Generalized-disorder collective-boson mode-softening universality principle

Journal of Non-Crystalline Solids 40 (1980) 453-467 © North-Holland Publishing Company

GENERALIZED-DISORDER COLLECTIVE-BOSON MODE-SOFTENING UNIVERSALITY PRINCIPLE Edward SIEGEL Queen Mary College, University o f London, London, England and C.N. Pq., S~o Jose Dos Campos, Brazil

A generalized-disorder coilective-boson mode-softening universality principle (GDCBMSUP) is is proposed as a dynamic description underlying all disorder induced properties in all types of disordered systems. Via the unique features of GDCBMSUP: isotropic non-monatonicity, existence of a unique set of disorder parameters at any one set of system variables (temperature, T, pressure, P, concentration of fth impurity, xj, grain size, r .... ), scaling of non-monatonicity and the derived disorder parameter set with T, P, Xl' r ..... a law o f corresponding states between similar collective bosons of different systems, or of different collective bosons in the same or similar systems, and Anderson localization of collective bosons and/or collective boson scattered fermions, we argue that the generalized disorder can be defined and certain qualitatively unique generalized disorder induced properties can be understood. A dynamic collective boson definition of generalized disorder is arrived at which qualitatively designates and classifies the plethora of disordered systems in nature, and in mathematical models. Application is made to phonons, plasmons and magnons (collective boson modes) in many diversely and ostensibly differently disordered systems of many diverse physical types. Application to less ordinately defined disorder: optical phonons in crystals, the Peierls transiton, the Jahn-Teller transition, one-dimensional magnets, random walk Brownian motion, classical and quantum optical and acoustical wave reflection off mirrors, and the classical mechanics of a ball bouncing off a wall are proposed to illustrate the broadness of the generalized disorder universality-principle and its collective-boson, mode-softened dynamic consequences on dispersion relations in the systems.

1. Introduction A rather unique universality principle appears to exist for different collectivebosons in distinctly different disordered systems. Various manifestations of this GDCBMSUP have existed since the pioneering work of Landau and F e y n m a n on phonons and rotons in q u a n t u m liquid helium, but a broader definition of this dynamic definition of disorder allows one to trace back this principle into the classical mechanics and electromagnetic theories of the eighteenth and nineteenth centuries. We here briefly attempt to point out this underlying physical order in the study of desparate disordered systems, both generally accepted as disordered in some nonequivalent way, or not generally considered as disordered, but obeying the same collective-boson dynamic properties.

453

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E. Siegel/ A universalityprinciple

2. Discussion

This universality principle, a manifestation of Haken [1] synergetics, is seen experimentally in many systems and derived theoretically by many authors [ 2 - 5 ] , especially pointed out in its generality by Hubbard and Beeby [2], but not well investigated in generality nor utilized generally. It is indigenous to some sort of generalized disorder causing mode-softening of the collective-boson mode dispersion relations, a dynamic result. This mode-softening, called by some negative-dispersion, exists at wavevectors above k = 0 but for small enough k such that the wavevector is still a well defined quantum number in the particular system under study. Damping of the collective-bosons, as seen by the growth of the imaginary part of the collective-boson frequency as the wavevector increases, lm w(k), still does not destroy the mode-softening with its concomitant properties to be detailed, but can impair detailed experimental corroboration of GDCBMSUP and detailed calculations of its specific physical manifestations. Such a collective-boson mode-softening can lead to a totally unique dynamical definition of disorder in very desparate different systems, identical in each system for each relevant collective-boson studied, in terms of the dispersion relation of the collective-boson mode excitations above the ground state of the system versus details of the system ground state, which vary qualitatively as well as quantitatively with the type and degree of disorder and type of system. This unique dynamical definition of disorder can be experimentally derived through the forward-scattering, low-angle-scattering structure factor of the system, rather than by a qualitatively different detailed, system by system, disorder definition in each case. The structure factor static limiting case, S(k), contains the generalized-disorder in its k = 0 forward scattering amplitude and its rise to the lowest wavevector peak in S(k). For time-dependent systems, the full S(k, w) must be considered, allowing inclusion of various random processes in space and time within this dynamic definition of generalized-disorder, with all the indigenous properties to be detailed here. Collective-bosons in disordered systems seem to all have the same five indigenous, universal, synergetic, general properties independent of collective-boson type, or disorder type or degree, or system type: (1) isotropic-non-monatonicity (negative dispersion); (2) existence of a disorder parameter set: (at any one variable set: temperature T, pressure P, concentration of the flh impurity xj, grain size r .... ) and at the first maximum in the collective-boson dispersion relation: (a) wavevector of the first maximum kc (equivalent to the maximum frequency we and thus to collectiveboson effective temperature hwc = kB0c). (b) curvature at the first maximum K(kc) - (this is maximum and negative) (c) negative group velocity beyond the first maximum, og(k >kc) < 0. [the threshold of which is vg(kc) = 0 at the first maximum]. If a subsidiary roton-like minimum exists, if k is a well defined quantum number at large k and Im w(k > k c ) < w ( k >kc), an identical three disorder parameter set also exists at the roton-like minimum kc2.

