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

When a plasma spheromak is formed via ionization of a gas by an electric field the free electrons and ions initially have similar but opposite linear momenta. These electrons and ions move in opposite directions along almost the same closed path. However, the electrons have much more kinetic energy than the ions. A plasma spheromak relies on free electron and ion linear momentum balance to form the spheromak wall with the radial electric field that provides spheromak stability.

Over time collisions between the circulating charged particles and neutral particles cause particle energy, not linear momenta, to be become more equally distributed among the particles. Hence presence of neutral particles in the same space as the plasma spheromak eventually causes a plasma spheromak to become a random plasma. Thus a plasma spheromak is only semi-stable. The plasma spheromak lifetime, which is typically of the order of 100 to 500 microseconds, can be enhanced by minimizing the neutral particle concentration in the space, especially neutral particle species that have high electron impact ionization cross sections.

This web page addresses issues related to plasma spheromak lifetime. As the name implies, Plasma Spheromak Lifetime Ts is the period of time during which a plasma spheromak provides useful energy storage. The energy is stored in the spheromak's electric and magnetic fields. This energy storage occurs as a result of the Spheromak's net charge and the spheromak's spiral circulatinng current. Loss of either net charge or spiral circulating current reduces the spheromak's stored energy and hence the spheromak's lifetime.

In order for there to be sufficient time to physically close the liquid lead shell around the injected plasma spheromak the spheromak must have a lifetime of at least 2 mS. Meeting this lifetime requirement together with the required spheromak energy is a major technical challenge.

In a plasma spheromak the electron and ion momenta are initially equal, which means that the electron kinetic energy per particle (temperature) is much higher than the ion kinetic energy per particle (temperature). The spheromak plasma becomes a random plasma when the electrons lose energy to the ions. The plasma spheromak life time is typically of the order of 80 us to 800 us in laboratory hydrogen, deuterium and helium plasmas.

A spheromak is an unstable state in which free electrons are trapped at an abnormally low potential energy. Impact ionization of neutral gas molecules reduces the electrostatic charge on the spheromak. That impact ionization also bleeds off energy from the spheromak's magnetic field. These two mechanisms acting together cause the spheromak trapping potential to gradually rise until the trapping barrier reaches zero at which point the spheromak ceases to exist.

If the charged particle motion is highly ordered most of the kinetic energy Ekb converts to magnetic field energy Em so that:
Epv > Epc + Ekc.

In state "c" the kinetic energy per particle Ekc is insufficient to allow an individual particle to escape the trap. Thus a spheromak is essentially a cluster of charged particles caught in an energy potential well that is a result of ordered particle motion.

The energy potential well is formed by the poloidal and toroidal magnetic fields caused by current within the plasma sheet and by the electric field resulting from the distributed positive charge on the plasma sheet. The current within the plasma sheet arises from the particle order where due to initial conservation of momentum the free electrons have much more energy per particle than the ions. In this situation the fields have absorbed sufficient energy from the particles that:
(Ekc + Epc) < Epv
and individual particles do not have sufficient kinetic energy to escape from the narrow region between the toroidal and poloidal magnetic fields where they are trapped.

When a spheromak is formed the voltage between the spheromak core and ground is set by the spheromak generator power supply. However, once the spheromak disconnects from the spheromak generator the voltage between the spheromak core and ground is a function of the net charge on the spheromak, This net charge reduces with time due to impact ionization of neutral atoms by energetic spheromak electrons and due to thermal electron emission from the surrounding enclosure walls.

While the spheromak is connected to the spheromak generator power supply any neutral atom that is ionized tends to become part of the spheromak. The electrons produced via impact ionization of neutrals are absorbed by the spheromak generator power supply. Similarly electrons emaitted by the enclosure walls are also absorbed by the spheromak generator power supply. Hence if a newly formed spheromak is held in contact woth the spheromak generator the ambient neutral gas pressure drops. However, once the spheromak disconnects from the spheromak generator the ambient pressure starts to rise due to electron absorption by the spheromak which causes formation of more neutrals.

