Chapter 4

Elastic constants and phase transition in Sulphamic Acid single crystal

The clu.stic pro~~ei-fies of orthorhombic Sulphuil7ic ucid crystril 110i~e been

preserited in thi.s chapter. The measurements ofelasf ic consfa17ts by ultrusonic

PEO technique ond tonperature variution of elustic conslunfs oiler /he runge

300K-JOOK iicn~r heen undertaken. All the nine eltrstic coi7.sttr17t.s ivei.e ilierisured

rwitrg /l'LO /c.chiiic[i~c by /i?easuring velocity i,? riifirunl .vyiiii~ret,?' rIireclior7s.

Slci:/iic.e ,111ot.c phuse velocify, s1ou~nes.s. Young's niodzil~i.~, li17enr

conij~re~sibili~y huve been ))lade and it reveuis the ani.sotro~~j~ in elasric

11ro11w'.iie.s. DS(' i~~eusure/~zents on phase trunsilion in this crystal at u l leq~ s lo~v

heu/i/cg !.cite u1.e pi.esenter1.

lIlii.siic I r , i , . ~ r o i i i \ (iir.1 1'11ii.sr /rii,r,si~io,~ I I ~ I , , .s:,!g/r (.,:ysio/

-. . - -- -. - - I43

Elastic constants and phase transition in Sulphamic Acid single crystal

4.1 Introduction

Sulphalnic acid is a very interesting crystal with chemical formula

NH;SOj possessing orthorhonlbic symlnetly and exhibiting Piezo-electric and

Non-liricnr optical properties [4.1]. NLO crystal finds wide application in

optical iiarmonic generation, optical modulation, telecommunication, computer

and optical signal processing etc. Only little inforlllation about cryslallograpliic

and iphysical properties of Sulphamic acid can be found in the literature. X ray

diffrac~ion technique [4.2, 4.31, IR [4.4] Neutron diffraction scattering [4.5,4.6],

Raman [4.5], l'hermal expansion studies [4.1], Thermal analysis [4.7] and

Thermo-elastic properties [4.1] by resonance ultrasonic technique of Sulphaniic

acid have been reported in the literature. Earlier reports reveals that it

crystallircs in the Orthorhombic symmetry [4.2] with space group Pbca with

lattice parameter j4.21 a = 8 . 1 1 5 4 b = 8.066A, c = 9 . 2 j j A The structure of the

clystal has been studied using X ray diffraction technique [4.2- 4.31. I t has eight

molecules per unit cell. Both determinations are in agreement with respect to

the general si~apc and disposition of the dipolar ion structure for the lnolecule in

thc crystal, but differ by as much as 0.1 A in solile of the atomic positional paranicters. Both authors have stated that molecule cxists in the cryslal as the

zwittcrio~i (N\'f13'S03-).

Ilaman 14.51 and infrared study [4.4] also supported the zwitterion

structure. The Ralnan spectra in the low frequency region indicatcd that the ion

is essentially tetrahedral. The infrared study of Sulphamic acid was well

interpreted in terlils of zwitterion structure. But Philip el al. [4.21] analyzed the

Surface Enhanced Ranian Scattering (SERS) and FTlR spectra of Sulpliamic

acid and suggested a structure between that of zwitterions and molecular form.

A possible hydrogen bonding system was discussed by Osaki [4.2], who lcd to

the conclusion that the NHj+group must bond to five-oxygen atoms resulting in

one single and two bifurcated hydrogen bonds. The asymmetry of the quasi

tetrahedral [NHjSO,]- ion suggests an urusual frequent formation of acentric

species having polar properties like piezo- electric and nonlinear optic effects,

which might be suited for technical applications.

