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By Charles Rhodes, P.Eng., Ph.D.

Conceptually a plasma spheromak is a closed plasma sheet formed by a closed spiral of plasma hose in the plane of the plasma sheet. The direction of the plasma hose axis within the plasma sheet conforms to the toroid surface curvature.

The plasma hose current causes both toroidal and poloidal magnetic field components.

A spheromak is cylindrically symmetric about the spheromak main axis and is mirror symmetric about the spheromak's equatorial plane. The plasma sheet has a net charge Qs that is uniformly distributed over the plasma hose length Lh. This distributed charge causes a spherically radial electric field outside the plasma sheet and a cylindrically radial electric field inside the plasma sheet. The center to center plasma hose spacing Dh is smaller in the spheromak core than at the spheromak periphery. Hence the charge per unit area on the plasma sheet is higher in the spheromak core than at the spheromak periphery.

In the central core of the spheromak the electric field is zero and the purely poloidal magnetic field points along the spheromak's main axis of symmetry. In the region enclosed by the plasma sheet the magnetic field is purely toroidal and the electric field is cylindrically radial. In the region not enclosed by the plasma sheet the magnetic field is purely poloidal and the electric field is spherically radial. The charge circulates in the narrow low magnetic field region at the interface surface between the toroidal and poloidal magnetic fields.

A spheromak can be mathematically modelled as a closed single layer plasma hose spiral in the shape of a toroid. This plasma sheet is formed from a long length Lh of plasma hose. The plasma hose axial direction gradually changes over the surface of the toroid. The behavior of plasma hose is discussed on the web page titled: PLASMA SHEET PROPERTIES.

Photographs of experimental spheromaks within cylindrical metal enclosures show that the experimental spheromak cross section is not round. The following diagram shows the approximate cross sectional shape of an experimentally observed spheromak.

A location in a spheromak is referred to in cylindrical co-ordinates (R, H) where R is the radius from the axis of symmetry and H is the height above the spheromak equatorial plane. On the equatorial plane:
H = 0.
Thus R takes only positive values whereas H takes both positive and negative values.

Rc = spheromak core radius on the equatorial plane
Rs = spheromak outside radius on the equatorial plane
Rf = the radius from main symmetry axis at which spheromak length is maximum
Rx = approximate radius of curvature at (Rf, Hf) and (Rf, -Hf)
Hf, -Hf = distances of the plasma sheet from the equatorial plane at R = Rf

A photograph of a typical laboratory spheromak gives:
Rs = 4.2 Rc
Rf = 3.0 Rc
|Hf| = 2.7 Rc
Rx ~ 0.3 Rcc

A spheromak is a cylindrically symmetric closed charge sheet formed from spiraling charge enclosing a toroidal shaped volume. However, for a real spheromak in a vacuum chamber the toroid cross section is not round. The radius of curvature on the outside of the spheromak near the equatorial plane is greater than the radius of a coaxial sphere. Near the equatorial plane the real spheromak core radius is nearly constant. A real spheromak exhibits distinct small radius corners on the charge sheet at the position of the maximum spheromak axial length.

A spheromak has important distinct geometrical dimensions.

An experimental spheromak photographed in a laboratory differs from an ideal spheromak in free space in two important respects.
1. The spheromak is usually in a cylindrical metal enclosure that has a wall radius Rw that may not satisfy the condition:
Rw > 2.71 Rs.
Consequently the field energy density outside the spheromak on the equatorial plane is cylindrical rather than spherical. This change in external field energy density causes a reduction in the spheromak outside radius Rs and reduces the position stability of the spheromak outside wall.

2. Outside the spheromak charge sheet but close to the spheromak the energy density approximation:
U = Uo[Ro^2 / (Ro^2 + R^2 + H^2)]^2
is not exactly true. This issue causes spheromak shape distortion, especially in the necks of the spheromak where the electric fields from opposite sides of the spheromak are partially cancelling and the consequently reduced energy density is supplemented by the poloidal magnetic field energy density.

