Summary of UCRL P/rotron (Mirror Machine) Program · peak of the mirror. This condition may also be...

P/377 USA

Summary of UCRL P/rotron (Mirror Machine) Program"

By R. F. Post*

Under the sponsorship of the Atomic Energy Com-mission, work has been going forward at the Universityof California Radiation Laboratory since 1952 toinvestigate the application of the so-called " magneticmirror " effect to the creation and confinement of ahigh temperature plasma. By the middle of 1953several specific ways by which this principle could beapplied to the problem had been conceived andanalyzed and were integrated into a proposedapproach which has come to be dubbed "The MirrorMachine".1 Study of the various aspects of the physicsof the Mirror Machine has been the responsibility of aLaboratory experimental group (called the"Pyrotron" Group) under the scientific direction ofthe author. During the intervening years of effort,experiments have been performed which demonstratethe confinement properties of the Mirror Machinegeometry and confirm several of its basic principles ofoperation. There remain, however, many basic quan-titative and practical questions to be answered beforethe possibility of producing self-sustained fusionreactions in a Mirror Machine could be properlyassessed. Nevertheless, the experimental and theoreti-cal investigations to date have amply demonstratedthe usefulness of the mirror principle in the experi-mental study of magnetically confined plasmas. Thisreport presents some of the theory of operation of theMirror Machine, and summarizes the experimentalwork which has been carried out.

PRINCIPLE OF THE MAGNETIC MIRROR

The modus operandi of the Mirror Machine is tocreate, heat and control a high temperature plasma bymeans of externally generated magnetic fields. Themagnetic mirror principle is an essential element, notonly in the confinement, but in the various mani-pulations which are performed in order to create andto heat the plasma.

Confinement

The magnetic mirror principle is an old one in therealm of charged particle dynamics. It is encounteredfor example, in the reflection of charged cosmic rayparticles by the earth's magnetic field. As here to be

understood, the magnetic mirror effect arises when-ever a charged particle moves into a region of magneticfield where the strength of the field increases in adirection parallel to the local direction of the field lines,i.e., wherever the lines of force converge towardeach other. Such regions of converging field lines tendto reflect charged particles, that is, they are " mag-netic mirrors ". The basic confinement geometry ofthe Mirror Machine thus is formed by a cage ofmagnetic field lines lying between two mirrors, so thatconfiguration of magnetic lines resembles a two-endedwine bottle, with the ends of the bottle defining themirror regions. This is illustrated in Fig. 1, which alsoshows schematically the location of the external coilswhich produce the confining fields. The central, uni-form field, region can in principle be of arbitrarylength, as dictated by experimental convenience orother considerations. For various reasons, it has beenfound highly desirable to maintain axial symmetry inthe fields, although the general principle of particleconfinement by mirrors does not require this.

The confinement of a plasma between magneticmirrors can be understood in terms of an individualparticle picture. The conditions which determine thebinding of individual particles between magneticmirrors can be obtained through the use of certainadiabatic invariants applying generally to themotion of charged particles in a magnetic field.

The first of these invariants is the magnetic mo-ment, p, associated with the rotational component ofmotion of a charged particle as it carries out itshelical motion in the magnetic field.2 The assumptionthat ¡i is an absolute constant of the motion is notstrictly valid, but, as later noted, it represents a verygood approximation in nearly all cases of practicalinterest in the Mirror Machine.

The magnitude of /i is given by the expression

¡j. = Wx/H = ^mvx2/H ergs/gauss (1)

* Radiation Laboratory, University of California, Liver-more, California.

which states that the ratio of rotational energy tomagnetic field remains a constant at any point alongthe helical trajectory of the charged particle. In thiscase the magnetic field is to be evaluated at the Une offorce on which the guiding center of the particle ismoving.

Now, at the point at which a particle moving towarda magnetic mirror is reflected, its entire energy of

245

246 SESSION A-9 P/377 R. F. POST

COIL

FIELD CONFIGURATION

A

DIRECTION OFFIELD LINESAT A-A

FIELD STRENGTHS

Figure 1. (a) Schematic of magnetic field lines and magneticcoils, (b) mirror loss cones, (c) axial magnetic field variation

motion, W, is in rotation, so that at this pointцН = W. The condition for binding charged particlesbetween two magnetic mirrors of equal strength istherefore simply that for bound particles

/¿Ям > W (2)

where Ям is the strength of the magnetic field at thepeak of the mirror. This condition may also be ex-pressed in terms of field strength and energies throughuse of (1). If Ho is the lowest value of the magneticfield between the mirrors, and Ям is the peak value atthe mirror (both being evaluated on the same fluxsurface) then (2) becomes

- WjW±(0) < RM (3)

where Дм = Ям/Яо (the "mirror ratio") andW± (0) is the value of W at the point where Я takes onthe value Щ.

It is to be noted that the condition that particlesbe bound does not depend on their mass, charge,absolute energy, or spatial position, nor on theabsolute strength or detailed configuration of themagnetic field. Instead, it depends only on the ratiosof energy components and magnetic field strengths.Such an insensitivity to the details of particle types ororbits is what is needed to permit the achievement of aconfined plasma; i.e., a gas composed of chargedparticles of different types, charges and energies.The simplicity of the binding conditions reflects theinsensitivity to detailed motion required to accomplishthis.

Further insight into the nature of confinement of a

plasma by mirrors can be obtained by manipulation ofthe reflection condition. By differentiation, holdingthe total particle energy constant, it can be shownthat a particle moving along a magnetic line of force(or flux surface) into a magnetic mirror always ex-periences a retarding force which is parallel to thelocal direction of the magnetic field and is given bythe expression3

" • dynes, (4)

where и is the axial distance from the plane ofsymmetry.

This force, which is the same as that expected on aclassical magnetic dipole moving in a magnetic fieldgradient, represents the reflecting force of the mirror.Integrated up to the peak of the mirrors, (4) yieldsagain the binding condition (3).

Since Ц has been assumed constant, (4) may alsobe written in the form

Fu = - (5)

This shows that the quantity цН acts as a potentialso that the region between two magnetic mirrors ofequal strength lies in a potential well between twopotential maxima of height /¿Я т а х.

Loss Cones

It is evident, either from (2), (3) or (5), that it isnot possible to confine a plasma which is isotropic inits velocity conditions—one in which all instantaneousspatial directions of motion are allowed—between twomagnetic mirrors: confinement of an isotropic plasmais only possible to contemplate in special cases wheremultiple mirrors might be employed. This limitationcan be understood in terms of the concept of losscones. The pitch angle, в, of the helical motion ofparticles moving along a Une of force is transformed inaccordance with a well-known relationship.4

sin в{и) = [R{u)]i sin 0(0) (6)

where the angle 6(u) is measured with respect to thelocal direction of the magnetic field. R(u) is themirror ratio, evaluated at u, R(u) = H(u)¡H(0).

Expression (6) bears a resemblance to Smell's Lawof classical optics, which relates refraction angleswithin optically dense media. Here -\/R is analogousto the index of refraction of Snell's Law. As in theoptical case, total internal reflection can occur forthose angles larger than a critical angle, 6e, found bysetting sin в equal to its maximum value of 1. Thus

sin вс = RM-i. (7)

This relation defines the loss cones for particlesbound between magnetic mirrors, illustrated in Fig. 1.All particles with pitch angles lying outside the losscones are bound, while all with pitch angles within theloss cone will be lost upon their first encounter witheither mirror (for equal mirror strengths). It should beemphasized that the concept of the mirror loss cone

UCRL PYROTRON PROGRAM 247

pertains to the velocity space of the trapped particlesand has nothing to do with the spatial dimensions ofthe confining fields.

Although the binding of particles between magneticmirrors can be accomplished with fields which are notaxially symmetric, there are substantial advantagesto the adoption of axial symmetry. One of theseadvantages arises from the fact that all " magneticbottles " involve magnetic fields with gradients orcurvature of the magnetic field lines in the confine-ment zone, the existence of which gives rise tosystematic drifts of the confined particles across themagnetic lines of force.6 If the particle orbit dia-meters are small, compared to the dimensional scaleof the field gradients or field curvature, these driftswill be slow compared to the particle velocities them-selves. If directed across the field, however, theywould still be too rapid to be tolerated and wouldeffectively destroy the confinement. Furthermore,drifts of this type are oppositely directed for electronsand ions and may thus give rise to charge separationand electric fields within the plasma. These electricfields then may induce a general drift of the plasmaacross the field to the walls. However, if the magneticfield is axially symmetric, as for example in the mirrorfield configuration of Fig. 1, the particle drifts willalso be axially symmetric, leading only to a rotationaldrift of the plasma particles around the axis of sym-metry, positive ions and electrons drifting in oppositedirections. However, this will not result in a tendencyfor charge separation to occur, since each flow closeson itself.

Another consequence of the use of axial sym-metry is that even though the confined particles maybe reflected back and forth between the mirrors a verylarge number of times, this fact will not lead to aprogressive " walking " across the field. In fact, it ispossible to show that all particles trapped betweenthe mirrors are also bound to a high order of approxi-mation to the flux surfaces on which they move andmay not move outward or inward to another fluxtube (apart from the normal slow diffusion effectsarising from interparticle collisions).

