PHYSICAL RE VIEW B VO LUME 12, NUMBER 12 15 DE CEMBE R 1975

Lattice dynamics of solid n- and y-N, crystals at various pressures

Eduardo HulerDepartment of Inorganic and Physical Chemsitry, Soreq Nuclear Research Centre, Yavne, Israel

Alex Zunger*Department of Theoretical Physics, Soreq Nuclear Research Centre, Yavne, Israel

and Department of Chemistry, Tel Aviv University, Tel-Aviv, Israel(Received 31 March 1975)

A previously published interaction potential between N, molecules has been utilized to compute the latticemode frequencies of solid a- and y-N, at the equilibrium crystal structure corresponding to various pressures.Dispersion curves and the density of states are given. These are then used to calculate the lattice heatcapacity, Gruneisen mode parameters throughout the Brillouin zone, linear thermal-expansion coefficient,Debye temperatures, and temperature-dependent root-mean-square amplitudes of vibrations. Whenever

comparison with experimental data is possible, good agreement is obtained.

I. INTRODUCTION

Lattice-. dynamics calculations, which have beencarried out extensively on atomic crystals, havebeen extended recently to the study of molecularcrystals. ':This is mainly due to the accumula-tion of extensive exyerimental data on propertiesof molecular solids that are determined by thephonon dynamics, such as Raman and infraredspectra, coherent and incoherent neutron scatter-ing, heat capacity, amplitude of vibrations, ther-mal expansion, and high-pressure Raman spectra.Using an explicit form for the intermolecular po-tential, it is possible to calculate not only lattice-dynamical properties but also the equilibrium crys-tal structure corresponding to various crystalphases and the lattice cohesive energy.

In a previous paper' it has been shown that anexplicit atom-atom potential is suitable for repro-ducing Raman and infrared frequencies, and- unit-cell parameters of n- and y-N~ crystals, cohesiveenergy of n-N~, and virial coefficients for gaseousN, which are in agreement with experimental re-sults. It has been shown that special care must betaken in the minimization of the crystal energy withrespect to structural parameters in order to avoiderroneous results in the calculation of lattice fre-quencies. The effect of zero-point energy andconvergence of the lattice sums of the calculatedstructure and lattice frequencies, were also inves-tigated. It is the aim of the present paper to ex-tend these studies to properties which depend onthe detailed dynamical spectrum through theBrillouin zone, such as mean-square amplitudesof vibration, heat capacity, dispersion curves,density of lattice modes, and thermal expansion.The effect of pressure on some of these propertieswill also be examined.

II. LATTICE-DYNAMICS CALCULATION

The 3or vibrational frequencies &u&(k) of a crys-tal containing 0 molecules in the unit cell, each

g 8 jk t:R(l )-R(l ')] (2)

where M, and M, . denote masses, l and /' label theunit cells (l = 1. ~ ~ N), and R(l ) denotes the positionof the unit cell relative to the origin. The forceconstants are given

tt„„,(lst, l s t ) —(l ) (P, ,

)

M„(let) denotes the displacement in the X directionof the atom t in molecule s located at unit cell /.The static interaction potential 4, is a sum of in-termolecular (C t,t„) and intramolecular (tf „„,)terms:

@g @inter+ @intra (4)

No assumptions about the rigidity of the moleculesare made. The intermolecular potential is given

oi'inter 2 inter(Dst, s't') y (5)

sf s't' l'

where V„„,(D, ", ,, t ) is the assumed atom-atom in-teraction potential between atoms t belonging to amolecule s located in a central unit cell (l =0) andatom t' of the molecule s', located in unit cell l'.Do,"„,, denotes the corresponding atom-atom dis-tance. The intramolecular yotential is

having v atoms, are obtained for a given value ofthe wave vector k, by solving the secular equations

ID „(ss'lf~'Ik) ~i(k)~11 ~tt I=0, (1)

where & and &' denote Cartesian components, s ands' label the molecules in the unit cell (s = 1. ~ ~ o),and t and t' denote the atoms (f = 1 v). Thebranch index j ranges from 1 to Sgv. The dynam-ical matrix has elements given

1D„,1,.(ss'

Itt'

I k) =(~ g i'„„,(1st, l's't')t' 1 ~ nt

LATTICE DYNAMICS OF SOLID n- AND y-N, CRYSTALS. . .

where Do„'o„. denote intramolecular atom-atom dis-tances. The self-term in the dynamical matrix iscalculated using the constraint of invariance of thepotential energy with respect to an over-all trans-lation of the particles in the system, ' i.e. ,

@„g (fsf, fst) = — Q P„„,(fst, 1's't').E'a't'Atilt

The constraints which have to be imposed on thedynamical matrix due to the lattice symmetry andthe invariance of the lattice sums with respect tothe choice of the origin, are automatically ac-counted for in our computational scheme owing tothe explicit calculation of the force constants fromthe interaction potential.

