# Multilevel Tunnelling Systems and Fractal Clustering in the Low-Temperature Mixed Alkali-Silicate Glasses

Authors: Giancarlo Jug; Maksym Paliienko
Journal: The Scientific World Journal
 (2013)

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## Abstract

The thermal and dielectric anomalies of window-type glasses at low temperatures (T<1 K) are rather successfully explained by the two-level systems (2LS) standard tunneling model (STM). However, the magnetic effects discovered in the multisilicate glasses in recent times, magnetic effects in the organic glasses, and also some older data from mixed (SiO2)1−x(K2O)x and (SiO2)1−x(Na2O)x glasses indicate the need for a suitable extension of the 2LS-STM. We show that—not only for the magnetic effects, but also for the mixed glasses in the absence of a field—the right extension of the 2LS-STM is provided by the (anomalous) multilevel tunnelling systems (ATS) proposed by one of us for multicomponent amorphous solids. Though a secondary type of TS, different from the standard 2LS, was invoked long ago already, we clarify their physical origin and mathematical description and show that their contribution considerably improves the agreement with the experimental data. In spite of dealing with low-temperature properties, our work impinges on the structure and statistical physics of glasses at all temperatures.


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## Body

## 1. Introduction

Glasses are ubiquitous materials of considerable importance for many practical applications; however, for physicists the nature of the glass transition and the ultimate microscopic structure of glasses determining their physical properties remain to this day issues of considerable intellectual challenge [1]. Glasses are normally regarded as fully homogeneously disordered amorphous systems, much alike liquids except for the glassy arrested dynamics close and below the glass transition temperature Tg, which leads to an increase of several orders of magnitude in the viscosity for T→Tg+. Nevertheless, this homogeneity is most probably only a useful idealization, for real glasses must always contain some small (in ceramic glasses not so small) concentration of tiny, ordered, or nearly ordered regions of variable size with their own frozen dynamics. Indeed the thermodynamically stable phase of an undercooled liquid would be the perfect crystal; thus, every substance in approaching the crystallization temperature Tc (Tc>Tg) from above would spontaneously generate local regions of enhanced regularity (RER) much like a system (a vapour or a paramagnet) approaching its critical temperature is known to develop regions (droplets) resembling the ordered low-temperature phase. These RER are of course to be distinguished from the concept of short-ranged atomic order which is typical of ideal glasses and is restricted to the first few atomic spacings. We are considering in this paper realistic glasses in which a degree of devitrification has occurred. The size and concentration of these RER will depend, for example, on the rapidity of the quench leading to the formation of the glass, but also on the chemical composition of the substance, the presence of impurities, and so on. However, on general grounds, even the purest of glasses should contain RER in non-zero concentration and size.This case has been demonstrated recently for the structure of the metallic glassZr50Cu45Al5 [2], where a combination of fluctuation electron spectroscopy (FEM) and Monte Carlo simulation (MC) has revealed the presence of crystalline regions of subnanometer size embedded in an otherwise homogeneously amorphous mass of the same composition. It is believed that other metallic glasses should present similar structural features and thus—on general grounds—one would expect that nonmetallic window glasses too, like pure SiO2 and all the more so the commercial multisilicates of complex chemical composition, should present a multiphased structure with the size and concentration of the near-crystalline regions, or RER, depending, for example, on composition, quench rate, and the presence of impurities acting as nucleation centres for the RER. Indeed, materials of the general composition (MgO)x(Al2O3)y(SiO2)1-x-y (MAS, in short) are termed ceramic glasses (one of the best known commercial examples being Schott’s Ceran where Li2O replaces MgO, and of course CaO or BaO can also replace or be added to MgO and still yield a ceramic glass). These materials are known to contain microcrystals embedded in an otherwise homogeneously amorphous matrix [3]. This is not surprising, for materials made up of a good glass-former (e.g., SiO2, Al2O3, etc.) and good crystal-formers (e.g., BaO, K2O, …) are known to be multiphased [4] with the good crystal-formers generating their own pockets and channels carved out within the otherwise homogeneously amorphous network of the good glass-former’s ions [5]. Within these pockets and channels, incipient nano- or even microcrystals may form, but the point of view will be taken in this work that on general grounds even the purest, single-component (e.g., As, SiO2) glass-former will be rich in RER unless the quench-rate from the melt is so large as to avoid the formation of crystalline regions or RER.These refined structural details of glasses are evidently hard to reveal in all and especially the near-ideal cases (no good crystal-formers, no impurities added, and rapid quenches) with the available spectroscopic techniques. For example, X-ray spectroscopy does not reveal nano-crystals below the nanometer size. However, at low and very low temperatures—where all said structural features remain basically unaltered—some recent experimental findings might now improve perspectives with what would appear set to become a new spectroscopy tool. Indeed a series of remarkable magnetic effects have recently been discovered in nonmagnetic glasses (multisilicates and organic glasses) [6–13] with, in the opinion of the present authors, a most likely explanation for the new phenomena stemming precisely from the multiphase nature of real glasses and the presence of the RER or microcrystalline regions in their microscopic structure. In turn, when the multiphase theory shall be fully developed, the magnetic effects could represent a valid new spectroscopic tool capable of characterizing micro- or nanocrystals or even incipient crystals and RER in the real glasses. The key to this possible development is some new exciting physics of the cold glasses in the presence (and even in the absence, as shown in the present paper) of a magnetic field. The magnetic effects in the cold glasses could become, eventually, the amorphous counterpart of the de Haas-van Alphen and Shubnikov-de Haas effects in crystalline solids in determining the real structure of amorphous solids.Systematic research on the low-temperature properties of glasses has been ongoing for more than 40 years, and some significant theoretical and experimental progress has been made in the understanding of the unusual behaviour of glasses and of their low-temperature anomalies [14–16]. This temperature range (T<1 K) is deemed important for the appearance of universal behaviour (independent of composition), as well as for the effects of quantum mechanics in the physics of glasses. However, to make progress in the understanding of the low-temperature physics of glasses, there remains a wide range of important questions that are still open or only partially answered, particularly in the light of some still poorly understood recent, and even older, experiments in cold composite glasses.It is well known that cold glasses show somewhat universal thermal, acoustic, and dielectric properties which are very different from those of crystalline solids at low temperatures (below 1 K) [17, 18]. The heat capacity Cp of dielectric glasses is much larger and the thermal conductivity κ is orders of magnitude lower than the corresponding values found in their crystalline counterparts. Cp depends approximately linearly and κ almost quadratically on temperature T (in crystals one can observe a cubic dependence for both properties). The dielectric constant (real part) ϵ′ and sound velocity at low frequencies display in glasses