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Problems of a thermonuclear reactor with a rotating plasma
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1980 Nucl. Fusion 20 579
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PROBLEMS OF A THERMONUCLEAR REACTOR
WITH A ROTATING PLASMA
A.A. BEKHTENEV, V.I. VOLOSOV, V.E. PAL'CHIKOV, M.S. PEKKER, Yu.N. YUDIN
Institute of Nuclear Physics,
Novosibirsk, Union of Soviet Socialist Republics
ABSTRACT. The authors consider the physical problems involved in the design of a thermonuclear
reactor with a rotating plasma. Detailed consideration is given to a version of the reactor in which the plasma
is stabilized mainly by radial variation of the rate of rotation of the plasma (electric shear). Such aspects as
the heating, longitudinal confinement, stability and equilibrium of the plasma as well as the problem of
impurities are considered and a calculation of the reactor's efficiency is made. The authors discuss the
engineering problems of the creation of a high-intensity radial electric field in the plasma and describe a
modification of this type of reactor — a system without magnetic mirrors (a 'centrifugal trap').
One possible way of solving the problem of controlled 1. DESCRIPTION OF THE REACTOR
thermonuclear fusion is to use modified open magnetic
traps, i.e. systems in which there is better confinement A schematic diagram of the rotating plasma reactor
of the plasma along the magnetic field than in the is shown in Fig. 1. It consists of an open magnetic trap
classical mirror trap [ 1 ]. Over the last few years, several with an axi-symmetric field, the mirrors of which contain
systems have been advanced for solving this problem, the co-axial electrodes in contact with the plasma. Each
for example, the ambipolar trap [2], the gas-kinetic electrode is fed a high voltage which shapes the required
trap [3], the reversed magnetic field trap [4] etc. profile of the radial electric field E(r) in the plasma.1
Another such system is the rotating plasma trap, the As will be demonstrated below, the working
main features of which as a possible thermonuclear temperature of the deuterium and tritium ions should
device were discussed in a number of papers [5—7]. be of the order of 30—100 keV, the energy of
A large number of experiments have been carried out rotation Wgo being of the order of 150-500 keV
in traps of this kind in the 1960s at relatively low ion = mic2E2/2H2). The size of the magnetic field
energy (10—100 eV), as described, for example, in the is determined by the plasma density n, the permissible
review paper of Lehnert [6]. However, comprehensive values of (3, and the ratio a/r , where a is the width of
0
studies on the possibility of using rotating plasma traps the plasma layer and r is the plasma radius in the
0
as thermonuclear reactors present us with a number of centre of the trap. For n a 3 X 1013 cm"3, 0 = 0.25,
problems that have not been studied before in detail. a/r = 0.1—0.3, the magnetic field intensity in the
0
Below we consider some of the problems arising in centre of the trap is of the order of 15—25 kG.
connection with one of the possible designs for this Accordingly, the electric field intensity should be
reactor. The characteristic feature of this design is ~ 100 keV • cm"1. Since the radial dimension of the
stabilization of the plasma based on the establishment plasma has to be greater than the ion Larmor radius
of an appropriate radial electric field profile by means (a/Pi ^10), for the given plasma parameters a ranges
of a system of ring electrodes in contact with the plasma. between 15 and 50 cm; the full potential applied to
This paper discusses the longitudinal confinement, the plasma is, accordingly, of the order of a few
stability and equilibrium of the plasma as well as the megavolts (1.5—5 MV). The radius of the trap and its
efficiency of the reactor and the possibility of improving length L are of the order of a few metres. Here we give
it by recovering the energy of escaping ions and alpha- typical parameters of the reactor but they could
particles; it also touches on a number of technical problems
involved in constructing a device of this kind. Possible
modifications of the magnetic system of the reactor are
1 The possibility of using the ring electrodes for shaping
discussed, reactor operation in the steady state being the
an electric field in a rotating plasma was noted for the first time
main consideration.
in Ref. [8] (see also Section 7).
NCULEAR FUSION, Vol.20, No.S (1980) 579
BEKHTENEV et al.
