NONLINEAR ACOUSTICS AND SOLID STATE PHYSICS

HAL Id: jpa-00219532

https://hal.archives-ouvertes.fr/jpa-00219532

Submitted on 1 Jan 1979

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

NONLINEAR ACOUSTICS AND SOLID STATE PHYSICS

M. Breazeale

To cite this version:

M. Breazeale. NONLINEAR ACOUSTICS AND SOLID STATE PHYSICS. Journal de Physique

Colloques, 1979, 40 (C8), pp.C8-164-C8-167. �10.1051/jphyscol:1979827�. �jpa-00219532�

JOURNAL DE PHYSIQUE Colloque C8, supplement au n°ll, tome 40, novemhre 1979, page C8-164

NONLINEAR ACOUSTICS AT® SOLID STATE PHYSICS M.A. Breazeale

Department of Physios, The University of Tennessee, Knoxville, Tennessee 37916, USA.

Résumé.- L'étude de l'acoustique non linéaire en plus de développements mathématiques et d'applica- tions pratiques pleines de promesses nous conduit â des progrès essentiels dans l'étude de la phy- sique du solide. La déformation non linéaire d'une onde ultra-sonore dans un solide est contrôlée par un paramètre non linéaire qui est à la fois fonction des constantes élastiques du troisième ordre {jOECj et des constantes élastiques -plus connues- du deuxième ordre 0>OECl. La mesure de la déformation des ondes ultrasonores permet la détermination des constantes du troisième ordre, les constantes du deuxième ordre pouvant être calculées à partir des vitesses des ondes ultrasonores.

Le nombre des constantes élastiques est déterminé par la symétrie du cristal. La grandeur et le signe des constantes du 3e ordre sont déterminés par les forces intermoléculaires. Si T o n considère un cristal cubique dans l'hypothèse d'un champ de force central entre premiers voisins, à des tempé- ratures voisines de 0°K on trouve que C ^ j - 2 0 ^ 2 = 2Ci66 e t c123 = c456 = c144 = °-

Au cours des dernières années nous avons mesure les constantes du 3e ordre en fonction de la tem- pérature, jusqu'à 3° K, dans des cristaux tels que le germanium et le cuivre et dans de la silice amorphe fondue. Nous présentons un résumé des résultats obtenus à l'heure actuelle et nous faisons une comparaison du comportement des forces inter-moléculaires pour des cristaux de types différents.

Récemment nous avons montré que dans le cuivre les forces centrales entre premiers voisins sem- blent jouer un rôle prépondérant dans la valeur des constantes du 3e ordre pour les températures de 40°K et de 190°K. Le résultat n'est plus vérifié à 0°K contrairement à ce que l'on pouvait attendre.

Abstract.- In addition to mathematical advances and very promising practical applications, the study of Nonlinear Acoustics now is leading to fundamental advances in Solid State Physics. Nonlinear dis- tortion of an Ultrasonic wave in a solid is controlled by a nonlinearity parameter which is a func- tion of the third-order elastic (TOE) constants as well as the second-order elastic (S0E)(the usual) elastic constants.Measurement of the waveform distortion, then, makes possible the evaluation of the TOE constants, because the SOE constants can be evaluated from ultrasonic wave velocities.

The number of elastic constants is determined by the crystal symmetry. The magnitude and sign of the TOE constants are determined by intermolecular forces. In a cubic crystal in which central for- ces and nearest-neighbor interactions exist one would find C ^ = 2Cji2 = 2Cl66 and C1 23 = C455 = C144 = 0 in the limit of 0°K.

Over the past several years we have measured TOE constants as a function of temperature down to 3°K in such crystals as germanium and copper and in amorphous fused silica. A summary of presently available data is presented, and a comparison of the behavior of the intermolecular forces for dif- ferent types of crystals is made. Recently we have found that in copper central forces and nearest- neighbor interactions seem to predominate in determining TOE constants, but this does not happen near 0°K as expected. It seems to happen near 40°K and again at 190°K.

When a sinusoidal ultrasonic wave of finite am- plitude propagates through a nonlinear medium it undergoes waveform distortion, as we all know. The amount of this distortion and its dependence upon frequency and propagation distance are determined by the nonlinearity parameter of the propagating medium. For gases this nonlinearity parameter is simply Y + 1 where y is the ratio of specific heats. For liquids it is B/A + 2, where B/A is the ratio of two terms in the equation of state. For solids the situation is somewhat more complicated.

