P/2446 USA
Astron Thermonuclear Reactor
By N. С Christofilos*
1. INTRODUCTION scattering, they have a finite lifetime. Hence in order to maintain the E-layer, a continuous injection of The basic requirements of an ideal scheme of con- electrons from outside is required. fining a plasma at an adequate density and tempera- ture to produce thermonuclear reactions on a large 2. ESTABLISHING THE E-LAYER scale can be summarized as follows: (a) A magnetic field pattern must be established The E-layer is established within a long cylindrical whereby the magnetic lines are closed onto them- vessel (Fig. 1). In this vessel a magnetic field is at selves within a vacuum vessel, before injection of first established by means of external coils. The direc- any plasma within this pattern. In addition, it is tion of this field is substantially parallel to the axis of desired that this pattern of closed magnetic lines be the cylinder but converging at both ends in order to axially symmetric with no field components in the azi- reflect the electrons. It is desirable, mostly for mathe- muthal direction. Thus the drifts resulting from field matical convenience, that the gyration radius of the and pressure gradients are all in the azimuthal electrons, as they move along the E-layer, remain direction. Hence, no plasma loss can result from such constant although their azimuthal momentum varies drifts. Since this field pattern must enclose the current as a function of position. This condition is satisfied if distribution, which creates this pattern, the only way the vector potential of the external field obeys the to establish such currents is through organized motion equation of charged particles. (b) Besides providing a pattern of closed magnetic lines, means must be provided for ionizing neutral gas to establish a plasma and, thereafter, for heating this where k = 1.84/г«, r\ is the desired radius of the E- plasma up to ignition temperature. layer, 2L is the length of the E-layer, Ji(kr) is the We hope to meet the above requirements in the Bessel function of first order and a is a constant Astron reactor by a long cylindrical layer of relativistic determining the ratio between the axial and azi- electrons. This layer of rotating relativistic electrons, muthal momentum of the electrons far from the ends hereafter called E-layer, is the key feature of the of the E-layer. Astron concept. It not only performs the above- The electrons are injected at r = r¿, z = L with mentioned functions but, in addition, its presence is a almost zero axial momentum. However, at the point of necessary condition, as will be shown later, in order to injection the electrons encounter a radial component obtain an equilibrium solution of the plasma in the of the magnetic field. Hence, their azimuthal momen- steady state, such an equilibrium solution satisfying tum changes as they move towards the center plane of hydrodynamic, diffusion and electromagnetic equa- the tank. The ratio of the total momentum, p, to the tions. azimuthal momentum, pd, far from the ends is The relativistic electrons ionize neutral atoms and Pipe = 1 + ajoikrt). (2) thus a plasma can be established as soon as the pat- tern of closed magnetic lines is created. The condition The value of Bo is given by the equation for the latter is that the number of relativistic electrons in the E-layer exceed a certain critical number. Then *®° "~ ~~ZT n i „т (h*, vi ' W the relativistic electrons as they rotate inside the plasma lose energy by Coulomb scattering to the where mo is the rest mass and у is the relativistic mass ratio of the electrons. electrons of the plasma. If this energy transfer, which 2 2 is an energy gain for the plasma, is higher than the The constant a must be selected so that 2pe /p > 1 plasma losses by diffusion or other loss processes, the (far from the ends), otherwise the defocusing action of the end fields, in the radial direction, would lead to temperature of the plasma increases. Since the elec- 1 trons of the E-layer are continuously losing energy by unstable electron orbits. This method of injection provides the electrons always with a considerable axial * University of California Radiation Laboratory, Liver- momentum; this constitutes an axial pressure and more, California. prevents the collapsing of the E-layer, in the axial 279 280 SESSION A-9 P/2446 N. С CHRISTOFILOS
2446.1 ELECTRON 6UN OR ACCELERATOR -Е-LAYER
ЛАХЛЛЛЛЛЛЛЛЛЛЛЛ^
УУУ V YVW Figure 1. Astron reactor, schematic direction, from the contractive forces of its own the ionization process have acquired also a kinetic magnetic field. The electrostatic charge of the E-layer energy of a few electron volts. Thus as the ionization is assumed to be neutralized by positive ions. continues and the plasma is being established its Since the injection of the electrons must continue, initial temperature is not zero but a few electron volts. after a steady state is established, in order to maintain However, this temperature further increases as the the E-layer, a technique is required allowing external relativistic electrons lose energy, by Coulomb scatter- injection to a region where the magnetic field is con- ing, to the plasma electrons. stant in time. This is accomplished by a combination The energy loss of relativistic electrons is given by of standing and traveling waves whereby electrons the well-known Bethe formula injected from outside, periodically in short pulses, are 2 y = —4ттг пос1пА, trapped in the trough of a traveling wave, forming a е (5) ring of rotating electrons, and irreversibly injected in where no is the density of the plasma electrons and  the region where the E-layer is to be established. is the ratio of the maximum to minimum interaction Details of this injection method are beyond the scope distance. In Eq. (5) it has been assumed that the of the present general description and will be discussed velocity of the electrons is equal to с Then the energy in a later paper. gain per plasma electron is With each bunch or ring of electrons injected in the E-layer region the charge per unit length of the E- и = (b) layer increases. The rotating charges constitute a cur- rent which creates within the volume enclosed by the where Vo = шос2/е, NQ is the number of electrons per 2 E-layer a magnetic field in a direction opposite to cm length of the layer, a0 = 4тгге In Â, r0 is the the external magnetic field. Thus the net value of plasma radius, the magnetic field within this volume decreases con- 2 Cr° tinuously as the charge of the E-layer increases. When a = —5— nrdr (6a) the number of electrons in the E-layer, per cm length of the layer, reaches a critical value No equal to and no is the plasma electron density near the E- layer; и and Vo are expressed in the same units. No = y ¡re (4) The temperature of the plasma continues to rise as (where re is the classical electron radius), then the long as the rate of losses due to diffusion across the magnetic field inside the volume enclosed by the E- magnetic Hnes and other processes is lower than û. layer far from the ends is reduced to zero. Then the The rate of these losses in comparison with the energy field within the enclosed volume can be reversed by gain will be discussed later in Section 6 of this paper. increasing the charge per unit length of the layer a We observe, however, that the energy gain is pro- little more. At that moment the combination of the portional to the charge per unit length of the layer, external field with the E-layer field provides the which in turn, by Eq. (4), is proportional to the pattern of closed magnetic lines. relativistic mass ratio of the E-layer electrons.
