Acoustic phonon mode softening in InCd alloy single crystals

Solid State Communications, Vol. 19, pp. 7 9 1 - 7 9 4 , 1976.

Pergamon Press.

Printed in Great Britain

ACOUSTIC PHONON MODE SOFTENING IN I n - C d ALLOY SINGLE CRYSTALS M.R. Madhava and G.A. Saunders School of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, U.K.

(Received 22 March 1976 by R. Loudon) To examine the effect of alloying Ir with Cd on the lattice stability, the elastic constants of single crystals of the tetragonal In 3.42 at.% Cd alloy have been determined. The shear modulus ½(C11 -- C12) is even less in the alloy than in In and decreases rapidly as temperature rises. It is concluded that the [110], qll [110] acoustic phonon mode is soft near the zone centre and softens even further as the phase boundary demarking the incipient diffusionless phase transition is approached. FOR A CRYSTAL to be stable, its energy must increase under a slight strain: the macroscopic deformation energy must be positive definite, a condition which imposes restrictions on the elastic stiffness moduli Cijkt. Should certain moduli become zero, the crystal will become unstable to the corresponding strain. Thus elastic constant measurements made in crystals near phase boundaries can provide fundamental information about the lattice instabilities which can lead to structural transformations. A diffusionless structural transformation - probably of the martensitic type from the tetragonal A6 In structure (point group 4/mmm) to the face centred cubic (f.c.c.) structure takes place in In rich, I n - C d alloys. 1 - a In the absence of any knowledge of the lattice vibration properties it has not been possible previously to discuss the transformation mechanism at the microscopic level in I n - C d alloys. To ascertain whether the incipient lattice instability in In itself is enhanced by alloying with Cd (as it is by T14-8), the elastic constants of single crystals of a tetragonal I n - C d alloy have been measured. Alloys containing less than 3.6 at.% Cd exist only in the A6 structure and do not undergo the phase transition. 2 Beyond this composition the alloys freeze in the f.c.c, structure; the transition temperature Tc decreases sharply with increasing Cd concentration, and while an alloy of about 3.7 at.% Cd would undergo the transition near the melting point (~ 150°C), Tc for a 4.85 at.% Cd alloy is ~ 79°Cfl 'a The approach taken here has been to study a tetragonal alloy (composition 3.42 at.% Cd) close to the phase boundary and thus to the stability limit, if any. A large single crystal (4 cm x 1.7 cm x 1 cm) of In 3.42 at.% Cd alloy was grown by the horizontal zone method starting from 99.999% purity elements. The lattice parameters at room temperature (295°K), obtained from Debye-Scherrer powder photographs, 791

were a = 4.625 -+ 0.009 A, c = 4.853 +- 0.013 A, c/a = 1.049 + 0.004, in agreement with published values. 9 The measured density at room temperature (p = 7.316 +- 0.001 g cm -3 ) agrees well with X-ray density. Transit times of 20 MHz ultrasonic waves were measured by the pulse superposition technique to a precision of 1 part in l0 s ; the ultrasonic wave velocities at 80 and 295°K of the chosen modes of propagation are presented in Table 1 ; the results have been corrected ( ~ 0.3%) for the apparent phase shift at the transducerbond-sample interface and those at 80°K also for thermal contraction. The elastic stiffness constants, calculated from the relationships in Table 1, and corrected for thermal contraction where necessary are presented in Table 2 in comparison with those for In and tetragonal I n - T l alloys. Indium has a small value of the elastic modulus ½(Cll -- Clz), which provides a measure of the resistance to deformation when a shear stress is applied across a (110) plane in a [1]-0] direction. For a tetragonal crystal to be stable ½(Cll - C12) must be positive. The effect of alloying with Cd - like that of T1 - is to reduce this elastic modulus to even smaller values but to have no physically significant effects on the other moduli. Following this finding, it became necessary to know the dependence of ½(Cll -- C12) on temperature - this is shown for the In - 3.42 at.% Cd alloy in comparison with those for In and tetragonal In--T1 alloys in Fig. 1 : in all cases as the temperature rises ½(Cll - - C12) decreases rapidly. Inspection of the phase diagram s reveals that, although this alloy remains in the tetragonal phase throughout, as the temperature is raised the phase boundary is approached from the tetragonal side (but it is not reached because melting ensues first). It can be concluded from the temperature dependence of ½(C11 - - C 1 2 ) t h a t as the phase transition is approached from the tetragonal side this modulus

792

ACOUSTIC PHONON MODE SOFTENING IN I n - C d

Vol. 19, No. 8

Table 1. Elastic stiffness constant equations and ultrasonic wave velocities in In 3.42 at. % Cd alloy single crystals. Propagation direction

Polarisation direction

Relationships between velocities and elastic stiffness constants

Measured ultrasonic wave velocities (10s cm.sec -1 ) 295 ° K

Measured ultrasonic wave velocities (10s cm. sec -1 ) 80 ° K

[100] [100] [100] [001] [001] [110] [1101 [110] [011] [011]

[100] [010] [001] [001] x - y plane [110] [1i'01 [001 ] [100] q~ 7r

04 Pv z pv~ pv] pv~ or26 pv~ pv~ pvg PV~o

2.476 1.240 0.966 2.455 0.970 2.721 0.514 0.968 1.114 2.598

2.620 1.455 2.563 2.891 1.068 2.730

-

-

¢+~-

I011]

= = = = =

C,, C66 C44 C33 C~ ½(ell "1- C12 -[- 2C66) ½(C,, - C,2) C44 ½(C44 + C66) ½(A + {.42 - - B + C 2 } '/2)

=

= = = =

PV]l = ½ ( A - - { A 2 - - B + C 2 } 1/2)

q~is an angle which depends upon the values of the elastic constants and is measured from [001 ] in the (100) plane. A = C44 + ½(Cll + C33)

B = (Cn + C44)(C33 + C44)

C = C13 + C44.

