Download Orbit theory study of electron confinement in a Polywell™ device

Analytical orbital theory analysis
of electron confinement in a
Polywell™ device
David Gummersall |PhD Student
Supervised by Dr Joe Khachan
Image: R. W. Bussard, 57th International Astronautical Congress, 2006
Aims
Aims for our analysis was to analytically derive expressions for:
1.
Electron confinement scaling laws.
2.
The average radial electron turning distances.
3.
Power loss expressions for maintaining equilibrium electron density.
These expressions were derived from an orbit theory analysis or first principles
approach. Comparisons will be made with previous orbit theory simulations
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Polywell™ a virtual cathode system
Magnetic null
Fig. 1 Magnetic field structure inside Polywell™,
highlighting the magnetic null.
Fig. 2 A three dimensional schematic layout
of Buzzard’s Polywell™ design.
• Central minimum magnetic field has certain plasma M.H.D. Stability properties.
• Virtual cathode may form due to space charge trapping of electrons.
• Ions are then electrostatically confined by the electron’s electric field.
Fig 1. Image: N. Krall, Physics of Plasmas, 2, 1995. Reformatted in colour by Mark Duncan, 2007
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Grid less IEC
Magnetic null
Fig. 1 Magnetic field structure inside Polywell™,
highlighting the magnetic null.
Fig. 3 Spindle cusp magnetic field line projection
along with guiding center particle orbits.
• Spindle cusp generates a circular line cusp.
• Polywell effectively ‘plugs’ this line cusp.
• 2 point and 1 line cusp VS 14 point cusps.
Fig 3. J. Egedal and A. Fasoli, Physics of Plasmas, 8, 2001.
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Trapped electron motion
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Orbit theory simulation and parameter set
Five main parameters of interest:
• Low Beta regime.
• Non-interacting condition.
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Magnetic cusp reflection
RT
Coils
r0
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Derivation of r0 and B0
Adiabatically invariant only if:
where
Two equations:
~
(1)
(2)
,
Eqn. 2: M. Carr, D. Gummersall, S. Cornish and J. Khachan, Physics of Plasmas, 18, 2011
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Confinement time model
Assume magnetic cusp acts like a 1D magnetic mirror.
System was adiabatically invariant a short distance from the central magnetic null.
where
Bm is the maximum strength of the magnetic field and B0 is the strength of the
magnetic field in the weakest section of the adiabatically invariant path.
Using
and
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Confinement time analysis
0
Fig. 4 A sample of fitted
curves to a 1 m radius
Polywell™ with 100 eV
electrons where the line data
were the best fit curves to
corresponding data.
Time constant/scaling law:
0
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4th order Magnetic field fit
Magnetic field for
R=I=1
where
Can invert B(ra) to get a close approximation of ra(B).
Use this to find a more accurate RT as BT is known via:
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Comparisons with simulation data
RT/R
RT/R
From simulation data we get the time-weighted average electron radial distance.
This can be interpreted as roughly RT as radial velocity is zero at turning point.
Used only the electrons which stayed
within the Polywell for 10xT.O.F. in
absence of magnetic fields.
For a given current and electron energy
can determine the average radius of
electron. Hence can predict if electrons
will be confined.
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Polywell power loss
Power loss defined by:
After substitutions:
Total power loss:
Power loss density:
Where Nmax and ρmax are the maximum electron total number and
density, respectively.
For a 1 litre volume
R = 4 cm
ρmax = 1016 m-3
K = 10 keV
I = 106 A.Turns
Pl ~ 200 kW
• No space charge effects
• No scattering
• No application of ChildLangmuir Law
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Current and future work
Current simulation is a 1D2V Gyrokinetic, leapfrog, PIC modeling of
point cusps.
Simulation improvements:
• Expand model to 2D3V dimensions.
• Include full 2D electron motion.
• Include electron-electron scattering.
• Explore electrostatic ‘plugging’ of cusps (Doland, 1994).
• Include ion interactions with electron plasma.
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