On the validity of the continuum description of molecular crystals

lnt J Engng Sci Vol. 19. No. 12. pp. 176~-1773, 1981 Printed in Great Britain

(~)20-7225/81/121765~9950~0~J/0 r~ 1981 Pergamon Pres, IId

ON THE VALIDITY OF THE CONTINUUM DESCRIPTION OF MOLECULAR CRYSTALS INGA FISCHER-HJALMARS Institute of TheoreticalPhysics, Universityof Stockholm,S-11346 Stockholm,Sweden model proposed by Askar for coupled rotation--displacement motion in certain molecular crystals is improvedin realismand studied in moredetail. Analyticallattice wave solutionsare obtained in both long wave and short wave limits of the first Brillouin zone, and numericalsolutions are givenfor the intermediate region. Continuumequations are derived,and their validityinvestigatedboth analyticallyand numerically.It is found that these equations are valid not only in the longwavethird of the Brillouin zone.a third usually assigned to the proper macro-continuum,but also in the next, middle third, where they thus give a pseudo-continuumdescription of the crystal. In the last third, however, the continuum description fails completely. Abstract--A

1. INTRODUCTION AND SUMMARY IN Srt~ARCHfor materials with properties to be expected for micropolar media, it has been suggested that certain molecular crystals may have such features. This proposition has been investigated by various authors. Askar[l,2] has made a particularly detailed study of lhe possibility that internal microrotational modes may be coupled to displacement modes. The present notes will be concerned with Askar's model and will develop some further aspects on it. In Section 2 the discrete lattice model is applied to the KNO3 crystal in a manner similar to the one suggested by Askar[l]. However, the model is made more physically realistic by allowing the directions of the bonds at the K + ion to move freely, as they should. Furthermore, a geometrical simplification is introduced which is not quantitatively essential, but makes it possible to find analytical solutions of the equations of motion, in both long wave and short wave limits. These asymptotic solutions facilitate a general discussion of the dispersion curves. It is found, as also shown numerically by Askar[l], that within a certain wavelength range a strong coupling exists between the displacement (u)-rotation (~), so called u~-mode, and the pure microrotational ~b-mode. After transformation of the displacement variables to internal and center of mass co¢~rdinates, pseudo-continuum equations are derived in Section 3. Analytical solutions of these equations are compared to those of the lattice model. Following Askar[1,2] we have in Section 4 assigned numerical values to the various parameters of the model. It is then possible to make a more detailed comparison between tae lattice and pseudo-continuum solutions and to find the range of validity of the latter. The result of this comparison is that within the long wave third of the first Brillouin zoae both frequency values and amplitude relations of the two models are very similar. This is what should be expected, since this frequency range usually is assumed to correspond to the proper macro-continuum. However, also within the middle third of the Brillouin zone the results from the pseudo-continuum model are in fair agreement with the lattice wave solutions. This is mcsI satisfactory, since in our model the strong coupling between the displacement u~-mode and tie microrotational ~-mode occurs within just this part of the Brillouin zone. The pseudocontinuum description of the crystal can, therefore, be used with reasonable confidence f,)r detailed study of the region of avoided crossing of the dispersion curves. In the short wave third of the Brillouin zone, as expected, the continuum description fails completely f)r displacement modes. 2. DISCRETE LATTICE MOTION 2.1 The model Consider the KNO3 crystal in the phase with aragonite structure. The unit cell, containi~lg four molecules, is orthorhombic. Therefore, waves along the three orthogonal crystals axes are uncoupled. The molecule consists of two ions, K + and NO~. The negative ion is planar, the O atoras 1765