The parameter of a laboratory spheromak that best indicates the value of:
(Rs - Rc) / 2
for an ideal spheromak is Hf because at (Rf, Hf) both of the aforementioned distortion effects are minimal. Using a photograph Rc is easily measured directly. Hence the best theoretical value of Rs is obtained using the relationship:
Hf = (Rs - Rc) / 2
Rs = 2 Hf + Rc

Hence the spheromak shape factor So is given by:
So = (Rs / Rc)^0.5
= [(2 Hf + Rc) / Rc]^0.5

The data from the spheromak photo indicates that:
Hf = 2.7 Rc
implying that:
So = [(2 Hf + Rc) / Rc]^0.5
= [6.4]^0.5
= 2.53

In terms of a closed plasma hose of length Lh a spheromak is described by the equation:
Lh^2 = (Np Lpf)^2 + (Nt Lt)^2
Np = integer number of poloidal turns around the spheromak axis of symmetry;
Lpf = average purely poloidal turn length;
Nt = integer number of purely toroidal turns;
Lt = length of each purely toroidal turn.

After an element of charge moving along the plasma hose has passed through the spheromak core hole Nt times and has circled around the main axis of spheromak symmetry Np times, it reaches the point in the plasma hose closed path where it originally started.

There are Nt parallel plasma hoses that go through the equatorial plane in the central core of the spheromak and hence form the spheromak core walls.

On the web page titled: PLASMA SHEET PROPERTIES it is shown that:
(Qi Ni + Qe Ne)^2 = ( 1 / C)^2 [Qi Ni Vi + Qe Ne Ve]}^2
and for a plasma spheromak this equation simplifies to:
|(Ni - Ne) / Ne| = |Ve / C|

This equation establishes a relationship between the number of plasma ions Ni, the number of plasma free electrons Ne and the free electron velocity Ve. This equation is a condition for plasma sheet existence and hence is a condition for plasma spheromak existence.

Note if the plasma hose charge per unit length is uniform then |Ve| is constant and hence the electron kinetic energy is constant along the plasma hose. Hence the plasma sheet is an equipotential surface.

Nt << Np
the spheromak magnetic field is almost entirely poloidal and the spheromak is known as a FRC (Field Reversed Configuration).

Inside the torus:
Bt = Btc (Rc / R)

The toroidal magnetic field can point either Clock Wise (CW) or Counter Clock Wise (CCW) around the spheromak's central poloidal magnetic field. Hence a spheromak has two possible distinct magnetic states of equal energy.

In the toroidal region where:
Rc < R < Rs
-Hf < H < HF
the poloidal magnetic field components cancel to zero.

Me = electron mass
Mp = proton mass
Mi = ion mass

Conservation of linear momentum in the plasma hose gives:
Me Ve + Mi Vi = 0
- Ve = (Mi Vi) / Me

Mi >>> Me
-Ve >> Vi

Thus in numerical evaluation of plasma sheet properties Vi is generally negligibly small.

Consider movement of electrons and ions along a charge hose within a magnetic field B and a radial electric field E. The positive ion path radius of curvature Ri is determined by a balance between the electromagnetic force of the form:
F = Q [(Vi X B) + E]
and the centrifugal force of the form:
F = Mi Vi^2 / Ri

Equating these two forces gives:
Q (Vi B + E) = (Mi Vi^2) / Ri
Ri = (Mi Vi^2) / (Q (Vi B + E))
Thus the effect of a positive radial electric field is to reduce the radius of curvature of ions orbiting the spheromak central axis.

The electron path radius of curvature is determined by a balance between the electromagnetic force of the form:
F = - Q [(Ve X B) + E]
and the centrifugal force of the form:
F = Me Ve^2 / Re

Equating these two forces gives:
- Q (Ve B) = (Mi Ve^2) / Re
Re = (Me Ve^2) / (- Q (Ve B + E))
Thus the effect of a positive radial electric field is to increase the radius of curvature of electrons orbiting the spheromak central axis.

