On the attenuation of anharmonic adsorbate vibrations
D. Bonart a, 1, R. Honke a, A.P. Mayer a,., p. Pavone a, U. Schr6der a, D. Strauch a, R.K. Wehner b a lnstitutfiir Theoretische Physik, Universitdit Regensburg, 93040 Regensburg, Germany
b Institutfiir Theoretische Festk6rperphysik, Universit~it Miinster, 48149 Miinster, Germany
Received 15 May 1997; received in revised form 11 August 1997
Abstract
Adsorbate vibrations on crystal surfaces are discussed as candidates for the existence of breathers in microscopic solid state physics. The attenuation of coherently excited adsorbate vibrations due to their anharmonic interaction with the quantum and thermal fluctuations of other phonon modes in the system is investigated. On the basis of ab initio calculations of harmonic and anharmonic force constants for the system Si(1 1 1) : H, the lifetime of the zone-center Si-H stretching mode is determined as function of temperature. The results compare well with available experimental data and yield a first estimate for the lifetime of localized modes with amplitudes smaller than the mean square displacements due to thermal and zero-point motion. A theoretical treatment of the attenuation of nonlinear localized adsorbate modes is given in the opposite regime, i.e. large driven amplitudes as compared to the thermal and zero-point motion. 6) 1998 Elsevier Science B.V.
In spite of the great interest that intrinsic localized modes (ILMs) have attracted since their discovery [1-4], no
clear experimental evidence has yet been found for their existence in microscopic solid state systems. Among the
various systems envisaged as possible candidates for an experimental discovery of ILMs [5], surfaces and edges of
crystals have been considered [6,7] and more recently also surfaces covered by adsorbates with the adsorbate particles
much lighter than the subtrate atoms [8]. In their phonon dispersion relation, these systems often exhibit very flat
branches associated with vibrations of the adsorbate atoms [9]. They indicate a weak inter-site interaction between
these atoms. Hence, localization in the directions parallel to the surface can be achieved with comparatively moderate amplitudes. An important question for the experimental detection of ILMs is obviously their lifetime. This problem
is addressed in the present paper. I f an external field couples linearly to the displacement coordinates of the atoms,
as is the case in the ILM excitation mechanisms suggested recently [10], the quantum-statistical expectation values of the displacement coordinates will be non-zero. Via the anharmonicity of the lattice potential, these expectation
* Corresponding author. Tel.: 4-49 941 943 2033; fax: 4-49 941 943 4382; e-maih [email protected]. 1 Present address: Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287-1504, USA.
values couple to the incoherent zero-point motion and thermal fluctuations of the adsorbate atoms as well as of the substrate atoms, which gives rise to damping. Small-amplitude ILMs, which are not strongly localized, may be regarded as envelope solitons [ 1]. For a first and rough estimate of their lifetime, one may take the damping constant of extended phonon modes. As a prototype example, we have studied the adsorbate system Si(1 1 1) : H, for which a number of experimental and theoretical investigations concerning harmonic and anharmonic vibrational properties have been carded out. In particular, a two-phonon bound state has been identified experimentally by sum-frequency generation [ 11] and confirmed theoretically on the basis of ab initio calculations [ 12]. Later, ILMs have been found as solutions of the classical equations of motion for the hydrogen displacements, again with the help of ab initio data for the force constants [9]. In Section 2, this calculation is summarized.
The intrinsic damping of the zone-center stretching mode of the system Si(1 1 1) : H had been determined exper- imentally as function of temperature by infrared absorption spectroscopy [13,14]. By combining density functional perturbation theory [ 15] and the frozen phonon approach [ 16], a parameter-free calculation has been performed of the anharmonic force constants needed to determine the contribution to the damping constant of the zone-center stretch- ing mode resulting from effective four-phonon scattering processes. These are dominant at elevated temperatures and are found to reproduce the experimental data quite well in the temperature regime between 200 and 400 K.
The situation analyzed in Section 3 corresponds to a regime, where the mean square displacements of the hydrogen atoms due to the zero-point motion considerably exceed the amplitudes of the ILMs. A theoretical approach to treat the problem of ILM damping in the opposite regime of large amplitudes and small quantum fluctuations is presented in Section 4. In this regime, nonlinear damping may become important. Effects of non-local and nonlinear damping on the temporal evolution of an ILM are demonstrated for the example of the one-dimensional discrete nonlinear Schr6dinger equation.