1166

I. FISCHER-HJALMARS

forming an equilateral triangle with the N atom at the center. The NOT-planes are parallel with the plane spanned by two of the crystal axes, x and y say. The detailed arrangement of the ions is described in Refs. [2,3]. For the present purpose we shall assume a simplified structure with the ions alternating and equidistance along an axis in the z-direction (Askar’s L, = Lz = a). Like Askar we shall study a transversal displacement mode in the z-direction and its coupling with a rotational mode. The l-dimensional model lattice is outlined in Fig. 1. The repeat distance in the r-direction is 2a. We assume the ions to become displaced in the x-direction and the NOT-ions to rotate around the y-axis. Such a rotation will give rise to a transversal force on K+ and, correspondingly, a displacement of K’ will give rise to a torque on NO;. In Askar’s model this coupling is represented by the bending of a flexible bar. Let K be the bending constant. In addition there is as usual the central interaction between the ions, represented by a spring with a stretching force constant, C say, cf. e.g. Ref. [4]. Let the NO;-ions (type A ions) have the mass MAand the moment of inertia J and the K+-ions (type B) the mass MB.The motion can be described in terms of transversal displacements f(“) (ion type A, number n) and g(“) (type B, number n) and rotations. The bars will be rigidly attached to the ions of type A, and the rotation angle 1+9 of the end of the bar will be the same as the rotation angle of the A ion. Since the ions of type B have no extension, it is not possible to define any rotation angle of this ion. This means that the ends of the bars will rotate freely at all points B, so that the angle of the 1.h.s. bar end, pj, will not be the same as the angle of the r.h.s. end, 9,. In the point 1 the transversal forces Tkland moments Mklfrom ion 1 acting on ion k will be T,* = K(f(‘)- gc2’)+ ;nK(g”’ + &‘), Tlo = K(g”’ -

f"') t ;oK(IL’” t cp;“);

(2.1)

M,2 = ;aK(f”’ - gc2’)+ &‘K(Z$“’ + (pj*‘), Ml0 = ;oK(g’” -f(i)) + ;n2K(24”’ t q;“‘).

(2.2)

In point 2 we have T23= K(gc2’- f”‘) + ;aK(p’:) + $‘3’), 1 T2, = KCf”’ - gc2’)t -aK(qj*’ t t+b”‘); 2

(2.3)

M23= ;oK(gc2) - fc3’)t +‘K(Zrp;*’ t $“‘), M,, = +K(f”

- gc2’)+ ;a2K(2rpj2’ + IL”‘).

Fig. 1. One-dimensional model lattice. Rigid NOT-ions (A) with finite extension, bass MA,and moment of inertia. J. K+-ions (B) are points with mass MB, but without moment of inertia. Non-central interaction between the ions A, B represented by bars, rigidly connected to ions A but freely rotating around ions B. Central interactions represented by springs.

(2.4)

Validityof the continuumdescriptionof molecularcrystals

i767

Since the ion number 2 is of type B and thus without extension, the moments M23 and M2~ must vanish. We can. therefore, use (2.4) to eliminate ~ and {Pr from (2.1)-(2.3). We obtain in point 1

TI2= ~K(f'"- g'2')+~oK~ ''~, T,o= ~K(g'°'- f'") + ~aKO'";

,2:.5)

MI2 = Jag(f'"-g~2))+~a Kt~Ill , 1

2 -

M,o= ~aK(g'°'- f'") + ~a2K~,b'";

,:.6,

and in point 2

1 {21 _ fo~) + ]aKOO,, T> = 71K(g

T2, = ~K(f"'- f2') +~aKOm 2.2

(2.7)

Equations of motion The equations of motion are

MAi''=

12.8)

MBff,(2'= ( C +}K )(f°' +fm- 2g'2))-~aK( O°~- ~lll),

JO;~l)=~aK(g 121-g'~)

--

] 2 -K~ ~1} , ~a

(29)

(2.101

With the assumption of travelling waves (f, g, g,) = (F, G, ",P')exp

[i(qz -

rot)]

we obtain

1 [2(C + ~K )- Maw2]F- 2(C +-~K ) cos (qa)G = O,

,*(C +JK)cos(qa)F + [2(C +~K)- MBw2]G+ iSaKsln"(qa)q~=O, , -~aKsin(qa)G+

2.3

a2K-jo 2)~ = 0 .