3. INITIAL PLASMA FORMATION 4. PLASMA EQUILIBRIUM As soon as the pattern of closed magnetic lines is As soon as the plasma temperature starts to rise, established, concentration of plasma is possible within the plasma pressure also rises. Since the last closed this pattern. By injecting neutral particles within the magnetic line constitutes the boundary of the plasma, vessel, these particles become ionized as they travel the pressure is zero at this boundary. Hence a pressure through the E-layer region. The electrons liberated by gradient is created across the magnetic lines. Because ASTRON THERMONUCLEAR REACTOR 281 of this pressure gradient, the plasma starts diffusing magnetic field have been computed. The results are outwards, and furthermore a Hall current is created shown in the graphs (Figs. 2 and 3). In Fig. 2 a cross which crossed with В (the magneticfield) balance s the section is shown on a plane through the axis of sym- pressure gradient. This Hall current in turn modifies metry. The magnetic lines (which are also equipressure the pattern of closed magnetic lines. This modification, lines) are shown as well as the shape of the plasma at however, tends to enhance the intensity of the mag- the ends. The line where p = 0 is the last closed netic field; thus the pattern of closed magnetic lines magnetic line. Hence it constitutes the plasma boun- becomes denser as the plasma pressure rises. Of course dary. A line just outside this boundary line is open during the buildup of the plasma pressure the and it goes out along the axis. Two of these Unes are external magnetic field must be increased so that, far shown in Fig. 2. The plasma diffuses out, being guided from the ends, the following equation is satisfied: through these lines to form two collimated beams (one at each end). In this way the diverters which proved necessary in the stellarator do exist inherently in the Astron. The solution given above is continuous inside where p is the maximum plasma pressure, Be is the magneticfield a t the surface of the layer and Bo is the as well as outside the plasma except for two singular external magnetic field (also far from the ends). Then points at the intersection of the line p = 0 with the the question arises as to whether or not a self- axis of symmetry. Since there is no plasma outside, consistent equilibrium of the plasma can exist under the necessary current, as required by the vector these conditions. In order to solve this problem, I potential solution, will be provided by material coils. started with the assumption that in a cross section At this point it should be noted that since the plasma of the plasma, on a plane normal to the axis of sym- is diverted out through the open lines, for example metry and far from the ends, the magnetic field goes between the line p = 0 and the outermost line shown through zero within the plasma at an unknown radius in Fig. 2, with finite velocity, it follows that the pres- П < ro, where ro is the radius of the plasma cylindrical sure in this region, however small, is also finite and boundary. Further, I assumed that the magnetic flux not exactly zero. Consequently even in this region the through this cross section between r = r\ and r = ro plasma provides some current; thus the material coils is equal and opposite to the flux beftveen r = 0 and can be located beyond the outermost line shown in r = n; the Larmor radius has been assumed negligible Fig. 2. in comparison with the physical dimensions of the In Fig. 3, a cross section of the plasma on a plane system. Then, using hydrodynamic, diffusion and normal to the axis of symmetry (far from the ends) is electromagnetic equations, a class of solutions has been shown. In the upper graph (Fig. 3a) the plasma obtained for the cylindrical part, far from the ends. density is plotted as a function of the radius. We One of these self-consistent equilibrium solutions has observe that the pressure is zero at the axis of sym- been worked out for the whole volume including the metry, as well as at the boundary, whereas its maxi- ends. The detailed derivation of this solution is given mum occurs in the region of the E-layer. In the lower in Appendix 1. The solution for the vector potential in graph (Fig. Щ the magnetic field distribution is this case is shown. Since the magnetic field is reversed within the plasma volume, we have p = 8ттр/Во2 = 1 by A, = ), (8) definition. In addition, because of the fact that the maximum pressure is not at the axis of symmetry, the where quantity YJ, ¿We2¿' for r < r%, which is proportional to the rate of thermonuclear for n < r < ro,
