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# Sputtering, Impact threshold energy

In this example we estimate the fraction of oxygen atoms in the atmosphere at 350 km altitude that have sufficient energy to induce sputtering on Ag and Al. Assume a Maxwellian distribution of oxygen. In order for sputtering to occur, the impact energy must exceed a threshold which is proportional to the binding energy of a surface atom to its neighbors  Assume the binding energy U of a surface atom is 48 eV for Ag and 3.4 eV for Al.

```clc;clear all;

% Given:
% Atomic Mass
M_Al = 26.98;           %[amu]
M_Ag = 107.87;          %[amu]
M_O  = 16;              %[amu]
amu  = 1.66053892e-27;  %[kg]
ev   = 1.60217657e-19;  %[J]
mu   = 398600;          %Standard gravitational parameter [km^3s-2]
kb   = 1.3806488e-23;   %[JK-1] Boltzmann constant
Alt  = 350;             %[km]
T    = 900;             %[K]
% Binding energy U of a surface atom
U_Al = 3.4;
U_Ag = 48;```

Because M_O/M_Al > 0.3 for impacting O of mass mi on a target Al atom of mass mt the energy threshold Eth to induce sputtering is

```E_th_Al = 8 *U_Al*(M_O/M_Al)^0.4;
fprintf('Energy threshold of oxygen atoms to induce sputtering on Al Eth = %4.2f eV\n',E_th_Al);```
`Energy threshold of oxygen atoms to induce sputtering on Al Eth = 22.07 eV`

Because M_O/M_Ag <= 0.3 for impacting O of mass mi on a target Ag atom of mass mt the energy threshold Eth to induce sputtering is

```gamma= 4*M_O*M_Ag/(M_O+M_Ag)^2;
E_th_Ag = U_Ag/gamma/(1 - gamma);
fprintf('Energy threshold of oxygen atoms to induce sputtering on Ag Eth = %4.2f eV\n',E_th_Ag);
% In order to sputtering to occur, the impacting atom need to have a 1D
% velocity such that
V_Al = (2*E_th_Al*ev/(M_O*amu))^0.5/1000;   %[km/s]
V_Ag = (2*E_th_Ag*ev/(M_O*amu))^0.5/1000;   %[km/s]
% Spacecraft velocity around Earth
V_sc = (mu/(Re + Alt))^0.5;
% Minimum impact velocity relative to spacecraft
V_Alr = V_Al - V_sc     %[km/s]
V_Agr = V_Ag - V_sc```
```Energy threshold of oxygen atoms to induce sputtering on Ag Eth = 193.94 eV

V_Alr =

8.6179

V_Agr =

40.6672```

Fraction of O that will induce sputtering on Al belongs to an interval [f_min, f_max]

```x = (M_O*amu/(kb*T))^0.5*V_Alr*1000;
f_min = 1/(2*pi)^0.5*exp(-x^2/2)*(1/x - 1/x^3);
f_max = 1/(2*pi)^0.5*exp(-x^2/2)*1/x;
fprintf('%4.3d < Oxygen fraction that will induce sputtering on Al < %4.3d \n',f_min, f_max)```
`1.036e-036 < Oxygen fraction that will induce sputtering on Al < 1.042e-036`

Fraction of O that will induce sputtering on Ag belongs to an interval [f_min, f_max]

```x = (M_O*amu/(kb*T))^0.5*V_Agr*1000;
f_min = 1/(2*pi)^0.5*exp(-x^2/2)*(1/x - 1/x^3);
f_max = 1/(2*pi)^0.5*exp(-x^2/2)*1/x ;
fprintf('%4.3d < Oxygen fraction that will induce sputtering on Ag < %4.3d \n',f_min, f_max)```
` 000 < Oxygen fraction that will induce sputtering on Ag <  000`