If a spheromak is compressed there is rapid formation of neutrals.

The greater the neutral density the faster the neutrals are ionized by energetic spheromak electrons and hence the faster that a spheromak loses its net charge.

Within a PIF system a spheromak exists within a quasi-cylindrical metal enclosure. There is spontaneous thermal ionic electron emission from the enclosure wall which discharges the spheromak and hence reduces the spheromak life time. In design of a PIF system it is essential to ensure that the electric field at the enclosure wall is below the level that causes rapid spontaneous electron emission.

The spontaneous electron emission from the enclosure walls is dependent on the work function of the liquid lead and the operating temperature of the liquid lead. To minimize this electron emission by the enclosure wall the wall temperature must be as low as practical. This wall emission problem is magnified if there is lithium alloyed into the liquid lead. The work function of lead is about 4.25 eV whereas the work function of lithium is only 2.9 eV. Thus it is essential that the fraction of lithium and other low work function impurities in the liquid lead be as small as possible.

Note that the rate of spheromak discharge increases with the surrounding neutral atom concentration which in turn gets larger as the spheromak loses its net charge. The larger the evacuated volue the longer it takes the neutral atom concentration to rise and hence the longer the spheromak lifetime.

Recall from SPHEROMAK PROPERTIES that the condition for spheromak stability is:
(- Epl) > Eke where:
(Epl) = depth of potential well
Eke = kinetic energy per electron.

Adding more free electrons to a spheromak reduces the spheromak's net charge and reduces the average electron velocity Ve which reduces both the height of the potential well (-Epl) and the average free electron kinetic energy Eke.

If an excess low energy electron of a spheromak collides with a spheromak positive ion recombination will occur which forms a neutral atom which escapes from the spheromak. Each such escape reduces both the number of positive ions in the spheromak and the spheromak's net charge.

If the energetic electrons of a spheromak collide with neutral atoms either the neutral atom is ionized or it is not. Non ionizing collisions lead to gradual loss of spheromak electron energy. Ionizing collisions form electron-ion pairs. If the ion motion is unfavorable, as is usually the case, the ion will escape the spheromak causing spheromak net charge reduction. The captured electron will recombine with an existing spheromak ion causing formation of another neutral atom. The concentration of neutral atoms strongly affects the spheromak lifetime.

At the end of spheromak life the spheromak ions interact with the enclosure wall, leading to rapid loss of spheromak net charge and spheromak electrons interact with the enclosure wall leading to rapid loss of stored magnetic field energy. To prevent such losses to the enclosure wall the spheromak diameter must be suitably small with respect to the enclosure diameter.

In a spheromak there is a stable closed spiral free electron path and a stable closed spiral ion path. These two paths are adjacent and parallel but physically slightly separated from one another. Hence in an ideal spheromak the rate of transfer of energy from the higher kinetic energy free electrons to the lower kinetic energy ions is very much smaller than in a random plasma in which random electron-ion collisions continuously occur. Thus, absent any surrounding neutral gas molecules, a spheromak can theoretically persist as a stable plasma configuration for a long period of time. However, the actual spheromak persistence time is limited by the presence of ionizable neutral gas atoms and is known as the spheromak lifetime.

A real spheromak, even if structurally ideal, will gradually undergo a parameter change due to energetic free electrons in the spheromak impacting neutral gas molecules that randomly move into the narrow zone between the poloidal and toroidal magnetic fields where the spheromak's energetic free electrons are trapped. When a neutral gas molecule is impact ionized it becomes an electron-ion pair.

Since the impact ionizations occur almost entirely in the narrow low magnetic field zone where the circulating free electrons and ions are trapped, there is an extremely high probability that, as long as the trapping potential is maintained, the newly formed electron-ion pair will also be trapped by the spheromaks electric and magnetic fields. Hence the newly formed electron-ion pair will be incorporated into the spheromak increasing the number of ions Ni and the number of free electrons Ne and reducing the number of neutral gas atoms Nn. However, the spheromak order will be disturbed.

