GRE Physics Companion: Differential Cross Sections and Rutherford Scattering

GRE Physics Companion: Differential Cross Sections and Rutherford Scattering

For axially symmetric classical scattering,

(1)

For repulsive Coulomb scattering,

(2)

so

(3)

1 High-value GRE facts

1. Rutherford scattering uses the relative or center of mass kinetic energy.
2. b = (k∕2E) cot(χ∕2) for repulsive Coulomb scattering.
3. dσ∕dΩ ∝ E⁻² at fixed angle.
4. dσ∕dΩ ∝ csc ⁴(χ∕2).
5. Large-angle scattering corresponds to small impact parameter.
6. The head-on closest approach is r_min = k∕E.
7. The Coulomb scattering length is a = k∕(2E).
8. The integrated cross section above χ₀ is πa² cot ²(χ ₀∕2).
9. The ideal unscreened Coulomb total cross section diverges at zero angle.
10. For a very heavy target, center of mass energy and angle approach their projectile laboratory counterparts.

Part I: Original GRE-style problems

Problem 1: energy scaling

At fixed scattering angle, the projectile energy is doubled. The Rutherford differential cross section changes by a factor of

1. 1∕4
2. 1∕2
3. 1
4. 2
5. 4

Problem 2: charge scaling

At fixed E and χ, one charge magnitude is doubled. The Rutherford differential cross section changes by a factor of

1. 1∕4
2. 1∕2
3. 1
4. 2
5. 4

Problem 3: impact parameter at 90^∘

Define a = k∕(2E). For χ = 90^∘, the Rutherford impact parameter is

1. 0
2. a∕2
3. a
4. 2a
5. infinity

Problem 4: head-on closest approach

For repulsive Coulomb scattering, the head-on closest approach is

1. a∕2
2. a
3. 2a
4. 4a
5. zero

Problem 5: small-angle behavior

For small χ, Rutherford scattering behaves approximately as

1. χ⁻¹
2. χ⁻²
3. χ⁻³
4. χ⁻⁴
5. e^−χ

Problem 6: differential cross-section units

The customary units of dσ∕dΩ are

1. m
2. m∕s
3. m²
4. m²∕sr
5. sr∕m²

Problem 7: barn conversion

One barn equals

1. 10⁻¹⁵ m²
2. 10⁻²⁰ m²
3. 10⁻²⁴ m²
4. 10⁻²⁸ m²
5. 10⁻³² m²

Problem 8: integrated angular cut

The cross section for Rutherford scattering through angles at least χ₀ is

1. πa² cot ²(χ ₀∕2)
2. πa² tan ²(χ ₀∕2)
3. 4πa²
4. πa² sin ²χ ₀
5. a²∕χ ₀

Problem 9: small angle and impact parameter

For repulsive Rutherford scattering, as b increases,

1. χ increases
2. χ decreases
3. χ remains fixed
4. E must vanish
5. the reduced mass becomes zero

Problem 10: center of mass energy

A projectile of mass m₁ with lab kinetic energy E_lab strikes a stationary target of mass m₂. The relative kinetic energy is

1. E_lab
2. E_lab
3. E_lab
4. E_lab
5. zero

Problem 11: pure Coulomb total cross section

If arbitrarily small nonzero scattering angles are counted for an unscreened Coulomb potential, the classical total cross section is

1. zero
2. πa²
3. finite and independent of energy
4. divergent
5. equal to one barn

Problem 12: heavy target limit

If m₂ ≫ m₁, then for a stationary target

1. E_CM ≪ E_lab and angles are unrelated
2. E_CM ≈ E_lab and 𝜃_lab ≈ χ
3. E_CM = 0
4. 𝜃_lab = 0 for all events
5. the reduced mass approaches m₂

Part II: Complete worked solutions

Solution 1

At fixed angle,

(4)

Therefore

(5)

Answer: (A).

Solution 2

Since

(6)

and

(7)

doubling one charge multiplies the cross section by four. Answer: (E).

Solution 3

(8)

Answer: (C).

Solution 4

For a head-on encounter,

(9)

Answer: (C).

Solution 5

For small angle,

(10)

Therefore

(11)

Answer: (D).

Solution 6

A differential cross section is an area per unit solid angle:

(12)

Answer: (D).

Solution 7

By definition,

(13)

Answer: (D).

Solution 8

All events with χ ≥ χ₀ correspond to

(14)

Thus

(15)

Answer: (A).

Solution 9

From

(16)

larger b corresponds to smaller χ. Answer: (B).

Solution 10

Since

(17)

and

(18)

(19)

Answer: (C).

Solution 11

As

(20)

(21)

Hence the ideal unscreened Coulomb integrated cross section diverges. Answer: (D).

Solution 12

When

(22)

the reduced mass approaches m₁, the center of mass energy approaches projectile lab energy, and the lab projectile angle approaches the center of mass angle. Answer: (B).

2 GRE checklist

1. Use center of mass relative energy in the exact Rutherford formula.
2. Keep the half-angle: cot(χ∕2) and csc ⁴(χ∕2).
3. Remember the E⁻² scaling.
4. Large angle means small impact parameter.
5. Distinguish a = k∕(2E) from r_min = k∕E for head-on scattering.
6. Use the barn conversion 1 barn = 10⁻²⁸ m².
7. A zero-angle cutoff is essential for a finite integrated pure-Coulomb cross section.
8. Check whether a laboratory-to-center of mass angle transformation is needed.

References

References