Chapter 5

Ultrasonic study of elastic properties and phase transitions in Potassium Sulphamate crystal

The siiidies currietl out on the elastic properties of or/horhor~tbic Potcrssiuin

.sulph(iirrcite cg,srti/ tire described in this chapter. The synthesis of tile iiiaferiul,

~1urificliiir1)7. oys fo i grow~rh, identification ofthe r~zorpho1og)i of the gro,l,ri cr~~s to l

cind /~re/~c~ratiorr ( i f the sj~ecimen are described All ihe nine indeperidei~t second

order elcistic .iiill,ies.s constants, compliance constant, Poissoi?'.~ i.ti/io.s [ire

estir~ici~ed A /ho~.oiigh st~idy ofthe temperalure vnriu/ioii of elasfic constnr~ts over

ti rui~go qf iOli h-400 K has heen undertuken. /;'or tile sub.s~cii~/iiitio,i of /lie

ii/ti.(i.~oiii(~ .t/ii(!). Oi/fireriliul cunning calorin7etric .stirdy a/ cr very .~loli' iiecitirig

id? i . ~ (i/.so coii~ii~c~eii iirid presented.

Ultrasonic study of elastic properties and phase transitions

in Potassium Sulphamate crystal

5.1. Introduction

I'otassium sillpharnate is a very interesting piezo electric crystal. The

structure of the ct-!'stal has been studied using x-ray diffractio~l technique by

Brown and Cox [ j . l ] and Jeffrey and Stadler [5.2]. In both cases authors

suggested that the nitrogen valency directions are co-planar and that hydrogen

atoms lie in the smne mirror plane of symmetry of the molecule. 01dy Little

infor~naiion about crystallographic and physical properties of Potassium

sulphamatc crystals can be found in the literature. Hence the elastic properties of

these saniples using ultrasonic Pulse Echo Overlap (PEO) technique are carried

out. From earlier reports [5.1,5.2] it. is obvious that it crystallizes in the

orthorhombic symmetry with space group Pbcm [ ~ 2 h l ' ] with lattice parameters

a = 5.907b., b = 8.333A and c = 8.302A, Z = 4 molecules/unit cell. Density =

2.2078 gmlcc. Molccular volume = 102.16 x 10." m3 The illtermolecular 0-H

distance of Potassiuln Sulphamate is found to be 2.15 which is sig~iificantly

less than the van der waais contact distance of 2.8 A and constitutes a weak

hydrogen bond. 'I'hc prominent hkl planes of the crystal are [-loo], [OlO], (1201,

[I lo], [ I 001, [I -201, [O-1 01 [-]-lo], [OO-11 and have perpendicular distances

iioni thc origin as I . 14.1.49,1.04,0.97,1.14,1.68,1.75,0.09,1.7~ respectively. The

investigation shows conclusively that the bond angles around the nitrogen atom

are S-N-lI 1 1 0 . 2 and H-N-H llO.lO. These are close to the ideal tetrahedral

angle of 109.5" ant1 it must be assumed that the nitrogen atom has a S P ~

configuration with a lone pair electron in the fourth tetrahedral position. The

conclusion is in disagreement with the expected relationship between bond length

and bond order for the S-N bond. In potassiu~n sulphamate the S-N bond is 1101

significantly longer but thc configuration is SF".

Cox et al. [5.4] carried out neutron diffraction study on the crystal

structure of Potassium Sulphamate. In tneir study it was observed that the

hydrogen bonds were so disposed throughout the structure that no well-defined

cleavage was found in any direction of .he crystals. They also observed that

sulphamate ion in potassium sulphamate was not switterionic. It has pinacoid in

the directions [loo], [OlO], [OOl], prisms iri the directions [120], [ l o l l , dipyramid

in the direction [ l 111.

The thermo elastic constant T, and anisotropy of the thermal expansion a,,

and Gruneisen tensor G were studied 3y Haussiihl el ul. 15.31. Potassium

Sulphamate possesses large values of Ti] since it has strong Hydrogen bond. Thc

ui, of potassium sulphamate is smaller than in perchlorates or tetrafluroborates

and G is having almost the same valuc: as that of perchlorates. The elastic

constants of this crystal were measu~ed by using Ultrasonic Resonance

Technique [5.3].

DSC studies by Rapp [5.6] suggested that the crystal exhibits a first order

phase transition at about 450 K.