Once bound, there is no tendency for particles toescape the confining fields, within the assumptionsmade to this point. However, in predicting the con-ditions for confinement of a plasma by means of con-ditions applying to the individual particles of theplasma, one makes the tacit assumption that co-operative effects will not act in such a way as todestroy the confinement. Such cooperative effects canmodify the confinement through static or time-varyingelectric fields arising from charge separation, orthrough diamagnetic effects which change the localstrength of the confining fields themselves. However,if the confining magnetic fields are axially symmetric,and if the presence of the plasma does not destroy thissymmetry then, as has already been explained, chargeseparation effects cannot give rise to systematic driftsacross the field. Similarly, in such a circumstance thediamagnetic effect of the plasma can only lead to an

axially symmetric depression of the confining fields inthe central regions of the confinement zone, but willleave the fields at the mirrors essentially unaltered(since the density falls nearly to zero at the peak ofthe mirrors, where particles are escaping). Symmetricdiamagnetic effects therefore tend to increase themirror ratios above the vacuum field value. Ofcourse, there will exist a critical value of /?, the ratioof plasma pressure to magnetic pressure, above whichstable confinement is not possible. This critical relativepressure will be dependent on various parameters of.the system, such as: the mirror ratio; the shape,symmetry and aspect ratio of the fields; and theplasma boundary conditions. Although the preciseconditions for stability of plasmas confined bymagnetic mirrors are not at this time sufficiently wellunderstood theoretically to predict them with con-fidence, there is now agreement that it should bepossible to confine plasmas with a substantial value ofjS between magnetic mirrors.6 This conclusion is borneout by the experiments here reported, which indicatestable confinement. It should be emphasized, however,that, encouraging as this may be, neither the theoreti-cal predictions nor the experimental results are yetsufficiently advanced to guarantee that plasmas of thesize, temperature and pressure necessary to produceself-sustaining fusion reactions in a Mirror Machinewould be stable. As in all other known " magneticbottles " the ultimate role of plasma instabilities hasnot yet been determined in the Mirror Machine.

LOSS PROCESSES

Non-Adiabatic Effects

Although it has been shown that trapping conditionsbased on adiabatic invariants predict that a plasmamight, in principle, be confined within a MirrorMachine for indefinitely long periods of time, it isclear that mechanisms will exist for the escape ofparticles, in spite of the trapping.

One might first of all question the validity of theassumption of constant magnetic moment, sincetrapping depends critically on this assumption. Itis clear that this assumption cannot be exactly satis-fied in actual magnetic fields, where particle orbitsare not infinitesimal compared to the dimensions of theconfining magnetic field.

It can be seen from first principles, as for exampleshown by Alfvén2 that the magnetic moment will bevery nearly a constant in situations where the mag-netic field varies by only a small amount in the courseof a single rotation period of the particle. It mightalso be suspected, in the light of the analogy betweenthis problem and the general theory of non-adiabaticeffects of classical mechanics, that the deviations fromadiabaticity should rapidly diminish as the relativeorbit size is reduced. Kruskal7 has shown that theconvergence is indeed rapid but his results are notreadily applicable to the Mirror Machine. Usingnumerical methods, however, it has been shown8: (a)that the fluctuations in magnetic moment associated


with non-adiabatic effects are of importance only forrelatively large orbits, (b) that these fluctuations seemto be cyclic in nature, rather than cumulative, and(c) that they diminish approximately exponentiallywith the reciprocal of the particle orbit size, so thatit should always be possible to scale an experimentin such a way as to make non-adiabatic orbit effectsnegligible. The results may be roughly summarizedby noting that, for the typical orbits which wereconsidered, the maximum amplitude of the fractionalchanges in the magnetic moment, which occurredafter successive periods of back-and-forth motionof trapped particles, could be well represented by anexpression of the form

= aerbh (8)

where v = Ъпш\Ь, i.e., the mean circumference of theparticle orbit divided by the distance between themirrors. The mirror fields were represented by thefunction Нг = Ho [1+a cos ul(p)] plus the appro-priate curl-free function for Hr. Here, in dimensionlessform, и = %rz¡L and p = 2ттг/Ь. With a = 0.25,typical values of the constants a and b were in therange 4 < a < 6, 1.5 < b < 2, for various radialpositions of the orbits. From these values it can beseen that provided v < 0.2, the variations in ¡x aretotally negligible, so that the adiabatic orbit approxi-mation is well satisfied. For example, if L = 100 cmthen all orbits with mean radii less than about 3 cmcan be considered to behave adiabatically. Thisimposes restrictions on the minimum values ofmagnetic fields which can be used, or on the minimumsize of the confinement zones, but it is clear that, asthe scale of the apparatus is increased, the assumptionof orbital adiabaticity becomes increasingly wellsatisfied.

Collision LossesEven under conditions where the adiabatic invari-

ants establish effective trapping of particles, there stillremains a simple and direct mechanism for the loss ofparticles from a mirror machine. This is, of course, themechanism of interpaiticle collisions, the mechanismwhich limits the confinement time of any stable,magnetically confined plasma. Collisions can inducechanges in either the magnetic moment or energyof a trapped particle, and thus can cause its velocityvector to enter the escape cone. The dominant collisioncross section in a totally ionized plasma is the Coulomb,cross section, which varies inversely with the squareof the relative energy of the colliding particles. Thusthe rate of losses (if dominated by collisions) canbe reduced by increasing the kinetic temperature ofthe plasma. The basic rate of these loss processes maybe estimated by consideration of the "relaxation"time of energetic particles in a plasma, as calculated byChandrasekhar or by Spitzer. In this case, the relevantquantity is the mean rate of dispersion in pitch angleof a particle as a result of collisions, since, if a giventrapped particle is scattered through a large angle invelocity space, the probability is large that it will have

been scattered into the escape cone and thus lost. As isusual in a plasma, distant collisions (within a Debyesphere) play a greater role than single scatteringevents. The time for the growth of the angular dis-persion in this (multiple) scattering process for aparticle of given energy is governed by the relation

02 = t/fa (9)

Ь = _„,А о»,.л 1__ А' (10)

where

(x) log Л'

M and v are the mass and velocity of the scatteredparticle, ж2 = W/kT = %Mv*/kT and log Л has theusual value5 of about 20. Ж{х) is a slowly varyingfunction of x and is approximately equal to 0.5 fortypical values.

In terms of deuteron energies in kev,

= 2.6 x W°W*/n seconds. (И)

For б2 = 1, the scattering time becomes equal to t.Thus ¿D represents a rough estimate of the confinementtime of ions in a Mirror Machine. It is clear that theconfinement time will also depend on the mirror ratioR, but detailed calculations show that for large valuesof this ratio, the confinement varies only slowlywith R.

Some numerical values are of interest in this con-nection. Suppose W = 0.01 kev, and n = 1014, amean particle energy and density which might beachieved in a simple discharge plasma. In this case¿D = 0.26 /usée. This means that the confinement oflow temperature plasmas by simple mirrors will be ofvery short duration, unless means for rapid heatingare provided, which would extend the confinementtime. On the other hand, for W = 150 kev, the sameparticle density would give to = 0.5 sec, which is longenough to provide adequate confinement for experi-mental studies and is within a factor of about 20 of themean reaction time of a tritium-deuteron plasma at acorresponding temperature. Since the energy releasedin a single nuclear reaction would be about 100 timesthe mean energy of the plasma particles, it can be seenthat the possibility exists for producing an energeticallyself-sustaining reaction, with a modest margin ofenergy profit, in this case about 4 to 1.

The problem of end losses may be more preciselyformulated by noting that these losses arise frombinary collision processes, so that it should be possibleat all times to write the loss rate in the form

A n\avyaf{R) (12)

where (ov\ represents a scattering rate parameter andf(R) measures the effective fractional escape cone ofthe mirrors for diffusing particles. If <<n>>8 remainsapproximately constant during the decay, then inte-gration of the equation shows that a given initialdensity will decay with time as

wheren = nor/it+т) (13)

(14)


The relaxation time approximation consists ofsetting f(R) = 1—i.e., ignoring the dependence onmirror ratio, thereby overestimating the loss rate—and, at the same time, approximating the true valueof no(ovya by 1/ÍD, which tends to underestimate theloss rate. Thus, in this approximation, the transientdecay of the plasma in a Mirror Machine is given by

n = notb/(t+tj)). (15)

In this approximation, fo represents the time forone-half of the original plasma to escape through themirrors. Now, the instantaneous rate of nuclearreactions which could occur in the plasma is pro-portional to пг. Integrating # 2 from (15) over all time,it is found that the total number of reactions whichwill occur is the same as that calculated by assumingthat the density has the constant value щ for time foand then immediately drops to zero; i.e., number ofreactions is proportional to щЧц.