In such a system the particles are acted on by a centri-
fugal inertial force m £2| r, for which reason the
particles escaping from the trap have to overcome the
centrifugal potential barrier m £2 (ro — rp/2 (the
E
subscript k relates to the magnetic mirrors). Here the
loss cone for ions in the velocity space is transformed
into a two-sheet hyperboloid (Fig. 2):
v| = vj (R-l) (r --rr22)) - *p - (4)
o
or, at Hr2 = const
_2
FIG.l. Diagram of a rotating plasma trap: (I) ring electrodes;
(5)
(2) inner liner; (3) outer liner; (4) magnetic field coils; the m
broken line shows the course of the magnetic field lines.
where R is the mirror ratio (R = Hk/H = ro/r2,), V
0 Eu
is the drift velocity in the centre of the trap, and <p is
0
the ambipolar potential between the centre of the
obviously vary somewhat when the problem of
trap and the electrode located in the magnetic
optimizing them has been solved.
mirror [9].
Let us note the most important characteristics of
Injection and heating in this type of trap can be
the behaviour of the plasma in this trap.
achieved most simply by introducing into the plasma
In a trap with crossed E and H fields, all particles,
relatively cold neutral atoms and then ionizing them.
apart from rotation in the Larmor orbits and motion
We shall consider these processes in a rotating system
along the magnetic field lines, exhibit an azimuthal
of co-ordinates (where the plasma as a whole is immobile).
drift with a velocity
Here, the flow of neutral atoms goes through the plasma
at a velocity V o; an ion-electron pair is generated in
FXrt E
VE = eH2 (1) tahned ptlhaes melae catfrtoenr iroontaiztea tiino na oLf aar mnoeur torarbl iat toatm a; vtehleo ciiotny
of V o. As all the ions at the centre of the trap are
E
and, correspondingly, with an angular velocity generated with the same energy (Wi = WEO> W|| = 0),
their distribution function is close to the 6-function
(2)
where
F = eE + (3)
The last term in Eq. (3) is normally small, hence,
allowing for the fact that the potential difference in
the plasma between two fairly close magnetic surfaces
is for all practical purposes constant along the magnetic
field lines, we find on the magnetic surface
i2 = — = const (2')
LE Hr
i.e. the so called 'isorotation law' [6].
The condition of confining the particles in this type
FIG.2. Plasma confinement boundary in the phase space:
of trap can be derived conveniently by a consideration (I) conventional mirror trap (T<Ti); (II) rotating plasma
e
of their motion in a rotating system of co-ordinates. trap; (III) sphere on which the ions are generated in the trap.
580 NUCLEAR FUSION, Vol.20, No.5 (1980)
ROTATING-PLASMA REACTOR
just after generation of the ions (Fig. 2). Because of effect, j|| = (3n/3r) (pgi/n) j (r)> i-e. a current equal
o
Coulomb collisions, the ions are thermalized, and their to j || flows in the quasi-neutral plasma to the ring
distribution function spreads in velocity space. This electrodes (here j|| is the current density, j (r) is the
o
approach to injection in a rotating plasma trap is more electron current density along the force lines [12]).
or less equivalent to fast-neutral-atom injection in The radial current corresponding to this current, from
conventional open traps, which was well studied in the condition div j = 0, heats the plasma up. Thus, in
experiments [10, 11]. this case the electric current heating the plasma is
Even if the region of ionization of neutral atoms entirely determined by the electron-ion flows from the
is spread along z, i.e. ionization also occurs near the plasma along the magnetic field due to the Coulomb
mirrors, the energy of the generated ions depends processes and is equal to their difference. Note that
only slightly on the position of the point of ionization. this current is a/pgi times as low as j .