a universal logarithmic dependence in T. These “anomalous” and yet universal thermal, dielectric, and acoustic properties of glasses are well explained (at least for T<1 K) since 1972 when Phillips [19] and also Anderson et al. [20], independently, introduced the tunnelling model (TM), the fundamental postulate of which was the general existence of atoms or small groups of atoms in cold amorphous solids which can tunnel like a single quantum-mechanical particle between two configurations of very similar energy (two-level systems (2LS)). The 2LS-TM is widely used in the investigation of the low-temperature properties of glasses, mostly because of its technical simplicity. In fact, it will be argued in this paper that tunneling takes place in more complicated local potential scenarios (multiwelled potentials) and a situation will be discussed where the use of a number of “states” greater than two is essential. Moreover, new insight will be given on the role of percolation and fractal theory in the TM of multicomponent glasses. We present in this paper the justification and details of the construction of an extended TM that has been successfully employed to explain the unusual properties of the cold glasses in a magnetic field [21], as well as in zero field when systematic changes in the glass’ composition are involved [22].The linear dependence inln(T) of the real part of the dielectric constant ϵ′(T) makes the cold glasses useful in low-temperature thermometry, and, normally, structural window-type glasses are expected to be isotropic insulators that do not present any remarkable magnetic-field response phenomena (other than a weak response in Cp to the trace paramagnetic impurities). For some multicomponent silicate glass, it has become possible to measure observable, much larger than expected changes in ϵ′(T,B) (δϵ′/ϵ′~10-4) already in a magnetic field as weak as a few Oe [6, 7]. A typical glass giving such strong response has the composition Al2O3-BaO-SiO2 thus, a MAS ceramic-glass, herewith termed AlBaSiO. The measurements were made on thick sol-gel fabricated films, a fabrication procedure favoring microcrystal formation [4], cooled in a 3He-He4 dilution refrigerator reaching temperatures as low as 6 mK. Magnetic effects have been reported for both the real and imaginary part of ϵ at low frequency (ω~1 kHz), for the heat capacity Cp (see, e.g., [21]) and for the polarization echo (where changes in the presence of a magnetic field have been the strongest [10–12]) as well. This behavior was confirmed in other multicomponent glasses, like borosilicate optical glass BK7 and commercial Duran [8], and, moreover, similar effects on ϵ′(T) have been confirmed in studies of the structural glass a-SiO2+xCyHz in the range 50<T<400 mK and B≤3T [9]. Although the dielectric magnetocapacitance enhancement is not dramatic (δϵ′(B)/ϵ′ is typically in the 10-6–10-4 range), the available measurements show that an unusual effect of the magnetic field is indeed present in the above glasses, yet not measurable in ultrapure SiO2 (Suprasil W) and cannot be ascribed to spurious agents (The presence of incipient- or microcrystals in real glasses (sometimes called devitrification) should not be considered a spurious effect. On the contrary, the magnetic effects should be a way of characterising the two-phase structure of real glasses.) or to trace paramagnetic impurities (always present in silicate glasses, although in <6 ppm concentration in the case of BK7). Polarization-echo experiments in the AlBaSiO, Duran, and BK7 glasses have also shown considerable sensitivity in the response of the echo amplitude to very weak magnetic fields, and the magnetic effects clearly do not scale with the concentration of paramagnetic impurities [8, 10–12]. Striking magnetic effects, the presence of a novel isotope effect, and remarkable oscillations in the dephasing time have also been reported in studies of the polarization echoes in organic glasses (amorphous glycerol) [13]. However, in terms of a detailed theoretical justification for all of the observed magnetic effects (and the lack of an observable magnetic effect in the acoustic response [23], so far) an explanation relying on a single theoretical model for all of the available experimental data is still missing. We believe the two-phase model reproposed in this paper to be the correct generalization of the standard 2LS-TM that is being sought and here we work out its predictions in zero magnetic field, but for different controlled concentrations of glass-forming and crystal-forming components. In this way, we put our approach to a new test.The essential behavior of the dielectric response of glasses at low temperatures is well known [17, 18]. According to the standard 2LS-TM (STM from now on), the dielectric constant is predicted to vary like -lnT due to the constant density of states of the TS. Above a certain temperature T0(ω), relaxational absorption of the TS becomes important, resulting in an increase of the dielectric constant with temperature proportional to +lnT according to the STM. This has been checked experimentally for several glasses. The temperature T0 of the resulting minimum depends on the frequency ω and occurs around 50 to 100 mK in measurements at around 1 kHz.Some more interesting behavior has been shown by some as yet unexplained data from experiments on the mixed(SiO2)1-x(K2O)x and (SiO2)1-x(Na2O)x glasses, studied as a function of the concentration x of the good crystal-former at low temperatures [24]. The heat capacity Cp(T) for these glasses is larger than that for pure vitreous silica and the behavior as a function of T is very peculiar for different molar concentrations x of potassium or sodium oxide and is not explained by the STM. The heat capacity decreases and then increases again with increasing molar concentration x of K2O. The minimum in the dielectric constant ϵ′(T) is observed for T0 near 100 mK as is typical for these glassy solids. The temperature dependence of ϵ′, both above and below T0, shows however a slope in ±lnT qualitatively increasing with increasing concentration x of K2O. One can notice, moreover, that above the minimum T0 the relaxation part of ϵ′ is increasing faster in slope than the resonant part below T0 for the same x [24], a feature completely unexplained thus far. This work is an indication that not only the magnetic and electric fields influence the properties of glasses, but the concentration of chemical species in the composite materials too (a fact not accounted for by the STM). In this paper, we show in detail how the very same approach that explains the magnetic properties in the multisilicates [21] also provides a quantitative explanation for the above-mentioned composition-dependent physical properties. The picture that emerges regarding the nature of the TS in the multicomponent glasses provides a novel and detailed description of the micro- and nanostructure of the glassy state. In turn, the linear dependence of the concentration xATS of anomalous TS (ATS)—that responsible for the magnetic and composition effects in our theory—on x fully corroborates the founding assumptions of our approach.The paper is organised as follows. In Section2, we present a detailed justification for the two-phase approach and the construction of the two-species TS model for the amorphous solids at low temperatures. In Section 3, we present the detailed predictions of this model for the dielectric constant ϵ′(T,x) as a function of temperature T and composition x of alkali oxide (good-crystal former) for the mixed glasses and we compare the predictions with the experimental data [24]. In Section 4, we present the detailed predictions of our model for the heat capacity Cp(T,x) for the mixed glasses and we compare the predictions with the available experimental data [24]. Section 5 contains our conclusions about the nature of the TS; namely, we show how the tunneling “particle” must in fact represent a whole cluster of N correlated real tunneling ions in the material. Finally, in the appendix, we work out how the effective tunneling parameters of our model are related, via N, to more standard microscopic tunneling parameters. A short preliminary account of this work was published in [22].