In the computational scheme employed the calcu-lation of the lattice dynamics is carried out in thefollowing way: An initial crystal configuration isgenerated by assuming the values of the unit cellparameters a, 5, c, n, p, and y and the positionsof the v atoms in one of the o molecules in thecentral unit cell. The positions of the rest of theatoms in the crystal are generated by applyingsymmetry operations of an assumed space groupto this initial configuration. The positions ofthe av. atoms in the unit cell and the values ofthe unit-cell parameters at static equilibrium areobtained from the solution of the set of equations

7; 4, =0, x=1 o'w, (8a)

V- 4 =0 P=1- 6 (8b)

where r„ is the position vector of atom x in the

unit cell while the R~ (p = 1 ~ ~ 6) are the 6 unit-cellparameters. This set of equations is solved itera-tively using steepest-descent and Newton-Baphsonminimization techniques without imposing any sym-metry .restrictions on the minimization path. Thelattice sums in Eqs. (5) and (6) are examined to beconvergent (interactions are summed up to 25 A).The dynamical eigenvalue problem [Eq. (1)] issolved at the unit-nell parameters and atomic po-sitions which satisfy Eq. (8). Since the crystalinteraction potential 4 [Eqs. (4)-(6)] is stronglyanharmonic, the elements of the dynamical matrixgiven in Eq. (3) depend on both the unit-cell pa-rameters (which determine the volume per mole-cule) and the atomic positions inside the unit cell.Relaxation of all the forces and torques exerted onthe molecules [Eq. (Ba)] is necessary for obtainingthe lattice frequencies ~~(k) in a way which is con-sistent with the potential adopted. In simple atomiccrystals (e.g. , fcc, bcc) the symmetry of the lat-tice already assures the vanishing of forces on theatoms.

The effect of zero-point energy on the latticeconformation is introduced in the second step ofthe calculation where we minimize the total inter-action potential 4„, rather than the static part 4, :

Cq, ~ =4, +C, .The minimizing conditions are given

(10a)

(10b)

&r @'~0~ =Oy

&R,@'~.~ = 0.

4„denotes the zero-point energy, calculated fromthe density of states D(a&):

00

O'„=— D(co) dr@.. o

(10c)

The density of states is obtained (see below) fromthe eigenvalues &u&(k) calculated in the previousstep, using the interpolation scheme of Gilat andRaubenheimer. A channel width of 0. 25 cm ~ isemployed and 800-k-grid points are used. At theconformation obtained after solving Eqs. (10), werepeat the dynamical calculation. This cycle isrepeated until the conformation and lattice spectrumare stabilized. The calculated dynamical proper-ties reported here for the zero-pressure Q.-N3phase, correspond to this final step in the compu-tation scheme. The calculations on the high-pres-sure y-Ka phase and on the e-Nz at-P10 are per-.formed by solving Eqs. (8a) and (10a) for a givenmolar volume. The value of —VR C„, is now re-lated to the external pressure that corresponds tothese volumes at a dynamical equilibrium.

In the previous work5 the intermolecular poten-tial Eq. (5) was approximated by a Lennard-Jonestype of interaction:

l/2 6-o, r'

1'&.&.,(D.g„~ ) —& Do, i— —

Do. r ~ (11)SC&8't~ stg8't'

The parameters a and & have been fitted througha least-square procedure to reproduce the experi-mental unit-cell dimensions, 9'o the cohesive en-ergy, ""and the infrared"'" and Raman" ' fre-quencies of u-Nz at k = 0. This yielded

0=3.30 A,

e = 0.30 kcal/mole,

where the intramolecular bond length was keptfixed at the experimental value of 1.OS8 A..

For the intramolecular interaction [Eq. (6)], wehave adopted the form suggested by Levine fromthe vibrational analysis of isolated Na molecule, '8

V (d) =D [1-dJd] [e-'""~4']' (l2)

where do is the intramolecular bond length, d=D, q ]„D~ is the molecular dissociation energy,and P and P are parameters given by Levine. Theexact form of the intramolecular potential does notaffect the lattice modes because of the large en-ergy separation between internal and externalmodes (the lowest molecular frequency is around2300 cm ' and the highest lattice frequency is 110cm~).