0
Let us use E and H to designate the fields in the For this injection to occur, the energy of the neutral
o o
centre of the trap and R = H(z)/H the ratio between atoms has to be sufficient for their path in the plasma
z 0
the magnetic field at the point of generation of the to be of the order of either the transverse dimension
ion and H , in which case the longitudinal and of the trap or the longitudinal dimension of the mirror.
o
transverse energies of the ion are For the reactor parameters considered above, these
conditions are satisfied at velocities = 107 cms"1
Wi = W /R ; W|| = W d-l/R ) (10—100 eV). The neutral atom energy is much less
EO Z E0 z
than WEO-
i.e. in velocity space the ions are generated on a sphere
of radius Vgo • Allowance for the longitudinal electric
field leads to a deformation of this sphere by the 2. LONGITUDINAL CONFINEMENT
quantity ~ &P /WE0 = T /Ti.
O e
Let us analyse what macroscopic processes are Before the results of our accurate calculations will
responsible for plasma heating, i.e. what is the source be presented, let us show, by a simple and crude estimate,
of plasma energy. In case the radial currents in a that the longitudinal plasma confinement time in a
rotating plasma trap are much higher than the longi- rotating plasma trap may be much longer than that valid
tudinal ones and the plasma density and transverse in a classical mirror trap. Coulomb processes only are
conductivity near the side walls are high enough, the taken into account in this estimate, the radial plasma
plasma is heated up by radial plasma currents between structure is regarded as homogeneous. We shall compare
the outside and inside walls of the installations. In the mean ion energy with the potential barrier for the
another case realized in the reactor under consideration ions Uj = WEO (1—1/R) — e»^, assuming that the dimen-
0
the longitudinal currents for the ring electrodes are sion of the mirror is much smaller than the length of
very significant. In this case, both density and the trap. Under steady-state conditions we make the
conductivity of the plasma near the side walls are low energy received by the ion and electron from the
or equal to zero (n(rj) = n (r2) = 0). In the stationary electric field equal to the mean energy removed by
dense rotating plasma in this trap (at a/pj > 1), the them from the trap (in the rotating co-ordinate system)
electrons and ions generated in the trap escape from
the plasma along the force lines during the ionization w (i- Q = W
E0 ie EO
time. Because of the radial density gradient, the
magnitudes of the electron and ion longitudinal flows
(6)
are not equal since the rates of generation of electrons
and ions at point r are not equal either. The rate of where W-* and W* are the mean transverse energies of
ion generation at r is determined by the rate of the ions and electrons when escaping from the trap,
ionization and charge exchange at point r + pgi and Qi is the energy transferred from the ions to the
e
(PEi = miVEOc/eHo)> because the ion, after ionization electrons.
(or charge exchange), shifts, on an average, by a Larmor Summing these expressions, we obtain
radius in radial direction; the rate of electron genera-
tion at point r is also determined by the ionization at W /R = Wf (7)
E0
point r — PEe- Accordingly, in the stationary regime,
the longitudinal currents of the outgoing ions and
Use is made here of an approximate relation linking the
electrons differ by a small quantity determined by this
energy carried off by the particle through the potential
581
NUCLEAR FUSION, Vol.20, No.S (1980)
BEKHTENEV et al.
barrier from the magnetic trap and its temperature, on where
the assumption that the barrier is much higher than T
[13]. The electric potential between the trap centre
x = V/VEO ; X = 0 at x <
and the mirror can be found from the condition of e
equality between the flow of escaping electrons and
ions [14]:
-e<A)/T
e e
(8)
IT X
from which we obtain — dx' d cos 0
/-* mfl
0 0
Ui/Ti=R-l-
2(T
e
or
+ J I f^x'dx'dcos0
Ui/TiSR-1 at (9)
0 x
Hence, the potential barrier confining the ions in the
trap may be several times higher than the ion temperature.
IT X
Correspondingly, the ratio of nr in the rotating plasma
J J f/j x ^1 + j
trap to nr in a conventional trap is of the order of dx'
exp (Ui/Ti) at the same ion temperature (here, r is the
0 0
confinement time for the particles in the trap). The
exponential nature of this relationship would appear to
have been demonstrated for the first time in Ref. [15].
