Growth and X-ray studies of (NaCl)x(KCl)y−x(KBr)1−y single crystals

Growth and X-ray studies of (NaCl)x(KCl)y−x(KBr)1−y single crystals

Journal of Physics and Chemistry of Solids 66 (2005) 1705–1713 www.elsevier.com/locate/jpcs Growth and X-ray studies of (NaCl)x(KCl)yKx(KBr)1Ky singl...

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Journal of Physics and Chemistry of Solids 66 (2005) 1705–1713 www.elsevier.com/locate/jpcs

Growth and X-ray studies of (NaCl)x(KCl)yKx(KBr)1Ky single crystals K. Jayakumari, C. Mahadevan* Physics Research Centre, S.T. Hindu College, Nagercoil-629 002, Tamilnadu, India Received 17 June 2003; revised 14 October 2004; accepted 11 July 2005

Abstract Ternary mixed crystals of NaCl, KCl and KBr were grown by the melt method for the first time. Densities and refractive indices of all the grown crystals were determined and also used for the estimation of the bulk composition in the crystal. Lattice parameters and thermal parameters like Debye–Waller factor, mean square amplitude of vibration, Debye temperature and Debye frequency were determined from the X-ray powder diffraction data. The observed lattice parameters showed the existence of two phases in crystals with NaCl content greater than 0.1 mole fraction. The thermal parameters show a highly non-linear composition dependence. The results are reported. q 2005 Elsevier Ltd. All rights reserved. Keywords: B. Crystal growth; C. X-ray diffraction

1. Introduction A mixed crystal is obtained by crystallizing together two isomorphous crystals like KCl and KBr with comparable lattice constants. For ionic crystals like alkali halides complete miscibility is possible only above a temperature T K given by TZ4.5 S2, S being the percentage difference in lattice constants [1]. Alkali halide mixed crystals are of the completely disordered substitutional type. Haribabu and Subbarao [2] have reviewed the aspects of the growth and characterization of alkali halide mixed crystals. Sirdeshmukh and Srinivas [1] have reviewed the physical properties. Several more reports are available on binary mixed crystals of alkali halides [3–5]. However, very limited reports are available on ternary and quaternary mixed crystals of alkali halides [6,7]. A study of literature has shown that there are broad miscibility gaps in several binary systems of alkali halides. In the case of KBr1KxIx crystals, Nair and Walker [8] observed that for the extreme concentration ranges x! 0.3 and xO0.7 the system could be characterized by a single f.c.c. lattice parameter, while in the intermediate region three f.c.c. phases characterized by three lattice * Corresponding author. Tel.: C91 4652 222 127. E-mail address: [email protected] (C. Mahadevan).

0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.07.008

parameters. NaCl–KCl system is another example of not forming a continuous series of mixed crystals. The crystalline solutions in this system provide a useful model for comparison with more complex solutions such as those considered in the present study (NaCl–KCl–KBr). Thompson and Waldbaum [9] have analyzed the two-phase region halite–sylvite in the system NaCl–KCl. Alkali halide crystals are widely used as laser window materials, neutron monochromators, infrared prisms, infrared transmitters, etc. But the uses are limited by their mechanical properties and hence there exists the need to strengthen these. The mixed and impurity added (doped) crystals of alkali halides are found to be harder than the end members and so they are more useful in these applications. In addition, mixed alkali halides find their applications in optical, opto-electronic and electronic devices. In view of this, it becomes necessary and useful to prepare binary and ternary mixed crystals regardless of miscibility problem and characterize them by measuring their physical properties. Mahadevan and his co-workers [6] obtained larger and more stable crystals from (NaCl)x(KCl)0.9Kx(KBr)0.1 solutions than from NaxK1KxCl solutions. They grew the crystals from aqueous solutions only. Though the miscibility problem was there, their study has illustrated that a KBr addition to NaCl–KCl system may yield a new class of stable materials. The present work is a systematic addition to this previous study.

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Growth of single crystals, density and refractive index measurements and determination of lattice and thermal parameters like Debye–Waller factor, mean square amplitude of vibration, Debye temperature and Debye frequency were undertaken. Results are reported here in this paper.

the Gladstone’s rule [10]: ½ðnK1Þ=dP Z ½ðn1 K1Þ=d1 P1 C ½ðn2 K1Þ=d2 P2

2. Experimental details

where n, n1 and n2 represent the refractive indices of solution, solvent and solute (crystal) respectively. d, d1 and d2 are the densities of solution, solvent and solute, respectively. P, P1 and P2 are the percentage weights of solution, solvent and solute, respectively.