2. Intrinsic localized modes in the system Si(1 1 1) : H
To search for ILM solutions of the classical equations of motion for the displacements of the hydrogen atoms on the (1 1 1) surface of a silicon crystal, the approximation of a rigid substrate is made. The equations of motion for the displacement vectors u(/) then have the simple form
m a/i~ (l) = K~ (u(l)) + Z tp~ ( l i ' )u~ (l ' ) + Jc, (l). (2.1) 1',,6
Here, l labels the lattice sites, ma is the mass of an adsorbate atom, and J(l) is an external force acting on the adsorbate atom at site 1. The anharmonic on-site force K = - V Vo is expanded up to third order in the displacement coordinates and the expansion coefficients have been determined by ab initio calculations using the frozen-phonon method as in [12] and also a combination of ffozen-phonon and density-functional perturbation theory.
I 3 1 4 "1"- a22u2] (u 2 -+- u 2) --t- 1 2 VO(U) = ga30u 3 q- ~--~a40u 3 q- l[a12u3 ~-~ao4(u 1 q- u2) 2. (2.2)
In (2.1), tp~ ( l l ' ) denotes harmonic on-site and inter-site force constants. The latter have been restricted to nearest neighbors only and, for simplicity, have been derived from a central potential of the form
VI (r) = at (r - ro) + b / ( r - r0) 2, (2.3)
where r is the actual and r0 is the equilibrium nearest-neighbor distance. The two parameters al and b, have been fitted to the dispersion curve of the stretching mode as obtained from the ab initio calculations. For the stretching mode, almost perfect agreement is achieved between the dispersion curve determined from (2.3) and the ab initio data. We note, however, that for a proper description of the bending mode dispersion curves, a central potential
58 D. Bonart et al. / Physica D 119 (1998) 56-67
between the hydrogen atoms would not be sufficient because of the lower symmetry of the substrate surface as compared to the triangular lattice of the hydrogen adlayer.
To seek ILM solutions in models of the type (2.1) with on-site interactions, it has been suggested to start with the so-called anti-continuous limit [ 19,20]. An alternative method, which is also applicable in systems with translational invariance, is to start from the continuous limit of small amplitude and large spatial width of the ILM. Here, ILMs may be regarded as envelope solitons associated with a carrier wave having wave vector q and frequency o~0 at a maximum or minimum of the phonon dispersion relation,
u ( / ; t ) = ei[q'R(l)-°J°t]w(q)A(r, r) , (2.4)
where R(/) is the equilibrium position of the hydrogen atom in the Ith surface elementary cell, w(q) is the modal displacement vector (in our case of stretching modes it is parallel to the surface normal), and the complex amplitude A is assumed to depend continuously on the space coordinate parallel to the surface, r, and on a slow time variable r.
The slow variations of A are governed by the two-dimensional nonlinear Schr6dinger equation
The indices/~ and A only run over 1 and 2. The parameters occurring in (2.5) are the curvatures of the phonon dispersion sheet of the corresponding branch (in our case the stretching modes) at the cartier wave vector q and the effective quartic anharmonic coupling constant/C4 which is easily calculated from the interaction potentials (2.2) and (2.3) [21]. The relative signs of the parameters in (2.5) decide on the existence of envelope solitons. For the /~-point at the boundary of the surface Brillouin zone, the conditions for their existence are satisfied. The envelope soliton solution of the two-dimensional nonlinear Schr6dinger equation has been used as an initial guess for a numerical search routine to solve the equations of motion (2.1) within an extended rotating wave approximation: The time-dependence of the displacement is of the form
u(/; t) = U0(/) + U1 (l) cos(wt) + . . . (2.6)
An example for an exact numerical ILM solution is shown in Fig. 1. Since the carrier wave is a zone-boundary mode, the vibrations of neighboring atoms are anti-phase, while in Fig. 1, only the modulus of Ul (l) is displayed. The ILM frequency w may be regarded as a function of the maximum vibrational amplitude U m = max{IUl(/)l}. For values of U m < 0.01r0, the relation between w and Um follows the predictions of the nonlinear Schr6dinger equation, while for Um ~ 0.01r0, a cross-over takes place to a behavior of the frequency as function of Um that follows from a simple Poincar6-Lindstedt calculation of the anharmonic frequency shift of a single hydrogen atom vibrating according to the equation of motion (2.1) with, however, all other hydrogen atoms kept fixed at their equilibrium positions.
