CHAPTER VI. L W TEMPERATURE THERMAL EXPANSION OF YBa2Cu307 AND. GdBa Cu

Transcription

1 CHAPTER VI L W TEMPERATURE THERMAL EXPANSION OF YBa2Cu307 AND GdBa Cu

2 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 GPs. 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; tb,@ 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

3 =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. ) da. 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 ij 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

4 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 L 1 L 0 the unstrained state and de in the strained state Then Here are the Lagrangian strain njk 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.

5 1 U - u = - (a2u/av 0 1 J " * L2 a "kl 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 IJ.kl 2 = ( a u / a I).. a Vkl 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 VP i Murnaghan [3] gives the following expression for

6 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

7 using the result B x k i a x 8 a P 6.11 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

8 -6 medium. Using (6.1) and (6.5) for u we get A t o ~k.pm the first order in s J k as we now substitute plane wave solutions in(6.12). write the solution in deformed coordinates as We may where w is the actual velocity of the wave in the deformed state and n are the direction cosines of 1 wave propagation. However, it 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

9 cosines are now n,. The frequency of the wave changes 1 from o to o. In.the unstrained state the velocity w 0 0 i the direction N is in w = w k / z n 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 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 (6.12) 2 0 = is Po v U N N U J jk,pm P m k 0 These three linear homogeneous equations corresponding to j = 1.2,s can be solved only if giving the three natural velocities for any

10 direction of wave propagation. The GPs y" (8,$) and 1 y'(8.@) th 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 straintv= dlogc along the c-auis, or (2) a uniform areal strain c'= dlog~ in the plane perpendicular to the c axis. The direction cosines are N = sin 8 X N = srn@sin@, N = cos8. To the first order in.&, the Y z '-1 uniform longitudinal strain t" corresponds to c 3 3 and

11 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 6.23 ~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.

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14 6.24 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

15 - - - and x the roots of the equal ion are x, x 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

16 (:Ps dep~ncl only on d and not on the azimuth 0. whelp (8. +) ~ives 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

17 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 In anorthorhombic crystal the velocity w 0 computation. is independent of 4 and we need consider the calcl~lation of w 0 only in the plane q5 = o. In this plane, for any direction 8 of propagation. the determinant lactorises into a quadrntir eq'lnt ion and a linear equation This is true even for v in the presence of the strains &' and 6" so that the calculation of yi (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~.

18 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 YI(-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 Cu 0 from tahle (3.1) of rhapter 111 and table (4.1) of chapter IV, 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 table(6.1). crystal. Po is the density of the unstrained

19 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 x. -9/2 '1-9/2-9/2 J Xj J and x. at angle8 is multiplied by --3/2 srn8 and the sum y, x srn Qn over all the 8 J J Y1'

20 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 Angle (degree) PIG.6.1. GPS Y ~ O R THE THREE ACOUSTIC BRANCHES AS A FUNCTION OP ANGLE FOR YBa2Cu3O7.

21 ).- : 0.20 E' 0 a 0.00 C al.- m al C 3 & A A A A A - Angle (degree) fl PIG.6.2. GPa FOR THE THREE ACOUSTIC BRANCHES AS A FUNCTION OF ANGLE FOR YBa2Cu307.

22 , POLAR DIAGRAM SHOWING THE PLOT OF THE GPS AND 'rs' PIG.6.3. FOR THE ACOUSTIC BRANCHES AS A FUNCTION OF THE ANGLE WHICH THE DIRECTION OF PROPAGATION MAKES WIT6 THE ORTHORHOMBIC AXIS FOR YBa2Cu307-

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24 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,. I,

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26 /2 values are taken to be proportional to fy,x do. J J - - Thus the low temperature limit ~~(-9) and y ll(-9)are obtained as /2 ~~(-9) J J = C ( C Yz. X. ) sin en - - yi (-9) and yii (-9) have been thus obtained for YBa Cu are 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

27 Here S., are the elastic compliance LJ coefficients; v is the molar volume and x is the LOO -B r isothermal compressibility ;LBr and yi, are the average Gruneisen functions used by Brugger and Fritz 151. From (6.30) we may calculate