2.1. Growth of single crystals

2.4. Estimation of bulk composition

(NaCl)x(KCl)y-x(KBr)1-y single crystals were grown from the melt, for the first time, by the Czochralski method. AnalaR grade samples of NaCl, KCl and KBr (with a minimum assay of 99.9% for NaCl and 99.5% for others) were used as the starting material for the growth of the crystals. One hundred grams of the substance, weighed according to the molecular ratio by weight, was thoroughly mixed and was placed in a silica crucible. The crucible was placed in a 1200 8C capable muffle furnace having a temperature controller (accuracy is G 2 8C) and heated till the whole substance was melted. The temperature was then increased to 900 8C, where it was held for 45 min to homogenize by convection-assisted mixing. The temperatures of melting and freezing were noted as Tm and TfK, respectively. Crystals were pulled from the melt by using a crystal puller at the rate of 1.2 cm/h. The pulled crystal was allowed to cool naturally to room temperature in about 12 h. In order to keep the composition unaffected no seed crystal was used. The seed rod (sufficiently thin) itself was used to pull the crystal. It was grooved at the end to prevent the pulled crystal from falling into the melt. A total of twenty six crystals [20 ternary mixed crystals, viz. (NaCl)x(KCl)yKx(KBr)1Ky with x varying from 0.1 in steps of 0.1 and yZ0.2, 0.4, 0.5, 0.6 and 0.8; three binary mixed crystals, viz. (NaCl)0.5(KCl)0.5, (NaCl)0.5(KBr)0.5 and (KCl)0.5(KBr)0.5; three end member crystals, viz, NaCl, KCl and KBr] were grown by the above method and under identical conditions. The binary mixed and end member crystals were grown for comparison purposes.

It has been found that the density and refractive index values form a linear relationship with composition for the binary mixed crystals [1]. Assuming that these values have a linear relationship with composition for the ternary mixed crystals also, the following relations may be written:

2.2. Density measurement Densities of all the grown crystals were determined to an accuracy of G0.008 g/cc by using the flotation method. Carbon tetrachloride of density 1.594 g/cc and bromoform of density 2.890 g/cc were used as the lower and higher density liquids, respectively. 2.3. Refractive index measurement Refractive index of an under saturated solution of the crystal in distilled water was measured using an Abbe refractometer. Refractive index of the crystal was determined (to an accuracy of G0.006) by using

d Z xd1 C ðyKxÞd2 C ð1KyÞd3 n Z xn1 C ðyKxÞn2 C ð1KyÞn3 Here, d, d1, d2 and d3 represent the densities of mixed crystal, NaCl, KCl and KBr, respectively; n, n1, n2 and n3 represent the refractive indices of mixed crystal, NaCl, KCl and KBr, respectively. Compositions of the grown mixed crystals were estimated by solving the above two equations for x and y values. 2.5. XRD data collection X-ray diffraction data were collected from powdered samples using an automated X-ray powder diffractometer with scintillation counter and monochromated Cu Ka (lZ ˚ ) radiation. The reflections were indexed following 1.5418 A the procedures of Lipson and Steeple [11].

3. Analysis of the X-ray data 3.1. Lattice parameter Sirdeshmukh and Srinivas [1] have explained in detail about KCl–KBr (KClxBr1Kx) and KxRb1KxI lattice constant values and stated that the Retger’s rule with coefficient 3 represents an ideal for mixed crystals of alkali halides. Slagle and McKinstry [12] shows that the coefficient is 3.26 instead of the expected three for the KCl–KBr binary. We consider here the coefficient claimed by Sirdeshmukh and Srinivas (relatively recent reference) as we deal with the ternary systems. Analysis of the X-ray diffraction peaks (for the mixed systems considered in the present study) by the available methods [13] shows that, for ternary mixed crystals with composition (taken for crystallization) xZ0.1, all the peaks can be indexed with a single f.c.c. lattice and the lattice parameter obtained almost obeys Retger’s rule extended to

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ternary crystals: a3 Z xa31 C ðyKxÞa32 C ð1KyÞa33 where a is the lattice parameter of the mixed crystal and a1, a2 and a3 are respectively, the lattice parameters of the NaCl, KCl and KBr crystals. However, for crystals with xO0.1, it has been found that all the X-ray diffraction peaks can be indexed with two f.c.c. lattices instead of one which shows the existence of two f.c.c. phases. The calculated lattice parameters show that one phase nearly corresponds to pure NaCl and the other phase corresponds to the mixed system. A similar result was reported for the KBr–KI mixed crystals [8]. Since no KCl or KBr lines are found to be separated out in the diffraction pattern, the mixed phase should contain all the KCl and KBr present in the crystal. Lattice parameters calculated by assuming the mixed phase to contain only KCl and KBr (as NaCl forms a separate lattice) in the appropriate molecular ratio and applying Retgers rule was found to be greater than the observed lattice parameter. This decrease in the observed lattice parameter showed that the mixed phase contained some NaCl also in addition to KCl and KBr. Composition in the mixed phase was estimated by assuming Retger’s rule (since KCl–KBr mixed system completely obeys Retger’s rule [1]). In the case of end member crystals and KCl–KBr binary crystal also, all the peaks can be indexed with a single f.c.c. lattice. But, in the case of NaCl–KCl and NaCl–KBr binary mixed crystals considered in the present study, all the peaks can be indexed with two f.c.c. lattices instead of one showing the existence of two f.c.c. phases.

