NaCl interstitial solid

With the approach to the defect-free Na1Cl1 structure established in paragraph7.1.2, the generalization to the Na‰Cl1 “interstitial solid” case is straightforward. The sublattice of the large “Cl” hard spheres is again assumed defect free. For the incompletely occupied Na sublattice holds ÷⁽¹⁾ = ‰ yielding an extra entropic contribution compared to Na1Cl1 (cf. eq. (7.6)). Study of a crystal at the desired occupancy can be achieved within a broad range by adapting the stoichiometry

‰¸ of the coexisting liquid. As a function of ‰¸ at constant Î = 0.3, figure 7.7 shows the equilibrium parameters of the Na‰Cl1 crystal at melting. With Lebowitz’s above-quoted liquid solution for c^(ab), crystallization is promoted by the addition of small spheres to the liquid below ‰¸ ¥ 1. This is reflected in a decrease of n^crit,(2)_s in the lowest ‰¸ range (cf. figure 7.7(a)). At the same time the Gaussian widths (figure7.7(b)) take their maximum near ‰¸ = 0.5. As mentioned above, the initial increase in Á(s) might be related to the increase in small-sphere diffusion with increasing interstitial (i.e. small-sphere sublattice) concentration observed by [Fil+11].

The occupancy/solid stoichiometry of‰s= 2/ 3studied therein is reached at‰¸ ¥3.6 in the present DFT approach (figure 7.7(c)). The corresponding dispersion relations will be compared to some obtained by MC simulations following [Fil+11] in figure7.9.

In the absence of the small species ‰s(‰¸ = 0) = 0 as expected. Moreover, comparison with figure 7.4 at Î = 1 shows that the single HS crystal freezing

parameters are reproduced (note the factor of 2 between both plots forn^crits/¸ due to the different reference crystal structures).

7.2. Dispersion relations from position measurements

For the single-component periodic crystal, Walz and Fuchs solved the Œ+ 3 dis-sipationless and isothermal Zwanzig–Mori e.o.m. — [WF10, eq. (25)] respectively eqs. (5.11) for B = 1 — by the following ansatz [Wal09; WF10]

”n_g(q, t) = ≠i(g+q)_–n_g”u_–(q, t)≠ n_g

n0”c(q, t). ([WF10, eq. 34]) It allows to relate the solution of each e.o.m. to a simple hydrodynamic relation whose validity is ensured. With the caveat in mind that the equilibrium fluctuations treated in eqs. (5.11) forB Ø2 decay on different time scales in the hydrodynamic limit, we generalize eq. ([WF10, eq. 34]) to that case, restricting ourselves toB = 2 for simplicity.

The vector components v_–,g^(s) (q) =i(g+q)_–n^(s)_g are easily generalized from [WF10, eq. (28a)] to rewrite the e.o.m. as

ˆt”n^(s)_g (q, t) =≠v_–,g^(s)

Í^(s)₀ (q)”j_–^(s)(q, t) , (5.11a) ˆt”j_–^(a)(q, t) =^ÿB

b=1

ÿ

g^Õ,g

v^(a)ú_–,gÕ(q)J_g^(ab)úÕ,g (q)”n^(b)_g (q, t) . (5.11b) The ansatz

”n^(s)_g (q, t) = ≠v_–,g^(s) (q)u^(s)_– (q, t)≠ n^(s)_g

n^(s)₀ ”c^(s)(q, t) (7.20a)

(3.45)

== ≠v_–,g^(s) (0)u^(s)_– (q, t) + n^(s)_g

n^(s)₀ ”n^(s)(q, t) (7.20b) turns the equations of motion (5.11a) into

v_–,g^(s) (q=0)

C

ˆtu^(s)_– (q, t)≠ 1

Í^(s)₀ j_–^(s)(q, t)

D

=

=n^(s)_g

C i

Í^(s)₀ q–”j_–^(s)(q, t)≠ˆt

A

iq–”u^(s)_– (q, t)≠ 1

n^(s)₀ ”c^(s)(q, t)

BD

’g. (7.21) The left hand side vanishes with eq. (3.56), the r.h.s. with both eq. (3.45) and the continuity equation for”n^(s) eq. (3.40). Concerning eq. (5.11b) we can follow [WF10, eq. (36) et seqq.]:

ˆ_t”j_–^(a)(q, t)^(7.20a)== ≠^ÿ

With the assumption of species-wise constant defect densities, ˆt”c^(s) © 0 and again eq. (3.56), eq. (7.22) becomes the wave equation (5.12),

ˆ_t²”j_–^(a)(q, t) = ≠1 Í0

ÿ

b

»^(ab)_–— ”j_—^(b)(q, t) .

