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Velocity of sound in solid iodine Leith, Marshall Montgomery 1966

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VELOCITY OF SOUND IN SOLID IODINE by MARSHALL MONTGOMERY LEITH B.Sc., The U n i v e r s i t y of B r i t i s h Columbia, 1964  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1966  In p r e s e n t i n g the  this  thesis  Columbia,  I agree that  the Library  a v a i l a b l e f o r r e f e r e n c e and s t u d y . mission  f o r extensive  p u r p o s e s may  of this  thesis  w i t h o u t my w r i t t e n  Department o f ,  It i s understood  u  thesis  by t h e Head o f my  for financial  permission.  PhyBJCS  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Columbia  per-  for scholarly  D e p a r t m e n t o r by  that  gain  of  s h a l l make i t f r e e l y  I f u r t h e r agree that  copying o f t h i s  be g r a n t e d  representatives  cation  Date  fulfilment of  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y  British  his  in partial  copying or  shall  n o t be  publiallowed  iii  Abstract  The p u l s e t e c h n i q u e was employed t o measure v e l o c i t i e s o f l o n g i t u d i n a l and t r a n s v e r s e m e c h a n i c a l waves i n s o l i d i o d i n e a t u l t r a s o n i c f r e q u e n cies.  P u l s e s were produced and t h e i r echoes d i s p l a y e d by a S p e r r y U l t r a s o n i c  A t t e n u a t o r Comparator.  T r a n s d u c e r s f o r g e n e r a t i n g the waves were: l o n g i t u d i n a l  waves, X-cut q u a r t z c r y s t a l s o f 5 and 12 Mcs.; t r a n s v e r s e waves, Y-cut q u a r t z c r y s t a l of. 5 Mcs. i  For  l o n g i t u d i n a l measurements the sample was immersed i n baths o f  l i q u i d h e l i u m , l i q u i d n i t r o g e n , s o l i d c a r b o n d i o x i d e , i c e , ambient t e m p e r a t u r e , and b o i l i n g water.  No d i s p e r s i o n was found.  room  Four samples were  used, but as i o d i n e i s a n i s o t r o p i c i n many o f i t s p h y s i c a l p r o p e r t i e s most a t t e n t i o n was p a i d t o two amorphous samples which were thought t o g i v e r e p r e s e n t a t i v e average v a l u e s o f the p r o p e r t i e s o f i o d i n e . From the v a l u e s o f l o n g i t u d i n a l v e l o c i t y , Young's modulus was calculated. T r a n s v e r s e v e l o c i t y measurements were made i n baths o f l i q u i d h e l i u m , l i q u i d n i t r o g e n , carbon d i o x i d e , and ambient room temperature. The shear, modulus was determined from the t r a n s v e r s e v e l o c i t y ; c o m p r e s s i b i l i t y was found from Young's modulus and the shear modulus. A l l these q u a n t i t i e s behave much as" would be expected o f a s o l i d 1  material. The Debye c h a r a c t e r i s t i c temperature c a l c u l a t e d from the l o n g i t u d i n a l and t r a n s v e r s e v e l o c i t i e s was compared w i t h v a l u e s c a l c u l a t e d from s p e c i f i c heat d a t a and Lindemann m e l t i n g p o i n t f o r m u l a .  iv Table of Contents I  II  Introduction History of E l a s t i c Constant Measurements  1  General Properties of Iodine  1  Relation of V e l o c i t y of Sound to Thermal Properties  2  Theory Hooke's Law  4  Technical E l a s t i c Moduli  5  E l a s t i c Moduli f o r Orthorhombic C r y s t a l  6  Isotropy Condition  6  R e l a t i o n between E l a s t i c Moduli and V e l o c i t i e s of Sound  7  Relation between Adiabatic and Isothermal Moduli  8  Debye Theory of S p e c i f i c Heat  9  Phase and Group V e l o c i t i e s III  11  Apparatus and Procedure Iodine Samples  12  Transducers  15  Measurements and Apparatus  IV  Sperry U l t r a s o n i c Attenuator Comparator  18  Resistance Thermometer  22  Cryostat  24  Results, Calculations and Conclusions Longitudinal V e l o c i t y and Young's Modulus  27  Transverse V e l o c i t y and Shear Modulus  37  V  V  Debye C h a r a c t e r i s t i c Temperature  37  Conclusions  38  Bibliography  40  vi L i s t of Figures Figure 1  Photograph of Iodine Samples  14  2  Photograph of Pulse Pattern from Sperry Comparator  14  3  Etalon f o r Iodine Measurements  17  4  Etalon f o r L i q u i d Helium Measurements  17  5  Block Diagram of Sperry Comparator  20  6  C i r c u i t Diagram f o r Resistance Thermometer  23  7  Cryostat  25  8  C i r c u i t Diagram f o r Matching Network  26  9  Longitudinal and Transverse V e l o c i t i e s as Functions of Temperature for Sample #2  29  10  Longitudinal and Transverse V e l o c i t i e s f o r Sample #1  30  11  Longitudinal V e l o c i t y f o r Sample #3  31  12  Young's Modulus and Shear Modulus f o r Sample #2  33  13  C o m p r e s s i b i l i t y f o r Sample #2  34  14  Young's Modulus and Shear Modulus f o r Sample #1  35  15  Young's Modulus f o r Sample #3  36  vii Acknowledgements I would l i k e to thank: my research d i r e c t o r , Dr. P. R. Critchlow, f o r h i s guidance and a i d i n a l l phases of t h i s p r o j e c t ; Dr. M. J . Crooks f o r h i s h e l p f u l d i s c u s s i o n s ; the B r i t i s h Columbia Research Council f o r the use of the Sperry U l t r a s o n i c Attenuator Comparator; Mr. S. Huff of the E l e c t r i c a l Engineering Department for p l a t i n g the quartz transducers; Mr. R. Weissbach and Mr. G. Brooks f o r t h e i r help i n b u i l d i n g and moving equipment; Mr. J . Lees f o r the glasswork; Mr. D. L. Johnson f o r help with the iodine samples; Mr. B. A. K e l l e y and Mr. D. Chung f o r t h e i r help with the photography.  