It is well known that in two dimensions the solitary wave solutions of the nonlinear Schr6dinger equation are unstable [ 17]. However, this does not imply instability of the ILM solutions in the discrete system. A linear stability analysis for some of these localized modes has been carried out along the fines of refs. [6,18,19]. It was found that the modes were either stable or their Floquet exponents had real parts of magnitude below the numerical accuracy. This is in agreement with simulations of the ILMs carried out by integrating numerically the equations of motion for the hydrogen atoms.
In addition to the ILMs of the type displayed in Fig. 1, ILM solutions have been found with two instead of one hydrogen atom having maximum amplitude. They turned out to be unstable in the linear stability analysis.
It should be noted that within the rigid substrate approximation a set of three nonlinearly coupled discrete nonlinear Schr6dinger equations may be derived for an approximate description of the classical dynamics of the hydrogen atoms.
D. Bonart et al./Physica D 119 (1998) 56-67 59
Fig. 1. Modulus of vibrational amplitudes IU1 (1) l of an intrinsic localized mode solution for H adsorbate vibrations.
3. Attenuation of the zone-center adsorbate modes
The considerations of the preceding section have been entirely in the framework of classical dynamics. When moving to a quantum-mechanical description, the displacement coordinates in the equations of motion for the atomic displacements have to be replaced by Heisenberg operators and the coupling to the substrate modes has to be taken into account, too. Performing the quantum-statistical average of the equation of motion for the atomic displacements, one is led to the following equation:
Here, l = (L, K) is now a composite index denoting both the elementary cell L and the sublattice K. In the absence of anharmonicity, the right-hand side of (3.1) would only consist of its first two terms. The external field would drive the system into a coherent state, and the classical equation of motion would be recovered with u replaced by (u). The anharmonicity of the lattice potential gives rise to the additional forces V[°°](/) which may be regarded as a functional of the expectation values of the atomic displacements, (u(/; t)) [22]. It can be expanded around the equilibrium in the absence of any external field as follows:
OG
v J(t; ,)= [ dC I',8
O~ O0
1 E / / n (3) , . t ' ) + ~ dt t dt F ~ × ( l l 1 ; t - t', t r - (u~(l'; t ' ))(u×(ln; tn)) + O((u)3). (3.2)
I ' , ln,flY-oO - 0 ~
The expansion coefficients are the self-energy H and the vertex functions F (n), n = 3, 4 . . . . . which are taken in real space and in the time domain.
60 D. Bonart et al. / Physica D 119 (1998) 56-67
For sufficiently small amplitudes (u), i.e. for sufficiently small driving forces J, one may neglect the higher order terms in (3.2) and consider the self-energy only. Transforming from displacement coordinates u~(LK) to normal coordinates A(qj) via
u(/K) = ~ eiq'R(L)w(Klqj)A(qj), (3.3) qJ
the potential energy can be written in the following form:
with the anharmonic coefficients Vn (ql jl . . . . . qn in) describing phonon-phonon interaction. The self-energy of the phonon mode q j , FIqj (09) as defined in [22], is related to the function FI in (3.2) in a straightforward way. The quantity Fqj (o9) = -Im(FIqj (o9)) is called the damping function of the mode q j .
Here, we are interested in the attenuation of the zone-center stretching mode (0s) for which experimental data are available [ 13,14]. At zero temperature, intrinsic attenuation of a phonon mode can only occur via spontaneous decay. Because of the large values of the stretching mode frequencies as compared to those of the bending and substrate modes, decay processes leading to attenuation of the stretching modes involve at least four phonons of lower frequency (e.g. three bending modes and one substrate mode). A quantitative evaluation of these processes does not seem feasible at present because of the difficulties in obtaining reliable anharmonic coupling constants up to fifth order.