Consequently, the eqs. (7.19) can be rewritten as known physical relations with the ansatz (7.20a). Bear in mind that this requires Working hypothesis 1 to hold.

Because the ansatz (7.20a) is a species-wise generalization of [WF10, eq. (34)] it is straightforward to solve it for ”u^(s) and ”n^(s) in the same species-wise manner:

”u^(s)(q, t) =iN^(s)≠1· ^ÿ

cf. [WF10, eq. (41)]. Equation (42) therein on the other hand generalizes to

”n^(s)(q, t) = n0

With eq. (3.45) and following [WF10, eq. 43], eq. (7.23b) is equivalent to

”c^(s)(q, t) =≠n0ÿ

It is insightful to compare eq. (7.23a) as as definition of u with that employed in the potential expansion method: The ideal, defect-free lattice referred to by the latter has been replaced by average particle densities n^(s) which in eq. (7.23) occur explicitly and via

”n^(s)_g (q, t) =fl^(s)(g+q, t)≠”(q)n^(s)_g .

· · · y

Figure 7.8.: Schematic illustration of the pre-processing step of particle position data in order to evaluate eq.(7.23). The curly brackets indicate a set of statistically independent measurements of the particle positions taken at fixed times (“snapshots”). The rightmost frame shows the average particle densities derived from that set.

With a suitable experimental or simulation method of obtaining statistically indepen-dent positional measurements fl^(s)_i (r, ti)at fixed times ti, a set of “snapshots” can be extracted.⁷ Averaging over this ensemble then yields the equilibrium densitiesn^(s) as

n^(s)(r) = lim

IæŒ

1 I

ÿI

i=1fl^(s)_i (r, ti) =:^efl^(s)_i (r)^f

M .

The subscript “M” indicates an average over a set of measurements whose size I needs to provide sufficient statistics of a system’s phase space. The procedure is schematically illustrated for a hypothetical 2d interstitial solid in figure 7.8.

Subsequently, the snapshots can be processed to measurements of ”u^(s) and ”n^(s) (or

”c^(s)) with eqs. (7.23). The displacement field correlations ^e”u^(a)_– ^ú”u^(b)_— ^f(q)obey

e”u^(a)_– ^ú”u^(b)_— ^f(q) = V k_BT

Í^(ab)₀ D^≠1(_–—^ab)(q) (7.24) where D ≥ _͹₀ is the physical dynamical (block) matrix whose eigenvalues give the physical eigenfrequencies ʇ of the phonon modes (cf. [Här13, eq. (66)]).

A set of positional snapshots was obtained with a Molecular Dynamics simulation in the N V T ensemble as described in [Fil+11] and references therein. The cubic simulation box contained an equilibrated Na‰Cl1 hard sphere crystal with N⁽²⁾ = 2048 large particles at a particle density of n⁽²⁾₀ = 1.35 and N⁽¹⁾ = 1366 small particles, matching an occupancy of approximately ‰s ¥ 2 /3. Effects of the box walls were eliminated by periodic boundary conditions. This setup was used to evaluate the dynamical matrix D from eq. (7.24) with È·Í © È·Í_M and I = 500 equilibrium configurations and assuming equal particle masses m⁽¹⁾ = m⁽²⁾. The eigenvalues of D(q) are plotted as crosses in figure7.9 for the three high-symmetry BZ paths À,∆ and» shown in the previous dispersion relations and continuously beyond the first Brillouin zone in the same directions. The lines in that figure were obtained for approximately the same occupancy‰s from the DFT approach outlined

7One needs to ensure that these measurements are taken from an equilibrated crystal.

0 28

≈ K X^Õ

ÊËÒ kBT/(m‡2)È

˾(110) Š^ƾ(100) X

DFT y-scaled ◊1.4

Š^Ⱦ(111) L

Figure 7.9.: Dispersion relations for the Na_2/3Cl1 system described in paragraph7.1.4obtained from MC simulations through eq.(7.24) (symbols) and the DFT approach (lines).