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(3) Scaling of isotropic-non-monatonicity (1) and disorder parameter set(s) (2) with respect to (T, P, Xj, r .... ). (4) Law of corresponding states existence between: [via (1) and (2)]: (a) different model systems with the same collective-boson; (b) different collective-bosons in the same model system; (c) different collective-bosons in different model systems. (5) Anderson localization of: (a) eollective-bosons [via (1) and (2c)] - existence of a collective-boson stopping band in wavevector form; (b) collective-boson scatteredfermions [via (1) and (2c)] - pseudo-umklapp backscattering. We examine briefly, and then illustrate with a wealth of disordered systems, theoretically and experimentally studied, in the literature, each of these features. This will reinforce the spirit of universality and Haken's synergetic we hope to excite here. (1) Isotropic non-monatonicity (negative-dispersion) simply means that the generalized-disorder causes the collective-boson dispersion relation to develop at least one maximum which qualitatively differs from an ordered analogue of the same system in that the non-monatonicity is isotropic in k-space. Typically [2],

co(k) = co(k) = h2 k2 /ZmS(k )

(1)

or Egelstaff [3] or Siegel et al. [6] variants of (1). The inflection point in the slope of the forward (low angle) scattering amplitude for radiation off the system in the system structure factor S(k) [and therefore in l/S(k)] leads to a maximum in the function k2/S(k) or co(k) at a different k (generally) from the S(k) inflection point. This property is dominated by the low.angle forward-scattering amplitude since S(k) is measured by scattering radiation off the system. The interval of wavevector from k = 0 up to k -~kinflection point yields the totality of dispersion relation of interest. (2) The disorder parameter set(s) is qualitatively seen from (1) and qualitatively derived from S(k) at any one (T, P, Xi, r, ...) system external variable set. It can be used to define the generalized-disorder in terms of its effect on S(k) during forward scattering, therefore on co(k) via (1), therefore on kc (or coc or 0c, all equivalent via the dispersion relation functional 1 : 1 mapping) on r(kc) and on vg(k >kc). If k exists and is weel def'med out to such large values that a roton-like minimum can be measured or calculated, then there exist three more identically defined disorder parameters defined at the roton.like minimum kc2. (3) Scaling with respect to (T, P, Xj, r .... ) the external variable set of (1) and (2) imply means that since experimentally S(k) =S(k; T, P, Xi, r .... ), via eq. (1) or its variants [3,6]

co(k) =co(k; T, P, Xi, r, ...)=h2k2/2rnS(k; T, P, Xi, r, ...). Thus, the disorder parameters also scale:

kc = kc(T, P, Xj, r .... ) ;