## 2. Building Up a Suitable Tunneling Model

The traditional picture [17, 18] viewed the TS, present in low concentration (~1016 g-1) in the material, associated with the nonequivalence of two (or more) bonding-angle configurations per atomic unit in the amorphous solid’s atomic structure. Each TS is represented in the standard case by a particle in an asymmetric (one-dimensional (1D)) double-well potential where, at low-T, only the ground states of the two constituent single wells are assumed to be relevant. Consequently, only the two lowest-lying double-well states are taken to determine the physics of each single TS. A 2LS simplified picture then applies, and one can describe the low energy Hamiltonian of each independent TS in terms of an equivalent notation with spin-1/2 pseudospin matrices σx and σz (Pauli matrices), leading to the compact notation H0(2)=-(1/2)(Δσz+Δ0σx) for the Hamiltonian of a single 2LS TS. In matrix form (the so-called well- or position-space representation, 〈i|H0(2)|j〉, |i〉 being the two unequivalent wells, i=1,2 or i=L,R), this then reads:
(1)H0(2)=-12(ΔΔ0Δ0-Δ).
Here, the phenomenological parameters Δ and Δ0 (known as the energy asymmetry and (twice) the tunnelling matrix element, resp.) represent a way of describing the essential low-T relevant features of the full, and yet unknown in its details, TS single-particle Hamiltonian in the effective single-well matrix representation. One obtains ℰ1,2=±(1/2)Δ2+Δ02 for the two lowest-lying energy levels, and the physics of the glass is then extracted by assuming (initially) the 2LS to be independent entities in the glass are averaging physical quantities over a probability distribution for the parameters Δ,  Δ0 of the standard form (P- being a material-dependent constant):
(2)P(Δ,Δ0)=P-Δ0.
This distribution reflects the generally accepted opinion that Δ and -ln(Δ0/ħΩ) (the latter proportional to the double-well potential barrier V0 divided by the single-well attempt frequency Ω, V0/ħΩ) should be rather broadly distributed in a homogeneously disordered solid. This leads to an almost constant density of states (DOS) and the above STM has been employed with considerable success in order to explain a wide range of physical properties (thermal, dielectric (ac and pulsed), acoustic, etc. [17, 18]) of nonmetallic glasses below 1 K.There are, however, several drawbacks with the STM as thoughtfully pointed out by Leggett and coworkers [25–27]. For a start, the nature of the TS (and of the two wells of a single 2LS) and that of the motion inside a single TS remain to date completely unknown (We remark that recently, thanks also to the efforts towards the explanation of the magnetic effects [21, 28] and from the study of quantum domain-wall excitations in the cold glasses [29], a picture is emerging of a correlated (or coherent) tunneling cluster of some N (charged) particles (atoms or molecules) which is being represented, in the TM, by a single fictitious tunneling particle. In this paper, we argue that N~200 in agreement with [29].). Much easier is the diagnostic for the nature of 2LS in the case of disordered crystals, such as Li-KCl or KBr-KCN solutions [30] (we shall come back to disordered crystals later). On general grounds, other types of (multilevel) excitations are always possible in glasses and it is not clear why their distribution of parameters should be so similar (and given by (2)) in all of the amorphous solids. Next, the STM has gathered great consensus for the explanation of many experiments at low temperatures, but in its simplest form (1)-(2) it fails to explain sound velocity shift and adsorption data at low-T and the origin of the “bump” in Cp (and “plateau” in κ) well above T0 that goes under the name of boson peak (see, e.g., the references in [25–27]). Moreover, the STM fails to explain the remarkable universality of the ultrasonic attenuation coefficient Q-1 (roughly, independent of every external parameter and glass chemical composition) below 1 K [31]. To resolve these (and other) difficulties with the STM, Leggett and collaborators have proposed a generic model in the context of anharmonic elasticity theory which can account for all of the significant features of glasses below 1 K, including the super universality of Q-1 [25–27].However, it is hard to see how this generic elastic model can be extended to account for the magnetic and composition-dependent effects in glasses, also considering that in the multicomponent (i.e., real, non model) glasses most of the said universality features (e.g., inCp(T,B) and ϵ′(T,B) [6, 7, 21] or in Cp(T,x) and ϵ′(T,x) [22, 24]) are lost. Therefore, here we adopt the strategy of resuming the TS approach by means of a completely different (and more modern) justification for the TM and then extend the STM to take the presence of a magnetic field into account and to explain composition-dependent features (this work). In a rather general fashion, the TS can be thought of as arising from the shape of the theoretical potential-energy landscape E({ri}) of a glass as T is lowered well below the glass freezing transition Tg. The concept of free-energy landscape was introduced, for example, by Stillinger [32, 33] and successfully employed in the study of glasses (e.g., [1]) and spin-glasses (e.g., [34, 35]). A large number of local and global minima develop in E({ri}) as T→0, the lowest-energy minima of interest being made up of nw=2,3,… local wells separated by shallow energy barriers. At low-T, these configuration-space local multiwelled potentials are our TS, and it seems reasonable to expect that the nw=2-welled potentials (2LS) should be ubiquitous in this picture. These should be thought of as an effective representation of local “tremblements” of the equilibrium positions {ri(0)} of some of the glass atoms/ions’ positions spanning over a large number of near-neighbors’ distances (unlike in the case of disordered crystals, where the TS are known to be rather well-localized dynamical entities). Hence, just as the nw=2-welled case is possible, so ought to be the nw=3,4,…-welled situations which would also be local rearrangements involving several atoms/ions/molecules. The concentration of these local potentials should not necessarily decrease exponentially with increasing nw, in glasses, as it is known to happen for the disordered crystals (2LS present with probability c2, 3LS with c3, 4LS with c4,… etc., c being the defects’ percent concentration).We can reason this out over the quantitative description of the glassy energy landscape of a model situation, as was studied by Heuer [36] who considered the molecular-dynamics (MD) simulation data of a toy glass made up of several (13 or 32) particles interacting through a Lennard-Jones potential and with periodic boundary conditions applied. Adopting a suitable 1D projection procedure, where a “distance” between two local total energy minima is (not completely unambiguously) defined, the 1D position of a local minimum is somehow attained and the energy landscape of the model system can be charted out. Figure 1 reports this chart for the total energy landscape for a given density (from [36]). Beside the deep minimum of the crystalline configuration, a large number of local minima are visualized and then a suitable definition of local double-welled potentials (2LS) is adopted to classify couples of adjacent minima constituting a single tunneling 2LS (highlighted in black, Figure 1). This definition guarantees that at low temperatures a “particle” subjected to any such local potentials will switch between both minima without escaping to a third minimum. Interestingly, the distribution of the tunneling parameters Δ,Δ0 (suitably defined) for these 2LS could also be evaluated from MD simulations of the above toy model, and this P(Δ,Δ0) turned out to be not so perfectly flat as a function of Δ as implied by (2). Rather, an increase (though no divergence) of probability for 2LS with Δ→0 was measured in previous MD simulations [37]. Still, Figure 1 also allows for tunneling multiwelled local potentials to be identified, and we have highlighted (in light blue) some of them (three- and four-welled local potentials). The requirement that a “particle” subjected to such multiwelled local potentials should not escape (at low-T) to foreign minima has been equally respected, and one can see that these multiwelled situations are not at all rare. We therefore believe that 3LS, 4LS, and so on should also be considered in the TM. The reduced Hamiltonians (well- or position-representation) for these local multiwelled potentials can be easily written down, as generalizations of (1). For nw=3 (3LS),
(3)H0(3)=(E1D0D0D0E2D0D0D0E3),
where E1,  E2,  E3 are random energy asymmetries between the wells chosen to satisfy ∑‍i=13Ei=0 and taken from an appropriate probability distribution (see below), together with the tunneling parameter D0>0 (see below). For nw=4 (4LS):
(4)H0(4)=(E1D1D2D1D1E2D1D2D2D1E3D1D1D2D1E4),
where E1,  E2,  and E3,  E4 are random energy asymmetries taken from an appropriate probability distribution, together with the tunneling parameters D1 (n.n. well hopping) and D2 (n.n.n. hopping, |D2|≪|D1|). These are simple, possible choices; clearly, other special-purpose generalizations of the 2LS matrix Hamiltonian are possible and we believe that the 3LS of (3) is the minimal generic multiwelled potential which can take the magnetic field into account [21] (the 2LS Hamiltonian of (1) could also be adjusted for this purpose; however, the energy spectrum would be totally insensitive to B). One can easily convince oneself, at this point, that as long as the energy parameters of the above multiwelled effective Hamiltonians obey the usual uniform distribution (see (2), suitably reformulated) as is advocated by the STM, the DOS g(E) will remain (roughly) a constant. It is then to be expected that all these multiwelled local potentials will give rise to the very same physics as in the nw=2 case and that thus, in practice, the 2LS choice represents the appropriate minimal model for all of the extra low-energy excitations characterising amorphous solids at low-T. It is clear from the above discussion, however, that the 2LS tunneling “particle” is not atomic particle at all, but, on general grounds, it rather represents the local rearrangements of a good number of real particles (ions or molecules).Figure 1