In this chapter, measurement of elastic stiffness constant by PEO

technique and investigation of the phas: transition of Potassium Sulphamate

crystal above room temperature of the cr),stal are presented, since no such study

using ultrasonic PEO technique is repor:ed in the literature. Measurements of

elastic constants have proved to be an excc:llent probe for phase transition studies.

Elastic constants are measured using PEO technique. The anisotropy in the elastic

properties of Potassium Sulphamate is well studied by measuring ultrasonic

velocity in the crystal in certain spe;ified crystallographic directions and

evaluating all the nine Elastic stiffness constants, Compliance constants and

Poisson's ratios and the results are presen.ed in Section 5.2.3. In Section 5.2.1 the

details of sample preparation are given and in Section 5.3.4 the surface plots in a-

h. a-c :incl I>-c pl;inL,, 01 ' phase velocity, slowness, Yo~ing's n~odulus ailti lincar

compl-cssihility art gilten.

5 .2 Experimental Technique

I a g c single crystals of size ( 35x30~12 ) mn13 have been gr01v11 from

supersat~~rated ailueous solution of the salt by slow evaporation technique ovei- a

period ~1 '60-65 days. The solution was prepared by dissolving equimolar fraction

of I(2CO; and Sulpl?;linic acid in double distilled water

The tcinl,erature of ~ h c bath was maintained constant at 305K and col~t~.olIed to

an accuracy 01' i 0 ' ' ' K . The details of the apparatus used for thc crystal growth

were ilcscribcd in scction 2.2. The photograph of the grown Potassiuln

Sulpha~ncitc crystal i s as shown in Figure 5.1. A drawing of tlie morpliology of'

the cr)stal is sliown in Figure 5.2. At 305 K the solubility is 70.4

g~~~l lOO~i i l . l - l~C) .~f l ie solubility curve is a s shown in Figure 5.4. The solubility

reaches a saturation value near 305 K. The positive temperature coefficient

i~~dica tcs that thc crystal can be grown by slow evaporation technique. The

arrangement of inolccules in the unit cell o f the crystal viewed along a- axis is

depicted in Figure 5 . 3

Figure 5.1 Photograph of grown Potessium Sulphamate Ctystal

Figure 5.3 Arrnngernent of molecule.^ in the Unil cell oJPota.vsium Sulyhanmte crvstnl ubozr~ a4mis .

Figure 5. B Morphology of Potassium Sulphamate Crystal

4 -

Figure 5.4 Soiubii~ty curve of Potassium sulphamate crystal

Tabfc 5 1 Comparison of the computed Mferfacial angles of the crystal with the

nieasured value.

I

( Crystal faces

1 100- 110

Interfacial angle between the faces

1 iO0-i l i

I- I r i00- i i i I -

102- i 02 t I 110-i10

021-02i t I 101-111

Computed

144.7

Measured

144 I-.

135

ioo-ilop 125 2

I35

70 4

70 4

104 5 --- 135

135

124

135

70 1 70

104

135

135 100-1 1 1 135

F i q u ~ 5.5 (a) Stereographic proaction of Potassium Subhamate crystal about c-axis

4

F@um 5 3 ( b ) S b ~ m p h i c pmjgcffun of Potassium Sulphamate crystal about h x i s

I'hc iirtcl-facial angles of the crystal were measured using an accurate

co~itact golriu~~lc~cr. Uy k110~~illg the lattice parameters, crystal systc111 and spacc

group one can construct a stereographic plot (Figures 5.5(a)-5.5(c)) by using thc

computer programme '.[crystal'. The natural faces of the sample have been

identified by tlie technique discussed in Section 2.2.3. The samples have been

cut using a slow speed diamond wheel saw. The samples have been cut so as to

have propagation directions along [loo], [OlO], [OOl], [ l o l l , [I 101 and 101 I].

The sample lengths along the measured directions are in the range of 0.8 to 1.2

cni. Since all c~lttings are niade very accurately, the error due to misorientation is

below -I 0.5". Thc samples are polished cautiously using ceriuni oxide powder to

optical reflection icvel so as to ensure good bonding of the transducer to the

sample surface.

