18 Collisionaland RadiativeProcesses
Chapter II
3. CROSSSECfIONS
Consider a uniform beam of test particles incident on a single field
particle supposedfixed as indicated schematically in Fig. 2. Let n1 denote
the number density of test particles in the beam, and let g denote the
relative speed of the test particles with respect to the field particle. A
measure of beam intensity is then provided by the magnitude of the test
particle flux density,
-
r 1 = n1g,
(3.1)
the number of test particles crossing unit area normal to the beam in
unit time. If Land -r denote length and time dimensions, then the
dimensions of flux density are [r 1] = (L- 3)(L-r-1) = L- 2-r-1.
The total collision cross section Q12(g) between a test particle (labeled 1)
and a field particle (labeled 2) is defined as follows:
Q12
( )=
-
g
No. of test particles colliding with
the field particle per unit time
r1
.
(3.2)
The dimensions of Q 12 are
[Q12 ] = -T -1
= 1;2 = area.
[r 1]
Since, from equation (3.2), (the number of collisions/unit time) = r 1Q12
= (number of particles/unit time, unit area) x (area), we may interpret Q12
as the effective geometrical blocking area which the field particle presentsto
the beam.The total cross section is an atomic quantity. It is a function only
of the two kinds of particles involved (in this case, independent of the
order of the subscripts),and of the relative speedg.
~
r,
~
,...
= n,y
~
,
t
,
,...
/
""
=..-
Test particle
beam
~
~
./
~
@
~
Field
particle
~
~
;,..
~
;,..
:..
Figure 2.
~
Definition of the cross section.
i
I
i
",;.~",J",,~_c
~c,"'-""-'">""
t I
"
Section3
CrossSections 19
The above definition of the total cross section makes no distinction
between the kinds of collisions that may be occurring. It is in fact a sum
over the elastic collision cross section and all the relevant nonelastic
collision cross sections,i.e.,
Q12
= Q\e! + L Q\"i).
(3.3)
Each of these more particular cross sections may be defined similarly to
equation (3.2) but with an appropriate change in the numerator. For
example, if the field particle were an atom in the ground level, then the
cross section for excitation of the field particle to the first excited level
by a test particle would be
No. of test particles causing a (1 -.2) excitation of the
Q~12-2J(g)==
field particl~~r ~nit time.
In this case the cross section dependson the order of the particle subscripts
and on the particular atomic process,as well as on g. If the test particle
were an electron,then Q\\--2)(g) would be zero unlessthe relative kinetic
energy (see Sec.6) between the electron and atom exceededthe threshold
energy £12of the first excited level. For sufficiently small g, it is clear that
Q = Q(e);when g becomes large enough to produce nonelastic collisions,
it is a matter of observation that the second term in equation (3.3) is
roughly of the same order of magnitude as Q(e).The spirit in which this
statement is to be taken is illustrated in Table 6, which is taken from
Massey and Burhop (1952, 1969).
Table 6 Relative Probabilities of Different Types of Collisions of
Electrons in Atomic Hydrogen
Energyof incident electrons(eV)
Type of collision
100
Elastic
Excitation of 2nd level
Excitation of 3rd level
Excitation of 4th level
Excitation of 5th level
Excitation of higher levels
All discretelevels
Ionization
12.2
33.5
5.9
2.2
1.0
1.7
44.3
43.5
200
400
1000
10,000
-,
Total crosssection
(in units of 10-16 cm2)
2.16
Percentageof all collisions
10.2
9.8
8.7
6.5
33.6
39.0
42.8
45.3
5.8
6.8
6.3
7.0
2.0
2.2
2.4
2.6
0.9
1.0
1.2
1.2
1.7
2.0
2.2
2.3
44.0
51.0
54.8
58.4
45.8
39.2
36.5
35.1
1.32
0.70
0.33
0.043
20
Collisional
and Radiative Processes
Chapter II
x
~
dO
~
~
r,
~
z
~
~
:
Figure 3.
Angles for the differential cross section.
For any particular collision process which causesa deflection of the test
particle, it is appropriate to break down the corresponding cross section
even further, similarly to what was done for the total cross section.
With referenceto Fig. 3, the differential cross sectionfor scattering into the
solid angle d.Q= sin X dXdl/Jdefinedby the sphericalpolar angles(X,l/J) is
denoted by /12(X, 4» and is defined as follows:
No. of test particles scattered by the field particle
I 12(X,l/J)d.Q=
into the solid angle d.Qper unit time.
r1
(3.4)
.
Of course, /12 will also be a function of the particular atomic process
and of g, and in fact it is related to the corresponding cross section
according to the equation
Q12=f
/12(X,l/J)d.Q.