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FðhklÞ Z4½x1 ðfNaC GfClKÞ C y1 ðfKC GfClKÞ C z1 ðfKC GfBrKÞ for the mixed phase in NaClKKClKKBr systems FðhklÞ Z4½x1 ðfNaC GfClKÞ C y1 ðfKC GfClKÞ for the mixed phase in NaClKKCl system FðhklÞ Z4½x1 ðfNaC GfClKÞ C z1 ðfKC GfBrKÞ for the mixed phase in NaClKKBr system Here, fNaC , fkC , fClK and fBrK are the respective scattering factors for NaC, KC, ClK and BrK ions taken from the literature [15]. x1, y1 and z1 are the respective fractional contents of NaCl, KCl and KBr present in the mixed phase. The plus sign applies to reflections with even values of hCkCl and minus sign to those with the odd values of hCkCl. The equation for Bragg intensity may be written in logarithmic scale as lnðIE =IC Þ Z ln K K2Bðsinq=lÞ2 : The mean Debye–Waller factor (B) was obtained from a least squares treatment of ln(IE/IC) against (sin q/l)2. This is called Bobs. IE is the experimentally observed intensity, IC is the calculated intensity, K is the scale factor, q is the Bragg angle and l is the wavelength of the radiation. In the case of mixed crystals, the presence of mixing of ions creates a static contribution (Bstatic). According to Dernier et al. [quoted in Ref. 16] Bstatic for binary mixed crystals of the form AxB1-xC is given by

3.2. Thermal parameters

Bstatic Z xð1KxÞðrA KrB Þ2

The mean Debye–Waller factor for all the twenty six grown crystals were determined by the Wilson plot method [14]. As the number of reflections are limited, only a common Debye–Waller factor was determined for all the atoms in every system whether the system has one or two f.c.c. phases. In the case of systems with single f.c.c. lattice, the structure factors were calculated using the relations:

where rA and rB are ionic radii of A and B ions, respectively. (NaCl)x(KCl)yKx(KBr)1Ky crystals contain NaC, KC, Cl- and Br- ions. Replacement of ions is possible only between NaC and KC ions and Cl- and Brions. Hence, Bstatic for the present system can be estimated by using the relation

FðhklÞ Z 4ðfNaC GfClKÞ for NaCl

FðhklÞ Z 4ðfKC GfBrKÞ for KBr

where x and 1Kx are the mole fractions of NaC and KC ions and rA and rB are the ionic radii, respectively. Similarly, y and 1Ky are the mole fractions of ClK and BrK and rC and rD are their respective ionic radii. For mixed crystals,

FðhklÞ Z 4½y1 ðfKC GfClKÞ C z1 ðfKC GfBrKÞ for KCl

Bobs Z Bthermal C Bstatic :

FðhklÞ Z 4ðfKC GfClKÞ for KCl

KKBr mixed crystal In the case of systems with two f.c.c. lattices, the structure factors were calculated using the relations: FðhklÞ Z 4½ðfNaC GfClKÞ for the NaCl phase

Bstatic Z xð1KxÞðrA KrB Þ2 C yð1KyÞðrC KrD Þ2

The Debye temperature (qD) was obtained from the Debye–Waller theory expression (for end member crystals, BobsZBthermal): Bthermal Z

6h2 WðxÞ mkT

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3.3. Compressibility and mean sound velocity

Fig. 1. Photograph showing the grown (NaCl)x(KCl)0.6Kx(KBr)0.4 crystals.

where m is the mean atomic mass, T is the absolute temperature at which the intensities were measured, h is the Planck’s constant and k is the Boltzmann’s constant. The function W(x) is given by WðxÞ Z

4ðxÞ 1 C x 4 x2

where xZqD/T and f(x) is an integral. The values of W(x) for a wide range of x have been tabulated by Benson and Gill [17]. qD, the Debye temperature, was evaluated by using the above expression. The mean square amplitude of vibration (!u2O) was obtained from [14]

Srinivas and Sirdeshmukh [19] have used an empirical relation suggested by Madelung to determine the Debye temperature from compressibility data for KCl–KBr mixed crystals and obtained values highly comparable to those obtained from specific heat and elastic constant data. The same empirical relation, qD Z Aj1=2 MK1=2 V 1=6 where A is a constant, j is the compressibility, M is the molecular mass and V is the molecular volume, was used to determine the compressibility of all the grown crystals from their Debye temperature estimated in the present study. Also the mean sound velocity (Cm) was determined for all the grown crystals using the relation [20] qD Z ðh=kÞ½3pNd=ð4pMÞ1=3 Cm where N is the Avogadro’s number, d is the density and p is the number of vibrating units in the molecule. 4. Results and discussion

B Z 8p2 ! u2 O : The Debye frequency (fD) was obtained from [18] fD Z qD ðk=hÞ:

The ternary mixed crystals are found to be harder and less transparent when compared to the end members. The transparency of the crystals is found to be reduced when the crystals are cooled from high temperature to

Table 1 Specific refractive energy, density and refractive index values together with the initial and final compositions System (with composition taken for crystallization)

Specific refractive energy, (nK1)/d

Density, d (g/cc)

Refractive index, n

NaCl KCl KBr NaCl0.1KCl0.7KBr0.2 NaCl0.2KCl0.6KBr0.2 NaCl0.3KCl0.5KBr0.2 NaCl0.4KCl0.4KBr0.2 NaCl0.5KCl0.3KBr0.2 NaCl0.6KCl0.2KBr0.2 NaCl0.7KCl0.1KBr0.2 NaCl0.1KCl0.5KBr0.4 NaCl0.2KCl0.4KBr0.4 NaCl0.3KCl0.3KBr0.4 NaCl0.4KCl0.2KBr0.4 NaCl0.5KCl0.1KBr0.4 NaCl0.1KCl0.4KBr0.5 NaCl0.2KCl0.3KBr0.5 NaCl0.3KCl0.2KBr0.5 NaCl0.4KCl0.1KBr0.5 NaCl0.1KCl0.3KBr0.6 NaCl0.2KCl0.2KBr0.6 NaCl0.3KCl0.1KBr0.6 NaCl0.1KCl0.1KBr0.8 NaCl0.5KCl0.5 NaCl0.5KBr0.5 KCl0.5KBr0.5

