Extension to a 3D Example - Ion Behavior: Scattering Angle Solutions and Stream- tube Divergenc

Computation of CEX-Ion Flux and Sputtering Rate

11. Output x, v, and h

4.6 Ion Behavior: Scattering Angle Solutions and Stream- tube Divergencetube Divergence

4.6.2 Extension to a 3D Example

cross-section, ie.,

σ₀= π/2

dσ dθ dθ=

_∞

−∞

dσ dθ

dz dθ

₋₁

dz. (4.35)

Since the integral over allz is bounded, the integral over any range of ﬁnitez will also be ﬁnite.

φu

π/2−θu

ˆ ˆ u

φ⁺ θ⁺ O

P(x)

Figure 4.17: Deﬁnition of the azimuthal and inclination scattering angles. Both the azimuthal and inclination scattering angles, φ⁺ and θ⁺, are measured with respect to the primary ion velocity vector, ˆu. The zero-point for φ⁺ is arbitrary, due to the averaged 2π azimuthal symmetry of the scattering process, and is of no consequence to the initial speed or energy of the scattered ion. The φ⁺ = 0 convention shown here was chosen due to the resulting simple rotation matrices (Equation 4.39). The initial speed of the scattered ion isv=ucosθ⁺.

velocity components,û= (ûx,uˆy,uˆz). Let us then define the two anglesθu andφu, such that

sinθu≡ ˆ

u²_x+ ˆu²_y, and (4.37)

cosφu≡ uˆx

ˆ u²_x+ ˆu²_y

. (4.38)

The initial velocity unit vector, vˆ = (ˆvx,vˆy,vˆz), of the scattered ion can then be found from the following rotation operation:

ˆ v=

⎛

⎜⎜

⎝ ˆ vx

ˆ vy

ˆ vz

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝

cosθucosφu −sinφu sinθucosφu

cosθusinφu cosφu sinθusinφu

−sinθu 0 cosθu

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝

sinθ⁺cosφ⁺ sinθ⁺sinφ⁺

cosθ⁺

⎞

⎟⎟

⎠

(4.39)

It can be veriﬁed thatuˆ·vˆ= cosθ⁺. The magnitude of the initial velocity of the scattered ion is described by Equation 4.36.

In Section 4.6.1 we examined the locations that scattered ions passed through a speciﬁed plane (˜x= 60 cm), and obtained sets of curves — one curve for each scattering event. A procedure similar to that used to construct the curves in Figures 4.14 through 4.16 can be used for the more general 3D case, however, instead of obtaining sets of curves, sets of surfaces will be obtained. Each point on these surfaces will represent some parameter deﬁning the position of a CEX ion scattered with the pair of scattering angles, (θ⁺, φ⁺).

In the 2D example, the downstream distancez at which the ion trajectory passed through the plane at ˜x= 60 was the quantity of interest, however this is inadequate for the 3D case. It is very easy to imagine situations in which the scattered particle would never have a trajectory passing through the plane at ˜x= 60; the value would not exist. Let us imagine a sphere that is centered on the scattering center, P(x), and such that the target point, S(˜x), lies on the surface of this sphere. The radius of this sphere is Rs = |˜x−x|. For any scattered ion with scattering angles (θ⁺, φ⁺), there exists a point on this sphere,T(x), through which the ion will pass. We will make the 3D analogy of the downstream distancez, from the 2D example, to be the length of the chord, D(θ⁺, φ⁺), joining the location that the trajectory passes through the surface of the sphere,T(x), and the target point, S(˜x), i.e.,D(θ⁺, φ⁺) =|x−˜x|. By this deﬁnition, the minimum attainable value occurs when the ion passes through S(˜x): D_min = 0. The maximum value occurs when the ion passes through the point diametrically opposed to S(˜x): D_max = 2Rs. Figure 4.18 graphically depicts the deﬁnition described.

For any scattering location and primary-ion velocity vector, a surface can be constructed depict- ing the relationship between the two scattering angles,φ⁺ andθ⁺, and the chord length described above,D(θ⁺, φ⁺). For example, let us assume that now the scattering center is not located in the radial plane passing through the point of interest, but is instead located at the general cartesian coordinates,P(x, y, z), and the target point is located, as before, atS(˜x,y,˜ z) = (60,0,0). In this ex-˜ ample, rather than primary ions moving parallel to the thrust axis, we will assume that the primary ions have the velocity anglesθu= 10^◦ andφu = 180^◦. The resulting surfaces and contours for scattering events occuring at four diﬀerent scattering centersP_I−IV(x) are shown in Figures 4.19 through

x

S T

O D

Ion trajectory

Figure 4.18: Scattering target interception sphere. A sphere is centered on the scattering centerP(x).

The target point S(˜x) is located on the surface of the sphere and deﬁnes the radius,Rs=|˜x−x|.

An ion scattered fromPwith the scattering angles (θ⁺, φ⁺) passes through the surface of the sphere at the positionT(x). The chord length,D(θ⁺, φ⁺) =|x˜−x|, is computed.

4.22. The scattering-angle solutions (those pairs of scattering angles which result in an ion passing through the speciﬁed target point) for any scattering event are those angles for whichD(θ⁺, φ⁺) = 0.

For example, the scattering event at P_II(x) has two solutions: (1) (θ⁺₁, φ⁺₁) = (110.0^◦,89.1^◦), and (2) (θ⁺₂, φ⁺₂) = (159.5^◦,86.1^◦). The scattering event atP_IV(x), on the other hand, has no solutions that result in scattered CEX ions passing through the speciﬁed target point,S(˜x,y,˜ z) = (60,˜ 0,0).

The 2π-periodicity inφ⁺ is apparent.

