Collision Model HS1

From SIMION

This SIMION user program implements a rather complete hard-sphere collision model. Collision models are useful for simulating non-vacuum conditions, in which case ions collide against a background gas and are deflected randomly.

Features and assumptions of the model:

Ion collisions follow the hard-sphere collision model. Energy transfers occur solely via these collisions.
Ion collisions are elastic.
Background gas is assumed neutral in charge.
Background gas velocity follows the Maxwell-Boltzmann distribution.
Background gas mean velocity may be non-zero.
Kinetically cooling and heating collisions are simulated.
Background gas as a whole is unaffected by ion collisions.

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Download

Download code: REV-4 (2007-02)

SL version: collision_hs1.sl
PRG version: collision_hs1.prg (compiled from collision_hs1.sl with the SL Compiler)
SIMION 8 includes a Lua version (collision_hs1 example). It is somewhat more updated and documented than the SL version above.

Note: REV-4 made this correction: Issue I362 (http://www.simion.com/issue/362) - HS1 collision model does not accurately thermalize.

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Evaluation and Comparison

The figure below shows ion trajectories in the HS1 collision model (dots mark collision events). Conditions: ions of mass 200 amu, 15 angstrom collision diameter, and initial velocity to the right at 24 eV colliding against a He background gas with 2 angstrom collision diameter, 4 mTorr pressure, and 275 K temperature. Collisions tend to kinetically cool ions initially. As ions slow down, mean-free-path decreases and the scattering effect increases.

The below figures area plots of ion speed per collision number using the HS1 collision model as well as the dahl_drag.prg and _Trap/INJECT.PRG in SIMION 7.0 models for comparison. The [Ling1997 (http://dx.doi.org/10.1002/(SICI)1097-0231(19970830)11:13%3c1467::AID-RCM54%3e3.0.CO;2-X)] paper (Figure 4) provides a similar graph for its collision model under the same conditions. The HS1 and dahl_drag.prg models are in fairly good agreement in this aspect. However, these differ from the Ling1997 and _Trap/INJECT.PRG graphs, which are similar to each other and show almost twice as rapid dampening. (Note: full details of Ling1997 are not available.) The reduced dampening in the former models seems partly due to the inclusion of heating collisions from behind the ions.

Figure: Dampening using Collision Model HS1

Figure: Dampening using Collision Model dahl_drag.prg

Figure: Dampening using Collision Model _Trap/INJECT.PRG in SIMION 7.0 (also resembles Ling1997)

Despite these similarities in the above regard, the models can still be quite different. For example, HS1 model supports a variable mean-free-path (unlike dahl_drag.prg), and this affects the frequency of collisions, especially as speeds change. Models can also handle angular scattering differently (e.g. _Trap/INJECT.PRG does not provide any angular scattering).

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Derivation of Mean Relative Speed

This program calculates mean relative speed between the ion and background gas in order to calculate mean-free-path. The following is a derivation of the equation for mean relative velocity.

Compute average relative speed c of a single particle (ion) against a background gas (gas). The background gas is assumed to be Maxwell distributed in velocity.

We start with

$c = \iiint_V |\mathbf{v_{ion}} - \mathbf{v_{gas}}| f(\mathbf{v_{gas}}) \, d\mathbf{v_{gas}}$

where f is the three-dimensional Maxwell distribution given by

$f(\mathbf{v}) = (A/\pi)^{3/2} \exp(-A\mathbf{v}^2)$

such that

A = (m g a s / 2 k T)

Substituting,

$c = \iiint_V |\mathbf{v_{ion}} - \mathbf{v_{gas}}| (A/\pi)^{3/2} \exp(-A\mathbf{v_{gas}}^2) \, d\mathbf{v_{gas}}$

Let $\mathbf{u} = \mathbf{v_{gas}} - \mathbf{v_{ion}}, \, d\mathbf{u} = d\mathbf{v}_{gas}$ .

$c = (A / \pi)^{3/2} \iiint_V |\mathbf{u}| \exp(-A\mathbf{v_{ion}}^2 - 2A\mathbf{v_{ion}}\mathbf{u} - A\mathbf{u}^2) \, d\mathbf{u}$

$= (A/\pi)^{3/2} \exp(-A\mathbf{v_{ion}}^2) \iiint_V |\mathbf{u}| \exp(-A\mathbf{u}^2 - 2A\mathbf{v_{ion}}\mathbf{u}) \, d\mathbf{u}$

Convert to spherical coordinates and let $r = |\mathbf{u}|, a = |\mathbf{v_{ion}}|, \mathbf{v_{ion}} \cdot \mathbf{u} = |\mathbf{v_{ion}}||\mathbf{u}|\cos \theta$ .