Table.?. 1 The general information about !he crystal can be tabulated as follows

Parameters I Neutrcn scattering X ray Diffraction l a

b

c

Space group

Molecular volu~ne

8 036 A 8.074 A

8 025 A 5.020 A

9 236 A 9 236 A

Calculated density

Neutron diffraction studies on th: crystal structure of Sulphamic acid

were carried out by Sass [4.6]. It confirn~t:d the zwitterion form of the molecule

but differs from that of X ray studies [4.2]. The study of the deformation

electron density in Sulphamic acid at 7E K by X ray and Neutron diffraction

technique was conducted by Bats el a1.[1.5]. Since the hydrogen bonds are so

disposed throughout the structure, Bats e t a / . suggested that there exists no well-

defined cleavage in any direction of the crystal.

2.165g1nicc

Michaut el 01. [4.20] canied out the temperature dependence of the ESR

spectra of the radical S03WH2 trapped in gamma irradiated Sulphamic acid single

crystal. His study described the potential well hindering the reorientations of the

N'H~ group.

2.167glnicc

0.7106 A h I.CO8 8, A'

I'lic ihcriiio elastic constant T,j, and anisotropy of the thcrmal ex l~~ns ion

ri,, and iir~~ncisen tcnsor G were studied by Haussiihl el cri. l4.31 and thcy fc>und

tilac Sulphamic acid possesses large value of T,, since i t has strong Hydrogen bond.

Maussiihl studied the anonlalous behavior of thennal expansion [4.3] of certain

si~lphamates. Studies of Rapp [4.7] by DSC suggested that Sulphamic acid exhibits

a first order phase transition in the temperature range of about 450 K.

Aim of the present study is to investigate the phase transition of Sulphamic

acid above roorn tempelatwe using PEO technique. Measurenlents of elastic

constants using ultrasonic Pulse Echo Overlap technique are proveci to be excellent

probe to investigate phase transition. The elastic propelties of Sulphanic Acid is well

studied by measuring ultrasonic velocity in the crystal in certain specified

crystallographic directions and evaluating the elastic stiffiiess constant, compliance

constant, Poisson's ratio. The surface plots of Phase velocity, Slowness, Yow~g's

modulns and Linear colnpressibility in a-b, a-c and b-c planes are made.

4. 2 Experimental Tecl~nique

4. 2.1 San~ple preparation

Large single crystals of Sulphamic acid of size (45~35x12) mm3 have

been grown from supersaturated aqueous solution of the salt by slow

evaporation technique over a period of 60-65 days. The temperature of the bath

was maintained constant at 305 K. The arrangement of moleculcs in unit cell of

the crystal is as shown in Figure 4.1. This is made with the help of the computer

programme "Atoms". The photograph of the grown crystal is dcpicted in Figure

4.2. 'l'he morphology of the crystal is shown in Figure 4.3. It is drawn with the

aid of the computer programme "Shape".

At 305 K thc soliibility is 21.7gm/lDOn~l.I-lz0.l'l~e solubility cut-ve is as

s h o w ~ ~ in Figure 1.4. Thc solubility increases slowly with rise o i tcnlperature

and reaches saturation value around 35'C.

Figure 4.4 So1ubil;ty curve of Sulpharnic acid crystal

Table. 4.2 Comparison of computed interfa.:ial angles of the Sulpharnic acid crystal wfth measured values.

Crystal faces Interfacial angles between faces

Cclnputed

100- T I T

010- 101 149.85

The interfacial angles are me~sured using an accurate contact

goniometer. By knowing the lattice pararreters, crystal system and space group

one can construct a stereographic plot by using the computer progranllnc

'Jcrystal'. Thc natural fices of the sample have been identified by the rnetliod

as discussed in Section 2.2.3 The stereograms of the crystal are depicted in

I:igurcs [4.5(a)-4.5(c)J.

