Scattering Geometry and Impact Parameter

Scattering Geometry and Impact Parameter

Scattering is the deflection of one object by an interaction with another.

In Classical Mechanics, a two-body scattering problem can be reduced to the relative motion of an effective particle of reduced mass

(1)

moving in an interaction potential U(r).

Far from the interaction region, the relative trajectory approaches a straight line. The perpendicular offset of that incoming line from the force center is the impact parameter b.

If the incoming relative speed is v_∞, then the relative angular momentum is

(2)

The outgoing asymptote is generally deflected through a scattering angle χ.

Thus a central scattering problem can be summarized by the mapping

(3)

The interaction potential determines that map.

1 Laboratory motion versus relative motion

Suppose two particles have positions

(4)

Define the relative coordinate

(5)

For an isolated pair,

(6)

Therefore the scattering geometry can be studied as a one-particle central-force problem in r.

The center-of-mass translation is dynamically separate.

This is why impact parameter, relative speed, relative angular momentum, and the relative scattering orbit are natural variables.

2 Asymptotic states

Assume the interaction becomes negligible at large separation:

(7)

Before the encounter,

(8)

After the encounter,

(9)

For elastic scattering in a conservative time-independent potential,

(10)

The speed is unchanged asymptotically, but the direction can change.

3 Impact parameter

The impact parameter b is the perpendicular distance between:

• the force center,
• the straight incoming asymptote that the relative particle would follow if no interaction acted.

It is not, in general, the actual closest distance reached during the encounter.

That distinction is essential:

(11)

in general.

For repulsive interactions,

(12)

can be greater than, equal to, or related nontrivially to b depending on the potential.

4 Impact parameter and angular momentum

Far from the target, choose the incoming relative velocity along the +x direction:

(13)

The incoming straight-line asymptote can be written

(14)

At large negative x, the relative angular momentum is

(15)

Its magnitude is

(16)

Equivalently, the specific relative angular momentum is

(17)

5 Head-on scattering

If

(18)

then

(19)

The motion is purely radial.

For a repulsive central force, a head-on projectile can reverse direction, giving

(20)

for complete backscattering.

For nonzero b, the angular-momentum barrier prevents purely radial motion.

6 Large impact parameter

If b is large compared with the characteristic range of the interaction, the force acts only weakly along the trajectory.

Then the deflection is typically small:

(21)

for ordinary short-range interactions.

This qualitative trend is one of the most useful checks on a scattering result.

7 Energy at infinity

If

(22)

then the relative energy is

(23)

The relative angular momentum is

(24)

Therefore the pair (E,ℓ) is equivalently specified by the asymptotic pair (v_∞,b).

This connection lets scattering geometry feed directly into the effective-potential method.

8 Effective potential in a scattering problem

For a conservative central interaction,

(25)

where

(26)

Substitute

(27)

and

(28)

The angular-momentum term becomes

	=	(29)
	= E.	(30)

Thus

(31)

9 Distance of closest approach

At the distance of closest approach,

(32)

the radial velocity vanishes:

(33)

Therefore

(34)

Equivalently,

(35)

Using E = μv_∞² and ℓ = μv _∞b,

(36)

This equation determines the turning point of the scattering orbit.

10 No interaction as a consistency check

If

(37)

then

(38)

Thus

(39)

That is exactly the geometry of a straight line passing a point at perpendicular distance b.

For a nonzero interaction, the actual trajectory bends and r_min generally differs from b.

11 Radial equation for scattering

From

(40)

solve for the radial speed:

(41)

Using the asymptotic variables,

(42)

Or, dividing by v_∞² and using E = μv_∞²,

(43)

12 Angular rate

Angular momentum conservation gives

(44)

Therefore

(45)

The orbit geometry can be obtained by combining the angular and radial rates.

13 From time evolution to orbit geometry

Use

(46)

For the incoming or outgoing branch,

(47)

Thus the angular change from closest approach to infinity is

(48)

For a symmetric central-force encounter, the total scattering angle is

(49)

Therefore

(50)

This is the general classical central-force deflection formula.

