Consider a glass in which some fraction of its constituent...

Consider a glass in which some fraction of its constituent atoms may occupy either of two slightly different positions giving rise to two energy levels $\Delta_i>0$ and $-\Delta_i$ for the $i$ th atom. (a) If each participating atom has the same levels $\Delta$ and $-\Delta$, calculate the contribution of these atoms to the heat capacity. (Ignore the usual Debye specific heat which will also be present in a real solid.) (b) If the glass has a random composition of such atoms so that all values of $\Delta_i$ are equally likely up to some limiting value $\Delta_0>0$, find the behavior of the low temperature heat capacity, i.e., $k T \ll \Delta_0$. (Definite integrals need not be evaluated provided they do not depend on any of the parameters.)

Consider a glass in which some fraction of its constituent atoms may occupy either of two slightly different positions giving rise to two energy levels $\Delta_i>0$ and $-\Delta_i$ for the $i$ th atom.
(a) If each participating atom has the same levels $\Delta$ and $-\Delta$, calculate the contribution of these atoms to the heat capacity. (Ignore the usual Debye specific heat which will also be present in a real solid.)
(b) If the glass has a random composition of such atoms so that all values of $\Delta_i$ are equally likely up to some limiting value $\Delta_0>0$, find the behavior of the low temperature heat capacity, i.e., $k T \ll \Delta_0$. (Definite integrals need not be evaluated provided they do not depend on any of the parameters.)

Key Concepts

Disorder and Energy Level Distribution in Glasses

In disordered materials such as glasses, structural irregularities can lead to a range of local environments and correspondingly a distribution of energy level splittings. This distribution affects the overall thermodynamic behavior of the system. Rather than a single, uniform energy gap, a continuous range of gaps alters response functions like heat capacity, particularly at low temperatures where the variety of energy scales leads to unique temperature dependencies.

Low Temperature Thermodynamics

Low temperature thermodynamics deals with the physical properties of systems when thermal energies are much smaller than the energy scales of the system, such as the energy gaps in two-level systems. In this regime, quantum effects dominate and the statistical mechanics can predict non-trivial temperature dependencies in quantities like specific heat, often exhibiting deviations from classical predictions (e.g., a non-Debye contribution in glasses) due to the thermal occupation of low-lying excitations.

Two-Level System

A two-level system is a foundational quantum mechanical model in which particles or atoms have only two accessible energy states. This simplified model allows for the study of thermal excitations and quantum transitions between a lower energy state and an upper energy state, providing insight into how discrete energy levels contribute to macroscopic thermodynamic quantities like heat capacity.

Heat Capacity in Quantum Systems

Heat capacity quantifies the amount of energy needed to raise the temperature of a system and is particularly influenced by the availability of excited states in quantum systems. In a two-level system, the thermal population of the excited state versus the ground state changes with temperature, resulting in a characteristic temperature-dependent contribution to the overall heat capacity that can reveal information about the energy gap between the states.

Transcript

00:02 This is the question number seven and this is the problem for two level system here there are two energy levels which are at suppose this is at even level and this is e0 and in this question their value of this e not are given that is e0 is equal to minus delta and e1 is equal to plus delta now we have to show that the total energy for this system is given by u is equal to minus delta tan hyperbolic delta by kb t and the capacity specific heat is kb delta by kb t whole square and sac hyperb delta by kb t by kb t so the is two problems we have to show now write down the total energy for this two level system e is equal to that is e0 e power minus a a not a not by kb t plus even e power minus even by k b t divided by e power minus e b b t by k b t divided by e power minus e b b b t by k b t plus e power minus even pi k b t so let us assume here is tau is equal to kb t this the energy kb is the bolzman constant is the temperature and this tau represent the energy here and substitute the value e0 is equal to minus delta and e1 is equal to plus delta then this total energy becomes e is equal to minus delta e power is this is minus e not and this is minus that it will be plus delta by tau plus e1 is plus delta delta e power minus delta by tau and the denominator is e power delta by tau plus e power minus delta by tau now in the numerator we can take comma delta and it is e power delta by tau and because this is minus is common and then will be minus e power minus delta by tau and this is same e power delta by tau plus e power minus delta by tau and this is the value of the sign hyperbolic delta by tau that is sorry this e is equal to minus delta sine hyperbolic delta by tau divide by this is cos hyperbolic delta by tau cos hyperbolic delta by tau now this energy in the question it is this is represent by e hence replace this e by u that is equal to minus delta tan hypervolic delta delta by and tau is the cavity we have assumed that and this will be kv t okay so this is the total energy for the stew energy level system which are at minus delta and the minus plus delta energy levels and the now find out the specific heat that is c is equal to del u by del t at conscent delta or sorry delta is equal to minus delta and then hyperbolic that is set hyperbolic delta by kb t and the differential of this is minus delta by kb t square now this will be that is c is equal to plus this minus minus plus and this is delta square by kb t square and sec hyperbolic delta by kb t if i multiply by this k b then i will get k b and this is delta by k b and this is delta by k b t whole square sec hyperbolic delta by k bt so this is the solution of the part a specific heat is given by this relation and the total energy by this relation now in the b part of the equation it is given that if the system has a random composition such that all values of the lambda are equally likely up to some limit that means there is some limit of this up to this value and they have they have random compositions and now in this case so that this specific heat is directly proportion to the temperature okay we have to show this problem and in the limit and kvt is less than this upper limit that is delta not we have to show that this specific heat is proportional to the temperature in the limit kb t less than delta not let us write this relation the probability p delta d delta that the upper limit lie it is the probability that the upper limit lie it is the probability that the upper limit lie between delta and delta plus ddil...

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