(2.11t

Solution in the long wavelengthlimit (qa small)

After series expansion of cos (qa) and discarding higher orders of qa the eqns (2.11) can be solved analytically. The highest w-value corresponds to an internal motion, often called tl':e optical mode. In this mode the NO3-ion and the center of mass of the two adjacent K+-ions move to and from each other. Therefore, we prefer to call this mode an internal vibration. The motion can be described by an internal coordinate ( scm=f m -~tgl",ol + g12~),

(2.12}

1768

I. FISCHER-HJAI_MARS

as will be further discussed in Section 3. Introducing the total mass M and the reduced mass m

M = MA + MB, m = MAMB/M,

(2.13)

we have

to~ = (2C + ~K )(m-' - q2a2M-') + O(q4a4), F : G = - MB" Ma + O(qZa2),

iqG'*=-(2C+~K)/(mR)+

1 + O(q2a2);

(2.14)

where

R = a2K/(2J).

(2.15)

The frequency tot, as well as the amplitude relation F : G are the same within the given accuracy as obtained without internal microrotation (0). The amplitude relation iqG : • shows however that the 0-mode has a finite coupling to the ~:-mode even when qa -~0. Therefore, we use two subscripts on to, both ~: and 0, the order indicating that the mode is mainly characterized by ~. Another root, tou,, is similar to the mode usually called the acoustical mode. We prefer to call it a displacement (u)-rotation (~b) mode. (We shall later introduce u as the center of mass coordinate of the displacement in the x-direction.) We find to~,

2Cf 22 q 4 a 4 [ a _ ~ l m =~-~[qa +3

MZA m

K

]}+O(q6a6),

m z(4C+K)

F : G = 1 + q2az(MA - MB)/(2M) + O(q3a3), : iqG = 1 + q2a24CJ](KMa2) + O(q4a4).

(2.16)

We have only given the amplitude relation F : G to the order q2a2 since higher order terms are so complicated that the interpretation is obscured. The expression of to~ is more transpicuous. It should be noted that the leading term is independent of K. (Askar [2] got a different result, not because of differences in our models but due to some mistake.) The coefficient of the q4a4-term in to~ contains a J-dependent term. It is interesting that a similar term also appears in the micropolar continuum theory (Brulin and Hjalmars[5]). But is should also be noted that the present l-dimensional lattice model leads to an amplitude relation iqG : • which differs by a factor 1/2 from that of the isotropic continuum theory. The present result is however in accord with the general expression given by Tauchert[6] in his eqn (19). The third root of (2.11) is essentially a microrotational mode

aZK

2 2K( 1 - MAP)

w~=-~--+ q a 2-M-(1---m-fi))~- O(q4a4)' where

P=-R/(2C +~K), 22J(1-MAP)

iqG : • = - q a M--~O - raP) + O(q4a4)' G : F= [1- M A R / (2C+~K) ][I+ 1 a 2.1j+O(q3 ).

(2.17)

With increasing qa, % will first increase slightly. As shown by the amplitude relations, the perturbation of ~, by u and ~ is very weak, vanishing in the limit qa ~ O.

Validity of the continuum description of molecular crystals

I "'69

2.4 Solution at the Brillouin zone boundary Since we have a discrete lattice model with the lattice parameter 2a we have qmax :=_+zr/(2a). At the zone boundary we find the following solutions of the eqns (2.11)

G : aq r = i(1 + 4C/K)J](aZMB); G > aq'; w~=(2C+~K)/Ma=-A;

(2.18) (2.19)

w] = aZK/(2J) - A ==_R - a,

a ~ : G = i(1 + 4 C / K - a2MR/J); a ~ > G;

(2.7:0)

A =- a2K 2 : [MBJ(B - R)].