E. Siegel/A universality principle

456 equivalently ~ c : ~ ( k c ) -- ~ ( T ,

e, X i, r . . . . ) ;

equivalently o~ = o~(r, e, X i, r . . . . ) ;

independently ~(kc) = r(T, P, X/, r, ...) ; independently

og(k >kc(T, P, X/, r, ...) = og(T, P, Xj, r.... ) and similarly at kc2 if a roton-like minimum exists and k is well defined for such large values, and if lm ~(kc2) < 6O(kc2). (4) The law of corresponding states simply means that in any system with generalized-disorder the relevant collective-bosons will resemble each other or in any systems with (different) generalized-disorder the relevant collective boson [of one type (e.g., phonons)] will resemble itself in the other system or in any systems with (different) generalized-disorder the relevant collective-bosons (of all types) will resemble each other. By resemble, it is meant that the indigenous, universal, synergetic properties (1), (2) and (3)will occur; viz: isotropic non-monatonicity, existence of disorder parameter set(s) and scaling. (5) Anderson localization means two distinct things depending upon what is Anderson localized. (a) For collective-bosons themselves, it means that at large wavevectors collective-bosons exhibit non-propagating (perhaps diffusive) behavior while at low wavevectors collective-bosons exhibit propagating behavior. In the semantics of disorder-parameter (c) this simply means that for low enough k, og(k) > 0, then at some critical k(= kc), og(k = kc) = 0, then at any k >kc a stopping band in the wavevector exists with respect to the propagation of that collective-boson, og(k > kc) < 0. This is not the usual way to consider Anderson localization is disordered systems, but in the original work there was no restriction that the localized object should be an electron. (b) For collective-boson scattered fermions (e.g., electrons, holes, positrons in a medium) an explanation is needed. Isotropically in k space a monatonic collective-boson (e.g. phonon) dispersion relation will forward scatter a fermion (e.g. electron) with the same sign as the group velocity. [In an ordered system with anisotropic collective-boson dispersion relations, some monatonic and some non-monatonic (e.g. phonons in graphite) there will exist some forward scattering and some backscattering of the fermions (e.g. electrons)]. Invoking scaling, as we further disorder the system (increase T, increase Xj, decrease P, decrease r...) the curvature and kc disorder parameters scale (the dispersion relation gets more "kinky", i.e. more isotropically non-monatonic at lower wavevectors) we come to a threshold where the fermion and collective-boson dispersion relations cross such that oF > 0 but Og c-B = 0. Here the fermion is still forward scat-

E. Siegel / A universality principle

457

tered, but off an Anderson localized collective-boson (the bottom of the wavevetor stopping band). Further invoking scaling, by further addition of disorder to the system, we force the intersection of the fermion with the collective-boson dispersion relations (at equal k and co to conserve energy and m o m e n t u m within each scattering event) to take place such that the group velocities have qualitatively opposite signs: Og F > 0, uC-B(k > kc) < 0. This is the regime of pseudo-umklapp backscattering isotropically in k space. The fermion will be isotropically backscattered in k space by each scattering event due to the qualitatively opposite signs of the collective-boson and fermion group velocities iff the generalized disorder (thermal, substitutional or interstitial, reduced pressure-expanded, gran s i z e . . . ) is sufficient (to make the dispersion relation " k i n k y " enough for the threshold to matter). I f f

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Fig. 1. The five qualitative collective-boson mode-softening properties indigenous to generalized-disorder: (1) isotropic non-monatonicity; (2) existence of disorder parameter set; (3) scaling with t~mperature, pressure, composition . . . of (1) and (2); (4) law of corresponding states between the same coUective-boson in different disordered systems, different coUeetive. bosons in same disordered system and different collective-bosons in different disordered systems; (5) Anderson localization of bosons and boson scattered fermions, [phonons and magnons are illustrated in part (2) only].

