A determination of the elastic constants of beta quartz

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Mf w m m m m m m

i

m

m

th e

m m

elastic cosstabts

quartz

^

by

llsifiia 1§» JBssas^w*

A £bftttl# Mmltfeaft tc fiat $*«£»*%* '3PMP lllC C# DOCTOR

OF fJJTLiSiflir

Itejor m a b $ M tt

physics

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UMI N um ber: D P 12017

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UMI Microform DP12017 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

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QC 535

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THEORETICAL • ♦ * »»*'•* • • * « * * • * • * • * ‘ T&« 6 © teimlmntal oqpatlon. » * • * • * # • * » '

6

Application# to beia^q^rt* ♦ * . * , * ♦ • * » * *

*7

TH© II d#t©rtalimntal ©Quation. Tb.® &@t©mainantal equation for tfca general T tm

IX

plat® {£»p} ♦ * * * ♦ # * * * » * * * # . * elastic moduli. • * • » » » • * * « . * * * *

BXPS&XMBBTA&

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18 19 8?

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The apparatus asat aotboA* * « * * * * * • * * * %aarts plat®® »©#t* . . . * »■ « * .* .* * * * * -«

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Temporaiui*© oorreetiom * * ■*>* ■* * * * ». * • .* 88 • fjjp RESULTS* * • * * ♦ • • * • * . . * » . • ♦ * * * * Obserwd resonances of th© plates * * . . . * . Oharaeterlstie m l w m of t&* plat#® * • * • * * Calculation of tlx© ©lastie ©octants* * * * » *■

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* •1 * *

iwsmmmim the ©aqperlmental techniques available for evaluat­ ing elastic properties of aelotropic or isotropic solids -can be divided into two general classes * ft» static method consists in observing tbs deformation of loaded bars or plates* fbft

measured im this way are- ti*ma& la#-*-

thermal sin#© tm the process Mi# small temperature changes the Bp0 &%msM bay#- i^pte- #pp#rtwai^r to- ’ be—-

by come equalised.

$& Its© dynamic aaethod, on 'tfe# other 'hand,

th© elastic properties' arc is^ffed M In th© medium.

$te# speed of souad.

Standing waves are set up between opposite

fa### of the- specl®^*. SSoewtiig tbs frequency, dimensions, density and mode of vibration, the elastic constants can be calculated* -«8£s$ M

:jh this case .small temperature differences will

mnmkb

regions of nodes and loops. fbe constants will

differ slightly from those computed by Its.# static method and termed lh© advantages of utilising high order harmonies in -calculations of adiabatic clastic cowtwtts f w » titrating plates -were first pointed out by Atanasoff sad Hart (1 )* fhQT hav# clearIv dexsoiistrated

as the- number of n^riai

planes between two opposite surfaces of a vibrating plate are IssrwASift*, the perturbing effect of tfe# ©d®e® has a diminish* ing influence #:» th® frequency *f/n*, wrtbMMS '*i*:la lb* #h» served frequency of the harmonic and *a* the order of fti®

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

harmonic * WmMmw the#©. conditions W m

®$m safely

fee regarded as a. .good apprMdJisablon be a plats- with isflmlbe. lateral dissensions* its# frequency *t/ziw then depends only on th® tl^skneae ef the plate, tk# :«©#©, of

an®. 'the

elastic properties of the m#diumj consequently, the theoreti­ cal trcatsisnt is greatly simplified* ■Bata Quarts, also known a m *®hightf quarts, m i s t s be tween 5 7 5 *0 * and S7 0 *e*

Although tbs static method of determining

©lastic constMtis 'Imhmm** if»r®*^li^iy difficult at tttiMMA ■■ temperatures, -am eaqperlassit using, this tschmiqtt© cm beta ^aarb*-. is dwwirib©# fey l«ffiif s»d ds Mendrot o © totoa * ^ * u »

toto©

mto toto® potato m a d e r ©oaslderatoioiu

~ S “ ~S ®

H w » - V

is

•

Ms© wotoor dtspl©e©m©nto ©3f tli© potato.

4£i© * $ u * b l o n

«& ssotioii

related .too © unit ipetai© t m m m m m C

/O

/

5 t *

g*.

