Thermal atomic displacements

-i I J

/ A

Dulong-Petit

/ i l l

0 50 100 150 200 T (K)

Fig. 7.2. The heat capacity of C^Q fullerite from experiments. After Jin Yu et al. (1994).

(Beyermann et al. 1992), from information about the atomic dynamics through neutron scattering (Pintschovius et al. 1992) and from the the-oretical modelling of those excitations (Jin Yu et al. 1994). Each C60

molecules contributes 3 x 60 modes vibration modes. When placed in a lattice, six of these modes refer to intermolecular motions (three vibrations and three rotations of the molecule). The remaining modes refer to intramolecular motions. The two sets of modes are well sepa-rated in energy. At ambient temperature only the intermolecular motions contribute significantly to C_p. The force constants for the intramolecular motions are almost as strong as in the diamond form of C. We therefore expect that Cp of fullerite (in a hypothetically stable state) would not reach the classical (Dulong-Petit) limit until around 2000 K.

5. Thermal atomic displacements

5.1. General relations

Much of the material in the following discussion of thermal atomic displacements has been developed in relation to thermal effects in X-ray crystallography. The monograph by Willis and Pryor (1975) covers these aspects in depth. Some additional formulae are given in Appendix D. Butt et al. (1988,1993) have tabulated displacements for many solids with cubic lattice structure.

Thermal atomic displacements 121

The instantaneous position R(/c, /; t) of the /cth atom in the /th unit cell is written

R(/c, /; 0 = R V , /) + u(/c, /; t), (7.33) where u is the displacement vector from the equilibrium position R°. We

wish to calculate the thermal averages (u²) and (w²), of u(/c, /) and its Cartesian components ua(K, /), respectively. Let there be Nc primitive cells in the lattice, with r atoms in each cell. From (D.10),

Here e(q, s; /c) is the /cth part of the total phonon eigenvector e, i.e. the component of e(q, s) which refers to the displacement of the /cth atom.

M_K is the mass of the atom of type /c, and E(q, a-) = ha)(q, s)[n(q, s) + 1/2]. The total mean-squared displacement is

(u²(/c, /)) = (a²(/c, /)) + (u²y(K, /)) + (M2

(/c, /)). (7.35) If the site (/c, /) has cubic symmetry, the average displacement is

isotropic, with

(u²(/c,/)) = 3(W2(/c,/)). (7.36)

Thus, it is worth stressing that in the approximation of harmonic lattice vibrations and cubic symmetry, the displacements are the same in all directions, e.g. the [100], [110] and [111] directions of an fee (or bec) lattice, although this may seem counter-intuitive.

The distribution function P(u_a, KI) for the displacement of the (/c/)-atom in the a -direction is Gaussian, i.e.

P(ua) = [27r(W2

(/c,/))]-^1/2exp[-W2

/2(W2

(/c,/))]. (7.37) This relation was proved in the classical limit by Debye (1914) and

Waller (1925), and shown by Ott (1935) to be true also in a quan-tum mechanical treatment. From the distribution functions P(ua), we can calculate the probability that an atom is to be found in a certain region near its equilibrium position. For instance, there is a 50% prob-ability that the a -component of the displacement vector u is larger

than 1.54(u²)^1/2. See Nelmes (1969) for details regarding displacement probabilities.

Anharmonic effects have been neglected here. They are pronounced for large displacements, and thus make P(u) less reliable when u² 3>

(u²).

5.2. Monatomic solid with cubic symmetry

In a monatomic solid with cubic symmetry, the index K is irrelevant.

Then

£ * « ( q , * K ( q , j ) = l, (7.38)

a

and eq. (7.34) gives

<"

^{(uz) = — /}

² >=^I

^{V ; J}F(co) dco. (7.39)

Thus, we have the important result that the mean-square thermal dis-placement in a monatomic solid (i.e. an element) with cubic symmetry can be calculated without knowledge about the eigenvectors e. We also note that eq. (7.39) converges at the lower integration limit only if

F((L>) ~ co^p for small &>, with p > 1. In a one-dimensional chain, which is often used to illustrate important concepts in lattice dynamics, F(co) tends to a constant for small co. Then (u²) diverges at all temperatures.