Bo is the external magnetic fieldfa r from the ends and Л is a parameter. This solution requires at r = r< afield discontinuit y Ar 2 г 2 or a field jump from -£O0~ * to +В0е-* г . This discontinuity can be realized only by the presence of a sheath of rotating charged particles where, within the thickness of the sheath, the field goes through zero. The presence, for other reasons, of the E-layer allows the existence of this solution. However, as it turned out, the investigation of this equilibrium solution could have led, eventually, to the discovery of the E-layer, had not this layer been postulated long before the above mathematical solution was obtained. о 2 z+L For a numerical example, the shape of the plasma, 2446 2 ^AXIS OF SYMMETRY the distribution of the plasma pressure and the Figure 2. Magnetic field geometry 282 SESSION A-9 P/2446 N. С CHRISTOFILOS reactions, is about three times higher than in a para- of a Poynting vector along the magnetic lines or bolic distribution. В Р = 0. (12) In the above solution V • (pv) is different from zero. Actually, V-(/>v) equals — S in the region 0 < r < Y% Since in the first case the Hnes are tilted and this can and +5 in the region Y i < r < г о. If we now assume be done at the expense of the energy of the perturba- that all the diffused particles are replaced by ionized tion, it seems that in the second case the perturba- neutral atoms in the region of the E-layer, we might tions are more dangerous. Consequently we confine question how the sources and sinks required by the most of our attention in the present paper to this solution would be realized. However, on closer exam- second case. ination we observe that these sources and sinks are Let E and H be the perturbed electric and magnetic automatically mutually cancelled or fields, respectively. Then Eq. (12) can be written PB = ExHB = E HxB = 0. (13) Srdr = 0. (10) Jo Equation (13) can be satisfied if E = 0. However, one finds that this condition is possible only in trivial Since no actual sources or sinks are present within the motions of the plasma along the lines or in trivial plasma, what actually will happen is this: Inside (in rotation of the plasma. Hence the case which makes the region 0 < r < r<), because of the lack of sinks the sense is density will tend to rise at a rate S. However, along BxH = 0. (14) the same magnetic line outside the E-layer the density will tend to decrease at the same rate. This rate of This case has been treated in normal mode analysis as density change is very slow and it is negligible during described in Appendix 2. For the purpose of this the time required by the particles to travel a complete analysis we assume that the perturbation is of the circuit along a line. As a result of this density change a type small pressure gradient is created along the lines and q = q(r, z) sin(me)e^, (15) the particles which are superfluous inside appear as a where $ is any of the perturbed quantities. Then it source on the same line outside. Consequently particles follows from th£ linearized hydromagnetic equations vanish from the inside region, thus creating virtual that sinks, and reappear outside as sources.
EQ = - ф(гАв) с (is) 5. PLASMA STABILITY and v — rf{rA ) sil Having now ascertained that an equilibrium solu- Q Q (16a) tion exists, we shall examine the stability of this where EQ and ve are the electric field and perturbed equilibrium solution with respect to small perturba- velocity, respectively, in the azimuthal direction, and tions. We can divide the possible perturbations into /and ф arbitrary functions of the flux ф = гАв. From two classes. further investigation (see Appendix 2) it was concluded 1. Perturbations where the magnetic lines are tilted that Ee and f(rAe) are zero at the axis of symmetry. or where Alfvén waves are propagated along the Since rAe is constant along a magnetic line, this magnetic lines. In this case the perturbation is means that EQ — 0 at any point in the boundary sur- characterized by a Poynting vector, P, traveling face. Thus one can observe that this kind of perturba- along the magnetic lines. This condition can be tion is an internal motion where the surface remains at expressed as rest. This is due to the fact that all boundary lines return through the axis of symmetry. As we move far B-P^O. (11) from the ends, the flux rAQ degenerates very soon to a 2. The second class is characterized by the absence function of the radius only. Then the same holds for/ and ф, and the perturbation degenerates to one of the X axis type q = q(r) sin(m0)¿?4 This perturbation has been investigated in more detail. If we define a new quantity w = p + (B • Н/4тг), where p is the perturbed pressure, then from the linearized equations we obtain (see Appendix 2) a second-order differential equation for w:
where p is the equilibrium density and
Figure 3. Density (a) and field strength (b) vs. radius the sound velocity of the medium. ASTRON THERMONUCLEAR REACTOR 283