0.2525 0.2498 0.2054 0.2374 0.2378 0.2387 0.2394 0.2394 0.2407 0.2403 0.2276 0.2288 0.2283 0.2283 0.2244 0.2238 0.2250 0.2243 0.2236 0.2199 0.2205 0.2198 0.2118 0.2532 0.2338 0.2279

2.1426(2.1641) 1.9785(1.9882) 2.7513(2.7505) 2.1441 2.1592 2.1750 2.1913 2.2133 2.2204 2.2527 2.2854 2.3048 2.3500 2.3674 2.4140 2.3896 2.3960 2.4136 2.4566 2.4578 2.4828 2.5062 2.6257 2.0219 2.3434 2.3337

1.5411(1.5443) 1.4942(1.4907) 1.5650(1.5596) 1.5091 1.5134 1.5192 1.5245 1.5298 1.5345 1.5414 1.5202 1.5274 1.5364 1.5405 1.5417 1.5347 1.5390 1.5415 1.5492 1.5405 1.5474 1.5508 1.5562 1.5120 1.5479 1.5326

Values reported in the literature are given in brackets.

Estimated bulk composition in the crystal

NaCl0.078KCl0.724KBr0.198 NaCl0.159KCl0.641KBr0.200 NaCl0.282KCl0.524KBr0.194 NaCl0.389KCl0.418KBr0.193 NaCl0.479KCl0.319KBr0.202 NaCl0.595KCl0.218KBr0.187 NaCl0.704KCl0.091KBr0.205 NaCl0.063KCl0.541KBr0.396 NaCl0.159KCl0.453KBr0.388 NaCl0.292KCl0.029KBr0.419 NaCl0.361KCl0.212KBr0.427 NaCl0.505KCl0.039KBr0.457 NaCl0.133KCl0.363KBr0.504 NaCl0.230KCl0.274KBr0.495 NaCl0.261KCl0.231KBr0.508 NaCl0.389KCl0.075KBr0.536 NaCl0.110KCl0.293KBr0.597 NaCl0.240KCl0.159KBr0.602 NaCl0.272KCl0.103KBr0.625 NaCl0.104KCl0.079KBr0.817 NaCl0.296KCl0.704 NaCl0.690KBr0.310 KCl0.499KBr0.501

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Table 2 Lattice constants. e.s.d.’s are given in parenthesis System (with composition taken for crystallization) NaCl KCl KBr NaCl0.1KCl0.7KBr0.2 NaCl0.2KCl0.6KBr0.2 NaCl0.3KCl0.5KBr0.2 NaCl0.4KCl0.4KBr0.2 NaCl0.5KCl0.3KBr0.2 NaCl0.6KCl0.2KBr0.2 NaCl0.7KCl0.1KBr0.2 NaCl0.1KCl0.5KBr0.4 NaCl0.2KCl0.4KBr0.4 NaCl0.3KCl0.3KBr0.4 NaCl0.4KCl0.2KBr0.4 NaCl0.5KCl0.1KBr0.4 NaCl0.1KCl0.4KBr0.5 NaCl0.2KCl0.3KBr0.5 NaCl0.3KCl0.2KBr0.5 NaCl0.4KCl0.1KBr0.5 NaCl0.1KCl0.3KBr0.6 NaCl0.2KCl0.2KBr0.6 NaCl0.3KCl0.1KBr0.6 NaCl0.1KCl0.1KBr0.8 NaCl0.5KCl0.5 NaCl0.5KBr0.5 KCl0.5KBr0.5

˚) Lattice constants (A For NaCl phase

For mixed phase

5.6507 (5.64009) 6.3164 (6.29294) 6.6052 (6.59820) – 5.6558 5.6576 5.6564 5.6596 5.6314 5.6175 – 5.6204 5.62610 5.62213 5.61810 – 5.5995 5.60410 5.60811 – 5.56423 5.60615 – 5.6239 5.59217 –

– – – 6.3672 6.3814 6.3726 6.3874 6.4086 6.4225 6.4611 6.38310 6.2685 6.4224 6.4485 6.4844 6.40810 6.4366 6.4605 6.4768 6.4529 6.4799 6.4966 6.5206 6.2753 6.5149 6.4126

Tm (8C)

Tf (8C)

1078 (1074) 1046 (1043) 1004 (1007) 971 963 953 933 898 961 1008 1033 973 907 868 863 963 913 903 893 978 940 945 963 1012 993 1083

1078 1046 1004 941 924 918 887 852 928 938 968 936 873 853 833 933 903 883 858 943 922 873 931 963 935 1028

The melting and freezing temperatures are also given. The literature values for the end members are given in parenthesis.

