Comments on some recent phenomenological potentials for the alkali halide crystals

1. Phya. Chm. Solid3 Vol. 39. pi. 129%-1299 6 Pcruamon Press Ltd.. 1978. Printed in Great Britain

0022-l697/7811?01-129?ilO2

0010

COMMENTS ON SOME RECENT PHENOMENOLOGICAL POTENTIALS FOR THE ALKALI HALIDE CRYSTALSt

Istitoto di Chin&a

Fisica. Universita

R. &XXNHOlTNER di Geneva, C&ova, Italy and GNSM-CNR.

Unibl di Cenova, Italy

and C. S. N. MURTHY$and F. G. FUM Istituto di Scienze Fisiche. Univcrsita. di Genova, Geneva, Italy and GNSM-CNR

Unit4 di Genova, Italy

(Received IO April 1978; accepted 17 Moy 1978)

Ahstra&--!%mecomments are presented on three phenomenologicalpotentials for the alkali halide crystals proposed in recent years by Narayan and Ramaseshan, by Woodcock, and by Romano. Margheritis the context a brief comment is made on the van der Waals coefficients reported by Hajj. 1.

lNTRODUCTKlN

to be given by the corresponding energy in the pertinent alkali halide molecule. This is written in the following form:

A

number of phenomenological potentials have been proposed in recent years for the alkali halide crystals. Here we will concentrate on three of these [l-3]. (a) The potential by Narayan and Ramaseshan[ 11.The repulsive energy contribution to the lattice energy per ion pair is written as follows

R(r)=$exp[-B(r”

(I)

where n, and nz are the numbers of nearest and nextnearest neighbors, while r and br are the corresponding distances. The parameters of the potential are A+, A-, p+ and p_, and are assumed to be characteristic of the ions and independent of the crystal structure: the “radii” r+ and r- are expressed, instead, in terms of r and of the parameters. The parameters for the various ions are determined through the Hildebrand equation of state and its volume derivative by a least-squares fit of crystal data on all the alkali halides at room temperature and at various pressures up to 45 kbar(a range in which some of the salts undergo a phase transition from the NaCl to the CsCl structure). The van der Waals interactions are included using the dipole-dipole coefficients reported by Hajj[4] and the dipolequadrupole coefficients reported by Mayer [S]. (b) The potential by Woodcock[2]. The crystal potential includes, besides the coulombic interactions, only the repulsive interactions between nearest neighbors. The repulsive energy between neighboring ions in an alkali halide crystal as a function of their separation is assumed tA preliminary report on this work was presented at the 1Ith National Conference of the Italian Association for Physical Chemistry in S. Marghcrita Lignre, Lkcember 9-l 1, 1976. *Present address: Department of Chemistry, Royal Holloway College, Egham, Surrey, England.

- I)]

(2)

(where distance is measured in units of r, and energy in units of e’/r, r. being the equilibrium interionic separation in the molecule). The molecular potential includes also a term -u/r’, describing monopole-induced dipole interactions. The. parameters A, B, n and a are all determined from molecular data. (c) The potential by Roman0 el a1.[3]. The repulsive energy contribution to the lattice energy per ion pair in an NaCl-type crystal is written as follows:

WLmP(r)= n,[A+ exp(- r+/p+) + A- exp(- r-/p-)1 + nJA+ expt- br/2p+) + A- expl- br/2p-)I

and Sinistri. In

WL’Yr) = 68. exp (- ar) + 6& exp [- V/(2)arl t 88.exp [- V(3)arl t 3B, exp (- 2ar) + 248, exp I- V(S)arl

(3)

where interactions up to fifth neighbors are included. The parameters B., B, and a are determined crystal by crystal from data at atmospheric pressure and at temperatures in the range 300-700K via the Hildebrand equation of state. The van der Waals interactions are considered using the coefficients reported by Mayer[S], as well as those reported by other Authors. Some comparisons will be made between these potentials and the Fumi-Tosi potential with variable p[6], whose parameters are determined from crystal data on the NaCI-type alkali halides in standard thermodynamic conditions.

We should like to point out first that the FT potential[6] reproduces rather well the experimental data 1295

R.