5.3. Results and discussions

5.3.1 Velocity measurements

X-cut and )'-cut quartz transducers of fundamental frequency 10 MHz are

rnountcd on the sa~iiple using suitable material like silicone grease. Silicone

grease is a good bonding material in the range 300 K-420 K. Absolute velocities

at room tempcraturc (303K) have been measured for the selected direction and

modes. Details of measurement of the elastic constants of orthorhonibic crystals

are reported in the Section 1.2.1. The ultrasonic velocities are measured using thc

PEO technique [5.71. MATEC model 7700 pulse niodulator and receiver systeni

with its associated subunits have been used for the velocity measurements and tlie

details are described in the literature [5.8]. The basic experimental set up is

described in Scctioli 2.1

'fhe McSkimin At criterion [5.9,5.10] for bond correction has been

applied ~ising conil~utcr programme [5.16] to correct for the phase lag introduced

by the bonding rnidiunr o n the RF echoes. The details of the McSkirnin At

criterion and the dc~ails of the lnethodology of tlie computer programme are

discussed in Scctioir 2.1. Taking into account the uncertainties in measuring the

length and various other experimental limitz.tions, one has obtained an absolute

accuracy better than 0.3% in velocity nleasurr:ments.

KNH2S03, being an orthorhombic clystal, has the following nine second

order Elastic stiffness constants C1j, C22, C3), C44, C55, C66, C12, C13 and C23. The

diagonal elastic constants C, ,, C 22, C]), C44, C55 and C66 have direct relationship

with the suitable ultrasonic mode velocity given by C,; = p ~ 2 . It is found by PEO

technique that no cleavage is present in an!, direction of the crystal as suggested

by Cox ei al. [5.4].

Table5.2 Measured ultrasonic velocities and elastic constants of Potassium Sulphamate crystal at 300K L, T and QL represent longitudinal,

trarlsverse and quasi longitudinal modes respectively

$1. No

Direction of propagation

Direction Velocity of measured

polarisation 1 (mls)

Elastic constalrt

( G P ~ ) V-C, ~relatiori

Table. 5.3 Elastic stiffness constants, Elastic compliance constants, and PoissonS

ratios of Potassium Sulphamate at 300K

Elastic siilliiess Elastic colnpliance 1 constallt (GPa) constant( l~~ '~~n~N") Poisson's ratio

I'hc relationship between elastic constants for relevant ultrasonic wave

velocity for the orthorhombic system is explained in Section 1.2.1. The off

diagonal constants have the following relationship.

1 2 c,, =t.. =j7le2cl, + c 2 c S 5 -PV,, )e2c5. +c'c,~ - p v I 2 )I -'55 . ( 5 . 3 ) C S r

where s = sin 0 c = cos 8 where v is the velocity of propagation of respective modes. The

angle 0 = 3 5 l 3 ' for. a-b plalie, 0 = 45' for b-c plane and I3 = 35' for a-c plane. 'The

density of potassiu~ii siilphaiiratc is p = 2.207gmlcc

Of thc 18 propagation modes, 12 .ire sufficient to evaluate all the ninc-

second order elastic constants and rest of t l ~ e modes can be applied to cross check

the measured constants. Considering all experimental uncertainties the absolute

accuracy of elastic value is estimated to bt: better than 0.2% for diagonal elastic

constants and 1% for off diagonal elastic cc~nstants.

Starting with the well-known Christoffel equation [5.16], one can deduce

the relationship between the elastic cor~stants. Velocities of propagation of

various ultrasonic modes measured along, selected directions in the crystal arc

listed in Table 5.2. By measuring ultrasoni,: velocity in the Potassium Sulphamate

crystal in certain specilied crystallographic directions, the anisotropy in the

elastic properties of the c~ystal is studkd and the elastic stiffness constant.

compliance constants, Poisson's ratios [5.14,5.15] are evaluated and tabulated in

Table 5. 3. There are nine values of compliance constant which are the

components obtained from the matrix invt:rse of elastic constants. Also there are

six values for Poisson's ratio in orthorhombic crystal. The equations for

Poisson's ratio are discussed in Section 1.4.5

Some elastic stiffness constants c~f this crystal measured by this study

(PEO method) show appreciable deviation from those measured using Resonant

Ultrasound Technique [5.3] (RUT) Out of the diagonal constants, the constants

C 1 1 (0.91%), C 33 (4.8%) and Cd4 (1.9Y0) show deviation below 5%, wliercas

constants C2? (7.7%), Cj5 (41.6%) andC(6 (60.7%)exhibit deviation above 5%.

While off diagonal constants C 1 2 (21.5%), C13 (375%) and Cz3 (268%) havc

exhibited large deviation fro111 the RUT values.