5880 g UNQEB

&I& D&S&ERSON CURVES

0Q

0

C4XXX X X X XX 0 X

0~pf

QS~pH'U

~ %

0O0

Of

o&

0~A

Q0'Q

0O

cd0C

Dispersion cur for &- a and y-N& were cae uations at 3osolving the dynamica equaculated by so ~

(BZ) along each of the~ .

n the Brillouin zonepoints ind. sion curves fordirections

lecules per unitonside red.orou Pa3, four mo e

t cel]. dimensions1at the ucell) are g "'1'b ium under normalcorresponding "o

3) red;cted by theto the equi i rium

pressure (mo~red with the

molar volume 27 cm prcomemploy p . l stic neutron scatteringntal coherent ine as ic "experxme

T ble l) The calculatedof Kjems and Dollingd greementenerally in goodispersion curves are gent betweenved values. The agreemewith the obser

d h hest optical-ntal and calculate lgthe expe»men. . t l) suggest that

~ lo itudinal op icamode frequencies ong'

e is reasonableroximation used here is rthe rigid-ion approx'not too im-olarization effects are noand electronic polari

ith the situationis is to be contrasted wi es where rigid-ion m e

l t}1

fre uency of t e igIn Fig. 2 the dispersion curv

olar volumes of 24. 1 andi1 direction and for mo ar vo um

27. 2 cms~ are g ven. T e exp

60

CL+f4+ INCFCL Q CL

0S

SQl04

CL)0 C

cd

—0 ~0

IV

0~r4K

SC4N

O

O0&:.3CTCO

50

120

90

60

(b)

00 05 I.O

tLj Q Q)—t +{,mg) Aouenbaag

OC0

Wave veQof

for n-N . (a) Molar volumeDis ersion curves for n-3I)"'1

1 .427. 2 cm3 (zero pressure;(4 kbars).

LATTICE DYNAMICS OF SOLID a- AND y-Ng CRYSTA LS. . .

TABLE I. Comparison between calculated and measured inelastic-neutron-scattering (Ref. 15) lattice frequencies at high-symmetry points in the Brillouinzone. The values in parenthesis correspond to the optical measurements (Refs.13-16). Values given in cm ~.

Calc.

I' Point

Obs.

M Point

Obs.

R Point

Calc.

X Point

Obs,

69. 547. 551, 847. 043. 738. 534, 5

69.60 (70. 0)60. 0 (60. 0)54. 2048. 56 (36.0)46. 9236.42 (36.0)32.37 (31.5)

61.2

53.742. 934. 724. 6

62, 7

55.046. 938.027. 9

68.4

42. 040. 832. 932. 2

68. 7

47. 343. 734. 833.9

60. 956.149.340. 840. 536.736.526. 524. 01

38.0

27. 5

phase transition was observed' '3 at a volume of24. 1 cm at 20. 5 'K. '

It can be seen that the dispersion of the curves isonly slightly affected by increasing the pressure.The frequencies of the optical branches are shiftedquite homogeneously to higher values, while thehighest acoustic mode is shifted strongly only fork & 0. 5 and much less for higher k-values. Thedispersion curves obtained in our calculations aresimilar in shape to those obtained by Ron andSchnepp using a different 6-12 potential, and tothose obtained by Raich et al. using a self-con-sistent phonon treatment. One significant differ-ence between our results and those of Ron andSchnepp is the removal of the accidental degeneracyat the R points in the present calculation. On theother hand, our calculations and those of Ron andSchnepp2~ and Raich et aE. 23 differ significantly fromthe recent calculations of Kjems and Dolling. 'The erratic behavior of the dispersion curves ob-tained by the latter probably result from the inad-equacy of the renormalization between translationaland librational modes at general k points.

Dispersion curves for y-N2 (space group D4„, twomolecules per unit cell) at the unit-cell parametersa=b=3. 94 A, e=5.08 A corresponding to the cal-culated equilibrium structure under an externalpressure of 4015 atm (experimental data' a=5= 3.957 A, e = 5. 109 A under the same pressure)are shown in Fig. 3 along several directions. [Inour previous work (Ref. 5), we have erroneouslyinterchanged the assignment of the B,~ and A~librational modes of a-Nm (Table VIII therein). Thecorrect order is revealed in Fig. 3 here. We aregrateful to Professor J. C. Raich for bringing thispoint to our attention. ] The assignment of thebranches were made by studying the transformationproperties of the eigenvectors and using the non-crossing rule for branches of the same symmetry.Up to the present time, no experimental neutron-

diffraction data is available for comparison withthe calculated results for y-N2. The agreementbetween calculated and optically observed frequen-cies at k=O is satisfactory. ~ Recently, Pawleyet aE. have reported a calculation on y-N3 at k=0using a e ~ potential. The results arequalitativelysimilar to our results, however better quantitativeagreement is obtained in the present calculationwith the 12-6 potentia, l.