Shown are plots along the three high symmetry pathsÀ,∆and», going beyond the first Brillouin zone. For the full set of equilibrium density parameters cf. table 7.1.

in paragraph 7.1.4. Table 7.1 shows the equilibrium density parameters employed in both approaches. Note that the match in ‰s between simulation and theory required different solid particle densities n^crit,(2)_s . The resulting different lattice constants mean that the q-axis in figure 7.9 needs to be read in terms of 2 differently–sized Brillouin zones. Furthermore, the DFT results were rescaled to eye to match the optical/upper MC modes, bringing these curves to a quantitative match within no more than 10%. The degeneracies of the transversal modes along the (100)- and the (111)-direction are well caught by the simulation results, too. The degeneracy at X^Õ in the leftmost part occurs where the wave vector q reaches another axis of 4-fold symmetry of the reciprocal lattice.⁸ The actual (unscaled) acoustic/lower DFT eigenfrequencies lie about 10% above the simulation results. Both approaches compare very well qualitatively and well for the lower modes quantitatively while the upper modes are underestimated in the DFT approach by a factor of 1.4. Even better agreement could probably be obtained with more advanced DFT approaches to crystals which desirably will reproduce a given sublattice occupancy at the same lattice constant as the simulation. Study of the small-q limit within the simulation approach is restricted by the size of the (cuboid) simulation box: The cartesian wave-vector components q– of q = q–ˆe– for which eq. (7.24) is evaluated need to be integer multiples of 2fi/L_– where L_– is the length of the–^th box edge. In other words,q can be chosen from a cubic lattice determined by the 3 edge lengths of the

8X^Õ is equivalent toX if the first BZ is constructed about the reciprocal lattice point at2≈.

simulation box. A further comparison proposed by A. Zippelius⁹ would be to extract the direct correlation function from the simulation statistics and evaluate eq. (7.14) with this input. In particular, this promises insight into the influence of isotropy and translational invariance in the analytical approximation made forc^(ab).

9personal communication, summer 2016.

8. Comparison of results

This chapter concludes partIIby an overview of its findings in section8.1. Section8.2 then confronts them with results of the perfect-lattice potential expansion approach from [Wal98, chap. 2.7] and presents a simple test case in the honeycomb lattice.

8.1. Dynamic and static approach

The dynamical matrix^P (q)from paragraph5.3.2could be shown to yield the desired d acoustic eigenvalue branches ⁄^acoustic_‡ (q) with ⁄^acoustic_‡ (q) = “‡(ˆq)q² +O(q³).

However, it remains open whether — for systems lacking inversion symmetry — the “_‡^tot(ˆq) obtained from exclusive consideration of ^P»^(tot)(q) are identical to the “_‡(ˆq) (cf. eq. (5.45)). This might not be the case if the blocks ^P»^(s)(q) and

P»(s)

⇡ (q) contain terms of O(q). The block ^P»^(tot)(q) gives the couplings of the total momentum-variable to itself and corresponds to [Wal09, eq. (27)] in the single-species case. At the same time, even for a single-single-species system, the ansatz [WF10, eq. (34b)] does not contain all terms present in eq. (6.4b). In the latter, the additional terms were attributed to internal (non-affine) strain. Note that [Wal09, eq. (2.34)]

starts off with a relation similar to eq. (6.4b)¹ before arguing against contributions at Bragg peaks other than g = 0. Indeed, internal strain cannot be seen as an independent thermodynamic variable like e.g. temperature. Elastic constants can be defined at constant temperature or entropy whereas internal strain inevitably assumes its equilibrium value at given external strains.

For internal strain in the form of sublattice displacements, the connection to optical modes is discussed in [MA67] by expanding the sublattice displacements in optical-mode eigenvectors. To get a better feeling for the role of the blocks

P»(s) and _⇡^P»^(s) in systems lacking inversion symmetry, the dispersion relations were calculated for the diamond structure distinguishing two sublattices from the Gaussian parametrization (7.2)². Albeit this ansatz suffers from conceptual problems, it seems

1For later discussion, the usage of generalized point defect densities rather than ”ng|_u,”c ”cgseems preferable because it is less redundant and more intuitive. The”cgwill give the counterpart to the generalized currents”j_g from eqs. (5.9).