Equation (17) is valid for any perturbation where instability and perturbations in general, where the the perturbed velocity is not constant in time. In the magnetic lines move parallel to themselves. These are latter case w = 0. However, the plasma is subject to considered as the most dangerous perturbations and rotation since angular momentum is continuously it is concluded that at least against such perturbations transferred from the E-layer to the plasma electrons. the Astron geometry is stable. This angular momentum cannot be transferred to the wall because the magnetic lines are closed onto them- 6. ENERGY BALANCE DURING THE PLASMA selves. Hence the pattern of closed magnetic lines HEATING PROCESS rotates together with the plasma. The angular momentum of the plasma is transferred outside as the The main energy losses of the plasma are diffusion plasma particles diffuse towards the open magnetic and bremsstrahlung. The first is predominant at low lines. The plasma angular momentum bears the same temperature (less than a kilovolt) whereas the second proportion to the angular momentum of the E-layer is the more important above ten kilovolts. The rate of as the mean lifetime of a plasma particle bears to the diffusion depends on the pressure and field distribution mean lifetime of an electron of the E-layer. Because of throughout the plasma, which in turn depend on the this rotation it makes no sense to consider velocities particular equilibrium solution selected. The solution constant in time, for any change of radial position discussed in Section 4 has been selected as being means change of the centrifugal force. However, simple and helpful in obtaining a clear picture of the otherwise this rotation is not taken into consideration plasma equilibrium. However, as an actual solution it in the stability calculations. is somewhat unrealistic. The reason is that it has been Equation (17) has been thoroughly investigated and assumed that the magnetic field just inside the E-layer is equal to the magnetic field just outside; this is not it has been found that in the case Ев = 0 at two dif- ferent values of r the admissible eigenvalues of a>2 are actually possible as it implies infinite charge in the always negative, which means stability. From the E-layer. Furthermore, it has been assumed that all the solutions of Eq. (17) one can compute all the new ions are produced in the region of the E-layer. perturbed quantities and then invert these solutions This is correct only at the very beginning of the plasma formation. Consequently, in order to calculate diffusion to functions of rAe. Thus, the general solutions can be obtained for all the perturbed quantities for the losses, a solution for the equilibrium is required with- type of perturbation В x H = 0. Thereafter one can out the above restrictions. write down the linearized hydromagnetic equations The following calculations are not considered in curvilinear coordinates and insert the solutions accurate; their purpose is to show the possibility of obtained into these equations. Then either the equa- heating the plasma with the relativistic electrons up tions are not satisfied, which means that perturbations to ignition temperature and not to compute the para- of the type В x H = 0 are not possible at all in the meters of an actual machine. In view of these approxi- Astron geometry, or they are satisfied, which means mations and considering that the cylindrical part of that the perturbation is possible but only stable the plasma is very long, the diffusion loss from the modes are admissible. Consequently, the conclusion ends is neglected. Thus in what follows we consider, as is that perturbations of type BxH = 0 either are far as diffusion is concerned, the plasma as being an possible in the Astron geometry and are stable or do infinite cylinder. not exist at all. Hence we are left with perturbations of In an equilibrium solution satisfying the above the type conditions, as well as the conditions considered in Appendix 1, the vector potential Ae and plasma BP^O. pressure^) are: In order to investigate this perturbation the linearized (a) Region 0 < r < equations have been written in curvilinear coordinates. One then would observe that terms are added in the (18) equations of motion resulting from the mechanism of the Alfvén waves, and their contribution is stabilizing. However, wherever the magnetic lines are concave - smh*[A(r* - (19) towards the plasma they contribute to instability. Direct normal-mode solutions of these equations (b) Region r% < r < appear extremely difficult if not impossible. Con- sequently, it appears that only through an energy Bu principle would it be possible to evaluate this class (20) of perturbations. The Princeton energy principle2 appears to be the most convenient for this investiga- tion. However, some adaption to the Astron geometry p = ^{ш is required. This problem is now under investigation (21) and the results will be reported later.