The dispersion of the Griineisenparameters y~(k)

(-) d In~, (k)(13)din V

were calculated numerically from the dispersioncurves at several volumes. Some representativeresults obtained for the three acoustical modes andtwo of the optical modes (one a pure translationalmode and the other a pure rotational mode at k= 0)along the [ill], [110], and [100]directions in u-N3,are shown in Fig. 4. The acoustical nondegenerateAu mode has the highest Gruneisen parameter atk=0. Around k=0. 5 along the [ill] and [110]andaround k =0. 7 along the [100]direction, theGruneisen parameter drops strongly, indicatingthe softening of this mode. The other acoustical-mode parameters as well as the optical-mode pa-rameters exhibit low dispersion. The mode pa-rameters of branches that are degenerate at %=0and nondegenerate at higher k values, differ onlyslightly. The agreement between observed ' andcalculated Gruneisen mode parameters at k =0 isquite poor. The addition of interaction terms fall-ing with distance slower than the 12-6 potential(e. g. , quadrupole-quadrupole) would lower the cal-culated values 7 thus bringing them to better agree-ment with experiment.

IV. DENSITY QF STATES

The k-space interpolation scheme of Gilat andBaubenheimers was used to obtain the normalized

IOQ

~W

CQ

cfCP

o Q

NQ

Gh

cj

Q'aC

clQc5

obS

oCh

Ou Q

oo

Ia

oQ

Cb

CgII

II

f~1

I I

LIJI

I

.i o

Scf

II

cd

cj

So6

~ W

Q0o

'U

cd

II

gf

cd

NQ

q3o

cfQ

~W

No

oc8

QQ

( ( UJ0) AJUBilbp)g

Q

N

S

6cd

c8Q

'QQl

'e t+4

o

0cd

V

CD

o ~ aMQ

Q

e ee

oQ eel

m

Ch ~

I'I

C3

~l

ll

sa~sujo&Dd uasIaunsg

' Dci

I o

o Q)

N

Q m

gQfs

Q.Q

op

Q

8 'Q

CO

eQ4Q

~ a

oQ

Q

~w

Q)

M

. e5

Q

n- AND y-N3 CRYSTA LS. . .

6xiO'

I'

I

'I

0-N2 (ajV = 27.2cm5

4xlO '0-N2

V = 24.I cd

I, I I . I

(b)

C3

DU)

VP

2xIO

I xIO —~ ~

I

Y- N2

V = 24.lcm

II

II

5xlO

00 20 00 60 80 IOO l20

FIG. 5. Density of states of &- and y-N2. (a) O'-N2 atV=27. 2 cm3; Q) O.'-N2 at V=24. 1 cm3; (c) y-N2 at V=24. 1 cms.

density-of-lattice-states function D(v) for n-Nzand y-Nz at various volumes. The standard meth-od was used for a cubic a phase while suitablemodifications were introduced into the BZ samplingscheme for treating the tetragonal y phase. Stan-dard channel widths of 0.25 cm ' were employed.The convergence properties of the calculated den-sity of states were examined by requiring that thecalculated nth moment of the frequency (~") will bestable within a prescribed tolerance with respectto an increase in the k-grid mesh used for sampl-ing the BZ. Since lattice contributions to thermo-dynamic properties are calculated by performingsuitable ensemble averaging over the density ofstates, such a convergence cheek will define the

accuracy of the calculated thermodynamic prop-erties. We thus compute

(aP),, =j D„,(~)aFd~, (14)

where D~, (~) denotes the normalized density ofstates calculated with a k-grid mesh indicated bynk, (i =x,y, and z in the reciprocal-space unit-cellaxis) for —2 &~ &20. We require that the largestdifference ((~")»|—(~")»,)/(aP)»c (where 4k',denotes the highest k-grid mesh) be less than 1%for all moments. Around k=O a fixed grid of 190k points is employed to avoid numerical errors in-troduced by the interpolation scheme. For low-temperature crystal phases such as n- and y-N~(transition temperature to the disordered high-temperature P phase being 36. 5-50 'K for pressurelower than 6 atm) such a stability of the -2 to 20thmoments against an increase in the k-grid mesh,should suffice t;o assure good accuracy in the cal-culated lattice free energy, heat capacity, androot-mean-square amplitudes of vibrations. It wasobserved that the use of 1007 inequivalent %pointsin the BZ and a mesh of 190 points around thevicinity of k=O assures the convergence of theDebye temperature to within less than 10~% for thefirst moment and 10~% for the 20th moment. A

similar accuracy was obtained for the y-N2 den-sity of states. For smaller unit-cell volumes asomewhat lower accuracy was obtained for the highmoments, e.g. , 4. 6 x 10~% for the 20th moment ofo.-N2 'at unit-cell dimension of 5.40 A (correspond-ing to a pressure of 4 atm at T=0 'K). This ac-curacy is considered sufficient for our purpose,and subsequently this grid was used in further cal-culations. The density of states calcul. ated withconvergent k grids for n-N, and y-N2 are given inFig. 5 for several densities.

