4
In this chapter it is convenient to define dΩ =
sin
θ
d
θ
d
φ
rather than d
2
Ω =
sin
θ
d
θ
d
φ
as in earlier chapters.
240
Chapter 12: Scattering Theory
12.3
Crosssections and scattering experiments
Children sometimes test their skill by taking turns to throw pebbles at a dis
tant target, perhaps a rock. If a pebble hits, it will bounce off in a different
direction, whereas a pebble that misses will simply continue undisturbed.
Each throw will not be repeated exactly, and after a long time we might
imagine that the children have thrown pebbles randomly, such that the dis
tribution of throws per unit area is uniform over a region surrounding the
target. If so, we can estimate the area of the target that the children see by
simply counting the number of pebbles that hit it – if
N
in
pebbles are thrown
in per unit area, and
N
sc
of them hit the rock, the rock has crosssectional
area
A
≃
N
sc
/N
in
.
(12
.
47)
With more care, we can measure the angle through which throws are de
flected. Pebbles that strike nearby points of a smooth rock will bounce off
in roughly the same direction, whereas a jagged rock may deflect pebbles
that hit closely spaced points very differently. Hence, counting the number
of pebbles that end up going in a given direction gives us information about
the rock’s shape.
We define the
differential crosssection
δσ
to be the
area of the target that deflects pebbles into a small solid angle
δ
Ω. If there
are
N
(
θ,φ
)
δ
Ω such pebbles, then
δσ
≡
N
(
θ,φ
)
δ
Ω
N
in
or
δσ
δ
Ω
=
N
(
θ,φ
)
N
in
,
(12
.
48)
and the total crosssection is
σ
tot
≡
integraldisplay
d
σ
=
integraldisplay
dΩ
d
σ
dΩ
=
integraldisplay
dΩ
N
(
θ,φ
)
N
in
=
N
sc
N
in
(12
.
49)
as above.
This may seem a rather baroque manner in which to investigate rocks,
but when you go out on a dark night with a torch, you probe objects in a very
similar way by throwing photons at them. A more complete analogy can be
drawn between pebblethrowing children and physicists with particle accel
erators: a beam containing a large number
N
b
of particles is fired towards
a target, and detectors measure the number of particles that scatter off into
each element of solid angle
δ
Ω. Long before the collision, a typical particle
in the beam looks like a free state

φ
)
, so the probability density of each
particle is
(
x

φ
)
2
and the number of particles per unit area perpendicular
to the beam direction is
n
in
(
x
⊥
) =
N
b
integraldisplay
d
x
bardbl
(
x

φ
)
2
,
(12
.
50)
where the integral is along the beam direction.
When

φ
)
is expanded in terms of momentum eigenstates, equation
(12.50) becomes
n
in
(
x
⊥
) =
N
b
(2
π
¯
h
)
3
integraldisplay
d
x
bardbl
d
3
p
d
3
p
′
e
i(
p
−
p
′
)
·
x
/
¯
h
φ
(
p
)
φ
∗
(
p
′
)
=
N
b
(2
π
¯
h
)
2
integraldisplay
d
3
p
d
3
p
′
δ
(
p
bardbl
−
p
′
bardbl
)e
i(
p
⊥
−
p
′
⊥
)
·
x
⊥
/
¯
h
φ
(
p
)
φ
∗
(
p
′
)
,
(12
.
51)
where the integral over
x
bardbl
produced the delta function of momentum along
the beam direction.
Experimental beams are highly collimated, so
φ
(
p
)
vanishes rapidly unless the momentum is near some average value ¯
p
.
In
You've reached the end of your free preview.
Want to read all 277 pages?
 Spring '15
 Unknow
 Physics, mechanics, The Land, probability amplitudes