$c = (A/\pi)^{3/2} \exp(-Aa^2) \,$

$\int_0^{\infty} \int_0^{2\pi} \int_0^{\pi} r \exp(-Ar^2 - 2Aar \, \cos \theta) r^{2} \sin \theta \, d\theta \, d\phi \, dr$

$= (A/\pi)^{3/2} \exp(-Aa^2) \,$

$\int_0^{\infty} r^{2} \int_0^{2\pi} \, d\phi \int_0^{\pi} r \exp(-Ar^2 - 2Aar \, \cos \theta) \sin \theta \, d\theta \, dr$

$= (A/\pi)^{3/2} (2\pi) \exp(-Aa^2) \,$

$\int_0^{\infty} r^{2} \int_0^{\pi} r \exp(-Ar^{2} - 2Aar \, \cos \theta) \sin \theta \, d\theta \, dr$

Let $s = -Ar^2 - 2Aar \, \cos\theta, ds = 2Aar \, \sin \theta \, d\theta$ .

$c = A^{1/2} a^{-1} \pi^{-1/2} \exp(-Aa^2) \int_0^{\infty} r^{2} \int_{-Ar^2 - 2Aar}^{-Ar^2 + 2Aar} \exp(s) \, ds \, dr$

$= A^{1/2} a^{-1} \pi^{-1/2} \exp(-Aa^2) \int_0^{\infty} r^{2} (\exp(-Ar^2 + 2Aar) - \exp(-Ar^2 - 2Aar)) \, dr$

To solve this integral, we use

$\int_0^{\infty} r^{2} \exp(-Ar^2 + Br) \, dr =$

$\frac{B^2+2A}{8A^3} \sqrt{\pi A} \exp(B^2/4A) (1 + \operatorname{erf}(B/2\sqrt{A})) + \frac{B}{4A^2}$

$\int_0^{\infty} r^{2} (\exp(-Ar^2 + Br) - \exp(-Ar^2 - Br)) \, dr =$

$\frac{B}{2A^2} - \frac{B^2+2A}{8A^3} \sqrt{\pi A} exp(B^2/4A) \operatorname{erf}(B/2\sqrt{A})$

$\int_0^{\infty} r^{2} (\exp(-Ar^2 + 2Aar) - \exp(-Ar^2 - 2Aar)) \, dr =$

$\frac{2Aa^2+1}{2A^2} \sqrt{\pi A} \exp(Aa^2) \operatorname{erf}(\sqrt{A}a) + a/A$

Substituting,

$c = A^{1/2} \pi^{-1/2} a^{-1} \exp(-Aa^2) \cdot$

$(\frac{2Aa^2 + 1}{2A^2}\pi^{1/2} \exp(Aa^2)A^{1/2}\operatorname{erf}(A^{1/2}a) + (a/A))$

$(2A^{-1/2} \pi^{-1/2}) \cdot \left[ \frac{\pi^{1/2}}{2} (A^{1/2}a + \frac{1}{2} A^{-1/2}a^{-1}) \operatorname{erf}(A^{1/2}a) + \frac{1}{2}\exp(-Aa^2) \right]$

Let

$\bar{c}_{gas} = \sqrt{\frac{8kT}{\pi m}} = \frac{2}{\sqrt{A\pi}}$ (mean gas speed)

$c^{*}_{gas} = \sqrt{\frac{2kT}{m}} = \frac{1}{\sqrt{A}}$ (median gas speed)

$s = c_{ion}/c^{*}_{gas} = a\sqrt{A}$

Substituting gives the result:

$c = \bar{c}_{gas} \left(\left(s + (2s)^{-1}\right)\frac{\sqrt{\pi}}{2}\operatorname{erf}(s) + \frac{1}{2}\exp(-s^2)\right)$

This result is in agreement with Ding2002 [1] (http://dx.doi.org/10.1016/S1387-3806(02)00921-1).

Note the following resuts:

As $c_{ion} \rightarrow 0, c \rightarrow \bar{c}_{gas}$ .

Also, as $c_{ion} \rightarrow \infty, c \sim \frac{\sqrt{\pi}}{2} \bar{c}_{gas} s = \frac{\sqrt{\pi}}{2} \frac{\bar{c}_{gas}}{c^{*}_{gas}} c_{ion} = c_{ion}$ .

Further, if $c_{ion} = \bar{c}_{gas}$ , then $c \approx 1.379 \bar{c}_{gas}$ , which is approximately the average relative speed between the gas particles themselves ( $\sqrt{2}\bar{c}_{gas}$ ).

The above results provide a rough justification for the approximation $c \approx \sqrt{c_{ion}^2 + \bar{c}_{gas}^2}.$ .

--david manura

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Info: Collision Model HS1

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Evaluation and Comparison

Derivation of Mean Relative Speed