E. Siegel I A universality principle

458

the fermion Fermi energy is at this threshold frequency (or beyond), these fermions only dominate the fermion transport coefficients of the system. Beyond the threshold (at kc) o f intersecting, energy and m o m e n t u m conserving, dispersion relations, each scattering event will backscatter the fermion. If there is some small energy loss in each scattering event (some slight inelastic component) (say 1 part in 1012 for e l e c t r o n - p h o n o n scattering) the fermion will encounter 1012 scattering events, hopping to 10 ~2 wells but not leaving some small region o f size 1012/1023 --10 -13 or 10-11% o f the system, after which it will truly become Anderson localized within one well and n o t execute propagation excursions any further. At typical coll e c t i v e - b o s o n - f e r m i o n collision frequencies (e.g. e l e c t r o n - p h o n o n collision fre-

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due to generalized-disorder. In the collective-boson case, (for phonons and magnons...) formation of stopping band in wavevector is stressed, even if Im k and Im to are finite. In the collective-boson scattered fermion case, scaling with temperature, pressure, composition,.., is shown to lead to isotropic pseudo-umklapp backscattering of the fermion via scattering off the stopping band regime of the collective-boson dispersion relation. This corresponds to a gap in electron density of states ge(to) forming, Agap, which may be partially filled in by localized electron states due to "phonon" anharmonicity and the dispersion relation anisotropy at high wavevector.

E. Siegel /A universality principle

459

quencies of 10 l~- Hz) this takes about 1 s; the dc transport measurements (e.g. electrical conductivity) will be zero (tic Anderson localization) but the ac transport measurements will be sampling the fermion at frequencies before it is truly Anderson localized i.e. at frequencies when it is only regionally localized (in say 10 -11% of the system site wells), so the ac transport measurements will be finite and nonzero, but not necessarily identical in value nor functional form to those obtained in the much less disordered ("kinky") system below the k c = k threshold [and with co(
Collectiveboson

Magnon

Plasmon

Magnon

Magnon

Magnon

Magnon

Magnon

System

Metallic glasses

Electron gas

Amorphous (metal) magnets modelled as random heisenberg models

Stoner model itinerant crys.tal magnet

Heisenberg-Mattis unfrustrated spin glass model

Toulouse totally frustrated spin glass

Anderson model spin glass

Table 1

Exchange bond e or It I [not sgn(t)] with Hamiltonian: n = ~.eia~ai + ~ t i ] a ~ a ]

Exchange bond r/i = ±1 with H = Jr~ rtiSi. Si + 1

Exchange bonds ~'i = ±1, J ordered in Hamiltonian: H = -~Ji]~i~]Si. $]

Magnon decay into Stoner excitations ( e - - e + pairs) randomly in space and time in Stoner gap

Configurational (exchange bonds invariant)

Plasmon decay into e - - e + pairs randomly in space and time

Configurational (+ substitutional in alloys)

Generalized-disorder

X

X?

×?

Experiment

×

X

×

X

×

X

Theory

[24, 27]

[24, 26]

[24, 25]

[19-23]

[13-18]

Be, Li, Na, K, A1, C [ 1 0 - 1 2 ]

PdxSi, Co8oP2o, Fe75 P15C1 o [7-9]

Reference

o~

Configurational (+diffusional translation) Configurational (+diffusional translation)(+alloy) Configurational (+alloy + anharmonicity) Configurational (+anharmonicity)

Magnon

Phonon

Phonon

Phonon

Phonon (+magnon)

Walstedt-Walker-Huber numerically simulated spin glass

Quantum liquid

Classical liquid (metal, semiconductor, insulator, ionic, polymer, noble gas)

Glasses (metal, ionic, polymer, semiconductor)

Powders + surfaces (incl. Heisenberg half chain)

Exchange bond disorder

Anharmonic magnon-magnon interactions random in space and time

Magnon

Halperin-Saslow spin glass + anharmonicity (non-linearized)

Generalized-disorder

Collectiveboson

System

Table 1 (continued)

X

×

X(?)

×

X

Experiment

X

×

X

X

X

X?

Theory

granular Aland Pb [ 4 4 - 4 7 ]

BeF2, Se, polymer glasses, molecular crystal glasses [40-43]

[2, 3, 6, 34-39]

[31-33] He

[24,29,30]

[24, 28]

Reference

4a.