(3.5)
41t
In almost all cases,scattering may be taken to be azimuthally symmetric
so that the integration over l/J may be performedto obtain
Q12
= 2n
f
It
/12(x)sin X dX.
(3.6)
0
The angle X is called the scattering angle.
When a test particle collides elastically with a fixed field particle, it
will lose momentum in the direction of its initial motion. Referring to
Fig. 4, for particles scattered into the angle between X and X + dX (any 4»,
the loss in the z-component of momentum is1
m1g - mtg cos X = m1g(1- cosX).
I
The fact that Ig/l = Igi follows from the discussionin Sec.6. The proof is required
in Exercise6.2.
cc_~,..~"-
..~~iiii"".~~-
,,~"
..
Section 3
Cross Sections
.(toy
/
./
m,g
./
./
/
..,.-/'
--""
~
/
/
//
/
/
/
/
21
/
/
/
/
/
.//~x
~
z
0
Figure 4. Momentum transfer in elastic scattering.
Since the number of test particles scattered elastically from a beam into
the solid angle d.Q.= sin XdXdcPis r 1I\e!(X, cf» d.Q.per unit time, the total
loss in the z-component of momentum per unit time is
f
mlg(1 - cos X)r lI\e!(X, cf»
d.Q.
=
(mlgr 1)
f
41t
(1 - cos X)Ir!(X, cP)d.Q..
41t
The quantity mlgr 1 is the momentum flux density in the test particle
beam, and so we may define
f
Q\11(g) ==
(1 - cos X)I\e!(X, cP) d.Q.
(3.7)
41t
as the cross sectionfor momentumtransfer.
Another interpretation of Q\11 may be obtainedas follows: Since
P (X .I.) d.Q.== I\e~(X, cf»
12
,
Of'
Q
d.Q.
(3 8
(e)
.
)
12
is the probability of elastic scattering into d.Q.,the average fraction of
momentumlost by a test particle in elastic collisions is
-1
mlg
f
PI2(X, cf»mlg(1 - cos X) d.Q. = Q\11
-w.
41t
(3.9)
Q12
Compared with Q\e!, the factor (1 - cos X) in equation (3.7) gives Q~11
more weight from large angle scattering events[i.e.,for X = 1t",(1 - cos X) = 2;
for X = 0, (1 - cos X) = 0]. For I\e! independent of X, Q\11= Q\e!, and so
these two cross sections will only differ when there is a pronounced concentration of scattering in the forward or backward directions. For elastic
electron atom scatteringat sufficientlylow energies,generallyQ\11~ Q\e! ,
but at higher energiesQ\11 can be less than Q\e! by a factor as large as
one half (see Fig. 24). The total elastic cross section Q\e! never appears in
the rigorous kinetic theory of the transport coefficients (i.e. the thermal
'"~,.
22 Collisional and Radiative Processes
Chapter II
conductivity, viscosity, electrical conductivity, etc.) for gases.The calculation
of the transport coefficients depends rather upon a set of generalized
momentum transfer cross sections defined by the relation
Q(l)(g)=
f
(1 - cos 'X)/~e~(X,<p)d.o.,
(3.10)
41t
where I is positive integer.
Exercise 3.1. Using the cross section concept, show that the probability
that a test particle travels a distance z or greater without a collision in a
uniform gas of field particles for which the number density is n2, is
e-Z/l1,
where 11= 1/n2Q12. (Suggestion: Set up a differential equation which
expresseshow the flux density of test particles r I(Z) is diminished in
traveling through field particles in a slab of thickness dz.) Using this
result, justify the designation of 11,as a mean (i.e. average)free path for
the test particle.
Exercise 3.2. The atmospheric number density n above the earth varies
with altitude z approximately as n = noe-zIH,where H is called the scale
height. For a constant collision cross section Q, calculate the probability
that a particle shot upward with very high speed (relative to the escape
velocity) will attain a height z without a collision. What is the probability
that the particle will escape? Make a numerical estimate and discuss.
(Assume H
10 km.)
'"
Exercise 3.3. Show for a rigid-sphere interaction, i.e. /(e) = a2/4, independent of angle, that
Q(I) = [ 1 - 1 + ( -1 )1] ( 2
2 1+ I
7t'a.)
!
Exercise 3.4. For the differential cross section /(e)(X)= a for 0 S X < 7t'S
and /(e)(X) = b for 7t'S< X S 7t',where 0 S S S 1, determine in as simple a
form as possible the condition on /(e)(X)in order that Q(I) > Q(e).