The highest-frequency peak in both a.- and y-N2is contributed by the high-energy modes that aretranslations at the I' point. The doublet in thea-N2 density of states at around 30-40 cm ' atzero pressure and at 55-60 cm-' Bt 4 kbars arecontributed by lattice modes that are pure rotationsat k=O. The increase in pressure tends to shiftthe whole spectrum almost uniformly to Nigherfrequencies although the low-frequency part isslightly less shifted.

It has been previously commented by severalauthors that it is desirable to obtain an effectiveDebye temperature from a detailed real-solid den-sity-of-states calculations in order to facilitatecalculations of properties such as the Mossbauerrecoiless fraction 3 and heat capacity. Figure6 reveals the dependence of the Debye temperatureGL)"' calculated from the nth moment of the fre-quency, on n.

5884 EDUARDQ HU LER AND A I EX ZUNGER 12

8'"'=(h/k )[-'(n+3)(uP)]'~", n &-3 nvO. (16)

GD"' is the cutoff temperature of a Debye distribu-tion having the same nth moment as the real D(e).Similar plots for rare-gas solids'~ indicate a vari-ation of less than 2% in Gn'"' in the range —2 &n & 4.In a molecular solid such as a-N~ the large num-ber of optical branches introduce strong deviationsfrom a Debye spectrum, resulting in a rapid vari-ation of ~"' for low e.

This "fitting" of an actual density of state to aDebye model in molecular crystals having- a rela-tively large number of optical branches requiressome comments. It can be easily seen that the"normalized" nth moment in the Qebye model witha characteristic cutoff frequency u~' is

20

l5

C=

O33ca

IO

(&n)/((ua)"= 3(n+3)(~)", (u)n) = s(n+ 3) (un (16)

and does not depend neither on the crystal spacegroup nor on the crystal density. These simpli-fications, inherent in the Debye model, are notsatisfied in actual calculations for molecular solidswhere a large part of the density of states is con-tributed by the optical branches that have an in-volved frequency dependence. Furthermore, the

fitting of an actual density of states to a Debye den-sity of states by minimizing the sum of differences

5 IO l5

I

20n

FlG. 7. Normalized moment ratio for the exact andDebye model calculations a-N2, a=5. 40 A; o.-N~, a

0 0=5.65 A, c.'-N&, a=5. 67 A; V-N2, a =b=5. 94 A, c=5.10

is divergent as n increases. This can be viewedfrom Fig. 7 where the ratio ((~")/(~')"/((&n)/

I

4I

l4I

200

FIG. 6. variation of 8D" calculated from the nth mo-ment of the real density of states D(~) for e-N2.

(&n)") vs n is plotted for several volumes of u-Nsand for y-N2. It is seen that while for the lowestvalues of n (which give more weight to the densityat low frequencies) the ratio is close to unity, itdiverges strongly for large n, the deviation in-creasing with decreasing volume. It is thereforeevident that in calculating thermodynamic prop-erties of molecular solids such as root-mean-square amplitude, zero-point energy, specificheat, etc. , by expanding the corresponding integralover the density of states in gower series of thevarious frequency momentss 'b' one should avoidusing a Debye density of states that strongly under-estimates high-frequency moments.

The calculations presented here also suggest thatthe Debye model might be used to determine prop-erties that depend on low and positive moments offrequency such as the high-temperature limit of thespecific heat, while it should be less adequate forcalculating the low-temperature heat capacity andthe high-temperature root-mean- square amplitudesthat depend on the negative frequency moments.The approximation made in the Debye model thatthe solid is an elastic continuum is thus unjustifiedfor solid properties that are related to high mo-monts where the discrete structure of the solidshould be explicitly considered.

To compare our calculated values of OD with ex-

LATTICE DYNAMICS OZ SOLID n- AND y-N, CRYSTALS. . . sass

periments the most common choice of Domb andSalter e» =(~ ) is employed.

Experimental determination of 8~ for the nphase are: 68 'K at T= 20 K, 73 'K at'4 10 K and80.6' at the lowest temperature measured T=4. 2= 4. 2 'K. ' Our calculation yields 8~ = 84. 4 K atthe unit-cell volume corresponding to equilibriumat 7. = 0 K in good agreement with the value ofBagatoskii et al. Using the linear expansion co-efficient determined experimentally by Bolz etal. , the calculated Q~ at the unit-cell volume cor-responding to 20 'K is 75. 3 'K, in reasonableagreement with the experimental data. '

Schuch and Mills' have estimated OD in comput-ing the experimental Debye-Wailer factor for y-N2,using the de Boer reduced equation of state' as aninterpolation scheme, obtaining OD = 94 'K. Ourcalculated value of 125.0 'K is considerably higherand provides a better approximation. A lowerDebye-Wailer factor is thus predicted by the pres-ent calculation. It should be noted that our calcu-lation of the volume dependence of the Debye tem-perature QD(V) predicts similar values for the nand the y phases at the same volume, in agree-ment with the approximation made by Raich'~ incalculating n-to-y phase transition.

