RUTHERFORD SCATTERING 
			K.5.2.4:
			Franz Michel,  Bernhard Schnizer, 
										Institut fuer Theoretische Physik
								Technische Universitaet Graz
								    A-8010  Graz, Austria
								Projekt Multimediale Lehre des
							Zentralen Informatikdienstes der TU Graz
												2002-04-22

1.   	Introduction

1.1	Fundamental Equations

Needs["Graphics`Arrow`"]
Needs["Graphics`Graphics`"]
Clear["@"]

The Laplace-Lenz vector   (Overscript[A,  →])    is  (the abscissa represents z, the ordinate  ρ ):

va  = {-1,  2 δ ϵ};

The numeric excentricity is:

ba  =  Sqrt[va . va]

The unit vector parallel to the Laplace-Lenz vector is:

ea  =  va/ba

ea . ea // Together

A unit vector in the trajectory plane  perpendicular to the Laplace-Lenz vector is:

el  =  {-2 δ ϵ, -1}/ba

el . el // Together

The trajectory in polar coordinates  ρ  and ϕ  is:

ρ  =   - 
          2 ϵ δ^2 /(1  -  
              ba  Cos[ϕ])

The range of the variable  ϕ  is determined by the zeros of the denominator:

s0  = Solve[Denominator[%]  ==  0,  ϕ]

A parametric vectorial representation of the trajectory is:

vr  =   ρ (ea Cos[ϕ]  -  el Sin[ϕ]) // 
        Simplify

The initial position determines the intial value ϕ1  of the variable ϕ :

vr[[1]] == - x1//Simplify ;

s1  =  Solve[%, ϕ]

These arrows give the Laplace-Lenz vector   (Overscript[A,  →])   and the unit vector  (Subscript[((Overscript[e,  →]) ), 1])  perpendicular to it:

aa = Graphics[{Arrow[{0, 0}, va], Arrow[{0, 0}, el]}];

2.	Scattering at Repulsive Potential, C > 0  ⇒  ϵ > 0.

2.1	Single Trajectory

lp0  =  ϕ  /.  s0 [[-1]] /.  de

lp1  =  ϕ  /.  s1 [[-1]] /.  de

vn  =  vr  /.  de;

an  =  aa  /.  de;

ParametricPlot[
        Evaluate[vn], {ϕ,  -lp1, 
          lp1},  
AspectRatio  ->  Automatic,  
        DisplayFunction  ->  Identity];

\(\(Show[%, an, \ PlotRange\  -> \ All, 
        AxesLabel\  -> \ {"z", "ρ"}, 
DisplayFunction\  -> \ $DisplayFunction\ , 
Epilog\  -> \ {Text[\*""(Overscript[A, →])"", 1.1 {\(-1\), 1}], 
            Text[\*""(Subscript[(Overscript[e, →]), l])"", {\(-1\), \(- .8\)}], \ PointSize[ .035], Hue[0], Point[{0, 0}]}];\)\)

2.2	Several Trajectories, Same Energy, Various Impact Parameters

th  =  Thickness[ .005];  ps  =  PointSize[ .02];

lpl2  =  ϕ  /.  s1 [[-1]] /.  
          del2;

vn2  =  vr  /.  del2;

pl  =  
        Table[ParametricPlot[
            Evaluate[vn2[[k]]], {ϕ, -lpl2[[k]] , 
              lpl2[[k]]},  
            PlotRange  ->  
              All, 
AspectRatio  ->  Automatic,  
            PlotStyle  ->  {Hue[Abs[k]  .1], th},  
            DisplayFunction  ->  Identity], {k, 
            Length[vn2]}];

\(\(pl2\  = \ 
        Show[pl, DisplayFunction\  -> $DisplayFunction, \ 
          ImageSize\  -> \ 600, 
          AxesLabel\  -> \ {"z", "ρ"}, 
PlotLabel\  -> \ 
            SequenceForm["ϵ = ", ϵ\  /. \ 
                del2[\([1, 2]\)], "
"], 
\ 
          Epilog\  -> \ {ps, Point[{0, 0}], th, \ Hue[0], 
              Line[{{\(-x1\), 0.0}, {\(-1\)/ϵ, 0.0}}] /. 
                del2[\([1]\)]}];\)\)

2.3	Several Trajectories, Same Energy (twice of above one), Various Impact Parameters

th  =  Thickness[ .005];  ps  =  PointSize[ .02];

lpl3  =  ϕ  /.  s1 [[-1]] /.  
          del3;

vn3  =  vr  /.  del3;

pl  =  
        Table[ParametricPlot[
            Evaluate[vn3[[k]]], {ϕ, -lpl3[[k]] , 
              lpl3[[k]]},  
            PlotRange  ->  
              All, 
AspectRatio  ->  Automatic,  
            PlotStyle  ->  {Hue[Abs[k]  .1], th},  
            DisplayFunction  ->  Identity], {k, 
            Length[vn3]}];