(2.21)

where

The eqns (2.18) and (2.20) are only valid when A is a small quantity. This condition is fulfilled in the example given below, and is likely to be fulfilled also in other applications. The root (2.19) corresponds to the displacement f of the type A ions without coupling to the g- and ~-modes. The solution (2.18) has the highest w-value. It is, therefore, reasonable to expect a connection between (2.18) and the vibrational solution (2.14) at the other limit. It is seen that the rotational mode is interacting with the vibrational mode for all values of qa. The w-value is seen to decrease with increasing qa. The lowest w-value, (2.20), refers to a motion where the rotational mode is dominating. When comparing with the w~ of (2.17) it is seen that ~o3 at qa = 7r[2 is lower than w, at small qa-values. However, it is also seen that there must be a range of intermediate qa-values where there is a strong coupling between the rotation mode 4' and the displacement-rotation mode t,~, since w.~ of (2.16) is smaller than w~ of (2.17) but to3 of (2.20) smaller than both the ~o-values of (2.18) and (2.19). This coupling was also found by Askar[1] from numerical calculations wth suitable chosen values of the force constants.

3. P S E U D O - C O N T I N U U M

EQUATIONS

3.1 Derivation of the equations To facilitate a comparison between the lattice model and continuum models we want to make a transformation to center of mass coordinates u and internal coordinates ~ and to rewrite the difference eqns (2.8)-(2.10) as differential equations. Since our model implies rigid connection of the bars to the NO3-ions but allows free rotation at the K+-ions, we choose the unit that should correspond to the micropole of a continuum as composed of an NO3-ion, number 1 say, and two halves of the nearest K+-ions, numbers 0 and 2 say. Then the center of mass will coincide with the center of the NO3-ion when the ions are at rest. We introduce for the center of mass motion: (MA + MB)U m = MAlta+ ~MB(g ~°~+ g(2~),

(3 I)

-~(g

I3 2)

and for the internal motion +g').

Expanding in Taylor series around the ion number 1 and retaining only second order terms we obtain from (2.8)-(2.10)

I. FISCHER-HJALblARS

1770

1

2

MB 2

12

~ a sc33+--~-a

(3.3)

u.33)

MA2 g~b3 + O(q4a4), +~--~a

(3.4)

JdJ: ~aZK(u,3 MA---~-~,3-qJ)+O(q3a3) •

(3.5)

Assuming travelling waves (u, ¢, ~b) : (U, E, ~) exp

[i(qz -

~ot)]

we obtain 1 . K [qZaE(2C +~K)- Mto2]U + q 2a 2MB --~-(2C +~K)= + iqa-~a* =O,

- q 22MA/ --~-~2C

- iqa~-~Ka~Ma 1 .MA,~.,, + (~K_~oE)a~ - iqa~KU + tqa~--~-,

= 0.

= O, (3.6)

The eqns (3.6) should be compared to the lattice eqns (2.11). It is seen that the pseudocontinuum equations are at least as complicated as the lattice equations. According to the derivation it is expected that in the long wavelength limit the solutions to (3.6) should be the same as the solutions to (2.11). As for the Brillouin zone boundary, qa = ~'12, the two first of the eqns (2.11) show that one solution, the •-mode, will be a pure displacement modein this limit. Equations (3.6), on the other hand, show that when qa = zr/2 we cannot expect any mode to become completely decoupled from the others. The problem is now to find the range of qa-values where (3.6) is a good approximation of (2.11). A complete analysis of this problem is rather complicated. We shall therefore restrict ourselves to part of the problem. 3,2

The displacement-rotation coupling

Since our main interest is to study the coupling between the u-mode and the ~O-mode, we shall seek a simplified form of (3.6) that may be adequate for this purpose. In order to estimate the magnitude of the various terms of (3.6) we shall make use of the solutions already found for (2.11). As we expect the motion of the center of mass coordinate u to represent the displacement-rotation mode in the long wavelength limit, the amplitude relations of (2.16) should be valid. Hence

[i(qz - ~ot]~ IF, G[ ~ ]UI, ~: = (F - G) exp [i(qz -tot)] ~ ]--,I, [~.1-~ q2a2(MA -- MB)I(2M)I U[. u : U exp

(3.7)

Since ~,33: _q2~ we find that the ~:33-term of (3.3) is of the order q4a4lU 1. Similarly, the ~(3-term of (3.5) is of the order q3a31U]. Thus, for the displacement.rotation soltJtion we have 1

2

1 2

1 2K(u,3 - ~b) + O(qSa3). Jq/= ~a

4 4

(3.8)

Validity of the continuumdescriptionof molecularcrystals In the Ion~ wavelength limit the solutions of (3.8) are one displacement-rotation mode w7,4, =

2CF

~'

9

4 4 J / + 3 K . ~-~)]+O(qa

U

dC ql . iqU = 1+ q ' a - ~ - M ~ ( I

,.