14 Check the deflection formula when U = 0

For no interaction,

(51)

Then

(52)

The integral evaluates to

(53)

Therefore

(54)

A nonexistent force produces no scattering, as required.

15 Signed versus unsigned scattering angles

Different texts use different sign conventions.

One convention treats

(55)

as the magnitude of the deflection.

Another assigns opposite signs to bending toward and away from a reference side.

In this article, χ is primarily used as a nonnegative deflection magnitude unless a signed angle is stated explicitly.

The geometry must be defined before interpreting the sign.

16 Families of trajectories

A beam contains particles with many impact parameters.

Each value of b generally produces a different trajectory and scattering angle:

(56)

For an axially symmetric central interaction, the incident beam can be thought of as a family of concentric rings of impact parameter.

This geometric picture leads directly to the concept of cross section.

17 Geometric cross section

Suppose an event occurs whenever

(57)

The set of incoming trajectories that produce that event occupies a circular area in impact-parameter space:

(58)

This area is called a cross section.

Its dimensions are those of area:

(59)

The cross section is not necessarily the literal physical area of an object. It is an effective area associated with a specified scattering outcome.

18 Annulus of impact parameters

Trajectories with impact parameters between b and b + db occupy an annulus with area

(60)

If those trajectories scatter into angles between χ and χ + dχ, then the same set of trajectories can also be counted by their outgoing solid angle.

19 Solid-angle element

For azimuthally symmetric scattering,

(61)

Therefore

(62)

Hence

(63)

This is the classical differential cross-section formula for a one-to-one impact-parameter-to-angle map.

It converts scattering geometry into an observable angular distribution.

20 Why the absolute value appears

For many repulsive interactions, larger impact parameters produce smaller deflections:

(64)

A cross section must be nonnegative.

Therefore

(65)

appears in the differential cross section.

21 Hard-sphere scattering

A useful purely geometric model is scattering from an impenetrable hard sphere of radius a.

The projectile travels freely until it reaches the surface, then reflects specularly.

For a grazing trajectory,

(66)

and

(67)

For a head-on trajectory,

(68)

and

(69)

The geometry gives

(70)

22 Hard-sphere differential cross section

Differentiate

(71)

(72)

Then

= sin .

(73)

Using

(74)

we obtain

(75)

The hard-sphere differential cross section is independent of scattering angle.

23 Hard-sphere total cross section

Any trajectory with

(76)

hits the sphere.

Therefore the total geometric cross section is

(77)

This is also recovered by integrating

(78)

over the full solid angle:

(79)

24 Example 1: angular momentum from impact parameter

A relative particle has

μ	= 2.0 kg,	(80)
v∞	= 5.0 m∕s,	(81)
b	= 0.30 m.	(82)

Then

ℓ	= μv∞b	(83)
	= (2.0)(5.0)(0.30)	(84)
	= 3.0 kg m2∕s.	(85)

Thus

(86)

The specific angular momentum is

(87)

25 Example 2: hard-sphere angle

A hard sphere has radius

(88)

A trajectory has

(89)

From

(90)

(91)

Therefore

(92)

and

(93)

26 Repulsive inverse-radius potential

Consider

(94)

At closest approach,

(95)

Multiply by r_min²:

(96)

Therefore

(97)

The physical positive root is

(98)

27 Head-on closest approach for the repulsive inverse-radius potential

For

(99)

the turning-point formula becomes

(100)

All asymptotic kinetic energy is converted into potential energy at the turning point:

(101)

This is the classical distance-of-closest-approach result for a head-on repulsive inverse-square force.

28 Example 3: repulsive closest approach

Let

	= 0.40 m,	(102)
b	= 0.30 m.	(103)

Then

rmin	=	(104)
	=	(105)
	≈ 0.561 m.	(106)

Thus

(107)

The straight-line miss distance was only

(108)

but repulsion increases the actual closest separation.

29 Small-angle scattering intuition

When the deflection is small,

(109)

the incoming path is nearly straight.