5 . 3 . 2 Temperature variation of elastic constants

The temperature variation of the velocity of longitudinal and shear waves

propagating along various directions in t ~e crystal have been deterluined in the

range 300K-400K by keeping the sample mounted on a suitable holder in a

temperature controlled chamber. The r.ite of temperature change i l l all the

measurements is in the range of 0.5 to 1 K per minute. Investigation beyond

400K was not conducted because of bonding problems. The thermal expansion

has been negiected while measuring the variation of ultrasonic wave velocities

with temperature.

Figure 5.6 Temperature variation of C,, and C,, of PS

Elnstic rr~tottinlies urounrl355K

I'he variations in the elastic constants, CI 1 , Czz, C;,, C j j and C66, with

tenlpcrature are investigated and displayed in (Figures 5.6-5.8) The transverse

elastic conslants C41.Cjj and C66 show weak a~~oma i i e s around 355K while the

longitudiilal elastic constants Cli, C22 and Cjj do not show any variations. This

rucans that shearing caused by tangential stress is considerably affected by

thermal energy. Sincc variations of shear elastic constants with thermal energy is

appreciable. From the observed anomalies in the constants a weak phase

transition in the crystal around 355K is suggested. The thermal hysterisis during

cooling and heating cycles through transition point is about 3K.

Present Differential Scanning Calorimetric spectru~u (Figure 5.9)

performed at a very slow heating of 1°/rnil~ shows minor feature centered on

340K (67 '~ ) . The energy associated with this weak feature is 0.8892J/g. The

DSC study do not support a phase transition in potassiu~ii sulpliamatc near 355K.

This may be due to the fact that the weak anomaly around 355K may not be

associated with appreciable thermal change, where as the sensitive ultrasonic

technique can detect such a weak anomaly. Studies of Rapp [5.7] in DSC found

that potassium sulpharnate exhibits a first clrder in the temperature range about

450K. Thermal expansion study of CsNIi2S03 by Haussiihl et,al. [5.3] along

[OIO] direction showed sharp increase i l the value of thermal expansioti

coefficient at about 350K. A similar pheno~nenon was found in betaine NI-L;SO;

where a steep increase in the value of thernal expansion coefficient was found

along [001] direction near 360K. Both cases indicated the occurrence of first

order phase transition. In view of the observations fro111 other family rne~ubers of

Sulphamates one can suspect a phase transision for Potassium Sulphamate crystal

around 355 K.

Figure 5.7 Temperature variation of C,, and C,, of PS

1 1 6

Figure 5 8 Temperature variation of C2, and C33 of PS

I050 300 320 340 360 380 400

Temperature (K)

5.3.3 Investigatio~~ of phase transition using IISC

The thermal changes linked with .his crystal by the inetiiod of Ilift'crential

Scanning Calorimetry (DSC) have been obscrved in the range 30-I0O0C at a slow

heating rate of lo/n~in.

I A - YJ 0 3 TtC , CC

Tmo.rrtun, ( ' E ,

F~gure 5.9 DSC scan of Pot 3ss1urn suipharnate crystal

5.3.4 Surface plots of Phase velocity, Slowness, Young's modulus and Linear compressibility

The anisotropy of elastic wave I,ropagation in KNHlSO, single clystal

can be made clear by drawing the phase velocity surface plots in the a-b, b-c

and a-c planes following a well known ~rocedure [5.13,5.15]. In Section 1.2.1,

the expression for the velocities in the symmetry planes of the orthorhombic

lattice is given. If the velocities are cal2ulated using the corresponding elastic

constants for different values of angle 0 :n the range from 0 to 360 degrees with

some small step like 0.Io, then the resulting velocities can be plotted to get a

phase velocity surface for the correspond ng plane. For plotting the curve on x-y

plane of thc paper, one has to convert the velocity angle H to s- and y- co-

ordinates. These convet-sions are perfo -med using Equations 3.13 and 3.14.

[Figures 5.10(a)- 5.10(c)] show the phase velocity surface in the respective

plants, in which (a) gives the ultrasonic mode corresponding to quasi -

longitudinal [QL] inode with higher velocity of propagation (b) and (c) represent

pure shear [PSI and quasi - shear [QS] modes respectively.

I:rom the above plots, the three velocities in any direction of the

symmctl-) plancs can be easily found out by measurilig tlie length of tlie straight

line drawn Srom llic center to the curve at the required angle from the symmetry

axis. Thi: graphs Iiave bcen plotted with a scale of 1 cm for 1 klii 1s of velocity.

