CHAPTER V I

L W TEMPERATURE THERMAL EXPANSION OF YBa2Cu307 AND

GdBa Cu 0 . 2 3 7

6.1 INTRODUCTION

A method of calculation of Grunei sen

parameters (GPs) from third order elastic constants is

given. A brief introduction to finite strain

elasticity theory and wave propagation in a

homogeneously deformed elastic medium is also presented

here to obtain the G P s .

6.2 INTRODUCTION TO FINITE STRAIN ELASTICITY THEORY AND

CALCULATION OF THE GENERALIZED GRUNEISEN PARAMETERS FOR

ACOUSTIC WAVES IN ORTHORHOMBIC CRYSTALS FROM THIRD ORDER

ELASTIC CONSTANTS.

Low temperature limits of the effective

- - Gruneisen functions yII '-3) and y1 - a of a uniaxial

crystal depend on the generalized GPs Y ;

tB,@ and Y ;

t B , @ of the acoustic modes propagating in different

directions in the crystal.

Let the position coordinates of a material

particle in the unstrained or natural state be a n

=r,z,s ) . Let the coordinates of the material particle

in the strained state be X (L= 1.2.3). Consider two 1

material particles located at a and a + da . Let their 1 L L.

coordinates in the deformed state be X and X + dx. L 1 1

The elements dx are related to da by the equation L L

dx = ( a x / a a . ) d a . L 1 J J

The convention that repeated indices indicates

summation over the indices will be followed here. 6 L J

is the kronecker delta symbol and c are the i j

deformation parameters. The Jacobian of the

transformation

J = Det ( a x . / a a ) , L j

is taken to be positive for all real

transformations. If dva is a volume element in the

natural state and dvx its volume after deformation

where p and p are the densities in the natural and 0

deformed states respectively . The square of the length

of arc from Q Lo a + da be de in 1 L 0

the L

unstrained state and de in the strained state

Then

Here nJk

are the Lagrangian strain

components. They are symmetric with respect to an

interchange of the indices J and k in terms of E ~k '

The internal energy function u ( 6 . 1 ) . for the 1 k

material is a function of the entropy and the Lagrangian

strain components. u refers to unit volume of the

undeformed state. u can be expanded in power of the

strain parameters about the undeformed state.

1 U - - u = - ( a 2 u / a V

0 1 J " *

L2 a " k l 10,s 1 )

The linear term in strain is absent because

the natural state is one where u isminimum. Following

Brugger [1] we define the elastic constants of different

orders referred to the natural state.

cS 2

I J . k l = ( a u / a I ) . . a V k l

L J

These are the adiabatic elastic constants of

second and third orders respectively. They are tensors

of fourth and sixth ranks. The number of independent

nonvanishing second order elastic constants and third

order elastic constants for different crystal systems

are tabulated by Bhagavantam [Z]. Starting from the free

energy F ( T, ) ,one can define isothermal elastic rs

constants as derivatives of F with respect to

V P i Murnaghan [ 3 ] gives the following expression for

stress:

The stress tensor T is referred to the deformed ~k

state of the medium . The conditions for equilibrium

require that the stress tensor be symmetric

A medium is said to be homogeneously deformed

if the components of the strain tensor Q do not vary Pq

from point to point in the medium . The homogeneously

deformed state is called the initial state and the

coordinates in this state are referred to by x;. When

the particles are given infinitesimal displacements u i

from this state , the resulting state , termed as the

final state , is-referred to by the coordinates

X - - x: + ui. The equationof motion is i I

using the result

i a x 6.11

B x k

8 a P

Thurston and Brugger [ 4 ] arrive at the

following wave equation in terms of the displacements u l

For a homogeneously strained medium,

, where

The denotes that the quantities have to

be evaluated in the homogeneously strained state of the

-6 medium . Using (6.1) and (6.5) for u we get A t o

~ k . p m

the first order in s as J k

we now substitute plane wave solutions in(6.12). We may

write the solution in deformed coordinates as

where w is the actual velocity of the wave in the

deformed state and n are the direction cosines of wave 1

propagation . However , i t is more advantageous to write

the displacements as

where v is called the natural velocity and N are the L

direction cosines of the wave in the undeformed state.