V. THERMAL EXPANSION

The density of states D(~) is used to compute thethermal-expansion coefficient of n-N~. This isdone by minimizing the total crystal free energyE(a, T) (where a denotes the unit-cell parameter),for various values of (a, 7). The free energy isgiven

E(a, T) = 4,(a)+ ~As D((u, a)(ed(u0

D(u, a) ln(1- e ""~ r)dv. (17)

4,(a) is the static energy calculated by performingthe appropriate lattice sums on the pair potentialV(r,.&) [Eqs. (4)-(6)]. It has been previously shown'

that summation of the contribution of 5' unit cellsrelative to a central, unit cell, suffices to assurethe convergence of C,(a) to within less than 0. I%%uq.

The second and third terms in Eq. (1V) denote thezero-point energy and the lattice thermal energy,respectively. The contribution of the intramolec-ular modes to these terms is neglected since thesemodes (of frequency 2300 cm ' for a-N~) are onlyslightly populated below the~a-j8 transition temper-ature and thus contribute negligibly to the thermalexpansion. The last two terms in Eq. (1V) arecalculated numerically from the density of states.Minimization of the static energy alone with re-spect, to the unit-cell volume has been shown toyield a unit-cell parameter of a= 5. 63 A while ad-

I.OI2

I.OI 0

0aC)

IQ06

I.004

I.002

l.0000 IO . , 20 : 40

T. ('K j

FIG. 8. Calculated temperature dependence of the 0.-N2 unit-cell parameter.

dition of the zero-point-energy term to the mini-mization procedure was shown to increase thisvalue to 5. 66 A. ' Zero-point-energy contribu-tions have been calculated self-consistently usingD(~, a) at the unit-cell parameter that minimizesthe first two terms in Eg. (17). Minimization ofE(a, T) yields the function a(T). The results areplotted in Fig. 6 in units of ao=—a(T=O). Fittingto the relation a/ao = 1+re yields an average linearthermal-expansion coefficient of q- 2. Sx10- 'K-'for 0 & 7.'& 20'. This agrees reasonably well withthe rough experimental estimation of g-2x10 4

oK-1 for 4 2&T&20It should be noted that the zero-point-energy

term could be calculated to a very good approxi-mation from the k=O mode frequencies without thedetailed knowledge of the density of states. ' Thiscould be done by using an Einstein model for eachof the optical branches with its appropriate fre-quency at k=O, and a Debye model for the acous-tic branches using an acoustic Debye temperaturethat is determined from the lowest optical frequen-cy. This yieMs a zero-point energy of 0.4138 and

0.3127 kcal/mole for n N~ at unit-ce-ll volumes of41 A~ and 44. 6 As/molecule, respectively, com-pared with 0.4097 and 0.3112 kcal/mole obtainedfrom a direct evaluation from the density of statesusing a low grid mesh of 226 points, and 0.4086and 0.3124 kcal/mole obtained with a higher gridmesh of SV1 points. Since the evaluation of D(~)for polyatomic molecular crystals is usually dif-ficult to implement in practice„such an approxi-mation to the zero-point energy and its volume de-pendence could be useful in calculating cohesiveenergies, conformations at dynamical equilibrium,and low temper atures.

5886 EDUARDO HU LER AND ALEX ZUNGER 12

aE

0

4)

OCL

2

0 20 30 40 50

V('K)

FIG. 9. Heat capacity. of e- and p-N2. The n-N2 val-ues are calculated at the equilibrium zero-pressure con-formation {V=27.2 cm ), and the y-N2 values are calcu-lated at equilibrium under a pressure of 4 kbars {V=24.1cm3). Points indicate the experimental data of Giauqueand Clayton {Ref. 11) for ~-N2.

VI. HEAT CAPACITY

Next, we turn to the calculation of the latticecontribution to the heat capacity of n- and y-N~.The heat capacity at constant pressure is

