Universality of ultrasonic attenuation coefficient of amorphous systems at low temperatures

The competition between unretarded dispersion interactions between molecules prevailing at medium range order length scales and their phonon induced coupling at larger scales leads to appearance of nano-scale sub structures in amorphous systems. The complexity of intermolecular interactions gives rise to randomization of their operators. Based on a random matrix modelling of the Hamiltonian and its linear response to an external strain field, we show that the ultrasonic attenuation coefficient can be expressed as a ratio of two crucial length-scales related to molecular dynamics. A nearly constant value of the ratio for a wide range of materials then provides a theoretical explanation of the experimentally observed qualitative universality of the ultrasonic attenuation coefficient at low temperatures.

The basic structural unit in a glass depends on the presence of various cations some of which act as network formers and others act as network modifiers (e.g. see pages 9-11 of [1]). Here we give the glass composition and the dominant structural units for 18 glasses used in tables I. The molar mass M 1 in Table I refers to the masses of these units.
(1) a − SiO 2 : The 3-d network in this case has the basic structural unit is Si[SiO4] with mass M 1 = 120.09 (see page 37 of [2], section 11.4.1 of [8]), also see section 2.2 and fig2.7(a) of [3]). (3) As 2 S 3 : The glass in this case forms chain like structure e.g. S − S, As − As or As − S (i.e S or As of one chain interacting with neighboring one). As S is dielectric, we use it as the basic unit participating in VWD interaction (see page 56 of [11], page 125 of [6], also see section 2. with 61% SiO 2 but with 35% P bO, this has a basic structural unit is Si 2 O 5 with its mass M 1 = (due to compound of type 2SiO 2 .P bO, 2 Si atoms get coordinated with 5O, (see page 17 of [9], page 14-15 of [5]).  [9], page 14-15 of [5]).
(7) V52: The constituents are 57.8ZrF 4 , 33.8BaF 2 and 8.5T hF 4 . Due to higher content of ZrF 4 , the main structural unit in this case is ZrF 4 tetrahedral with main role of cations Ba and T h is to cause 2-d structure (network modifiers) (page 35 of [7], page 157 of [5]).
Thus M 1 in this case is used as the molar mass of ZrF 4 . and N a is that of network modifier (i.e to cause 2-d structure) (page 35 of [4], page 157 of [7]). Thus M in this case is used as ZrF 4 .
(9) LAT: 60 ZrF 4 , 33 T hF 4 , 7 LaF 3 . Due to higher content of ZrF 4 , the main structural unit in this case is ZrF 4 tetrahedral with main role of the cations Ba, Al and N a is that of network modifier (i.e to cause 2-d structure) (page 35 of [7], page 157 of [5]). Thus M in this case is used as ZrF 6 .
(10) a − Se: Se atoms form chains or 8 atom rings through covalent/ionic bonding. The atoms on neighboring chains or rings interact by lone-pair electrons (VWD). So M is that of Se atom (page 43 of [11], page 115 of [6]). (15) PMMA: The monomer here has a phenyl group C 6 H 5 which appears as a side unit while the units along the main chain strongly connected by covalent bonds. As VWD interaction occurs between molecules on different chains, the main unit playing role here is C 6 H 5 . So M 1 taken is that of C 6 H 5 .
(16) PS: The monomer here has a phenyl group C 6 H 5 as a side unit as well as CH = CH 2 unit while the units along the main chain strongly connected by covalent bonds. As VWD interaction occurs between molecules on different chains, the main unit playing role here seems to be CH − CH or CH = CH 2 . The former could be part of Phenyl group. Note unlike other polymers, the monomer of P S is small and therefore only part of Phenyl group may be loosely held and participate in VWD.
(17) PC: as the monomer here is a big molecule, the Phenyl group may be loosely held and participate in VWD. So M 1 taken is that of C 6 H 5 (18) ET1000: here again the monomer is a big molecule, the Phenyl group may be loosely held and participate in VWD. So here again M 1 taken is that of C 6 H 5 .

II. RELATION BETWEEN γ AND γ m
Consider the linear response of a basic block, labeled as s containing g 0 molecules, to an external strain field. The existence of long wavelength phonons at low temperatures leads to a phonon-mediated pair-wise interaction among molecules, decaying as inverse cube of distance between them. Consider two molecules, labeled as "1" and "2" with their centers at a distance r within the block. Following the same formulation as in case of blocks, and with T αβ as the stress tensor component for the molecule, the corresponding interaction energy can be written as with α 0 = 4, r = |r 1 − r 2 |, κ (12) αβγδ as in the case of block-block interaction (given by eq.(??)), ρ m as the mass-density of the molecule, c as speed of the sound waves. The ensemble averaged interaction energy can then be approximated as with γ m as the average strength of the phonon induced r −3 coupling of the two molecules.
The interaction parameter γ m can be determined as follows. As Tr(V stress ) = 0, one can write, with . e as the ensemble average But Tr(V 2 stress ) e ≈ 1 α 0 πρ m c 2 r 3 2 te,te κ (12) αβγδ κ (12) α β γ δ Tr T Further as with T αβ;nm ≡ n|T (1) αβ |m , state |n referring to one of the N single molecule states (unperturbed). Following similar ideas as in the case of a block, we have T (1) αβ;mn T (2) αβ;kl = 0 ∀m, n, k, l and T (1) αβ;mn T 1) α β ;kl = τ 2 δ αα δ ββ (δ nk δ ml + δ nl δ mk ). On ensemble average, the above leads to The above on substitution in eq.(3) leads to Comparison of eq.(8) with eq. (3) gives where K 2 = te (κ (12) αβγδ ) 2 . To relate the above to basic block property γ 2 , we proceed as follows. The stress-operator for a basic block can be written in terms of those of molecules: The subscripts m, n now refer to an arbitrary pair chosen from N = N g 0 many body states of the basic block (e.g. the product states |e 0 m and |e 0 n of single molecule states). Further T Further assuming homogeneous interaction within a basic block, the variances of all matrix elements of the basic block can be approximated as almost equal. The left side of eq.(10) is then equal to N 2 ν 2 (with (Γ (s) αβ;mn ) 2 e = ν 2 ) which leads to Taking κ (12) αβγδ from eq.(8) of main text, we have K 2 = 18 1 + 4 1 −