\(\(pl3\  = \ 
        Show[pl, DisplayFunction\  -> $DisplayFunction, \ 
          ImageSize\  -> \ 600, 
          AxesLabel\  -> \ {"z", "ρ"}, 
PlotLabel\  -> \ 
            SequenceForm["ϵ = ", ϵ\  /. \ 
                del3[\([1, 2]\)], "
"], 
\ 
          Epilog\  -> \ {PointSize[ .014], Point[{0, 0}], th, \ Hue[0], 
              Line[{{\(-x1\), 0.0}, {\(-1\)/ϵ, 0.0}}] /. 
                del3[\([1]\)]}];\)\)

2.4	Preceeding Two Examples Together

Show[GraphicsArray[{{pl2}, {pl3}}],  
        ImageSize  ->  350];

2.5	Scattering Angle Versus Impact Paramter .
	Differential Cross Section

θ  =  
      2  ArcCot[ϵ δ]
dsig = (1/(2 ϵ))^2 1/
          Sin[ϑ/2]^4

2.5.1	ϵ = 1

\(\(Plot[180  tn/π, {δ, 0, 12}, \ 
        Ticks\  -> \ \ {Automatic, 
            Range[0, 180, 
              30]}, 
AxesLabel\  -> \ {"δ", "θ"}];\)\)
LogPlot[ts, {ϑ, 0, π},  
        PlotRange -> {{0, π}, {0.1, 10000}}, 
        Ticks  ->   {Range[0, π, π/4], 
            Automatic}, 
AxesLabel  ->  {"\
ϑ", "dσ"}];

2.5.2	Several Relative Energies

tnl  =  (Thread[  θ  /. ϵ  ->  
              Range[1, 8] ]);  
    cl  =  Table[
        Hue[(k  -  1)  .1], {k, 8}];
xt  =  1.05;
tel  =  {Text["ϵ", {xt, 105}], 
          Text["1", {xt, 90}], Text["2", {xt, 53}], 
          Text["3", {xt, 37}], 
Text["5", {xt, 
              22}], Text["8", {xt, 12}]};
\(\(Plot[Evaluate[180  tnl/π], {δ, 0, 1}, \ 
        Ticks\  -> \ \ {Automatic, 
            Range[0, 180, 
              30]}, 
AxesLabel\  -> \ {"δ", "ϑ"}, PlotStyle\  -> cl, 
        PlotRange\  -> \ {{0, 1.1}, {0, 
              180}}, \ 
AspectRatio\  -> 1.6, \ 
        Epilog\  -> \ tel, 
        PlotLabel\  -> \ "Rutherford scattering: Scattering angle
", ImageSize -> 300];\)\)
tsl = (Thread[  dsig /. ϵ  ->  
                Range[1, 8] ]);
xs = π +  .2;
\(\(tpl\  = \ {Text["ϵ=1", {xs, \(-0.5\)}], 
          Text["ϵ=2", {xs, \(-1.1\)}], 
          Text["ϵ=3", {xs, \(-1.45\)}], 
Text["ϵ=4", {xs, \(-1.7\)}], 
          Text["ϵ=5", {xs, \(-1.94\)}], 
          Text["ϵ=6", {xs, \(-2.1\)}], 
          Text["ϵ=7", {xs, \(-2.28\)}], 
          Text["ϵ=8", {π/
                2.2, \(-2\)}]};\)\)
\(\(LogPlot[Evaluate[tsl], {ϑ, 0, π}, \ 
        PlotRange -> {{0, π +  .5}, {0.004, 
              11000}}, 
Ticks\  -> \ \ 
{{{0, "0"}, {π/6, "30"}, {π/
                  3, "60"}, {π/
                  2, "90"}, {4  π/6, "120"}, {5  π/
                    6, "150"}, {π, "180"}}, Automatic}, 
        PlotLabel\  -> \ "Rutherford scattering: Differential cross section


", 
AxesLabel\  -> \ {"ϑ", \*""(1/(R^2))d(σ/d)Ω""}, AspectRatio\  -> 1.6, PlotStyle\  -> cl, \ 
        Epilog\  -> \ tpl, ImageSize -> 300];\)\)