At finite temperatures, scattering processes become possible. The lowest order diagrams giving rise to a non-zero contribution to the imaginary part of the self-energy due to such scattering processes [22,23] are shown in Fig. 2. The first, fourth and fifth of these diagrams contain quarfic vertices. The quartic coupling constants V4 (0s, 0s, q j , - q j ) are visualized in Fig. 3, where the function i~4(q, 09) = ~-~j V4(0s, 0s, qj , - q j ) x e/[(09 - 09qj)2 + e2] with e
corresponding to 2 cm -1 has been plotted for wave vectors on the edge of the irreducible triangle of the surface Brillouin zone. These anharmonic coupling coefficients have been obtained by a combination of density functional perturbation theory (DFPT) and the frozen phonon technique in the following way: Dynamical matrices have been calculated for a repeated slab geometry with a "frozen" zone-center stretching mode displacement with different amplitudes A. With these data, the second derivative of the dynamical matrices with respect to A at A = 0 has been carded out numerically. Contracting the second derivatives of the dynamical matrices for wave vector q with the modal displacement vectors w(qj) and w ( - q j ) gives the quartic anharmonic coupling coefficients shown in Fig. 3. (For more details see [24].) We would like to emphasize that these calculations do not contain any parameters. The electronic system is treated within the local density approximation [25,26] and the Kohn-Sham equations are solved with a basis set of plane waves. For the interaction between the ions and the valence electrons, soft norm-conserving pseudopotentials have been used [27,28].
Fig. 2. Lowest order diagrams contributing to the intrinsic lifetime of the H-stretching modes.
D. Bonart et al. / Physica D 119 (1998) 56--67 61
V4(q ,o) (arb. units)
T
or i _ _
o i
|
I
Fig. 3. Function Q4 (q, w) corresponding to quartic anharmonic coupling coefficients as defined in the text.
Fig. 3 shows that the quartic anharmonic coupling of the zone-center stretching mode is predominantly to other stretching modes and to bending modes. This suggests that one may restrict the propagators in the diagrams of Fig. 2 to those of the adsorbate modes.
The value of the damping function f q j (09) at the frequency of the phonon mode q j can be defined as the damping constant of this mode, if the damping function is slowly varying over frequency intervals having width of the order of f q j (Ogqj). This would not be the case, if the diagrams in Fig. 2 were evaluated with bare propagators, because of the flatness of the stretching and bending mode branches in the phonon dispersion relation. However, when taking account of the finite lifetime of the scattering products, the damping function FOs (09) becomes sufficiently smeared out that the definition of a damping constant in the way described above becomes meaningful.
In writing (3.5) we have assumed that the damping constants of the stretching modes (Fs) and of the bending modes (Fb) (as well as the thermal occupation numbers nb of the bending modes) are approximately independent of the wave vector.
62 D. Bonart et al./Physica D 119 (1998) 56--67
The effective quartic coupling constants V# ff between stretching and bending modes have been evaluated within the rigid substrate approximation, using the potential (2.2). However, the frequencies of all modes and the polarization vectors e(qb) of the bending modes in the x-y-plane have been taken from the full DFPT calculation with the adsorbate and substrate atoms mobile. The following expression is obtained:
where N is the number of unit cells and ms is the mass of a hydrogen atom. For the intrinsic attenuation of the bending modes, two-phonon decay processes via cubic anharmonicity are
allowed by energy conservation and may be regarded as dominant. They have been evaluated quantitatively for the zone-center bending modes from the well-known formula [22]
The cubic anharmonic coupling coefficients in (3.7) have been determined fully ab initio in the same way as the quartic coefficients of Fig. 3. Eq. (3.5) has then been solved self-consistently for Fs with the result shown in Fig. 4 and compared to experimental data by Dumas et al. [13] obtained by infrared absorption spectroscopy. Recently, these experiments have been repeated and extended to lower temperatures [14]. For temperatures down to 200 K, good agreement with the experimental data of ref. [13] and even better with those of ref. [14] has been achieved, when the effective quartic coupling coefficients in (3.5) are evaluated using in (3.6) the experimental values for the frequencies Ws and Wb instead of the values obtained from the ab initio calculations. If the latter were used, the
10.0
5
I 2 E 0 1.0
~ 0.5
~ 0.2 "1- ~ 0.1
1.1_ 0.05
0.02
i i i
• 10 \ \ 8
o ,oo
~ Temperatur (K,
\\ \\\
, \ ~ ,
5 10 15
1/T X 1000 (K -1)
60(
20
Fig. 4. Damping constant of the zone-center H stretching mode on the (1 1 1) surface of silicon as function of temperature (FWHM corresponds to 2/'s). Calculations with experimental frequencies (solid) and bare frequencies (dashed) in the effective quartic vertex. Experimental data taken from ref. [13]. Inlay: damping constant of the zone-center bending modes as function of temperature.
