857

l =23-210l =26-975

0 0-2 0-4 0-6 0-8Z2

Gra ph 4



430 W. Mandell

length with increase of breadth is very much less than for the A beams Thus on reducing the breadth of the B beams the maximum percentage change of wave-length is rather more than 3 %, while for the A beams of similar dimensions the change is about six times as much, so that the percentage changes of wave-length vary almost directly as the two Poisson’s ratios for the length-breadth directions for the two sets of beams. Graph 3 shows that on reducing the breadth to an infinitely small value, the wavelength per millimetre length is 143-0 m. for all three beams. On substituting values found by the statical method, equations (3) and (15), the wavelength per millimetre length is 147-8m. by calculation. Thus the wavelengths agree to within about 3 % by the two methods for these beams. The lower part of graph 4 is obtained by pi otting (A2 — A§ )/A2 as ordinates and b2/l2 as abscissae. The points lie fairly closely to a straight line, up to a point where the ratio of breadth to length is about two-thirds, but for broader beams the slope of the curve increases rapidly. The slope of the straight portion of the graph is 0-08 which therefore gives the value of k2 for these beams.

The results obtained with the C beams have already been referred to. A unit tensile stress along the direction of length of a centimetre cube gives an extension of 47-8 x 10-10cm., a contraction along the breadth of 10-2 x 10~10cm. and a contraction along the thickness of 19-25 x 10-10 cm. Owing to the anomalous nature of the results a curve is not plotted on graph 4. From graph 1 it is seen that long beams of small lateral dimensions have a wave-length of 158-6m./mm. length as compared with 166-8 m. obtained from the statical experiment, equations (2) and (15), where broad beams were employed. There is thus a difference of about 5 % by the two methods. In comparing the values obtained for the three sets of beams by the dynamical and statical methods which differ by 3-5 %, it should be pointed out that broad beams were used in the statical experiments and infinitely long ones in the dynamical experiments. Furthermore no corrections are applied to the dynamical experiments for the “ converse Piezo-electric effect” . When a steady potential difference is applied to the crystal it extends along its length with the result that electric charges are developed due to this extension. These produce stresses which prevent the crystal from extending to its full amount. This is equivalent to a decrease in the extension modulus so that the wave-length measured by the dynamical method is slightly too small. For a static field the decreases in the extension moduli vary as the square of the piezo-electric moduli which are large for the crystal beams under test. If this correction were applied then the values obtained by the statical and dynamical experiments for two of the sets of beams would be in still closer agreement.




The dynamical method is limited in its application for the determination of elastic constants since these are the only three differently oriented beams that can be employed for the production of longitudinal oscillations for this particular crystal, excepting the L-cut described by Cady (1937). In the statical experiment many difficulties arose since the beams were rather short, while corrections had to be applied for the depression of the supports and the “ bite” of the knife-edges into the crystal face on adding the loads. With one beam there was a depression of only ten interference fringes per load, so that it was necessary to read to one-tenth of a fringe, but the greatest difficulty arose owing to the fact that after applying the load a slight depression continued for a considerable time. In spite of these difficulties the results by the two methods are in good agreement.

The writer is pleased to acknowledge his indebtedness to the Trustees of the Dixon Fund for a grant for the purchase of apparatus.

His best thanks are due to Mr W. B. Medlam for advice on coupled circuits and to Dr Lownds for his kind interest in this work.

Summary

Experiments are made with longitudinal oscillations in crystal beams of sodium ammonium seignette salt. For a square beam a theory is deduced for the interaction of two longitudinal oscillations at right angles. These are considered as analogous to two coupled electrical circuits. If the natural frequencies in the two directions agree to within a small number of cycles per second, the theory requires that resonance shall occur with the production of two new fundamental frequencies. This result is demonstrated experimentally.

Experiments are also made to determine the change in wave-length with change in breadth and thickness. Theoretical calculations are made of the lateral contractions along the breadth and along the thickness, accompanying extensions along the length. The effect of thickness on the wave-length is found to be very small, in spite of the fact that the lateral contractions are large in two types of beams used. A change of breadth may change the wave-length by 20 %. The experimental results for two types of beams agree closely with an empirical formula, while one type of beam behaves in an anomalous manner.



432 W. Mandell

R eferences

Cady 1937 Proc. Phys. Soc. 49, 648. Chree 1889 Quart. J . Math. 23, 317-42.Davies 1933 Phil. Mag. 16, 104.Lack 1929 Proc. Inst. Radio Engrs, N .Y ., 17, 1123. H itchcock 1930 Rev. Sci. Instrum. 1, 13-21.Mandell 1928 a Proc. Roy. Soc. A, 121, 130.

— 1928 h Proc. Roy. Soc. A, 121, 122.Pierce 1923 Proc. Amer. Acad. Arts Sci. 59, 82. Rayleigh 1894 Sound, 1, 251.Voigt 1910 “ Lehrbuch der K ristallphysik.” Leipzig.

The alkaline perm anganate oxidation of organic substances selected for their bearing upon

the chemical constitution of coal

B y R. B. R andall, Ph .D., M. B enger, Ph .D., and C. M. Groocock, Ph .D.

Of the Bone Research Laboratories

{Communicated by Professor W. A. Bone, F.R.S.—Received 3 December 1937)

In Part VIII of the previous papers upon the Chemistry of Coal from these laboratories (Bone, Parsons, Sapiro and Groocock 1935) it was shown that on oxidation by means of boiling alkaline permanganate the main organic substance of the lignin -* peat -* coal -> anthracite series yields carbon dioxide, acetic, oxalic and benzene carboxylic acids, the proportions of the last named to the oxalic acid produced increasing with the maturity of the coal substance. Also, while every benzene carboxylic acid, except benzoic acid itself, was isolated from the oxidation products of some coal or other, in each and every case examined the penta- or hexacarboxylic acid predominated. Another notable result was that whereas no phthalic acids were detected among the benzenoid acids produced from lignins, and comparatively small proportions among those yielded by peats and brown coals, larger proportions thereof were yielded by bituminous coals and anthracites.

The results as a whole, while strongly supporting the view of the essential chemical continuity of the lignin -> peat -> coal -> anthracite series, and of



Resonance in crystal beams of sodium-ammonium...

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