Note that in effect a spheromak acts like a vacuum pump and, subject to spheromak energy and expansion constraints, gradually absorbs all the available neutral gas atoms into its structure.

If a newly formed electron-ion pair is fully incorporated into the spheromak the spheromaks Ne and Ni values will both equally increase and the quantity:
(Ni - Ne) will be unchanged.
However, the ratio:
[(Ni - Ne) / Ne]
will decrease due to the increase in Ne. This ratio change will cause a proportional increase the spheromak's linear dimensions. When the spheromak's linear dimensions have increased by a factor of 2.7 the spheromak plasma sheet equatorial radius Rs will reach the cylindrical enclosure radius Rw and the spheromak will be randomized by ion meutralization by the enclosure wall.

We know that spheromak compression causes an increase in the ratio:
[(Ni - Ne) / Ne]
via a decrease in Ne which is accompanied by a proportional decrease in spheromak linear size. Hence full incorporation of newly formed ions into a spheromak will cause an increase in Ne which will cause an increase in spheromak linear size. However, we also know that when the spheromak is inside a rigid cylindrical enclosure of radius Rw its normal initial radius Rs is given by:
Rs = (Rw / 2.718).
Hence, if the quantity (Ni - Ne) remains constant the maximum increase in Ne is 2.718 fold before the spheromak plasma sheet particles interact with the enclosure wall causing immediate spheromak randomization.

The impact ionization rate at any instant in time is given by:
(impact ionization rate) = (Ne Ve Sigma Nn / Volv)
Ne = number of free electrons in the spheromak
Ve = free electron velocity in the spheromak
Sigma = electron impact ionization cross section of neutral gas atoms
Nn = number of neutral gas atoms in vacuum chamber
Volv = vacuum chamber volume

If there are multiple species of neutral gas atoms the impact ionization rate is given by:
(impact ionization rate) = (Ne Ve / Volv)[sum of all (Sigmai Nni)]

The spheromak lifetime Ts is the time from initial spheromak formation until the spheromak randomizes.

The rate of change of Ne due to impact ionization of neutral gas atoms is given by:
(dNe / dT) = (Ne Ve Sigma Nn / Volv)
Nn = (Nno - Ni)
Nno = number of neutral atoms at time T = To before any atoms are ionized. The neutral atoms spread through the vacuum system volume Volv creating a neutral gas atom concentration:
(Nno / Volv)

Nn = Nno - Ni
(dNe / dT) = (Ne Ve Sigma [Nno - Ni] / Volv)

However for a spheromak:
Ne ~ Ni
(dNe / dT) = (Ne Ve Sigma [Nno - Ne] / Volv)

Rearranging this formula gives:
dT = dNe / {(Ne Ve Sigma / Volv) [(Nno - Ne)]}
= {(Volv / Ve sigma) [dNe / (Ne (Nno - Ne))]

The compressed spheromak lifetime Ts is the time period from compressed spheromak formation at T = Tb, Ne = Neb until the spheromak randomizes at T = Tc, Ne = Nec.

Integrating from T = Tb to T = Tc gives: Ts = (Tc - Tb)
= (Volv / Ve sigma) Integral from Neb to Nec of:
[dNe / (Ne (Nno - Ne))]
= (Volv / Ve sigma)(- 1 / Nno) {Ln[(Nno - Nec) / Nec] - Ln[(Nno - Neb) / Neb]}
= (Volv / Ve Sigma Nno){Ln[(Nno - Neb) / Neb] - Ln[(Nno - Nec) / Nec]}
= (Volv / Ve Sigma Nno) Ln{[(Nno - Neb) / (Nno - Nec)][Nec / Neb]}
= (Volv / Ve Sigma Nno){Ln[Nec / Neb] + Ln[(Nno - Neb) / (Nno - Nec)]}

Nec = 2.718 Neb
so the formula for Ts simplifies to:
Ts = [Volv / (Ve Sigma Nno)]{1 + Ln[(Nno - Neb) / (Nno - 2.718 Neb)]}