12%

EGGENHOFFNER etd.

used by Narayan and Ramaseshan[l] to determine the parameters of their potential: these data are the nearestneighbor distance r and the second derivative of the lattice energy d*W,Jr)/d?, at room temperature and at (generally) two different pressures (in the range from 0 to 45 kbar), for each of the NaCl and CsCl type phases of the alkali halides which occur under these conditions. Since the FT potential is known Only for 17 of the 20 alkali halides, we have restricted our attention to these. The calculated values of r and of d*WJdfl (at the calculated value of r) are reported in Table 1 for the FT potential and for the NR potential with the two sets of parameters given by the Authors[ 1, 71: the differences

between the calculated values and the experimental values adopted by Narayan and Ramaseshan[l] are also reported, together with the resulting root-mean-square deviations. The value of r is calculated by solving numerically the equation for dW&)/dr given by the Hildebrand equation of state (using for the isothermal compressibility and the coefficient of volume thermal expansion the values adopted by Narayan and Ramaseshan [ 11). We should note, on the other hand, that the NR potential involves unfortunately the values of the van der Waals dipole-dipole coefficients reported by Hajj[4], which are incorrect. Indeed the anion-anion coefficients

Table 1. A comparison between the FT and NR potentials? Solld

Pressure (Kbnr)

0

LIP

0

LlCl

45

Nearest-Neighbor

Phase NR:

1st

NaCl

1.974(-0.035)

Set

Distance

2nd

r(w)

Set

1.997(-0.012)

FT 2.015(0.006)

N‘lCl

2.570(O)

2.559(-0.011)

2.561

NdCl

2.481(-0.001)

2.468(-0.014)

2.466(-0.016)

(-0.009)

d2WL/dr2 NR: 2.471

1st (0.079)

(105ergs/cm2

set

2nd

molecule)

Set

FT

2.493(0.101)

2.394(0.002)

1.518(0.119)

1 .485(0.086)

1.410(0.011)

2.232(0.151)

2.151(0.070)

2.055(-0.026) 1.213(0.001~

0

NaCl

2.778(0.028)

2.760(0.010)

2.749(-0.001)

1 .236(0.024)

1 .205(-0.007)

45

NaCl

2.659(0.025)

2.639(0.005)

2.629(-0.005)

2.008(0.249)

1 .921

0

NaCl

3.081(0.078)

3.056(0.053)

3.010(0.007)

0.923(0.123)

0.881(0.081)

o.794(-0.006)

45

LIt?I‘

LII

NaF

(0.162)

1 .936(0.177)

NdCl

2.9@8(0.065)

2.8?6(0.033)

2.818(-0.025)

1 .767(0.333)

1 .662(0.228)

1.481

C

NJCl

2. 323(0.013)

2.325(0.015)

2.307(-0.003)

1.598(-0.014)

1 .662(O.C50)

1.630(0.018)

0

NJCI

2.810(-0.010)

2.811

2.836(0.016)

1.160(-0.108)

1 .I81

45

NaCI

2.682(-0.023)

2.684(-0.021)

2.712(0.007)

1.900(-0.048)

1 .897(-0.051)

2.005(0.057) 1 .036(-0.002)

N
(-o.oog)

(0.047)

(-0.087)

1 .230(-0.038)

0

NaCl

2.958(-0.031)

2.959(-0.030)

2.992(0.003)

1.056(0.018)

1.059(0.021)

45

NaCl

2.809(-0.040)

2.809(-0.040)

2.837(-0.012)

1.843(0.084)

1 .814(0.055)

1 .848(0.089)

0

NaCl

3.261

3.258(0.022)

3.223(-0.013)

0.806(0.037)

0.790(0.021)

0.788(0.019)

-1.672(0.207)

NJBI‘

NaI

(0.025)

45

NaCl

3.052(0.004)

3.044(-0.004)

3.012(-0.036)

1.614(0.149)

1 .610(0.145)

KF

0

NJCl

2.693(0.020)

2.697(0.024)

2.670(-0.003)

1 .205(-0.067)

1 .262(-0.010)

1.279(0.007)

KC1

0

NaCl

3.126(-0.021)

3.145(-0.002)

3.142(-0.005)

0.923(-0.032)

0.954(-0.001)

0.959(0.004)

15

NaCl

3.048(-0.025;

3.068(-0.005)

3.066(-0.007)

1 .220(-0.063)

1.262(-0.021)

20

CSCI

3.214(0.025)

