146
3 Ensembles: Systems of Particles
populations come at the cost of an approximately 40% reduction in the fractional
population of the ground level of Ge.
Comparison between the fractional population values for Si ∗ and Ge ∗ illustrates
the role of the excitation energies of excited states, with the energy levels for the
two excited states about 2.75 times higher on average for Ge ∗ than for Si ∗ , with
a consequent relative increase in the fractional population of the ground state of
Ge ∗ and decreases in the fractional populations of both excited states. Finally,
a comparison between the fractional populations for Si and Ge shows that the
degeneracies of the excited levels of Ge more than offset the increased excitation
energies to provide overall increases in the fractional populations of the excited
electronic levels of the Ge atom.
Formula (3.2.14) holds if we are dealing with discrete energy states. What
happens if the accessible states are part of a continuum as, e.g., turns out to be
an excellent approximation when we consider translational motion? Note that the
treatment of translational motion as a special case is particularly justified by the fact
that it exclusively gives rise to the volume dependence of the partition function for
any ensemble.
3.2.1 Translational States: Continuum Approximation
Let us assume that we are given the volume V and the number of particles N, and
that we measure the energy E for this system, and find ≤ energy ≤ + . Let
us now define ((, V ))) to be the number of microstates of the system of interest
having energies between and + . In principle we can always calculate this
quantity for any system. Let us do so now for a particle-in-a-box. Consider
V ) ≡ Number of microstates having energy ≤
+ V ) ≡ Number of microstates having energy ≤ + .
The number of microstates having energy E lying between and + is therefore
given by the difference between these two quantities. If we invoke a limiting process
whereby is taken to zero, we obtain the partial derivative
lim
→0
+ V ) − V )
≡
∂∂((, N, V )
∂∂
V
≡ ((, V ) ,
(3.2.15)
so that ((, V ) is called the density of states for continuum states.
Let us now apply the particle-in-a-box model to obtain an explicit expression for
((, V ) for translational motion. The expression for the energy for a particle in a
3-D box has the form
3 Ensembles: Systems of Particles
populations come at the cost of an approximately 40% reduction in the fractional
population of the ground level of Ge.
Comparison between the fractional population values for Si ∗ and Ge ∗ illustrates
the role of the excitation energies of excited states, with the energy levels for the
two excited states about 2.75 times higher on average for Ge ∗ than for Si ∗ , with
a consequent relative increase in the fractional population of the ground state of
Ge ∗ and decreases in the fractional populations of both excited states. Finally,
a comparison between the fractional populations for Si and Ge shows that the
degeneracies of the excited levels of Ge more than offset the increased excitation
energies to provide overall increases in the fractional populations of the excited
electronic levels of the Ge atom.
Formula (3.2.14) holds if we are dealing with discrete energy states. What
happens if the accessible states are part of a continuum as, e.g., turns out to be
an excellent approximation when we consider translational motion? Note that the
treatment of translational motion as a special case is particularly justified by the fact
that it exclusively gives rise to the volume dependence of the partition function for
any ensemble.
3.2.1 Translational States: Continuum Approximation
Let us assume that we are given the volume V and the number of particles N, and
that we measure the energy E for this system, and find ≤ energy ≤ + . Let
us now define ((, V ))) to be the number of microstates of the system of interest
having energies between and + . In principle we can always calculate this
quantity for any system. Let us do so now for a particle-in-a-box. Consider
V ) ≡ Number of microstates having energy ≤
+ V ) ≡ Number of microstates having energy ≤ + .
The number of microstates having energy E lying between and + is therefore
given by the difference between these two quantities. If we invoke a limiting process
whereby is taken to zero, we obtain the partial derivative
lim
→0
+ V ) − V )
≡
∂∂((, N, V )
∂∂
V
≡ ((, V ) ,
(3.2.15)
so that ((, V ) is called the density of states for continuum states.
Let us now apply the particle-in-a-box model to obtain an explicit expression for
((, V ) for translational motion. The expression for the energy for a particle in a
3-D box has the form