The Differential Cross-Section

Computation of CEX-Ion Flux and Sputtering Rate

11. Output x, v, and h

4.4 The Differential Cross-Section

5. Ifn=N, evaluatehk =min δs

vk, h_max

and go to 6; otherwisehk=hk−1 and go to 7.

6. n= 0; If xk−x₀> R go to 11.

7. Evaluatex_k+1=x_k+hkv_k+^h₂^2kf(xk).

8. Evaluatevk+1=vk+^h₂^k[f(xk) +f(xk+1)].

9. k=k+ 1;n=n+ 1.

10. ifk=K, go to 11; otherwise go to 5.

ψ δx P

δA

δΩ₀

¯ ϕˆu θ⁰

ˆ n

Figure 4.8: CEX neutrals scattered from a small volume. A flux of collimated, monochromatic ions,

¯

ϕˆu, pass through a small volume,δV, around the point P that contains some number of neutral atoms,N₀. The number of CEX ions,δN_s⁰, scattered into a small solid angleδΩ₀about a scattering angleθ⁰ is determined by the charge-exchange differential cross-section, dσ⁰/dΩ.

from the volume, within some solid angle, δΩ₀, of the scattering angle, θ⁰, measured with respect to the direction of motion of the ions,ˆu, is [23]

N˙_s⁰(θ⁰) =δN_s⁰(θ⁰)

δt ≡N₀ϕ¯dσ⁰ dΩ

θ⁰

δΩ₀=n₀( ¯ϕˆu·ˆn)dσ⁰ dΩ

θ⁰

δΩ₀δA δx. (4.19)

Similarly, the number of CEX ions due to a charge-exchange collision, δN_s⁺, measured per unit time, at a large distance from the volume, within some solid angle,δΩ₊, of the scattering angle,θ⁺, measured with respect to the direction of motion of the ions,u, isˆ

N˙_s⁺(θ⁺) =δN_s⁺(θ⁺)

δt ≡N₀ϕ¯dσ⁺ dΩ

θ⁺

δΩ₊=n₀( ¯ϕuˆ·ˆn)dσ⁺ dΩ

θ⁺

δΩ₊ δA δx. (4.20)

Since every collision results in one CEX ion and one CEX neutral, for every CEX ion measured there must be a corresponding CEX neutral scattered (see Figure 4.9). Therefore, from any specific volume, the two measured rates of CEX-ion and CEX-neutral production must be equal; i.e., the quantities in Equations 4.19 and 4.20 must be the same. Therefore

dσ⁺ dΩ

θ+

δΩ₊=dσ⁰ dΩ

θ0

δΩ₀. (4.21)

The quantitiesδΩ₀ andδΩ₊ are not independent, but are related to each other in a way such that the kinematic constraints of conservation of momentum and energy are met. In the infinitesimal limit:

dΩ₀= sinθ⁰dθ⁰dφ⁰, (4.22)

dΩ₊= sinθ⁺dθ⁺dφ⁺, (4.23)

since the elastic-collision dynamics between any individual ion and atom can be completely described in the plane of motion of the two particles, dφ⁰= dφ⁺. Additionally, for elastic collisions involving identical species, conservation of momentum and energy dictate that the two particles scatter at right angles to each other, as measured in the lab frame [57];

θ⁰+θ⁺=π/2. (4.24)

Therefore, in the case of charge-exchange collisions between identical species, Equation 4.21 reduces to

sinθ⁺dσ⁺ dΩ

θ⁺

= sinθ⁰dσ⁰ dΩ

θ⁰

= cosθ⁺dσ⁰ dΩ

π/2−θ⁺

,¹ (4.25)

or

dσ⁺ dΩ

θ⁺

=cosθ⁺ sinθ⁺

dσ⁰ dΩ

π/2−θ⁺

. (4.26)

If the charge-exchange differential cross-section between identical species can be measured for either of the collision products (ion/neutral), the cross-section for the other product (neutral/ion) can be obtained from Equation 4.26.

Measurements of the charge-exchange differential cross-section were performed at JPL [25]. A monochromatic, collimated beam of singly charged xenon ions was passed through a target cell containing neutral xenon atoms at a known pressure and temperature. The density of the xenon

1The pesky negative sign from the relation dθ⁰=−dθ⁺has been ignored, since we recognize that each quantity on either side of the equal sign must be positive. The negative sign only affects the order of the limits of integration.