INTRODUCTION History Before the development of short pulse technique, e l a s t i c moduli of various cubic c r y s t a l s , i e : NaCl, KC1, MgO, were measured by the composite o s c i l l a t o r method (Balamuth, 1934; Rose, 1936; Durand, 1936).  The composite  o s c i l l a t o r method involved cementing a p i e z o e l e c t r i c c r y s t a l to the sample and then varying the frequency of the voltage applied to the transducer u n t i l resonance was achieved.  Then information on the v e l o c i t y and e l a s t i c con-  stants was obtained from the frequency of response.  Using t h i s method the  above workers were able to c a l c u l a t e both adiabatic and isothermal e l a s t i c constants. With the advent of e l e c t r o n i c pulsing c i r c u i t s , simple and d i r e c t measurements could be made of v e l o c i t y and attenuation.  Values of the e l a s t i c  constants obtained by the pulse technique were i n good agreement with those previously measured by the composite o s c i l l a t o r method. (Gait,. 1948) The two techniques are equally good for v e l o c i t y measurements, provided the sample under i n v e s t i g a t i o n shows no d i s p e r s i o n .  But the pulse  method does have the advantage of e l i m i n a t i n g standing waves and i s superior for absorption measurements.  Because the pulse technique reduces the e f f e c t s  due to l o c a l heating and minimizes the t o t a l heat input i n t o the sample, i t i s i d e a l l y s u i t e d f o r low temperature i n v e s t i g a t i o n s . However, as Rosenburg (1951) has pointed out, the moduli measured with the pulse technique are adiabatic moduli and must be corrected i f isothermal moduli are desired. General Properties of Iodine Iodine forms an orthorhombic molecular c r y s t a l with 4 1^ molecules per u n i t c e l l .  In bulk form i t i s a s o f t b l u i s h - b l a c k s o l i d , while the  resublimed m a t e r i a l c o n s i s t s of almost opaque, strongly doubly r e f r a c t i n g  2 c r y s t a l s with a m e t a l l i c l u s t e r and high index of r e f r a c t i o n . Iodine i s a simple element, that i s , i t e x i s t s i n only one stable atomic species, of mass number 127.  I t s atomic number i s 53 and i t s atomic  weight i s 126.92. Previous papers have shown that iodine i s a n i s o t r o p i c i n d i e l e c t r i c constant (Simhony, 1963), e l e c t r i c a l c o n d u c t i v i t y (Bersohn, 1962), and c o e f f i c i e n t s of l i n e a r expansion (Straumanis, 1943). There i s l i t t l e i n the l i t e r a t u r e on the resistance of a l l o y s and metals to c o r r o s i o n by iodine.  But platinum, gold, and tantalum among the  metals, and such non-metallic materials as g l a s s , enamelled s t e e l , and t e f l o n seem to r e s i s t attack. R e l a t i o n of V e l o c i t y of Sound and Thermal Properties In accordance with the t h i r d law of thermodynamics,  the v e l o c i t y of  sound i n any m a t e r i a l begins to f l a t t e n o f f below l i q u i d n i t r o g e n temperature and becomes constant at l i q u i d helium temperature. the v e l o c i t y of sound measured at 1.2°K  Therefore, the value of  w i l l be a good approximation to use  for a l l lower temperatures. The heat capacity ( c ) , the thermal c o n d u c t i v i t y (K), the v e l o c i t y of sound ("V"), and the mean free path ( ^ ), of phonons or l a t t i c e v i b r a t i o n s are related.  Thus measurements of c, K, and ^ provide a means of c a l c u l a t i n g A ;  knowledge of A gives an i n s i g h t i n t o the mechanisms f o r i n t e r a c t i o n of phonons w i t h each other and with the sample w a l l s . At low temperatures there i s a very abrupt decrease i n the thermal c o n d u c t i v i t y of pure c r y s t a l s .  Under these circumstances, the phonon mean  free path should be constant and should be of the order of the diameter d, of the sample.  Then K — c v d where the only temperature dependent term on  3 the r i g h t i s c, which should vary as  at s u f f i c i e n t l y low temperatures.  So a knowledge of AT i s important i n the study of the behaviour of thermal properties of materials at low temperatures.  THEORY Hooke's Law Hooke's Law states that f o r small deformations the s t r a i n i s proport i o n a l to the s t r e s s , so that the s t r a i n components are l i n e a r functions of the stress components.  Stress r e f e r s to the i n t e r n a l forces occuring when  a body i s deformed and s t r a i n r e f e r s to the change i n an element of length involved i n the deformation. If  £ xx,  Syy,  £zz,  6 x y , £yz,  e z x are s t r a i n components  and Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz are s t r e s s components, Hooke's Law may be expressed: £xx = s„Xx + s Yy + s, Zz + s Yz + s -Zx + s, Xy (l  3  ls  l5  t  £yy = s^Xx + s Yy + s^Zz + s Yz + s Zx + s Xy 2J  1H  2J  2(  £ zz = s ,Xx + s Yy + s Z z + s yYz + s^Zx + s^Xy 3  3s  n  3  <2yz = s„Xx + s MJ Yy + s g i Zz + s^Yz + s v f Zx + s^Xy  £ zx = Sj,Xx +  SjjYy +  s^Zz + s^Yz + s Zx + s^Xy SJ  C y = s^.Xx + s jYy + s Zz + s^Yz + s Zx + s^Xy X  t  w  ts  A l s o , the stress components may be expressed as l i n e a r functions of the s t r a i n components: Xx = c„ £ xx +• c u S yy + c ^ £ zz + c , £ yz + c 15 6 zx + c M  Yy =  Cj, C  Zz = c  3l  xx + c  £ xx + c  2 l  3 l  1  l 6  C xy  £ yy + c^ 6. zz + c ^ £ yz + c C zx + c e xy 2 S  £ yy + c ^ 6 zz + c @-yz ZH  + c ^ £ zx + c  2 6  3 6  € xy  Yz = c exx + c ^ y y  + c  v  tyz + c ^zx.+ c,,<2xy <|y  t  6xx + C j 2yy + c (?zz + c £yz + c^-CZzx + c ^ C x y  Zx =  Si  SH  Xy = c ,exx + c^fcyy + -c £zz + c £.yz + c ^ f i z x . + ' C ^ €xy fc  w  4+  The c i j are c a l l e d the e l a s t i c s t i f f n e s s constants or moduli of e l a s t i c i t y and s i j are c a l l e d the e l a s t i c constants or the e l a s t i c compliance constants. In i o d i n e , an orthorhombic constants.  (Hearmon, 1946)  system, there are 9 independent e l a s t i c  These are-. s  u  8  <*  o  Technical E l a s t i c Technical  (G).  0  0  0  0  0  0  0  0  0 0  o  o  o  o  0  o  o  0  0  s f  o  o  0  0  0  literature,  SJ3  S  S  <o<o  however, does not always use the e l a s t i c  stiffness  which i s adequate o n l y f o r  of r i g i d i t y  0  Moduli  s t a n t s or the e l a s t i c  These a r e :  0  constants,  but a d i f f e r e n t  a body of orthorhombic  Young's modulus ( E ) , Young's modulus i s  or h i g h e r  Poisson's ratio  the r a t i o  ity  of  transverse  strain  to l o n g i t u d i n a l  i s d e f i n e d as the change i n s h e a r i n g s t r e s s  strain.  constants  symmetry.  £/*), and the modulus  of the change of compression  to the change of d e f o r m a t i o n caused by the c o m p r e s s i o n . the r a t i o  s e t of  con-  strain.  Poisson's ratio The modulus of  to the change i n  is  rigid-  shearing  6  Orthorhombic Crystal Zwikker (1954) considers an orthorhombic crystal with the coordinate axes along the crystallographic axes.  A normal stress Xx applied in the x-  direction gives rise to an elongation Cxx.in the x-direction and contractions 6 yy and Szz in the transverse directions. exx = Xx_ E,  <2.yy = -ytt'jgxx  tzz =-yU £xx  Gyy =_Yy_  Gxx = -/A^ £yy  <2zz =  &zz = Zz E  Cxx = -A »G zz  Gyy =-A £ zz  13  S  -n^dyy  5i  3  So there are 3 Young's moduli and 6 Poisson ratios, but only 3 of the Poisson ratios are independent because of the reciprocity relations: E, =  E  'E  ?  A shear stress  x  = _E_3  E  s  Yz in the yz plane produces a shear  G, (Lyz and similarly Yx = G 6 yx and Zx = G 2  Q zx.  3  =  E,  Syz, so Yz =  G ,, G  x  , G  3  are the  moduli of rigidity; these are identical with c^ , c ,, c . s  b b  Isotropy Condition The previous relationships are of use only for a single crystal.  For  by the isotropy condition, G = E 2  1 yU+  1 '  (1)  a multicrystalline agglomerate has only 2 independent moduli. The bulk modulus (K) is defined as the isotropic pressure divided by the volume change produced by the pressure. Young's modulus and Poisson's ratio:  It can be expressed in terms of  7  •  K -  E 3(1-2/1)  (2)  So by f i n d i n g E and G, K and/A. may be c a l c u l a t e d .  In the next sec-  t i o n , i t w i l l be shown that E and G may be found from measurements of the transverse  and l o n g i t u d i n a l sound v e l o c i t i e s .  V e l o c i t y of Sound To derive the r e l a t i o n between the e l a s t i c moduli and the v e l o c i t i e s of sound i n a m a t e r i a l , f i r s t consider a l o n g i t u d i n a l mechanical wave moving i n the x - d i r e c t i o n through a bar of f i n i t e c r o s s - s e c t i o n .  Let the l o c a l  elongation be (Bxx and the l o c a l s t r e s s be Xx, then by d e f i n i t i o n of Young's Modulus,  E  A  =  Xx exx  By Newton's law of force, the force a f f e c t i n g a layer of thickness dx and u n i t c r o s s - s e c t i o n equals a c c e l e r a t i o n , £Xx x  dx  5  _ jj .  OUA.  dx  J  where f i s the density of the m a t e r i a l and /U. i s v e l o c i t y i n x - d i r e c t i o n . The  forward and rear sections of the layer dx do not have the same  v e l o c i t y , but the mass remains constant by the equation of c o n t i n u i t y  But f i s connected with £ xx by  e * - v- OCOC  ^ j)  so e l i m i n a t i n g Xx and A f y i e l d s , (a)  E, ^)gxx. „  f  3 x  ^  2rt:  and  (b)  ^ xx  J) x  _2Q x X  ~  8  Now d i f f e r e n t i a t i n g (a) with respect to x and (b) with respect to t and equating the r e s u l t s gives E, ^ € xx / ^ x  o^gxx  =  L  which i s a wave equation with v e l o c i t y  For a transverse wave moving i n the x - d i r e c t i o n : consider a shear constant i n any plane x = constant, but dependent on x.  The determining  equations are: Xz = G (?xz  tiexz  s  =  *M-  ^ xz  x  3x  where A»J"is the z-component of the p a r t i c l e v e l o c i t y . equation  v  P =  J  From these comes the  x  d £xz  G ^ exz  =  which i s a l s o a wave equation with v e l o c i t y  Conversion of A d i a b a t i c to Isothermal Moduli From the 2 well-known thermodynamic r e l a t i o n s : P and  "  i t may be shown  ibility,  V  C_g _ k_r C " k V  where k  C  T  k  T  s  -  k. _ 5 =  Tp v  ,  V  (5)  Cp  i s the isothermal c o m p r e s s i b i l i t y , k  s  i s the a d i a b a t i c compress-  |2> i s the volume e x p a n s i v i t y , and V i s the molar volume.  9  Landau and Lifshitz (1959) show Gy  =  GS  Es  =  ET 1-ETT  (6)  (7)  P*V  cp  Debye Theory of Specific Heat The feature common to a l l theories of the specific heat of a solid is that the vibrational energy of a crystal containing N atoms is equivalent to the energy of a system of 3N harmonic oscillators.  The distinction  between the various theories is based on the differences, in the proposed frequency spectrum of the oscillators.  _,  The Debye theory considers the vibrational modes of the crystal as a whole, assuming i t to be an isotropic elastic continuum.  As long as the  wavelength of the wave is large compared with the interatomic distance, then the crystal appears to be a continuum to the wave.  The fact that the  crystal is actually made up of discrete atoms is taken into account by limiting, the total number of vibrational modes to 3N. quency spectrum is cut off at some maximum frequency,  That i s , the fre,  The distribution function for the Debye model is derived in a similar manner to that of the electromagnetic cavity oscillator. Z ( v ) ^ 0 is the number of possible modes of vibration in the frequency interval between 0 and  "0 + clO .  