D. Bonart et al./Physica D 119 (1998) 56-67 63
results for the damping constant would be higher by roughly a factor of 4. The reason for this is the following: The strong zero-point motion of the hydrogen atoms gives rise to large anharmonic shifts of the frequencies of the stretching and bending modes at zero temperature. They can be estimated from a comparison between the ab initio values for the frequencies (i.e. the bare frequencies) and the experimental values (i.e. the renormalized frequencies). This difference amounts to -~40 c m - 1. A perturbation calculation for the anharmonic shift of the stretching mode frequency up to first (second) order in h gives a value of - 1 0 5 cm -1 ( - 6 7 cm -1 ). The latter calculation involves
anharmonicity up to sixth order and its result is very sensitive to the numerical values of the anharmonic coupling coefficients. At the same time, strong compensations occur between the different contributions to the effective
quartic coupling coefficients in (3.6), as was predicted by Burke et al. [23] on the basis of model assumptions. These problems render an accurate determination of the damping constant of the stretching mode very difficult.
4. Attenuation in the regime of large amplitudes and small quantum fluctuations
The system discussed in the preceding section is characterized by strong zero-point motion of the adsorbate atoms.
In fact, the maximum amplitude of the ILM of Fig. 1 is smaller by a factor of 2-3 than the root mean square of the hydrogen displacements normal to the surface due to zero-point motion. In this regime, neglecting higher orders of
the coherent displacement amplitudes (u) should be a reasonable approximation. In this section, we consider the opposite regime of high amplitudes (u) and small quantum fluctuations. To characterize this regime, one may define dimensionless parameters
where Cn, n = 2, 3 . . . . . are effective force constants of nth order and 8u = u - (u). In the regime considered in this section, Pc1, Pc2 are of order 1, while pq 1, Pq2 << 1. Now, higher order terms have to be taken into account in expansion (3.2). On the other hand, in a diagrammatic evaluation of the self-energy and the vertex functions, only those diagrams have to be taken into account that are of lowest order in h at zero temperature. It may be shown
that the diagrams of first order in h all consist of one ring. Examples of such ring diagrams for the vertex functions F (3) and F (4) are shown in Fig. 5. Summation of this class of diagrams corresponds to the solution of the following coupled system of equations:
02 (u,~(l)) = Ka((u(l))) + y ] ¢c¢(ll')(Ul~(l') ) + Jc~(l) Ot 2
l',fl
1 x-, 02 +
~, a(u~(l))O(u×(1)) Ka( (u(l) ) ) F#y (ll),
-~ Fa~ (ll') = Su~ (ll') + S~c~ (l'l),
0 0 -~Sc¢(ll') = Gu~(ll') + Z Cay(ll")F~,~(l"l') + y ~ a(uy(l))
Derivative with respect to time is denoted by a dot. Equations of the type (4.2)-(4.5) have been considered earlier in a study of finite number of oscillators coupled by cubic anhannonicity [29]. Recently, Hizhnyakov [30] has studied a model of a classical oscillator with large amplitude coupled to a system of quantum oscillators in such a way that the Hamiltonian of the quantum oscillators is bilinear in the displacement coordinates and momenta. For this model, the approximation underlying Eqs. (4.2)-(4.5) becomes exact.
If the driving fields Ja (l; t) are acting at times t > 0 only, (4.2)-(4.5) have to be solved with the initial conditions (u(l; 0)) = 0, Sag(ll'; 0) = 0, while F~g(ll'; O) and G~g(ll'; 0) have to be set to their static equilibrium values in the absence of a driving field. By solving (4.3)-(4.5) by iteration and inserting into (4.2), one may generate the form (3.1) with (3.2) for the equation of motion for the coherent amplitudes with the ring diagram contributions in the self-energy and vertex functions.
We may note that the Eqs. (4.2)-(4.5), which involve equal-time correlation functions only, are amenable to numerical simulations. Another application of these equations has been to describe, within their framework, a purely vibrational form of self-induced transparency [31]. This has been done for the special model with equations
of motion (2.1), where
K~(u) = ~b3 ~ ugu× leggy I (4.9) g,y
with (Eag×) being the Levi-Civith tensor and ~3 being a cubic anharmonic coupling constant. Furthermore,
The first term corresponds to linear damping of each individual oscillator, while the second one corresponds to a
wave vector-dependent (non-local) part of the self-energy, which would not be felt by an extended zone-center mode.