0.6

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0.~

In 3.42at ~Cd o.

~-~ B

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Fig. 1. The softening of the [110], ql[ [1]-0] transverse acoustic phonon mode near the Brillouin zone centre as the temperature is increased is shown by the temperature dependence of ½(Cll -- C12). The data for In 3.42 at.% Cd alloy are shown by the large circles; Tm is the melting point, obtained by differential scanning calorimetry by B. Chapman, of this alloy. The data for In and the In-T1 alloys are from reference 8. tends towards z e r o The curve extrapolates to zero not at the melting point Tm but at a higher temperature at which we can surmise the crystals would undergo the transition - if they did not melt first. For the In-T1 alloys this behaviour of ½(Cll - - C12 ) is not a prelude to

fusion but rather evidences an instability which would lead to a military type phase change; s the results in Fig. 1 show that I n - C d alloys also possess this incipient instability. The temperature dependence of ½(Cll -- Cl2) is much larger than those of the other elastic constants

3v (volume) /3z 3xy

Compressibilities (in units of 10 -12 cm 2 dyne -1)

Sn S12 S13 $33 $44 S66

Elastic compliance moduli (in units of 10 -1o cm 2 dyne -1 )

Cn C12 C13 C33 C44 C66 ½(Ci1 -- C12 )

Elastic stiffness moduli (in units of 1011 dyne cm -2)

Temperature '

2.36 0.583 0.888

0.150 --0.0395 --0.101 0.208 0.154 0.083

4.54 4.01 4.15 4.51 0.651 1.21 0.26

300°K

Indium Ref. 10

2.48 0.915 0.781

0.198 --0.080 --0.109 0.227 0.147 0.095

4.27 3.91 3.93 4.22 0.682 1.05 0.18

290°K

In 11.5 at.% T1 Ref. 7

2.49 0.913 0.787

0.27 --0.134 --0.124 0.257 0.133 0.093

4.20 3.95 3.93 4.18 0.752 1.08 0.12

290°K

In 15 at.% T1 Ref. 7

2.38 0.954 0.714

0.174 --0.0857 --0.0811 0.172 0.146 0.0889

4.485 4.100 4.054 4.411 0.686 1.125 0.193

295°K

2.19 0.732 0.727

0.0900 --0.0176 -- 0.0652 0.138 0.118 0.0633

5.120 4.191 4.408 4.900 0.85 1.579 0.465

80 ° K

In 3.42 at.% Cd

Table 2. Elastic constants o f In and its tetragonal alloys with Cd and T1

0.025 0.048 0.069 0.21

0.028

{Co(280 ° K) -- C0(320 ° K)}/Co(300 ° K)

Temperature dependence of Cij at 300°K:

ta3

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¢3

I

z

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zZ o Z o Z

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794

ACOUSTIC PHONON MODE SOFTENING IN In-Cd

(see Table 2). The slope ACiffATreflects the anharmonicity of the binding forces corresponding to each propagation mode. The large A C~i/A T found for ½(CI1 -- C12) indicates that the atomic displacements associated with the [110], qll [11-0] are particularly subject to anharmonicity. Previously it has been shown for the In-T1 alloys that ½(Cll --C12) tends to zero as the martensitic transition is approached from either the cubic 4'6'7 or the tetragonal phase. 4'a This result indicated that the onset of instability of both phases is associated with softening of the [110], q [[[1i-0] acoustic phonon mode. Optimised model potential calculations 5 of the phonon dispersion curves for In-Tl alloys have shown theoretically that the [i 10], q II[1 TO] acoustic phonon mode softens close to the Brillouin zone centre: the slope

0co/0Q which gives the mode velocity of this [110]1 transverse mode is much less than that of the other branches and decreases towards zero as the phase boundary is neared. The dispersion curve shows this anomalous, alloy composition dependent behaviour only near the zone centre - throughout the rest of the zone this branch (and the other branches) behaves normally and is not sensitive to the alloy composition. As evidenced by the elastic constant data presented here, the lattice dynamical behaviour near the zone centre of the tetragonal In-Cd alloys must be closely similar to that of the In-Tl alloys; the acoustic mode softens as the phase boundary is neared and both kinds of alloy become unstable to a shear stress applied across a {110} plane in a (1]-0) direction.

REFERENCES 1.

BETTERIDGE W., Proc. Phys. Soc. 50, 519 (1938).

2.

HEUMANN T. & PREDEL B., Z. Metallkde. 53,240 (1962).

3.

POLOVOV V.M. & PONYATOVSKII E.G., Soy. Phys. JETP 37,476 (1973).

4.

PACE N.G. & SAUNDERS G.A., Proc. R. Soc. A326, 521 (1972).

5.

GUNTON D.J. & SAUNDERS G.A., Solid State Commun. 12, 569 (1973).

6.

GUNTON D.J. & SAUNDERS G.A., Solid State Commun. 14,865 (1974).

7.

GUNTON D.J. & SAUNDERS G.A., Proc. R. Soc. A343, 63 (1975).

8.

CHUNG D.Y., GUNTON D.J. & SAUNDERS G.A., Phys. Rev. (in press).

9.

STRAUMANIS M.E., RAO P.B. & JAMES W.J., Z. Metallkde. 62, 493 (1971).

10.

Vol. 19, No. 8

CHANDRASEKHAR B.S. & RAYNE J.A.,Phys. Rev. 124, 1011 (1961).