For plasma hose to exist the radii of curvature of ions and free electrons must be equal, or:
Ri = Re
(Mi Vi^2) / (Q (Vi B + E)) = (Me Ve^2) / (- Q (Ve B + E))
(Mi Vi^2) / (Me Ve^2) = (Vi B + E) / (- Ve B - E)

When the spheromak is initially being formed the electric field E = 0 and conservation of momentum requires that:
Mi Vi + Me Ve = 0
at which time the radial electric field E is zero. Hence:
Re = - (Me Ve) / (Q B)
= (Mi Vi) / (Q B)
= Ri

Thus as long as momentum of the individual particle types is conserved within the plasma hose the electrons and ions flow in opposite directions along the same curved path with radius:
Re = Ri.
and there is no radial electric field.

However, when the electrons and ions are following the same path in opposite directions collisions between electrons and ions cause the electrons to lose a bit energy and the ions to gain a bit energy. This change in particle energy causes the electron and ion paths to separate sufficiently that the electron and ion streams no longer collide. However, an electric field forms that tends to increase the electron path radius of curvature and decrease the ion path radius of curvature, causing maintenance of plasma hose. Over the life of the spheromak, as the free electrons lose energy to the ions the radial electric field E increases to maintain the plasma hose.

During the lifetime of the spheromak:
Vi B << E << (- Ve B)
so that the equation:
(Mi Vi^2) / (Me Ve^2) = [(Vi B + E) / (- Ve B - E)]
simplifies to:
(Mi Vi^2) / (Me Ve^2) ~ [(E) / (- Ve B)]

During the lifetime of the spheromak (- Ve B) decreases and E increases. When:
(- Ve B) = (2 E)
The spheromak ceases to exist.

During the lifetime of the spheromak the electron kinetic energy drops from its initial value of:
(Me / 2) Vea^2
to a final value of:
(Me / 2) Veb^2
(Me / 2) Vea^2 ~ 2 (Me / 2) Veb^2
Veb = Vea / (2)^0.5

Hence the maximum value of E is given by:
(2 E) = (- Vea B / (2)^0.5)
E = [- Vea B / (2)^1.5]

The electron kinetic energy Ekea is given by:
Ekea = (Me / 2) Vea^2
Vea = - (2 Ekea / Me)^0.5

Hence the maximum value of E is given by:
E = [- Vea B / (2)^1.5]
= [(2 Ekea / Me)^0.5 B / (2)^1.5]
= [(Ekea / Me)^0.5 B / 2]

The value of B is this equation is at a magnetic field minimum location where it is impractical to directly measure the magnetic field. However, we may be able to approximately infer the value of B from the spheromak size.

When the spheromak is first formed:
Re = - (Me Ve) / (Q B)
B = - (Me Ve) / Q Re
= (Me (2 Ekea / Me)^0.5) / Q Re
= (2 Ekea Me)^0.5 / Q Re

E = [(Ekea / Me)^0.5 B / 2]
= [(Ekea / Me)^0.5 [(2 Ekea Me)^0.5 / Q Re] / 2]
= [Ekea / (Q Re 2^0.5)]

Ekea = 25 eV = 25 X 1.602 X 10^-19 J = 40.05 X 10^-19 J
Q = 1.602 X 10^-19 coul
Re = 0.3 m giving:
E = 40.05 X 10^-19 J / (1.602 X 10^-19 coul X 0.3 m X 1.41)
= 59.1 volts / metre

This low value suggests that for a plasma spheromak the magnetic field energy completely dominates the electric field energy. Experimental evidence suggests that the same is true in an atomic particle spheromak.