^

# * * * * # * * *S *

"

Hkm. solution ©1* tototis equation under Ms© speeial eoisittioias t© to® applied* ©ustomarily pro©©©#® toy mmmm$m^ sinusoidal stetotoa 1©** iioiatos la to%e asdttsa* Xlrt

• - * -■

XOJ t

—

>d ^

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•

*

•

*

*■ ■# ♦ & »

i S M © a * im -Wb& b®ear :i©Xi*©iiy, tota* € X J * © p « t o i A «$ h k& U bi toos

f>(xu>) 'ytj

Qi^'cL't- ’2x^x4.

* • * * *

Us#. modes of vibration antieipated «r* independent of' Mi*

lateral diMensions along ae* ©aft 3%*- m m m # *1* and “to* may to* set pcMmmeitXy equal to© unity.

time s

/ ° ' y ^ ' ~t~ C/j'o./ vtlfe ©a -ton?-toSta*-

^

# * * • * »*%/A toto©

preoedlng ##mtol©a siMpltfl#® toot C ’, / CL/ — /ICL ^/J

y 4Jy' -— ^

» • » * ♦ * • • #& *

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

1

quantity X* **■

1nwlvee to© thlelm©#® **d*

of tt*. «*£*ta& sp#oiii©n

two opposite

faces perpsssdiemlar to toe % m

ji«s« 1X©X

direction and also to© order

liarraonic »j# fop to© vibration of frequency *f**

00%

*wp~

3&rfitsgfr»

this oxperlaMtMfc to® specimen® were prepared in to© for® Tim sound wave® m t up travel

of rsotangalar imraXX@Xeptp®dii#

.parallel -to. to© smallest or iaxieie®ssNt, aameneien ^dP-* standing ■it ffc- i f ' .....—-

-A*.* ■—»-

- Him'

|H»

-4ft- - mm' aLUl .^^4;

*»- -- --»

-^-.'^•••-

^

^

--■----.■

..i3 t O f --- - ~-.r. •-». -

L~.*. jjp • —mT.

_

.g © .'©T^ © - .

rfWfl- A :. ^-- ..^

jBE ©S. .--- -

—

waves a m ton® Maintained between -to* toe faces of largest area# fbe sat' of tore© femegeiMNKii* linear otpotiofic t6 > has .HlimltoBfWH* ©eiutlos* ppovideit to©- fctoto&*wefc of ISso #o» efficient© of to© &*« la ®es*o« BaepXXeib value© for I* nan bo ©btoiiied. in tors© of too elastic constants by ©diving too, cubic aquation which mssiito from to® detopsiiaanbf « U -• k ®, e„A, o«»« o*rt*in e d i t i o n ,

cm tfeia elastle ©anstsnts, some of wlileli &eeo»a* sero* w h ic h »jaat»e m

displayed in 3 i 'I X

a

v lg 0

’1 1

. .

t

v13

following laetriief #

3.

.

#

■3

#• .3 -

°5 S

Shoe*

#

»

#•- -nr- »

#:

•1 0

‘

3

C44° 1 / 2 (C

XL

°X2 *

As a result, the determinants! ©quations w r i t t e n up t© t M $ . .

point redue© to the o s p M M & o s * * C% 1

*C * )

(e 6 6

~K * 5

{0 « 3 .~E* ) ^ i i | S

4 « E *>

44

Cd

-«4

~ & *}

!||*

8®-

*® i ^ *

-& *}

*

0

4&e throe fa©tors in eaeh equation represent the three pr@«*-

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

■ In tfa* px*@«#»fs problem It is 2 ja&weMI jsbd®#.

& m a tfa# -'Vi©*!®®®

vibration be excited pi«#o#l«ctri#»Xly*

therefor#, sa»#««iii*y bo losow the

it 1«*

MBamtn th#

ilatlns set up In m crystal and the polarisation produeed by tb* ikjiplloatt^ *f .«& .®3&s$rlo field. ¥©lgt |f| sad others (8) h a m #stpg###«d tSbijt. rolatton^xlp In tensor form as#

%

* ®ua ^k i

8 & m m :.p f.,.

is

*

«nnb&»»# tb a t

relationship Is oao utilising a. third order

mmkmm. MN^asas&ts of

piezoelectric constants, t h is

lev the

te n s o r -m& b e re p re s e n te d b y its # a rm y *

*11

*12 *13

*14 %m•

%X

*2S *23

%4

*28 *26

* m

* m

* m

*z&

* m

*16 is

* m

ft®. last .##» subscripts in. each b»p«ai@mt «iiii are r#» lated to the strain* have been abbrevlabed In ©OJiformlty with.