(Color online) The energy landscape (forρ=1 Lennard-Jones density, adapted from [36]) of a toy glass model, with highlighted multiwelled potentials (black the 2LS, light blue the 3LS, 4LS, …).All changes if the glass is made up of a mixture of network-forming (NF) ions (like those of the good glass-formingSiO4 or (AlO4)- tetrahedral groups) as well as of network-modifying (NM) ions (like those of the good crystal-forming K+ or Na+, or Ba2+,… from the relative oxides) which, these last ones, could act as nucleating centres for a partial devitrification of the glass, as is known to occur in the multicomponent materials [38–41]. Indeed, the NM-ions of the good crystal-formers are termed “glass modifiers” in the glass chemistry literature [42] since they do not become part of the interconnected random network but carve out their own pockets and channels within the glassy network [5, 43]. Figure 2 (courtesy from Meyer et al. [5]) shows a snapshot of a MD simulation of the glass having composition Na2O·3(SiO2) (or (Na2O)0.25 (SiO2)0.75) at 2100 K (above Tg, in fact) in which the nonnetworking NM Na-atoms are put in evidence (big blue spheres). Simulations and experiments in the multisilicates definitely show that the NM-species in part destroy the networking capacity of the NF-ions and form their own clusters inside the NF-network [5]. The chance for these NM-clusters to be the nest of RER, incipient- or actual microcrystals is obviously very good, considering that these clusters are made of good crystal-forming atoms. However, on general grounds and as discussed in Introduction, we shall take the attitude that even the purest single-component glasses will contain RER in some measure. Figure 3 (from [2]) shows one such RER within a snapshot from a joint MC-simulation/FEM-measurement on the metallic glass Zr50Cu45Al5. The picture clearly shows an embryo crystal which could not grow to macroscopic size due to the arrested dynamics below Tg; such structures are expected to ubiquitous in all glasses, metallic and nonmetallic [44], except that they are difficult to observe with the available spectroscopic tools when subnanometric in size. The concentration and size of these RER will dictate whether magnetic- or composition-effects become measurable in the low-T experiments. a-SiO2 in its purest form (Suprasil W) revealed no measurable magnetic effects [6, 7, 9–12].Figure 2
(Color online) Molecular dynamics snapshot of the structure of sodium trisilicate at 2100 K at the densityρ=2.2g cm-3. The big blue spheres that are connected to each other represent the Na atoms. The Si–O network is drawn by yellow (Si) and red (O) spheres that are connected to each other by covalent bonds shown as sticks between Si and O spheres (from [5], by permission).Figure 3
(Color online) A region including the crystal-like supercluster from a snapshot of the model simulation—incorporating fluctuation electron microscopy data—of theZr50Cu45Al5 metallic glass at 300°C (from [2]). The atomic separation distances of the middle zone are about 0.25 nm. This is a first realistic image of a crystal embryo in a glass; this object should not be confused with the concept of short-range order in ideal glasses.It goes without saying that TS forming in the proximity and within these RER or microcrystalline regions will require a completely different mathematical description, in particular the possibility of having more than two wells affords a more realistic description of the energy landscape. Hence,nw>2 multiwelled systems inside the glass-modifying NM-pockets and -channels should follow some new energy-parameters’ distribution form when some degree of devitrification occurs, leading to entirely new physics. One of the present authors has proposed that precisely this situation occurs inside the magnetic-sensitive multicomponent glasses [21], and in this paper we show how this theory explains the B=0 composition-dependent dielectric and heat capacity data of [24] as well. Instead of the standard 1D double-welled (W-shaped) potential, leading to (1), which continues to describe the ordinary tunneling 2LS inherent to the homogeneously disordered a-SiO2 network, we take for the TS nested in or near the RER, crystal embryos or micro-crystals, the model of a “particle” having charge q and moving in a nw-welled 3D potential of the shape displayed, for nw=3, in Figure 4 for the 2D (x,y)-space. The hopping Hamiltonian of a single, non interacting tunneling 3LS has therefore the form (for a fictitious second-quantization particle in the well-coordinate representation)
(5)H0(3)=∑‍i=13Eici†ci+∑‍i≠jD0ci†cj+hc.,
and is described in matrix form by (3) (where in fact 〈i|H0(3)|j〉 is displayed, |i〉 (i=1,2,3) denoting the single-well ground states). This is our minimal generic model for a multiwelled TS. The parameter D0 is chosen positive (contrary to custom in the STM, indeed -(1/2)Δ0<0 in (1)) for a good number of reasons. First, due to the possible softness of the local NM-potential, since indeed in general [17, 18] D0≃aħΩe-bV0/ħΩ, a and b being numbers such that for V0≳ħΩa>0 and b=O(1) can arise [17, 18, 21]. This choice is still compatible with the concept of tunneling and at the same time yields rather large values of D0≈ħΩ. On more general grounds, however, one should take into account that the tunneling “particle” is not moving in a vacuum, but is embedded in a solid that is, for the most part deprived of microscopic dynamics, at low-T. Thus, the surrounding frozen atoms are taking a part in the determination of the tunneling particle’s lowest stationary states. In the case of a perfectly D3-symmetric local nw=3 welled potential of the type depicted in Figure 4, Hamiltonian (3) leads to a doubly degenerate ground state and a first excited nondegenerate state (as is easily verified from (3) if E1=E2=E3). This may seem unphysical and yet Sussmann has demonstrated, in a remarkable paper [45], that for electrons trapped in a crystal (or equivalently in a glass) the situation above described is realised whenever the trapping potential is multiwelled with a triangular (nw=3) or tetrahedral (nw=4) well-centers geometry. The binding of the seemingly antibonding ground state is then guaranteed by the TS interaction with the rest of the solid. This reasoning is irrelevant for the STM-2LS parameter Δ0, since both positive and negative signs for this parameter yield the same physics. If nw>2, the sign will matter and Sussmann’s work shows that the choice D0>0 is physically justified for an embedded particle (or vacuum) in the glass. Finally, it will be shown in Conclusions that in fact the tunneling “particle” cannot be considered a single atom, ion, or molecule, but rather it represents a cluster of N-correlated tunneling atomic-scale particles, with N≈200. Then, it is reasonable to expect that the ground state of such a cluster might be near-degenerate; so our choice D0>0 for the effective single tunneling “particle” is sound and not in conflict with any general quantum-mechanical principle. This D0>0 is the major assumption for the multiwelled TS theory. It should be mentioned, however, that multiwelled potentials appear also in the Jahn-Teller quantum phenomena [46] and that in that context degenerate ground states are also commonplace. In the present situation, however, the disorder inherent in glasses does not allow for a detailed symmetry analysis.Two-dimensional representation of the plausible source of magnetic-field sensitive (anomalous) tunneling systems in, for example, the AlBaSiO glass. The tight vitreous-SiO2 structure (a) is broken up by the Al- and large Ba-atoms (b), thus leaving many metal ions free to move in a nw-minima (soft) tunneling potential, with n≥3 (c). The unbroken Si–O–Si bond dynamics, if any, is of the usual 2LS-type.
(a)
(b)
(c)At this point, we make a choice for the probability distribution of the parametersE1,  E2,  E3, and D0 of a tunneling 3LS nesting in the proximity of a RER, crystal embryo, or micro-crystal (one could also work with a nw=4 model potential; in the appendix we show that essentially the same results can be attained). This is dictated by the fact that near-degeneracy (E1=E2=E3) must be favored, yet not fully attained for the wells’ energy asymmetries of one such 3LS. We thus choose, assuming again the tunneling potential barriers to be broadly distributed,
(6)PATS(E1,E2,E3;D0)=P*(E12+E22+E32)D0.
which has the advantage of making use of a dimensionless material-dependent parameter P*: PATS(E1,E2,E3;D0), multiplied by the concentration xATS of these anomalous (multiwelled, and now near-degenerate) tunneling systems (ATS), is the probability of finding one such ATS per unit volume. In the following, xATS will be absorbed in the parameter P*. This choice for PATS has provided a good description of the experimental data for the multisilicates in a magnetic field [21], when in the Hamiltonian (3) (or equivalently (5)) D0 at position (i,j) is replaced with D0eiϕij (ϕij being the appropriate Peierls phase). As was shown in [21], the spectrum of this B>0 modified 3LS Hamiltonian (3) is formally given by (using Viète’s formula for the cubic equation’s solutions):
(7)ℰkD0=21-∑i≠jEiEj6D02cos(13θ+θk)cosθ=(cosϕ+E1E2E32D03)(1-∑i≠jEiEj3D02)-3/2,
(with k=0, 1, 2 and θk=0,+(2/3)π,-(2/3)π distinguishing the three lowest eigenstates) and for a choice of E1,E2,E3 and D0≫E12+E22+E32 (near-degenerate limit); this is shown in Figure 5. One can see that for very small ϕ (the Aharonov-Bohm phase proportional to the magnetic field B: ϕ=2πΦ(B)/Φ0, Φ(B)=SΔ·B being the flux through the single ATS (see also the appendix)), the spectrum consists of an isolated near-degenerate doublet which is well separated from the higher excited states. We shall exploit the ϕ=0 limit of this description for an explanation of the composition-dependent experiments.Figure 5
Variation within the magnetic Aharonov-Bohm phaseϕ of the energy spectrum (units D0=1) for a choice of E1,E2, and  E3 with D/D0=0.01. In this work, we are interested in the ϕ=0 limit of this spectrum, which can be treated, at low-T, as that of an effective 2LS.It should be stressed at this point that in the absence of a magnetic field, like in this work, one could make use of a 2LS minimal model for the description of the ATS,H0(2)(E1,E2;D0), and with the distribution P(E1,E2;D0)=P*/D0E12+E22 ensuing from the proximity of RER or incipient microcrystallites. It was shown in [21] that, at least for the heat capacity, this leads to the same physics as obtained from the 3LS multiwelled model. There is no harm in using, for the ATS nesting in the incipient crystalline regions, a more realistic minimal generic multiwelled model like the above 3LS Hamiltonian H0(3) which better approximates the physical reality of the energy landscape. Moreover, the model for the composition-dependent effects remains the very same used for the magnetic effects, and many results already obtained for that theory can be exploited by setting simply B=0. We remark, also, that a distribution of the type (6) for the energy asymmetry was already proposed for the explanation of low-T experiments with mesoscopic Au and Ag wires [47], where TS (of standard 2LS type) were advocated and where the polycrystallinity of metals must be accounted for.In summary, we have fully justified the extended TM which we have used in [21] and which we exploit also in this paper. The realistic glass is recognized to have a structure resembling that of chocolate [48] (or of opals) and as is pictured in the cartoon in Figure 6: a homogeneously-disordered networked solid in which (at low-T in the glass) only standard 2LS are present with their own concentration x2LS and in which incipient crystallites are embedded (for chocolate, these would be sugar crystals). In the proximity or within these crystallites are nested the ATS, with their own concentration xATS in the solid and with their own quantum mechanics and statistics defined by the minimal generic model represented by (3) and (6). This is by no means an ad hoc model, since the very same model would describe TS in all types of real metallic and nonmetallic glasses and quantitatively explain all of the low-T experiments in nonmetallic glasses tackled so far.Figure 6
A 2D cartoon of the chocolate-like, ceramic-glass structure of a real glass, in which partial devitrification has occurred, with the location of its low-T, two-species TS. In the randomly networked bulk of the material sit the STM-2LS, with their own concentration x2LS, whilst within and in the proximity of the incipient crystallites nest the ATS, with their own bulk concentration xATS, each being described by (3) and (6). We expect xATS<x2LS and that xATS→0 in the best glasses.