Actually, although they correctly portray the basicphysical processes involved in end losses, estimates ofconfinement time based on simple relaxation con-siderations are likely to be in error by factors of twoor more. To obtain accurate values of the confinementtime more sophisticated methods are required. Judd,McDonald and Rosenbluth,9 and others,8 have appliedthe Fokker-Planck equation to this problem andhave derived results which should be much closer tothe true situation, even though it was necessary tointroduce simplifying approximations in their cal-culations. They find that calculations based on thesimple relaxation considerations tend to overestimatethe confinement times. They are also able to calculateexplicitly the otherwise intuitive result that the meanenergy of the escaping group of particles is alwayssubstantially less than the mean energy of the re-maining particles, a circumstance which arises becauselow energy particles are more rapidly scattered thanhigh energy ones.

The most detailed results in these end loss calcula-tions have been obtained by numerical integration.However, before these results were obtained, D. Judd,W. McDonald, and M. Rosenbluth derived an approxi-mate analytic solution which, in many cases, differsonly slightly from the more accurate numerical cal-culations. Their results can be expressed by two équa-tions which describe the dominant mode of decay ofan eigenvalue equation for the diffusion of particlesin the velocity space of the Mirror Machine.

The first equation, that for the density, is:

и и2[|тг(е4/ж2)<г>-1><г;-2> log A]X{R), (16)

where the angle brackets denote averages over theparticle distribution and log Л is the screening factor.6

The eigenvalue X(R) is closely approximated by theexpression X(R) = l/logio(.R), showing that the con-finement time varies linearly with the mirror ratio, atsmall mirror ratios, but that it varies more slowly atlarge values of R. We see immediately from the formof the equation that one can define a scattering timeas in Eq. (14):

(17)

The term in the first bracket is almost identical withthat for the relaxation time of a particle with themean velocity vo. The velocity averages give rise tosmall departures from the simple relaxation valuesbut, as noted, the corrections are usually not large.

In a similar way, an equation can be written for therate of energy transport through the mirror by par-ticle escape. The basic rate for this is, of course,simply given by the value of nffîB; but l^s, the meanenergy of escaping particles, as noted, is not the sameas the mean energy of the remaining particles. Theyfind, for the rate of energy loss:

nWB = -п*[Ьт{е*1т)<ь-1у logA]X{R). (18)

Since the value of <w~x> is not particularly sensitiveto the velocity distribution, one may calculate thisfor a Maxwellian distribution without committingexcessive error.

The mean energy of the escaping ions, Wa, may beevaluated from the equations. Dividing (18) by (16),W8 is seen to be ^ж<г;~2>~1. Approximating the actualdistribution by a Maxwellian, Ws turns out to be\kT, or \ of the Maxwellian mean energy. This isroughly the same value as was obtained by the moreaccurate numerical calculations.

As far as eventual practical applications are con-cerned, the over-all result of the detailed end losscalculations is to show that although a power balancecan in principle be obtained, both with the DT andDD reactions, by operation at sufficiently high tem-peratures, the margin is less favorable than the roughcalculations indicated. Thus, any contemplated appli-cation of the Mirror Machine to the generation ofpower would doubtless require care in minimizing oreffectively recovering the energy carried from theconfinement zone by escaping particles. Some possibleways by which this might be accomplished will bediscussed in sections to follow. Other methods to re-duce end losses through magnetic mirrors have beensuggested and are under study. One of these, involvinga rotation of the plasma to "enhance" the mirroreffect, is under study at this Laboratory and at theLos Alamos Scientific Laboratory.

In the present Mirror Machine program, scatter-ing losses impose a condition on the experimental useof simple mirror systems for the heating and confine-ment of plasmas. This condition is that operationssuch as injection and heating must be carried out intimes shorter than the scattering times. These require-ments can be readily met, however, in most circum-stances so that end losses do not present an appreciablebarrier to the studies.

Ambipolar Effects

Since the scattering rate for electrons is more rapidthan for ions of the same energy, the intrinsic end lossrate for electrons and ions will be markedly different.In an isolated plasma this situation will not persist,


since a difference in escape rates will inevitably resultin establishing a plasma potential which equalizes therate. In ordinary discharge plasmas, where this effectalso appears, this " ambipolar " loss rate is alwaysdominated by the slower species of the plasma. Thissituation seems to apply also to plasmas confined in theMirror Machine. Although many different cases arepossible, not all of which are understood, the role ofambipolar effects seems mainly to be to introduce a(usually) small correction to the end loss rate as cal-culated from ion-ion collisions. For example, in theimportant case where the electron temperature issmall compared with the ion temperature, Kaufman10

has shown that ambipolar phenomena establish apositive plasma potential which, in effect, changes themirror ratio for the ions to a somewhat smaller valuegiven by the expression

i?eff = 2?[l+y(2yri)]-i, (19)where y is a constant of order unity. For cases ofpractical interest, the resulting correction to the ionloss rates is small. However, the precise role ofambipolar diffusion effects and the electric potentialsto which they give rise is not well understood, especi-ally with regard to plasma stability, and must, there-fore, be labeled as part of the unfinished business ofthe Mirror Machine program.

Confinement of ImpuritiesIn the study of magnetically confined plasmas, and

in their eventual practical utilization, contaminationof the plasma by unwanted impurities presents aserious problem. Since the plasma particle densitieswhich are of present and future interest are aboutÎQ^-IO15 cm"3, the presence of impurities to the ex-tent of even a small fraction of a microgram per literrepresents a sizeable amount of contamination, per-centage-wise. Impurities usually originate fromadsorbed layers on the chamber walls, the release ofeven a small fraction of an atomic monolayer corre-sponding to a large degree of contamination.

Impurities of high atomic number greatly increasethe radiation losses from the plasma. At temperaturesof a few million degrees the radiation loss mechanismis principallylrom electron-excited optical transitionsof incompletely stripped impurity ions. In this case,the intrinsic radiation rates per impurity ion can be ashigh as one million times the radiation rate of a hydro-gen ion. At thermonuclear temperatures, the strippingbecomes more nearly complete, so that the radiationrate from impurities becomes less intense, finallyapproaching the bremsstrahlung value, Z2 times theloss rate for hydrogen. But even in this case, becauseof the relatively narrow margin between the thermo-nuclear power production and the radiation loss ratesfrom even a perfectly pure hydrogenic plasma, theconcentration of high-Z impurities must be kept to aminimum. The actual level of impurities in anymagnetically confined plasma will be determined by acompetition between their rate of influx and thedegree to which they are confined by the magneticfields.

In the light of these facts, it is important to notethat in the Mirror Machine, the confinement of im-purity ions should be much poorer than for energetichydrogen ions. Thus there tends to exist a natural"purification" mechanism discriminating againstthe presence of impurities in the plasma.

This happy circumstance arises from three generalcauses. First, the scattering rate for stripped high-Zions is more rapid than for hydrogen ions of thesame energy. Second, impurity ions originating fromthe walls will always have much lower energies thanthe mean energy of the confined ions. This willincrease their loss rate by scattering. Third, in the"normal" high temperature confinement conditionin the Mirror Machine, where ion energies are muchgreater than the electron temperature, the sign of theplasma potential should be positive, so that low energypositive ions cannot be bound at all, but will beactively expelled. These circumstances could be of animportance equal to that of adequate confinement inthe eventual problems of establishing a power balancefrom fusion reactions.

BASIC OPERATIONS

In order to create, heat and confine a hot plasma inany magnetic bottle a series of operations is alwaysrequired. In the Mirror Machine a somewhat differentphilosophy of these operations has been adopted fromthat used in most other approaches, for example,those utilizing the pinch effect. In those approaches,one starts with a chamber filled with neutral gas andthen attempts to ionize the gas, heat and confine theresulting plasma. By contrast, in the Mirror Machine ahighly evacuated chamber is employed, into the centerof which a relatively energetic plasma is injected,trapped and subsequently further heated. Since theMirror Machine possesses "open" ends through whichthe plasma can be introduced by means of externalsources, injection methods can be employed which arenot possible in systems of toroidal topology.

By using the magnetic mirror effect in various ways,it is possible to perform several basic operations on aplasma. These are employed in the Mirror Machine tocreate, heat, control and study the plasma. In additionto siniple confinement the following operations areused.

1. Radial compression—adiabatic compression ofthe plasma performed by uniformly increasing thestrength of the confining fields.

2. Axial compression—adiabatic compression per-formed by causing the mirrors to move closer together.

3. Transfer or axial acceleration—pushing theplasma axially from one confinement region toanother by moving the mirrors.

4. "Valve" action—controlling the direction ofdiffusion of the plasma by weakening or strengtheningone mirror relative to the other.

In the experimental study of these operations mostof the emphasis has been placed on achieving condi-tions where the assumption of adiabaticity is valid and


where collision losses and ambipolar effects are mini-mized by control of the heating cycle. This operationalregime is a relatively simple one to understand, isreadily analyzed theoretically, and presents somesubstantial practical advantages.

The primary method of heating used in the Mirf orMachine is adiabatic compression; i.e., the heatingoccurs when the confining magnetic fields are changedat a rate which is relatively slow compared with theperiods of rotation of the plasma ions.