At high temperatures E(co; T) -> £B7\ and (u²) becomes

(u²) = - ^ — . (7.40)

V } Mco²(-2)

Because co(—2) ~ M~^1/2 for a monatomic solid, we see that (u²) at high temperatures does not depend on the atomic mass M.

When T -> 0 only the zero-point vibrations remain. Then E(co; T)

= ha)/2 and eq. (7.39) gives

(u²) = ™ n. (7.41)

In this case, the mass dependence 6L)(—1) ~ M^{_ 1 / 2} does not cancel in eq. (7.41), and a heavy mass tends to have a low zero-point vibrational

Thermal atomic displacements 123

displacement. We note that the high and low temperature limits of the displacement do not depend on the detailed shape of the density of states F(co) but only on a single parameter, co(—2) and co(— 1), respectively.

5.3. Thermal displacements in a Debye model With a Debye model for F(co), eq. (7.39) becomes

9h²T

<^u2> = 777-^2 [ * 0 D / D + 0D/4T], (7.42)

where O is the Debye integral function 1 f^x z

*(*) = ~ / -r—r dz. (7.43) For small JC, O(x) = 1 — JC/4 + x²/36 — • • •, which yields the high

temperature expansion 9h²T

<^u*> = 777^2 t¹ + (V36)(0_D/r)² + •••]• (7.44) In the low temperature limit, O -> 0 and hence,

9 9h²

4 M £B# D

5.4. Debye-Wallerfactor

The mean-square displacements can be obtained from the temperature dependence of the intensity in X-ray or neutron scattering experiments (Debye 1914, Waller 1923). Let / be the actual intensity, 70 the intensity when the lattice is rigid, ki and k2 the wave vectors and A = 27r/|k|

the wavelength of the photon (or neutron) before and after the (elastic) scattering, q = ki — k2 and 26 the angle between ki and k2. One has

/ = /0exp(-2MD W). (7.46)

In the Debye-Waller factor exp(—2MDW) of a cubic lattice, M_DW = (l/2)((u.q)²) = ( l / 6 ) ( u V

= (l/3)(u²)(87r²sin²#A²) = Bsin²0/k², (7.47)

where B is called the B-factor of the atom. According to eq. (7.46), a plot of ln(///o) versus sin² 6/X² (a Wilson plot) should give a straight line with a slope which, at high temperatures, is linear in 7\ but anhar-monic effects give rise to some non-linearity (Hahn and Ludwig 1961, Maradudin and Flinn 1963, Cowley and Cowley 1966, Mair 1980).

5.5. Interatomic distance

The displacements discussed above refer to the deviation of a particular atom from its equilibrium position. Sometimes one is more interested in how the distance between two specific atoms varies due to the vibra-tions. Let the instantaneous positions of two atoms, labelled 0 and j , be R0 = RQ + uo and Rj =R°. + u7. The distance d between them differs from the distance do = Ry — R$ in the static lattice, such that

d = (R° + uj) - (R° + uo) = d₀ + (u; - u₀). (7.48) The mean-square relative displacement is

OT² = ( ( d - d0)²) = <(U;-Uo)²)

= <u²) + ( u²) - 2 ( u;- u0) . (7.49)

Consider now the relative displacement along the direction R_; — Ro, for instance the distance between near-neighbours. Thus, we seek

tfR = < [ e - ( u ; - u₀) ]²} , (7.50)

where e is a unit vector along R; — Ro. If the atoms vibrate indepen-dently, like in an Einstein model, one has (u7 • uo) = 0. In another extreme limit, that of acoustic long-wavelength vibrations, all atoms move in phase and d = do. In a real solid, we expect correlations between the atomic motions to be significant when 0 and j are near neighbours, but to be small for atoms far apart. A general expression for OTR, in a monatomic lattice with atomic mass M, is (Griineisen and Goens 1924, Warren 1969)

2 _ & y ^ [e(q,s) -e]² r / t o ( q , s ) 1

a R

" NMJ^ a>(q,s)

^C

° L 2£

B

r J

x { l - c o s [ q . ( R ; - R ° ) ] } . (7.51)

Thermal atomic displacements 125

Without the cosine term in eq. (7.51), a^ = 2(u²)/3 (cf. eqs. (7.11), (7.34) and (7.49)). That would correspond to Ry — Ro being so large (i.e.

the atoms 0 and j being so far apart) that the cosine term averages to zero when one sums over q. An accurate treatment of eq. (7.51) requires numerical calculations. Not even in a Debye model is it possible to get a closed-form algebraic expression for a£. The quantity a^ is accessible in experiments on extended X-ray absorption fine structures, XAFS (e.g.

