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Momentum transfer cross section

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In physics, and especially scattering theory, the momentum-transfer cross section (sometimes known as the momentum-transport cross section) is an effective scattering cross section useful for describing the average momentum transferred from a particle when it collides with a target. Essentially, it contains all the information about a scattering process necessary for calculating average momentum transfers but ignores other details about the scattering angle.

Contents

The momentum-transfer cross section σ t r is defined in terms of an (azimuthally symmetric and momentum independent) differential cross section d σ d Ω ( θ ) by

σ t r = ∫ ( 1 − cos ⁡ θ ) d σ d Ω ( θ ) d Ω

The momentum-transfer cross section can be written in terms of the phase shifts from a partial wave analysis as

σ t r = 4 π k 2 ∑ l = 0 ∞ ( l + 1 ) sin 2 ⁡ [ δ l + 1 ( k ) − δ l ( k ) ] .

Explanation

The factor of 1 − cos ⁡ θ arises as follows. Let the incoming particle be traveling along the z -axis with vector momentum

p → i n = q z ^ .

Suppose the particle scatters off the target with polar angle θ and azimuthal angle ϕ plane. Its new momentum is

p → o u t = q ′ cos ⁡ θ z ^ + q ′ sin ⁡ θ cos ⁡ ϕ x ^ + q ′ sin ⁡ θ sin ⁡ ϕ y ^ .

For collision to much heavier target than striking particle (ex: electron incident on the atom or ion), q ′ ⋍ q so

p → o u t ≃ q cos ⁡ θ z ^ + q sin ⁡ θ cos ⁡ ϕ x ^ + q sin ⁡ θ sin ⁡ ϕ y ^

By conservation of momentum, the target has acquired momentum

Δ p → = p → i n − p → o u t = q ( 1 − cos ⁡ θ ) z ^ − q sin ⁡ θ cos ⁡ ϕ x ^ − q sin ⁡ θ sin ⁡ ϕ y ^ .

Now, if many particles scatter off the target, and the target is assumed to have azimuthal symmetry, then the radial ( x and y ) components of the transferred momentum will average to zero. The average momentum transfer will be just q ( 1 − cos ⁡ θ ) z ^ . If we do the full averaging over all possible scattering events, we get

Δ p → a v g = ⟨ Δ p → ⟩ Ω .

where the total cross section is

σ t o t = ∫ d σ d Ω ( θ ) d Ω .

Here, the averaging is done by using expected value calculation (see d σ d Ω ( θ ) / σ t o t as a probability density function). Therefore, for a given total cross section, one does not need to compute new integrals for every possible momentum in order to determine the average momentum transferred to a target. One just needs to compute σ t r .

Application

This concept is used in calculating charge radius of nuclei such as proton and deuteron by electron scattering experiments.

To this purpose a useful quantity called the scattering vector q having the dimension of inverse length is defined as a function of energy E and scattering angle θ:

q = 2 E ℏ c sin ⁡ ( θ / 2 ) [ 1 + 2 E M c 2 sin 2 ⁡ ( θ / 2 ) ] 1 / 2

References

Momentum-transfer cross section Wikipedia
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