The adiabatic magnetic heating of a plasma can beunderstood either from an individual particle view-point, or from general thermodynamic principles.Consider, first, the behavior of trapped particles in aconfining field which is increasing with time. In sucha field, the energy of each trapped particle will in-crease, simply because of the constancy of themagnetic moment, ¡л = W±/H = constant (Eq. (1)).Thus, rotational energy will increase in directproportion to the magnetic field:

Wx(t) = W±(0)H(t)IH(0) s Wj.(0)a. (20)

At the same time it can be shown that adiabaticmagnetic heating of this type proceeds so that theflux through each elementary orbit remains constantand also so that the guiding centre of each particleremains on the same flux tube of the increasing mag-netic field, i.e., the plasma is uniformly compressedtoward the magnetic axis of the confining field.11

This leads to an increase in the plasma density aswell as an increase in its mean energy. This is the radialcompression Usted above. Since energy is imparted tothe rotational component of particle motion, thistype of heating leads to more effective binding of theparticles.

Axial compression, accompUshed by mechanicallyor electrically moving the mirror fields toward eachother, results in the same type of heating as discussedby Fermi in his theory of the origin of cosmic rays.4

Axial compression leads to an increase in the parallelenergy component of each trapped particle by anamount proportional to the square of the compressionfactor. If the mirrors are separated by a distanceЩ then

In this case the heating leads to poorer trapping, sothat it can only be used to a limited extent.

The increase in density which results from these twocompression processes is given by the product of theradial and axial compression factors. The radial com-pression is determined by the constant flux condition,which impUes that the radius of the plasma will varyas H~k and its area therefore as Я" 1 . The totalcompression is then given by

•M. - ШМК - акn{0) ~ Я(0, О)

Both types of adiabatic compression heating canbe related to simple thermodynamic gas laws. In radialcompression, the gas behaves as a two-dimensional

gas, since there are two degrees of freedom associatedwith rotational energy. For this case, the gas constant,y, has the value 2 and the heating varies Unearly withthe density T\ ~ nv1 ~ n. In the case of axialcompression, there is only one degree of freedom(motion along the Unes of force) so that y = (2+/)//= 3 and T¡ ~ и2. If colUsions become importantduring the compression heating, all three degrees offreedom will be coupled and the heating will proceedas in an ordinary gas; T ~ n .

CompressionMagnetic compression processes, as used in the

operation of the Mirror Machine, can be calculated bythe use of a single integral which incorporates all ofthe adiabatic assumptions. The adiabatic invariantswhich apply to the motion of trapped particles of theplasma moving between the mirrors are (1) themagnetic moment, ц = W±/H and (2) the actionintegral of the particle momentum along magneticUnes of force, taken over a period of the bound motionalong a Une of force, i.e., §р$и = A. The constancyof A is dependent on the assumption that the mag-netic field shall change only by a small relativeamount during the time of one period, a conditionwhich is usually satisfied in the experiments. Whenthis is true, the total energy of each particle is alsoslowly varying so that the angle transformationrelationship, Eq. (6), appUes to the motion.

All three of the above conditions can be combined ina single equation which predicts the essential characterof the compression process.12 Since we are notinterested in the detailed motion of each particle, butonly in the salient features of its motion, it is con-venient in this equation to represent the orbit of eachtrapped particle solely by the location of its end pointor "high water mark" in the confining mirror fields.The subsequent fate of the plasma can then bepredicted by following the evolution of these endpoints. The equation derived is:

[Я(0)]* Г % [Rm-R{u)]b du = S = constant (22)

where u\ and u% are the extreme values of и reachedby the particle, and R(u) is the function, Я(и)/Я(0) ;i.e., the field strength at any point u, relative to itsvalue at и = 0, where the field has its weakest value.Rm is the value of R(u) at u\ or u% (i.e., at the highwater marks). The value of the constant S is to bechosen from the initial conditions.

Equation (22) may also be written in other formswhich are sometimes more convenient to use. Writtenas an integral over the variable R, it becomes

( 2 3 )

where du/dR = (dR/du)-1 expressed as a function ofR. Here, of course, Rm{l) and Rm(2) have the samenumerical value. When the confining fields have aplane of symmetry at и = 0 the lower Umit of (22)


may be taken as zero, and the upper limit as wm, withcorresponding limits 1 and Rm for Eq. (23).

By use of the compression equation, the motion ofthe end points, Um or Rm can be determined. Com-bining this motion with the constant flux condition,the transformation of the radial distribution can alsobe obtained. To show that this last statement is true,it is necessary to demonstrate that, in the adiabatic,collision-free limit here assumed, there is no " mixing "of the distribution from one flux surface to another,even after many reflections have occurred. This canbe shown using the compression equations. Considerthe possible orbits which a particle of given fixedmagnetic moment, ц, energy W, and action integralA can execute in the field. These orbits will be describ-able by the compression equation. Now we mayevidently write (23) in the. form

Consider two possible orbits (a) and (b) for the aboveparticle, characterized by integrals Sa and S& respec-tively. Only du/dH may differ between the two orbits.Therefore, we may subtract integral Sa from Sb toarrive at a condition which these derivatives mustsatisfy. This condition is, since Sa = Sb,

JHma) a dH\\dn. = 0. (25)/

If the field system is axially symmetric, then thisequation can be satisfied only for those orbits for whichthe value (du/dH)\a and {du¡dH)\b are the same atevery point along u, i.e., on the surface of flux tubes.Thus, in this case, the particles are constrained tomove on flux surfaces so that mixing does not occur,as was to be proved. But even if the magnetic field isonly approximately axially symmetric, then, aspointed out by Teller,13 Eq. (25) still defines a set ofseparate closed surfaces on which the particles areconstrained to move. These surfaces are generatedfrom any initial given orbit by finding adjacent orbitsalong flux Unes for which (25) is satisfied.

It is evident that, if the magnetic field changesadiabaticallyT" (25) is still valid quasi-statically, so thatthe motion of the particle orbits is still constrained tolie on fixed but slowly collapsing flux tubes of thesystem. This allows the prediction of radial compres-sion effects by use of the compression equation. Thuseach initial end point of an orbit say [Мщ(0), r(0)] willbe transformed to a new point, [#m(¿), r{t]\ such that

Г Г'<°)LJo «m(0)

= 2ттгНМ . (26)Uo JUm(í)

To be used with the compression equations is therelationship giving the particle energy as a function oftime. From the constancy of fi it follows immediatelythat the final kinetic energy of any particle is relatedto its initial energy by the equation

W(t) = W{0) IRn,Rm(0)\

It is important to note that, under the assumptionsmade, particle energies do not appear in the integrals(22) or (23), so that if the trapped particle density isrepresented by its distribution in u\, u% or Rm, thisdistribution transforms in time in accordance withthese equations, independent of the energy distribu-tion. The transformation of the energy distributionwhich results from the compression can then be foundby using (27) together with the results from the com-pression equation.

When the external applied magnetic field is sub-stantially altered by the pressure of the plasma, it isnecessary to use self-consistent field values in com-puting the compression, found by iteration of thesolutions to the compression equation, or by othermeans. For many purposes, however, this difficultprocedure would not seem to be necessary, since itsomission does not lead to large errors.

Radial CompressionThe heating and the increase in density resulting

from a simple radial compression can be found fromthe compression equations above, if the shape of theconfining field is known.

In some cases the strength of field in the centralregion can be represented by a parabolic distribution,i.e., by the function R(u) = l + (#/A)2. Inserting thisinto the compression equation one finds that, inaddition to radial compression, even with the mirrorsheld stationary, an axial compression occurs inaccordance with the relationship

[#(0)] (Rm-1)2 = constant;

i.e., Я(0)^щ4 = constant, so that for the axial com-pression factor one obtains

_ Î ^ O ) _ ГЯ(0, Щum(t) - |#(0,0)J

(28)

This results in a uniform axial compression of theentire trapped-plasma. In those cases where very largecompression factors are used, the extra increase indensity which results is substantial. Combining theradial compression with the axial compression factorabove, the density is seen to vary as

n(t) = п{0)ак = w(0)a*. (29)

(27)

Taking an example from some of the experiments, ifa = 103, then the axial compression amounts to abouta factor of six, a very noticeable effect.

Combined Radial and Axial Compression

Although the trapping is always improved by asimple radial compression, the trapping may be madeless effective, when both radial and axial compressionare employed, unless certain restrictions are observed.The effect of a combined radial and axial compressioncan be predicted from the compression equations (22)or (23) by allowing R(u) to be a function of the time.Consider the case where the behavior of R(u) is


representable by a simple scaling of the axial co-ordinates of the field. This represents, approximately, apractical situation in which the mirror ratios are leftfixed, but the mirrors are moved together as a functionof time.

To represent this situation, let R(u) be given by thefunction g(u/A) where A changes with time (A decreasingcorresponds to axial compression, A increasing cor-responds to expansion); then, if g~x{R) is the functioninverse to g(u¡X), so that Xg-^R) = и, then du/dR= M(g-i)ldR. In this case, g-^R) is not, of course,a function of the time, but is constant during thecompression. Thus, (23) becomes .

"R)i^)dR=s> (30)

but now the integrand is no longer an explicit functionof the time.

