We have investigated the second order elastic constants of orientationally disordered NH4I using an Extended Three Body Force Shell Model (ETSM) in the temperature range 250K≤T≤350K.The second order elastic constants (C11, C12 and C44) obtained by us show an anomalous behaviour with the variation of temperature. The variation of the second order elastic constants with temperature is in good agreement with the measured data

Tiwari, Alpana

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Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.45, 2015         44  Thermoelastic Behavior of Orientationally Disordered Ammonium Iodide  Alpana Tiwari Maharani Laxmibai Government Girls P.G.College, Bhopal-462002 .India.  Abstract We have investigated the second order elastic constants of orientationally disordered NH4I using an Extended Three Body Force Shell Model (ETSM) in the temperature range 250K≤T≤350K.The second order elastic constants (C11, C12 and C44) obtained by us show an anomalous behaviour with the variation of temperature. The variation of the second order elastic constants with temperature is in good agreement with the measured data.   Introduction The thorough studies of the lattice properties of ionic solids have long been the region of interest for experimental and theoretical workers. Their simple structures and versatile applications generate the ideas for the understanding of the complex systems. Pure ammonium halides have always been the systems of interest for the experimental and theoretical investigators due to their polymorphism and orientational disorder introduced by the fast reorienting ammonium ion. An enormous amount of experimental and theoretical efforts have been devoted to study the various aspects of lattice properties of these materials. Ammonium iodide undergoes order-disorder transition (λ-transition) at Tλ=231K.At room temperature NH4I exhibits NaCl-type fcc structure with space group Fm3m. This transforms to Pm3m structure at about 257K.It is interesting to note that its stability range of CsCl type structure is relatively small, narrowed down to 25K[1-11]. In the dynamically disordered phase, distortions are introduced due to non-equivalence of N-H bonds, which induce elastic quadrupolar moments of NH4+ions in NH4I. Although the average symmetry of these crystals is cubic, but the orientational disorder locally breaks the cubic symmetry and anomalous softening of sheer elastic constant C44 is observed [12].   This is an attempt to investigate the lattice static and dynamic properties of ammonium iodide in their orientationally disordered phase by taking into account the effect of translation-rotation (TR) coupling, which has turned out to be responsible for important effects in orientationally disordered materials. Singh and Gaur [13] incorporated the effects of TR coupling in the framework of a realistic three body force shell model (TSM) [14] to develop ETSM. It has been applied successfully by Gaur et.al. and others to explain the elastic, thermal and dynamical properties of the orientationally disordered alkali cyanides, their mixed crystal counterparts, sodium superoxide, ammonium halides and ammonium-alkali halide mixed crystals [15-23]. The present theoretical approach is motivated from the versatility and considerable success of the ETSM, we have thought it pertinent to apply this ETSM to investigate the temperature dependence of the second order elastic constants of NH4I.  The essential formulation of the present model is given in the next section and the results obtained by us are presented and discussed in the subsequent section.  Formulation of ETSM The interionic interaction potential of ETSM has been expressed as [13]-                               Here, k(k’) denote the positive (negative) ions and sum is taken over all the (kk’) ions. In the above expression, the first two terms represent the long-range Coulomb and three body interactions (TBI) [14]. The third and fourth terms are the additional van der Waals (vdW) attraction terms due to the dipole–dipole (d-d) and dipole–quadrupole (d–q) interactions [24]. The fifth term is the Hafemeister and Flygare (HF) type short-range (SR) overlap repulsion [25] extended upto the second neighbour ions. βkk’ are the Pauling coefficients. The last term, ΦTR is the new contribution due to the translation-rotational (TR) coupling. The TR coupling coefficients are obtained on the lines of Sahu and Mahanti [26, 27]. The expressions for TR coupling coefficients are given a little later. Here, b and ρ are the hardness and range parameter.   Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.45, 2015         45  The TR coupling effects have been introduced through the second order elastic constants (SOECs) derived from the Eq. (1) and expressed as [13]- TRijijij CCC δ+= 0                                         (2) with Cij0 as their unrenormalized values and δCijTR are the TR coupling contributions to the SOECs. For the calculation of δCijTR, we have used the expressions as given in ref. [28]. )(8 11211 TAaC effTR χδ −=                                 (3) )(4 11212 TAaC effTR χδ =                                     (4) )(2 44244 TBaC effTR χδ −=                                  (5) Here, a is the lattice constant. The effective coupling constants Aeff and Beff are given by Aeff =AR+AQ and Beff = BR +BQ                    (6) with AR and BR as the parameters associated with the short-range repulsive interactions and AQ and BQ are associated with the quadrupole electric field gradient interaction given by Michel and Naudts [29] and Sahu and Mahanti [30] as- AR = √5π (b/ρ) (d2+a2) -1/2 [a (3f2–f0) +d (f1 –3f3)]                                                               (7)  BR=-√30π (b/ρ) (d2+a2)-1/2[d (f1–3f3)]           (8) With            (9) Where,       h= (1/ρ) (d2+a2)1/2    and    g = 2da/ (d2+a2) The various symbols appearing in the above eqns. (3) to (9) are explained in ref. [30]. The quadrupole electric field gradient interaction AQ and BQ and the SR repulsive interaction parameters AR and BR are calculated using their expressions from [30] and the method [21]. The susceptibility χ11 is related to C11 and C12 through Aeff, while χ44 is related to C44 through Beff. The computed results on the elastic constants (C11, C12, C44) of NH4I are presented and discussed below.  