Calculations have been made of the principal para-
meters of a plasma in an 'ideal' trap, with consideration 0 x
only of Coulomb collisions. A two-dimensional Fokker-
Planck equation was solved numerically in velocity space.
a, j3 are the kinds of particles, In AQ^ is the Coulomb
It was assumed that the plasma was homogeneous with
logarithm. In these equations, we use the zero-order
respect to r and located in a magnetic well rectangular
terms of the expansion of Rosenbluth potentials g
with respect to z. This set of equations in dimension- a
and ha in Legendre polynoms. The boundary conditions
less variables is given as follows (normalization to unity
for this problem have the form:
was made for source, rotation velocity and ion mass):
dfj _LJL —- = 0 at x = 0
dx
3t x2 3x 3x
agi a
X x '3x2 2x3sin0 90 = 0 at 0 = 0 and 0 = TT/2 (11)
fi = 0 on a surface x2 (R sin2 0 - 1) + (1 - 1/R - e^ ) = 0.
0
The ambipolar potential ip is determined from the
0
quasi-neutral conditions. The method of computation
and the evaluation of its accuracy are described in
3t 3x
Ref. [16]. For the sake of simplicity in the calculations
_L 9 of DT-plasma, the plasma was supposed to consist of
_: r_ { 2f
2x2 \ Y ax2 / one kind of ions with a mass of 2.47 (see, e.g. Ref. [2]).
As shown in Fig. 3a, nr increases considerably at
8(x) R > 3, as compared with nr for the conventional
(10)
2x3 3x magnetic mirror trap. Figure 3b shows the dependence
582 NUCLEAR FUSION, Vol.20, No.5 (1980)
ROTATING-PLASMA REACTOR
of Tj T on R. These plots are shown in dimensionless
; e
form since
to
with an accuracy of up to 5% over the 10-200-keV
injection energy range. Normally, the efficiency of a
thermonuclear reactor is defined by the quantity Q,
which is the ratio of the energy released in the plasma
in the thermonuclear reaction to the energy introduced
into the plasma.
Figure 4 shows Q as a function of R; the equation for
determining Q, i.e. Q = TIT. <OV> WR/8WE0 was
r
derived with allowance for the features of a trap with
a rotating plasma. Here a is the thermonuclear
reaction cross-section, and W is the energy released
r
as a result of one reaction event. It is normally
considered that W = 22.4 MeV, with allowance for
r
the energy released in the lithium blanket [17]. In
this trap the energy expended on generating an ion-
electron pair is not equal to 2WEO> but to 2WEO/R,
since when the ion and electron escape from the trap
along the magnetic field some of their energy returns
to the plasma.
3. EFFICIENCY OF THE REACTOR
The above-derived values of Q and nr for a rotating
plasma trap show the advantages of this system over
the classical open trap, but these calculations are valid
only for 'ideal' traps. To find the actual reactor
parameters we have to take into account a number of
important physical processes affecting its operation
and the Q factor values attainable, such as ionization
and charge-exchange processes, the possibility of
partial recovery of the ion and a-particle energy at the
0.2
end electrodes, and the possibility of recycling the
nuclear fuel to the trap ('re-injection'). Accordingly,
the corrections to the Fokker-Planck equations caused
by these processes have to be taken into account. It is o.l =1
1
clear that they only lead to a change in the source
function in Eq. (10). The character of the corrections
associated with the processes mentioned above as well
as the results of the calculations are given below.
Effects associated with the radial structure of the plasma
FIG.3. Dependence of nr and T/WEO on R for a deuterium
in a rotating plasma trap have been omitted here on the
plasma. Injection energy W = 20 keV; source power
E0
assumption that the transverse dimension of the plasma S = 10!Sparticles • cm'3; (I) conventional mirror trap [ 17];
is fairly large (see also Section 6). (II) rotating plasma trap.
583
NUCLEAR FUSION, Vol.20, No.S (1980)
BEKHTENEV et al.
Q where OQ[ is the charge-exchange cross-section, and
(7j, a are the cross-sections for ionization by the ions
e
and the electrons, respectively; v is the velocity of
charged particles in the laboratory frame; here, the
(8 velocity of the neutral particles is much less than v.