The condition that all plasma particles remain atleast as tightly bound after the compression as before,is simply that Rm does not increase for any particle.From (30) this is seen to be possible only if А[Я(0)]&is constant, or at least does not decrease with, time,i.e.,

ХЩН(0, t) > А2(0)Я(0, 0) (31)or,

к2 < a, (32)

since A(O)/A(¿) = к, the axial compression factor. If themirror ratio is also a function of time, then, in simplecases, this condition takes the form

(33)

The case, ХШ = constant, is interesting because itrepresents a uniform compression of the plasma in allthree dimensions. For this case, since the values ofRm remain constant for all trapped particles, theenergy of each of the particles, as given by Eq. (27),is simply proportional to a. However, the volume ofthe plasma varies as («к)~х and therefore as or*.Thus, the mean particle energy varies inversely with(volume)*, i.e., W ~ n«. This is the same as the lawof adiabatic heating of an ideal gas, as derived fromthermodynamic considerations. Although this corre-spondence with the classical result might also havebeen foreseen from Liouville's equation, it is interestingto see it emerge from the equations of a magneticallyconfined plasma.

If only an axial compression is carried out, withЯ(0) remaining constant, the compression equation(30) shows that the particles must climb higher on themirrors, so that the actual amount of axial compressionof the plasma which occurs will be less than the geo-metrical compression ratio. For example, if thestrength of confining fields varies parabolically, sothat R(u) = 1 + (и/А)2, then it is found that X(Rm-1)= constant, so that for every particle, ,um—Á&.Although the compression is uniform, it proceedsonly as \i rather than linearly with the geometriccompression.

Competing Effects

Collisions

When properly scheduled, magnetic compressionresults in improved trapping of the plasma, and theordering effect of the compression acts in oppositionto the disordering effect of collisions. Since the effectof heating the plasma is to reduce the mutual Cou-lomb collision cross section of the heated particles, itis clear that it should be possible to carry out thecompression at such a rate that it continues to over-come the collision losses throughout the cycle andthus inhibits the onset of losses through the mirrors.The condition for this to occur can be estimated fromthe relaxation time considerations already discussed.It is required that, during the compression, the gainof perpendicular (rotational) energy shall exceed thegain of parallel energy (kinetic energy of motion in thedirection of the field lines). Taking as an examplethe simple case of radial compression, from (20) thegain of rotational energy is obtainable from Wj_(t) =Wx(0)a(t), Eq. (20).

On the other hand, the mean instantaneous rate ofchange of parallel energy of the ions, owing to col-lisions, will be given approximately by the expression:

(dWJdty = KdW/dty = Wlb.

Now from (11), ¿D = AW*/п sec, where A =2.6 x 1010 for deuteron energies expressed in kev. Ifcompression heating dominates over dispersion dueto collisions, throughout the compression, then weobtain WJL > PFD, so that we may set Wx » W. Ifthe duration of the heating operation is equal to t,then we require that Wx > W¡, for all times lessthan t. This requires, therefore, from the relationsabove, that

aW{0)n(0) ¡\a(t')]idt'.

Jo(34)

If H increases linearly with time, so that a{t') = 1+t'/r,then (34) becomes (dropping the zeroes in n(0), etc.)

If the compression factor, a, is large, then this can beexpressed as a simple condition on the time withinwhich the compression must be accomplished to avoidexcessive scattering losses. This condition is

9ta < л

seconds. (36)

It is apparent that if the particle energy is low it ismore difficult to satisfy this condition. For this reason,even when it is satisfied for most of the constituentparticles of an initial distribution, the lowest energyparticles of the distribution are still likely to be lost.

To illustrate a case which might typically beencountered in an experiment, suppose that W± X W= 0.1 kev and the initial density, n, is 1012 cm"3. Ifit is desired to compress by a factor of 100, so that the


a | HptAPPROX.) FOR ADIABATIC COMPRESSION

10" 10

t a - TOTAL COMPRESSION TIME

A

i '2|O l 3 IO14

¿Hf" FÍELO IN GAUSS BELOWWHICH COMPRESSION REMAINSNON-ADIABATIC

IP 2 IO3 IO4 IOs IOe

§¡2

о10 Ю2 Ю3 IO4IO°

* Ю*

I 10 Ю 2 Ю 3 Ю 4

юг» I0"4 IO'3 I0" 2

r a - TOTAL COMPRESSION TIME

- I 10 IO2IO3

1С-'1 sec

377.2 B

Figure 2. (a) Allowed operation zones for adiabatic magnetic compression to suppress collisionaleffects; (b) deviations from adiabaticity, fraction of compression pulse during which Ions are not

compressed adlabatically

final mean energy becomes 10 kev and the final density Figure 2(a) illustrates the condition impospd by1014, then (36) shows that the compression time must Eq. (36) for various values of compression rates andbe less than about 0.04 sec. This is not an unduly initial energy. To satisfy the condition, compressionrestrictive requirement. Raising the initial energy times must be shorter than the values given by theallows even slower compression rates to be used. boundary lines. The final particle densities reached


are shown along the right margin, for various valuesof otmax- If the condition of adiabaticity, i.e.,

(the gyromagnetic period), is to be satisfied through-out the compression, then the initial magnetic fieldfor this to be true can be specified for each totalcompression time. These values of initial field, Ho, areindicated along the upper margin of the plot. It canbe seen that this condition could not conceivably besatisfied for the shortest compression times andlargest compression ratios indicated. It is worthnoting, however, because of the dependence of rg onmass, that the adiabatic condition is likely to be wellsatisfied at all times for electrons, even when it maynot be satisfied for ions.

However, in many cases of practical interest, thecompression will depart from adiabaticity only for ashort time at its very beginning. In this case theheating and compression may not deviate significantlyfrom the adiabatic values. Because the gyromagneticperiod, тд, rapidly diminishes, as the field starts toincrease, while the doubling period of the field, тд,increases; the ratio of these two quantities rapidlyincreases with time. This ratio is given by theexpression

x = rH¡Tg = eWftirMcÉ, (37)

where x >̂ 1 corresponds to the adiabatic limit.If evaluated for a linearly rising field, and for

M = the mass of a deuteron, this becomes

X =O.8xlOWota[a(t)]2

(otmax-1)(38)

Figure 2(6) shows the fraction of the total magneticcompression during which the adiabatic assumptionfails to be satisfied, i.e., the value of H](amB,xHo) =«(¿)/«max at which x rises to unity for various values ofamax, ta and Ho. When this fraction is small, the adia-batic result will be approached.

The general effect of compressing at too slow a rateis to retain the most energetic components of theparticle distribution, while losing the lower energygroups before the compression is completed. Thistends, therefore, to "upgrade" the energy distri-bution to a somewhat higher mean energy than wouldhave been expected, albeit at a lower density thanpredicted by the simple compression equation.

If axial compression as well as radial compression isincluded, the compression time requirement becomessomewhat more restrictive, because of the increaseddensity and because of the change in parallel energywhich accompanies axial compression.

Ambipolar DiffusionAmbipolar diffusion effects must also play a role in

the compression of a plasma. However, as has alreadybeen indicated, this role seems to be a secondary onein the experiments conducted to date, not interferingappreciably with the observed heating effects. More

sophisticated experimental work will, however, haveto be done to determine the precise influence ofambipolar diffusion on the plasma history. The exactway in which the ambipolar effects will appear willdepend on questions of the relative initial ion andelectron mean energies. For example, when the initialion energies are higher than the electron energies, theenergy of these electrons can be expected to lagincreasingly behind the ion mean energy, since thecollision effects will cause the heating energy to beshared among all degrees of freedom of these electrons,so that their heating rate will tend to be dependent onthe f power of the compression, rather than the firstpower.12 On the other hand, those electrons whichstart with energies sufficiently above the ion energieswill be heated at the same rate as the ions and so willremain at a higher energy than the ions throughout thecompression. The ratio of electron and ion energies atwhich the scattering times become comparable can beestimated from the relaxation time, Eq. (10). From thisequation it can be seen that

We/Wi = (M¡m)i X 15,

if evaluated for M = mass of a deuteron.

Radiation from ImpuritiesIt has already been noted that, especially at re-

latively low plasma temperatures, radiation fromimpurities can represent a potent mechanism for lossof energy from the plasma through the agency ofelectronic excitation of incompletely stripped im-purity ions. The rate of this radiation loss is thusproportional to the product of the electron particledensity and the particle density of the impurity ions.But in many cases the impurity density is roughly aconstant fraction, q, of the plasma ion density itself,so that the radiation rate varies as qne

2. The absoluterate of impurity radiation per unit volume is thenrepresentable by an expression of the form (for eachtype of ion)14:

Pt = qne^Te-if{hvlkT). (39)

On the other hand, the rate of energy input to theplasma depends linearly on the electron density andthe rate of rise of the magnetic field. For example,in the collision-free limit for a simple radial com-pression, Eq. (20) gives for the energy gain:

ph = nedH/dt. (40)

In order to heat the plasma it is obviously necessaryto put in energy from the magnetic compression at arate more rapid than the radiation rate. Since theenergy input rate varies linearly with density, whereasthe radiation varies as the square of the density forany given rate of increase of the field, there exists aninitial density above which the temperature will notincrease, the energy from compression being dis-sipated, by radiation. However, since the density canbe arbitrarily set by controlling .the injection, it isalways possible to find a low enough initial densitywhich leads to heating. Fortunately, it has been found


possible to control the impurity levels to the pointwhere radiation imposes no more stringent a limita-tion on the initial density than does the requirementof collision losses discussed previously: in the experi-ments, electron heating occurs essentially as pre-dicted by the compression equations. If the electronswere not heated, but were maintained at a lowtemperature by radiation losses, the ion energieswould also tend to be quickly damped by collisioncooling.