3.206(0.017)

3.208(0.019)

1.197(-0.014)

l.ZOO(-0.011)

1.267(-0.016) 1.254(0.043)

KBF

1 .640(-0.104)

45

CSCl

3.135(0.018)

3.128(0.011)

3.132(0.015)

1.569(-0.175)

1.571(-0.173)

0

NaCl

3.261(-0.039)

3.287(-0.013)

3.307(0.007)

0.846(0.014)

0.870(0.038)

0.886(0.054)

15

NZ.Cl

3.171(-0.041)

3.198(-0.014)

3.219(0.007)

1 .158(-0.056)

1.190(-0.024)

1 .224(0.010)

20

CSCl

3.350(0.004)

3.340(-0.006)

3.357(0.011)

1.140(0.161)

1.128(0.149)

1 .233(0.254)

45

CSCl

3.262(0.005)

3.252(-0.005)

3.275(0.018)

1 .524(-0.073)

1.506(-0.091)

1 .649(0.052)

N&l

3.495(-0.038)

3.520(-0.013)

3.535(0.002)

0.711

0.717(-0.017)

0.733(-0.001,

15

N&l

3.378(-0.042)

3.403(-0.017)

3.420(O)

1.050(-0.103)

1.058(-0.095)

1 .096(-0.057)

20

CSCl

3.579(O)

3.559(-0.020)

3.575(-0.004)

1 .025(-0.017)

0.988(-0.054)

1.119(0.077)

45

CSCl

3.471(-0.004)

3.449(-0.026)

3.476(0.001)

1 .432(-0.095)

1.379(-0.148)

1.558(0.031)

RbF

0

NaCl

Z.S12(-O.OOfJ)

2.817(-0.003)

2.829(0.009)

1 .346(0.078)

1 .356(0.088)

1.249(-0.019)

RbCl

0

NaCl

3.273(-0.018)

3.290(-0.001)

3.288(-0.003)

0.959(-0.023)

0.973(-0.009)

0.983(0.001)

10

CSCl

3.365(0.022)

3.363(0.020)

3.376(0.033)

1.125(0.031)

1.119(0.025)

1.160(0.066j

45

CSCl

3.247(0.025)

3.246cO.024)

3.261

1 .738(-0.060)

1 .734(-0.064)

1.788(-0.010)

0

NaCl

3.439(-0.005)

3.460(0.016)

3.435(-0.009)

0.831(-0.015)

0.846(O)

5

CSCl

3.562(0.035)

3.564(0.037)

3.543(0.016)

0.902(-0.023)

0.884(-0.041

45

CSCl

3.393cO.025)

3.393(0.025)

3.381(0.013)

1 .634(0.145)

1 .615(0.126)

,.681(0.192)

0

NaCl

3.611(-0.060)

3.635(0.036)

3.667(-0.004)

0.760(0.005)

0.760(0.005)

0.755(O)

5

CSCl

3.749(-0.020)

3.746(-0.323)

3.769(O)

0.825(0.006)

0.793(-0.026)

0.871(0.052)

45

CSCl

3.553(-0.028)

3.544(-0.037)

3.581(O)

1 .587(0.028)

1.544(-0.015)

1.699(0.140)

0

NaCl

2.994(-0.010)

2.996(-0.008)

3.004(O)

1.345(-0.099)

1 .380(-0.064)

l-442(-0.002)

0.030

0.022

KI

RbBr

RbI

CSF R.M.S.D.

o

(0.039)

0.014

(-O.G23)

0.107

0.086

0.851(0.005) )

0.941(0.016)

0.077

tone compares the values of the nearest-neighbor distance r and of the second derivative of the lattice energy d* W,/d? for the NaCl and CsCl type phases of the alkali halides at room temperature and at various pressures calculated with the Ff potential and with the NR potential (for 2 sets of parameters). The values in parenthesis give the differences between the computed values and the experimental values adopted by Narayan and Ramaseshan[l]. It should be noted that the NR potential is fitted by least squares to all these experimental data, while the FT potential is fitted only to the data at atmospheric pressure.