Finally I would like to note that the first class of where Bw and В i are the magnetic field just outside perturbations (B x H = 0) includes the so-called flute and inside the E-layer, respectively; TQ and r% the 284 SESSION A-9 P/2446 N. С CHRISTOFILOS plasma boundary radius and E-layer radius, respec- where cos S is the ratio of the azimuthal to the total tively, and momentum of the E-layer electrons. The plasma tem- perature can rise to any desired value as long as (22) W > ccW (30) L e a> where a is the ratio of the rate of the plasma losses (23) from all possible processes to the rate of diffusion. The rate of diffusion is proportional to F, which in and fto is the maximum value of the plasma pressure. turn is a function of the pressure. Assuming щ fixed, The diffusion velocity of the plasma is we observe that for у <^ 1 the quantity F is propor- 2 tional to the square root of the pressure. This condi- m0c Vp v = —• (24) tion (y <^ 1) is met at the very beginning of the plasma formation. Thereafter, as the plasma pressure where n is the plasma density. After substituting increases, F reaches a maximum, and for further ¿2/ 2 re = WQC and the value of Vp/B* from Eqs. (18) pressure increase the quantity F (and hence the through (21), Eq. (24) becomes diffusion loss) decreases almost as l/po. For BW\B% =10 this maximum occurs at y — 1, and jPmax = 2.75. (24a) Since the diffusion loss goes through a maximum for fixed plasma density, this density can be calculated This equation is valid in both (inside and outside) from Eq. (27) for F = 2.75. In view of condition (30) regions. We are interested in the energy loss due to and Eqs. (27), (28) and (29), the plasma density must diffusion at r = ro. This is be less than the value given by the relation
Wa = 27rro-2eu-v(ro), (25) no < — (31) where и is the plasma temperature. The time, т, between collisions is not constant 2 2 For Bw\Bi = 10 we have s (s + l) = 7.5, so that throughout the plasma since it is dependent on the Eq. (31) becomes density; however, we consider in the following the 2 y4 COS! S minimum value of т, which occurs at n = щ. This (32) value of r is We observe that the allowed density increases very (JL\ (26) \VQJ fast with the value of y. The temperature where the maximum diffusion loss occurs is where c± & 1. Substituting in Eq. (24a) and from 2 Eq. (25) we calculate the energy loss per cm length Щ > Vo{Fmax/sy) . (33) of the plasma due to diffusion: For Bw/Bi = 10 we have 5 = 1.22, and Eq. (33) becomes Wu = ^ {167rnoeVo)ï ergs/cm sec, (27) щ > 2.5 x 106a2/y2 ev. (33a) where For a numerical example we assume F = , cos2 S y = 100 and • = Ю-5- y = Then and ^o is the density of the plasma electrons in the 14 2 n0 <2x 10 /a , region of the E-layer. 2 The energy transferred to the plasma by Coulomb щ > 250a ev. scattering with the relativistic electrons is The two other main loss processes are the excitation loss and bremsstrahlung. However, the first one is We — eVoNarocno ergs/cm sec, (28) rather negligible in the present case. This loss results where N is the number of electrons per cm length of from orbital excitation of partially ionized atoms of the E-layer. This can be written high atomic number contaminating the plasma. In the present case this loss is small if a moderately good (28a) N = sy/re, vacuum can be provided. For example, for y = 100 a where contamination influx of about one micron-liter per B + В i second is tolerable. w (286) s = The bremsstrahlung loss per cm length of plasma is and ergs/cm sec, (34) Wb By) = (29) n where и is expressed in esu and r¡ is defined in Eq. (9). ASTRON THERMONUCLEAR REACTOR 285
We shall now evaluate this bremsstrahlung loss for 7. EXPERIMENTAL PROGRAM the values of no and щ as given in relations (32) and Although the concept of the E-layer for confining (33), at the temperature where the maximum diffusion and heating the plasma had been proposed several loss occurs, namely, at у = 1, or p = Хб^повщ = B 2. w years ago (early 1953), several features of the E-layer Substituting this value in Eq. (34) we obtain as described above were not well understood at that time. Consequently, only after almost three years of
Wb = 2.6 x 10-24^1^» фг#и (35) theoretical work, the Atomic Energy Commission decided that an experimental program was warranted Then from Eqs. (27), ) and (35) we can derive the on the Astron proposal. This program started early in value of a: 1957. The first purpose of the Astron group was to design a model where it would be possible to demon- 24 strate the feasibility of the basic feature of the Astron 2.6xlO- FoV 2 a= 1 + e 'JL\ cos S concept; namely, that the E-layer can be established and can create the pattern of closed magnetic lines. (36) The electron energy for this model will be of the order or of 3 Mev. Because of this low energy no plasma is expected to be established in this first model except (37) under exceedingly good vacuum conditions. The electrons are provided by means of an electron gun 2 The parameters of the numerical example give a « 1.2. capable of producing an electron beam of 1 Mev Thus, we conclude that in order to raise the tempera- energy and a peak current of 100 amp under pulsed ture beyond the value where the maximum diffusion conditions. This electron gun is now undergoing tests. 