## 3. Predictions for the Dielectric Constant

The 2LS-STM has been successful in the semiquantitative explanation of a variety of interesting thermal, dielectric, and acoustic anomalies of structural glasses at temperaturesT< 1 K [14–18], the physics of cold glasses being important not only for their universalities, but also because of their link with the physics of the glass transition (see, e.g., [49, 50]). Beside the linearity in T behavior of the heat capacity Cp, it is believed that the linearity in ±lnT behavior of the real part of the frequency-dependent dielectric constant ϵ′(T,ω) represents a cogent characterization of the glassy state at low temperatures. We begin by deriving this behavior and putting it to test on data for ϵ′ for pure amorphous silica (i.e., no measurable ATS effects).In the presence of an applied electric fieldF, we must add the dipole energy -F·p0 to the parameter (1/2)Δ in the expression (1) for the low-energy Hamiltonian H0(2). We can express the permittivity (strictly speaking, the polarization) as ϵ=-∂2f(F)/∂F2|F=0, where f(F)=-(1/kBT)lnZ(F) represents the free energy per unit volume. The statistical average implies also an integration over the two parameters of the 2LS, Δ and Δ0, according to the distribution given by (2). We can write the partition function in terms of the energy levels E1,2: Z=e-ℰ1/kBT+e-ℰ2/kBT.Figure7 (inset) shows the behavior of the T-dependent part of ϵ′(T,ω), Δϵ′/ϵ′=[ϵ′(T)-ϵ′(T0)]/ϵ′(T0), (where T0(ω) is a characteristic minimum) for pure vitreous SiO2 (Spectrosil). It can be seen that linear regimes in -lnT for T<T0 and +lnT for T>T0 are observed, and roughly with slopes S-=-2S and S+=+S>0, or in a −2 : 1 ratio. According to the 2LS-STM, in fact, we have the expressions [14–18, 51]
(8)Δϵ′ϵ′|2LS=Δϵ′ϵ′|2RES+Δϵ′ϵ′|2REL,Δϵ′ϵ′|2RES=2P-p02¯3ϵ0ϵr∫zminzmax‍dzz1-(Δ0min2kBTz)2tanhz,Δϵ′ϵ′|2REL=P-p02¯3ϵ0ϵr×∫zminzmax‍dz∫τminτmax‍dττ1-τminτ×cosh-2(z)11+ω2τ2,
where we neglect (for low ω) the frequency dependence in the RES part, where zmin,max=Δ0min,max/2kBT and where τ is the phenomenological 2LS relaxation time given by (with E=2kBTz) [17, 18]:
(9)τ-1=EΔ02γtanh(E2kBT).
In these expressions, Δ0min and Δ0max are Δ0’s phenomenological bounds, γ is an elastic material parameter of the solid, and τmin-1=E3/γtanh(E/2kBT), τmax-1=EΔ0min2/γtanh(E/2kBT). P- (containing the 2LS volume concentration, x2LS) is the probability per unit volume and energy that a 2LS occurs in the solid (it appears in (2)) and p02¯ is the average square 2LS electric dipole moment. Moreover, the strategy of dielectric relaxation theory has been adopted, whereby the full complex dielectric constant ϵ(T,ω) has been written as, for ωτ≪1 [51, 52],
(10)ϵ(T,ω)=ϵRES′(T)+ϵREL′(T)11+iωτ;
the subscripts RES and REL refer to the zero relaxation-time resonant and relaxational contributions to the linear response ϵ′ at zero frequency, respectively.Figure 7
Dielectric signature of purea-SiO2 (inset) and AlBaSiO (main) glasses. SiO2 data [53], fitted with (8), display a −2 : 1 2LS-TM behavior. AlBaSiO data [54] display rather a −1 : 1 behavior, yet could be fitted with (8) (dashed line) [54] with a large Δ0min=12.2 mK 2LS tunneling parameter. We have fitted all data with a more realistic Δ0min=3.9 mK and best fit parameters from Table 1 using (8) and (14) (driving frequency ω=1 kHz).Presently, from expressions (8) we deduce that: (1) the so-called resonant (RES) contribution has the leading behavior
(11)Δϵ′ϵ′|2RES≃{-23P-p02¯ϵ0ϵrln(2kBTΔ0max)ifT<Δ0max2kB,0ifT>Δ0max2kB;
(2) the relaxational (REL) contribution has, instead, the leading behavior
(12)Δϵ′ϵ′|2REL≃{0ifωτmin≫1,13P-p02¯ϵ0ϵrln(2kBTΔ0min)ifωτmin≪1.
Thus, the sum of the two contributions has a V-shaped form, in a semilogarithmic plot, with the minimum occurring at a T0 roughly given by the condition ωτmin(kBT)≃1, or kBT0(ω)≃((1/2)γω)1/3. ϵ0ϵr is here the bulk of the solid’s dielectric constant, and we see that a −2 : 1 characteristic behavior is justified by the STM with the T>T0 slope given by S=P-p02¯/3ϵ0ϵr.This behavior is observed in purea-SiO2 [53] (Figure 7 (inset), with the fitting parameters of Table 1, x=0, from our own best fit to (8)). However, in most multicomponent glasses one more often observes a V-shaped curve with a (roughly) −1 : 1 slope ratio. Figure 7 (main) shows this phenomenon for the multisilicate AlBaSiO glass (in fact, a MAS-type ceramic-glass), which has been extensively investigated in recent times due to its unexpected magnetic field response [6, 7, 9–12, 21]. Also, Figure 8 shows the remarkable behavior of the dielectric constant versus T for the glasses of composition (SiO2)1-x(K2O)x containing a molar concentration x of alkali oxide [24]. It is seen that a S-/S+ slope ratio of roughly −1 : 1 is observed, with the slope definitely changing with x (and faster for T>T0). These data from the Anderson group [24], thus far unexplained by the 2LS-STM, call for an extension of the accepted STM, and we show below that a simple explanation can be given in terms of the very same ATS that have been justified in Section 2 and advocated by one of us in order to explain the magnetic response of AlBaSiO and other multicomponent glasses [21]. In view of the interest for these materials in low-T metrology, and on fundamental grounds, such explanation appears overdue to us. Moreover, “additional” TS (beside the standard 2LS) of the type here advocated were already called for in [24] and other theoretical papers [55–57].Table 1
Extracted parameters for the glasses; K-Si stands for the(SiO2)1-x(K2O)x glasses. In all of the best fits, we have employed the values Δ0min = 3.9 mK and Δ0max = 10 K extracted from fitting the pure SiO2 data of Figure 7 (inset).
Glass type
x
A
2LS
γ
A
ATS
D
min
Γ
mol
10
-
5
10
-
8 sJ3
10
-
5
K
10−6 sK5
SiO
2
0
47.2
5.30
—
—
—
AlBaSiO
—
116.2
13.40
264.7
0.65
69.73
K-Si
0.05
104.1
1.33
75.5
0.87
3.55
K-Si
0.08
146.5
1.23
130.0
0.87
3.97
K-Si
0.10
158.5
1.15
160.0
0.87
5.08
K-Si
0.20
239.5
0.82
281.9
0.87
6.44Figure 8
Dielectric signature of mixed(SiO2)1-x(K2O)x glasses as function of T and x [24]. Fitting parameters from Table 1 using (8) and (14) from our theory (driving frequency ω=10 kHz).For the multiwelled (3LS, in practice) Hamiltonian (3), we have nw=3 low-lying energy levels, with ℰ0<ℰ1≪ℰ2. In the Ei→0 and D≡E12+E22+E32≪D0 limits (due to the chosen near-degenerate distribution, (6)), we can approximate the nw=3-eigenstate system through an effective 2LS (though sensitive to all three well asymmetries and their distribution) having gap Δℰ=ℰ1-ℰ0:
(13)limΔℰ≃E12+E22+E32≡D.
We have also exploited the condition E1+E2+E3=0. Using the theory of [21] to work out the 3LS contributions to ϵRES′ and ϵREL′, we arrive at the following expressions for the contribution to the dielectric anomaly from the advocated ATS:
(14)Δϵ′ϵ′|ATS=Δϵ′ϵ′|ARES+Δϵ′ϵ′|AREL,Δϵ′ϵ′|ARES=πP~*p12¯3ϵ0ϵrDmin∫1∞‍dyy2tanh(Dmin2kBTy),Δϵ′ϵ′|AREL=πP~*p12¯2ϵ0ϵrDmin(Dmin2kBT)×∫1∞‍dyycosh-2(Dmin2kBTy)11+ω2τAmax2.
Here, we have again neglected, for low-ω, the frequency dependence in the RES part; we have put y=D/Dmin, and τAmax is the largest phenomenological ATS relaxation time given by [59]
(15)τAmax-1=D5Γtanh(D2kBT).
Moreover, Dmin is the lowest energy gap of the multilevel ATS, Γ is another appropriate elastic constant, and P~* is the (slightly renormalised) probability per unit volume (after inclusion of xATS) that an ATS occurs within the NM-pockets and channels, with p12¯ the average square ATS dipole moment. P~* and P* are so related:
(16)P~*=P*ln(D0maxD0min).D0min and D0max being D0’s lower and upper bounds, respectively. This description is intimately linked to the chosen distribution function, (6), for these ATS which is favoring near-degenerate energy gaps D bound from below by Dmin. In turn, this produces an overall density of states given by ([21], for B=0):
(17)g(E)=g2LS+gATS(E)≃2P-+2πP~*Eθ(E-Dmin),
and that is now roughly of the form advocated by Yu and Legget [25–27] and by some other preceeding authors (e.g., [60]) to explain anomalies not accounted for by the standard 2LS-TM. θ(x) is the step function.Manipulation of the expressions in (14) shows that (1) the RES contribution from the ATS has the leading behavior (note that for T<Dmin/2kB,  ϵ′|ARES is roughly a constant)
(18)Δϵ′ϵ′|ARES≃{0ifT<Dmin2kB,πP~*p12¯6ϵ0ϵrkBTln(2kBTDmin)ifT>Dmin2kB;
(2) the REL contribution is, instead, characterised by the leading form
(19)Δϵ′ϵ′|AREL≃{0ifωτAmax≫1,πP~*p12¯ϵ0ϵrkBTln(kBTDmin)ifωτAmax≪1.
Thus, the V-shaped semilogarithmic curve is somewhat lost. However, adding the 2LS (8) and ATS (14) contributions together, one does recover a rounded V-shaped semilog with a slope S-≃-2S basically unchanged for T<T0 and an augmented slope S+=S+SATS for T>T0 with SATS=7πP~*p12¯/6ϵ0ϵrkBT  that for  T<Dmin/kB may approach 2S and thus (qualitatively) explain a −1 : 1 slope ratio.We have fitted the full expressions (8) and (14) to the data for AlBaSiO in Figure 7 (main) and to the x-dependent data for (SiO2)1-x(K2O)x in Figures 8 and 9, obtaining in all cases very good agreement between theory and experiments [24]. Figure 9 shows the fit of our theory to the frequency-dependent data for x=0.2. In all of these best fits, we have kept the value of Δ0min=3.9 mK fixed, as obtained from our pure SiO2 fit, and the value of Dmin also independent of x and ω. The idea is that these parameters are rather local ones and should not be influenced by NF/NM dilution. Table 1 gathers the values of all the (2LS and ATS) parameters used for our best fits and Figure 10 shows the dependence of the prefactors (containing x2LS in P- and xATS in P~*) with x. It can be seen that, as expected, the ATS prefactor AATS=πP~*p12¯/ϵ0ϵrDmin scales linearly with x, an excellent confirmation that the “additional”  TS of [24, 55–57] are those ATS, proposed by us and modelled as 3LS, forming near and inside the microcrystallites that may nucleate within the NM-pockets and channels. It can be seen, instead, that the 2LS prefactor A2LS=P-p02¯/ϵ0ϵr of our fits also increases, though less rapidly, with increasing x (a decrease like 1-x would be expected). We propose (adopting a NF-, NM-cluster percolation picture) that new, “dilution-induced” 2LS form with alkali mixing near the NF/NM interfaces of the NF percolating cluster(s) as x is increased from 0. This reasoning leads to the expression A2LS=Abulk(1-x)+Asurfxf for the 2LS prefactor, with Abulk,  Asurf,  and f fitting parameters. Our best fit leads to the value f=0.81, in fair agreement with the euristic expression
(20)f=1-(D-Ds)ν,
(where D is the fractal dimension of the percolating cluster, Ds with Ds≤D is that of its “bridging” surface (not necessarily the hull) and ν is the connectedness length’s exponent) that one would deduce from elementary fractal or percolation theory (see, e.g., [61, 62]). Ds is the fractal dimension of that part of the NM random-cluster’s surface where formation of TS takes place and we expect 2≤Ds≤D. It is indeed reasonable to expect new TS to be forming at these NM/NF random interfaces, for these are surfaces of chemical discontinuity in the material. The above expression is derived as follows. Imagine (as is shown in the cartoons in Figure 11) the NM-clusters percolating through the NF-bulk with a site concentration x, so that their volume scales like 𝒱~ℓD, where ℓ~xν is their typical linear size. The number of 2LS on the surface of these clusters will scale like N2LS(s)~xℓDs and so their density like N2LS(s)/𝒱~xx(Ds-Df)ν=xf with the given expression, (20), for f. If we consider clusters of 2D percolation and assume Ds=Dh=7/4 (the fractal dimension of the hull of the spanning cluster), then with D=91/48 and ν=4/3 [61, 62] we would get f=29/36=0.8055. More realistically, on the assumption of percolating 3D NM-clusters in the mixed glasses, we can make use of the values [61–63] D≃2.52, Ds=Dh=2.14, and ν≃0.88 to arrive at the value f=0.67 using (20) (We are well aware that the quoted fractal dimensions apply to percolation clusters, strictly, only at the percolation threshold x=xc. Our attitude is that fractal-type clusters can be used to model the NM-lumps from the good crystal-formers even for x<xc and with non-integer fractal dimensions. This assumption failing, we have no explanation for the extracted non-zero value of the f-exponent and Asurf-prefactor.). It is however not at all clear where, at the NM/NF fractal interfaces, the new 2LS will form (i.e., what the exact definition of Ds ought to be: hull surface sites, screening sites, dead-end sites, etc.). If all of the hull sites are involved, then for 3D x=xc percolation Ds=D and one then expects f=1. Thus, this new phenomenology opens a tantalizing new investigation avenue for research on the applications of fractal theory to low-T physics. At the same time, the knowledge of which type of NM/NF fractal interface sites are involved in the TS-formation would greatly improve our understanding about the microscopic nature of the TS (see also [28]).Figure 9
Dielectric signature of mixed(SiO2)1-x(K2O)x glasses as function of T and ω for x=0.2 [24]. Fitting parameters from Table 1 using (8) and (14) from our theory.Figure 10
The 2LS and ATS dimensionless prefactor parameters (×105) for all glasses (from Table 1) as a function of x. Our data fit well with our theoretical expectations (full lines).Figure 11
(Color online) A cartoon of the fractal (presumably percolating) geometry of the NM-pockets and channels (green); these NM-clusters grow with increasingx.


