Self-consistent approximation

The original aim of this thesis was understood as the retrieval of well-defined acoustic phonon modes for quasicrystals. Their existence is suggested by inelastic scattering

5That the transform(T,P)can be written in the form in (9.6) is a physical requirement. Point groups of superspace groups (unlike point groups of generalD-dimensional space groups) are reducible into a d-dimensional and a p-dimensional representation by orthogonal matrices (cf. [JCB07, sec. 2.4]).

experiments of e.g. de Boissieu et al. [De +07]⁶. This was believed to involve a wave equation like eq. (5.12) with a somehow reinterpreted dynamical matrix ». [JCB07, paragraph 6.4.1] states that such an exact form — given there in eq. (6.9) for periodic crystals — does not exist in quasicrystals. Considering the potential-expansion ansatz, the authors point out the impossibility to reduce the number of coupled equations to Bd at given BZ wave vectorq like in periodic crystals. A connection to experimental observations can be drawn by their subsequent discussion: They make a continuum approximation in the long-wavelength limit to obtain an e.o.m.

(d=p= 1 w.l.o.g.)

ˆ²

ˆt²u(r, t) = T(r) ˆ²

ˆr²u(r, t) (9.7)

with a space-dependent force constant per mass density T. The physical quantities can be embedded in superspace (D= 2, arbitrary but fixed internal space component), and with

u(r, t) =u¹r^Î, r^‹2e^≠iÊt, T (r) =T¹r^Î, r^‹2 eq. (9.7) becomes

≠ʲu¹r^Î, r^‹2=T¹r^Î, r^‹2 ˆ²

ˆr^Î2u¹r^Î, r^‹2. (9.8) u¹r^Î, r^‹2is then written as a Bloch function [MS66] for a given, sufficiently small wave vectorq and expanded in a Fourier series

u¹r^Î, r^‹2= ^ÿ

gœG

“^q(g)eⁱ[(^gÎ+q)^rÎ+g‹r‹] and similarly

T ¹r^Î, r^‹2= ^ÿ

gœG

S(g)eⁱ[^{gÎrÎ+g‹r‹}]. Inserting this into eq. (9.8) finally leads to

ʲ“^q(g) = ^ÿ

˜ g

S(˜g)¹q+g^Î≠g˜^Î22“^q(g≠g)˜ . (9.9) This eigenvalue problem has some similarity to the discussion of electron wave functions in an external potential [MS66, chap. 8 et seqq.]. There, as in eq. (9.9), strong coupling occurs between modes whose wave vectors differ by a reciprocal lattice vectorg. The coupling constantsS(g) are strongest ifg corresponds to a prominent Bragg peak. A “pseudo-Brillouin zone” is defined by considering just strongest Bragg peaks near the origin as reciprocal lattice points [Nii89]. [JCB07, Figure 6.2] shows how such a “pseudo-acoustic mode” reappears when the masses of a periodic diatomic harmonic chain are weakly modulated over a larger unit cell. Figure 9.1 tries to capture the same idea from the schematic dispersion relations of the linear chain

q[a^≠1] Ê[K/m]

fi 2fi

Ô5

0

optical mode average mode acoustic mode

Figure 9.1.: Dispersion relations for a linear chain. The solid lines show the acoustic and optical mode obtained from alternating spring constantsK_1/2 up to the second BZ boundary. The dashed line shows the acoustic mode from the same linear chain with only one average spring constantK˜ = (K1+K₂)/2.

shown in figure6.1 with identical particle masses mand alternating spring constants K_1/2. For the calculation, those were chosen to K₁ = 1 and K₂ = 1.5 in units of an arbitrary spring constantK. The acoustic and optical mode are plotted up to the boundary of the second Brillouin zone. The dashed acoustic mode was obtained for a monoatomic linear chain with nearest neighbor interactionsK˜ = (K1 +K₂)/2 and a lattice constanta/2. This can be understood as an average⁷ structure with a BZ twice as big as the original structure. One can of course work in the reverse direction and introduce larger and larger unit cells with weakly modulated spring constants.

The resulting BZ will only be a fraction of the original. But a “pseudo-acoustic”

mode — in the sense of the dashed line in figure9.1and [JCB07, Figure 6.2] — could certainly still be identified. Bearing in mind that a quasicrystal can be described as the limit of a series of “periodic approximants”⁸, this seems to give a qualitative explanation for the measurements from [De +07, fig. 3]. Their identification of the

“acoustic character” of a mode by a normalized inelastic intensity is explained in the methods part of that publication (see also [De 11] for a detailed review).

Now what could be done if acoustic-like modes were indeed not obtainable from the Zwanzig–Mori eqs. (5.9) under consideration of all couplings? How subsets of the reciprocal lattice can be excluded in a systematic and easy way will require further studies. In some quasicrystals exhibiting scaling symmetry — including the Fibonacci chain — points in the reciprocal superlattice can be mapped onto each other by discrete hyperbolic rotations [Jan92]. This allows to restrict the reciprocal

6See [JCB07, paragraph 6.5.1] for the theory behind these measurements.

7Note that this proceeding differs from the definition of areducedspring constantK1K2/(K1+K2) which would yield the correct long wavelength limit.

8Term for a periodic structure created when approximating irrational length ratios in the con-struction of quasicrystals by rational numbers [JCB07].

lattice subset and the q-integrals in eq. (5.9) in a self-consistent way: Figure 9.2 shows part of the reciprocal lattice projection in the Fibonacci chain with a cutoff criterion^---g^‹---Æk_cut^‹ for lattice points in the subsetG_∆µG. The hyperbolic envelope

k^Î k^‹

k_cut^‹

≠k_cut^‹

Figure 9.2.: Construction of a subsetG_∆µG of the reciprocal lattice. The hyperbolic zone is shown for two different lattice points. The projection boundaries are marked by dotted horizontal and the projections by dashed vertical lines. The size of the projected points was obtained from eq. (9.5) with Î = 5^1/4 to illustrate the hierarchy in Bragg peaks.

-

--(k^Î≠g^Î)(k^‹≠g^‹)^---= 1 is drawn for two lattice points. We conjecture that the set

Ó(k^Î, k^‹)œR˜^2--_-^--_-(k^Î≠g^Î)(k^‹≠g^‹)^---<1^Ô

contains no other lattice points than (g^Î, g^‹)and can therefore be used at given k^‹_cut to restrict the projection in eqs. (5.9) by

|q|< 1

2k_cut^‹ . (9.10)

Then eqs. (5.11) are recovered with the subset G_∆ of G obtained from the cutoff.

The case that|q| is much smaller than the boundary set in eq. (9.10) can then be reestablished as the long-wavelength limit. Connections of this ansatz to the concepts of periodic average structures [SH99] and the average unit cell [SBW16] should be investigated.

Finally recall that the Bogoliubov argument (5.6) gives a lower bound to the divergencies of the autocorrelationslimqæ0

e|”flg(q)|^2fproportional to the scattering intensities. This raises the question whether a hierarchy of the (in any case diverging) decay times among the slow variables ”n_g(q, t) might exist.