C (T)=9Nof C((o)C((a, T)d(a+99BTT0

where C(~, T) is the heat capacity of a harmonicoscillator having a frequency (d, N0 is Avogadro'snumber, 5%0 is the total number of degrees offreedom (two librations and three translations foreach molecule), 8 is the bulk modulus, and V is themolar volume. Using the experimental estimate3to 8 (10' -10"erg. cm ~) with the experimentallinear-expansion coefficient g and molar volumeV, the second term in Eq. (13) is estimated to besmall relative to the first term, and consequentlycan be neglected. The thermal-expansion effectson D(&o) are likewise small and may be safelyneglected. The calculated heat capacity as a func-tion of temperature of n-N& at zero pressure, andof y-N2 at a molar volume of 24. 1 cm correspond-ing to a pressure of 4 kbars, are shown on Fig. 9,together with the experimental points of Giauqueet al. i' for n-N2 above 16'K. The calculated andexperimental results agree fairly well below 27 'K.Above this temperature the agreement becomesworse owing to the increase in lattice anharmonicityaccompanying the a-to P phase transition at 35.6'K. The y-N~ heat capacity is considerably lower

than the corresponding Q. -N2 values owing to itshigher density of states at the high-frequency partof the spectrum that is relatively unpopulated atlow temperatures. Presently, no experimentaldata on the y-N~ heat capacity and on the low-tem-perature n-N& heat capacity, are available.

VII. AMPLITUDE OF VIBRATION

The amplitudes of vibrations of n-N~ were cal-culated directly from the displacement-correlationfunction

(u„u„) = (u„u, ) = (u,u, ),(u„u„) =(u,u, ) =(u,u, ).

(20)

The values of (u,.uj) at different temperatures aregiven in columns 2 and 3 of Table II. Column 4shows the components of the amplitude of vibrationalong the molecular axis, which is given by

(u) ) = (u) ) + 2uga)

column 5 gives the component of the displacementalong each one of the two principal axis perpendic-ular to the molecular axis

(u, ) = (u„—u„)'i'.It is difficult to estimate how much of the ther-

mal motion is due to rotation of the molecule and howmuch is due to the translation of its center of mass.However, assuming, as did La Placa and Hamil-ton, ' that the translational motion is isotropic,

TABI.K II. Thermal parameters, root-mean. -squaredisplacements, and estimate of amplitude of libration ofn-N2.

T{'K)

4 K10'K15 'K20'K25'K36'K

(u2;)(A&)

0.03150, 04090.05140, 06140.07260.1020

—0. 0080—0. 0124—0. 0150—0.0190—0.0230—0.0301

0.1240.1290.1460.1530.1630.204

(ut)g)

0.1980.2300.2570.2830.3090.360

{deg)

17.721.824. 327 330.134. 4

(u), (lt)u), .(lt')) = g) ~ e),(t I kj)t t'

~ J

x e„,(f' I kj) cothk&u, .(k)/2ksT, (19)

using the eigenvectors e(tl k, j) of the lattice fre-quencies calculated at N points in the Brillouinzone. The summation over j in Eq. (19) is per-formed over all the branches, and the acousticalmodes at k=0 are excluded. A dense sampling ofpoints in the Brillouin zone was taken so as toassure the convergence of the sum in Eq. (6). Theresults of the calculations using Eq. (19) satisfythe relation

LATTICE DYNAMICS OF SOLID a- AND y-Na CRYSTA LS. . .

u, can be associated with the translation, and

(8) = 2(v'SU„/f), where f is the length of the Nzmolecule, represents the root-mean-square angu-lar displacement (in radians) for each one of thetwo librational degrees of freedom.

The values for librational motion given in TableII are in agreement with the x-ray results of LaPlaca and Hamilton, who reported a value of 17'at 20'K.

In conclusion, our results for the mean-square-amplitude displacements confirm the approximatedcalculation of Cahill and LeRoi and Goodings andHenkelman, and point out that anharmonicitycouM be important in the librational motions ofo.-N~.

VIII. COMPARISON PATH SELF-CONSISTENT PHONONTREATMENT

Raich et a/. have recently performed a self-consistent phonon (SCP) treatment44 of solid u-N~using a different 12-6 potential. As a comparisonbetween the SCP and the quasiharmonic (QH) re-sults the above-mentioned authors presented thelattice frequencies at k= 0 as computed by both-methods using the same potential, however theexperimental unit-cell dimensions were used forthe QH treatment (a = 5. 649 A); while for the SCPtreatment the predicted theoretical value (a= 5. '714

A), was employed. Previous SCP calculations wewere performed only on atomic crystals. %'e thusbriefly comment on the relation between QH andSCP results for the molecular solid studied hereat T=O'K.

The basic methodology used in SCP theory is toincorporate anharmonic effects to first order byreplacing the QH force constants of Eq. (18) by anappropriate ensemble average over a function re-lated to the displacement-correlation tensor. Onthe other hand, anharmonic effects are partiallyintroduced in the QH scheme by calculating thelattice-dynamical properties at the atomic positionsthat minimize the frequency-dependent free energy.At T= 0 'K, this reduces to the modification in thelattice frequencies owing to the changes in atomicpositions induced by zero-point effects. Thus inthe QH approach one assumes that the collectiveeffect of all lattice mode is to renormalize thedynamical variables generating thereby a new setof fixed atomic positions at which the force can-stants are to be evaluated; in the SCP method, anensemble average is used. Since the ensembleaverage yields a larger force constant than thediscrete force constant obtained at a given unit-cell parameter a, one gets ~~~op(a) &~)"(a) for agiven value of a." On the other hand, the equilib-rium lattice parameter aacp calculated by minimiz-ing the SCP free energy, is usually larger than thecorresponding value ao„calculated in the QH ap-

TABI,K III. Comparison between self-consistent pho-non and quasiharmonic frequencies each calculated at theminimum of the corresponding free energy at O'K. Val-ues given in cm ~.

