58
2 Experimental Methods for Determination of Nucleation Rates
A more realistic example is a mixture of radioactive isotopes. For example,
suppose the radioactive decay of a mixture of isotopes of Yttrium was measured
with time. Suppose the Laplace transform of the time evolution profile showed that
18 mol% of the decay signal had a half-life of 3.35 days, 32 mol% had a half-life
of 106.6 days, 21 mol% had a half-life of 2.67 days and 29 mol% had a half-life of
58.5 days. This information can be expressed in the following form:
|mixture >= 0.18|3.35 days > + 0.32|106.6 days > + 0.21|2.67 days > + 0.29|58.5 days >
(2.3.14)
Meanwhile, suppose it turned out that
87 Y has a half-life of 3.35 days,
88 Y has
a half-life of 106.6 days,
90 Y has a half-life of 2.67 days, and
91 Y has a half-life of
58.5 days. Then, using the orthogonality of the Laplace transform,
| mixture >= 0.18|
87 Y > + 0.32|
88 Y > + 0.219
0 Y > + 0.29|
91 Y > (2.3.15)
The orthogonality of the Laplace transform assures the uniqueness of each exponential component that appears in Eq. (2.3.14), and one can conclude that each
component in the unknown mixture must correspond to the known isotopes of
Yttrium, as shown in Eq. (2.3.15).
At first sight, the orthogonality of the Laplace transform might appear of remote
relevance to the statistical mechanics or the nucleation theory. This is not so.
The Laplace transform is highly relevant to both the statistical mechanics and the
nucleation theory.
In statistical mechanics, the partition function of a canonical ensemble can be
given by the Laplace transform of the energy density of states, Ω(E):
Z (β) ≡
∞
∫
0
Ω(E)e
−β E d E
(2.3.16)
Here, the variable t in Eq. (2.3.11) corresponds to energy, E, and the variable
s corresponds to β ≡ (k B T )
−1 . The partition function thus “samples” the number
of energy states of the system of interest that is “parallel” to the inverse of the
temperature, β.
As for the nucleation theory, the expected value of the nucleation probability
density,
2 Experimental Methods for Determination of Nucleation Rates
A more realistic example is a mixture of radioactive isotopes. For example,
suppose the radioactive decay of a mixture of isotopes of Yttrium was measured
with time. Suppose the Laplace transform of the time evolution profile showed that
18 mol% of the decay signal had a half-life of 3.35 days, 32 mol% had a half-life
of 106.6 days, 21 mol% had a half-life of 2.67 days and 29 mol% had a half-life of
58.5 days. This information can be expressed in the following form:
|mixture >= 0.18|3.35 days > + 0.32|106.6 days > + 0.21|2.67 days > + 0.29|58.5 days >
(2.3.14)
Meanwhile, suppose it turned out that
87 Y has a half-life of 3.35 days,
88 Y has
a half-life of 106.6 days,
90 Y has a half-life of 2.67 days, and
91 Y has a half-life of
58.5 days. Then, using the orthogonality of the Laplace transform,
| mixture >= 0.18|
87 Y > + 0.32|
88 Y > + 0.219
0 Y > + 0.29|
91 Y > (2.3.15)
The orthogonality of the Laplace transform assures the uniqueness of each exponential component that appears in Eq. (2.3.14), and one can conclude that each
component in the unknown mixture must correspond to the known isotopes of
Yttrium, as shown in Eq. (2.3.15).
At first sight, the orthogonality of the Laplace transform might appear of remote
relevance to the statistical mechanics or the nucleation theory. This is not so.
The Laplace transform is highly relevant to both the statistical mechanics and the
nucleation theory.
In statistical mechanics, the partition function of a canonical ensemble can be
given by the Laplace transform of the energy density of states, Ω(E):
Z (β) ≡
∞
∫
0
Ω(E)e
−β E d E
(2.3.16)
Here, the variable t in Eq. (2.3.11) corresponds to energy, E, and the variable
s corresponds to β ≡ (k B T )
−1 . The partition function thus “samples” the number
of energy states of the system of interest that is “parallel” to the inverse of the
temperature, β.
As for the nucleation theory, the expected value of the nucleation probability
density,
, can be given by the Laplace transform of the nucleation probability
density with time, p(t):
p ≡
∞
∫
0
p(t)e
−ckt dt
(2.3.15)
Here,
is essentially a nucleation curve. At a constant subcooling, the
inner product of p(t) and e
–ckt has only one surviving exponential component that
corresponds to the nucleation rate at that subcooling.