2.6	Points on Screen for Various Impact Parameters

2.6.1	ϵ = 1

tn  =  θ  /. ϵ  ->  ev ;

bk = Graphics[{Thickness[ .009], Circle[{0, 0}, a]}];
vs =  .89;   a  =  1;
tp = 
        Prepend[{Point[{0, 0}], Line[{a {Cos[vs], Sin[vs]}, {0, 0}, {a, 0}}], 
            Circle[{0, 0},  .2 a, {0, vs}], 
            Text["θ",  .3 a {Cos[vs/2], Sin[vs/2]}]}, 
          Table[{Hue[δ/5.5 // Abs], 
              Point[a {Cos[tn], Sin[tn]}]}, {δ, -5, 
              5,  .2}]];
\(\(sc1\  = 
        Show[bk, AspectRatio -> Automatic, Axes -> False, 
          Epilog -> tp, 
          Prolog -> 
            PointSize[ .025], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotLabel\  -> \ 
            SequenceForm["ϵ = ", \ ev]];\)\)

2.6.2	ϵ = 2

tn  =  θ  /. ϵ  ->  ev ;

bk = Graphics[{Thickness[ .009], Circle[{0, 0}, a]}];
vs =  .89;   a  =  1;
tp = 
        Prepend[{Point[{0, 0}], Line[{a {Cos[vs], Sin[vs]}, {0, 0}, {a, 0}}], 
            Circle[{0, 0},  .2 a, {0, vs}], 
            Text["θ",  .3 a {Cos[vs/2], Sin[vs/2]}]}, 
          Table[{Hue[δ/5.5 // Abs], 
              Point[a {Cos[tn], Sin[tn]}]}, {δ, -5, 
              5,  .2}]];
\(\(sc2\  = \ 
        Show[bk, AspectRatio -> Automatic, Axes -> False, 
          Epilog -> tp, 
          Prolog -> 
            PointSize[ .025], 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotLabel\  -> \ 
            SequenceForm["ϵ = ", \ ev]];\)\)

2.6.3	ϵ = 0.5

tn  =  θ  /. ϵ  ->  ev ;

bk = Graphics[{Thickness[ .009], Circle[{0, 0}, a]}];
vs =  .89;   a  =  1;
tp = 
        Prepend[{Point[{0, 0}], Line[{a {Cos[vs], Sin[vs]}, {0, 0}, {a, 0}}], 
            Circle[{0, 0},  .2 a, {0, vs}], 
            Text["θ",  .3 a {Cos[vs/2], Sin[vs/2]}]}, 
          Table[{Hue[δ/5.5 // Abs], 
              Point[a {Cos[tn], Sin[tn]}]}, {δ, -5, 
              5,  .2}]];
\(\(sch\  = \ 
        Show[bk, AspectRatio -> Automatic, Axes -> False, 
          Epilog -> tp, 
          Prolog -> 
            PointSize[ .025], \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotLabel\  -> \ 
            SequenceForm["ϵ = ", \ ev]];\)\)

2.6.4	All screens together

Show[GraphicsArray[{sch, sc1, sc2}], 
        ImageSize  ->  700];

3.	Scattering at Attractive Potential, C < 0 ;  ϵ >  0.

3.1	Single Trajectory

lp0  =  ϕ  /.  s0[[2]]  /.  de

lp1  =  ϕ  /.  s1[[-1]] /.  de // Chop

2  π  -  lp0

2  π  -  lp1

vn  =  vr  /.  de;

an  =  aa  /.  de;

ParametricPlot[
        Evaluate[vn], {ϕ,  lp1, 
          2  π  -  lp1},  
AspectRatio  ->  
          Automatic, 
 
        DisplayFunction ->  Identity];

\(\(Show[%, an, \ PlotRange\  -> \ All, \ 
        DisplayFunction -> \ $DisplayFunction, 
Epilog\  -> \ {Text[\*""(Overscript[A, →])"", \(-\ 1.1\) {1, 1}], 
            Text[\*""(Subscript[(Overscript[e, →]), l])"", {1, \(- .8\)}], 
\ PointSize[ .025], Hue[0], Point[{0, 0}]}];\)\)

3.2	Several Trajectories, Same Energy, Various Impact Parameters

th  =  Thickness[ .005];  ps  =  PointSize[ .02];

lpl2  =  ϕ  /.  s1 [[-1]] /.  
          del2;

vn2  =  vr  /.  del2;

pl  =  
        Table[ParametricPlot[
            Evaluate[vn2[[k]]], {ϕ, lpl2[[k]] , 
              2  π  - lpl2[[k]]},  
            PlotRange  ->  
              All, 
AspectRatio  ->  Automatic,  
            PlotStyle  ->  {Hue[Abs[k]  .1], th} , 
            DisplayFunction  ->  Identity ], {k, 
            Length[vn2]}];