Let ho be the wavelength of a given wave in the

undeformed state travelling along a direction having

direction cosines N . After the deformation the wave i

length of the wave changes to A and the wave-

propagation direction is also changed. The direction

cosines are now n , . The frequency of the wave changes 1

from o to o . In.the unstrained state the velocity w in 0 0

i the direction N is

w = w k / z n 0 0 0

6 . 1 7

In the strained state the actual velocity w

of the wave is

w = w X / z n

and the natural velocity of the wave is

v = a h / 2 n 6 . 1 9 0

ie the ratio o / w directly gives v / o without 0 0

involving the changes in the dimensions of the specimen

Substituting (6.16) in ( 6 . 1 2 )

2 0 = is 0

Po v U N N U

J j k , p m P m k

These three linear homogeneous equations

corresponding to j = 1.2,s can be solved only i f

giving the three natural velocities for any

direction of wave propagation. The GPs y " (8 ,$) and 1

t h y ' ( 8 .@) for the J acoustic mode propagating in the

J

direction I , are

and

- y ; 8 , = -a Log Uq.)) / a Log A

= - 8 Log V (8,@ / a Log A J

v (8.4) is the natural velocity of the J th

J

acoustic mode propagating in the direction (8 ,@).

Referring to the c- axis as the z-axis when the

medium is homogeneously deformed by

(1) a c~niform longitl~dinal s t r a i n t V = dlogc along

the c-auis, or

(2) a uniform areal strain c'= d L o g ~ in the plane

perpendicular to the c axis.

The direction cosines are N = s in 8 con @, X

N = srn@sin@, N = c o s 8 . To the first order in .& , the Y z '-1

uniform longitudinal strain t" corresponds to c 3 3

and

the uniform areal strain c, is equivalent to c = 11

The rest of the c are zero . Thp expression for ' J - 6

D : A N N 6 . 2 3 ~k ik ,pm P m

under the strains c., and C , are writ ten down

below for the orthorhombic system taking into account

the nonvanishing second order elastic constants and

third order elastic constants .

6 . 2 4

Putting po v2 = X, the determinantal equation can he

expanded to give t h ~ ci~hic equation

9 2 X - A X + I I x - - c = 0

where A = C Oii

L

The A ,B , c are function of &, and r. . When c,

- - - and r'. are zero , their val~les are A , E. and c and

- - - the roots of the e q u a l ion are x , x and x

1 2 3

The derivatives of A,B and c are to be taken at

zero values of c and 6". After getting the individual

GPs of the acoustic modes, the low temperature limits

- - y,,'-3) and yL'-3) are calculated using (6.28). In

orthorhombic crystals the acoustic wave velocities and

( :Ps d e p ~ n c l only on d and not on the azimuth 0. whelp (8.

+ ) ~ i v e s the direct ion of wave prop~gat ion for the

elastic waves. The low temperature limits for thr

orthogonal crystals are then calculated using the

following expressions

Herr 1 is the index of the acoustic branch

Burgger and Fritz [ 5 ] have described a

+ . perturbation method for calculating ylm(q,~) from the

third order elastic constants suitable for machine

computation. A part from the fact that we do not need

i to know a1 1 the components YL,(q,j) c1.m = i to 9) the

application of the strain a ' and c " in the present

method does not alter the symmetry of the crystal and

hence the factorizability of the determinant in certain

directions which reduces the lahour of the computation.

In anorthorhombic crystal the velocity w is independent 0

of 4 and we need consider the calcl~lation of w only in 0

the plane q5 = o. In this plane, for any direction 8 of

propagation. the determinant lactorises into a quadrntir

eq'lnt i o n

and a linear equation

This is true even for v in the presence of the strains

& ' and 6" so that the calculation of y i ( 6 , o )and y ;

( 8 , o ) are considerably simplified. The numer i ca 1

integration suggested by Ramji Rao and Srinivasan [6,7]

and Rao and Menon [8,9] has been used to calculate the

- - low trmperattlrr limit y (-3) and yL(-3) of the thermal

I1

expansion of tho orthorhombic crystals in this thesi~.