Adiabatic Expansion—Energy RecoveryBefore leaving the subject of adiabatic compression,

it is important to note that, under the assumptionsmade, all of the operations discussed are reversible:i.e., the same equations apply to a magnetic expansionof the plasma, carried out by appropriate manipula-tion of the confining fields. In this case, the roles ofradial and axial decompression, in their competitionwith collision effects, are just reversed from the caseof compression. Axial decompression leads to improvedconfinement of the remaining plasma whereas radialdecompression leads to poorer trapping. An appro-priate combination of the two operations is thereforeindicated. In an adiabatic expansion the energy of theplasma will decrease, corresponding to a transfer ofenergy back to the confining fields, and (usually) anaccompanying flow of electrical energy out of thesystem.1 This fact offers a possible avenue to the directelectrical recovery of plasma internal energy, eitherthat associated with charged reaction products orunreacted plasma. In the Mirror Machine, efficientrecovery of the kinetic energy of escaping particles,which might be accomplished by adiabatic expansion,could be a very important means of improving thepower balance. It enables one to visualize a con-tinuously operating "plasma engine" which wouldutilize the compression and expansion operations,which have been described above, to create, heat,confine and recover energy from a reacting plasma.The feasibility of such a machine has, of course, notyet been proved.

INJECTION

One of the most difficult technical problems of theMirror Machine program has been the problem ofinjection. Because of the nature of plasma confine-ment by magnetic mirrors, it is not often satisfactoryto create the plasma by ionizing a cold gas inside theconfinement volume. Instead, other methods havebeen employed which involve the injection of plasmaor of streams of energetic particles. The "open ended"topology of the Mirror Machine allows one to employsome unusual methods for injection.

The dilemma which is faced in all injection prob-lems is the essential impossibility of trapping chargedparticles within a static magnetic field under conserva-tive conditions; i.e., in order to inject, something mustchange. Various methods for injecting into a MirrorMachine have been proposed15 and several of them

are under study. They employ one or more of thefollowing effects: (a) time-varying fields, (b) collision orcooperative-particle interactions and (c) change of chargestate of the injected particles.

Time-Varying FieldsIn the first category lie several of the methods which

have been applied in the program to date. In earlyexperiments,18 beams of energetic particles were cap-tured between the mirrors by injecting them throughregions where they were subjected to radio-frequencyfields. These fields produce an irreversible energy gainwhich leads to trapping. This method has not beenactively pursued because of problems of field pene-tration but it remains a potentially interesting process.

External InjectionAnother method which has been successfully

exploited might be called "external adiabatic trap-ping". In this method a plasma is captured byinjecting it through the mirrors into a rapidly risingmagnetic field.17 This method is similar to thatemployed in a betatron to accomplish injection. It canbe understood qualitatively from the binding equa-tions (2) or (3). Suppose a plasma is injected througha mirror from outside the confinement zone. Thennone of its ions will be initially bound. For these ions,an inequality which is the inverse of the binding con-dition wiH apply, i.e.,

W > fj.Hu. (41)

If, however, during the transit of the plasmathrough the confinement zone, the strength of themagnetic field is increased sufficiently, then because ofthe constancy of ¡i, the mirror field strength Ни mayincrease sufficiently to reverse the inequality so thatmany of the plasma ions are captured. This can beaccomplished either by strengthening the entire field,or simply by changing the field at the mirrors.

The condition that particles should be capturedunder these conditions has been worked out.1 Itshows, as might be expected, that injection can beaccomplished only as long as the magnetic fieldchanges by a sufficient fractional amount during onetransit of the particles. If the injected ions of theplasma flow over the top of the first mirror, making asmall helix angle, e, with respect to the plane of theMirror (e = fyt—B, where в is the pitch angle) theninjection can be accomplished as long as

Ш R \i-í) (L/vo)

(42)

where L is the separation between the mirrors (as-sumed large compared to length of the mirror regions),VQ is the velocity of the injected ions, and R is themirror ratio. If the field is assumed to rise linearlywith time, then the maximum acceptance time overwhich injection can be accomplished will be given by

-1\H (43)


When only the mirrors are strengthened, ratherthan the entire field, a similar expression is found.Also, if the far mirror is made much larger than thenear mirror, the acceptance time is doubled.

It can be seen that this injection mechanism is mosteffective for particles entering with small angles, e.Thus, if a stream of plasma with a wide range ofhelix angles is injected through the mirrors, then theinjection condition will continue to be satisfied longestfor the smallest angles e.

Internal Injection

Injection can also be accomplished by sourceslocated partially within the confinement volume. Herethe problem is one of preventing the plasma particlesfrom returning and hitting the source. One of themethods studied has been the use of compact ionsources located just inside the confinement volume,emitting energetic ions in a direction perpendicular tothe local direction of the field lines. The great advan-tage of using such sources is that the plasma is initiallycreated with a very high ion energy—many kilovolts—thus over-leaping most of the heating problems.There are serious problems, however, in reducing themethod to practice. As injected, the ions miss thesource after their first turn, because of the fieldgradient in which they move. From the force equation(4) it can easily be shown that, in the adiabaticapproximation, an ion injected at right angles to afield gradient advances in one turn by a distance

Lu = 7гр2/Л (44)

where p is the mean radius of curvature of the particlein the field and A is a characteristic distance associatedwith the field gradient, defined by

1_8H"Я du'

(45)

and not to be confused with the eigenvalue, X(R),used earlier.

Having missed the source on the first turn, the ionsmay then travel to the far end of the confinementregion, where they are reflected (since they were"born" inside the mirrors). While moving along thefield lines, the ions will precess slowly around the axisof the machine, because of their finite orbit size. It ispossible to arrange conditions so that this precession,though necessarily small at each reflection, since theions are nearly adiabatic (small orbits), can still belarge enough to cause the ions to miss the source ontheir first return. If this is the case, then they maycontinue to be reflected back and forth in the volumeuntil they have precessed completely around the axis.This might take, typically, 50 to 100 traversais. Atthis time they would be likely to hit the source andbe lost, thus limiting the injection and confinementtime to about the period of precession around theaxis, too short a time to be of much interest. If, how-ever, the magnetic field is increased while this isgoing on, then by the time each ion has precessedaround to a position where it might hit the source, the

axial compression which occurs in the increasing fieldmay prevent it from returning to the source position,so that it becomes trapped. In this way the time overwhich injection can be accomplished may be substan-tially extended over that of a single precession period.Injection will cease either [a) when, because of theincreasing field, the orbits become too small to missthe source, on the first turn, because of the pz depen-dence of (44), or (b) when they are not sufficientlycompressed axially to miss the source. The mostefficient use of the injection time occurs when con-ditions (a) and (b) fail simultaneously.

The amount of axial compression which occurswhen the field is increased can be calculated readilyfrom the compression equation (22). Assuming thatthe mirrors are alike, this may be written as

wheremj =

J=

On differentiating, we obtain

= dS = 0;but

Hence, if dj/dt = 0 (field shape does not changewith time), we find that

where Xj is the characteristic distance given by theexpression in brackets and тц = H¡(dH¡dt) representsthe instantaneous doubling time of the field as itincreases. If the field increases linearly with time,тя is simply the time in seconds subsequent to theinitiation of the field rise. In the equations, 8мщrepresents the distance which the end point of motionof the particle progresses during time St, if St corres-ponds to one of the times of return of the particle tothe first mirror.

In simple cases, for example where the mirrors arewidely separated, Aj = A so that (46) reduces to thesimple expression,

(47)

where A is defined as in Eq. (45). If the angle ofemission of the ions with respect to the normal to thefield lines is not exactly zero, but is + e then (47) and(44) become

(48)

(49)

In order for injection to be accomplished, bothand Lum must simultaneously be larger than Lz,

the length of the source in the direction of the mag-netic field. The maximum injection time availablemay be estimated by setting e = 0 and 8#m = Lum >Дг and solving for т т а х :

= {X/vp)zSt. (50)


Here Si is the total transit time in one precessionperiod, which might be many times the one-waytransit time of the ion between the mirrors. Thus, fortypical values of A and p which are compatible withthe condition ÙMm > Дгг, Tmax can be of the order of athousand times the single transit time, so thattheoretical injection times of about a millisecondcorrespond to confinement volumes of averagedimensions. More detailed calculations have beenmade which, however, do not materially change theexpected times.

Despite the extended injection times which can beachieved, the method of internal injection with con-ventional ion sources has serious drawbacks. First,even if it functioned exactly in accordance with theory,the density which could be achieved with currentdensities as limited by known ion source technology issmall, so that a substantial degree of compressionwould have to be used, subsequent to the injectionphase, in order to raise the density to a high enoughvalue. Second, there are serious problems in achievingspace-charge neutralization of the injected beams earlyin their history, so that the trapping can proceed.These problems are now being studied experimen-tally.18 However, in addition to the use of conventionalion sources, there is reason to believe that plasmainjectors being studied in other parts of the programcan be adapted for use in this high energy injectionapproach, with substantial advantages in the way ofachievable densities.