l3ulk samples havc been cut using a slow speed diamond wliccl saw

(niodcl-650. Soutll lJay technology, USA). The diamond saw consists of a thin

metal disc wit11 tiiicro sized diamond powder embedded on the outer edge. The

blade fixed to n rotation mechanism is driven by a speed controlled motor. 'l'iie

crystal to be cut is glued to a precision movable arm with a goniorneter and

counter weight. Once the directions are carefully adjusted with respect to the

blade. the arm can be lowered so that the crystal rests on the rotating blade

edge. 'l'lic blade is continuously cooled and cleaned by a coolant, whicli is kept

below tile tray. One advantage of this diamond wheel saw is that samplcs can bc

cut easily to havc parallel faces. This is very important for ultrasonic work

because the non-parallelism will bring in a non- exponentially decaying echo

pattern and thereby additional errors in velocity measurements. Saniples with

pairs ui' parallel planes perpendicular to [loo], [OIO], [OOI], [ I 101, [OI I] and

1 1 011 tlirectioris lia\:c been prepared for ultrasonic wave velocity measureinents.

All cuttings arc made very accurately. The error due to misorientation is below

i 0.5". No clcavage is present in any direction of the crystal.

I'hc edges o f the samples have been polished carefully using cerium

oxide powder to optical reflection level so as to ensure proper bonding of the

transducer to the sample surface. Polishing of the sample and cleaning of the

transduccr are vcry important for successful bonding. To make the bond thin,

thc transducer lias to be held pressed to the sample under spring action.

A r r ~ m p m m l qfmoleculrs in rhe Ul~i l cell of'Sulpha~nic ocid crysl~d rrhou~ cr uxis

-

4.2.2 1)cnsity tneasurcments

'fhe density ol'tl~e material is measured by Archimedes' principle by lii~cli11~

the loss of weight of the solid in liquid. Carbon tetrachloride is used in this

measurement. The density of CC4 is 1.67gdcc. It is measured to be 2.162gnlicc. In

the elastic stiff~ess constant measurement density has vital role [C, = p v2]. So it should be measured with great accuracy. Results of this study are in well ageernent

wit11 calculated valuc lorn X-ray diffraction study [2.167gldcc][4.3] a11d that fi-om

neutron ditrraction study [2.165gn/cc][4.5].

4.2.3 Velocity nieasurements

The ultrasonic velocities are measured using the PEO technique [4.10].

The details of measurement technique are by Papadakis [4.1 I]. A MATEC

modcl 7700 pulse modulator and receiver system with its associated subunits

has been used for the velocity measurements. X- Cut transduccrs of resonant

frequency 10 MHz and 6 mm in diameter are used for the measurement of

longitudinal velocity and Y-Cut transducer of resonant frequency 10 MHz and

6mm diametcr are used for the measurement of transverse velocity. Large

number of clear cchoes indicated that the grown samples are free from defects.

The same transducer has been used to detect the echoes generated by successive

reflection of the waves from the rear end of the sample. Absolute velocities at

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

modes. The McSkirnin At criterion [4.12-141 has been applied to correct the

phase lag introduced by the bonding medium on the RF echoes. The

temperature variation of the velocity of longitudinal and shear waves

propagating along the various directions in the crystal (Figures 4.6-4.8) have

been deternlincd in the range 300K-400 K by keeping the salnple mounted on a

suitable holder in a temperature controlled chamber. Thc rate of temperature

change in all the measurements is in the range of 0.5 to 1 K per minute. The

variation of velocity with tetllperature beyond 400 K was not invcsligatecl

because of bonding problems. The thermal expansion has been neglected while

measitring lhc variation of ultrasonic wave velocities with temperature.

Eiusrlc cwmrrrs a d phase amsl(luns In &lphmlc Add single t~ysrof

X

Figuri? 4.5 (8) Stereographic prajedbn of SuEphamic acid about a-exis

.&

Figurn 4.5 (b) stemgraphic pmjtxtion Sulphamk acid about &xis

Hmfk canmAl caod Ekw t a w s t a n in cWphmi~ Add sin& m n b

1

Fgure4.5(c) Sfereographk projestion of the Sulphamic acid crystal about c-axis

4.3 Results and Discussio~~s

1.3.1 Measurcmcnts of elastic constants

NH,SO,, being an Orthorhombic crystal, has the following ninc second

order elastic stiffness constants C I I , C22, C33, C44, Cjj, Cb6, Ci2. C i 3 and Cz;