Not only are there both longitudinal and transverse waves to consider, but also the fact that solids can be anisotropic means that the longitudinal and transverse waves in general are coupled, so that one cannot define a nonlinearity parameter for eve- ry possible direction in a crystalline solid. Our

solution to this problem has been to consider the propagation of finite amplitude ultrasonic waves only along the principal crystal!ographic direc- tions, and to restrict our attention for the pre- sent to crystals of cubic symmetry. In the princi- pal directions pure mode propagation exists for the longitudinal wave, even though the transverse wave is always coupled to a longitudinal wave. The pure- mode longitudinal wave is coupled to its own second harmonic in exactly the same way a longitudinal wave in a fluid is. For these waves a nonlinearity parameter can be defined, and it is found that the nonlinearity parameter is a function of the second- order elastic (SOE) constants and third-order elas- tic (TOE) constants. The study of nonlinear distor- tion of ultrasonic waves in solids, then, has pro- duced a means of measuring the TOE constants (the

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979827

SOE constants can be determined from t h e v e l o c i t y c i t i v e d e t e c t o r has worked w e l l f o r us, so t h a t we o f u l t r a s o n i c waves i n t h e sample). Furthermore, have been a b l e t o determine B f o r several samples t h e technique can be adapted t o measurements as a - n o t o n l y a t room temperature /3/, b u t a l s o a t low f u n c t i o n o f temperature, which produces r e s u l t s temperatures- down t o 3OK. The behavior o f 6 as a which a r e o f fundamental importance t o s o l i d s t a t e f u n c t i o n o f temperature and the behavior o f t h e physics. A t present t h i s i s t h e o n l y technique TOE constants as a f u n c t i o n o f temperature a r e o f which i s c o n s i s t e n t l y producing such r e s u l t s . Some most i n t e r e s t a t present. L e t us c o n t r a s t t h e i r o f our r e c e n t r e s u l t s a r e q u i t e e x c i t i n g ; however,

f i r s t l e t me remind you o f t h e technique.

I n t h e p r i n c i p a l d i r e c t i o n s i n a cubic crys- t a l l o n g i t u d i n a l waves o f f i n i t e amplitude a r e des- c r i bed by /1/

where K2 and K3 are l i n e a r combinations, r e s p e c t i - v e l y , o f t h e SOE and TOE constants, as shown i n Table I. The n o n l i n e a r i t y parameter /2/ i s t h e ne- g a t i v e o f t h e r a t i o o f t h e n o n l i n e a r term t o t h e l i n e a r term i n Eq. 1, namely,

3K2

+

K3

= B ( 2 )

K2

For an i n i t i a l l y s i n u s o i d a l disturbance a t a = 0, t h e s o l u t i o n s takes t h e form :

A: k2 aB

u = A1 S i n (ka-wt)

+

cos 2 (ka-wt)

+ . . .

8

(3) i n which t h e second harmonic amplitude :

contains t h e n o n l i n e a r i t y parameter 8. Our measu- rements, then, c o n s i s t o f absolute measurement of t h e second harmonic amp1 i tude, t h e fundamental amplitude, t h e frequency, and t h e sample l e n g t h . The o n l y r e a l problem i n these measurements i s t h e f a c t t h a t t h e second harmpnic amplitude i s so low ( o f t h e o r d e r o f

8,

a t 60 MHz) t h a t care must be exercised i n g e t t i n g an accurate value. A capa-

behavior i n germanium, i n which c o v a l e n t bonding among t h e atoms e x i s t s , w i t h t h a t i n copper, i n which exchange energy r e s u l t i n g from over1 qppi ng o f closed e l e c t r o n i c she1 1 s seems t o predominate i n determining t h e e l a s t i c constants.

N o n l i n e a r i t y parameters measured as a func- t i o n o f temperature f o r germanium a r e shown i n F i g . 1 ; those f o r copper a r e shown i n Fig. 2.

I I

12 GERMANIUM

Fig .

¹ : Temperature dependence o f t h e n o n l i n e a r i t y parameters o f germanium.

I

^COPPER

I

Fig. 2: Temperature dependence o f t h e n o n l i n e a r i t y parameters of copper.