2A metastable diamond structure was obtained from appropriate superposition of the GEM4 potential described in [Här13, and references therein].

worthwhile mentioning that 0<“‡(ˆq)<“_‡^tot(ˆq) was found which corresponds to a lowering of the elastic constants from the coupling to optical modes. A first suggestion for future work is to calculate dispersion relations for a zincblende structure, obtained from diamond by decoration of one sublattice with a different species. This promises more decisive insights: On the one hand, phenomenological hydrodynamics of binary mixtures [Das11, sec. 5.1.5] also predicts a wave equation for the total momentum densityj. On the other hand,0<“‡(ˆq)<“_‡^tot(ˆq)is expected to persist to zincblende (or equivalent). From that, one could conclude that eitherWorking hypothesis 2 or Working hypothesis 1 , at least in binary systems, does not hold.

The mentioned conceptual problems in the treatment of diamond restrict the finding of optical modes fromchapter 5to systems of several species. Diamond, while forming a non-primitive lattice, consists only of a single species. But as there is no correspondence between particle and lattice site by the nature of the approach, the total momentum density j cannot be decomposed into single species contributions and relative currents are hard to define. Yet, non-hydrodynamic currents occurred in the general e.o.m. (5.9) in the form of the{”j_g(q, t)}. Based on the parametrization

n_g =^ÿB

s=1

e^≠ig·b(s)n^(s,0)_g

with n^(s,0)_g characterizing the density peaks on sublattice s, M. Fuchs proposed the following ansatz:

”j_g(q, t) = 1 n₀

ÿB

s=1e^≠ig·b(s)n^(s,0)_g ”j^(s)(q, t) . (8.1) It aims to identify currents ”j^(s) associated with the different basis positions. An inversion of the relation (8.1) in the style of eq. (7.23) would recover a set {”j^(s)}, defined in terms of quantities that are microscopically well-defined for any number of species. Inserting a relation”j^(s)({”j_g})into the e.o.m. (5.9) is expected to recover a dynamical matrix of the form (5.13).

Note that this problem is not encountered in the perfect-lattice regime which we identi-fied with the classical, defect-free low temperature limit in section2.3. Sublattices can be clearly identified which motivates a final stretch back to the potential-expansion method and eq. (1.2). Provided that the classical low temperature limit for a given equilibrium density n is well-defined and leads to a defect-free structure, one would expect to recover the same dispersion relations from as fromD.

8.2. Potential expansion method

A central objective of this thesis is the description of phonons in crystals with an arbitrary amount of point defects at any temperatureT below melting and compatible

a

0

x

y

Figure 8.1.: Honeycomb lattice. Equilibrium particle positions are marked by dots, nearest-neighbor interactions by lines. Coordinate axes are shown for later reference.

with eq. (2.1). In a classical treatment, this should still include systems without point defects in the perfect-lattice regime (introduced on page 16) for which the mechanical equilibrium densities n^(s) are described by,

n^(s)(r) = ^ÿ

RœL

”¹r≠R≠b^(s)2 (8.2)

with b^(s) the basis vector of the s^th species (cf. eq. (2.14)). The equilibrium den-sity (8.2) is nothing else than the reference configuration of the potential-expansion method. The absence of point defects restores the one-to-one mapping of particles to lattice-sites in equilibrium. In a crystal with a basis occupied by only one species, we can hence still refer to each sublattice separately. From the result (6.20) we conclude the necessity to consider a crystal without a center of inversion. A well-known binary example of that kind is the “honeycomb structure” shown in figure 8.1. We will use it as a test case to compare the results of [Wal09] to those of chapter 5 within the perfect lattice regime below. With the same example in mind, we first draw a more formal comparison to Wallace’s presentation of a static potential approach [Wal98, section 2.7]. Crucially, the paragraph on “nonprimitive lattices” therein must hold as the honeycomb structure features more than one particle per unit cell.

A generalization of [Wal98, section 2.7] is presented in [Kan95] which overcomes a limitation of the “Wallace approach”: The mechanical equilibrium condition used by Wallace to determine “internal strains” at a given external strain has to be replaced by a thermodynamic one at finite temperature. Yet, in the perfect-lattice regime — where thermal fluctuations are neglected — both approaches become equivalent.