10  Where v  i s volume o f the c r y s t a l ,  the t r a n s v e r s e v e l o c i t y .  £jL i s the l o n g i t u d i n a l v e l o c i t y , fi-% i s  F o r each f r e q u e n c y o r wavelenjgth t h e r e e x i s t two  t r a n s v e r s e modes and one l o n g i t u d i n a l mode. W i t h the Debye c u t - o f f p r o c e d u r e ,  J o  Z ( 0 ) a O = 3 IS/  T h i s assumes t h a t v e l o c i t i e s Cj_ and  a r e independent o f the f r e q u e n c y .  A t t h i s p o i n t , i t i s c o n v e n i e n t t o i n t r o d u c e a parameter, the c h a r a c t e r i s t i c t e m p e r a t u r e , c a l l e d the Debye temperature g i v e n by  Q  n  =  K I t i s now o b v i o u s t h a t i f the l o n g i t u d i n a l and t r a n s v e r s e v e l o c i t i e s o f sound a r e known, 8Q c a n be c a l c u l a t e d . The v i b r a t i o n a l energy o f a c r y s t a l i s g i v e n by > j  o Then  fi-sj  which g i v e s  ZU)  AJ/KT  _  (  ^  There a r e two u s u a l a p p r o x i m a t i o n s :  =  /C^ =. 3 N K.  gives  AO  C  v  one f o r T  >  >  £\>  which  i n agreement w i t h c l a s s i c a l t h e o r y , and another f o r T« ©  = J l TT^ M>k  /X  \3  T h e r e f o r e the Debye temperature may  a l s o be o b t a i n e d from s p e c i f i c heat measurements. A l t h o u g h the Debye t h e o r y i s q u i t e s u c c e s s f u l f o r some t h e r m a l l y s i m p l e s u b s t a n c e s , d i s c r e p a n c i e s do o c c u r i n the low temperature r e g i o n . Blackman  (1941) has suggested t h a t the s p e c i f i c c r y s t a l p r o p e r t i e s a r e o f  much g r e a t e r s i g n i f i c a n c e than had been r e a l i z e d , and t h a t the v i b r a t i o n a l spectrum o f a c r y s t a l c a n be v e r y d i f f e r e n t from the Debye spectrum. More r e c e n t l y , f r e q u e n c y spectrums have been o b t a i n e d from x - r a y and n e u t r o n s c a t t e r i n g experiment d a t a .  These e x p e r i m e n t s have a l s o shed  on the i n t e r a c t i o n s i n the c r y s t a l l a t t i c e  (Walker,, 1956).  light  fi  11 Dispersion:  Phase and Group V e l o c i t i e s  For any wave motion there  is  frequency jj- and the wave l e n g t h X . velocity.  The phase v e l o c i t y ,  frequency,  is  the v e l o c i t y  velocities,  altered.  If  velocity,  V  r  =  which may be c o n s t a n t or a f u n c t i o n of each wave  the  profile  the p r o f i l e  the group  the phase of  travels.  the components o f a wave packet move w i t h  so the phase r e l a t i o n s  between the components are frequencies,  due to d i s p e r s i o n may be slow.  remain w e l l  the  Then w e l l marked  d e f i n e d and so the packet has a d e f i n i t e  velocity.  Let k = " 5 T > and so  where V i s  the packet c o n s i s t s of a narrow range o f  a l t e r a t i o n of f e a t u r e s of  r e l a t i o n between the  In g e n e r a l ,  w i t h which the c r e s t  In a d i s p e r s i v e medium, different  a definite  =  be the group v e l o c i t y .  But  ATM  or + k 4r".  ^ If  there  In  this  (10) i s no d i s p e r s i o n  = nr.  experiment no d i s p e r s i o n was o b s e r v e d ,  experimental accuracy, velocities.  ^j^- - 0 and so  and the v e l o c i t i e s  so V  &  = ATwithin  used i n the formulas are  group  12 PROCEDURE AND EQUIPMENT Iodine Samples For these i n v e s t i g a t i o n s , four samples were used ( F i g . 1).  Samples  I and I I I were grown i n a Bridgman furnace and samples I I and IV were amorphous specimens formed by heating reagent iodine ( l i s t e d i m p u r i t i e s : Chlorine, Bromine, 0.005%; N o n - v o l i t i l e matter, 0.010%) i n a glass tube. The Bridgman furnace technique of c r y s t a l growing i s e s s e n t i a l l y growth by crossing a temperature gradient.  The temperature gradient must  be steep; f o r iodine, from over the melting temperature (114°C)to about 30°C i n about a centimeter. Iodine used i n the Bridgman furnace was reagent grade which had been further r e f i n e d by condensation of the sublimed vapour. The ends of the sample were cut with a razor blade and then sandpapered f l a t and smooth.  With care and p r a c t i c e specimens could be formed  that gave good pulse patterns on the Sperry o s c i l l o s c o p e .  The end surfaces  had to be smooth and p a r a l l e l to prevent wedging and to provide good coupling between the quartz c r y s t a l , used as a transducer, and the sample. A f t e r each experiment, the sample was washed to remove the stopcock grease and was then stored i n saturated iodine water s o l u t i o n .  These pre-  cautions were taken to prevent the samples from d e t e r i o r a t i n g between experiments. As iodine i s an e l e c t r i c a l i n s u l a t o r , a ground connection f o r the quartz transducer was provided by a s t r i p of aluminum f o i l . ( F i g . 1) s t r i p was 0.5 mm.  t h i c k and about 4 mm. wide,  The  A length of very fine gold  wire was a l s o used, but i t gave the same v e l o c i t i e s and was more d i f f i c u l t to work with.  Even though 'No-naq' stop-cock grease i s s o l i d at l i q u i d  nitrogen temperature i t was found to be a good bonding agent f o r sample-foil  13 and f o i l - t r a n s d u c e r bonds.  I n a d d i t i o n , i t i s easy t o handle and i s a good  c o u p l i n g agent f o r shear waves even a t room temperature. As the s t o p - c o c k grease e t c h e s the i o d i n e , the sample had to be reshaped a f t e r e v e r y experiment.  