However the ion gun efficiency Fg gives:
Fg = (Nea / Nno)
= (G Neb / Nno)

where typical ion gun efficiencies are in the range:
Fg = 0.10 to Fg = 0.3
and the typical plasma injector gain G is about:
G = 5.
Neb = (Nea / G)
= (Fg Nno / G)

Hence for practical purposes:
Ln[(Nno - Neb) / (Nno - 2.718 Neb)]
= Ln[(Nno - (Fg Nno / G)) / (Nno - 2.718 (Fg Nno / G))]
= Ln[(1 - (Fg / G)) / (1 - 2.718 (Fg / G))]
< Ln[(1 - .06) / (1 - .163)]
= Ln[.94 / .837]
= Ln[1.123]
= .116
<< 1

so the formula for Ts simplifies to:
Ts = [Volv / (Ve Sigma Nno)]
This formula applies to both uncompresssed and compressed spheromaks.

If there are multiple gas species present:
Ts = [Volv / (Ve (Sum of all (Sigmai Nnoi)))]

Recall that spheromak lifetime is given by:
Ts = [Volv / (Ve Sigma Nno)]

Comparing compressed spheromaks to uncompressed spheromaks where subscript "a" indicates an uncompressed spheromak and subscript "b" indicates a compressed spheromak:
(Tsb / Tsa) = [Volvb / (Veb Sigmab Nnob)] / [Volva / (Vea Sigmaa Nnoa)]
= [Volvb / Volva] [Vea / Veb][Sigmaa / Sigmab][Nnoa / Nnob]

Volvb = Volva
Nnoa = Nnob
(Tsb / Tsa) = [Volvb / Volva] [Vea / Veb][Sigmaa / Sigmab][Nnoa / Nnob]
= [1] [Vea / Veb][Sigmaa / Sigmab][1]
= [Vea / Veb][Sigmaa / Sigmab]

Experimental data show that if Eke is much greater than the gas ionization energy Sigma changes slowly with electron velocity, so to a good approximation:
(Tsb / Tsa) = [Vea / Veb].
Hence after compression a spheromak has a shorter lifetime than before compression. This conclusion is confirmed by experimental data.

For the purposes of this analysis it is helpful to tabulate certain properties of common gases. This data is provided by hyperphysics and NIST.
GASIonization Energy in eVSigma at 337 eV in units of 10^-20 m^2
Hydrogen 13.598 0.30
Helium 24.587 0.27
Nitrogen 14.534 0.965
Oxygen 13.618 0.92
Carbon 11.260 1.12
Neon 21.564
Fluorine 17.422
Chlorine 12.967
Argon 15.759
Bromine 11.814
Krypton 13.999
Iodine 10.451
Xenon 12.130
Carbon Dioxide 19.0 2.7
Lithium 1.847 5

For free electrons with kinetic energies in the range 20 eV to 500 ev data available from NIST shows that for neutral hydrogen gas molecules the impact ionization cross section Sigmab lies in the range:
0.4 X 10^-20 m^2 < Sigmab < 1.0 X 10^-20 m^2
The NIST data further shows that for electrons with kinetic energies in the range:
300 eV to 500 eV
impacting neutral hydrogen molecules:
Sigmab ~ 0.5 X 10^-20 m^2.

The NIST data further shows that for for free electron kinetic energies below 15 eV the hydrogen molecule electron impact ionization cross section Sigmaa rapidly falls to zero. Hence if the spheromak free electron kinetic energy is less than 15 eV and if the only gas species present are hydrogen and helium the spheromak lifetime becomes very long.