Greegor and Lytle 1979, Bohmer and Rabe 1979 and Marcus and Tsai 1984). Displacement correlation functions have also been calculated from the phonon spectrum, for instance by Beni and Platzman (1976) for Zn, Sevillano et al. (1979) for Cu, Fe and Pt and Zywietz et al. (1996) for SiC. See Appendix D for further comments.

5.6. General expression for the thermal displacement

We now seek a general but tractable expression for (u²(/c, /)) in a lat-tice of arbitrary structure. It is convenient to introduce a matrix B with elements (do not confuse B with the B-factor in eq. (7.47))

Btj = {uiUj). (7.52)

Let the r atoms in the primitive cell be numbered by K = 1 , . . . , r.

Then ut is a component of the vector (wlx, u\y, u\z, U2X,. •., urz). From a result in matrix theory (Born 1942) we can write (for brevity restricted here to high temperatures)

B(KK) =

NCMK

E

q

1 720

D~\q; KK) + 1 h 12 \knT

kuT D(q; KK) + (7.53)

D is the 3r x 3r dynamical matrix and I is a unit matrix. B{KK) refers to the displacement of atoms of kind K. This expression is useful since it allows the evaluation of (u²) directly from the inverted dynamical matrix D.

5.7. Two atoms per primitive cell

An interesting special case is a lattice with a primitive cell contain-ing two atoms, denoted by 1 and 2. Let their masses be M(l) and Af(2). From eq. (7.34) we cannot obtain the displacements of each kind of atom without knowing the 6-component eigenvectors e(q, s) = [e(q, s; 1), e(q, s; 2)]. However, we can obtain the weighted sum

M(l)(u?> + M(2){u²2) = 2 / [E((o; T)/(o²]F(co) dco. (7.54) Jo

Here E(o), T) is the Einstein thermal energy. To obtain the displacement of each atom we turn to eq. (7.53). D(q) is now a 6 x 6 matrix. It can be blocked into four parts, D(KK') = D₀(/c/c^/)/[M(/c)M(/c^/)]^1/2 (cf. eq.

(C.3)). Then eq. (7.53) yields the high temperature result

M(l){nj)

a

= (k

B

T/N

c

)J2Va

^l

(* ID, (7.55)

q

M(2)(uj)_a = (k_BT/N_c)J^D-^l(q; 22). (7.56)

q

The index a refers to a Cartesian component in the displacements and in the block D(/c/c). At high temperatures, we only keep the first term on the righthand side of eq. (7.53). Then, since D^_1(q, KK) ~ M(K), the atomic masses cancel on each side of eqs. (7.55) and (7.56), respec-tively. It follows that (Uj) and (u^> do not depend on the masses but only on the forces between the atoms.

If the force-constant part obeys D₀(ll) = Do(22), one has the important result that the mean-square thermal displacements in a di-atomic solid at high temperatures are the same for both kinds of atoms, irrespective of their mass ratio. The condition Do (11) = Do (22) is ful-filled if there are only nearest-neighbour interactions, but also for direct Coulomb forces in an ionic compound A⁺B~. In the low temperature limit, on the other hand, only zero-point vibrations remain. Then the average displacement amplitudes are unequal for the two kinds of atoms (cf. the analogous result in eq. (7.41)).

The monatomic hexagonal close-packed lattice is a special case of diatomic lattices. The B-matrix has the form

((ul) 0 0 \

B = 0 (u²a) 0 , (7.57)

V 0 0 {u])J

Thermal atomic displacements 127

where ua refers to vibrations along an a-axis, and uc to vibrations per-pendicular to that axis. D0(ll) = D0(22) and M(l) = Af(2) = M.