The s a n d p a p e r i n g i n v o l v e d i n r e s h a p i n g  would cause s t r a i n s which c o u l d change the v e l o c i t y ; a l t h o u g h t h i s e f f e c t was not n o t i c e d . D i f f i c u l t i e s may  o c c u r when a sample i s c o o l e d .  c r a c k ; h a r d e n i n g and d i f f e r e n t i a l c o n t r a c t i o n may  The  sample can  r u p t u r e the f i l m  cementing  the t r a n s d u c e r t o the sample; the f r e q u e n c y o f the t r a n s d u c e r i s temperature dependent; and the l e n g t h o f the sample changes. I f the sample does c r a c k then the p u l s e s s h o u l d d i s a p p e a r or become extremely d i s t o r t e d .  I f the h a r d e n i n g and d i f f e r e n t i a l  contraction  a c t u a l l y r u p t u r e the f i l m , then q u i t e a d i s t i n c t i v e p a t t e r n r e s u l t s , (Morse, 1951)  The change i n the l e n g t h o f the sample was  corrected for.  F i n a l l y , as t h e r e i s no d i s p e r s i o n e v i d e n t i n t h i s e x p e r i m e n t ,  the change  o f f r e q u e n c y p r o b a b l y w i l l have no e f f e c t on the v e l o c i t y measurements.  14  FIG. 1 Iodine Samples used i n t h i s i n v e s t i g a t i o n . Sample #4 i s shown with aluminum f o i l e l e c t r i c a l contact. Samples #2 and #4 are about 1.5 cm. i n diameter and 0.70 cm. high. Samples #1 and #3 are about 0.80 cm. i n diameter and 1 cm. high.  m  FIG. 2 Echo pattern as shown by Sperry U l t r a s o n i c Comparator. Sample #1 at room temperature.  15  Transducers Three q u a r t z c r y s t a l s were used as t r a n s d u c e r s .  Two were X r c u t f o r  l o n g i t u d i n a l waves o f 5 and 12 megacycles p e r second (Mcs.), and one was Y-cut f o r t r a n s v e r s e waves o f 5 megacycles per second.  They were a l l  p l a t e d on one s i d e f i r s t w i t h t a n t a l u m and then w i t h p l a t i n u m so t h a t the r e s i s t a n c e o f the s u r f a c e was about 100 ohms. Both the t a n t a l u m and the p l a t i n u m were d e p o s i t e d by s p u t t e r i n g i n an atmosphere o f 30 m i c r o n s o f argon. as an u n d e r c o a t ,  f o r without  The t a n t a l u m was found  t o be needed  i t the p l a t i n u m c o u l d be s c r a t c h e d from the  quartz surface. The bandwidth c h a r a c t e r i s t i c s o f a t r a n s d u c e r a r e determined combined e f f e c t o f the m e c h a n i c a l quality factor Q ' .  Q  q u a l i t y f a c t o r Q and the e l e c t r i c a l  F o r good t r a n s i e n t o r p u l s e r e s p o n s e ,  band i s d e s i r a b l e and c o n s e q u e n t l y  by the  a low Q and Q . a r e  a wide  »  frequency  necessary,  i s g i v e n by, Q  =  TT.  J/v*. &  2  fc  f o r t h e f i r s t harmonic mode, where  0  + ?,£.,  f i s the d e n s i t y and  i s the v e l o c i t y  o f sound o f a m a t e r i a l and f <~ i s c a l l e d the c h a r a c t e r i s t i c impedance o f the m a t e r i a l , f  0  ^ i s the c h a r a c t e r i s t i c impedance o f the l o a d , i n t h i s  case i o d i n e , f,\. £ i s the c h a r a c t e r i s t i c impedance o f the b a c k i n g o f the s  u  transducer..' f ^ C ^ ; i s  the c h a r a c t e r i s t i c impedance o f the t r a n s d u c e r .  A l o w Q r e q u i r e s a l o a d i n g medium w i t h h i g h c h a r a c t e r i s t i c impedance and a b a c k i n g o f h i g h impedance and a c o u s t i c l o s s .  F o r these  reasons,  l u c i t e w i t h a t a p e r e d b r a s s e l e c t r o d e was used f o r b a c k i n g . As i o d i n e i s w e l l matched, t h a t i s , has n e a r l y the same a c o u s t i c impedance as an X-cut q u a r t z c r y s t a l , the e x p e r i m e n t a l  system,has a s a t i s f a c t o r y  mechanical  Q.  0/ i s given by,  ;  Tr  f <_ 0  where Jt.^ i s the electromechanical coupling f a c t o r , f 0 <- again r e f e r s to the load,rand  /«A^to  the transducer.  Q' increases with i n c r e a s i n g load, whereas Q decreases with increasing load.  Therefore, minimizing Q and Q' presents a problem.  A low Q' insures the l e a s t possible change i n the r e l a t i v e and phases of the pulse throughout i t s frequency spectrum.  amplitudes  17  L u c i t e Backing  Tapered Brass Electrode  Quartz C r y s t a l Transducer Brass Container  *  •«  FIG. 3  Copper Former f o r Resistance Thermometer  Resistance Thermometer  Etalon f o r Iodine V e l o c i t y Measurements  l  0  1/ Tapered Brass Electrode Quartz C r y s t a l Transducer  2  LP  L u c i t e Backing Hole to admit L i q u i d Helium  Brass Container  FIG. 4  Etalon f o r L i q u i d Helium V e l o c i t y Measurements  18 Measurements and A p p a r a t u s The  l e n g t h o f the sample was measured w i t h a micrometer b e f o r e and  a f t e r an experiment. carefully.  As i o d i n e i s s o f t , these measurements had t o be made  Because the c o e f f i c i e n t o f l i n e a r e x p a n s i o n  v a l u e o f 83.7 x 1 0 "  6  (°K)  - 1  i s l a r g e , (an average  from 78°K t o 290°K, Dewar, 1902) the l e n g t h o f  the sample was c o r r e c t e d down t o l i q u i d n i t r o g e n temperature.  A t tempera-  t u r e s below l i q u i d n i t r o g e n temperature the sample's l e n g t h was taken as constant. The S p e r r y U l t r a s o n i c A t t e n u a t o r Comparator ( F i g . 4) d i s p l a y s the echoes o f the p u l s e d i r e c t l y on a cathode r a y o s c i l l o s c o p e . The time o f f l i g h t o r t h e l e n g t h o f time between echoes c a n be measured w i t h a b u i l t - i n delay.  The a c c u r a c y o f the d e l a y i s g i v e n i n the s p e c i f i c a t i o n s a s ' 0 . 