Environmental Engineering Reference
In-Depth Information
where
2.4 eV. This manipulation shows that the
systemmay have a stable configuration of bound ions and electrons (i.e.,
ε
D
2.5/(3
5/3
π
4/3
)(
m
e
e
4
/
„
2
)
D
0
ε
<
0).
min
The stable distribution of charged particles corresponds to
. The system
so described is called a Wigner crystal. It can be seen that a Wigner crystal, like real
metals, is characterized by an electron number density of the order of the typical
atom number density
a
0
. Note that the above results are based on simple models
which do not account for details of interaction in metals, and hence the results have
a qualitative character.
D
1.9/
1.3.6
Ideal Electron-Gas and Ion-Gas Systems
The gaseous state condition for a weakly ionized gas relates not only to the inter-
actions among charged particles or among neutral particles as separate groups but
also to interactions between charged and neutral particles. The interaction between
neutral atomic particles has a short-range character, whereas the interaction of a
charged particle with neutral particles may be long range and is stronger than the
interaction between neutral particles. Therefore, one can expect a violation of the
condition for the gaseous state to occur in the interaction of one charged particle
with surrounding particles in a dense gas. That is, in a system of atoms and a single
charged particle, where the interaction of the atoms satisfies the gaseous state cri-
terion, the interaction of the charged particle with the atoms does not have gaseous
character, that is, the charged particle interacts with many atoms simultaneously.
We now consider this phenomenon in detail.
We begin by examining the behavior of electrons in a dense gas. The gaseous
character of the interaction between electrons and atoms implies the condition
)
1
λ
D
(
N
σ
r
,
(1.98)
where
is the cross section for
electron-atom scattering,
N
is
t
he number density of atoms, and
r
is the mean dis-
tance between atoms. We take
r
to be the Wigner-Seitz radius, so
r
λ
isthemeanfreepathofanelectroninthegas,
σ
N
/3)
1/3
.
D
(4
π
L
2
,where
L
is the scat-
The electron-atom cross section is represented by
σ
D
4
π
tering length for slow electrons scattered by atoms.
We can write the condition (1.98) as
N
N
cr
,
(1.99)
L
3
p
3)
1
.Table1.8listsvaluesof
N
cr
and the critical pressure
where
N
cr
D
(4
π
p
cr
300 K. It is seen that the gaseous state
condition for interaction of electrons with atoms can be violated in a dense gas.
The other gaseous state conditions for the electron-atom interaction in a dense
gas require that positions of neighboring atoms should not influence electron-
atom scattering. These relationships give
D
N
cr
T
for electron interaction at
T
D
L
r
,
p r
/
„
1 .
(1.100)
