A modification of the soft potential model of glasses

Replacement of the cubic term in the soft potential model of glasses with a linear one can be interpreted as an introduction of the internal pressure (coefficient of the linear term), additive to the applied external pressure. It makes the potential unique and removes the ambiguity, present in the conventional representation (different choices of the origin). Pressure-induced transformation of soft systems becomes in the new representation extremely simple – they evolve with the pressure along straight lines. The “sea-gull” distribution of the soft potential parameters transforms into a uniform one in these new variables.


INTRODUCTION
* Anomalous low-temperature thermal properties of glasses have been an active field of research since their discovery in the early 1970s [1]. The soft potential model (SPM) [2][3][4][5][6] has turned out to be a versatile method for capturing essential features of glasses: two-level systems, soft vibrations (the boson peak), and relaxation systems. In particular, effects of hydrostatic pressure on lowtemperature glasses can be modelled in this way [7][8][9]. While the effect of pressure on crystalline solids is mainly expressed by increase in vibrational frequencies, the pressure effects in glasses are more diverse. Besides the increase in vibrational frequencies, transitions between different types of excitations (two-level systems and soft vibrations) can be induced by the applied pressure. The pressure can also affect barriers between the two equilibria in SPM and, as a result, influence the irreversible relaxation in glasses [9]. On the other hand, studies of these effects can yield valuable information on the local structure and dynamics in glasses -problems still gaining much attention. In what follows, we demonstrate that a slight modification of SPM, introduction of "internal pressure" as a linear term in the potential instead of the cubic one, enables one to handle both pressurized and unpressurized situations on a * Corresponding author, jaak.kikas@ut.ee similar footing and gives, in our opinion, a more physical interpretation of the parameters of the potential.

THE SOFT POTENTIAL
In what follows, the quantities with the subscript "0" refer to the situation with no external pressure applied.
The form of the soft potential is given by [2][3][4] where the quantities 0 ξ and 0 η are random dimensionless parameters, subject to distribution to fit the experimental data. For the sake of simplicity, in what follows we take the energy factor 0 1.
differ from (1) only by the choice of the origin [5]. An alternative way of representing the potential (1) is the replacement of the cubic term by a linear one [6]. With the change of variable x z x t → = − and taking In what follows, the quantities with the asterisk (*) correspond to such a linearized form of the soft potential.
with the single-well potentials located above this curve (see Fig. 1b). Notice that only single-well potentials with * 0 0 η < can be transformed into double-well ones with pressure.
In order to match the experimental observation, a distribution of the soft potential parameters of the "seagull" type is usually adopted: Using (4), it is easy to show that the distribution for the variables *

PRESSURE EFFECTS
The applied hydrostatic pressure P is usually described as an additional linear term in the soft potential [7-9]: In Eq. (8) pressure is introduced as a dimensionless quantity; in physical units the pressure 1 P = is of the order of the Young modulus. With a proper choice of the origin, the potential can always be brought to a form where there is no linear term. For this one must take .
The cubic term can be removed from the "pressurized" potential (8), (10) as well. This is reached by trans- The result is The value of * * 0 η η ≡ remains, however, independent of pressure and is determined by the "unpressurized" values 0 ξ and 0 η only. In Fig. 4 the dependence of the points * * ( , ) P η on the initial parameter 0 ξ is presented.
Notice that the coefficient of the linear term * ( ) P P now sums up from two contributions: external pressure P and the quantity * 0 , P with the value given by (4). Hence it is reasonable to interpret the latter quantity as an "internal pressure", additive to the applied external pressure and subject to variations from location to location. Pressure-induced transformation of soft systems becomes in the

CONCLUSIONS
We have demonstrated that the replacement of the cubic term in the soft potential model with a linear one can be interpreted as an introduction of the internal pressure, additive to the applied external pressure. It also makes the potential unique and removes the ambiguity present In a further paper we are going to calculate the distribution of the internal pressure proceeding from the postulated ( , )-statistics ξ η of glasses in order to compare it to some microscopic elastic models of glasses [10].