28 At low temperature,equation ( 6.31) give The low temperature limit of the volume Gruneisen function is then obtained as ~r - Br The Values of Y1'-3', YII 9. (-3, YL y (( (-9)- - Y~ for YBa Cu 0 are reported in table (6.2) LOW TEMPERATURE THERMAL EXPANSION OF GdBa Cu Using the second and third order elastic

29 constants of GdBa Cu 0 from table (3.2) of chapter 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). unstrained crystal. Po is the density of the 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 yi1(-3) obtained for GdBa Cu are and.i75 respectively. The values of yl (-3),

30 -2.00 &--,--,,,,,,,,,,,,,,,, I,,,,,,,,, I,,,,,,,,,, Angle (degree) FIC.6.7. GPO r FOR TRE TRREK ACOUgRC BRANCRW AS A FUNCTION OF ANGLE FOR Cd~Cu30,.

31 Angle (degree). rig.6.8 GPs r FOR THE THREE ACOUSTIC BRANCHES AS A FUNCTION OP ANGLE FOR Gd-Cu.,O,

32 CIG.6.9. POLAR DUCRAIl SROWMG TEE PLOT OF TALI GPs f4 AND 7;' OR THE Acommc BRANCRW AS A rmcnon or rn~ ANGLE WRICA THE DIRECTION Or PROPAGATION MAKES WITH TRI! ORTAORflOIlBIC AXIS roll Gd~ClL,O7-

33 FIG POLAR DIAGRAH SAOWMG THE PLOT OF THE GPs yr FOR THE ACOUSTIC BRANCH AS A FUNCTION OF THE ANGLE WAICA TI DIRECTION OF PROPAGATION MAKES WITH TEE ORTAORAOMBIC AXIS FOR GdBa2Cu307.

34 PIG POLAR DIAGRAM SHOWING THE PLOT OF THE GPs \rn FOR THE ACOUSTIC BRANcHx AS A FUNCTION OF THE ANGLE WHICH %HE DIRECTION OF PROPAGATION UAKES WITH THE ORTHORAOMBIC AXIS FOR GdBa2Cu30,.

35 ,, rig.6.u. POLAR DIAGRAM SHOWING TIIE PC., OF FIE GPO rs AND r' FOR THE AcOaSnC BRANCHES AS A FUNCTION OF TEE ANGLE *HXCn THE DIRECTION OF PROPAGATION MAKW WITH TFIE ORTROREOMBIC AX= FOR GdBa2CU,0,.

36 - - Er - Br - Yil (-3), Y 1 '-3'' 11-3, and L also collected in tnhle (6.4). of GdBa Cu arr 6.5. RESULTS AND DISCUSSIONS. From the fig~~(6.1) and (6.7) of YBa CU 0 and GdBa CII 0 we can see that anisotropy in are quite 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 it can he seen that y; and y arc negative whilr y 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 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

37 at high temperature all 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 YBa Cu 0 and GdBa Cu in

38 tu Llw orlhorhonlbic axis in YBa. Cu d is in the ur~-ztrained state the density ol' Llle ct-ystal

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40 " 7 d -@ +.I n 3 i,! a % j,. - j i? L E > " 3 3 m 4 3 a 0 v l rl 3. 9 & -XI +- N ro i "'4 1,: 2 *. kr. h? k Q [ - i. " C " T 2 5 Y?? Y? E F! s - L o 3 N? j N? j Y N N? i I, I I I I I I I I I I I O N m N N f m r - 3 m m t - 0 ~ 2 m m ~ - s ( 9 c q? c q c q c q r : 1.,? O Q Q 7 t I I Y m o N m % U 3 I Y*. -8!n i q0 m ~ c r t - t ; ) $! m m s - N N N N N ~ ~? D ~? d k 2 m 2 c 3. 2 Y N????? + N? - * 2 2 $ $ 2

41 'I'aL7le<6.4.~ R I CalculaLr~.I Values 01 yl(-3). 1,, (3). li (-3), - - Ur ).,, (-3) alid y,,i (3dE)n Cu

42 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).