14 loss occurs the allowed density is of the order of 10 . The other components of the model have been designed As the temperature rises above this value the brems- and will be constructed as needed for the program. strahlung loss increases whereas the diffusion loss After completion of the tests with the low-energy becomes less important. Thus, at very high tempera- electrons, and provided that the E layer can be ture we shall compare the energy gain We with the established, it is planned to increase the E-layer bremsstrahlung loss only. Then Eqs. (28) and (34) electron energy in successive steps so that, eventually, yield positive power gain would be demonstrated. Finally the possibilities of the Astron concept lead- (38) ing to an economical power reactor will be briefly discussed. An engineering group started several where и is expressed in kev. Substituting in this months ago to investigate possible designs and para- equation the numerical values and и = 100 kev the meters of an Astron power reactor under the assump- value of the density is tion that the basic principles are sound. The results of this study, thus far, are that it appears to be feasible to build and operate an Astron power reactor n0 < 3x1014. competitively with conventional power plants. It Thus we can conclude that, provided the energy of should be noted that the particular components the E-layer electrons is high enough (about 50 Mev), associated with the acceleration and injection of the we can maintain a plasma density of 1014 at any electrons in the E-layer constitute less than 15% of desired temperature. This can be done without the the total cost. The balance of the system involves help of fusion energy; thus, it is possible to heat a high- familiar equipment such as turbogenerators, magnets, density plasma up to a temperature much higher than power supplies and switchgear, for which the costs are the ignition temperature. more or less amenable to reliable computation. This In addition to satisfying the requirements of plasma engineering study was carried out for an Astron power heating, the distribution of high energy electrons must "reactor with the following parameters: provide for equilibrium of the E layer. This requires the E layer to be longer than the active plasma vol- E-layer electron energy 50 Mev ume ; i.e., the pattern of closed magnetic lines. As E-layer radius 50 cm the E-layer electrons enter this pattern their azimu- Plasma radius 70 cm thal momentum changes in proportion to the change External magnetic field 40,000 gauss of vector potential, along the E layer, from the tip of Length of the E layer 30 m the plasma to the region of highest pressure. Conse- Diameter of the reaction tank 150 cm quently the value of у must satisfy the condition Plasma temperature 25 kev Net electric power output 500,000 kw Bon У > (39) The above results must be considered at this time as being only indicative, although encouraging, for For a power reactor this value is of the same order as many difficulties are anticipated in the effort to in the above numerical example. materialize the Astron concept to a power-producing 286 SESSION A-9 P/2446 N. С CHRISTOFILOS thermonuclear reactor. Consequently, one must antici- radius of the plasma), whereas 2L (the length of the pate that many years will pass before this ultimate plasma) is assumed to be at least 20r0. Thus the second goal is achieved. function vanishes very fast as one moves from the end inward, parallel to the z axis. Consequently in this APPENDIX 1 internal region the magnetic lines are parallel to the As mentioned above, the vector potential governing axis of symmetry. It is then possible to derive first the the equilibrium of the plasma, in the proposed reactor, solution for the infinite cylinder, namely the function should satisfy the requirement within the plasma that ф(г) only, and thereafter to modify the solution by adding the ¿-dependent function. In the cylindrical part (far from the ends) we observe that the pressure BzdF = О on any plane z = constant, where z is the axis of ,_*££ ,.„ symmetry and by F is understood the area which, on (А any such plane, is occupied by the plasma. The pressure is constant along a magnetic line as well The equations that should be satisfied by a steady- as the function ф = rA. In order to satisfy the basic state equilibrium solution are requirement that (Al.l) Г r B = VxA, Bzrdr = 0 Jo (A1.2) —J = VxVxA, it is obvious that in a region where the radius is less than a value ri, the value of В is positive and in the (A1.3) remaining region (where r% < r < ro) the value of В is negative. M From Eq. (A1.7), solving for B, we obtain } 9 9 (A1.4) ¿ fna) rp 2 Bz= ±(Во -8ттр)К (A1.7a) (A1.5) which indicates that the same line has a value 2 and for a steady state + (Bo —8ттр)% in the region where r < r% and the value — (2?o2 — 8ттр)% in the regions r% < r < ro- = V-(pv). (Al. Sa) From В = VxA, Eq. (Al.l), we have The equilibrium solution is assumed symmetric d(rA) B = (Al.la) about the z axis; the coordinates are cylindrical: z rdr ' г, 0, z; the vector potential A and the current j have Let components only in the azimuthal direction. in the region r < г%, Further definitions are: фо = rAo in the region r% < r < В = the magnetic field having components in For every value of r = r , where ф = фг(г ) in the the radial and z directions, m т internal region, there corresponds a value rn in the ex- p = the scalar pressure of the plasma, ternal region, where r is determined from the equation p = the mass density of the plasma, n v = the diffusion velocity, ФОЫ = ФГЫ, (А1.8) M, m = the ion and electron mass respectively, where rm < r\ and r% < rn < ro. r = the mean time between Coulomb collisions, From Eqs. (A1.7) and (Al.la) it follows that дф(г ) т (Al.8a) the electron gyrofrequency, and rndr S = the strength of sources or sinks within The above conditions, (A1.8) and (A1.8a), are satisfied the plasma volume. for any value of r if фо and ф% are solutions respectively It is assumed that diffused particles are replaced by of the differential equations neutral atoms at the region of highest pressure. Con- sequently, within the plasma volume the condition -1£-гфо = 0, (Al.9)