the room temperature which may be due to the introduction of thermal defects. Increase in KCl content also reduces the transparency [NaCl and KBr are colourless while KCl is white in colour (reported in the literature [21])]. Another problem is related to the degree of solubility of impurities introduced into the crystal. The concentration of impurities ‘dissolved’ in the lattice increases as the temperature of the crystal increases. If for a certain temperature T the concentration of impurities is higher than allowable due to the solubility limit, then the substance excess precipitates to form a new phase—the precipitate. The precipitate tends to form on dislocations which may be revealed by electron microscopy; the crystal becomes ‘milky’ [22]. All the grown crystals exhibit cleavage property which shows that the grown crystals are single crystals. Crystals up to a size of 3.5 cm could be grown. Photograph of few grown crystals is shown in Fig. 1 as an illustration. The necks formed are due to the temperature fluctuations while pulling the crystal (the temperature was controlled to an accuracy of G2 8C only). The observed specific refractive energy, density and refractive index of all the mixed as well as pure (end member) crystals are given in Table 1. Observed density and refractive index of the end members compare well with

those reported in the literature [23] (reported values are given in brackets). Composition of the starting material and estimated bulk composition of the grown crystals are also provided in Table 1. The lattice parameters obtained in the present study are provided in Table 2. The melting and freezing temperatures Table 3 Composition of the mixed phase of ternary mixed crystals System (with composition taken for crystallization)

Composition NaCl

KCl

KBr

(NaCl)0.2(KCl)0.6(KBr)0.2 (NaCl)0.3(KCl)0.5(KBr)0.2 (NaCl)0.4(KCl)0.4(KBr)0.2 (NaCl)0.5(KCl)0.3(KBr)0.2 (NaCl)0.6(KCl)0.2(KBr)0.2 (NaCl)0.7(KCl)0.1(KBr)0.2 (NaCl)0.2(KCl)0.4(KBr)0.4 (NaCl)0.3(KCl)0.3(KBr)0.4 (NaCl)0.4(KCl)0.2(KBr)0.4 (NaCl)0.5(KCl)0.1(KBr)0.4 (NaCl)0.2(KCl)0.3(KBr)0.5 (NaCl)0.3(KCl)0.2(KBr)0.5 (NaCl)0.4(KCl)0.1(KBr)0.5 (NaCl)0.2(KCl)0.2(KBr)0.6 (NaCl)0.3(KCl)0.1(KBr)0.6

0.001 0.003 0.003 0.003 0.004 0.008 0.028 0.009 0.009 0.014 0.010 0.008 0.014 0.010 0.010

0.761 0.727 0.682 0.610 0.537 0.304 0.523 0.405 0.329 0.077 0.353 0.311 0.121 0.206 0.140

0.238 0.270 0.315 0.387 0.459 0.688 0.449 0.586 0.662 0.909 0.637 0.681 0.865 0.784 0.850

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parameters between NaCl and KCl (11.796%) and NaCl and KBr (17.091%). This is shown in the equimolar binary mixed crystals also. For KCl and KBr, the percentage deviation in lattice parameters is only 4.735% and this system can form a continuous series of solid solutions. Hence, for higher concentrations of NaCl, NaCl is forming a separate lattice and KCl–KBr with some NaCl is forming another lattice. Composition (taken for crystallization) dependences of lattice parameter of the mixed phase for NaCl mole %Z10, 20, 30, 40 and 50 are shown in Fig. 2 together with Venudhar et al.’s [24] results for the KCl–KBr solid solution (NaCl mole %Z0). Systems with NaCl mole %Z60 and 70 are not included as there is only one value in each case. The lattice parameter observed for (KCl)0.5(KBr)0.5 in the present study is in essential agreement with the work of Venudhar et al. [24]. The composition dependence of lattice parameters in a ternary mixed crystal series can be expressed by a general relation of the type

6.6

Lattice parameter (Å)

6.55 6.5 6.45 6.4

0 mole % NaCl [24] 10 mole % NaCl 20 mole % NaCl 30 mole % NaCl 40 mole % NaCl 50 mole % NaCl

6.35 6.3 6.25 0

20

40

60

80

100

KBr (mole %) Fig. 2. Lattice parameters of mixed crystals at constant mole % of NaCl.

(Tm and Tf, respectively) are also provided in Table 2. Lattice parameters of the end member crystal compositions are in reasonable agreement with those reported in the literature [23] (literature values are given in brackets). Estimated values of the composition in the mixed phases of the ternary mixed crystals are given in Table 3. Separation of NaCl phase from the mixed lattice may be due to the large percentage deviation in lattice

an Z xan1 C ðyKxÞan2 C ð1KyÞan3 If the volumes are assumed to be additive, we get a3 Z xa31 C ðyKxÞa32 C ð1KyÞa33

Table 4 ˚ 2), mean square amplitude of vibration (!u2O A ˚ 2), Debye temperature (qD K), Debye Values of mean Debye–Waller factors (Bobs, Bstatic and Bthermal A frequency (fD!1012 Hz), compressibility (j!10K11 m2/N) and mean sound velocity (Cm!103 m sK1) System (with composition taken for crystallization)a

Bobs

Bstatic

Bthermal

!u2O

qD

fD

j

Cm

A B C A0.5B0.5 A0.5C0.5 B0.5C0.5 A0.1B0.7C0.2 A0.2B0.6C0.2 A0.3B0.5C0.2 A0.4B0.4C0.2 A0.5B0.3C0.2 A0.6B0.2C0.2 A0.7B0.1C0.2 A0.1B0.5C0.4 A0.2B0.4C0.4 A0.3B0.3C0.4 A0.4B0.2C0.4 A0.5B0.1C0.4 A0.1B0.4C0.5 A0.2B0.3C0.5 A0.3B0.2C0.5 A0.4B0.1C0.5 A0.1B0.3C0.6 A0.2B0.2C0.6 A0.3B0.1C0.6 A0.1B0.1C0.8

