GRE Physics Companion: Scattering Geometry and Impact Parameter

GRE Physics Companion: Scattering Geometry and Impact Parameter

The core scattering relations are

(1)

(2)

and

(3)

1 High-value GRE facts

1. impact parameter is defined from the incoming asymptote.
2. In general, b≠r_min.
3. Relative angular momentum is ℓ = μv_∞b.
4. Head-on scattering has b = 0 and therefore ℓ = 0.
5. For U(∞) = 0, the asymptotic relative energy is E = μv_∞².
6. Closest approach is a radial turning point.
7. A cross section has dimensions of area.
8. An impact-parameter annulus has area dσ = 2πbdb.
9. For axial symmetry, dΩ = 2π sin χdχ.
10. Hard-sphere scattering has dσ∕dΩ = a²∕4.

Part I: Original GRE-style problems

Problem 1: angular momentum

A reduced particle has μ = 2m, incoming speed v, and impact parameter b. Its angular-momentum magnitude is

1. mvb
2. 2mvb
3. mv∕b
4. 2mv∕b
5. zero

Problem 2: head-on collision

For a head-on central-force scattering event,

1. b = 0
2. b = r_min always
3. ℓ≠0
4. χ = 0 necessarily
5. v_∞ = 0

Problem 3: no interaction

If U(r) = 0, then the distance of closest approach is

1. 0
2. b∕2
3. b
4. 2b
5. infinity

Problem 4: closest approach

At the distance of closest approach in a conservative central scattering problem,

1. ṙ = 0
2. 𝜃 = 0
3. ℓ = 0
4. E = 0
5. U = 0

Problem 5: geometric cross section

An event occurs whenever b ≤ b₀. Its cross section is

1. 2πb₀
2. πb₀
3. πb₀²
4. 2πb₀²
5. 4πb₀²

Problem 6: annulus area

The area associated with impact parameters between b and b + db is

1. πbdb
2. 2πbdb
3. 4πbdb
4. πb² db
5. db∕b

Problem 7: solid angle

For azimuthally symmetric scattering, the solid-angle element between χ and χ + dχ is

1. 2π dχ
2. π sin χdχ
3. 2π sin χdχ
4. 4π sin χdχ
5. sin ²χdχ

Problem 8: hard-sphere grazing

For hard-sphere scattering from radius a, a grazing trajectory has

1. b = 0, χ = π
2. b = a, χ = 0
3. b = a, χ = π
4. b = 0, χ = 0
5. b = 2a, χ = π∕2

Problem 9: hard-sphere differential cross section

For hard-sphere radius a,

(4)

equals

1. a²
2. a²∕2
3. a²∕4
4. πa²
5. 4πa²

Problem 10: inverse-radius head-on turning point

For repulsive potential U = k∕r and a head-on trajectory, the closest approach is

1. k∕E
2. E∕k
3. 
4. 2k∕E
5. zero

Problem 11: large impact parameter

For an ordinary short-range interaction, increasing b far beyond the interaction range generally makes the scattering angle

1. larger
2. smaller
3. exactly π
4. undefined
5. independent of b

Problem 12: frame issue

The reduced-coordinate scattering calculation most naturally gives

1. the center-of-mass or relative scattering geometry
2. the laboratory angle of particle 1 in every case
3. the laboratory angle of particle 2 in every case
4. only total linear momentum
5. no angle information

Part II: Complete worked solutions

Solution 1

Use

(5)

With μ = 2m,

(6)

Answer: (B).

Solution 2

Head-on means the incoming asymptote passes through the force center:

(7)

Therefore ℓ = 0. Answer: (A).

Solution 3

For U = 0, the turning-point equation gives

(8)

Thus

(9)

Answer: (C).

Solution 4

Closest approach is a radial turning point, so

(10)

Answer: (A).

Solution 5

The allowed incoming trajectories fill a disk of radius b₀:

(11)

Answer: (C).

Solution 6

The annulus area is circumference times width:

(12)

Answer: (B).

Solution 7

For axial symmetry,

(13)

Answer: (C).

Solution 8

A grazing trajectory just touches the sphere:

(14)

It experiences vanishing deflection:

(15)

Answer: (B).

Solution 9

Hard-sphere geometry gives

(16)

Answer: (C).

Solution 10

For b = 0, energy conservation gives

(17)

Thus

(18)

Answer: (A).

Solution 11

A large impact parameter keeps the trajectory far from a short-range interaction, so the deflection tends toward zero. Answer: (B).

Solution 12

The reduced-coordinate problem is naturally a center-of-mass or relative-motion calculation. Laboratory particle angles require an additional frame transformation. Answer: (A).

2 GRE checklist

1. Distinguish b from r_min.
2. Use ℓ = μv_∞b.
3. Treat closest approach as a radial turning point.
4. Count impact-parameter area with dσ = 2πbdb.
5. Use dΩ = 2π sin χdχ for axial symmetry.
6. Keep the absolute value in the differential cross section.
7. Check hard-sphere limits: b = 0 → χ = π and b = a → χ = 0.
8. Distinguish relative or center-of-mass scattering angles from laboratory particle angles.

References

References