Comments on some recent p~nomenol~c~

potentials for the alkali halide crystals

I297

Table 2. Results for the W potential+ Alkali

Holecular Rotational

n

Halide

camtant

:

cc (t o-4cin-1

e

Cry&11 )

Cohesive Energy

Nearest-Neighbor Distance

(real/mole)

xn the Crystal (8,

LIF

4

184.5(-12.6)

X%.4(9.3)

-262.4(-17.5)

-220.9l24.0)

1.971(-0.043)

2.223(0.209)

LICl

6

25.6(-54.5)

81.8(1.7)

-21?.7(-16.3)

-146.8t54.6)

2.448(-0.122)

3.058(0.488)

LlBr

6

14.1(-42.3)

52.9(-3.5)

-202.0(-9.2)

-113.8(79.0)

2.671

3.444cO.693)

LiI

6

7.6(-33.3)

38.6(-2.3)

-182.7(-3.2)

-86.2193.3)

2.9766-0.024)

NaF

6

18.1(-27.5)

45.2(-0.4)

-224.3(-7.3)

(-O.OSO)

3.897(0.89?)

2.393cO.076)

NdCl

6

5.6(-10.5)

14.3(-1.8)

-195.31-10.2)

-158.1(27.0)

2.7?6(-0.044)

3.142(0.322)

NaBr

6

2.91-6.5)

8.3(-1-l)

-183.0(-6.4)

-142.9(33.7)

2.972(-0.017)

3.393(0.404)

Na:

6

1.7(-4.8)

5.8(-0.7)

-166.6(-1.6)

-116.6c48.4)

3.265cO.028)

3.840(0.603)

KF

3

20.7(-2.7)

23.0(-0.4)

-189.7(2.?)

-160.2(32.2)

2.734fO.060)

3.017fO.343)

KC1

6

3.2(-4.7)

7.2(-0.7)

-175.4(-7.1)

-152.8cb5.5)

3.088(-0.059)

3.375(0.228)

K?T

6

1.3(-2.7)

3.5(-0.5)

-163.4(-l

-128.2c33.3)

3.327tO.029)

3.754(C.456)

KI

6

1.2(-l

2.9(0.2)

-155.3(-3.6)

-138.4(13.3)

RbF

2

16.8(1.6)

15.4cO.2)

-187.8(-3.8)

-195.0(-11

.5)

.P)

.O)

3.505(-0.028)

3.768(0.235)

2.754(-0.061)

2.665(-0.150) 3.551(0.260)

RbCl

6

1.7(-2.8)

4.1(-0.4)

-167.56-5.7)

-144.5(17.3)

3.235(-0.056)

RbEW

6

0.7(-1.2)

1.7(-0.2)

-158.7(-3.1)

-135.6(20.0)

3.423(-0.022)

3.753cO.308)

RbI

6

0.4(-0.7)

l.O(-0.1)

-146.8(-0.1)

-122.5c24.2)

3.707(0.036)

4.077(0.406)

CsF

1

16.0(5’.0)

10.4(-0.6)

-180.1(-5.0)

-189.1(-14.0)

2.797(-0.207)

2.607(-0.397)

R.X.S.D.

7.8

41.0

0.074

0.443

tColumn 2 gives the values of the parameter n reported by Woodcock(2j and columns 4.6 and 8 give in order the corresponding computed values of the rotational constant a, for the various alkali halide molecules and of the cohesive energy and n~est-nei~~r distance in the various NaCI-type alkali halides in standard thermodynamic conditions. Column 3 gives our values for the parameter n and columns 5,7 and 9 give the corresponding computed values for the pertinent quantities. The values in parenthesis give the differences between computed and experimental values. The experimental values of B, are those adopted by WoodcockI21; the experimental values of the nearest-neighbor distance are those adopted by FI’[6], while the experimental values of the cohesive energy are obtained from those adopted by musing more recent values of the electron al%nities of the halogens [ IO].