A greater insight into the elastic anisotropy of a crystal can be obtained

by plotting the in\,crse phase velocity (slowness) surfaces. The surfacc plots of

slowncss for KNH2S03 clystal are plotted in [Figures 5.1 l(a)- 5.11(c)]

Phase velocity-XY Plane 6000 I I I I

L 1 x x

. n o L I I I I I I -60U0 -4000 -2000 0 2000 4300 6000

Phase velocity (r/s)

Figure 5.10 (a) Surface plots of phase velocity ~n the XY plane of PS

Phase velocity-XZ plane

6000 7 I L 1 xx

- -,

600%00u - a 0 - lorn o idm $00 adou

Phase velocity (r/s)

Figure5.l0(b)Surface plots of pha se velocity in the XZ plane

Phase veloc:~ ty-YZ Plane 6000 7 ' L xx 1

- I- 600!600~l -3000 0 3000 6000

Phase velocity (=Is)

Figure 5.10 (c) Suflace plots of phase velocity in the Y Z plane

U11rii.soiiic . , / i i<(~ ~,/Elo.\iic properlies o~ldphosc iratlsiiio~is i t 7 I'oio.s.~iiit~i Szrlplioi~~ure crysrul -- 201

Slovness-XY Plane o .lo-+ I I ' I. xx

QL ++

- 3 10.' - rl \ Vi - L'? Vi IU U CI b 0 -.

Ln . -4 -I 113

Figure 5.17 (a) Surface plots of jnverse phase velocity (slowness) in the XY plane

I I I I -4 .lo-' -3 0 3 10.' 6 10.'

Slovness (sf=)

Figure 5.11 (b) Surface plots of inverse phase velocity (slowness) in the XZ plane

S 1 o w n e s : ; - Y Z P l a n e

F~gure 5.11 (c) Surface plots of inverse phdse velocity (slowness) in the Y Z plane

Young's Hodulus (x 0.1 GPa)

-4 -2 0 2 4 Young's Hodulus

Figure 5.12 Surface plots of Young's moduli in the XY, XZ and YZ planes

~ ~ Linear Compressibility Id I I

I XY xx I

Linear compressibility

Figure 5.13 Surface plots of l~nearcompressibility in the XY, XZ and YZ planes

The velocity surface plots alone cannot completely describe the

anisotropy of tlic elastic properties of a crystal. Young's niodulus s ~ ~ r f a c e plots

are vcry inipor~ant in this regard. The Young's modulus E [5.13] in thc direction

of unit vector i i i for an orthorhombic crystal is given by the equation

Tlie cross scctionj of Young's moduli.. surfaces of KNH2S03

plotted in the a-b, b-c and a-c planes are shown i n (Figure 5.12). The

linear conipressibility of an orthorhombic c~ysta l in matrix form can be

writtct~ as

[i = [SI , + ~ , ~ + s l 3 ] n l ~ + [ ~ 1 2 + ~ 2 2 + ~ 2 3 ] n 2 ~ + [ ~ 1 3 + ~ 2 3 + ~ 3 3 ] n 3 ' (5.5)

The lineal- compressibility of KNH2S03 crystal in the a-b, b-c and a-c

planes Ihave bccn plotted. The plots are as sl?owli in (Figure 5.13). The

I'oissoii's ratios [5.14,5.15] have been evaluated using the equations as csplained

in Section [1.4]. The vol~iii~e cornpressibilit:j Siikk is an invariant parameter for

a crystal. In matrix notation i t is given by

Where S,, 's are the corresponding compliance constants

Hence bulk lnodulus of the crystal is given by K = i/Si,kl (5.7)

Volulne Compressibility of the cryst.11 = 0.45 ~ 1 0 " ~ ~ . ' n r n ~

Bulk modulus of the crystal = 22.22 G Pa.

5.4. Conclusions

PEO technique has been effectivel:! iniplelnented for evaluating all the

nine elastic constants, compliance constants, Poisson's ratios, bulk modulus and

volume compressibility of Potassium sulphamate crystal. The anisotropy in

elastic properties are well studied by the surface plots of phase velocity,

slowness, lincar compressibility and Young s modulus.

PEO technique is successfully implemented in studying the phase

transition in Potassiurn sulphamate single crystal. The variations in the elastic

constants, C I I , Cz2, C33, C44, CSS and C6(,, \vith temperature are investigated. The

transverse elastic coiistalils Ci14 C j j and Ch6 have exhibited iniilor anomalies

around 355 K suggest a possible weak phase transition. But DSC does not

support this possibility. More work is in progress about the nature of the phase

transition occurring in the crystal.