('fable 4.3). l'he diagonal elastic constants C I I , C 22, C33, C44. Csj and C66 have

direct relationship with tlic ultrasonic mode velocity given by C,, = p ~ 2 . Tile

relationships between clastic constants for relevant ultrasonic wave velocities

for the orthorhonibic system are reported in literature [4.8]. The off diagonal

elastic constants can be found out from the following equation. The elastic

constant C l z can bc calculated by measuring the velocity perpendicular to a-b

planes; here tlic angle is measured from a- axis.

The elastic constant Cz3 is measured by propagating the sound with

tlic velocity normal to the b-c plane. The angle is measured from b-axis

l h e elastic constant C l j can be measured by propagating the waves

perpendicular to a-c plane where angle 0 is measured from c-axis

where s = sin 0 c = cos 0 and 8 is the angle of rotation for I-espective axes. The

angles are 44.83 ", 48.93 and 4 1.25 "for fab, fb, and f, and are measured from the a, b,

kind c axcs respcctivcly. 7'11~11 vlo , is the velocity of the wave in [I 101, v l l in the [OI I] and

\ I ? in the [I01 / direction, a ~ i d p is tlic density of the sample (= 2.162gmlcc. for Sulfal~iic

t~cid).

Elcisfic coiuiorils u~ id Pilase rru~lsiriui~ i , ~ S~ii/~iiciiiiic Acid sir~gle C ~ s l a l 161

01' the 18 propagation modes, velocity ~iieasuremellts of 12 modes are

sut'ticie~it to evaluate all the nine second- order elastic constants with cross

checks possible using the remaining modes. Considering all experimental

uncertainties, the absolute accuracy of elastic constant value is estiiiiated to be

bcttcs than 0.2% for diagonal elastic constants and 1% for off diagonal elastic

const:~nts. 111 all velocity measurements, the correct overlap identification is

made and McSkirnin At criterion [4.12, 4.131 for bond correctioll has been

applied using computer programme [4.15]. By measuring uitraso~iic velocity in

the Sulphamic acid crystal in certain specified crystallographic directions, the

anisotropy of elastic properties of the crystal is studied and the elastic stiffness

constants, compliance constants and Poisson's ratios [4.16-4.181 are evaluated

(Table 4.4). There are nine values of compliance and constants wliich are tlie

components obtained from the matrix inverse of elastic constants.

Table 4.3 Measured velocities and elastic constants of Sulphamic acid crystal at 300K

Velocity Elastic Direction of' ( Direction of 1 measured 1 rollstallt V-C. , 1 1 ' I N o I*lodo a i o poIarizatio~i Vlmls) C..(GPa) relationsh~p

The abbreviation used have the following meaning: L-Longitudinal, T-Transverse. QL-quasi- longitudinal QT-quasi-transverse s = sin 8 c = cos 0 and 0 is the angle of rotation with the respective axes. Tlic angles are 44.83 ,48.93 and 41.25 for tlie f,, fhc atid f,, and are ~nieasurcd fro111 tlic ;I, b, and c axes respectively Then v is tlie velocity of propagation of respzctivc lnodz and p is the density of the sample (=2.162gm/cc. for sulfaiiiic acid).

I62 ( ' I?opier 4

Also. here arc six values for Poissc~n's ratio in 01-thol.hombic crystal.