6.3 LOW TEMPERATURE THERMAL EXPANSION OF YHa2C1~307

The third order elastic constants ohtained in

tahle (4.1) have been used to obtain the Gruneisen

parameters (GPS) for the acoustic modes and.the low

- - temperature limits Y I ( - 3 ) and y , ,<-3) has been

calculated. In orthorhombic crystals the acoustic wave

velocities and the GPs depend only on 8 and not on the

azimuth $ where (8, $ ) gives the direction of wavc

propagating in the crystal. Using the second and third

order elnstic constnnts of YRa C u 0 from tahle (3.1) of 2 3 7

rhapter 1 1 1 and table (4.1) of chapter I V , the wav?

velocities and GPs for the elastic waves propagating at

different angles 8 to the z-axis in the xz planes are

calc~~lated using the procedure described in detail in

sertion (6.2) of this chapter. The calculated values

are given in table(6.1). Generalized GPs lor the

elastic waves propagating at different angle 8 to the

orthorhomhic axis in YBa Cu 0 are also given in 2 3 7

table(6.1). Po

is the density of the unstrained

crystal.

Fig.(6.1) gives the plot of the variation of

y . for the three acoustic branches as a function of the

angle (8). Fig.(6.2) gives the variation of ).,, for the

three acoustic hranches as a function of 8 , which is the

angle the direction of propagation makes with the

c-axis. Also the variation of the generalized GPs

).,and y " of these elastic modes with the direction of

propagation are shown hy polar diagrams. Fig.

. 3 , . 4 , ( 6 . 5 , and (6.6) are the polar diagram

showing the variation of GPs y for the three acoustic

branches as a Pltnction of the angle 8 which the

direction of propagation makes with the orthorhomhic

axis.

- - The low temperature 1 imi ts yL(-3) and y,,(-3)

are obtained using equation (6.28) hy numerical

integration procedure using the data from tahle(6.1).

Since the solid angles of the cone of semi-vertical

angle 8 is proportional to srn8 , the value of Y1'

-9/2 -9/2 -9/2 x . '1 X j

and x . at a n g l e 8 is multiplied by J J

--3/2 srn8 and the sum y , x srn Qn over all the 8

J J

PIG.6.1. GPS Y ~ O R THE THREE ACOUSTIC BRANCHES AS A FUNCTION OP ANGLE FOR YBa2Cu3O7.

- - -

-1.20 - - - - - - - - - -

- 1.60

0 0 0 0 0 - f+ I * * * * - ri' A A A A A - r;

~ ~ I I I I I ~ ~ ~ I I I I I I I I I ~ I I I I I I I I I ~ I ~ ~ ~ ~ ~ ~ ~ ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

0.00 20.00 40.00 60.00 80.00 100.00 Angle (degree)

0.80

0.60

0.40

0.20 0) .- : E' 0 a 0.00 C al m .- al C 3

& -0.20

-0.40

-0.60 A A A A A -

Angle (degree)

fl

PIG.6.2. GPa FOR THE THREE ACOUSTIC BRANCHES A S A FUNCTION OF ANGLE FOR YBa2Cu307.

, PIG.6.3. POLAR DIAGRAM SHOWING THE PLOT OF THE GPS AND 'rs'

FOR THE ACOUSTIC BRANCHES AS A FUNCTION OF THE ANGLE WHICH THE DIRECTION O F PROPAGATION MAKES W I T 6 THE ORTHORHOMBIC AXIS

FOR YBa2Cu307-

I ,

FIG.6.5. POLAR DIAGRAM SHOWING THE PLOT OF THE GPS 'r, FOR THE ACOUSTIC BRANCH AS A FUNCTION OF THE ANGLE WHICH THE

DIRECTION OF PROPAGATION MAKES WITH THE ORTHORAOMBIC AXIS FOR YBa2C?0,.

133

--9/2 values are taken to be proportional t o fy,x do.