## 4. Predictions for the Heat Capacity

We now come to the explanation of the, also rather anomalous, heat-capacity data for the mixed glasses(SiO2)1-x(K2O), reported in [24] as a function of  T and for different x. The heat capacity’s low-temperature dependence in zero magnetic field is, for pure glasses, usually given by the following expression:
(21)Cp(T)=BphT3+B2LST.
The first term accounts for the Debye-type contribution from the acoustic phonons and dominates above 1 K; the second term is usually attributed to the low-energy excitations specific of all vitreous solids—the tunneling 2LS. Bph and B2LS are material-dependent constants. This expression describes well the experimental data for pure silica glass at zero field (Figure 12, black circles: x=0 with fit parameters from Table 2), but it fails for the multicomponent glasses, like AlBaSiO, BK7, Duran (see, e.g., [21] and references therein) and for the mixed glasses (SiO2)1-x(K2O)x for x>0 [24].Table 2
Extracted parameters for fits to the heat capacity data forSiO2 and (SiO2)1-x(K2O)x glasses, with Dmin = 0.87 K and V0 = −0.42 K as fixed.
Glass type
x
B
ph
×
1
0
8
B
2LS
×
1
0
8
B
ATS
×
1
0
8
x
j
mol
Jm−3 K−4
Jm−3 K−2
Jm−3 K−4
ppm
SiO
2
0
245.55
70.65
—
—
K-Si
0.05
260.92
155.23
22.77
29.86
K-Si
0.08
266.36
196.11
36.44
18.15
K-Si
0.10
269.46
221.62
45.55
10.54
K-Si
0.20
281.42
337.19
91.11
3.00Figure 12
The temperature dependence of the heat capacity fora-SiO2 (black circles) and for the (SiO2)1-x(K2O)x glasses [24]. The full lines are our theoretical curves, as generated by (31).Typically, the heat capacity’s experimental data for the multicomponent glasses in zero field denote a kind of “shoulder” at intermediate-low temperatures. This suggests a density of states, for at least some of the independent TS in the glass, of the formg(E)∝1/E, in contrast to the standard 2LS-TM prediction, g(E)≃ const., which ensues from the standard TM distribution of parameters. Indeed, this 1/E contribution to the DOS was the very first observation that has led to the hypothesis of the ATS formulated in [21].To find out the precise expression for the heat capacity due to the ATS, we make use of the 3LS formulation for the ATS described in [21] and in more detail in Section 2. The heat capacity is determined from the second derivative of the free energy with respect to temperature:
(22)CpATS(T)=-T∂2FATS(T)∂T2,
where FATS(T) is the free energy of the ATS given by, if we neglect the third, highest energy level in the spectrum of Hamiltonian (3) (effective 2LS approximation):
(23)FATS(T)=-kBTln(e-ℰ0/kBT+e-ℰ1/kBT)=-kBTln(2cosh(E2kBT)),
with E=ℰ1-ℰ0. The heat capacity is then obtained by averaging over the parameter distribution, or, equivalently, by a convolution with the DOS:
(24)CpATS(T)=kB∫0∞‍dEgATS(E)(E2kBT)2cosh-2(E2kBT),
where density of states gATS(E) has the following form [21]:
(25)gATS(E)=∫‍dD∫‍dD0P(D,D0)δ(E-D)≃{2P*EifE>Dmin,0ifE<Dmin
and Dmin is the lower cutoff. The final expression for the ATS heat capacity results in [21]:
(26)CpATS(T)=BATS[ln(2cosh(Dmin2kBT))-Dmin2kBTtanh(Dmin2kBT)],
where the prefactor for the ATS is BATS=2πP~*kBnATSρ(x), P~* as in Section 3, nATS being the ATS mass concentration, and ρ(x) the glass’ mass density. Of course, xATS=nATSρ(x). For kBT≳Dmin, this is indeed roughly a constant and gives the observed “shoulder” in Cp(T) when the contribution BphT3 (from virtual phonons) as well as the STM linear term B2LST are taken into account.Both prefactors, for the 2LS and ATS contributions, are dependent on the molar concentrationx of alkali-oxide, just as we found in Section 3 for the prefactors of the dielectric constant: B2LS≃Bbulk(1-x)+Bsurfxf, BATS≃Bx. Also Bph requires to be reevaluated. With increasing K2O molar concentration x for the (SiO2)1-x(K2O)x glass, the number of phonons from the NM-component (K2O in this case) increases linearly with the concentration x, and for the NF-component (SiO2) it should also decrease linearly, like (1-x). Just as we assumed in the previous section, there are fractal/percolation effects between the NM- and NF-clusters, which makes room for some percolation clusters’ interfaces where the phonons also might contribute somehow with a term proportional to Cphxf (Cph being an x-independent constant).For these glasses, moreover, a nonnegligible concentration ofFe3+ (or, according to coloring, Fe2+) impurities is reported, a side effect of the industrial production process. Estimates give 102 ppm for AlBaSiO and 126 ppm for Duran, 6 ppm for BK7, 100 ppm for Pyrex 7740, and 12 ppm for Pyrex 9700 (see, e.g., the discussion in [21]). All glasses may, indeed, contain some [FeO4]0 impurity-substitution F-centers (in the glass, similar to a liquid, in concentrations however much, much lower than the nominal Fe bulk concentrations [43]). The Fe3+ cation and the O2- anion, on which the hole is localized (forming the O- species, i.e., the O2- + hole subsystem), form a bound small polaron. In this configuration, the Fe3+ cation is subject to a crystal field with an approximate C3 symmetry axis along the Fe3+-O- direction. This axis plays a quantization role for the Fe3+ electronic spin. The hole is assumed to be tunneling between two neighboring oxygen ions, switching the quantization axis between two directions, and therefore entangling its spin states. This is likely to give some tiny contribution to the heat capacity, and we should, therefore, also take it here into account [64]. The spin Hamiltonian of the [FeO4]0 F-center is Hs-S=VzszSz, where Vz is the principal value of the dipole interaction matrix and sz and Sz are the spin operators of the hole and of the Fe3+ ion, respectively. In the absence of a magnetic field, there are only two low-lying energy levels: E1,2=±(5/4)|Vz|. The unknown distribution function G(Vz) must approach zero when its argument approaches either zero or infinity and have a maximum at a definite argument value V0. The simplest one-parameter function displaying such properties is a Poisson distribution:
(27)G(Vz)=4Vz2V03exp(-2VzV0),Vz∈(-∞;0],V0<0.
The contribution from the [FeO4]0 ensemble to the heat capacity is, as usual,
(28)CFe3+(T)=-T∂2FFe3+∂T2,
where FFe3+(T) is the free energy of the [FeO4]0 ensemble, that one evaluates as
(29)FFe3+=-kBTln(e-E1/kBT+e-E2/kBT)=-kBTln(2cosh(E2kBT));
here E=(5/4)|Vz|. Using said distribution function for G(Vz), (27), as well as the expression for CFe3+(T) from FFe3+(T), one can obtain an expression for the heat capacity from the trace [FeO4]0 centres in the glass, and which should be added to the total heat capacity Cp:
(30)CpFe3+(T)=ρ(x)njkB×∫0∞‍dVz(E2kBT)2cosh-2(E2kBT)G(Vz)=ρ(x)njkB∫0∞‍dVz25Vz416T21V03e(-2Vz/V0)×cosh-2(5Vz8kBT),
where nj=xj/ρ(x) is the mass concentration of the tiny amount of Fe3+ ions (a very small fraction of the total bulk Fe-concentration) substituting the Si4+ in the network.Hence, the total heat capacity will be the sum of all these contributions: (21), (26), and (30) (Regarding the role of the Fe-impurities at zero magnetic field, this was totally overlooked in [21] where, to provide a good fit of the data, the existence of a weak, stray magnetic field was wrongly advocated.):
(31)Cp(T)=BphT3+B2LST+CpATS(T)+CpFe3+(T).
Making use of expression (31), we have fitted the experimental data for the heat capacity of the (SiO2)1-x(K2O)x  glasses from [24]. In order to fit the pure a-SiO2 data, we use only formula (21), that fits the pure silica’s data well within the 2LS-STM.The heat capacityCp(T,x) data [24] for the (SiO2)1-x(K2O)x glasses were obtained using a signal-averaging technique and for these samples the data are presented in Figure 12. As one can see, the heat capacity for the (SiO2)1-x(K2O)x glasses at low temperatures is larger than that for pure silica glass, as is typical for the multicomponent glasses, already with the smallest 5% concentration of K2O. The heat capacity decreases and then again increases with increasing molar concentration x of K2O. The additional heat capacity arises from the addition of ATS in the K2O NM-clusters and also from the presence of Fe3+ impurities, contained in small (and unknown) concentrations, but contributing to the low- and middle-range of the temperature dependence.Both prefactors, for 2LS and ATS, are indeed dependent on the molar concentrationx from our data analysis, and in the same way as we did in Section 3 we have fitted the extracted prefactors with the forms: B2LS≃Bbulk(1-x)+Bsurfxf, BATS≃Bx (B being some constant). These dependencies are shown in Figure 13. Also Bph is found to change by increasing the concentration x of the good crystal-former, K2O, and in the way we anticipated.Figure 13
The 2LS and ATS prefactor parameters(×108) for all glasses (from Table 2) as a function of x. The experimental data fit well with our theoretical expectations with f=0.81 (full lines).With increasing concentrationx, for the (SiO2)1-x(K2O)x glass, the number of phonons from the NM-component (K2O) increases linearly with the concentration, and for the NF-component (SiO2), it should be decreasing linearly like (1-x). As we reasoned for the dielectric constant, there are percolation mixing effects between the NM- and the NF-systems, which create percolation clusters and their NF/NM interfaces where phonons also might be populated in a way proportional to Cphxf. As it turns out, the very same value f=0.81 can be extracted from all our fits, just as was done in Section 3 for the dielectric constant data.