.I'he equation for Poisson's ratio when ur~iaxial stress is applied along a-

direction is

Siniilarly equations for Poisson's ratio when uniaxial stress is applied along b -

and c- directions are

Table 4.4 Elastic stiffness constants. Elastic ~ompliance constants, and Poisson's ratios of Sulphamic acid crystal at 303 K

Elastic stiffness Elastic corn 1 ance $ 2 .I 1 Poisson~s ratio / constant(GPa) constant(xl0- n N )

Eluslic ciiti.si~~1~1.~ o,,,/ P / ~ ~ , v c i~.o,~siiiwi i,,/i0117l""uic lci(/.\i i,g/e Cry"u1 163

Soine elastic stiffness constants of this crystal measured by this s t ~ ~ d y

(PEO neth hod) show appreciable deviation from those measured sing resonant

ultrasound technique [4.1]. Out of the diagonal constants, the constants C i I

(5%), C 3, (8.7%)) and C j j (9.8%) show deviation above 5%, whereas constants

C22 (7.8%), C 4 ~ (2.1%) andC66 (4%)exhibit deviation below 5%. While off

diagolial constants C l z (43%), CI, (19.8%) and Cz3 (17.1%) have exhibited

large deviation fro111 the RUT values.

4.3.2 'l'emper:iture variation of Elastic constants

The temperature variation of the elastic constants in specified directions

in the clystal arc plotted in [Figures 4.6 - 4.81 in the range 300 K-400 K by

keeping the sample mounted on a suitable holder in a temperature-controlled

chamber. Tlic rate of temperature change in all the measurements is in the range

of 0.5 K to 1 K per minute. The variation of velocity with temperature beyond

400 K was not investigated because of bonding problenls

Figure 4.6 Varfat~on of C33 and C55 of SA with temperature

Figure 4.7 Variation of C,, and C,, of SA with temperature

Figure 4.8 Variation of C,, and I:,, of SA with temperature.

E l a s ~ i ~ coiisiarir.\ ~ 1 i 0 i'iiuse tro,~silion 111 .S~ i I / i /~ t i i i i i c :Ici(/ v i i g i e Ctyslai ~- 165

Elastic crtrut7rrrlie.s it1 tire regiort of325K to 345K

Longitudinal elastic constants C i l , Czl, and trallsversc elastic

constants C44, C;< C6f, were subjected to temperature variation study. It can be

seen from I:igu1.%(4.6 - 4.8) that a number of elastic constants are showing

a~~oi~ia lous belia\,iours in the temperature region 325K-345K. Thc most

pronounced anomalies are shown by the elastic constant Cd4. It shows a dip at

3?5K and pcak at 335K and a small dip at 340K. Tlic constant C C , ~ shows a

small step decrcasc at 345K. The constant C Z ~ shows ~ilinor anomalies in the

rangc 335K-34SK. The constant Css exhibits anomalies in the range 335K-

340ti. The constants Cl and Cj3 do have significant anomalies. The anomaly in

Cq4 is also sho\vn in a cooling experiment and small thermal hysterisis of nearly

2K \vas j. . , ,. obscrved. These anon~alous behaviours of elastic constants of the

crystals can be attributed to a phase transition in the crystal around 335K. The

thel-ma1 expansion has been neglected while measuring the variation of

ultrasonic wave velocities with temperature.

4.3.3 Investigation of phase transition using DSC

'l'he present DSC study on sulphamic acid, in the temperature range

238K-473K has been carried out at a very slow heating rate of l0/min. Annealed

saml)les arc usetl for this study. Differential scanning calorimetric scan (Figure

4.9) shows that there is a clear change in the specific heat appearing as a sharp

dip near 33 1 K (58 '~) . The endothermic thermal anomaly is well correlated with

the suspected phase transition near 335K as suggested from the present

ultrasonic study. The energy involved in the transition is 1.599Jlg. DSC data

also shows a possible transition at 1 2 . 9 3 ~ ~ . But this region was not scanned

using ultrasonic. Transition already reported by Rapp [4.7] at 450K is also

prescnt in our data. In the thermal expansion studies Haussuhl ct al. [4.1]

obscrved anomalies for the sulphamate family members CsNH2S03 and betaine

NHiSO; around 350K.

I:?