The four cases shown present the analogous 3D extension of the four cases discussed in the 2D example:

(I) One unique pair of scattering angles, (θ⁺, φ⁺), results in the scattered ion passing through the target point; the streamtube area at the target point is greater than zero (d ˜A/dΩ>0).

(II) Two unique pairs of scattering angles result in a scattered ion passing through the target point.

(III) One unique pair of scattering angles results in the scattered ion passing through the target point; the streamtube area at the target point is equal to zero (d ˜A/dΩ = 0).

(IV) No ion from this scattering event passes through the target point.

-20 0

0 20

20 40 60

xPosition (cm)

yPosition(cm) S

IV III

II I

-60 0 0.1

1 10

7140 8883 90 100

100 200 300

Inclination Angleθ⁺ (^◦)

Azimuth Angleφ⁺ (^◦)

ChordLengthD(cm)

-60 0

40 71 83 88 90

100 200 300

0.5 9 18

27 35.9 76 56

96 InclinationAngleθ+(◦)

Azimuth Angle φ⁺ (^◦)

Figure 4.19: Chord-length surfaces and contours for the scattering event atP_I(x) = (12,6,2). The chord length between the target point S(˜x) and the point T(x) that an ion passes through the target sphere,D(θ⁺, φ⁺), is a function of the scattering angles of the ion, (θ⁺, φ⁺). The primary-ion velocity vector angles are speciﬁed to be θu = 10^◦ and φu = 180^◦. Top figure: Top view of the thruster, indicating the relative position of the four example scattering centers P_I−IV(x) and the target point S(˜x) = (60,0,0). Middle figure: Chord-length surface as a function of the pair of scattering angles (θ⁺, φ⁺). Bottom figure: Constant value contours of the chord-length surface.

One scattering solution (whereD(θ⁺, φ⁺) = 0) exists for this scattering event.

-20 0

0 40 60

xPosition (cm)

yPosition(cm) S

IV III

II I

-60 0 0.1

1 10

7140 8883 90 100

100 200 300

Inclination Angleθ⁺ (^◦)

Azimuth Angleφ⁺ (^◦)

ChordLengthD(cm)

-60 0

40 71 83 88 90

100 200 300

6.1 20 35 52.8 72 91 110

InclinationAngleθ+(◦)

Azimuth Angleφ⁺ (^◦)

Figure 4.20: Chord-length surfaces and contours for the scattering event at P_II(x) = (5,6,2). The chord length between the target point S(˜x) and the point T(x) that an ion passes through the target sphere,D(θ⁺, φ⁺), is a function of the scattering angles of the ion, (θ⁺, φ⁺). The primary-ion velocity vector angles are speciﬁed to be θu = 10^◦ and φu = 180^◦. Top figure: Top view of the thruster, indicating the relative position of the four example scattering centers P_I−IV(x) and the target point S(˜x) = (60,0,0). Middle figure: Chord-length surface as a function of the pair of scattering angles (θ⁺, φ⁺). Bottom figure: Constant value contours of the chord-length surface.

Two scattering solutions (whereD(θ⁺, φ⁺) = 0) exist for this scattering event.

-20 0

0 20

20 40 60

xPosition (cm)

yPosition(cm) S

IV III

II I

-60 0 0.1

1 10

7140 8883 90 100

100 200 300

Inclination Angleθ⁺ (^◦)

Azimuth Angle φ⁺ (^◦)

ChordLengthD(cm)

-60 0

40 71 83 88 90

100 200 300

0.5 30 55 81.7 93 105 118 InclinationAngleθ+ (◦ )

Azimuth Angleφ⁺ (^◦)

Figure 4.21: Chord-length surfaces and contours for the scattering event atP_III(x) = (0.95,6,2).

The chord length between the target pointS(˜x) and the pointT(x) that an ion passes through the target sphere,D(θ⁺, φ⁺), is a function of the scattering angles of the ion, (θ⁺, φ⁺). The primary-ion velocity vector angles are speciﬁed to be θu = 10^◦ and φu = 180^◦. Top figure: Top view of the thruster, indicating the relative position of the four example scattering centers P_I−IV(x) and the target point S(˜x) = (60,0,0). Middle figure: Chord-length surface as a function of the pair of scattering angles (θ⁺, φ⁺). Bottom figure: Constant value contours of the chord-length surface.

One scattering solution (whereD(θ⁺, φ⁺) = 0) exists for this scattering event.

-20 0

0 40 60

xPosition (cm)

yPosition(cm) S

IV III

II I

-60 0 0.1

1 10

7140 8883 90 100

100 200 300

Inclination Angleθ⁺ (^◦)

Azimuth Angle φ⁺ (^◦)

ChordLengthD(cm)

-60 0

40 71 83 88 90

100 200 300

9 20

118

65 95 104.4 124

124

InclinationAngleθ+ (◦ )

Azimuth Angleφ⁺ (^◦)

Figure 4.22: Chord-length surfaces and contours for the scattering event at P_IV(x) = (-14,6,2).

No scattering solutions (whereD(θ⁺, φ⁺) = 0) exist for this scattering event.

Given a particular scattering event with a primary-ion velocity,u, a scattering center location, P(x), and a target point location, S(˜x), we must solve a boundary value problem in order to determine the solution angles, (θ⁺, φ⁺). The goal is to locate all scattering angle pairs for which D(θ⁺, φ⁺) is a minimum. Any pair for whichD(θ⁺, φ⁺) = 0 is a solution. Computing the surfaces, of which Figures 4.19 through 4.22 are examples, for every possible scattering event is impractical.

Instead, a Gauss-Newton with an inexact line search root-ﬁnding algorithm was developed [35]. The pseudo-code for the method is presented in the following section.

4.6.3 Pseudo-Code: Scattering Angle Partial Gauss-Newton Search Al-

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