For a spheromak with 20 eV free electrons:
Ve = (2 Eke / Me)^0.5
= (2 X 20 eV X 1.602 X 10^-19 J / eV) / (9.1 X 10^-31 kg)]^0.5
= 2.653 X 10^6 m / s

For a compressed spheromak with 500 eV free electrons:
Ve = (2 Eke / Me)^0.5
= (2 X 500 eV X 1.602 X 10^-19 J / eV) / (9.1 X 10^-31 kg)]^0.5
= 13.268 X 10^6 m / s


FIND Volv:
Assume that there are two spheromak injectors. For the apparatus under consideration the vacuum chamber volume Volv is the volume of a 7.5 m long cone that varies from a maximum radius of 1.5 m to a minimum radius of 0.30 m plus the ball valve assembly volume of:
Pi (.3 M)^2 (~ 2 m) = .565 m^3
plus half the volume of a pressure vessel with a contained volume of:
(4/3) Pi [Ri^3 - Rob^3]
= (4 / 3) (3.14159) [(2.2 m)^3 - (.8772 m)^3]
= (4 / 3) (3.14159) [10.648 - .675]
= 41.775 m^3

Application of geometry gives the volume of the cone as:
Cone Volume = Integral from Z = 0 to Z = 7.5 m of:
Pi R^2 dZ
R = (K Z) + .30 m

Assume that the cone axial length is 7.5 m.

Cone walls are defined by:
1.50 m = K (7.5 m) + .30 m
K = (1.50 m - .30 m) / 7.5 m
= .16

dZ = dR / K

Integrating gives:
Cone volume = Pi (Ra^3 - Rb^3) / (3 K)
= [3.14159 / 3(.16)] (1.5^3 - .30^3) m^3
= 6.545 (3.375 - .027) m^3
= 21.913 m^3

Hence the total vacuum chamber volume Volv associated with one spheromak generation/compression system is:
Volv = (41.775 m^3 / 2) + 21.913 m^3 + .565 m^3
= 43.365 m^3


As is shown on other web pages, for liquid lead aspects of the PIF system to work as intended:
Ts = 6 X 10^-3 s

Recall that:
Ts = [Volv / (Ve Sigma Nno)]

If Volv is insufficient, it is impossible to achieve an adequate compressed spheromak lifetime.

Clearly a key issue is achieving the required value of (Volv / Nno). Rearrange this formula to get:
(Volv / Nno) = Ts Ve sigma

Substitution into this formula for a compressed spheromak with Eke = 500 eV gives:
(Volv / Nno) = Tsb Ve sigma
= (6 X 10^-3 s) X (13.268 X 10^6 m / s) X (0.5 X 10^-20 m^2)
= 39.804 X 10^-17 m^3 / atom
= 3.9804 X 10^-16 m^3 / atom

For a good vacuum system that achieves 10^-10 bar of atmospheric molecules using wall heating for outgasing and a cryogenic trap:
{Volv / Nno}
= (22.4 lit / 6.023 X 10^23 molecules) X (1 m^3 / 1000 lit) X (1 / 10^-10)
= (22.4 / 6.023) X 10^-16 m^3 / molecule
= 3.719 X 10^-16 m^3 / molecule

Thus the vacuum system must be rated to pull a vacuum of < 10^-10 bar. The vacuum rating for other gases must be even better due to the larger electron impact ionization cross sections of non-hydrogen gas atoms. Ideally the vacuum system should be able to attain 10^-11 bar.

Thus the first requirement for a functional PIF energy system is for the vacuum system to be able to reach and hold < 10^-11 bar before deuterium injection for spheromak formation. Then the main sources of neutral gas molecules are neutral deuterium molecules emitted by the ion gun during the spheromak formation process and neutral deuterium molecules emitted by the spheromak during the spheromak compression process.

Note that the PIF system relies on inter atomic attraction between lithium and a binding element to keep the lithium vapor pressure at less than 10^-11 bar. This vacuum level requirement effectively prohibits the use of lead-lithium alloy because after each fusion pulse with that alloy the lithium vapor pressure will be too high to allow the spheromak lifetime necessary to support the next fusion pulse.