Figure 4.9 Differentfal Scanning Calorimcitric spectrum of Sulphamic acid

4.3.4 Surface plots of Phase velocity, !Slowness, Young's modulus ant1 linear compressibility

With the aim of getting an insight into the anisotropy of elastic wavz

propagation in NH3SOj single crystal, the vc:locity surface plots in the a-b, b-c

and a-c plane have been made following a well-known procedure [4.17,4.19].

Figures 4.10 [a, b and c] show the ptase velocity surface plots in the

respective planes (a) corresponds to quasi - ongitudinal [QL] mode with higher velocity of propagation (b) and (c) represer t pure shear [PSI and quasi shear

[QS] modes respectively. A greater insigk.t into the elastic anisotropy of a

crystal is obtained by plotting the inverse phase velocity [slowness] surfaces.

Slowness surface plots provide a better pictorial representation of elastic

anisotropy in a crystal. The slowness surface plots [4.17,4.19] for NHj SO?

crystal are plotted in Figures 4.1 1 (a, b and c)

-61300 -6000 -4000 -2000 0 2000 4000 6000

velocity (r/s)

Figure 4.10 (a) Surface plots ofphase velocity along the xy plane

nO00 1 I I I I I -6000 -4000 -2000 0 2000 4000 6000

Phase velocity (r/s)

F~gure 4.10 (b) Surface plots ofphase velocity in the x-z plane

P h a s e velocity- YZ plane

-5000 0 5000 v e l o c i t : ~ (m/s)

Figure 4.10 (c) Surface plots of phase velocity in the y-z plane

-2.10-' 0 2.10-' 410-" Slowness (s/m)

Figure 4.11 (a) Surface plots of slowness in the x-y plane

:- az Plane

1

. .. Slowness (s/.)

Figure 4.11 (6) Surface plots of slowness in the x-z plane

Slovness -YZ plane 1 I I I

F~gore 4.11 (c) Surface plots of slowness in the y-z plane

Young's modulus

Figure 4.12 Surface plots of Young's 61oduli in the x -y, x-z and y-z planes

Figure 4.13 Surface plots of linear conpressibility in the x -y, x-z and y-z planes

The velocity surface plots are unable to describe the anisotropy of

the elastic properties of a crystal completely. Young's ~nodulus [4.17.4.19]

surface plots are very important in this regards. The Young's lnodulus E in

Eluslic i , ~ , i i ~ ~ r i , i i s o,id 1'11ii.sc i r u n s i ~ i o ~ ? i t i S i i / p l ? ~ u u ~ ~ f c ! ~ / . s ! ~ , ~ : / ~ , ~. ( ' ! :~~.v lo / 171

the direction of ~lnit vector ni for an orthorhombic crystal is discussed in

the literature and is given by

?he cross sectionSol' Young's moduli.., surfaces of NH3SO3 plotted in tlie a-b,

b-c aid a-c planes are shown in Figure 4.12. The linear co~npressibility of an

o~?horiiombic ciyst:il in matrix fonn can be written as

'l'he surface plots of linear compressibility [4.17] of HNH2S03 crystal in

thc a-h, b-c and ,I-c planes are shown in (Figure 4.1 3).

The Volun~e compressibility [4.17] Sllkk is an invariant parameter for a

crystal. In matrix notation it is given by

Where S,,'s are t l ~c corresponding compliance constants. Hence Bulk rnodulus

of the crystal is given by

The \olumc coml)ressibility of this crystal is evaluated to be 0 . 4 3 7 x 1 0 ~ ~ ~ ~ ~ ~ n 1 ~

and bulb n~odulus is 22.88 GPa.

I'EO technique has been successfully implemented for evaluating all thc

nine elastic constants, Compliance constants, Poisson's ratios, Bulk l~lodulus

and Volume conlpressibility of Sulphamic acid crystal. The anisotropy in elastic

propcrtics are well studied by the surface plots of Phase velocity, Slowness,

L.incar coinpressihility and Young's modulus.

The variation: of elastic constants C 11, C22, C33, C.I.I. Cj j and C66 in the

temperature range 300K- 400K have shown a clear indication of weak phase

transition around 335K. DSC study also exiibited a weak anomaly near 331 K.