5 %  from 5 - 90 y U s e c . I n the S p e r r y Comparator, a t r i g g e r p u l s e s t a r t s s i m u l t a n e o u s l y the h o r i z o n t a l sweep o f the o s c i l l o s c o p e and a s w i t c h i n g c i r c u i t which a l t e r n a t e l y a c t i v a t e s e i t h e r the p u l s e d o s c i l l a t o r o r a comparator p u l s e .  The  comparator p u l s e i s used i n a t t e n u a t i o n measurements, and f o r m e a s u r i n g the time between echoes.  When the t r i g g e r a c t i v a t e s the p u l s e d o s c i l l a t o r , the  l a t t e r d r i v e s t h e q u a r t z c r y s t a l w i t h a p u l s e o f 0.5, 1, o r 2 m i c r o - s e c o n d duration.  The f r e q u e n c y  o f the o s c i l l a t o r can be v a r i e d from 1 t o 200 Mcs,  By the p i e z o e l e c t r i c e f f e c t , the q u a r t z c r y s t a l generates  a pressure  v i b r a t i o n , t h a t i s , a sound p u l s e , i n the sample and t h i s p u l s e echoes between t h e p a r a l l e l f a c e s o f the sample.  Each time the p u l s e r e t u r n s t o  the q u a r t z c r y s t a l i n t e r f a c e a s m a l l v o l t a g e i s g e n e r a t e d  which i s r e c e i v e d ,  a m p l i f i e d , and a p p l i e d t o the v e r t i c a l d e f l e c t i o n p l a t e s o f the o s c i l l o s c o p e . The o s c i l l o s c o p e then d i s p l a y s a d e c a y i n g  t r a i n o f echoes ( F i g . 2 ) , each one  19 of which has made one more round t r i p through the sample than the preceding echo. As a check on the time delay of the Sperry Comparator, the v e l o c i t y of sound i n l i q u i d helium was measured at 1.90°K.  The test was c a r r i e d out  with a brass etalon ( F i g . 4) and a 12 Mcs. quartz c r y s t a l transducer s i l v e r plated on both sides.  The bottom support f o r the transducer was made as  p a r a l l e l as possible to the r e f l e c t o r of the etalon. The v e l o c i t y was found to be 233 - 7 m sec"'''. with the accepted value of 229 m s e c " . 1  This i s i n agreement  (Atkins, 1959)  As a second check on the time delay of the Sperry and also on the bonding agent, the v e l o c i t y of sound i n a cadmium specimen was measured as 2.80 ± 0.05 x 1 0 cm s e c " . 5  1  The accepted value i s 2.78 x 10 cm s e c " . 5  1  20  Pulsed Oscillator  Switch  Sample  100 Pulses Per Second Trigger  Pulse Comparator  Exponential Curve Generator  Sweep Generator  FIG. 5  Mixer  Local Oscillator  I.F. S t r i p and Detector  Oscilloscope Screen j  Block Diagram of Sperry U l t r a s o n i c Attenuator Comparator  21 Two methods of getting, the sample to a temperature were employed. The f i r s t was to immerse the sample i n a bath and then wait f o r thermal e q u i l i b r i u m to be achieved. appeared on the o s c i l l o s c o p e .  This was taken to be when a stable pattern Baths used were:  warm water, room temperature,  melting i c e , f i n e l y chipped s o l i d carbon dioxide,, l i q u i d nitrogen, and l i q u i d helium. The second method was slow warming.  The sample was cooled i n the  dewar system to l i q u i d nitrogen or helium temperature and then allowed to warm slowly.  From 4.2°K to 40°K took about 5 hours and i t was necessary to  have a l i t t l e l i q u i d helium i n the inner dewar otherwise the temperature rose too q u i c k l y .  The warming process from 78°K to 200°K also took about  5 hours. Because iodine i s a poor thermal conductor, the question of whether the  sample i s i n thermal e q u i l i b r i u m during these warming processes a r i s e s .  One check on t h i s i s that the value of the v e l o c i t y obtained at 200°K f o r the warming process agrees with that obtained f o r the s o l i d carbon dioxide bath.  A second check, i s that a rough c a l c u l a t i o n of the maximum energy  needed to warm the iodine shows that much more energy can be provided to the  sample than i s required. Temperatures were mainly measured with a copper wire resistance  thermometer which consisted of #44 enamelled copper wire wound on a copper former.  The room temperature resistance was 33.97 ohms.  The copper former  was used to eliminate stress i n the wire due to d i f f e r e n t i a l  contraction.  The r e s i s t a n c e of the thermometer was measured at 273°K and at 4.2°K. Then from the expression  =  RT ~  22 and from tables on page 115 i n White r e l a t i n g values of Z to temperature, the temperature of the thermometer could be found by measuring i t s resistance (Fig. 6 ) . Unfortunately, although copper can be obtained i n very pure form and can be drawn i n t o f i n e wire, i t has a rather high Debye temperature and i s e a s i l y strained.  Therefore, a copper wire resistance thermometer i s i n s e n s i -  t i v e below 20°K, and must be handled c a r e f u l l y . Theory of Resistance Thermometer In a metal, the free electrons responsible f o r e l e c t r i c a l are  conductivity  scattered by imperfections i n the c r y s t a l l a t t i c e and by thermal v i b r a -  tions of the l a t t i c e .  These processes l i m i t the mean free path and so  determine the e l e c t r i c a l  conductivity.  Matthiessen's r u l e i s w r i t t e n  f = fx f; . Where ?/^ i s the r e s i s t i v i t y +  due to s t a t i c imperfections and i s temperature independent, and / r e s i s t i v i t y due to thermal v i b r a t i o n s and i s temperature  A  i s the  dependent.  