2.115 2.121 1.742 1.509 2.251 2.151 2.270 1.740 2.190 1.980 2.029 2.055 2.357 2.501 1.648 1.672 1.545 2.468 2.200 2.306 1.927 2.260 1.403 2.092 2.299 1.765

0.030 0.036 0.006 0.014 0.023 0.033 0.038 0.040 0.038 0.034 0.014 0.025 0.035 0.039 0.042 0.022 0.031 0.033 0.040 0.020 0.032 0.034 0.017

2.115 2.121 1.742 1.479 2.216 2.145 2.256 1.718 2.157 1.943 1.990 2.017 2.324 2.487 1.623 1.637 1.506 2.426 2.178 2.275 1.893 2.220 1.384 2.060 2.265 1.750

0.027 0.027 0.022 0.019 0.028 0.027 0.029 0.022 0.027 0.025 0.025 0.026 0.029 0.032 0.021 0.021 0.019 0.031 0.028 0.029 0.024 0.028 0.018 0.026 0.029 0.022

237.5 209.7 182.7 260.3 201.4 182.5 193.4 223.8 202.2 215.7 214.6 216.8 203.2 174.7 219.0 219.0 229.6 181.4 183.1 181.0 198.5 182.6 225.0 185.7 176.5 190.5

4.949 4.369 3.807 5.423 4.195 3.803 4.030 4.662 4.212 4.494 4.471 4.518 4.235 3.639 4.563 4.563 4.784 3.779 3.814 3.771 4.136 3.805 4.688 3.869 3.678 3.970

5.708 6.388 5.523 4.303 6.399 6.708 6.859 5.163 6.429 5.720 5.816 5.822 6.651 7.618 4.938 4.932 4.504 7.222 6.709 6.957 5.763 6.812 4.265 6.316 6.958 5.446

2.256 2.219 2.024 2.675 2.037 1.994 2.057 2.363 2.110 2.229 2.199 2.195 2.039 1.884 2.339 2.317 2.416 1.889 1.972 1.933 2.116 1.929 2.434 1.992 1.890 2.083

a

AZNaCl, BZKCl and CZKBr.

K. Jayakumari, C. Mahadevan / Journal of Physics and Chemistry of Solids 66 (2005) 1705–1713

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Table 5 Debye–Waller factors (B) and Debye temperatures (qD) of NaCl, KCl and KBr obtained by different workers by different methods (i) B (A˚2) values for NaCl 2.115 – 1.514 1.514 1.560 – – – – –

KCl 2.121 2.0 1.777 2.061 – 1.98 – – – –

KBr 1.742 1.6 1.917 2.212 – – 2.14 2.23 2.45/2.23 2.20/2.28

2.37 2.264/2.458 2.105/2.158 2.25/2.38 2.175/2.201

Method X-ray X-ray X-ray X-ray X-ray X-ray X-ray X-ray X-ray cation/anion Neutron (single crystal) cation/anion Neutron (single crystal) cation/anion Theory cation/anion Theory cation/anion Theory cation/anion Theory cation/anion





2.39/2.36

1.53 1.557/1.348 1.539/1.292 1.65/1.41 – (ii) qD (K) values for NaCl 237.5 – 281 – 278 278 – – – 320-322 322 292 283 281

2.17 1.929/1.995 1.771/1.813 2.20/2.19 – KCl 209.7 215 231 – 206 – 220 217.8 – 236 236 229 223 249

Sirdeshmukh et al. [23] Reid and Smith [30] Sirdeshmukh et al. [23] Deganello [31] Buyers and Smith [quoted in 29]

KBr 182.7 185 – 157 155 – 160 – 162 172 172 181 177 197

Method X-ray X-ray X-ray X-ray X-ray/neutron X-ray X-ray X-ray X-ray Elastic constants Elastic constants Compressibility Theory Mossbauer effect

Obtained by [Reference] Present work Venudhar et al. [24] Canut and Amoros [32] Meisalo and Inkinen [28] Sirdeshmukh et al. [23] Geetakrishna et al. [4] Pathak and Trivedi [25] Srinivas and Sirdeshmukh [26] Srinivas and Sirdeshmukh [27] Konti and Varshini [33] Sirdeshmukh et al. [23] Sirdeshmukh et al. [23] Reid and Smith [30] Shepard et al. [34]

which is the Retger’s rule extended to ternary mixed crystals representing an ideal ternary mixed crystal. Retger’s rule, ˚ , is not accurate or appropriate with deviations up to 0.5 A for calculating the lattice parameters in these systems. The thermal parameters obtained in the present study, viz. Deby–Waller factors (Bobs, Bstatic and Bthermal), mean square amplitude of vibration, Debye temperature and Debye frequency are given in Table 4. Compressibility and mean sound velocity values are also provided in Table 4. The Debye–Waller factors and Debye temperatures obtained for the end member crystals by different authors are compared in Table 5. Large differences in these values obtained by different authors may be attributed to the difference in the preparation of crystals and method of estimation of these parameters. Also, defects created during cooling of the grown crystal from melting temperature to room temperature and natural impurities (available in the starting materials) getting into the crystal may play an important role in this. The bulk composition dependence of Debye–Waller factor is highly non-linear and in some cases it even exceeds that for the end members. Even after making