Traveling Mirror InjectionTo conclude the discussion of injection into time-

varying fields, mention should be made of the use oftraveling mirrors to accomplish injection. If a plasmasource is immersed in a moving mirror confinementzone, a capture process similar to the one mentionedin the previous discussions can occur. This processhas been observed experimentally.19

In this case, the velocity of translation and theshape of the moving mirror field can be chosen tomaximize the injection time. Just as in the case ofinternal adiabatic injection, if necessary, this timecan, in principle, be made much larger than theprecession time.

Capture by Collision or Cooperative ParticleInteractions

Collision effects can lead to capture of a plasma aswell as to losses. This effect has been successfullyexploited in some of the experiments. For example, ifa relatively dense, low temperature plasma is suddenlydischarged into a mirror confinement chamber thenthere is an appreciable probability that some of theplasma will be captured by mutual collision effectswithin the volume. The rest of the plasma will thenescape through the second mirror, leaving behind atrapped component. Since the capture arises fromnonconservative processes, there is no requirementthat the magnetic field be changed to accomplish this

injection, but the confinement time will necessarilybe limited if heating is not employed.

The approximate conditions for collision capturecan be established from relaxation time considerations.If the helix angle of particles crossing the mirrors is e,then particles will tend to be captured which approxi-mately satisfy the condition16

£ 4 2.5xl0-»In/f2 (51)where L is the distance between mirrors, W is themean particle energy in kilovolts and n is the plasmaion density. It is clear that this condition stronglyfavors small helix angles and low particle energies.

Another method which involves cooperative pheno-mena to effect injection has been successfully demon-strated and studied qualitatively. This is the methodof injecting plasma streams across the confinementvolume and causing them to intersect in the center ofthe confinement volume. It is well known, from astro-physical theory and other evidence, that a stream ofplasma can move across a magnetic field, sustained inan evacuated region, by the process of setting up astate of charge separation on opposing surfaces of thestream. This is the same mechanism as that whichoccurs in a flute instability,20 for example, where itenables the transport of certain unstable plasma con-figurations across a magnetic field. Such effects havebeen reported in the literature.21

When such streams enter a conducting medium, forexample another plasma disposed along the magneticfield lines, they can no longer freely cross the field andwill tend to mix with the other plasma. This is theclue to the injection mechanism. This process has beenqualitatively demonstrated in some of the experi-ments to be reported, but the potentialities of themethod have not been fully explored.

In principle, any cooperative mechanism such ascollision and polarization effects, just described, orplasma diamagnetic effects, can lead to capture of atleast a part of an injected plasma. The utilization ofcooperative processes seems to represent a very pro-mising attack on the problem of injection.

Change of Charge StateAnother method by which energetic particles can be

trapped in a magnetic confinement volume is bychange of charge state. For example, beams of ener-getic neutral atoms can be produced by establishedtechniques. Such beams can freely penetrate a mag-netic confinement volume, and would normally passcompletely through it. If, however, a mechanismexists which will ionize a part of all of the atoms of thebeam as it passes through the chambers, the ions andelectrons thus produced could be trapped. One of themain difficulties of the method is the establishment ofan effective breakup mechanism. G. Gibson, W. Lamband E. Lauer22 are studying the possibility of usinga very low residual neutral gas pressure as a means forionizing very energetic atoms, so that a high energyplasma would slowly accumulate in a dc magneticconfinement zone. Their method is potentially applic-able to any steady-state magnetic bottle. Another


method, which might be especially applicable to theMirror Machine, would be to use the plasma createdby one of the injection processes already described toprovide efficient breakup for neutral beanis of some-what lower energies than those considered by Laueret al.

Molecular ions can also be used in the same way toaccomplish injection. Studies at Oak Ridge NationalLaboratory are now being carried forward using anunusual arc breakup mechanism discovered by Luce.23

In this method, capture of energetic molecular ionsresults when they are broken up into their constituentatomic ions, because of the sudden change of radius ofcurvature of the orbits in the field. The potentialitiesand difficulties of this method have not been fullyexplored, but intensive work is being carried out atOak Ridge to generate a high temperature plasma bythese methods. Although Luce's method is, in prin-ciple, applicable to any dc magnetic bottle, thepresent experiments are being done in magneticmirror geometry.

SUMMARY OF EXPERIMENTAL RESULTS

The experimental study of the confinement ofplasma by magnetic mirrors began on a small scale inearly 1952. Preliminary experiments conducted byPost and Steller 24 with transient plasmas produced byrf excitation within dc magnetic mirror fields showedqualitatively that confinement was possible. This wasfollowed up by the experiments of Ford18 and otherson rf trapping of a plasma. In 1953, a serious effortwas undertaken to investigate compression heating ofthe plasma. In the early experiments of Coensgen in1953 and 1954,25 plasma produced by means of anelectromagnetic shock tube was magnetically com-pressed by pulsed mirror fields which rose to peakvalues of about 200,000 gauss in a period of a hundredmicroseconds or so. Time-resolved pictures of thecompression process were obtained and showed thatthe plasma was being stably compressed. It was soonrealized, however, that the magnetic field technologythen developed did not allow effective use of themagnetic compression process with the high densitiesbeing used. Recently, the Los Alamos group26 andKolb27 at NRL have performed similar experiments,but with greatly improved energy storage and pulsedfield technology and have obtained more strikingresults. The early experiments of Coensgen, however,pointed out the basic feasibility of the magnetic com-pression process and provided encouragement forcontinuation of the research.

Because of a realization that, under the conditionsthen achievable, the compression process wouldfunction more effectively with less dense, more highlyionized plasmas, a search was made for improvedmethods of injection. This led to the development, byFord and Coensgen, of early forms of the occluded gassource, a source in which a burst of ionized hydrogenwas released by an electrical discharge on the surfaceof a "hydride" of titanium. Using this source, it was

soon possible to show that substantial heating wasoccurring and that the plasma was being confined forperiods of the order of a millisecond, in rough agree-ment with the expected diffusion loss rates. In theexperiments of Coensgen, escaping energetic radiationand particles were observed that gave direct proof ofthe heating. Confinement was established both by theanalysis of the escaping radiation and by directmeasurement with microwave interferometric tech-niques.26

Theoretical calculations based on certain hydro-magnetic models were being made at about this time.These seemed to indicate that plasmas in MirrorMachines should exhibit hydromagnetic instabilitiesand, under the conditions of the experiments thenbeing conducted, might be expected to be lost in avery few microseconds. The fact that plasmas weredemonstrably being confined, for periods perhaps athousand times as long as the theoretical instabilitytimes, led to the realization that the model used in theearly calculations did not correspond closely enoughto the real situation. Since that time, it has beenrealized that stability may be influenced by: (a) thefinite nature of an actual plasma in a Mirror Machine,as opposed to the infinite periodic plasma of the earlytheory; (6) the existence of conductors or quasi-con-ductors at the ends of the machine; (c) the noniso-tropic nature of the plasma pressure; (d) the non-zerosize of the particle orbits and, perhaps, (e) the existenceof ambipolar effects. Although there is general agree-ment that these factors may explain the observedstability, and calculations have been performed8

(with simplifying assumptions) which predict stableconfinement, the problem is so complicated that nowholly satisfactory theoretical treatment has yetbeen made.

DC Mirror Machine "Cucumber"

In 1954, an experiment of another type was alsotried, specifically aimed at the problem of stabilityand diffusion across a magnetic field. It was desirablefor many reasons to obtain a situation in which theparticle pressure constituted as large a fraction aspossible of the magnetic pressure. Aside from theobvious method of pushing the temperature anddensity up, one way to accomplish this is to reducethe magnetic pressure to the lowest practicable value,so that a low temperature, low density plasma mightsuffice. Thus a dc Mirror Machine, called "Cucumber",was built, with a large-diameter central chamber(eventually 18 in.) and very low confining fields (aslow as 20 gauss). The machine was "capped" withstrong mirror coils (6000 gauss) so that very largemirror ratios could be achieved. In Cucumber theprocess of collision trapping, described in the previoussection, operated very effectively, so that it waspossible to trap a substantial fraction of an injectedlow energy plasma by this process alone. Stable andlong time confinement of the plasma was observed,under conditions where the plasma pressure wasestimated to be several percent of the (low) magnetic


pressure. These experiments helped qualitatively toconfirm the stability picture and to show the influenceof high mirror ratios in reducing end losses. They alsoshowed that, under the conditions of the experimentat least, there seemed to be no anomalously large rateof transverse diffusion of the plasma across the mag-netic field. An example of the confinement data takenin this machine is shown in Fig. 3. The traces representsignals received on a very thin wire probe. The plasmasource was pulsed in all cases, but in only one casewere all the confining fields turned on.

Recent Experiments

The recent experimental work in the Mirror Machineprogram has concentrated on the objectives of (1)studying the heating and confinement process indetail with relatively simple injection techniques,(2) increasing the total usable compression factors toreach higher temperatures and longer confinementtimes and (3) exploration of new high energy injectionmethods.