Lattice periodicity changes Lattice periodicity changes (Kohn anomaly)

Magnon

One-dimensional magnets (embedded in three-dimensional medium)

Classical acoustic wave Classical analog of phonon-like ball in wall Quantum waveparticle

Classical acoustic wave reflection off surface

Classical mechanics of ball bouncing off wall

Quantum wave-particle at interface

Random scattering

So,tons? Elec~omagneticwave

Nonlinear systems? (incl. muscles)

Classical and quatum optical wave reflection in mirror

Propagating wave (e.g. density)

Random walk brownian motion

Half-space surface changing periodicity from the int'mite suddenly in space. Wave develops negative group velocity beyond reflecting surface

Half-space surface changing periodicity from the infinite suddenly in space. Wave develops negative group velocity beyond reflecting surface

Half-space surface changing periodicity from the infinite suddenly in space. Wave develops negative group velocity beyond reflecting surface

Half-space surface changing periodicity from the infinite suddenly in space. Wave develops negative group velocity beyond reflecting surface

Lattice periodicity changes Random (in space and time) scattering

Phonon Phonon

Peierls transition

Jahn-Teller transition

Different vibrational periodicity from acoustic phonons (unit cell)

Optical phonons

Optical phonon in crystals

Generalized-disorder

Collective-boson

System

Table 2

WKB approx, calcs.

Any reflection

[52]

1-D systems CDWs

(CHa)4NMnC13 =TMMC, CPC=CuC12 • 2 NCsD s [48]

Any crystal

Reference

Randomness in space and time of gas or liquid medium Configurational, vibrational excitations of peptide groups on c~-helix protein macromolecules of sacromeres of muscle fibers

Chemical reaction waves catalysis waves Excitons

Chemical reactions(?) and catalysis (?)(Zhabotinskii reaction)

Polarons, polaritons, ripplons

Polarons, polaritons, ripplons configurational

Existence of surface

Surface plasmons

Surface plasmons

Glasses, liquids, muscles

Existence of metal particle near infinite surface

(Local) surface plasmons

Light emission from small metal particle-dielectric layer Josephson tunnel junctions

Symmetry breaking

Generalized-dirorder

Optic waves (lasing), magnons (magnetic transitions), superfluid waves (helium), cooper pairs (superconducting transition)

Collective-boson

Analogous phase transitions that involve "symmetry breaking" (incl. lasing)

System

Table 2 (continued)

[52]

[51 ]

[50]

[49]

Reference

4~ O~ t;o

t~

464

E. Siegel / A universality principle la)

kc &)

Icl

co

q"

fb)

k~

q" {dl

/ kc

kc

q-

(f)

(el

q~

k~

q,

Fig. 3. Magnon dispersion relations exhibiting mode-softened, negative-dispersion universality: (a) one-dimensional magnet; (b) three-dimensional anharmonic magnet; (c) Stoner itinerant electron magnet; (d) amorphous metallic magnet; (e) random Heisenberg magnet; (f) anharmonic magnon magnet, Toulouse spin glass, Heisenberg-Mattis spin glass, Anderson spin glass.

3. Conclusions In conclusion, the universality of this generalized-disorder collective-boson mode-softening is most startling. It emphasizes the generalized definition of disorder that is required to unify the physics of many such diverse systems. By invoking the dynamic effects on the collective-boson dispersion relation, ( 1 ) - ( 5 ) , such a unified description, or perhaps dynamic definition of generalized disorder can be realized. The application to many other, non-conventionally disordered systems, which exhibit similar collective-boson mode-softening (table 2) is very speculative, but dynamically can unify much of the physics of many diverse, diversely disordered or aperiodic systems which contain many different kinds of collective-boson modes. Whether this generalized-disorder definition in the dynamic sense replaces, or is merely redundant with, the usual definition of disorder as(l-order parameter) remains to be seen, however, its unifying viewpoint is a valuable principle. Whether one or more disorder parameters is necessary and/or sufficient to completely dynamically describe the generalized-disorder in a fashion to replace the (1-order parameter) non-dynamic conventional definition is still an open question; perhaps

E. Siegel / A universality principle

465

the necessary/sufficient question varies from system to system. However the utility of this generalized-disorder, collective-boson, mode-softening, universality-principle, especially in explaining scaling, laws of corresponding states, collective-boson or collective-boson scattered fermion Anderson localization and unifying the diversity of disorder induced phenomena cannot be disputed. In a subsequent paper it will be demonstrated that the GDCBMSUP leads naturally to the low temperature two- (or multi-) level system in glasses and amorphous solids, as well as powders, whose basic orgin has escaped elucidation to date.

