atomSurfaceInteractionsSI

• A Mathematica package for carrying out temperature dependent atom-surface interactions for Silicon

• Documentation for functions can be found in atomSurfaceInteractionsSI.m

• This package was made to coencide with a paper currently submitted.

Example usage of "atomSurfaceInteractionsSI"

A mathematica package can be called using the following commands

<< "atomSurfaceInteractionsSI`"

ParallelNeeds["atomSurfaceInteractionsSI`"]

Coefficients C₃(T), C₄(T) can be calculated in atomic units

• Input is the temperature in kelvins output is in atomic unis

Input

Column@{C30[300],c30[1200],cFour[300],cFour[1200]}

Output

0.049157474910736720926696931099180
0.048069552198726293996164215985588
15.342546111691665341578420821219
15.880743713011591292711286267971

atomSurfaceInteractionsSI supports many atoms.

• The atom used for the atom-surface interaction can be selected as an option
• If no option is selected the default is "hydrogen"

Input

c30[300, element -> "hydrogen"]

Output

0.10565340763052804213727070168717

• Below is an example which creates a table of C₃(T) and C₄(T) values for multiple atoms at room temperature

Input

els = {"hydrogen", "helium", "lithium", "beryllium", "neon", "sodium","rubidium", "krypton", "argon", "xenon"};
MatrixForm@Prepend[
  Table[{
    Capitalize@els[[i]],
    c30[300, element -> els[[i]]],
    cFour[300, element -> els[[i]]]}, {i, Length[els]}],
  {"element", "\!\(\*SubscriptBox[\(C\), \(3\)]\)(T=300K)", 
   "\!\(\*SubscriptBox[\(C\), \(4\)]\)(T=300K)"}]

Output

Element	C3(T=300K)	C4(T=300K)
Hydrogen	0.1057	50.0004
Helium	0.0492	15.3425
Lithium	1.0570	1820.61
Beryllium	0.5506	418.675
Neon	0.1084	29.5214
Sodium	1.1678	1804.94
Rubidium	6.4872	3547.80
Krypton	0.5870	182.779
Argon	0.3925	122.951
Xenon	0.9182	303.079

Output units can be changed to be given in electron volts and angstroms

• C₃(T) -> [eVų] and C₄(T) -> [eVÅ⁴]
• Input is the temperature in kelvins output is in eVÅⁿ

Input

Column@
 {c30[300, outputUnits -> "eV"],
  c30[1200, outputUnits -> "eV"],
  cFour[300, outputUnits -> "eV"],
  cFour[1200, outputUnits -> "eV"]}

Output

0.198218
0.193831
32.738
33.8864

C₃(T) and C₄(T) can easily be plotted as a function of temperature

Input

GraphicsRow[{
  ListLinePlot[
   Evaluate@ParallelTable[{T, c30[T]}, {T, 10, 1500, (1500 - 10)/35}],
   PlotTheme -> "Scientific",
   FrameLabel -> {"Temperature", TraditionalForm@Subscript[C, 3]}],
  ListLinePlot[
   Evaluate@ParallelTable[{T, cFour[T]}, {T, 10, 1500, (1500 - 10)/35}],
   PlotTheme -> "Scientific",
   FrameLabel -> {"Temperature", TraditionalForm@Subscript[C, 4]}]}]

Output

The dielectric functions derived using the Dirac - Lorentz fit, the Clausius-Mossoti fit, and the tabulated data given in M.A. Green can be quickly calculated and compared

• input is temperature in Kelvins and ω in atomic units. output is in atomic units

Input

N@{diFunSiSL[300, .07], diFunSiCM[300, .07], diFunSiGr[300, .07]}

Output

{15.081 +0.123618 i,14.6614 +0.0965488 i,14.7832 +0.113017 i}

• Plots of the dielectric function at various temperatures can be created

Input

GraphicsRow[{
  Plot[
   Evaluate@
    ParallelTable[diFunSiSL[iT, I*\[Omega]], {iT, 300, 1500, 300}],
   {\[Omega], 10^-6, 10^2},
   ScalingFunctions -> {"Log", "Linear"},
   PlotTheme -> "Scientific",
   FrameLabel -> {"i\[Omega]", "Dielectric Function"}],
  Plot[
   Evaluate@
    ParallelTable[Re@diFunSiSL[iT, \[Omega]], {iT, 300, 1500, 300}],
   {\[Omega], 0, .25},
   PlotTheme -> "Scientific",
   FrameLabel -> {"\[Omega]", "Re Dielectric Function"}],
  Plot[
   Evaluate@
    ParallelTable[Im@diFunSiSL[iT, \[Omega]], {iT, 300, 1500, 300}],
   {\[Omega], 0, .25},
   PlotTheme -> "Scientific",
   FrameLabel -> {"\[Omega]", "IM Dielectric Function"}]}]

Output

3d plots showing the dielectric function as a function of temperature and frequency can be made easily.

Input

GraphicsRow[{
  Plot3D[Re@diFunSiSL[iT, i\[Omega]], {i\[Omega], 0, .25}, {iT, 300, 
    1500},
   PlotTheme -> "Scientific",
   BoxRatios -> 1,
   AxesLabel -> {"\[Omega]", "T [K]", "Re\[Epsilon]"}],
  Plot3D[Im@diFunSiSL[iT, i\[Omega]], {i\[Omega], 0, .25}, {iT, 300, 
    1500},
   PlotTheme -> "Scientific",
   BoxRatios -> 1,
   AxesLabel -> {"\[Omega]", "T [K]", "Im\[Epsilon]"}]}]

Output

Dynamic polarizability as derived in paper. inputs is ω output is in atomic units default element is helium but element -> "rubidium" will change atom used to rubidium

Input

GraphicsRow[{
  Plot[
   dynPol[I*\[Omega]],
   {\[Omega], 10^-6, 10^3},
   ScalingFunctions -> {"Log", "Linear"},
   PlotTheme -> "Scientific",
   FrameLabel -> {"i\[Omega]", "Dynamic Polarizability"}],
  Plot[
   dynPol[I*\[Omega], element -> "rubidium"],
   {\[Omega], 10^-6, 10^3},
   ScalingFunctions -> {"Log", "Linear"},
   PlotTheme -> "Scientific",
   FrameLabel -> {"i\[Omega]", "Dynamic Polarizability"}]}]

Output

Dynamic polarizability plotted on the real axis

Input

Plot[
 dynPol[\[Omega], element -> "hydrogen"],
 {\[Omega], 0, 0.475},
 PlotRange -> {Full, {-60, 60}},
 AspectRatio -> 1,
 ScalingFunctions -> {"Linear", "Linear"},
 PlotTheme -> "Scientific",
 FrameLabel -> {"\[Omega]", "Dynamic Polarizability"}]

Output

the full Casimir-Polder potential as described in Eq.(17) of Phys. Rev. A 70, 053619

Input

datF[T_] := 
  datF[T] = 
   ParallelTable[{\[Omega], CPfull[T, \[Omega]]}, {\[Omega], 10^6, 
     10^7, (10^7 - 10^6)/75}];

ListPlot[
 {datF[300], datF[900], datF[1500]},
 Joined -> True,
 FrameLabel -> {"z/\!\(\*SubscriptBox[\(a\), \(0\)]\)", 
   "V/\!\(\*SubscriptBox[\(E\), \(h\)]\)"},
 PlotLegends -> {"300K", "900K", "1500K"},
 PlotTheme -> "Scientific"]

Output