J J - -

Thus the low temperature limit ~ ~ ( - 9 ) and y ll(-9)are

obtained as

9 - --3/2 ~ ~ ( - 9 ) = C ( C Yz. X . ) sin e n

J J

- - yI (-9) and yII (-9) have been thus obtained for YBa Cu 0

2 3 7

are - . 3 6 1 and . 2

The lattice thermal expansion coefficients at

various temperatures can be expressed in terms of the

- - effective Gruneisen function Y ~ ( T ) and yII(~) as

follows

Here S . , are the elastic compliance LJ

coefficients; v is the molar volume and x is the LOO

-B r isothermal compressibility ; L B r and y I , are the

average Gruneisen functions used by Brugger and Fritz

1 5 1 . From (6.30) we may calculate

A t low temperature,equation ( 6.31) give

The low temperature limit of the volume

Gruneisen function is then obtained as

- - The Values of

- ~r - Br Y1'-3', Y I I 9 . YL ( - 3 , y (( (-9)-

- Y~

for YBa Cu 0 are reported in table (6.2) 2 3 7

6.4 LOW TEMPERATURE THERMAL EXPANSION OF GdBa Cu 0 2 3 7

Using the second and third order elastic

constants of GdBa Cu 0 from table (3.2) of chapter111 2 3 7

and table (4.2) of chapter IV respectively the wave

velocities and the GPs for the elastic waves

propagating at different angles 8 to the z-axis in the

xz planes are calculated using the procedure given in

section 6.2 of this chapter. The calculated values are

given in table (6.3). Po is the density of the

unstrained crystal.

Fig.(6.7) gives the plot of the variation of y ,

for the three acoustic branches as a function of the

angle (8). Fig.(6.8) gives the variation of y., for the

three acoustic branches as a function of 8 , which is the

angle the direction of propagation makes with the

c-axis. Figs.(6.9), (6.10), (6.11) and (6.12) are the

polar diagrams showing the variation of the GPs y for

the three acoustic branches as a function of the angle e

which the direction of propagation makes with the

orthorhombic axis.

- - y1(-3) and y I1( -3) obtained for GdBa Cu 0

2 3 7 are

- -.404 and .I75 respectively . The values of yl (-3),

-2.00 &--,--, , , , , , , , , , , , , , , , I , , , , , , , , , I , , , , , , , , , , 000 20.00 40.00 60.00 80.00 100.00

Angle (degree)

FIC.6 .7 . GPO r FOR TRE TRREK A C O U g R C B R A N C R W A S A FUNCTION OF ANGLE FOR C d ~ C u 3 0 , .

- 0 80 0 0 0 20.00 40.00 60.00 80 00 100.00

Angle (degree)

. rIG.6.8 GPs r FOR THE THREE ACOUSTIC BRANCHES

A S A FUNCTION OP ANGLE FOR Gd-Cu.,O,

CIG.6.9. POLAR DUCRAIl SROWMG TEE PLOT OF TALI GPs f4 AND 7;' OR THE A c o m m c BRANCRW AS A r m c n o N or r n ~ ANGLE WRICA

THE DIRECTION O r PROPAGATION MAKES WITH TRI! ORTAORflOIlBIC AXIS roll Gd~ClL ,O7-

FIG.6.10. POLAR DIAGRAH SAOWMG THE PLOT OF THE G P s yr FOR THE ACOUSTIC BRANCH A S A FUNCTION OF THE ANGLE WAICA TI

DIRECTION OF PROPAGATION MAKES WITH T E E ORTAORAOMBIC AXIS FOR GdBa2Cu307.

PIG.6 .11 . POLAR DIAGRAM SHOWING THE P L O T O F THE G P s \ r N FOR THE ACOUSTIC B R A N c H x AS A FUNCTION O F THE ANGLE WHICH %HE

DIRECTION O F PROPAGATION UAKES WITH THE ORTHORAOMBIC A X I S FOR GdBa2Cu30,.