References

-5. I ( ' . . i .Brow~~ a ~ ~ c l E.G.Cox : J . Chern. Society, 1, 1-10 (19401 I('r:~?sl~i/ .si~.ii(.f~ir.c r!i ~ J O / i i . S , ~ ~ l ~ ~ ~ ~ . ~ ~ l ~ / l / l i l ~ ~ l C i / ~

5.2 (;.,I .icfVrey ; I I I ~ t1.P Stadler : J. Chem. Soc., 1, I467 (195 I) I X-rm rI$fi(icliorr /or q ~ . s t ~ i / sii.iic/ure (?fporussiurn sulphairrure

5.3 l<.lklaussLihl :111d S. tlaussiilll : Z. fur Kristallogr., 210, 269-275(1995)IElo.stic /v(~pw/ie.s I!/ srilfiirriic acid o i ~ d sulpharrrrites ofNu, K.

5.4 ( ; W . C o s , I .M. Sahine, V.M.Padnlanablian, M.K.Chung and A.J.Stii:iadi : Acta Cr? s, 23. 578.58 I ( I 967) / A rreu1ro11 d ~ ~ ~ c t i o t i .YIIJL(), ~ ? f ~ ~ r ~ / ~ r , s . s i i i ~ ~ ~ . s i~ /~~/ r i i r r~~i f t

5.5 1 . A Ka1cl;i;ir and E.1- tlcilmann : Z.fur Kt-istallogr.. 103, 41-53 (1041)l Die Ki.i.rfiii/.~ii.iik,iir r1e.s firliurirnn~ido suljorlurs K(NH>SO;J

5.6 K L \ / I<ap11 : Kristallographie einiger Amidosulfate eiti-und zwei~\~crt iger ha[ioncn. L)iplo~narbeit Universitat Munchenl992 1i)1'jj'erei1fiii/ sco~iriirrg c/iIoriirrerric irre~rsurenien~s ofsulphair~a~es

5.8 t.I'.Papadakis in 'Plrysical Acouslics ' VolXII Eds. W P.ibIrisorr nrid X.A'. T11iir.stoii (Academic I'rcss New York 1976) p.227

5.9 HJMcSkilniti : Acou. Soc. Am.,33, 12 (1961)

5.10 H.J.McSki~n~li. and I'.Andreatch : Acou. Soc. Am., 34, 609 (1962)

5. I i Ii.J.McShi~i~iii in 'Pl~ysical Acoustics ' Volutire I, Purl A Ed 1P'P. Mrrsori (Academic I'rcss Ncw York 1964) p.271

5.12. L..Godfic) a11d J.Pllilip : J . Appl. Phys., 75, 5, 2393-2397 (1994) / Eiiistic c11ri.siurr1~ ~irr11 i1ig11 /etriperulure ~ I I I O I I I ~ ~ I ~ S rlecir 425K iri I , i f l ~ i~ i~~ i l r j '~i i~~izerr~~~ri i .sii/ij/~iife.

5 . 13. iil J . P . Musgi-;ive : L',yslui Acoustics, irrlr.odllctior7 fo sflid) r?/'c/u.sfic i!,ul~r.s aird ~~~hr.oiioii 111 c.~:i~.\/ol.s: Holden-Day 1970

5.14 N.I'.Nyc . l'iiysic(ri properlies of cryst~ils,(Oxford university press. London 1957)

5.15 A.V.Ales aiid J.Pliilip : Material Science and Engineering B, 90, 241-245 (2000)/Elu.s/ic prol~erties ofdi-uniiiroi~iunr liyd,oger~ cirruie sirigle cr:~'.siu/s

5.16 l..C;odfrc) a~ id J.Pliilip : Acoust. Letters, 19,111-14, (1995) 1 / I riiirrro.icu1 7i.~~/iriic/iie,for horid correclion in ullrasorric r~rea.suretiieirrs

5.18 L.Codt'sey a ~ ~ d J.Philip : J. Phy. E. Sci. Inst:, 2 2 5 16(l989) 1.4 feiriperirliir.e c o ~ i r ~ - o l / r ~ ~ , / r ~ r ~ cry.s/ol groi~,/h experir11er1t.s oflorrg diir.u/ioir.