Table 1. K2 and K3 f o r p r i n c i p a l d i r e c t i o n s o f a cubic c r y s t a l

F

D i r e c t i o n K2 K3

[loo1

^c11 C l l l

11 103 q ( ~ l l 1 + ~ 1 2 + 2 ~ 4 4 ) i ( ~ 1 1 1 ⁺ 3C112 ⁺ lZC166)

C l l l l

^:(cll

+

2C12 + 4C44) V ( C 1 l l ¹ + 6C112 ⁺ 12C144 + 24C166 ⁺ "123 ⁺ 16'456)

*

I

c8-166 M.A. Breazeale

I n each case a smooth curve i s drawn connecting the data p o i n t s . As can be seen, t h e n o n l i n e a r i t y para- meter i s d i f f e r e n t i n t h e t h r e e p r i n c i p a l d i r e c t i o n s

i n both cases, b u t t h e d i f f e r e n c e i n the magnitudes between germanium and copper i s n o t g r e a t . F u r t h e r - more, t h e magnitude increases i n t h e o r d e r [I001

,

[Ill], and [110] i n b o t h cases. The v a r i a t i o n w i t h temperature i s n o t g r e a t , t h e g r e a t e s t v a r i a t i o n being o n l y 20 % f o r t h e

[IIOJ

d i r e c t i o n i n germa- nium. One can conclude, t h e r e f o r e , t h a t t h e n o n l i - n e a r i t y parameter o f s o l i d s , l i k e t h e Griineisen parameter, should be nominally independent o f tem- perature.

There i s much more i n f o r m a t i o n t o be obtained from these data, however. Even though we cannot i s o - l a t e a l l s i x o f t h e TOE constants, we can i s o l a t e c e r t a i n combinations o f them. These are t h e q u a n t i - t i e s l i s t e d i n Table I as K3. Using Eq. ( 2 ) , and known values o f K2 = p ~ 2 , we can determine t h e com- b i n a t i o n s K3 p l o t t e d i n Figs. 3 and 4. The value o f

TEMPERATURE ( K) 2 0 0

2-20b

^GERMANIUM

1

F i g . 3: TOE constant combination K3 f o r germanium c a l c u l a t e d from n o n l i n e a r i t y parameters.

TEMPERATURE ( K )

F i g . 4: TOE constant combination K3 f o r copper c a l - c u l a t e d from nonl i n e a r i ty parameters.

K3 f o r t h e

[loo]

d i r e c t i o n i s t h e s i n g l e TOE cons- t a n t CIl1. The values o f K3 i n the o t h e r d i r e c t i o n s a r e t h e more complicated l i n e a r combinations i n Table I. Upon examination i t i s c l e a r t h a t t h e c u r - ves i n F i g s . 3 and 4 s t i l l a r e n o t t h e s i m p l e s t combinations o f TOE constants a v a i l a b l e from t h e data. Clll appears i n each expression f o r K3, so i t can be s u b t r a c t e d o u t . I n t h i s manner one can ob- t a i n the curves shown i n Figs. 5 and 6, which a r e t h e s i m p l e s t combinations o f TOE constants a v a i l a - b l e from o u r data on t h e n o n l i n e a r i t v Parameters.

Fig.5 : Comparison o f s i m p l e s t TOE constant combi- n a t i o n a v a i l a b l e from germanium data w i t h p r e d i c - t i o n s o f c e n t r a l forces, nearest-neighbor model : C112+4C166'5/2 Clll and C123+6C144+8C456' 0.

TEMPERATURE (

K )

^'

50 1 0 0 1 5 0 2 9 0 2 5 0 3 0 0

TEMPERATURE ( K) 5 0 100 150 2 0 0 2 5 0 3 0 0

Fig.6 : Comparison of s i m p l e s t copper TOE constant combination and p r e d i c t i o n s o f model.

.- qE - +2-

.

^a, V ⁰ K

>, 2 -2-

- 9

^-4-

\

JOURNAL DE PHYSIQUE

F u r t h e r i n t e r p r e t a t i o n o f these curves i s faci- l i t a t e d by comparison w i t h a model. We w i l l use t h e simplest model a v a i l a b l e : a model i n which we assu- me t h a t t h e atoms i n t e r a c t by c e n t r a l forces, and i n which t h e r e a r e o n l y nearest-neighbor i n t e r a c - t i o n s . Such a model has proved t o be inadequate t o e x p l a i n t h e magnitude o f SOE constants, as we know.

I t p r e d i c t s t h a t

Cll

=

2C12

=

2C44,

which i s n o t s a t i s f i e d by any of t h e c u b i c c r y s t a l s . On t h e o t k r hand, we a r e concerned w i t h t h e TOE constants which are determined by t h e n o n l i n e a r i n t e r a c t i o n s . For t h e TOE constants the c e n t r a l forces, nearest-neigh- b a r i n t e r a c t i o n model p r e d i c t s t h a t

Clll

=

2C112

=

2C166

and

C123

=

C456

=

C144

= 0. We can use these expressions t o see how w e l l those q u a n t i t i e s we can measure a c t u a l l y f o l l o w t h e p r e d i c t i o n s o f the mo- d e l . I n Figs.

5

and

6

t h e combination

ClZ3

t

6C144 +

8C456

should vanish. Fig.