Obtained by [Reference] Present Work Venudhar et al. [24] Lonsdale [quoted in 25] Linkoaho [quoted in 25] Geetakrishna et al. [4] Srinivas and Sirdeshmukh [26] Baldwin et al. [quoted in 27] Srinivas and Sirdeshmukh [27] Meisalo and Inkinen [28] Pryor [quoted in 29] Bacon et al. [quoted in 29]

correction for the static component, the Debye–Waller factors for some mixed crystals are found to be larger. Kumaraswamy et al. [3] made similar observations in the case of RbBr–RbI mixed crystals and attributed the enhancement of the Debye–Waller factor to an increase in the vibrational entropy due to mixing. Similar reasons may be responsible for the enhancement of Bthermal observed in the present system. Composition (taken for crystallization) dependences of Debye temperature of the mixed phase for NaCl mole %Z 10, 20, 30, 40 and 50 are shown in Fig. 3 together with Venudhar et al.’s [24] results for the KCl–KBr solid solution. Systems with NaCl mole %Z60 and 70 are not included as there is only one value in each case. The Debye temperature observed for (KCl)0.5(KBr)0.5 in the present study is in essential agreement with the work of Venudhar et al. [24]. Debye temperatures of the mixed crystals show a nonlinear variation with composition, deviating highly (except for few samples) from the Kopp–Neumann relation [4] extended to ternary mixed crystals: 3 K3 K3 qK3 Z xqK 1 C ðyKxÞq2 C ð1KyÞq3 . In some cases, it is

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K. Jayakumari, C. Mahadevan / Journal of Physics and Chemistry of Solids 66 (2005) 1705–1713

240 0 mole % NaCl [24] 10 mole % NaCl 20 mole % NaCl 30 mole % NaCl 40 mole % NaCl 50 mole % NaCl

Debye temperature (K)

230 220 210

found to vary non-linearly with bulk composition. Values for some mixed crystals even exceed those for the end members which may be attributed to the increase in vibrational entropy due to mixing.

200 190

Acknowledgements

180 170 160 150 0

20

40

60

80

100

KBr (mole %)

One of the authors C.M. thanks the University Grants Commission, Hyderabad for the grant of a Minor Research Project and the Tamilnadu State Council for Science and Technology, Chennai for the grant of a Major Research Project. The author K.J. thanks the University Grants Commission for the FDP Award.

Fig. 3. Debye temperatures of mixed crystals at constant mole % of NaCl.

References even less than that of the end members. This happens because of the enhancement of the Debye–Waller factor which may be related to the increase in vibrational entropy due to mixing. Debye frequencies obtained for all the grown crystals lie in the infrared range. Variation of the Debye frequency, compressibility and mean sound velocity with composition is not in any order similar to that for the Debye temperature. Very recently, Sirdeshmuch et al. [5] have reported that impurity hardening is more effective than solid solution hardening. In our studies also we have found that (though not studied systematically) the multiphased systems are harder than the other ones. Hence, these multiphased systems may be treated in par with the impurity added systems in gaining more mechanical strength.

5. Conclusion Ternary mixed crystals of NaCl, KCl and KBr were grown by the Czochralski method. Composition of the grown crystals were determined using the measured density and refractive index values by assuming the additive rule satisfying for them. Lattice parameters were estimated by the X-ray diffraction method. The present study shows that, for lower concentration of NaCl, the system exhibits only a single f.c.c. phase and the lattice parameters almost obey the Retger’s rule extended to ternary mixed crystals. But, for higher concentration of NaCl, the system exhibits two separate f.c.c. phases aggregated to form the crystal of which one nearly corresponds to the pure NaCl and the other corresponds to the mixed system. Thermal parameters, viz. Debye–Waller factor, mean square amplitude of vibration, Debye temperature and Debye frequency were determined for (NaCl)x(KCl)yKx(KBr)1Ky single crystals from the X-ray powder diffraction data. The thermal parameters are