θ⁰ θ⁺

+

+

δΩ₊

δΩ₀

Figure 4.9: Elastic charge-exchange collision angles in the lab frame. The rate at which ions are scattered intoδΩ₊ must equal the rate at which neutral atoms are scattered intoδΩ₀. In a collision involving identical particles, the two particles must scatter at right angles to each other,θ⁰+θ⁺= π/2.

ions in the target cell was obtained from the ideal gas law. Located downstream of the target cell was a detector which collected the neutrals produced as a result of the charge-exchange collisions occuring in the target cell. The remaining ions left in the beam were deflected away from the collector plate using charged deflection plates. The detector had an active area of 16 cm², with 256×256 pixel resolution. In order to differentiate between different scattering angles, the array of collecting pixels was divided into rings centered around the beam axis. The scattering angle to any ring was assumed to be the average of the angles to the innner and outer pixels that define the ring. The maximum measurable angle was 3.6^◦.

The results of measurements (black dots) made for beam ions with an energy of 1100 eV, and an extrapolated curve fit of the data (red line) up to a scattering angle of 90^◦, are shown in Figure 4.10. For identical species, no scattering events can occur at angles greater than 90^◦, thus requiring the differential cross-section to drop off to zero at 90^◦. The measurements did not extend up to scattering angles of 90^◦, and we have no further information as to what angle this drop off actually occurs. The curve fit neglects the presence of a drop off, so we anticipate the extrapolation of the data to be higher at larger angles than in reality.

The total charge-exchange cross-section is defined to be

σ₀=

Ω

dσ

dΩ dΩ. (4.27)

10⁻¹

0.01 0.1 1

10¹ 10³ 10⁵ 10⁷

10 90

DifferentialCrossSectiondσ0 dΩ(˚A2/str)

Neutral Scattering Angleθ⁰ (deg)

10⁻¹ 0 10¹ 10³ 10⁵ 10⁷

80 89 89.9 89.99

DifferentialCrossSectiondσ+ dΩ(˚A2 /str)

Ion Scattering Angleθ⁺ (deg)

Figure 4.10: Xe-Xe⁺ charge-exchange differential cross-section. Black dots in the left figure show data taken from measurements of CEX neutrals from collisions involving primary ions with an energy of E₀ = 1100 eV. The red line in the left figure is a curve-fit to the measured data (black dots). The right figure is the corresponding differential cross-section for the CEX ions obtained by transformation of the data in the left figure, according to Equation 4.26.

For charge-exchange collisions between xenon ions and atoms, the total cross-section is finite and has a value of approximately 55 ˚A²for ion energies of 300 eV [58]. A cut-off angle very close to zero,θ⁰_c, was imposed where the curve fit for the differential cross-section is constant for all angles less than this value. This was necessary in order to maintain a finite total cross-section. The extrapolating curve fit for the data has the functional form

dσ⁰ dΩ = 7.2

θ⁰ 10^◦

₋₅/2

˚A²/str, θ⁰_c = 0.038^◦. (4.28)

The cut-off angle,θ_c⁰, was chosen such that the total cross section was equal to the measured value, σ₀ = 55 ˚A². Though this value for the total cross-section is the value for 300 eV ions, it has been found that at low energies, the total cross-section changes little [58]. The functional form for the differential cross-section can easily be modified to conform to accurate values of the total cross- section by adjusting the cut-off angle; however for modest changes in the total cross-section the

cut-off angle will change very little.

The differential cross-section for CEX ions is shown on the right of Figure 4.10. The rapid increase of the differential cross-section at anglesθ⁺<∼30^◦ is due to both the sin⁻¹θ⁺ conversion from Equation 4.26, and that, as mentioned earlier, the differential cross-section is artificially high as θ⁰→90^◦. The decrease in the differential cross-section asθ⁺→90^◦ is due to the cosθ⁺ conversion from Equation 4.26, and the imposition of a constant value for θ⁰ < θc⁰. Charge-exchange ion quantities obtained from scattering angles near these two extremes may be suspect, however it will be shown later that the scattering angles of relevance to this work do not lie close to either extreme.