The third term is a nonlinear on-site damping. This type of nonlinear damping is distinct from the one considered
in [33] as the latter conserves Q while the nonlinear damping term in (5.3) does not.
Results of a numerical integration of (5.2) are shown in Fig. 6 with initial conditions corresponding to a static
ILM solution localized at the site l0 with ratio U(lo) /U(lo ± 1) ~ 100. (It may be noted that via the transformation
V(l; t) = U(l; t) exp(i/zt), a static ILM solution of (5.2) can be transformed into an oscillating ILM solution of
the nonlinear Schrrdinger equation with the parameter I2 replaced by S-2 - / z . ) In addition to a reduction of the
Q
100
8O
60
4O
20
0 0
i I !
\, \.'x "'-..
, " ~ 5 10 15 t 20
3
W
2
0 i |
0 5
/'1 /
/ / 10 15 t 20
Fig. 6. Temporal evolution of the quantity Q (left panel) and the ILM width W (right panel) for three different sets of parameters for local linear (F0), non-local linear (FI), and local nonlinear (F) damping: F 0 = 0.1, FI = 0, y = 0 (solid); F0 = 0.1, FI = 0.01, y = 0 (dashed-dotted); F0 = 0.01, F 1 = 0, F = 0.0009 (dashed). Further parameters: U(lo) = 10, ~b =/C4 = 1.
66 D. Bonart et al. / P hysica D 119 (1998) 5 ~ 6 7
energy and the quantity Q, the damping terms in the nonlinear equations of motion also lead to a broadening of the solution that was initially highly localized. This broadening is characterized by the width W defined as
W = N -1 IU( l )12( l - 10) 2 , (5.4) l=l
where N is the number of sites in the chain. In Fig. 6, the evolution of W and Q for the case of non-local damping (/'1 = 0.01,/ '0 = 0.1, ~/ = 0) and of nonlinear damping (F1 = 0, F0 = 0.01, ~/ = 0.0009) are compared to the situation of purely local linear damping (/-'1 = 0, F0 = 0.1, ~/ = 0). In the latter case Q decays exponentially, while this is not the case for non-local and nonlinear damping. In the non-local case, the parameters are chosen such that the initial damping rate 0 Q / O r is approximately the same as for purely linear and local damping. The results displayed in the figure show that linear damping is much more effective than nonlinear damping over long time scales, as would have been anticipated, and additional non-local damping leads to fast broadening of the ILM.
6. Conclusions
Intrinsic localized modes have been predicted to exist as solutions of the classical equations of motion for the adatom displacement coordinates in adsorbate systems. To make predictions about the observability of such modes, knowledge of their lifetime is paramount. The problem of intrinsic attenuation of adsorbate modes has been discussed in the regime of strong quantum fluctuations and moderate amplitudes of the ILMs (I) and the regime of small quantum fluctuations and large amplitudes of the externally excited vibrational mode (II). A realization of the first regime is the system Si(1 1 1) : H, for which quantitative results have been presented concerning the damping constants of (extended) zone-center stretching and bending modes. These calculations in the framework of perturbation theory face the problem of strong compensations of different terms and therefore require high precision in the determination of harmonic and anharmonic force constants. In addition, due to the strong zero-point motion of the adatoms, low order perturbation theory can become unreliable.
For the second regime, a system of coupled equations has been derived for the coherently driven "classical" displacement coordinates and equal-time correlation functions. From these equations, an effective equation of motion may be derived for the "classical" displacement coordinates of the adsorbate atoms with nonlinear damping terms. First results of numerical simulations suggest that the evolution of the width of an ILM is sensitive to the wave vector dependence of the self-energy, i.e. the nonlocality of the damping, and to the extent to which the damping is nonlinear. A more detailed study of the attenuation of anharmonic adsorbate vibrations is underway.
Acknowledgements
We would like to thank P. Jakob for having communicated to us data prior to publication. Financial support by the Deutsche Forschungsgemeinschaft (grant no. Ma 1074/5 and Ma 1074/6) is gratefully acknowledged.
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