SdV = 0 g-.-г* = 0. (Al.9a) should be satisfied by the solution. The solution for the vector potential is assumed to be of the form Solutions satisfying these differential equations are 2 ф0 = сое*? , (Al. 10) = ф(г)+Дг).—JV^> (Al .6) v ; cosh(^L) ф± = Cle-*r\ (Al. 10a) and k to be of the order of ro'1 (ro is the boundary Taking into consideration that the field should ASTRON THERMONUCLEAR REACTOR 287 decrease as one moves away from the axis of sym- The strength of the required distributed sources inside metry, we observe that the second solution corresponds the plasma are (assuming т constant) to the inside region. At r = Yo, p — 0 and fi = — Bo, then — = — So for Y < Yi, (A1.17) (Al.ll) and the second equation becomes for Yi < Y < Yo, (Al. 17a)
= -^*-лг* (Al.lla) and Yi* = 0. At г = ri it is obvious that Jo Srdv = -So-^ Фо = Фи The complete solution with the ¿-dependent function is whence assumed in the form П2—ro2 = —ri2 or Y i = го/л/2. The magnetic field is B = Вое-*2 iovr
c1k1Jo{kiYi)+C2hJo{k2Yi) = 0; Be = 2B0e-^i\ (A1.12b) (3) atr = r<, The only way to create such a field jump at Y = Y% is dp ¡BY = 0 (since B = 0). to provide a current sheath at that radius. Since this z Then current sheath is inside a high-temperature plasma, 2 such a current can only be created by an organized f° d£ f °° . Во I 5— dz = — I JBYUZ = -Q— motion of charged particles. Hence the relativistic JL àZ Jo O7T electron layer will fulfill this requirement and create (4) at Y = n% * = b, the field jump. Let Л = *e/2r,. (A1.13) where € = k/ke. Then Furthermore, the arguments k\Yi and U2YÍ should be (A1.14) selected so that Br is always finite for every value of Y between zero and Yo except Y = 0 where it becomes and zero. The complete solution for keYi = 10 and e = 0.184 is: (A1.15) ke Ho) where j is the current per cm of the electron layer. e = 5.435 x [1.666/I(2.12A;) +2.136/i(5.385*)] The rate of diffusion is „_|_05(я2-2) for Y > Y Mme2 cosh(1.84L/f¿) = a if
(A1.16) B0Yi = 5.435 x [1.665/i(2.12*) +2.136/i(5.385#)]
x = a for Y < (A1.16a) cosh(1.84i/ri) 288 SESSION A-9 P/2446 N. С CHRISTOFILOS where x = r¡r%9 a = 1 at the boundary line where In the present appendix our attention will be restricted 2 p = 0, and a = 0 for ^ = ^o = B0 ¡8TT. The field to the second category. configuration and the plasma shape have been com- puted from the above numerical example and plotted Equations, Assumptions and Approximations in Fig. 2. The equations of the problem are The solution is continuous for a > 1. This implies that the current distribution as given by said vector potential is finite beyond the plasma boundary where no plasma exists to create such a current as required by the vector potential. E+vxB = О, (A2.3) Consequently, in order that the above solution can be physically realized, material coils should be placed VxE = --H, (A2.4) beyond the plasma boundary. Such coils should be energized in such a way that the current distribution с inside these coils is as given by the vector potential. (A2.5) At a certain distance (towards negative z) from the plasma boundary, the vector potential function should dp dp be substituted by another function of the form (A2.6) pdt A = !+V.(pv)=O (A2.7) so that the vector potential vanishes at infinity. ( As mentioned above, the solution given here has been obtained in the frame of certain approximations. where B, p, p are the equilibrium (undisturbed) values This approximation breaks near the boundary within of the magnetic field, plasma pressure, the last few Larmor radii. and plasma density, respectively, In a further approximation there should be taken E, H are the perturbed values of the electric and into consideration that: magnetic field, respectively, j is the Hall current, (1) the mean time between collisions, r, is propor- J is the perturbed current, tional to I//), p} p are the perturbed values of the plasma pres- sure and density, respectively, and v is (2) the temperature decreases as the electrons the perturbed mass velocity of the approaching the boundary and encountering higher plasma. field lose energy by radiation, From Eqs. (A2.6) and (A2.7) we eliminate др/dt and obtain (3) the Hall current density at the boundary line goes to zero, since there are no particles there to (A2.8) create such a current. A further approximation in the cylindrical part can and we introduce a new variable be expressed in the form w = (A2.9) ф = rA = 2 c e-^nñn for 0 < r < n, г t n From the system of six Eqs. (A2.2), (A2.3), (A2.4), 2 (A2.5), (A2.8) and (A2.9) we can determine the six ф0 = гА0 = 2 cne+ttn(r -robr for rt < г < r0 п = ъ unknown quantities
