Closed form solution: non-resonant ion cyclotron resonance heating​
​by Robert Mendelsohn

Particle Driven at Cyclotron Resonance Frequency

For a particle driven by a circularly polarized electric field whose frequency matches the driven particle’s cyclotron resonance frequency, we have, in polar coordinates:​θ =
q
B
m
d
t
​​r =
E
B
t
​​or, alternatively in Cartesian coordinates (note: subscript d denotes the resonant species):​x =
E
B
t
cos(
q
B
m
d
t
)y =
E
B
t
sin(
q
B
m
d
t
)
We can plot this function:
In[]:=
ParametricPlot3D[{t*Sin[t],t*Cos[t],t},{t,0,40}]
Out[]=

Closed-Form Expression for Non-Resonant Species, Circularly Polarized

E
-Field

But what about the non-resonant species? To my knowledge no closed-form expression has been published before, and quite interestingly it can be found by using the DSolve function, as will be shown below.

Equation of Motion: Derivation

For a particle driven by a circularly polarized electric field whose frequency does NOT match the driving frequency, we have, in polar coordinates:(note that subscript p is for non-resonant species)​θ =
q
B
m
p
t
​​r =
1
B
∫
E
t
Translate these to Cartesian coordinates and use them as arguments to DSolve:
In[]:=
DSolve[{x'[0]0,y'[0]0,x''[t]n*e*Sin[k*t]-n*y'[t]*b,y''[t]n*e*Sin[k*t]+n*x'[t]*b,x[0]0,y[0]0},{x[t],y[t]},t]
Out[]=
x[t]
1
bk(-k+bn)(k+bn)
(e
2
k
-
2
b
e
2
n
-e
2
k
Cos[kt]+
2
b
e
2
n
Cos[kt]-e
2
k
Cos[bnt]+e
2
k
Cos[kt]
2
Cos[bnt]
+bekn
2
Cos[bnt]
Sin[kt]-e
2
k
Sin[bnt]+e
2
k
Cos[kt]
2
Sin[bnt]
+beknSin[kt]
2
Sin[bnt]
),y[t]
1
bk(-k+bn)(k+bn)
-e
2
k
+
2
b
e
2
n
+e
2
k
2
Cos
kt
2

-
2
b
e
2
n
2
Cos
kt
2

+e
2
k
Cos[bnt]-e
2
k
Cos[kt]
2
Cos[bnt]
-e
2
k
2
Sin
kt
2

+
2
b
e
2
n
2
Sin
kt
2

+bekn
2
Cos[bnt]
Sin[kt]-e
2
k
Sin[bnt]-e
2
k
Cos[kt]
2
Sin[bnt]
+beknSin[kt]
2
Sin[bnt]

And that is the closed-form solution for x and y position as a function of time, for a single particle. Real plasmas experiencing ion cyclotron resonance heating have other physical interactions to account for, including Landau damping, Doppler broadening, neutral collisions, etc.
In[]:=
fnonresonant=SimplifySimplifySqrtDD
1
bk(-k+bn)(k+bn)
(e
2
k
-
2
b
e
2
n
-e
2
k
Cos[kt]+
2
b
e
2
n
Cos[kt]-e
2
k
Cos[bnt]+e
2
k
Cos[kt]
2
Cos[bnt]
+bekn
2
Cos[bnt]
Sin[kt]-e
2
k
Sin[bnt]+e
2
k
Cos[kt]
2
Sin[bnt]
+beknSin[kt]
2
Sin[bnt]
),t,t^2+DD
1
bk(-k+bn)(k+bn)
-e
2
k
+
2
b
e
2
n
+e
2
k
2
Cos
kt
2

-
2
b
e
2
n
2
Cos
kt
2

+e
2
k
Cos[bnt]-e
2
k
Cos[kt]
2
Cos[bnt]
-e
2
k
2
Sin
kt
2

+
2
b
e
2
n
2
Sin
kt
2

+bekn
2
Cos[bnt]
Sin[kt]-e
2
k
Sin[bnt]-e
2
k
Cos[kt]
2
Sin[bnt]
+beknSin[kt]
2
Sin[bnt]
,t,t^2,Assumptions{k>0,n>0,b>0,t≥0}/.{n(q/mn),k->2*Pi*((q*b)/(2*Pi*mr))},Assumptions{k>0,n>0,b>0,t≥0}
Out[]=
2
q
-
2
e
2
q
-
2
mn
-3
2
mr
+2mnmrCosb
1
mn
-
1
mr
qt+4
2
mr
Cos
bqt
mn
Cos
bqt
mr
+
2
mn
Cos
2bqt
mr
-
2
mr
Cos
2bqt
mr
-2mnmrCos
b(mn+mr)qt
mnmr

2
mn
2
mr
mnmrAbs-
1
2
mn
+
1
2
mr

2
q

Let’s plot the motion of a resonant and non-resonant ion (no mutual interactions considered):

Equation of Motion: Force on Non-Resonant Particle


Equation of Motion: Equations and Plot

Equations


Plots Functions


Plots

In[]:=
SI[x_]:=x//UnitConvert//QuantityMagnitude//N​​HydrogenNucleusMass=Quantity[1.00727646,"Daltons"];​​DeuteriumNucleusMass=Quantity[2.01594137,"Daltons"];​​ElementaryCharge=Quantity[1.602`*^-19,"Coulombs"];
In[]:=
CircularlyPolarizedPlotTwoMasses[1,92,235,238,1,16.04,0,1250]
Out[]=
In[]:=
CircularlyPolarizedPlotTwoMasses[1,92,235,238,1,16.04,0,3*1250]
Out[]=
The resonant particle’s radius increases linearly whereas the non-resonant particle experiences a beat-like phenomenon, as expected.