Me = free electron mass

The relationship between Ve and the free electron kinetic energy Eke is:
Eke = (Me Ve^2) / 2
- Ve = (2 Eke / Me)^0.5

Recall that the existence requirement for a plasma sheet gives:
|(Ni - Ne) / Ne| = |Ve / C|

Recall that The net charge Qs on a spheromak is given by:
Qs = Q (Ni - Ne)
(Ni - Ne) = Qs / Q

Combining these three equations gives:
|(Ni - Ne) / Ne|
= |Ve / C|
= (2 Eke / Me)^0.5 (1 / C)

(2 Eke / Me) = (C Qs / Q Ne)^2
Eke = (Me / 2) (C Qs / Q Ne)^2

This equation gives the free electron kinetic energy Eke in a plasma spheromak in terms of the physical constants Me, C, Q, the net charge Qs and the number of free electrons Ne.

An important result of the equation:
|(Ni - Ne) / Ne| = |Ve / C|
is that it gives insight into what happens during spheromak compression. During an ideal spheromak compression the net charge:
Qs = Q (Ni - Ne)
remains constant while Ne and Ni both decrease due to recombination of spheromak electrons and ions. In order for the spheromak existence condition to continue to be met, Ve increases.

Hence, a compressed spheromak has a smaller number of free electrons than an uncompressed spheromak. The neutral gas atoms that are products of recombination tend to reduce the lifetime of a compressed spheromak. This is a major issue in the Plasma Impact Fusion process.

During the process of spheromak compression the recombining electrons and ions lose their kinetic energy by emission of UV radiation. Hence the process of spheromak compression is not perfectly adiabatic. A further complication is that this UV radiation, when it interacts with the enclosure wall, causes emission of electrons. These free electrons, if caught in the radial electric field around a spheromak, will tend to discharge the spheromak. Hence the inside walls of the charge injector need a coating that minimizes UV triggered electron emission. Possibly a high UV reflectivity internal surface can be used to guide most of the UV energy back to the upstream end of the charge injector where it can be absorbed with minimum consequence.

There is an experiment that can be used to demonstrate the applicability of plasma hose theory to a spheromak. In essence the plasma hose theory assumes that a stable spheromak can only exist if:
- Me Ve = Mi Vi
Vi = (- Me Ve / Mi)

Hence in a spheromak both electrons and ions have the same momentum magnitude of opposite sign so that the net momentum is zero, In a random plasma the energy rather than the momentum is equally distributed over the particles.

However electron kinetic energy Eke is given by:
Eke = (Me / 2) Ve^2
Ve = - (2 Eke / Me)^0.5

Vi = (Me / Mi)(2 Eke / Me)^0.5
= (2 Eke Me)^0.5 / Mi

Note that under plasma sheet theory the entropy of a spheromak is less than the entropy of a random plasma, so a spheromak is at best only semi-stable and will eventually decay into a random plasma. In a spheromak the particle kinetic energy is primarily in the free electrons whereas in a random plasma, in accordance with the equipartition theorem, free electrons and ions have approximately the same kinetic energy. In a spheromak:
(- Ve / Vi) = (Mi / Me),
whereas in a random plasma:
Me Ve^2 / 2 = Mi Vi^2 / 2
(-Ve / Vi) = (Mi / Me)^0.5

Thus during a spheromak decay:
(Mi / Me) > (-Ve / Vi) > (Mi / Me)^0.5
and during spheromak decay the free electrons lose energy to the ions as well as to neutral particles.

For a laboratory spheromak Me and the possible values of Mi are known, Ve can be reliably measured via Thomson Scattering and Vi can be measured via the Ion Doppler technique. Hence in principle it should not be difficult to experimentally distinguish whether the formula:
(- Ve / Vi) = (Mi / Me)
or the formula:
(-Ve / Vi) = (Mi / Me)^0.5
is experimentally valid.

However, note that the ion doppler signal may be confused by the thermal component of ion motion, which will generate emission line broadening. Further confusion will occur if there is any uncertainty with respect to the ion mass.