,, r IG .6 .U . POLAR DIAGRAM SHOWING T I IE PC., OF F I E GPO rs AND r'

FOR THE A c O a S n C BRANCHES AS A FUNCTION OF TEE ANGLE *HXCn THE DIRECTION OF PROPAGATION M A K W WITH TFIE ORTROREOMBIC AX=

FOR GdBa2CU,0,.

- - E r - B r - Y i l (-3) , Y 1 '-3'' 11 - 3 , and L

of GdBa Cu 0 arr 2 3 7

also collected in tnhle ( 6 . 4 ) .

6.5. RESULTS AND DISCUSSIONS.

From the fig~~(6.1) and (6.7) of YBa CU 0 and 2 3 7

GdBa CII 0 we can see that anisotropy in are quite 2 3 7

large compared to Y ; and Y ; , Y; reaches a minimum

0 at po , which means that the value of y in this branch

increases continuously as we move towards the c-axis.

Also i t can he seen that y; and y arc negative whilr y . 2 6 4

is positive .

Fips.(6.2).and (6.8) give the variation of y.

of YHa Cu 0 and GdBa Cu 0 . The anisotropy in y - and 2 3 7 2 3 7 4

y5" are large compared to y , ' . Half the gammas in the 6

y branches are positive while all the gammas in y6 5

branches are negative.

The symmetry in y , , is quite marked even though 4

all the y ' s are positive. One may infer from the

negative values of the large number of elastic y 's that

a t high temperature a l l the plastic y ' s may hecome

negative when the crystal is subjrctrd to a transverse

thprmal strain perpendicular to the c-axis which may

lead to lattice instability and phase transformation in

YBa Cu 0 and GdBa Cu 0 . 2 3 7 2 3 7

tu Llw orLhorhonlbic axis in YBa. Cu. 0-. 2 .3 d

is the density ol' Llle ct-ystal

in the ur~-ztrained state

" 7

d

n 3 i,

@

+ -

.I !! a

% - j j

,. i ?

L

E >

" 3

0

3

m v

ll

4

rl 3

3

a

3%

-

13

1

- U

3

.. 3

2

YN

??

??

?+

N?

-

*2

2$

$2

. . 9 &

-XI

+-

N rO

i

"'4

1,: 2

* . kr

. h? k -8

!n i

I Y* q0

.

Q[

-i

."

C"

T

2 5

Y?

?Y

?E

F!

-

11

11

11

11

11

1

s-

Lo

3N

?j

N?

jY

NN

?

i I,

I

I

I

I

II

II

I I

I

ON

mN

N

fm

r-

3

mm

t-

0~

2m

m~

-s

(

9c

q?

cq

cq

cq

r:

1.

,?

OQ

Q

7

t-

11

11

II

Y

m

o

N

m

11

11

11

11

11

1

m~

cr

t

-t

-0

71

;)

$!

m m

s-

N

NN

NN

~~

?~

?,

D

dk

2m

2c

'I'aL7le<6.4.~

- - - R I CalculaLr~.I Values 01 yL(-3). 1 ,, (3) . l i (-3),

- Ur - ). ,, (-3) alid y ,,I (3dE)n Cu. 0

2 . 3 7

REFERENCES

1. K.Bruggcr Physical Review A.133, 1611 (1964).

2. S.Bllagavantarn Crystal symmetry and Physical

Properties Chapter 11 , Academic Press (1966).

3. F.U.Mur11agha11 , Finite deformation of an elastic

solid . John Wiley and Sons N.Y.( 1951).

4. R. N. Thurston and K. Brugger Physical Review A

133, 1604 (1964).

5. K.Brugger and T.C. Fritz Physical Review 157.

A524 (1967).

6. K.Rarnji Kao and K.Srinivasan Physics Status Solidi

29, 865 (1968).

7. R.Ramji Rao and R.Srinivasan Proceedings of indian

National Academy of science ,A36,97 (1970).

8. R.Ramji Rao and C.S. Menon, Journal. of.Low.Temp.

Phys.20,563(1975).

9. R.Ramji Rao and C.S.Menon, Journal. of

Low.Ternp.Phys. 22,325(1976).