Wm

i§#ntl©it#ft

Ssai# «a»» %to© eyiiiseiry ©f the

crystal structure imp©©#© certain condition© ^> -ar» -Mil - a t*

IS*#®!# w"Wl6w3#«08|l'" ■CrdT ©mwlw i%it%iTi i f *

.

**.. iu±.-itk a i - a .

%* tota and to© o M 3* axis.

- jr t t # #

a * f e * i uMk

'« h M

IJSww^©XX t5J3i€l ^3tw*p.

’ii-»-'l^niWTrTif1» m i-fig .

A«* © pr IX ta ^sasral., the spso.lara.. for beta *pas*rts nibratioiMi becomes wary simple*

5he thickness modes are easily distinguished on the

oscilloscope acresn In

k

the lateral nods* by ibeiy:spacing

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* 2 4 ?*

along the fretp m m j mas^m I t

the thickness a n d lateral

dimensions o f the b&g*flc art sufficiently different*

Sv@u

the thickness modes earn usually fee discriminated f r o m one another fey the Individual ehar&etorlsties of their oseilloscope patterns*

f h# pattern differences are m o s t evident in Observation o f

the m t © at w h ic h the oscillations damp out*

their intensities, w h i c h Is translated fey the t®to4Iioto#p# as amplitude of defiestlon, is also helpful i n this eotttjeetlon* A fter a series o f measured frequencies t o t o t o w n totolhtod, it is a simple matt er to allocate the various responses to N

their p r o p er sequence#

A slide rule Is a convenient^ aid, fo r

under each eonseeutlv© odd numbe r o n the slid© should appear a response o f a selected mode w hen the slide h a s b e e n adjusted to the one po s ition where

this

correlation m m happen*

ttna

half o f the difference In fre^xeneies h e tween successive responses is the frequency *f/h* utilised. I m to# #omputati«m

of &*♦

Stag* toe speei»#a. hetoeem to# electrode® SB mus t to at a temperature above £73*0 if observation* are to he made o n beta quarto, a special holder was devised whi c h iar did** played i n Figure S.* rods

as

liefcel electrodes w e r e used w i t h iron

leads to feto exterior o f the furnace.

A

sheet o f

m l e a placed o v e r the face o f e a c h electrode prevents oorrtact w i t h the specimen a n d reduces likelihood of breaking

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Fig. 5

(a) (b) Crystal holder and thermocouple assembly, (a) Electrode exposed, (b) ready for insertion into furnace tube.

the plat© due to nomxnl£otm eonduefcloa of beat. from the eleetrod#*

llngla&ed porcelain plates and cylinders mak©

tip fee, /felaae# o f structural m t e r t a l *

©a,# holder assembly

fits sn ugly In & stealstanoo furnace S e a t e d lay d i re c t current

to avoid Induction olgft&o. toeing, pisfcsd up. toy. t$». detestor*-

4 ©liroBisl-almitel fesrtaoeonpl© was- used to aajfpare fee ©f the holder adj&@«ii to .the ifMlsMb^ Soma of the plate® need In tills ea^riattnt wane oeleeto& &MR* feos©.prepare#;. toy atanasoff and. Sart Cl) for their work on alpha Quarts. Several additional plates were made mp wife orientation® net found among their ©o!3e ©tion. m m material from which these near oats were obtained was•ear©fully examined for optleal feinaing by -fee use of p»3jfc*iso&. light*

Moreover, fee surfm m m of '.£$>• ttnal spuei**.