5

shows t h a t t h i s a c t u a l - l y happens a t approximately 15°K f o r germanium. The f a c t t h a t t h i s combination becomes p o s i t i v e below t h i s temperature cannot be explained a t t h e moment.

Another t e s t o f the agreement between t h e data and t h e c e n t r a l forces, nearest-neighbor model comes from the f a c t t h a t , according t o t h e model the com- b i n a t i o n

C112

t

4C166

=

5/2 Clll.

On Figs.

5

and

6

t h e value

5/2 Clll

i s i n d i c a t e d by a d o t t e d l i n e . For germanium t h e combination

C112 + ^4C166

^indeed

i s approaching

5/2 Clll

near 0°K. This means t h a t near 0°K t h e c e n t r a l forces, nearest-neighbor model can be used t o p r e d i c t q u i t e w e l l t h e r e l a t i o n s h i p among t h e TOE constants o f germanium. This i s con- s i s t e n t , because the model i s expected t o be most accurate a t O°K, and l e s s accurate as t h e tempera- t u r e i s increased.

L e t us now t u r n t o t h e data f o r copper. The same combinations o f TOE constants a r e shown i n Fig.

6. Clll

has more o r l e s s t h e same behavior f o r copper as f o r germanium, w i t h o n l y a s l i g h t l y l a r - g e r v a r i a t i o n w i t h temperature. Likewise, t h e r e i s reasonable agreement between

C112

t

4C166

and

5/2 Clll,

except t h a t f o r copper t h e agreement r e - mains e s s e n t i a l l y the same, a d i f f e r e n c e o f appro- x i m a t e l y

6

%, f o r a l l temperatures. For copper t h e combination

Clp3 + 6C144 + 8C456

i s more n e a r l y zero f o r the e n t i r e temperature range. I n f a c t , i t vani- shes completely a t two temperatures : 40°K and again a t 200°K.

L e t us assume f o r t h e moment t h a t t h e TOE

I n t h a t case we would be l e f t w i t h t h e n e c e s s i t y o f e x p l a i n i n g t h e d i p s i n t h e data i n Fig.

6

which occur between 40 and 200°K. I f we examine known physical mechanisms we f i n d t h a t d i s l o c a t i o n i n t e r - a c t i o n , f i r s t r e p o r t e d by Bordoni

/5/,

can produce i n copper a peak i n a t t e n u a t i o n which has a s i m i l a r temperature behavior. Seeger and Mann

/6/

have p o i n t e d o u t t h a t n o n l i n e a r i t y i s v e r y l a r g e near a d i s l o c a t i o n . Therefore, we a r e l e d t o speculate t h a t we a r e viewing f o r t h e f i r s t time t h e e f f e c t o f Bordoni r e l a x a t i o n on the measured value o f the TOE constants o f copper. The samples we used f o r these measurements were neutron i r r a d i a t e d t o r e - duced d i s l o c a t i o n i n t e r a c t i o n s . Thus, i f t h i s i s the c o r r e c t i n t e r p r e t a t i o n o f the data, we can say immediately t h a t our measurements a r e e s p e c i a l l y s e n s i t i v e t o t h e presence o f d i s l o c a t i o n s . Addi- t i o n a l measurements a f t e r f u r t h e r neutron i r r a d i a - t i o n o r a f t e r annealing w i l l be r e q u i r e d t o con- f i r m whether t h i s i s t h e c o r r e c t i n t e r p r e t a t i o n o f the temperature dependence o f t h e TOE constants o f copper.

Acknowledgment : Research supported by t h e Urlited States O f f i c e o f Naval Research.

References

/1/

Breazeale,

M.A.

and Joseph Ford, J.Appl.Phys.

36 (1965) 3486.

-

/2/

The n o n l i n e a r i t y parameter B i s t h r e e times as l a r g e as t h e dimensionless parameter B previous- l y used by W.T.Yost and M.A.Breazeale i n Phys.

Rev.B

9 (1974) 510

and by James A.Bains and M.f4.Breazeale i n Phys.Rev.B

- 13 (1976) 3623.

/3/

Gauster,

W.B.

and Breazeale, M.A.,Phys. Rev.

168 (1968) 655.

-

/4/

Bains, James A. and Breazeale, M.A.,Phys. Rev.

B

2 (1976) 3623.

/5/

Bordoni, P i e r o Giorgio, J. Acoust. Soc. Am.

26

(1954) 495.

/6/

Seegar,

A.

and Mann E., Z. Naturforschung

14a

(1959) 154.

constants o f copper a c t u a l l y do f o l l o w t h e p r e d i c - t i o n s o f t h e c e n t r a l forces, nearest-neighbor model

.