In summary, fro111 the observed anomalies in the elastic constants and from tile

DSC investigation a new weak phase transition in sulphamic acid crystal near

335K is proposed. Further studies are required to establish the nature of this

transition.

References

4.1 E.Haussulil and S. Haussiihl, 2. fur Kristallogr., 210, 269 (1995) )lElrtsiic

~ ~ r ( ~ / ~ e ~ i i e . s i!/' S~tlphatttic acid and sulphomu~es ofNri, K . .. ...

4.2 K.Osaki, H.l'adokoro and i.Nitta, Bull. Chem. Soc. Japan, 28, 524 (1955) / X -

rriy dv$~tctioii un o.ys/al structure of sulpharnic acid crystal

4.3 F.A.Kanda and A.J.King, J . Am. Chem. Soc., 73, 23 15, (195 1) / X-ru~j

rltffr.uciio~i 111i crys~ul struclure of sulfarnic ucid

4.4 . M . V u a g i i ; ~ t and i'..L.Wagner, J . Chem. Pliy., 26, 1,77 (1957) ll'ihrmtiot~cil

spect1.u ~irirl .s~ruciu~e ofsolid sulphai~iic ucid a11d the S I I / / J ~ U I I I L I ~ ~ jot/

4.5 J.W.Bats and I'.Coppens, Acta Crystallogr. B, 33,37 (1977) /A stud!, o f the

e j r t t ~ t i i ~ electron de~uiry in S u l p h a ~ ~ ~ i c acid at 78K by X-ray roi~l tie~itroti

Diifructioti

4.6 R.L Sass: Acta Crystallalogr., 13,320-324 (1960) /A ~ ~ e u t r o n dffr.rrciioti study

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kationen.Diplo~~?arbeit Universitat Munchen 1992 iDfferetltial sca~~nirlg

ci110~itrieh.1~ ~i ieasurer~ze~~~s of sulphan~ates

4.10 .I.E.May JI-: IRE. Natl. Conv. Rec., 6 part 2, 134 (1958)

4.1 1 E.I'.Papadnhis: in 'P/~ysical Acoustics' Vol. ,YII Eds. IV.P.Musoti L I I I ~

RIV Tirirrsii~~~ (Academic Press New York 1976) p.227

4.12 tl.J.McSkimin: Acou. Soc. Am., 33, 12 (1961)

4.14 H.J.McSkitnin: irr ~Plry.sica1 Acoustics' Voluri7e I, Part A Erl. WP.Musol7

(Academic Press New York 1964) p.271

4.15 L.Godfrey and J.Philip : Acoust. Letters, 19,111-14,(1995) / A nuliiericui

Teclririque for bond correciion in zrltrasc~~ic nreasureii~en/.s

4.16 J.F.Nye : Physicul properties of crysials, (Oxford university press ,London

1957) p. 143

4.17 A.V.Alex and J.Philip: Material Scielce and Engineering B, 90, 241-245

(2000). / Elnsric j~i.r~/~crtics of di-ariiiii~~niurii hydrogen cifrrrte .single cr~,s/ci/.\~:

Ail ult~.a.so~~ic sllrdy

4.18 N.V.Pcrelomova and M.M.Tagieva : in 'Problems in Crystal P11)~sics' (Mir

Publishers, Moscow. 1983) p.118

4.19 J.P. Musgrave : Ciy.sta1 Acoustics, Infroduction to study of elaslic waves ci~itl

vibration in crysra1.s Holden-Day ( 19:'0 ) p.61

4.20. P.Michaut,J.Roncin and K.Leibler : Ctlemical physics letters, 40,1,28-3 1 (!976)

4.21. D. Philip, A. Eapen and G.Aruldas ; J.Solid State Chemistry, 116,217-223

(1995) /Vibrational and surface ee,ihanced Ra~nuii scalterii7g speciru of

S1rlphan7ic acid.