The r e s i s t a n c e of a conductor i s given by A where JL i s the length, A the cross section area, and / i s the r e s i s t i v i t y of the conductor. Combining the above formula f o r resistance with Matthiessen's r u l e £  gives, So  _  A T - fx  i s the thermometric property which i s important to the  resistance thermometer and —r~ Now, as i n the region  6  0  determines the s e n s i t i v i t y  < T <  8o  P-eLT^  where  of the  3*H\*S  thermometer. , i tis  apparent why the copper wire thermometer ( B = 315°K f o r copper) becomes a  i n s e n s i t i v e below 20°K.  23  i  I-  Rheostat • v AMAMAAAA/VV Milliammeter  Standard Resistor ^—\AAAA/NA/—f~ Current Leads Monitoring  Resistance — A W W Thermometer  Potentiometer  P o t e n t i a l Leads Potentiometer  FIG. 6  C i r c u i t f o r Measuring the Resistance of the Resistance Thermometer  24 The c r y o s t a t , ( F i g . 7) i n which the helium and nitrogen temperature measurements were made, i s of standard design, c o n s i s t i n g of an outer dewar for l i q u i d nitrogen and an inner dewar f o r l i q u i d helium.  The e t a l o n and  thermometer former were supported from a s t a i n l e s s s t e e l tube.  The c o a x i a l  cable was a V' s t a i n l e s s s t e e l tube with #32 copper wire as a center conductor and polystyrene as an i n s u l a t o r . At the top of the c r y o s t a t was a matching network ( F i g . 8) c o n s i s t i n g of a v a r i a b l e inductor and c a p a c i t o r .  The e f f e c t of the f i x e d capacitance  of the c r y s t a l and the s t a i n l e s s s t e e l c o a x i a l cable was tuned out by the v a r i a b l e inductance.  Then any excess inductance i n the c i r c u i t was tuned out  by the v a r i a b l e c a p a c i t o r .  25 Coaxial Cable Syphon Cap  n  Leads to Resistance  Thermometer  To Return Line To Pumping Line To Manometers  L i q u i d Nitrogen  L i q u i d Helium  Etalon f o r Iodine V e l o c i t y Measurements  FIG. 7  Dewar System f o r a L i q u i d Helium Temperature Bath  26  To Transducer  FIG. 8  Matching Network  27 RESULTS, CALCULATIONS AND CONCLUSIONS Longitudinal V e l o c i t y and Young's Modulus The experimental r e s u l t s f o r sample #2 have been given most a t t e n t i o n . Sample #2 was an amorphous specimen and so the r e s u l t s for i t should be representative of the average acoustic and e l a s t i c properties of iodine. As a check, the transverse and l o n g i t u d i n a l v e l o c i t i e s were measured i n another amorphous sample, sample #4.  The v e l o c i t i e s for the two samples  agreed w i t h i n the experimental e r r o r . As samples #1 and #3 appeared to be composed of several large c r y s t a l s the v e l o c i t i e s found i n them are probably nearly the v e l o c i t y along a particular crystal axis. In figures 9, 10, and 11, the experimental value of the l o n g i t u d i n a l v e l o c i t y i s represented by a s o l i d l i n e .  The dotted curve i n d i c a t e s the  v e l o c i t y corrected for thermal c o n t r a c t i o n . This c o r r e c t i o n was made using a l i n e a r expansion c o e f f i c i e n t of 83.6 x 10" (°K) 6  i s an average value over the temperature  _1  (Dewar, 1902).  range 290°K to 78°K.  This  The length  was corrected to 78°K and then was taken as constant. The main source of e r r o r i n l o n g i t u d i n a l v e l o c i t y arose i n the measurement of the delay time f o r the pulses.  The pulses were not sharp  spikes but rounded humps ( F i g . 2), due to the specimens not being perfect u l t r a s o n i c samples.  For l o n g i t u d i n a l waves i n sample #2 the e r r o r at room  temperature was 0.2 /Ksec  i n 7,0  yUsec  or 37»; at 4.2°K, 0,2 /H-sec i n 5.0  yusec or 47 . c  The other source of e r r o r l i e s i n determining the length of the sample.  At room temperature, for sample #2 t h i s amounted to 0.01 cm, i n  0.70 cm. or about 1%. This e r r o r would be larger for lower temperatures as., the value f o r the l i n e a r expansion i s unsure.  28 The e r r o r s from these two sources are much the same for the other two samples. The c o r r e c t i o n due to the c o n t r a c t i o n of the sample i s less than the possible e r r o r at 4.2°K. room temperature  i s 2%.  The percentage change i n length with reference to So the trend has been dotted to i n d i c a t e that the  v e l o c i t y should be less than the experimental value. The l o n g i t u d i n a l v e l o c i t i e s were measured at 5 and 12 Mcs. and w i t h i n the experimental e r r o r , no d i s p e r s i o n was discovered.  Therefore, the pulse  technique was acceptable for v e l o c i t y measurements. In sample #1 two l o n g i t u d i n a l v e l o c i t i e s were noticed.  The lower  appeared at the e a r l y stages of the experiment, but could not be generated during the l a t e r stages.  The experiments went on over 5 months and the  sample was smoothed several times. 5 and 12 Mcs. transducers.  The lower value was noted with both the  The r a t i o of the higher to lower v e l o c i t y was  approximately constant (1.3*0.1) over the temperature range 290°K to 1.2°K. R e f l e c t i o n from a g r a i n boundary, that i s , the pulse being r e f l e c t e d back and f o r t h i n one of the large c r y s t a l s , may have caused the lower v e l o c i t y value.  Thermal d i f f u s i o n and thermal c y c l i n g would destroy the  boundary and t h i s would e x p l a i n why the lower v e l o c i t y value could not be found i n the l a t e r stages of the i n v e s t i g a t i o n .  32  Young's modulus (E) was c a l c u l a t e d from E = tyl J where f density of iodine. Nm"  f o r sample #2,  i s the  At room temperature, E was found to be 2 . 0 ± 0 . 2  x.10  1 0  In t h i s case / was taken as 4 . 