were obtained by an improper application of the technique originally proposed by MayerfS], using only the optical absorption data by Eby et a1.[8] which stop at 11.4eV for alJ crystals, and thus include only a fraction of the pertinent anion absorption spectrum, different for the different anionst (and also for a given anion in the different metal salts). The truncated absorption spectrum is involved also in the evaluation of the anion-cation coefficients. We will not discuss in detail the model on which the NR potential is based, which deviates somewhat from the standard Born model. We note, however, that some of the basic ideas of the model are certainly open to question: we refer, in particular, to the separability of the repulsive energy between two ions into the sum of two terms, each pertaining only to one of the interacting ions. We note also that the model does not seem to be completely self-consistent; e.g. the ions are apparently considered non-spherical in their repulsive interactions, and yet one adopts the usual Madelung form for the coulombic contribution to the lattice energy which assumes spherical ions. In the same context, one should note also that the expression adopted for the repulsive energy contribution to the lattice energy per ion pair omits the factor i in front of the next-nearest neighbor contribution (which is necessary to avoid counting twice these interactions in the total lattice energy). iIn a careful application of Mayer’s technique, LynchpI has shown that reasonable truncation limits for the anion absorption spectrum range from 16 to 21 eV in passingfrom Kf to KBr and KCI.

3.THEWPOTEUTlAL

The potential as reported in the paper by Woodcock121 compares already somewhat unfavourably with the FT potential&]. We have computed with the W potential as reported the cohesive energy in standard thermodynamic conditions for the family of NaCl-type alkali halides for which the FI’ potential is available (using the vibrational internal energies of the crystals adopted by FT). We have also computed the pertinent nearest-neighbor distances with the same method as in Section 2 (using the values adopt& by FT for the relevant experimental quantities). These computed values are reported in Table 2, together with the differences between computed and ex~~rnen~ values and the ensuing R.M.S.D.‘s. The R.M.S.D.‘s for the cohesive energy and for the nearestneighbor distance are 7.8 kcallmole and 0.074 A, respectively. The corresponding r.m.s.d’s for the FT potential are, instead, 3.9 kcallmole and 0.008A: the computed values of the nearest-neighbor distance in the individual salts are included in Table 1 (in the P = 0 lines), while the computed values of the cohesive energy are reported in Table 2 of Ref. [61 (under the heading “1st set of data”). We should like to note, however, that the values of the repulsive parameters reported by Woodcock[21 do not appear to correspond to a best fit of the data for the alkali halide molecules which are used in their determination; specifically, we find that the values reported for the parameter n do not give good values for the rotational constants a,‘~. The values of n reported by Woodcock[Z] and the values of a, that we calculate from

R.

1298

EGGENHOFFNER eta/.

themt are given in Table 2. Table 2 gives also our best common value of n (restricted to be an integer) for all the salts except the fluorides, and our best individual values of n for the 5 fluorides, together with the values of a, that we calculate from them. The use of the improved values of the parameter n increases significantly the discrepancy between the computed cohesive energy and nearest-neighbor distance of the NaCI-type alkali halides and the experimental values. This is apparent from Table 2. For NaF the repulsive potential actually becomes attractive, while the new R.M.S.D.‘s for the cohesive energy and for the nearest-neighbor distance for the other 16 salts become 41.0 K&/mole and 0.443 A, respectively. These results support the intuitive notion that it is inadvisable to derive a repulsive potential for an alkali halide crystal from data on the alkali halide molecule, tThese values differ from those reported by Woodcock[Z]. We use in Woodcock’s equation (4) for (I, the following explicit expressions of @(I) and d”(1) in terms of the parameters A, B, n and a: &‘(1)= - 200 - 2 t A[n*B* - nB(n - 9) + 201 d’“(t)= 120+o:20q_A[n’B3-3n*B’(n-5)tnB(n*-

lSnt7-t)

These expressions are not reported in 121.It should be noted that the parameters A, B and a are all expressible in terms of the parameter n and of molecular data (different from a,) bv means of equations given in [2].

I

owing in particular to the significant differences in the electronic charge distributions in the molecule and in the crystal. 4. THE RMS. POTRRTM

We restrict our attention to the potential reported in the paper by Roman0 et a/.[31 for NaCl, KCI and KBr using Mayer’s van der Waals coefficients: only in these

cases one can make a comparison with the FT potential 161. The comparison shows that the FT potential and the R.M.S. potential as reported give quite comparable values for the total repulsive energy contribution to the lattice energy per ion pair as a function of the nearestneighbor distance around the equilibrium value in standard thermodynamic conditions, but that the R.M.S. potential gives much smaller values than the Ff potential for the repulsive energy contribution due to the nearestneighbors. The point is illustrated in Figs. l(a) and l(b) for KBr. (It should be noted that the repulsive energy contributions due to the third to fifth neighbors, which are considered in the R.M.S. potential but not in the FT potential, are completely negligible on the scale of the figure (being less than 0.01 x lo-‘* ergs/molecule at 3.1 A,., To investigate whether this discrepancy between the two potentials is a necessary consequence of the different parameters adopted and of the different fitting procedures used for their determination, we have reexamined the determination of the parameters for the

an

.

b 2.5 ^,

l

t O.