From eq. (7.54) we obtain the average (u²) over all directions, (u²) = 2(u²_a) + (u²_c)

= (1/M) / [E(co; T)/(o²]F(o)) dco. (7.58) Jo

In an arbitrary direction q, making an angle 9 with the oaxis, one has

((u • q)²) = (u²c) cos² 9 + (u²a) sin² 9. (7.59)

Example: mean-square displacements in NaCl-type lattices. Since the essential interatomic forces in alkali halides, A⁺B~, are Coulomb forces plus interaction from overlap of atomic wave functions on neighbouring ions, we expect that (u²(A)) & (u²(B)) at high temperatures. Cal-culations confirm this result (Huiszoon and Groenewegen 1972). For work on thermal displacements in NbCo.95 (which has NaCl-structure), see Kaufmann and Meyer (1984), citing Milliner, Reichardt and Chris-tensen. Butt et al. (1993) have tabulated atomic displacements in 52 diatomic cubic compounds, based on diffraction experiments.

Example: anisotropic vibrations in zinc. Zinc has unusually anisotropic thermal displacements. According to eq. (7.57) it suffices to find (u²) and (w²). A strict treatment requires not only the frequencies &>(q, s) but also the corresponding eigenvectors. However, it is a useful ap-proximation (Griineisen and Goens 1924) to introduce two effective density-of-states functions of the Debye type, Fa(co) and Fc(co), and choose Debye temperatures 9_a and 9_C such that the temperature depen-dence of the displacements is well described. Figure 7.3 shows that this gives a good account of (u²) and (w²) in zinc. The normalisation of Fa (co) and Fc(co) is defined by fitting theory and experiment at T = 0 K.

The inset shows the actual F(co) and the two Debye spectra used here.

5.8. Combined static and dynamic displacements

In a random alloy AXB\-X, the atoms may take the positions of an ordered lattice but show displacements relative to the ideal lattice posi-tions, because of the randomness and different atomic sizes. Such static

Ch. 7. Thermal properties of harmonic lattice vibrations

Fig. 7.3. Anisotropic thermal displacements in zinc, calculated with separate Debye models (shown in the inset together with the full F(co)) for directions parallel (||) and perpendicular (±) to the hep c-axis. Filled and open circles are measured values.

After Potzeletal. (1984).

displacements are superimposed on the dynamic (vibrational) displace-ments. Static displacements affect the vibration frequencies, but that would enter as higher-order corrections in this case. Then it is reason-able to consider static and dynamic displacements as uncorrected and get, for the total displacement,

(U )total ^ (U )static + ( u )dynamic- (7.60) For an example of this approach, see Dernier et al. (1976) in an analysis of the mixed-valence compound Smo.7Yo.3S.

5.9. Vibrational velocity

The instantaneous velocity va(ic9 /; t) of an atom (KI) in the direction a is obtained from dua(K, /; t)/dt (cf. Appendix D). Then, in analogy to eq. (7.34), we get the thermal average

(VI(K, /, 0) = TTTT" J2 ^E^ ^s)s<*^ ^{5; K}K(1> ^s> *)• (⁷-⁶¹)

NCMK

q,s

In a monatomic lattice with cubic symmetry, (v²) = (v*) + (vh + (u²) is isotropic;

(v²) E{fl), T)F(co) da), (7.62)

Temperature and pressure induced polymorphism 129

with the low and high temperature limits

(v²) = 3heo(l)/2M = (9kB/8M)9D(l), T = 0, (7.63)

(v²) = 3kBT/M, kBT » ha)max. (7.64)

See Chapter 19, §3 for comments on bounds to (v²).

Example: Heisenberg's uncertainty relation in solids. In a monatomic lattice with cubic symmetry, the zero-point vibrational displacement and velocity are given by

(u²x)T=0 = h/2McD(-l), (7.65)

{v²_x)_T=o = ha)(l)/2M. (7.66)

The inequality co(l) > &>(—1), eq. (6.23), is equivalent with the Heisenberg uncertainty relation

(u²x)T=0(v²x)T=o > h²/4M². (7.67) For a single one-dimensional oscillator the relation (7.67) becomes an

equality, while the Debye model gives (9/8)(/j²/4M²) for the lefthand side of eq. (7.67). An analogous inequality holds for the displacement and velocity of a particular atom in a lattice with several different atoms (Housley and Hess 1966).

6. Temperature and pressure induced polymorphism

In document THERMOPHYSICAL PROPERTIES OF MATERIALS (pagina 141-150)