\(\(pln2\  = \ 
        Show[pl, DisplayFunction\  -> $DisplayFunction, \ 
          ImageSize\  -> \ 600, 
          AxesLabel\  -> \ {"z", "ρ"}, 
PlotLabel\  -> \ 
            SequenceForm["ϵ = ", ϵ\  /. \ 
                del2[\([1, 2]\)], "

"], 
\ 
          Epilog\  -> \ {ps, 
              Point[{0, 0}] /. del2[\([1]\)]}];\)\)

3.3	Several Trajectories, Same Energy (twice of above one), Various Impact Parameters

th  =  Thickness[ .005];  ps  =  PointSize[ .02];

lpl3  =  ϕ  /.  s1 [[-1]] /.  
          del3;

vn3  =  vr  /.  del3;

pl  =  
        Table[ParametricPlot[
            Evaluate[vn3[[k]]], {ϕ, lpl3[[k]] , 
              2  π  - lpl3[[k]]},  
            PlotRange  ->  
              All, 
AspectRatio  ->  Automatic,  
            PlotStyle  ->  {Hue[Abs[k]  .1], th} , 
            DisplayFunction  ->  Identity ], {k, 
            Length[vn2]}];

\(\(pln3\  = \ 
        Show[pl, DisplayFunction\  -> $DisplayFunction, \ 
          ImageSize\  -> \ 600, 
          AxesLabel\  -> \ {"z", "ρ"}, 
PlotLabel\  -> \ 
            SequenceForm["ϵ = ", ϵ\  /. \ 
                del3[\([1, 2]\)], "

"], 
\ 
          Epilog\  -> \ {ps, 
              Point[{0, 0}] /. del2[\([1]\)]}];\)\)

3.4	 Preceeding Two Examples Together

Show[GraphicsArray[{{pln2}, {pln3}}],  
        ImageSize  ->  550];

del4  =  
        Table[{x1  ->  
              4, ϵ  ->  -1,  δ  ->   kk}, {kk, -1.9, 1.9,  .2}];

lpl4  =  ϕ  /.  s1 [[-1]] /.  
          del4;

vn4  =  vr  /.  del4;

pl  =  
        Table[ParametricPlot[
            Evaluate[vn4[[k]]], {ϕ, lpl4[[k]] , 
              2  π  - lpl4[[k]]},  
            PlotRange  ->  
              All, 
AspectRatio  ->  Automatic,  
            PlotStyle  ->  {Hue[Abs[k]  .1], th} , 
            DisplayFunction  ->  Identity ], {k, 
            Length[vn4]}];

\(\(pl2\  = \ 
        Show[pl, DisplayFunction\  -> $DisplayFunction, \ 
          ImageSize\  -> \ 600, 
          AxesLabel\  -> \ {"z", "ρ"}, 
PlotLabel\  -> \ 
            SequenceForm["ϵ = ", ϵ\  /. \ 
                del4[\([1, 2]\)], "

"], 
\ 
          Epilog\  -> \ {ps, 
              Point[{0, 0}] /. del4[\([1]\)]}];\)\)

3.5	Several Trajectories, Same Energy, Various Impact Parameters >=< 0

lpl4  =  ϕ  /.  s1 [[-1]] /.  
          del4;

vn4  =  vr  /.  del4;

pl  =  
        Table[ParametricPlot[
            Evaluate[vn4[[k]]], {ϕ, lpl4[[k]] , 
              2  π  - lpl4[[k]]},  
            PlotRange  ->  
              All, 
AspectRatio  ->  Automatic,  
            PlotStyle  ->  {Hue[Abs[k]  .1], th} , 
            DisplayFunction  ->  Identity ], {k, 
            Length[vn4]}];

\(\(pln4\  = \ 
        Show[pl, DisplayFunction\  -> $DisplayFunction, \ 
          ImageSize\  -> \ 600, 
          AxesLabel\  -> \ {"z", "ρ"}, 
PlotLabel\  -> \ 
            SequenceForm["ϵ = ", ϵ\  /. \ 
                del4[\([1, 2]\)], "

"], 
\ 
          Epilog\  -> \ {ps, 
              Point[{0, 0}] /. del4[\([1]\)]}];\)\)

3.6	Scattering Angle Versus Impact Paramter. 
	Differential Cross Section

3.7	Points on Screen for Various Impact Parameters

The analyses of the above to subsections need not be repeated, as the correponding formulas are the same
as in 2.5, 2.6 respectively.