and accordingly, thereafter, to determine the z- v, ф, w, H, E, j. dependent functions. The plasma has been assumed of infinite conductivity insofar as the ohmic term (not shown here) in APPENDIX 2 Eq. (A2.2) is concerned, so that this term can be con- sidered negligible. However, Coulomb collisions are In the investigation of the stability of the equi- assumed to bring about a Maxwellian distribution in librium state of the plasma, I classify possible per- plasma, and their frequency is assumed to be such as turbations into two categories, the criterion being to allow the consideration of the stress tensor to be whether or not a Poynting vector (P) travels along the isotropic. magnetic lines. This can be expressed as The solutions are assumed in the form
(1) PB^O (Alfvén waves) (A2.1) q = q(r, z)e*mo+*°*, (A2.10) (2) P.В = 0 (A2.1a) where q is any of the perturbed quantities. Further it ASTRON THERMONUCLEAR REACTOR 289 is assumed that \co\ < eB/Mc, where M is the ion (rAe) being practically a function of r only in most of mass. the plasma volume, far from the ends. The expression (A2.1a) can be written Consequently the perturbed quantities vQ and EB degenerate in this region to a function of r only, and ExH В = E HxB = О, (А2.11) in general the perturbation is degenerated to the form which implies ime+ùit E = 0 (A2.11a) q = q(r)e . or Since rAe is constant along a magnetic line, it follows HxB = 0. (A2.«llb) thai f (rAB) and ф(гАе) are constants along a magnetic The first case yields only trivial motion along the line. Then if we find the solutions of Eqs. (A2.2), magnetic lines or trivial rotation of the plasma and (A2.3), (A2.4), (A2.5), (A2.8) and (A2.9) in the region is not of interest. Consequently the case which makes where rAB is practically a function of r only, then the sense is the second, namely values of Ee, vB and the other perturbed quantities can be easily obtained for the whole volume. Con- HxB = 0. sequently it is enough to solve the system of the six Expanding this equation we obtain Eqs. (A2.2), (A2.3), (A2.4), (A2.5), (A2.8) and (A2.9) for a perturbation of the type
Hr Br (A2.12) q = q(r)eime+iü)t. Hz and This form of perturbation, after eliminating from H = 0. (A2.12a) e the system of six equations the quantities p, j, h and Combining Eqs. (A2.12) and (A2.12a) with Eqs. E, yields the equations (A2.3) and (A2.4) we obtain two sets of equations: 3w ptov = -—, (A2.17) 8E r r (A2.13) 8z pcovB = — г— w> (A2.18) Er = —vQBZ} (A2.13a)
Ez = veBr,^ (A2.13b) (А2Л9) which, after elimination of Er and Ez, yield the partial differential equation where (2^+_L\ = S2, where S is the velocity of (A2.14)
sound in the medium. By eliminating vQ from Eqs. (A2.18) and (A2.19), we obtain tlr &r (b) ^ (A2.15)
£lz XJZ 0 { Е А2Л5а ^Hr = |§-^- = т ^-^ ( ) Finally from Eqs. (A2.17) and (A2.20) by eliminating
vr we obtain _сон _ 3(гЕв) 3Er _ 3(rEe) . m £
(A2.15b) which in turn yield the partial differential equation (A2.21)
3Ee 3Ee Ee From Eqs. (A2.14a), (A2.16a), (A2.17a) and (A2.18) ^r-^r-r^z-^- = —Яг—9 we have The general solutions of the partial differential (А2.21а) equations (A2.14) and (A2.16) are, respectively,
ve = rf(rAe)efi™°+h>*, (A2.14a) • - w = ptorf(rAe)ei™°+t«>t. (A2.21b)
EQ = -ф(гАву™°+^, (A2.16a) The last two equations impose restrictions on the behavior of w along the radius. Our task now is to determine boundary conditions satisfying these where / and Л *» m* " (A2.22) yielding solutions By this substitution we not only simplify n Eq. (A2.26a) but also we can represent piecewise, by w oc r , (A2.22a) appropriate selection of the order and argument of the where modified Bessel functions, any density distribution n = (A2.23) where the density is zero at the axis of symmetry, and at a boundary radius r = r^. From Eq. (A2.17) we have Then Eq. (A2.26a) becomes 1 d\ (A2.22b) _ 0.25 — ОС \u = 0. дг (A2.30) Since vr should be finite at the origin it follows that 2 du n-3 = (ж + 1)*-2 > 0 (A2.24) At r = 0, we have и = — = 0 (from Eqs. (A2.22a), от or 2 (A2.24a) and (A2.25)). m > 3. J From Eqs. (A2.20) and (A2.25) we have Thus the modes m = 0 and m = 1 are excluded. The first admissible mode is m = 2, resulting in дт n > 3.26, (A2.24a) The function 8(туг)/дт is alternating in the region 0 < T < To, as and by Eq. (A2.21a) If we now assume that w2 is positive, then by л to *V = 0, — at Y = Eq. (A2.31) и should be an alternating function. How- ever, for positive io2, it results from Eq. (A2.30) that и Now we have established the boundary conditions and is a function starting from zero at r = 0 and increasing we can proceed to determine the admissible eigen- monotonically with increasing radius, as d2u/dr2 values of со2 in Eq. (A2.21). remains always positive. This contradicts the require- By changing the variable w to и where ment imposed by Eq. (A2.31). Consequently, solutions are possible only for nega- и = {r/p)*w, (A2.25) tive eigenvalues of со2, indicating that all the perturba- Eq. (A2.21) becomes tions of the type HxB = 0 л о "™ \ rtn I "^^^ are positively stable. An investigation of the lowest possible value of \o)\ gave M > Sofa, which can be written as where So2 = yp/p, the sound velocity in the plasma in the absence of a magnetic field. /CO REFERENCES (A2.26a) where 1. J. Killeen, Orbital Stability of the Electron Layer in the Astron, University of California Radiation Laboratory, UCRL-5101 (24 January 1958). 2. I. B. Bernstein, E. A. Frieman, M. D. Kruskal and R. M. Kulsrud, An Energy Principle for Hydromagnetic Stability (A2.27) Problems, Princeton Univ., NYO-7315 (4 March 1957).