However, even if condition (12) is known to be
satisfied, we have to take into account the fact that
16
the charge exchange leads to additional volume energy
losses, which reduces the efficiency of the reactor.
Ionization leads to generation of particles with an
energy of W|| = 0, Wi = m Vg /2, while charge exchange
Q
creates ions with an energy W|| = 0; Wi = WEO> instead
of ions with an energy of the order of Tj if either
process is analysed in the rotating system of co-ordinates.
10
Allowance for effects associated with the charge-
exchange process introduces additional collision terms
into the Fokker-Planck equation. Then, the equation
for fj in dimensionless variables has the form:
<ajv) + <av>
e
(13)
Here, account is taken of the fact that the intensity of
WO ZOO 300 HOO
(keV) the ion source is given by
FIG.4. Dependence ofQ on the deuteron injection energy
WED = mp- V£0/2 for an ideal DT reactor; at nD = nT; So = n n0 <aev>)
W = (3/2)W ; (I)R = 5; (II) R = 3.
ET ED
S is assumed to be independent of time. In these
o
expression, n is the ion density, n the density of
0
neutral atoms, and < > denotes an average over
Charge exchange and ionization the distribution functions of the colliding particles.
The additional terms in Eq. (13) are due to charge-
exchange processes. The second term in the parenthesis
The ionization and charge-exchange processes in a
of Eq. (13) describes the generation of ions with an
rotating plasma create radial plasma flow and, accordingly,
energy of WEO- The third term on the right-hand side
radial electric currents. The occurrence of the flow is
of this equation describes the loss of ions with an
due to the fact that after each ionization or charge-
energy of the order of Ti. Note that allowance for
exchange event in the rotating plasma trap, the ion is
the charge-exchange process does not change the
displaced along the electric field by a value of
equation for the electron distribution function. For Q
the order of the Larmor radius. The condition for
to be defined, the fact should be taken into account
which this flow does not affect the escape of ions from
that charge exchange leads to volume energy losses
the trap along the magnetic field, i.e. it need not be
in addition to those occurring in 'ideal' traps. The
taken into account when solving the Fokker-Planck
energy flow removed from the plasma by the fast
equations, reduces to the condition of smallness of the
'charge-exchange' atoms escaping from the trap is
radial ji currents compared with the longitudinal j
0
given by
(seeRef. [12]):
<a v3>
(12) Pi = 1 <a0iv3> = So WE0 0i (14)
JO <aev> VE0
584 NUCLEAR FUSION, Vol.20, No.5 (1980)
ROTATING-PLASMA REACTOR
The additional energy flow heating up the plasma and,
accordingly, carried off by the ions escaping along the
magnetic field is equal to
H
m;
iv> V| - <a v|v + V 12» n n
o Oi E0 0
)iv> - <aoiv x2>
(15)
Jiv) + <av>
ft
The expression for P is derived after integration of
2
Eq. (13) over d3x with the factor x2 due to additional
charge-exchange terms. The efficiency of the thermo- FIG.5. Diagram illustrating the motion of an ion near the end
nuclear reactor, taking charge exchange into account, electrodes.
is determined by the expression
nr <ffv> W X is the mean free path of the ion in the trap, wj « vj..
r
Q = (16)
Hence
2W
E0
R
2irp* /L
4
"" A
and for
Recovery of charged-particle energy
O ^ T
"7*-=4; -=10"5; we have ^s 30°
A A
The ion and electron escaping from the trap along
the magnetic field overcome the corresponding potential
barriers and transfer a portion of their energy equal to The mean energy of the ion in its collision with the
WEO (1-1/R) to the electric field. This effect was taken electrode is determined by the expression
into account above in the calculation of Q.