Initially use a spheromak formed from He-4 because, unlike hydrogen that can exist as either molecular or atomic hydrogen ions, there is no uncertainty about the ion mass. Assume a He-4 spheromak with a free electron kinetic energy of 25 eV.
Assume that the atoms are singly ionized (He+ ions).
Mi = 4 X 1.67 X 10^-27 kg
= 6.68 X 10^-27 kg

Me = 9.1 X 10^-31 kg
Eke = 25 eV X 1.602 X 10^-19 J / eV
= 40.05 X 10^-19 J.

- Ve = {[(Me / 2) Ve^2] / [Me / 2]}^0.5
= {Eke / [Me / 2]}^0.5
= [2 Eke / Me]^0.5

Thus if:
(- Ve / Vi) = (Mi / Me)
Vi = (- Me Ve / Mi)
= (- Me / Mi)[2 Eke / Me]^0.5 = (2 Eke Me)^0.5 / Mi
= (2 X 40.05 X 10^-19 J X 9.1 X 10^-31 kg)^0.5 / 6.68 X 10^-27 kg
= 404 m / s

This velocity causes a small frequency shift, above or below Fe, but this frequency shift may easily be lost in the thermal broadening.

However, if:
(- Ve / Vi) = (Mi / Me)^0.5
Vi = (Me / Mi)^0.5 (-Ve)
= (Me / Mi)^0.5 [2 Eke / Me]^0.5
= [2 Eke / Mi]^0.5
= [2 X 40.05 X 10^-19 J / 6.68 X 10^-27 kg]^0.5
= 3.463 X 10^4 m / s
= 34,630 m / s

If ions with this velocity are present they should be easy to detect.

If Eke is increased to 400 eV via spheromak compression the ion velocity should increase from 404 m / s to:
(404 m / s) X (400 eV / 25 eV)^0.5 = 1616 m / s
This frequency shift is at least comparable to the thermal line broadening.

If the plasma is random with Eke = 400 eV there should be ions present with:
Vi = 4 X 34,630 m / s
= 138,520 m / s

By comparison the room temperature molecular thermal velocity Vit from the equipartition theorem is given by:
(Mi / 2) Vit^2 = (.025 eV)
Vit = [(.05 eV X 1.602 X 10^-19 J / eV) / (6.68 X 10^-27 kg)]^0.5
= .1095 X 10^4 m / s
= 1095 m / s

Thus the underlying assumption that a spheromak is composed of a plasma sheet can be experimentally demonstrated. At low free electron kinetic energies it may be difficult to separate the ion doppler frequency shift due to spheromak particle motion from the line broadening due to thermal ion motion. However, for compressed spheromaks with 400 eV free electrons the ion doppler frequency shift due to spheromak particle motion should be four fold larger and comparison of ion doppler and Thomson Scattering data should give clear results.

Note that in principle this technique can also be used to determine the dominant ion mass in a compressed spheromak. The spheromak ion velocity Vi measured via the Ion Doppler technique is inversely proportional to the ion mass.

H+1 Mp
H2+2 Mp
D+2 Mp
D2+4 Mp
DH+3 Mp
TH+4 Mp
TD+5 Mp
TT+6 Mp
He-4+4 Mp


In executing the above experimental measurements it is essential to remember that the Thomson scattering and ion doppler measurement techniques were originally developed for random plasmas, but the particle motion in a spheromak is not random.

The motion path of the electrons and ions is a spiral within the plasma sheet and has a substantial axial component within the spheromak core. These issues will seriously affect the calibration of some instruments. Also both Thomson Scattering and Ion Doppler instrumentation sample the plasma density near the core of the spheromak where the spacial density of free electrons and ions is much higher than the average spacial density in the spheromak. Failure to properly take these issues into account can lead to a computed average free electron/ion density that is much higher than the actual average free electron/ion density. Moreover, this erroneous calculation may appear to be confirmed by data from an ion probe, which will also indicate too high due to sampling the outer surface of the spheromak.

This web page last updated April 2, 2015.

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