A useful approximation is to estimate the transverse impulse along the unperturbed trajectory:

(110)

For small deflection,

(111)

This impulse picture is approximate, but it provides physical intuition:

• stronger force gives larger deflection,
• lower incoming speed gives larger deflection,
• smaller impact parameter usually gives larger deflection.

30 When the map bχ is not one-to-one

The simple differential cross-section formula

(112)

assumes one impact parameter corresponds to the chosen scattering angle.

Some potentials can produce several impact parameters that lead to the same χ.

Then the contributions must be added:

(113)

This behavior appears in more advanced classical scattering phenomena such as rainbow scattering.

31 Scattering angle in the center-of-mass frame

The reduced-coordinate calculation naturally produces the center-of-mass-frame scattering angle.

For two finite masses, the laboratory scattering angle of a particular particle need not equal the relative or center-of-mass scattering angle.

Transforming between frames requires the velocity relations

v1	= VCM + r,	(114)
v2	= VCM −r.	(115)

Thus:

(116)

32 Connection to collisions

A collision is a short-range interaction.

Scattering language emphasizes:

• the asymptotic incoming state,
• the asymptotic outgoing state,
• the impact parameter,
• the scattering angle,
• the distribution of outcomes over many trajectories.

The same conservation laws used in collision mechanics remain valid.

Scattering theory reorganizes those laws around trajectory geometry and measurable angular distributions.

33 Connection to Rutherford scattering

For a repulsive inverse-radius potential,

(117)

the deflection integral can be evaluated analytically.

The result leads to a simple relation between b and χ, and then to the celebrated inverse-fourth-power angular dependence of the Rutherford differential cross section.

That derivation is the subject of M04-11.

34 Common mistakes

1. Confusing the impact parameter b with the actual closest approach r_min.
2. Forgetting that b is defined from the incoming asymptote, not from the bent trajectory.
3. Using either physical mass instead of the reduced mass in the relative scattering problem.
4. Using particle speed instead of relative speed v_∞ in ℓ = μv_∞b.
5. Forgetting that U(∞) = 0 is an energy-reference choice used when writing E = μv_∞².
6. Conserving speed asymptotically for an inelastic or nonconservative scattering problem.
7. Forgetting the angular-momentum barrier in the closest-approach condition.
8. Using χ = π − 2𝜃₀ without defining the scattering-angle convention.
9. Forgetting the absolute value in dσ∕dΩ.
10. Treating cross section as necessarily equal to the literal physical area of the target.
11. Assuming the simple differential cross-section formula is sufficient when several impact parameters produce the same scattering angle.
12. Confusing center-of-mass-frame scattering angle with the laboratory angle of one particle.
13. Assuming zero torque means zero work. A central force can change speed through radial work while conserving angular momentum.

35 Practice exercises

1. Define the impact parameter geometrically and explain why it is generally not equal to the distance of closest approach.
2. Derive ℓ = μv_∞b from the incoming asymptotic trajectory.
3. Show that the specific relative angular momentum is h = v_∞b.
4. Starting from the relative energy equation, derive

   (118)

5. Show that U = 0 implies r_min = b.
6. Derive the radial equation

   (119)

7. Derive the central-force deflection integral for χ(b).
8. Verify that the deflection integral gives χ = 0 for U = 0.
9. Show that trajectories in an impact-parameter annulus have area dσ = 2πbdb.
10. Derive

    (120)

11. For a hard sphere, derive b = a cos(χ∕2).
12. Show that hard-sphere scattering has dσ∕dΩ = a²∕4.
13. Integrate the hard-sphere differential cross section to recover σ_total = πa².
14. For U(r) = k∕r, derive the positive-root expression for r_min.
15. Explain why the relative scattering angle and laboratory scattering angle are generally different for finite masses.

36 Summary

The impact parameter is the perpendicular offset of the incoming asymptote:

(121)

For relative speed v_∞,

(122)

With U(∞) = 0,

(123)

The closest approach satisfies

(124)

The central-force scattering angle is

(125)

Impact-parameter geometry gives

(126)

and, for an azimuthally symmetric one-to-one scattering map,

(127)

These formulas provide the foundation for the inverse-square scattering calculation in M04-11.

References

References