Let us consider one more mechanism for the recovery
of the ion energy. Since the energy of an ion moving
in a cycloid varies along its trajectory, we can arrange
the geometry of the end electrodes such that the ions
hit the electrode with minimal kinetic energy. For this mj VE0 2viVE0
cos v? (18)
purpose the end surface of the electrodes is inclined
towards a plane perpendicular to the z axis at the angle
i// (Fig. 5) so that the ion collides with the most
where we average with respect to r and the transverse
'positive' electrode (the top one in Fig. 5).
ion energy in the mirror. To evaluate Wj we apply the
The angle \p is found from the condition that over Pastukhov formulas [13]. In the case where 2p* > A
the time taken to revolve around the Larmor orbit the we obtain
displacement of the ion along z is less than d, i.e.
^jfi- 2 <> V ) ^ ^ (2 VS
Vi E0
(17)
(19)
where p? is the ion Larmor radius in the mirror. The A more accurate evaluation taking into account the
quantity v|| in the mirror (as in Ref. [18]) is given by dependence of Wj on 2p*/A takes the form
II = (l-exp[-(A/2pf)2]))
v
(20)
NUCLEAR FUSIDN, Vol.20, No.S (1980) • 585
BEKHTENEV et al.
Transfer of a-particle energy to the plasma
Alpha particles generated outside the loss cone
remain inside the trap and transfer energy to the
plasma by means of Coulomb collisions; a-particles
generated inside the loss cone move to the electrodes.
Like the ions escaping along z, they leave some of then-
rotation energy in the plasma, i.e.
(22)
0.25
where Wga = m V| /2. The energy WE<* is the
a 0
portion of the energy of rotation of the centre of
90 gravity of the DT system that is distributed, after the
thermonuclear reaction, between the a-particle and
FIG.6. Function I(ty).
the neutron.
In the reactor with 'skew' electrodes the a-particles
transfer some of their transverse energy to the electric
field (5W ) and, correspondingly, to the plasma, in the
a
same way as the ions. When considering this effect it
should be borne in mind that the longitudinal energy
In Eqs (19) and (20), it was assumed that of the a-particles is comparable with their transverse
<vl> = V£ /R. energy (as distinct from the case of the ions).
0
Electrons escaping from the trap are accelerated by This energy, after averaging over all a-particles
the electric field in the inter-electrode gap and reach generated (isotropic distribution function;W = const)
a
the positive electrode (see Fig. 5) with an additional and over all possible angles of collisions between
energy equal to W — eE A/2, from which we obtain the a-particles and electrodes, is given by
e
mean energy removed from the trap by the ion-electron
pair W:
(23)
4pf
w = w w ^
i + e
where W is the kinetic energy received by an a-particle
a
in the thermonuclear reaction. Figure 6 shows a plot
of I(i//). When calculating 5W we did not take into
a
277WEQ
X(l-exp[-(A/2pf)2]) -P)f/ = R (21) account effects associated with the discreteness of the
electrodes, since p > A. For a-particles the efficiency
a
with which the energy is transferred to the field is
The contribution of the secondary electrons to the much lower than for ions since Wa > WE<*/R- AS
energy balance in this system is fairly small. Indeed, is clear from this calculation, for Q > 50 (taking into
since the electrons escaping from the plasma reach the account the heating of the plasma by the a-particles),
positive electrode (see Figs 5 and 9), the secondary the reactor becomes self-sustaining, i.e. an emf sufficient
electrons generated by them cannot go into the plasma to sustain the reaction is created at the end electrodes
and, because of the retarding potential, return to the through the slowing-down of the a-particles in the
same electrode. A small number of secondary electrons electric fields.