The connection of the potential expansion (1.1) to the dynamical matrix »(q) from [WF10, eq. (27)] is discussed in section V.C. therein and in more detail in [Wal09, sections 3.3 and 3.4]. We quote the Zwanzig–Mori wave equation at vanishing defect

fluctuations”c= 0 from [Wal09, sec. 3.3]

mn₀”u¨_– =≠»_–“(q)”u_“ [Wal09, eq. (2.42)]

=. . .=A⁽²⁾_–—“”q—q”u“

where the “wave propagation tensor” A⁽²⁾_–—“” can be obtained from » by [Wal09, eq. (3.43)]. The discussion in [Wal09, sec. 3.4] then shows that in the (classical) low-temperature limit and with the mean-spherical approximation for the direct correlation function,c(r,r^Õ) = ≠—Õ(r,r^Õ) [MA97],

A⁽²⁾_–—“” =≠ 1 2V

ÿ

i,jœL

ˆ²Õ ˆu⁽ⁱ⁾– ˆu^(j)“

1R⁽ⁱ⁾ ≠R^(j)2

—

1R⁽ⁱ⁾ ≠R^(j)2

” [Wal09, eq. (3.56)]

= ˆA(primitive) –“—”

whereAˆ(primitive)

is the wave propagation tensor defined forprimitivelattices in [Wal98, eq. (7.20)] and employed in [Wal98, eq. (12.11)]. The case of nonprimitive lattices is also considered in [Wal98, section 2.7] and the generalization to Aˆ is introduced in [Wal98, eq. (7.60)]:

Aˆ–“—” =≠ 1 2V

ÿ

i‹,µ

Õ–“(0µ,i‹) [R—(i‹)≠R—(0µ)] [R”(i‹)≠R”(0µ)] + + 1

2V

ÿ

i‹,µ

ÿ

Ÿ

ÕŸ“(0µ,i‹) [R”(i‹)XŸ,–—(µ) +R—(i‹)XŸ,–”(µ)] (8.3) with Õ–—(i‹,jµ) = ˆ²Õ

ˆu⁽ⁱ–^‹)ˆu^(j_—^µ)({R(kfl)}) and R(kfl) =R^(kfl) the equilibrium positions.

The coupling tensorX is defined in [Wal98, eqs. (7.50) and (7.51)] by the mechanical equilibrium condition

S–(µ) = ^ÿ

—“

X–,—“(µ)u—“ (8.4)

where the constant vector S(µ) is the sublattice displacement of the µ^th sublattice.³ In a binary system S(2) can be identified with ”b from chapter 6. Equation (8.4) corresponds to eq. (6.19a) and X to . Further, Wallace discusses the vanishing of X [Wal98, p. 85] — also found for in eq. (6.20) — in crystals with a center of inversion. On that condition, the simpler form Aˆ(primitive)

–—“” identified by [Wal09, eq. (3.56)] is recovered. However, this simplification does not hold for lattices such as honeycomb or diamond as will be seen in the following model calculation:

3The uniqueness of that solution is ensured by fixing the w.l.o.g. first sublattice in space,S(1) :=0.

Honeycomb model calculation

An exemplary dispersion relation will be derived in the harmonic potential approxi-mation [BH88;LW70] already mentioned in chapter 1. The displacementu^(ia)(q, t) of the a^th basis particle in the i^th unit cell w.r.t. its equilibrium position obeys the e.o.m.

the second equilibrium derivative of the pair interaction potential Õ^ÓR^(is)Ô w.r.t.

displacements of the i^th_a and thej^th_b particle (cf. eq. (1.1)). A plane-wave ansatz u^(is)(q, t) =^Ú m

m^(s)u^(s)(q)e^{i(q·Ris≠Êt)} (8.7)

— withqrestricted to a setB µ1^st BZ compatible with periodic boundary conditions

— recovers from eq. (8.5b) the e.o.m. (1.2) ʲu^(a)_– (q) = ^ÿ

The reformulation of the e.o.m. (8.5b) resembles the transform eq. (5.24) in that it allows to formulate the eigenvalue problem with a self-adjoint and q-antisymmetric dynamical block matrix

D(q) = D^†(q) = D^T(≠q) . cf. eq. (5.26) .

Under the assumption of centrosymmetric nearest-neighbor interactions, the harmonic potential energy Õh

with f: spring constant, a: lattice constant.

4Prof. R. Schilling, personal communication (lecture notes on “Quantentheorie makroskopischer Systeme”), summer term 2009.