+ 3 K ) + O(q~a~).

t3.gt

and ¢~ne microrotational mode

( q2a2(2C+~K)/(MR)]+O(q6a~).

.)~,= ~ f - + q - a s ~ + q 4 a 4 j ~ ~ J f. l + iqU " q~ = - q'a'~--~[

2C+ K

(M2a2)+O(q6a~),

(3.101

l,et us nm~ compare the results from the pseudo-continuum equations with the lattice model, i.e. 0.9) with (2.16) and (3.10) with (2.17). The leading term of ¢0,~ is the same in the two cases and the first term of the q4a4-coefficient, J/(a2M), is also the same. The following part of the q4a'Lcoefficient looks different but is expected to have the same sign in the two cases. The amplitude relations $ : iqG and • : iqU are also the same. (iomparing ~o~ of (2.17) and (3.10) it is seen that the constant term and the leading part of the q:a2-term are the same. The leading parts of the amplitude relations iqG : 'Jr and iqU q~ are also ihe same. The approximate pseudo-continuum eqns (3.8) are, therefore, expected to give a good representation of the lattice model at least over part of the qa-range. However, :he analytical solutions in the abbreviated forms of (2.16), (2.17) and (3.9), (3.10) are not sufl~ciertly accurate to allow an evaluation of the real size of the qa-range. To this end it is necessary to resort to numerical calculations. 4. NUMERICAL APPLICATIONS For a numerical verification of the conclusion, drawn from the analytical expressions of Sections 2 and 3, it is necessary to find suitable numerical values of the various parameters of the model. For this purpose we shall use data for the specific system we had in mind when constructing our model, i.e. the KNO3 crystal in its aragonite structure. The masses MA, M,~ are known and the moment of inertia J can easily be calculated from the molecular geometry. The lattice parameter / =- 2a can be estimated from crystallographic data. These data show that the ions are arranged in a zigzag chain with alternating distances between the ions. In our model the chain is straightened out and the ions equidistant. Since our present purpose only is to find the orde~ of magnitude of the various terms this simplification is not serious. As suggested by Askar[2] IR and Raman frequencies may be used to estimate the force constants C and K A detailed discussion of the lattice dynamics has been given by Balkar,~ski and Teng[3] for the interpretation of their measurements of Raman spectra. They interpret a line at 133 cm ~ (~o -- 2.5t x 10 ~3sec -~) as related to an ionic mode and most likely a transverse optical mode. The symmetry assignment shows that the displacement of this mode is along our ~--axis [3, 2]. A line at 50 cm ~ (~o = 0.94 x 10~3sec ~) is interpreted as a rotational mode, cour,led to this displacement[3,2]. Adopting these results o2~6 of (2.14) with qa = 0 is put equa to 2.5 i x 10 j~ sec ~and o2~,of (2.17), qa = 0, equal to 0.94 x 1013s e c i. The numerical values obtained in this manner are collected in Table 1. Our values are similar to but not identical with those used by Askar[l. 2], who studied a chain with alternating distances. Fhe ,,~-values obtained from the lattice model by numerical solution of eqns (2.11) are shown as dispersion curves in Fig. 2 (solid lines). When the coupling between the microrotation and the displacements is neglected, the dispersion curves will follow the dotted lines of Fig. 2. t is seen that the range of strong coupling is found for 1 <_ ql <_2. Our curves deviate somewhat from those published in Ref. [1], Fig. 6. The main reason for this deviation seems to be some numerical error in Ref. [1]. Fhe mJmerical solution of the pseudo-continuum eqns (3.8) gives a u~-mode and a ~b-mode which almost coincide with those of the lattice model when q1<2. When q l > 2 , the lower

1772

I. FISCHER-H.IALMARS Table I. Numerical values of the parameters M A = 1. 029 • 10 - 2 5 kg MB = 0 . 6 4 8 J

• 10 - 2 5 kg

= 7.37

10 - 4 6 kg m 2

t = 2a= 6.46 .