Even if the vacuum system is of excellent quality Nia is limited by:
Nia = Fg Nno
and for a compressed spheromak:
Nib = (Nia / G)
= (Fg / G) Nno

G = (Rwa / Rwb) = plasma injector gain

However, Nno is constrained by the equation:
Nno = [(Volv / (Ve Sigma Ts)]
= [( 43.365 m^3) X (10^16 atoms / 3.9804 m^3)]
= 10.89 X 10^16 atoms

The corresponding upper limit on Nia for an uncompressed spheromak is:
Nia = (Fg Nno)
= (0.1) X 10.89 X 10^16 atoms
= 1.089 X 10^16 ions

However, the smallest value of Nia that will convey sufficient energy to the random plasma is about:
Nia = 3 X 10^16 ions.
Hence to make the PIF system functional the product:
(Fg Volv / Ts) must be increased three fold. The most elegant way to increase this product is to increase the ion gun efficiency Fg. If for any reason Fg cannot be increased then the vacuum system volume must be increased.

The corresponding upper limit on Nib for a compressed spheromak with G = 5 and with the required lifetime Ts is given by:
Nib = (Nia / G)
= (1 / 5) X 3.0 X 10^16 atoms
= 0.6 X 10^16 ions

THIS IS A HARD UPPER LIMIT ON THE NUMBER OF SPHEROMAK IONS WHICH ARE INJECTED INTO THE VACUUM CHAMBER. This hard limit constrains the amount of energy that can be delivered to the random plasma by a spheromak.

The number of neutral gas atoms in the spherical portion of the vacuum chamber is approximately given by:
(Nno / 2) ~ 5 X 10^16 atoms
Note that this number is four orders of magnitude below the number of hydrogen isotope atoms needed for providing energy for the fusion energy pulse.

General Fusion experimental spheromaks with moderate free electron kinetic energy (25 eV free electrons) have typical lifetimes of about 500 X 10^-6 s. Spheromaks need at least that much free electron kinetic energy in order to ionize neutral gas atoms, which is a necessary step to incorporating these gas atoms into the spheromak.

General Fusion's experimental compressed spheromaks (300 eV to 500 eV free electrons) have typical lifetimes of about 80 microseconds. The volume Volv of General Fusion's experimental vacuum chamber is too small to adequately dilute the number of neutral gas atoms that are injected into the vacuum chamber by the ion gun of the spheromak generator.

After spheromak compression it may be possible to marginally increase the effective spheromak lifetime by combining two suitable spheromaks in the reaction chamber to form a Field Reversed Configuration (FRC). There are claims in the scientific literature that an FRC exhibits a longer lifetime than a spheromak. However, this author is not convinced that FRC formation will make a material improvement to spheromak lifetime as compared to simply increasing the vacuum chamber volume as described herein.

Improving the ion gun efficiency Fg is not easy, but is important for the commercial success of the Plasma Impact Fusion (PIF) energy generation process. Ion guns are dominated by complex space charge, surface recombination, magnetic field geometry, sputtering and RF coupling issues that are beyond the scope of this web page. The larger the fraction of ion gun injected atoms that are initially incorporated in the spheromak the lower is Nno for a particular value of Ni and hence the longer is the spheromak lifetime.

For the PIF process to work as contemplated herein the vacuum chamber must have sufficient volume to rapidly and sufficiently dilute neutral hydrogen isotope gas atoms that are injected by the spheromak generator ion gun.

The vacuum pump must reduce the partial pressure of neutral atmospheric gas molecules that do not originate from the spheromak formation and compression processes to about 10^-11 bar. Achieving that target requires a high performance vacuum system with no leaks. To maintain that quality of vacuum either the liquid lead-lithium gun piston seals must be perfect or the liquid lead-lithium gun pistons must be magnetically coupled to the compressed gas based gun piston acceleration system. Providing gun piston magnetic coupling requires significant redesign of the liquid lead guns.

If the radial electric field at the enclosure inside surface in the neck of the plasma injector is too large electrons will be emitted from the enclosure metal surface. This electric field enhanced electron emission will reduce the compressed spheromak net charge and hence will reduce the compressed spheromak lifetime. Hence this author believes that it is essential to make the plasma injector neck inside diameter about 0.6 m to maximize the compressed spheromak lifetime.

This web page last updated November 2, 2014.

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