## 5. Summary and Conclusions

We have demonstrated that there is direct evidence in zero magnetic field already for the existence of multiwelled ATS (modelled as tunneling 3LS) and with the new distribution function advocated to explain the magnetic field effects in the multicomponent glasses (see [21]). The relevance of near-degenerate multiwelled TS in glasses is a new and unexpected finding in this field of research. Our work predicts, in particular, that the magnetic response of the mixed alkali-silicate glasses should be important and scale like the molar alkali concentration x. At the same time, the −1 : 1 slope-ratio problem of the standard TM in comparison with experimental data for ϵ′(T) has been given a simple explanation in terms of our two-species tunneling model. The main result of this work is that the concentration xATS (absorbed in P~* and thus in the AATS- and BATS-prefactors) of ATS indeed scales linearly with x for both ϵ′(T,x) and Cp(T,x). This is supported by our analysis of the existing experimental data, [24], very well indeed. Our analysis is, in our view, strong evidence that the ATS are nesting in the NM-clusters of the good crystal-formers. Our fractal modeling of the phase-separation NF/NM cluster interfaces in the multicomponent glasses gives a strong indication that the TS are forming in correspondence to the chemical discontinuities in the structure of amorphous materials. The justification of our mathematical modeling implies the existence of incipient crystallites in all amorphous solids, where the relevant degrees of freedom appear to be correlated over decades or even hundreds of atomic spacings. This cooperativity now seems to be a commonplace occurrence in the glassy state at all temperatures below Tg.Using the results of this analysis (and for AlBaSiO the results of the experimental data analysis in a magnetic field [21]), we can estimate the value of the dipole moment associated with the ATS, peff=p12¯. For AlBaSiO, using the value of P~* extracted from Cp [21] and that of AATS given in Table 1, we extract peff=0.41D. For (SiO2)1-x(K2O)x, we notice from the definitions in Section 3 that the ratio of the dielectric and heat capacity prefactors
(32)AATSBATS=ρ(x)2ϵ0ϵrkBDminp-12,
is almost independent of the K2O concentration x. From our extracted values in Tables 1 and 2 and the measured values of ρ(x) [24], we estimate peff=0.045 D for the mixed glasses, independently of x! Considering the elementary atomic electric-dipole’s value is ea0=2.54 D, these small values of peff for the ATS confirm that their physics must come from the coherent (or correlated) tunneling of small ionic clusters (the very same origin for the large values of Dmin and for D0min,max; see the appendix). Indeed, a cluster of N coherently tunneling particles has a dipole moment peff=|∑‍i=1Npi| that can become much smaller than ea0 (the order of magnitude of each |pi| in the sum) as N grows large. The fact, that we extract values of peff much smaller than ea0, confirms the picture of a correlated tunneling cluster in the B=0 case already.It is noteworthy that several papers from the Anderson group have proved that the addition of any NM-species in a networking pure glass causes significant (and thus far unexplained) deviations from the predictions of the 2LS-STM [24, 65, 66]. We have explained the origin of these deviations for Cp(T,x) as well as for ϵ(T,x). However, experiments do show that the thermal conductivity κ(T,x)∝T2 remains (below 1 K) remarkably universal and composition independent [24]. This is connected with the superuniversality of the internal friction coefficient, Q-1, in the cold glasses; these and other remarkable findings will be addressed elsewhere within the context of our approach.In summary, we have shown that there is direct evidence in zero magnetic field already for the multiwelled ATS advocated to explain the magnetic field effects in the multicomponent glasses. Similarx-dependent phenomena are to be expected for the low-T anomalies of the MAS-type ceramic-glass of composition (SiO2)1-x(MgO)x, which should also respond to the magnetic field (Experiments on these glasses, using different isotopes Mg24-Mg26 and Mg25, could also serve to confirm the nuclear quadrupole explanation for the magnetic effects.) (just like the mixed alkali-silicates of this work should). One may remark, at this point, that any extension of the 2LS-STM enlarging the adjustable-parameter space is bound to improve agreement with the experimental data. In this paper, we have shown that it was not just a matter of quantitative agreement, but qualitative as well. Whilst agreeing that the TM remains unsatisfactory, we stress that it is the only approach we know of, which is versatile enough to allow for an interesting explanation of rather puzzling phenomena at low-T in the real glasses. Furthermore, our two-species, multilevel TS model has been able to consistently explain a good number of different experimental data [21, 59]. It cannot be a mere coincidence that the same phenomenological model, with rather similar material parameters in different experiments, is capable of explaining so much new physics. Far from being an ad hoc model, our approach reveals the intimate microscopic structure of the real glasses, which cannot be considered as being homogeneously disordered anymore, and this must have some important consequences also for a better understanding of the mechanisms underlying the glass transition.As for the possibility of estimating the size and density of the incipient crystals in glasses from our theory, we remark that the simplified geometric-averaging procedure adopted for the physics of the ATS so far [21] does not allow anything more than an estimate of the P* parameter (~1.97×1017 cm-3 for AlBaSiO [21], this being in fact the value of xATSP*, P* being the unknown dimensionless parameter of the ATS distribution in (6)). However, the geometric-averaging procedure should be performed in two stages (within the incipient microcrystals first and then within the glassy matrix in which the crystallites are embedded) at the price of making the theory considerably more complicated. When this is done, with a more efficient and complete theoretical formulation, then information on the size distribution of the incipient crystallites could be gained from further low-T experiments in magnetic fields and at different controlled compositions.



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