.

. .

0 .

.

0. 0. 0 .

0

0.

a. 0 . 0 . 0

.

0.

Fig. 1. Variation of total (0) and first neighbor (0) repulsive energy contributions to the lattice energy with the nearest-neighbor distance in KBr. (a) FT (b) R.M.S.as reported (c) R.M.S.with new set of values of parameters.

Comments on some recent phenomenological potentials for the alkali halide crystals

1299

Table 3. Values of the R.M.S. parameterst Solid.

RMS Potential Parameters+

BU

NaCl

KBr

cx

(10-8ergs/molecule)

(8-l)

RMS like:

0.043

0.417

2.700

:

0.200

0.484

3.165

RMS llke:

0.087

1.120

2.680

New

0.278

0.913

2.971

RMS llke:

0.105

1.634

2.650

NEW

0.429

1.072

2.990

New:

KC1

El

tit may be noted that the new values of the parameters B. and (I in the three salts are quite close to the values that one can estimate for them from the pertinent FT parameters. This applies also to the BI parameter in KBr but not in NaCl and KCI. SThis set of values for the parametersin NaCl is less accurate than the other sets since it was obtained from the SIMPLEX part of the MINLJITroutine.

In each salt the new set of values for the R.M.S. R.M.S. potential. Specifically, we have re-examined for each of the three salts in question the determination of parameters essentially eliminates the discrepancy bethe R.M.S. parameters B., & and Q from the least- tween the m potential and the R.M.S. potential dissquares fit of the experimental values of the first deriva- cussed above. This is illustrated in Fig. l(c) for KBr. The R.M.S.-like sets of values for the parameters give, of tive of the lattice energy dW&)/dr at various temperacourse, almost the same results as the values reported in tures, as given by the Hildebrand equation of state.? The least-squares fit was performed with the MINUIT [31. routine[ll], using the experimental values of dWL(r)/dr (with their estimated uncertainties) at seven temperatures Acknowledgements-We are indebted to Group II of the Genova section of INF’N for the use of their HP 2100. One of us between 300 and 6OOK,spaced by a 50”interval. The values (C.S.N.M.)wishes also to thank the GNSM-CNR for the award were recalculated from the pertinent experimental quanof a research contract. tities reported in the original sources quoted in [31. In each of the three salts we find, besides a minimum corresponding rather well to the one reported in [3], REFERENCES another fully comparable minimum in a different region I. Narayan R. and Ramaseshan S., J. Phys. Chem. Solids 37, 395 (1976). of the parameter space. The two set of values of the 2. Woodcock L. V., I. Chem. Sot. Faraday Trans. II 70, 1405 parameters for each of the salts are reported in Table 3. (1974). It has been checked that the reported sets of parameters 3. Roman0 S., Margheritis C. and Sinistri C., Z. Naturforsch. lead to computed values of ?d2WJd? at room 29a, 1202(1974). 4. Hajj F., L Chem. Phys. 44, 4618 (1966). temperature in agreement with the experimental values 5. Mayer J. E.. J. Chem. Phys. 1,270 (1933). given by the volume derivative of the Hildebrand equa6. Tosi M. P. and Fumi F. G.. J. Phvs. Chem. Solids 25. 45 tion of state, within their estimated uncertainty of about mw. 5%. 7. Narayan R. and Ramaseshan S., Curr. Sci. 46, 359 (1977). tWe have not reexamined the determinationof the parameters from the experimental values of d*W,(r)/d? at various temperatures, since we find that the estimation of values from those of dW,(r)/dr, following the approach adopted in 131, leaves considerable uncertainties.

8. Ebv J. E.. Teeearden K. J. and Dutton D. B., Phvs. Reu. 116. lti (1959). 9. Lynch D. W., I. Phys. Chem. Solids 28, 1941(l%7). 10. Berry R. S. and Reimann C. W., J. Chem. Phys. 38, I540 (1%3). Il. James F. and Roos M., CERN Program Library Routine D506/D516(1976).