Mitt were ©toiied with hydrofluoric acid in a ifarther effort to deteot aonunif©mitty «f structur®-* $&» sables surwivtiig these tests are desoribed Is fable 1* fh# preparation of outs with Y or B orientation Is relatively easy, by well known opt teal methods, when 'the natural facets- of fee crystal are available* ##»S9» type the proses®: is oeiMMtaftfc. n w

for the 75*310*#

©©aplfcabed*. Mt e r

a Y fa©@ ha® been esfeablished. and .fee his# of fee rough . crystal juMpssKtely ground perpwdiculasr to fee opti® axis,

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

it third plan© 1* out so that It is at right angles with the has# tmt mfc** any desired aisgi® with tins ¥ fsc©* in this ease 73*30**

A fourth surface is now created A I A

Is perpendicular to this last plans*

The aorssl 'to the

fourth surface makes the angle with the optic eiei# which is desired for the cut, for imb*tta®ts#. 59*29**

&k% m m

throng the block ..parallel to this fourth surface will yield blanks from which parall©lopipsds may be shaped* each, stage of to*

ttm

amrfaces

mm

At

polishid ■to an

optical finish* !Ph© angles can Mien b® aie*ysiir®d by using the prim*, table of a. sptc^oscope*, fable I the Dimensions of the Quarts plates la centimeters.

Orientation

'i^^lckness.-

¥' 20*00*'

*4444

.2*127

ft' 10*, 61* 47 1/2*

•4500

2.176

2.178 2.222

73* 30*, 69* 22*

*8081

1*460

2*214

*4442

2.227

2.200

0*, 46*

J*et®ral Dimensions in xi xj

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m already pointed out, t!m direction cosine® of each out mat is# inserted in the equations fresi.which the elastic, constants arc to be obtained * ■fable II display© the munarleal mines for these cosine®.,,' Thus far the measurements relating to

liseer

dlaaasloa®' ;* a & angle® of ©ri#atiitioi»

h a m 'been

mAm on the speeis®« as alpha, quartz at. room temperature. Since, the saftc«€A la to/be heated tm d0G*&*# expansion will chinige.. the dimension© of .tbs plat® as well .a© the ori#n.tatl©m* S t a r t s .espani* a t a different r a t e pa r al l el to

the

optic axis

as collared to that at any direction perpendicular to this .#»!#, Mae .latter expansion being, im m

the

la^ar,

coefficients

1B m $ m

been measured b y racheteller (9) east also by lay (10),,

The m i n e s published b y lay will b e w m € In Use ealculi^lene to follow. To cosapute the thtels*#® o f the plate a s b e t a quarts f& m .

f^BMk jposm t#«p@iwitwre MaMaa***- ® d %

n e e is made o f the

equations d p * d|j(l e The superscripts « and f that Miey m s or x £

axis

of

s a t be.

the direction oos.lnos signify

the cos lass o f the angles b e t w e e n the b m b k I Mi#

plate

quarts t e i p a c t l w l y . mmm

on

* *■ *> ■* * SIS,

m m

a n d Mss a% aacl* a s a l p h a a n d b e t a

B y ■alpha quarts la aslant quarts at m l u © dbr if* is n ot measureable a n d

from Mae expansion coefficient®.

If

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

p la

-

29

-

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

*

30

-

feh# anglo between the .norwl fem the plat# .sad the . 3%... axis

at spews® temperature* then the correction is radios# to be added to this angle da# fee th# offset of inerostaod tempera­

ture will to# irery nearly* :«

»* Tg/ 2 ) sis ip

* * rn'*- * * * 80*

Tfai quantity ry or % 1* tfa* ohango in langtto per unit length • u p the fe#«imfai®re range eeiMidered for direefeloMi perpendleular or parallel to the xm axis respeeti m ly * As a eense— f t * ® # o f fehe «feaa»8* l a a u g l s p #. fell# dlr®efei©&

through-

ost the de [email protected] equations mentioned earlier aost be ad** Justed aeeme&tag&r* *dtol® 3dC ale# eontains the eorreeted mines.

for

thickness

.sad direction ©eslnse

of the cuts

based

os Jay*® mimes for ^ ®hd '%♦ ,: lo #os®X#ta tfe*.

of fette eaperliwsnfeisX method*,

several advantages and disadvantages of the technique will fe# #©i®llli@p#d* "fSbS; as«#te»®Btafc of a. roaoaaafto frequeiiey i a m l m s

matching the minima of two oscilloscope' patfeeraas each of which .are traooa of boat fiwfaenc ies*