The symmetric sum “_(n.n.)” goes over all pairs of lattice vectors that belong to nearest neighbors. Inserting this expression for Õ into eq. (8.6) yields

ˆ²Õh

1 iajb belong to nearest neighbors,

0 else. (8.11)

Note that the first term in the expression (8.10) is zero if a ”=b whereas the second term vanishes, because of the definition (8.11) if a=b. Here we tacitly assumed a bipartite lattice where nearest neighbors to a given sublattice must belong to another sublattice. In the case of two sublattices, we can identify diagonal and off-diagonal blocks of the dynamical matrix as follows:

D^(ss)_–— (q) = 2

Here the ”₁^(s) are nearest neighbors of a particle on sublattice s. For the honeycomb lattice we have identical species m⁽¹⁾ =m⁽²⁾, and the set ^Ó”₁^(s)Ô takes the cartesian

with Ï⁽²⁾≠Ï⁽¹⁾ = fi in addition. Without loss of generality, we choose the coordinate system shown in figure 8.1and call “sublattice 1” the one centered at the origin 0 with Ï⁽¹⁾ = 0. For our purpose, we choose specifically ˆex as the direction of wave propagation, i.e. q=qˆex. This puts us in a position to evaluate the entries (8.12) of

the dynamical matrix for the given honeycomb model:

which holds for b= 1,2, irrespective of q. For the off-diagonal blocks we obtain 1

The s = 2 elements are the corresponding complex conjugates. In summary, we obtain for the Dynamical matrix

D(qˆex) = f

As each of the 4 blocks in eq. (8.13) has diagonal formx- and y-direction decouple, and we can immediately identify a longitudinal and a transversal subspace alongeˆ^x/y. We consider the longitudinal eigenmodes and obtain as criterion for the dimensionless eigenfrequencies ʘ := (m/f)^1/2Ê

Now how is this to be compared with the result from [Wal09] in the perfect-lattice limit? The wave propagation tensor from eq. ([Wal09, eq. (3.56)]) is obtained by summing over all lattice vectors without distinction of sublattices. But dropping that distinction is equivalent to summing over all 4 blocksD^(ab) of the dynamical matrix from eq. (8.9). Before obtaining a low q limit for the acoustic mode from that sum, we give a second argument for that approach by working back from the microscopic to the field-theoretical approach: The plane-wave ansatz (8.7) made for the displacement vectorsu^(is)(t)of single particlesis at specific wave vectorsqneeds to be related to the continuous symmetry-restoring hydrodynamic fields u^(s)(r, t) from paragraph 3.3.3in the perfect-lattice regime. First we assume that the u^(s)_i (t) can be expressed as a normalized superposition of the ansatz (8.7) for every qœB, B µ 1^st BZ being the set of wave vectors compatible with the periodic boundary conditions, Now we insert this into the microscopic particle density,

fl^(s)(r, t) =^N making use of the small-displacement limit and the argument of”(. . .)to approximate Ris by r. This recovers a continuous displacement field

fl^(s)(r, t) = ^N

for which we assume the equations from paragraph 3.3.3to hold within the perfect-lattice regime. Under this assumption we can draw a comparison with the wave equation (3.54b). Having Í⁽¹⁾₀ =Í⁽²⁾₀ = Í0/ 2for the honeycomb lattice With the relation (3.41) for the reversible components of˜j^(s) follows

ˆ²

ˆt²˜j(q, t) =≠^ÿ

a,b

D^(ab)(q)·˜j(q, t) .

The sum^q_a,bD^(ab)(q) over the 4 blocks from eq. (8.13) yields for q=qˆex the low q eigenfrequency limit

˜

ʲ = 6≠2 [cos (aq/ 2) + 2 cos (aq)] = 5

2(aq)² +O¹q⁴² (8.17) which lies above the result ʘ²_≠ from eq. (8.14). Provided that the identification of this second result withA⁽²⁾_xxxxq² is valid⁵,A⁽²⁾ cannot be the correct wave propagation matrix in the perfect-lattice limit. Possibly, the classical low-temperature limit was not performed correctly in [Wal09, sec. 3.4]. It seems then natural to ask which other limit could be obtained for A⁽²⁾ that matches Wallace’sAˆ(primitive)

in primitive lattices but contains additional terms in non-primitive ones.