Mode

TuEuTfTgA„Tg

~s&eP (asc

70. 152. 047. 648. 844. 340. 836.7

cuj~" (a~„)

70. 853.247. 847. 844. 139.335. 2

proximation. 48 This tends to decrease the ~u& (a~cd, )relative to &a~@"(a~„). To investigate in more de-tail these conflicting effects we repeated our QHcalculation using the potential employed by Raichet al. ~3 A careful minimization of the total latticeenergy at T =0' yielded az„= 5. 62 A as comparedwith as~~ = 5. V14 A. The frequencies at k=0 ob-tained by us at this a~„value are compared withthe frequencies obtained by Raich et a/. in a SCPtreatment at a«p in Table III. The weighted devia-tion

~QH ~SOP

The previously published, interaction potentialbetween Na molecules was used to compute thelattice-mode frequencies of n- and y-N3 crystalsat the equilibrium structure corresponding tovarious pressures. Dispersion curves as well asdensity of lattice modes are given for both phases.These are used to compute the lattice heat capacity,root-mean-square amplitudes of vibrations, dis-persion of Gruneisen mode parameters, linear

is only 0. 95 cm-". 47 The dispersion curves calcu-lated by the QH method have exactly the sameshape as those obtained by SCP calculation, sincethe averaging procedure used by the latter methodremoves almost completely all the k dependence ofthe anharmonic corrections. Similarly, the bind-ing energy calculated by the SCP method is 1.58kcal/mole, while that calculated by the QH methodis 1.56 kcal/mole.

Vfe thus conclude that if one follows consistentlythe QH scheme, reasonably accurate frequenciescan be obtained owing to the nearly cancelling ef-fects of the increase in the lattice frequencies atfixed volume and the increase in equilibrium vol-ume in going from QH to SCP approximation. Formore accurate determination of the lattice param-eter at equilibrium and behavior at higher temper-atures, SCP methods can not be avoided.

IX. SUMMARY

5888 EDUARDO HU LER AND A LEX ZUNGER

thermal-expansion coefficients, and the Debyetemperatures. Reasonable agreement is obtainedwith experimental data whenever experimental dataare available. More intensive examination of theproposed calculations would be possible when moreexperimental data such as neutron scattering (co-herent elastic and incoherent inelastic) and crystal-lographically determined amplitude of vibrationwould be available.

The calculation of the dynamical properties ofmolecular crystals do not pose any serious diffi-culty relative to the more familiar atomic crystalscalculations. The marked differences in the cal-culation schemes are the following.

(a) In calculations on molecular crystals of ar-bitrary conformation, one should take special carewith the problem of relaxing all forces and torquesexerted by the crystal field on the molecules. Be-cause of the anharmonicity of the crystal potential,residual forces and torques have a marked effecton the calculated lattice frequencies.

(b) Because of the presence of a relatively largenumber of optical modes in the molecular crystallattice spectrum, the lattice zero-point energycontributes a significant part to the free energy atlow temperatures. Optimization of the crystalstructure for a given interaction potential, should

take zero-point effect into account. Because of thehigh ratio between the number of optical to acout-tical branches characterizing molecular crystals,the lattice spectrum deviates significantly from aDebye-type spectrum. A Debye model is thus inad-dequate for computing thermodynamical propertiesthat involve high-frequency moments.

In view of the practical possibility of computinglattice properties for a given molecular potentiaI,it seems that such calculation should provide agood test to the quality of the interaction potential.Substantial effort has been lately devoted to thecalculation of interaction potential from first prin-ciples, ~ and from combinations of first principleand empirical data. ' The quality of these poten-tials has been examined by calculating virial coef-ficients and viscosity ' or lattice cohesive ener-gies. ' Vfe believe that calculations of lattice-dynamical properties in the corresponding crystalswould provide a much better examination of theanisotropy and detailed characteristics of such in-

teractionn

potentials. '~

ACKNOWLEDGMENTS

The authors are grateful to Professor G. Gilatand Professor M. l. Klein for helpful discussionsand comments.

*Present address: Dept. of physics, Northwestern Uni-versity, Evanston, Ill. 60201.

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12 LA TT ICE DYNAMICS OF SO LID n- AND y-N~ CBYSTA I S. . . 5889

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