10 - 1 0 m

C

= 12. 16 N / m

K

= 1.25 N/m

mode (continuation of the u¢-mode) remains indistinguishable from the lattice result. However, the upper mode (continuation of the C-mode) begins to deviate from the lattice result and is shown by the broken line of Fig. 2. Some numerical values of amplitude relations from both lattice and pseudo-continuum models are collected in Table 2. The table shows that the amplitude relations of (2.11) and (3.8) are very similar for the two models when ql < 2. For ql > 1.5, i.e. after the avoided crossing between the W-mode and the u@mode, it is seen that these two modes exchange properties. At

T

~ ¢ - mode

2 xtO'3 //

=

~-rnode

1 xtO 'L

• ..o..'o .

----~ q t Fig. 2. Dispersion curves calculated for a simplified K N O r m o d e l . Solid lines show the results for the lattice model and broken lines for the pseudo-continuum model. When ql < 1.5, solid and broken lines coincide. The lowest curve is in the whole range the same for the two models. Dotted lines refer to a lattice model without uO coupling. Table ?. Amplitude relations for the lattice model, eqn (2.11), and the pseudo-continuum model, eqn (3.8)

q~

$ - mode (2.11)

u y-mode

-mode (2 11)

iqG Y

E G

0.17

-6. 1

0.52

(2.11)

(3.8) ~

(3.8') iqG Y

iqU

~

iqU Y

1.6

-3- 10 - 4

- 3 ' 10 - 4

1.0

0.99

0.99

-6.0

1.5

-3" 10 - 3

-3" 10 - 3

1.0

0.90

0.89

1.05

-5.9

1.4

-0.03

-0.02

1.0

0.59

0.56

1.57

-5.7

1.2

-0.16

- 0 . 18

1.0

0.17

0.14

2.09

-5.4

1.3

-1.04

-0.92

0.8

0.06

0.05

3.14

-5.3

=

- =

-3.29

0

0.05

0.03

Validity of the continuum description of molecular crystals

1773

the Brillouin zone boundary (2.11) the microrotational ~-mode has been transferred to a pure displacement/-mode describing the motion of the NO3-ions only. Simultaneously, the u~-mode describes displacement of the K+-ions coupled to the microrotation of the NO3-ions. Usually it is assumed that the upper limit for validity of continuum models is ql ~ 1. It is interesting that our pseudo-continuum eqns (3.8) give a fair description of the lattice model for a considerably larger range. In particular, it is gratifying that (3.8) can describe the str3ng coupling region. It must however be remembered that closer to the Brillouin zone boundary, q l 2. we cannot expect the pseudo-continuum equations to be adequate. Ackt~owledeement--Thanks are due to Prof. Stig Hjalmars for suggesting the improvement of the model to describe free K~-honds, REFERENCES II] A. ASKAR, J. Phys. Chem. Solids M, 1901 (1973). [2] A, ASKAR, Int. J. Engng Sci. 10, 293 (1972). [3] M. BAI,KANSKI and M. K. TENG, In Physics of the Solid State (Edited by S. Balakrishna, B. Krishnamurti arid B. Ramachandra), p. 289. Academic Press, New York (1969). [4] C. KITTEL, Introduction to Solid State Physics, 4th Edn. Wiley, New York (1971). [51 O. BRULIN and S. HJALMARS, Int. J. Engng Sci. ES 71SK. 16] I. R. q AUCHERT, Rec. Adv. Engng Sci, Part I 5, 325 (1970).

(Received 29 October 1980)