References [1] [21 [3] [4] [5]

H. Haken, Synergetics (Springer-Verlag, New York, 1977). J. Hubbard and J. Beeby, J. Phys. C: Solid St. Phys. 2 (1969) 556. P. Egelstaff, An Introduction to the Liquid State (Academic Press, New York, 1968). E. Siegel, Bull. APS 24 (January, 1979) 1; ibid. (March, 1979). E. Siegel, J. Phys. Chem. Liquids, 4 (1975) 4,217, 205,211,233,259; ibid. 5 (1976) 1, 9; M. Omini, Phil. Mag. 26 (1972) 2,287; J. Percus and G. Yevick, Phys. Rev. 110 (1958) 1. [6] E. Siegel, Int. Conf. Lattice Dynamics, Paris (1977) to be published; Statphys-13, Haifa (1977) to be published in: Ann. Israel Acad. Sci.; The Physics of SiO2 and Its Interfaces, Proc. IBM Conf. (Pergamon Press, New York, 1978); IBM Conf. Amorphous Solids, unpublished. [7] H.A. Mook and C.C. Tsuei, ORNL Report 77-31 (March, 1977). [81 H.A. Mook, N. Wakabayashi and D. Pan, Phys. Rev. Letters 34 (1975) 1029; H.A. Mook, D. Pan, J. Axe and L. Passel, AlP Conf. Proc. 24, eds. C.D. Graham Jr., G.H. Lander and J.J. Rhyne (ALP,New York, 1975). [9] Y. Takahashi and M. Shimizu, Phys. Letters 58A (1976) 419. [10] P.M. Platzmann and P. Eisenberger, Solid St. Comm. 14 (1974) 1; P.M. Platzmann and P. Eisenberger, Phys. Rev. Letters 33 (1974) 3, 152; P.M. Platzman, P. Eisenberger and K. Pandy, Phys. Rev. Letters 31 (1973) 1,311. [111 J.T. Devreese, F. Brosens and L.F. Lemmens, Phys. Stat. Sol. (b)91 (1979); ibid. (b)74 (1976) 45; ibid. (b)80 (1977) 99; ibid. (b)81 (1977) 551. [12] G. Barnea, preprints, Technion, Haifa (Ph-78-60). [13] D.G. Hall and J.S. Faulkner, in: Proc. Neutron scatt. Conf. II, ed. R.M. Moon, ERDA Rept. CONF-760601-P2. [14] R. Alben, in: Magnetism and Magnetic Materials, 1975, AIP Conf. Proc. 29, eds. J.J. Becker, G.H. Lander and J.J. Rhyne (ALP, New York, 1976). [ 15 ] L. Roth, private communication. [ 16] N. River, private communication. [17] R. Elliot, private communication. [181 J. Morkowski, Int. Conf. Magnetism, Amsterdam (1976). [19] L.C. Barrel, Phys. Rev. B8 (1973) 5316; ibid. B7 (1973) 3153; H.A. Mook, J.W. Lynn and R.M. Nicklow, Phys. Rev. Letters 30 (1973) 556. [20] E. Siegel, J. Mag. Magn. Mat. 5 (1977) 84; T. Arai and M. PaxrineUo,Phys. Rev. Letters 27 (1971) 1226; A. Kutsuki and E.P. Wohlfarth, Proc. Roy. Soc. A345 (1969) 231. [21] G. Barnea and G. Horowitz, J. Phys. C: Solid St. Phys. 6 (1973) 738; ibid. C 8 (1975) 2124; G.C. Windsor, R.D. Lowde and L. Allan, Phys. Rev. Letters 22 (1969) 16, 849. [22] H.A. Mook and R.M. Nickelow, Phys. Rev. B7 (1973) 336; J.F. Cook and H.L. Davis, AIP Conf. Proc. 10 (1972) 1218.

466

E. Siegel/A universality principle

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