More insight into the discrepancy to the binary result eq. (8.14) can be obtained from an approximate calculation of the corresponding eigenvector.

A 3≠ ³₄(aq)² ≠e^iaq/2≠2e^≠iaq

≠e^≠iaq/2≠2e^iaq 3≠ ³₄(aq)²

B A u⁽¹⁾_x (q) u⁽²⁾_x (q)

B

=

A O(q³) O(q³)

B

…

53≠3

4(aq)²⁶u⁽¹⁾_x (q)≠

53≠i3

2aq≠9

8(aq)²⁶u⁽²⁾_x (q) = O¹q³² ,

≠

53 +i3

2aq≠ 9

8(aq)²⁶u⁽¹⁾_x (q) +⁵3≠3

4(aq)²⁶u⁽²⁾_x (q) = O¹q³² . (8.18) We are looking for leading order corrections in the eigenvector ¹u⁽¹⁾_x (q), u⁽²⁾_x (q)²and make the ansatz

u^(s)_x (q) = “^(s)+”^(s)aq+Á^(s)(aq)²+O¹q³² .

Subsequently, we determine the parameters {“^(s),”^(s)} by eliminating the leading orders in q in the equations (8.18).The q-independent terms yield

3¹“⁽¹⁾≠“⁽²⁾²= 0 … “⁽¹⁾ =“⁽²⁾. For the linear order we obtain

3”⁽¹⁾≠3”⁽²⁾+i3

2“⁽¹⁾ = 0 … ”⁽¹⁾≠”⁽²⁾ =≠i1

2“⁽¹⁾ (8.19a) and finally, in the quadratic order

1

2“⁽¹⁾≠i2”⁽¹⁾≠4¹Á⁽¹⁾≠Á⁽²⁾²= 0, (8.19b)

≠1

2“⁽¹⁾+i2”⁽¹⁾+ 4¹Á⁽¹⁾≠Á⁽²⁾²= 0. (8.19c)

5Note that this could be easily verified from an explicit evaluation of[Wal09, eq. (3.56)] given on page96.

Within the 3 equations (8.19a) to (8.19c) occur 5 unknowns. The symmetric choice Á⁽¹⁾ =Á⁽²⁾ simplifies theO(q²) equations and yields

”⁽¹⁾ =≠i1 4“⁽¹⁾. Combining this with equation (8.19a) implies

”^(1/2)=ûi1 4“⁽¹⁾. For the normalization wechoose

“⁽¹⁾ = 1

Ò1 + (¹/4aq)²

and obtain, up to linear order in q, the normalized longitudinal eigenvectors within our approximation

u⁽_x^1/2)(q) = 1

Ò1 + (¹/4aq)²

31ûi1 4aq

4

. (8.20)

For finiteq, aq-depenent phase shift occurs between the displacement fields at the two basis positions apparently. This is plausible as e.g. an in-phase oscillation would imply zero strain for the spring connecting the two basis atoms. A treatment of the system with a single displacement field however assumes fixed q-independent phase shifts among the basis positions. This is also visible from the summation overu^(a) in eq. (8.16) where the O(q)terms from eq. (8.20) cancel out. A final comment to these findings will be given in the conclusions on pp.123.

Part III.

Aspects of Quasiperiodic crystals

9. Extensions to chapter 5

Due to the characteristic structure of their reciprocal lattice, the generalization of the Zwanzig–Mori approach from chapter 5to quasicrystals is a nontrivial task. This chapter points out major challenges of that endeavor and marks some strategies to deal with them.

9.1. Reciprocal lattice and Bogoliubov inequality

The restriction to periodic crystals in section 2.2 was obtained by requiring the set {G1, . . . ,GI} in eq. (2.7) to consist of linearly independent vectors. In the following, we will examine the consequences of relaxing this restriction, allowing for quasiperiodic crystals with indexing dimension D > d. More precisely, a basis {G1, . . . ,GD} of Gnow has to be integrally independent,

The restriction to periodic crystals in section 2.2 was obtained by requiring the set {G1, . . . ,GI} in eq. (2.7) to consist of linearly independent vectors. In the following, we will examine the consequences of relaxing this restriction, allowing for quasiperiodic crystals with indexing dimension D > d. More precisely, a basis {G1, . . . ,GD} of Gnow has to be integrally independent,

Im Dokument Phonons and Elasticity in Disordered binary Crystals (Seite 93-0)