generated on the end surface of the electrode (the
thickness of the electrode is A') enter the plasma and
are an additional source of electrons which differs from Re-injection
the main source of electrons by the factor 5* A'/(A + A');
6* is the secondary emission coefficient. This effect An ion escaping from the trap and recombining on
has not been taken into account in the calculations the end electrodes may then, with a fairly high degree
because it is assumed that A'/(A' + A) for the reactor of probability, return to the space occupied by the
can be of the order of 0.1, and 5* < 1 at T = 5-20 keV. plasma in the form of a neutral atom, re-ionize and heat
e
586 NUCLEAR FUSION, Vol.20, No.S (1980)
ROTATING-PLASMA REACTOR
up in the crossed fields. In this way, the D and T nuclei We can then evaluate the mean distance covered by
escaping from the reactor are collected and returned to the atom before it is ionized:
the plasma, i.e. there is a sort of 're-injection'. This
process may substantially improve the actual burnout 1 2X (z )( /R-l)
i 0 N
coefficient (c^) of the fuel, provided that the losses = Z0 11 + In (26)
2(R- /R)
N
of deuterium and tritium nuclei through incomplete
return to the plasma are less than the 'losses' due to
where Xj(z ) is the mean free path of the neutral atom
0
occurrence of the fusion reaction:
at the centre of the trap, which is determined by ioniza-
tion and charge-exchange processes (here, Xj(z ) < z ).
0 0
(24)
Applying Eq. (26), we find <p(z) and H(z)/H ; then,
0
substituting these values in Eq. (5) and assuming
Here a0 is the burnout coefficient without allowance v|| = 0, we determine the angular co-ordinate of the
for re-injection and K is the coefficient for return region of the phase space in which generation of the
of the fuel to the plasma. It is clear that in a ions is most probable:
steady-state reactor K -• 1 and, correspondingly,
Oil ""*" 1 on account of saturation of the walls by the
plasma ions. sin9(Zi) =
V Ifo)
Re-injection may affect the reactor parameters by ~
altering the shape of the plasma source function in the
phase space. If the range of the neutral D and T atoms For a reactor with n s 3 X 1013 cm"3, z s 60 cm,
0
returning to the plasma is shorter than the transition assuming that the mean energy of the neutral atoms
region between the electrodes and the central part of is s 100 eV (see Ref. [19]), we get zi s 30 cm. This
the trap (the dimension of the mirror), then the ion corresponds to an angular distance of the order of 20°
source associated with re-injection will be located in at R = 5 between the ion re-injection region and the
the phase space close to the loss cone and the para- 'loss cone' (A0). It should be emphasized that
meters nr and Q may vary appreciably by comparison although a mean free path of the returned neutral
with the K = 0 case. atoms can be short compared to z , the ions generated
0
Ions that are formed through ionization of returned during ionization of these atoms (at Zj < z ) are
0
neutral atoms are located in the phase space on a captured in the trap as well as the injected ions are
sphere with energy Wgo- The ion distribution on the captured in the trap centre because the ion potential
sphere may be found if the probability of neutral-atom well covers the whole volume of the trap including the
ionization along the length of the trap is known. area near the electrodes. The only difference from the
Unfortunately, the data available at present on the ions injected into the trap centre is that the injection
velocity distribution of the neutral atoms escaping from region in phase space shifts close to the surface of the
the wall as a function of the energy of the ions 'loss cone'. The energy-of these ions is of the order of
bombarding the surface are not sufficiently accurate, Wgo (see Section 1).
so we shall consider the simplest case, in which all the The calculation of nr, Ti made for the case in which
atoms entering the plasma are monoenergetic and move the ion source is in the phase space at different points
along the magnetic field lines. Let us assume that the relative to the plane v|| = 0 and the loss cone surface
ratio H(z)/H depends on z in the following way: (see Fig. 7) shows that these parameters do not strongly
0
depend on the position of the source. Thus the 're-
injection' makes its possible to attain a high degree of
(25)
burnout of the fuel with virtually no impairment of
the plasma parameters.
Here, z is read off inside the trap; H is the field at
o
the centre of the trap, and z is the length of the mirror. Plasma heat conductivity
0
The ion density and electrostatic potential are found
from the condition of quasi-neutrality: Heat losses due to the longitudinal heat conductivity
are absent in the system under consideration, in
n(z) = n exp(-e contrast to common mirror traps. This is explained
0
by the fact that the mean free path of the particles is
e viz) = W o(Te/(Ti + T ))(l - H /H(z)) here much longer than the system's dimension as well
e e 0
NUCLEAR FUSION, Vol.20, No.S (1980) 587