Results and Discussion The values of the vdW coefficients (ckk’ and dkk’) and their overall coefficients (C, D) for NH4I have been evaluated by us using their expressions reported elsewhere [24]. Their estimated values are listed in Table 1. The model parameters (b,ρ,f(r)) have been evaluated from the equilibrium condition and using the input data (ro, B) from Table 2 and the values of vdW and TR-coupling coefficients listed in Table 1. The values of the model parameters thus obtained have also been depicted in Table 2. The values of the first and second order space derivatives of f(r) have been evaluated from the relation [31]. f(r) = fo exp (-r/ρ)                                        (10) The novelty in our calculations lies in the fact that we have not used those properties as the input data which are intended for the prediction. In order to obtain such input data at different temperatures, we have used the thermal expansion relation. Experimentally determined values of r0 have been used [32].   Table 1: The van der Waals coefficients [ckk and dkk in the units of 10-60erg/ cm6 and 10−76erg/ cm8, respectively] of NH4I and TR coupling parameters of NH4I at 300K.  ∫+−−+−−=ggyhnn dyeygf112)1(0)1(vdW coefficients NH4I TR-coupling Parameters Values ckk 101.07 AR [10-4dyn] -0.080 ckk’ 336.71  BR [10-4dyn] 0.798  ck’k’ 1123.0 Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.45, 2015         46               Using the expressions given in [21], we have computed the values of the second order elastic constants (C11, C12 and C44) as a function of temperature 250K≤T≤350K. While computing these elastic constants, the contributions from the translational-rotational coupling are taken into account. The values of the second order elastic constants obtained from ETSM have been plotted as a function of   Table 2: The input data [r0, B (1010dyn.cm2)] and the model parameters for  NH4I .temperature and concentration (x) as displayed in fig.1.                                        dkk 785.02 AQ [10-4dyn] 0.221 dkk’ 244.10 BQ [10-4dyn] -0.187 dk'k' 76.00 C 5101 D 2927 r0[Å]  Model parameters b [1012erg] ρ [Å] f [r] 3.620 4.227 4.997 -7.815 2.42.452.52.552.62.65250 270 290 310 330 350T(K)C 11(1011 dyne/cm2 )Calc.Expt.4.464.484.54.524.544.564.58250 270 290 310 330 350T(K)C 12(1010 dyne/cm2 )2.402.502.602.702.802.903.00250 270 290 310 330 350T(K)C 44(1010 dyne/cm2 )CalcExptFig 1: Variation of second order elastic constants of NH4I as a function of temperature.  [a] [b] [c] Advances in Physics Theories and Applications                                                                                                  www.iiste.org ISSN 2224-719X (Paper) ISSN 2225-0638 (Online) Vol.45, 2015         47   It can be seen in fig.1a that C11 shows decreasing trend with increase in temperature and in fig.1b that C12 shows increasing trend with increase in temperature this feature is usual behaviour of the rocksalt structure materials. Whereas C44 (fig.1c) becomes almost constant for the entire temperature range. It is totally softened.It has been observed that the agreement improve quite significantly in all the cases by the inclusion of TR coupling. An anomalous softening of the shear elastic constant C44 has is due to the TR coupling of the quadrupole moments of the NH4+ ions with the shear modes. Our results for C11 and C44 are fairly in agreement with Ultrasonic experiment data [33].It is interesting to note that the observed anomalous softening occurring in C44 is reproduced well by the ETSM results. It is interesting to note that the Cauchy relation (C12=C44) which are believed to be fulfilled by the alkali halides are violated by all the ammonium halides.  Conclusion  The present ETSM has been applied, probably for the first time, to explore the elastic properties of the orientationally disordered ammonium iodide. It can be concluded on the basis of the above discussions that our theoretical results for elastic properties show the similar behaviour as experimentally observed for the orientationally disordered mixed crystals of same category and are reasonably good. This is not unexpected as the ETSM framework is capable to account for the effects of the Cauchy violations (elastic properties), the realistic interaction potential, the polarizibilities (dielectric properties) and the TR coupling effects for the present system of orientationally disordered materials. A detailed description of the theoretical investigation of the second order elastic constants of the orientationally disordered ammonium iodide crystals has been presented in this paper.   References 1. Perry C H and Lowndes R P, J. Chem. Phys. 51, 3648 (1969). 2. N. Personage and L. A. K. Staveley, “Disorder in Crystals” (Calarendin, Oxford 1978). 3. Seymour R.S. and Pryor A.W. Acta Crystalloger, B 26, 1847 (1970).  4. F. Simon, C. V. Simon and M. Ruhemann, Z. Physik Chem. 129, 344 (1927);  A.G.Cole, Ph.D.thesis, MIT, (1952). 5. H.A. Levy and S. W. Peterson, Phys. Rev. 83, 1270 (1951); 86, 766 (1952); J. Am. Chem. Soc. 75, 1536 (1952). 6. A.C. Menzies and H.R. Mills, Proc. Roy. Soc. 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Thermoelastic Behavior of Orientationally Disordered Ammonium Iodide

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Thermoelastic Behavior of Orientationally Disordered Ammonium Iodide

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