[1] D.B. Sirdeshmukh, K. Srinivas, Physical properties of mixed crystals of alkali halides, J. Mater. Sci. 21 (1986) 4117–4130. [2] V. Haribabu, V. Subbarao, Growth and characterization of alkali halide mixed crystals, Prog. Crystal Growth Charact. 8 (1984) 189– 260. [3] T. Kumaraswamy, K.G. Subhadra, D.B. Sirdeshmukh, X-ray diffraction studies of RbBr-RbI mixed crystals, Pramana-J. Phys. 43 (1994) 33–39. [4] P. Geetakrishna, K.G. Subhadra, T. Kumaraswamy, D.B. Sirdeshmukh, Debye-Waller factors and Debye temperatures of alkali halide mixed crystals, Pramana-J. Phys. 52 (1999) 503– 509. [5] D.B. Sirdeshmukh, T. Kumaraswamy, P. Geetakrishna, K.G. Subhadra, Systematic hardness measurements on mixed and doped crystals of rubidium halides, Bull. Mater. Sci. 26 (2003) 261– 265. [6] X. Sahaya Shajan, K. Sivaraman, C. Mahadevan, D. Chandrasekharam, Lattice variation and stability of NaCl–KCl mixed crystals grown from aqueous solutions, Cryst. Res. Technol. 27 (1992) K79–K82. [7] S.K. Mohanlal, D. Pathinettam Padiyan, X-ray studies on quaternary system RbxK1-xBryCl1-y, Cryst. Res. Technol. 38 (2003) 494–498. [8] I.R. Nair, C.T. Walker, Raman scattering and X-ray scattering studies on KBr1-xIx, KCl1-xIx, and K1-xRbxCl 7 (1973) 2740–2753. [9] J.B. Thompson Jr., D.R. Waldbaum, Analysis of the two-phase region halite-sylvite in the system NaCl–KCl, Geochim. Cosmochim. Acta 33 (1969) 671–690. [10] A.E.H. Tutton, Crystallography and Practical Crystal Measurement, vol. II, Today and Tomorrow’s Book Agency, New Delhi, 1965. p. 1014. [11] H. Lipson, H. Steeple, Interpretation of X-ray Powder Diffraction Patterns, Macmillan, New York, 1970. [12] O.D. Slagle, H.A. McKinstry, The lattice parameter in the solid solution KCl–KBr, Acta Cryst. 21 (1966) 1013. [13] B.E. Warren, X-ray Diffraction, Addison Wesley, California, 1969. [14] G.H. Stout, L.H. Jensen, X-ray Structure Determination—A Practical Guide, second ed., John Wiley & Sons, New York, 1989. [15] D.T. Cromer, J.B. Mann, X-ray scattering factors computed from numerical Hartree-Fock wave functions, Acta Crystallogr. A24 (1968) 321–324. [16] M. Theivanayagom, C. Mahadevan, Lattice variation and thermal parameters of NixMg1-xSO4.7H2O single crystals, Bull. Mater. Sci. 24 (2001) 441–444.

K. Jayakumari, C. Mahadevan / Journal of Physics and Chemistry of Solids 66 (2005) 1705–1713 [17] G.C. Benson, E.K. Gill, Tables of Integral Functions Related to DebyeWaller Factor, National Research Council of Canada, Ottawa, 1966. [18] C. Kittel, Introduction to Solid State Physics, fifth ed., Wiley Eastern Limited, New Delhi, 1976. [19] K. Srinivas, D.B. Sirdeshmukh, Empirical estimates of the Debye temperatures of KCl-KBr mixed crystals, Indian J. Pure Appl. Phys. 24 (1986) 95–97. [20] K.G. Subhadra, D.B. Sirdeshmukh, Debye characteristic temperatures of some crystals with NaCl structure, Indian J. Pure Appl. Phys. 16 (1978) 693–695. [21] J.A. Dean (Ed.), Lange’s Handbook of Chemistry12th ed., Mc Graw Hill Book Company, New York, 1979. [22] I. Bunget, M. Popescu, Physics of Solid Dielectrics, Elsevier, New York, 1984. [23] D.B. Sirdeshmukh, L. Sirdeshmukh, K.G. Subhadra, Alkali Halides— A Handbook of Physical Properties Springer Series in Materials Science-49, Springer, Berlin, 2001. [24] Y.C. Venudhar, L. Iyengar, K.V. Krishna Rao, Thermal expansion and Debye temperatures of KCl–KBr mixed crystals by an X-ray method, J. Mater. Sci. 21 (1986) 110–116. [25] P.D. Pathak, M. Trivedi, Debye q of some alkali halides by X-ray diffraction and the law of corresponding states, J. Phys. C: Solid State Phys. 4 (1971) L219–L221.

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[26] K. Srinivas, D.B. Sirdeshmukh, X-ray determination of the mean Debye-Waller factors and Debye temperatures of KCl-RbCl mixed crystals, Pramana-J. Phys. 31 (1988) 221–224. [27] K. Srinivas, D.B. Sirdeshmukh, X-ray determination of mean DebyeWaller factors and Debye temperatures of the KxRb(1-x)Br mixed crystal system, Pramana-J. Phys. 25 (1985) 71–73. [28] V. Meisalo, O. Inkinen, An X-ray diffraction analysis of potassium bromide, Acta Crystallogr. 22 (1967) 58–65. [29] P. Geetakrishna, Studies on some alkali halide crystals, PhD Thesis, Submitted to Kakatiya University, Warangal, 1997. [30] J.S. Reid, T. Smith, Improved Debye-Waller factors for some alkali halides, J. Phys. Chem. Solids 31 (1970) 2689–2697. [31] S. Deganello, Temperature dependence of the root-mean-square displacements of atoms in the range 228 to 500 8C for NaF, NaCl, KCl and KBr, Zeitz. Kristallogr. 142 (1975) 45–51. [32] M.L. Canut, J.L. Amoros, On the inversion temperature function of the first order (one phonon) scattering and the determination of Debye characteristic temperatures, Proc. Phys. Soc. 79 (1961) 712– 720. [33] A. Konti, Y.P. Varshini, Debye temperatures of alkali halides, Can. J. Phys. 49 (1971) 3115–3121. [34] C.K. Shepard, J.G. Mullen, G. Schupp, Debye-Waller factors of alkali halides, Phys. Rev. B57 (1998) 889–897.