9 4 gm cm" . This r e s u l t  i s i n q u a n t i t a t i v e agreement with an i n t e r p o l a t e d value of E. (Zwikker, 1 9 5 4 ) Zwikker i n d i c a t e s that on the basis of a p l o t of E against atomic number, E 10  for atomic number 53 should, be 2 . 5 x 10  Nm  -2  .  For c a l c u l a t i o n s of E below 200°K, / was taken to be 5 . 0 0 gm cm"  as  J* w i l l increase with decreasing temperature due to thermal c o n t r a c t i o n . The conversion from a d i a b a t i c Young's modulus to isothermal Young's modulus i s achieved by  6  P  At room temperature using: V ~ 52 cm mole" , p. =» 264 x 1 0 " 3  (Straumanis, 1 9 4 3 ) , and  C  1  . 6 . 5 7 c a l (mole)" ( V ) " 1 (Giauque, 1 9 5 9 ) , 1  p  6  i s found to be 1.3 1 0 , 2 x 1 0  1 0  Nm" , 2  (°K)  E  T  _ 1  34  o  o  35  15 -  i,o • 0.5  I 0  \  50  FIG IS  .—i  100  1  ISO  VOUNtfs MODULUS FOR SAMPLE *3  1  100  1—.  2.50  1  300  J —  3S0  37 Transverse V e l o c i t y and Shear Modulus The transverse v e l o c i t y ,  , was, measured at 5 Mcs.  were conducted on samples #1, #2, and #4. temperature.  At room temperature,  Experiments  was found to be constant with  - 0.&>1  , a r e s u l t which i s i n agreement  with the e m p i r i c a l r e l a t i o n i n Zwikker (1954). At room temperature i n sample #2, the error i n £•£ consisted of 0.2 ^isec. i n l O ^ s e c . , i e . 2% f o r pulse .width and 1% f o r length.  Again the error  i n length would be larger f o r lower temperatures. However,; there were not as many echoes f o r the transverse measurements, 2 or 3 compared w i t h 3 to 5 f o r the l o n g i t u d i n a l pulses.  So these v e l o c i t y  values should not be considered more s i g n i f i c a n t than the l o n g i t u d i n a l v e l o c i t y values. The shear modulus (G). i s c a l c u l a t e d from G  =/C^~f  . In t h i s case,  G i s the a d i a b a t i c shear modulus, but according to equation (6) the a d i a b a t i c shear modulus equals the isothermal shear modulus. The a d i a b a t i c c o m p r e s s i b i l i t y ( M.% ) i s c a l c u l a t e d from equations (1) and (2), L  - 9 - 3 E/G E  .  As, i s found to be 0.77 ± 0.08 x 10" • m N 10  and f l a t t e n s o f f to 0.1 ± 0.1 x 1 0 " m N 1 0  Debye Temperature f o r Iodine; The Debye 6 was found from .  2  _1  2  -1  at 290°K f o r sample #2  at 4.2°K.  38 22 N i s the number of atoms per cubic centimeter (2,35 x 10 ), k i s Boltzmann's constant and  i s Planck's constant divided by 2 T ,  The possible e r r o r i n QQ f o r sample #2 at 290°K i s approximately 87o, The © f o r sample #2 at 290°K was 116 - 9°K, i n reasonable agreement w i t h D  106°K c a l c u l a t e d by Schroedinger (1919). 9 o can also be c a l c u l a t e d from the Lindemarin melting point formula.  C i s a constant (320).  T ^ i s the melting temperature (387°K).  the molar volume 51.3 crn^ mole" . 1  V = /j? i s M  M i s the, molecular weight 254 gm.  The 8^  i s found to be 106°K. Conclusions; For t h i s s e r i e s of i n v e s t i g a t i o n s , the l o n g i t u d i n a l and transverse v e l o c i t i e s have been measured over temperature ranges of 1°K to 373°K and 4°K to 295°K r e s p e c t i v e l y . metallic solid.  They behave much as would be expected for a  The l o n g i t u d i n a l v e l o c i t y rose from 2,0 x 10^ cm s e c  - 1  at  290°K to 2.6 x 10 cm s e c " at "78^ and then f l a t t e n e d o f f to 2.8 x 10 cm 5  sec"  1  at 4°K.  1  5  The shape of the curve i s s i m i l a r to those of m e t a l l i c s o l i d s ,  and the f l a t t e n i n g o f f at low temperatures i s i n accordance with the t h i r d law of thermodynamics. The transverse v e l o c i t y , found to be 1.4 x 10^ cm s e c " at room temp1  erature, was constant down to l i q u i d helium temperature, w i t h i n the experimental error. In the v e l o c i t y measurements, the main source of e r r o r l a y . i n the samples not being perfect u l t r a s o n i c specimens. From the l o n g i t u d i n a l and transverse v e l o c i t i e s , the adiabatic Young's  39 modulus (2.0 x 1 0  1 0  Nm"  2  at 290°K), shear modulus (1.0 x. 1 0  1 0  Nm" ), and the 2  Debye c h a r a c t e r i s t i c temperature (116 i 9°K) were c a l c u l a t e d . The Debye temperature according to Schroedinger, and the Lindemann melting point formula i s 106°K. approximately 2.5 x l O ^ Nm" . 1  2  Young's modulus according to Zwikker i s  40 Bibliography A t k i n s , K. R. , (1959), L i q u i d Helium, Cambridge U n i v e r s i t y Press. Balamuth, L., (1934), Phys. Rev. 45  715.  Bersohn, R. , (1962), J . Chem. Phys. 36  3445.  Blackman, M., (1941), Repts. Prog. Phys. 8 Dewar, J . , (1902), Roy. Soc. Proc. 70 Durand, M. A., (1936), Phys. Rev. 50 Gait, J . K., (1948), Phys. Rev. 73  11.  237. 449.  1460.  Hearmon, R. F. S., (1946), Rev. Mod. Phys. 18  409.  Landau, L. D., and L i f s h i t z , E. M. , (1959), Theory of E l a s t i c i t y , Addison Wesley Publishing Co., Inc. Morse, R. W. , (1959), Progress i n Cryogenics, V o l . 1, Edited by K. Mendelssohn London Heywood & Company. Nernst, W., (1911), Annalen der Physik 36 Rose, F. C , (1936), Phys. Rev. 49  395.  50.  Rosenburg, H. M. , (1959), Progress i n Cryogenics, V o l . 1, Edited by K, Mendelssohn, London Heywood & Company, Schroedinger, E., (1919), Physik Z. 20  450.  S h i r l e y , D. A., And Giauque, W. F. , (1959), J . Am. Chem. Soc. 8_1 Simhony, M., (1963), J . Phys. Chem. S o l i d s 24  4778.  1297.  Straumanis, M., and Sauka, J . , (1943), Z. Phys. Chem. B53  320.  White, G. K., (1959), Experimental Techniques i n Low Temperature Physics, Clarendon Press. Walker, C. B., (1956), Phys. Rev. 103  547.  Zwikker, C., (1954), P h y s i c a l Properties of S o l i d M a t e r i a l s , Pergammon Press.  

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