High Compression

The most fruitful experiments to date have beenthe so-called "high-compression" or "plasma injec-tion" experiments, in which a burst of relatively lowenergy plasma is injected into an evacuated chamberimmersed in an initial weak magnetic mirror field.The field is then rapidly increased to a high final value(in a few hundred microseconds) compressing and

heating the plasma in the manner already described.Results of these experiments are being reported1 7 ' 2 8

so that a brief summary only will be given here.Figure 4 shows one of the devices with which these

experiments have been performed. The machineshown has been used to perform plasma injectionstudies in which the plasma bursts were trapped andstrongly compressed.

These high compression experiments have shownthat an initial plasma burst filling the confinementvolume with a hydrogen plasma to a density of 1011

to 1012 cm" 3 can be stably compressed to a finaldensity of ~ 1014 ст~ 3 by using very large magneticcompression ratios. The compression times used werein the general range of 200 to 500 microseconds. Themagnetic compression factors used ranged to 1000 to 1or even more. The diameter of the compressed plasmacolumn was determined in some of the experiments bymeans of fine wire probes, and by study of the radial

Figure 3. Plasma confinement experiments in dc Mirror fields.Traces represent, starting from the top: (1) Mirror at sourceposition off. (2) Mirror at far end off. (3) All fields on. (4) Mirrorfields on, central field off. (5) Central field on, Mirror fields off.

Tr ce time is 8 msec

Figure 4. Mirror machine used in plasma injection and transferstudies (F. H. Coensgen17)

distribution of escaping electrons. It was found thatthe radial compression of the plasma followed thepredicted behavior with acceptable accuracy. Axialcompression effects were also observed, in roughagreement with the predictions of the compressionequations.

Particle EmissionThe effective confinement time of the plasma was

deduced by analysis of escaping particle fluxesthrough the mirror and by other means, and was foundtypically to be characterized by half-value times ofthe order of a millisecond, with energetic componentsof the plasma being observed to remain as long as20 milliseconds. In some of the experiments, involvingvery large compression factors, the electron energydistribution of' escaping electrons was measured byabsorber techniques, over a range of about 3 kev to120 kev. Figure 5 shows an oscilloscope trace of theescaping electron flux as measured by a collimatedscintillât or. The sweep time is 5 milliseconds. By


Figure 5. Flux of escaping electrons as measured by scintillationcrystal located on magnetic axis, 5 cm outside mirror peak,¿-in. collimator. Sweep time is 5 msec; compression time,

0.5 msec

weighting the observed distributions to take accountof the influence of particle energy on the escapingparticle fluxes, it was possible to fit the distribution toa Maxwellian distribution corresponding to electrontemperatures between 10 and 20 kev. These highelectron temperatures were confirmed by the use ofmicrowave radiometry, which also served to establisha lower limit on the plasma density and to yieldinformation on the axial compression which hadoccurred.

As might have been expected, the peak of the fluxof escaping electrons occurs almost simultaneouslywith the peak of the plasma compression. This is in

Figure 6. High energy X-ray flux at midplane. Compression timeis 0.5 msec; magnetic field decay time to 1/e, ~ 20 msec; centralfield, 40,000 gauss; compression factor, ~ 1000:1 ; sweep time,

10 msec

contrast to the time of X-ray emission, as shown inFig. 6 and discussed below. Although the curves inFigs. 5 and б are generally similar, Fig. 5 does showsome small periodic fluctuations for which the ex-planation is not entirely clear.

It was realized that, in the experiments where thehighest compressions (lowest initial fields) were used,the ion mean energies could not have been as high asthe indicated electron temperatures. In these experi-ments, it was not at that time possible to measure theion energies directly. These could be inferred, however,from the confinement data and from the initial con-ditions, and are believed to have been about 1 kev.In separate experiments, where the conditions weremore favorable for measurement, compression heatingof ions was observed, with final energies as high as 2 to3 kev being observed. The small volume of plasma usedin all of these experiments, together with the relativelyslow time scale of the compressions precluded the

possibility of appreciable numbers of neutrons beingobserved.

In these high compression experiments, as normallycarried out, there seem to be no gross plasma insta-bilities. Measurement of the total energy content ofthe plasma by scintillator and calorimetric techniquesleads one to infer that the plasma pressure at the endof the compression cycle was in some cases about 8percent of the applied magnetic pressure. This is stillnot a sufficiently high value, nor was it obtained undersufficiently general circumstances to conclude thatlarge volumes of a plasma at useful temperature andpressures could be stably confined in a Mirror Machine.

X-ray EmissionAlthough the behavior of the major part of the

plasma was, on the whole, explicable in terms of theconcepts of compression heating and simple collisionlosses, there remain some interesting effects to beexplained. An example of one of these is the fact thatit is possible to detect the radial escape of a smallfraction of the most energetic electrons of the plasmaby the X rays which they produce at the chamber wallin the middle of the machine. This escape rate, thoughslow, seems too rapid to be explained in terms ofordinary scattering because of the high energies of theescaping electrons. At the midplane of the machinethe flux tubes have their largest diameter, so thattrapped electrons which are drifting radially will con-tact the chamber wall first at this point.

In the experiments, a collimated and filtered X-raysignal was detected by a scintillation crystal locatedat the midplane of the machine. This crystal respondedonly to X-ray quanta with energies above about 100

Figure 7. Quenching of electron thermal radiation by introductionof impurity gas (argon). Starting at the top: (1) no argon—basepressure = 5 x 1 0 ~ 7 mm Hg; (2) argon density, ^ 2 x 1 0 n cm" 3 ;

and (3) argon density, ^ 1 0 1 2 cm"3


Figure 8. Mirror machine used for radial injection studies

kev. Examples of the signals observed are shown inFig. 6. The sweep time used was 10 msec. Apart froman initial transient (mostly of electrical origin) it canbe seen that the signals are small for a time of theorder of a millisecond and reach their maxima atabout 2 to 3 msec after the start of the compressioncycle. The compressing magnetic field, however, roseto its maximum amplitude of about 40,000 gauss (inthe center of the machine) in 0.5 msec then decayedexponentially with a time constant of about 20 msec.Thus, the maximum X-ray signal was received muchlater than the time of maximum compression. This isevidence of a slow outward radial motion of an ener-getic component of the heated electrons. Part of thisradial motion, and the subsequent decay of theamplitude of the observed signal, can be identifiedwith the decay and consequent slow radial expansionof the confining field. But there remains a slow driftwhich is not easily explained in this way. Now, in3 msec, a 100-kv electron will have traveled about6xlO 7 cm, and will have executed some 3xlO 8

revolutions in the confining field. In the experiments,the radial distance between the chamber wall and thesurface of the compressed plasma is about 7 cm. Onecan therefore estimate that the radial drift velocityof 100 kev electrons could have a maximum value ofabout 2.5 x 103 cm/sec. Although this inferred radialdrift velocity is very small compared to the electronvelocity itself it is still too large to explain by ordinaryscattering, both since the Coulomb scattering crosssection of 100-kev electrons is very small (about6 x 10~23 cm2 on hydrogen) and because multiplescattering would provide a very strong selectivemechanism against any particle reaching the wallprior to being scattered into the escape cone. On theother hand, a quite small electric field, fluctuating at

a frequency comparable to or slower than the electrongyromagnetic frequency, could induce a randomlydirected crossed electric and magnetic field driftwhich would cause the observed radial transport.Such a drift would not be expected to interfereappreciably with the normal mirror binding of theenergetic electrons. Since it is known that the numberof electrons reaching the wall is small compared to thetotal number, and that the plasma density outsidethe compressed column is much smaller than in thecolumn it appears that the mechanism causing thetransport operates primarily in this region, and not inthe main body of the plasma. It is tempting, there-fore, to ascribe the transport to fluctuation phenomenaassociated with the low density of this exterior plasma.

In the high compression experiments it was possibleto show that the heating effects were quenched by theintroduction of relatively small particle densities ofhigh-Z gases, such as argon. This quenching effectbecame effective at about the density predicted bythe theory of the radiation losses discussed earlier;i.e., at about 3 x 1011 cm~3 for a field rising in 0.5 msec.Figure 7 illustrates this quenching effect as measuredby an 8-mra microwave radiometer. Shown are tracesfor three argon pressures. Peak signal is proportional

t o T e .The quenching is noticeable at 2 x 1011 cm"3 and is

complete at 1012. It is possible to conclude from thesemeasurements that the normal impurity level hadbeen reduced to the point that it did not substantiallyinterfere with the compression heating process.

Low-Energy InjectionTwo injection methods were used in these high

compression experiments. In the most commonlyused method, a low-density, 5- to 10-/xsec burst of


hydrogen plasma was injected in an axial directionthrough one of the mirrors from a source located justoutside the mirror. Capture of the plasma was accom-plished, in the manner described in the previoussection, by starting with a weak initial field andrapidly increasing it. Continued increase of the fieldled to the compression effects which have beendescribed.

In another injection method used, synchronizedbursts of plasma were injected across the chamber,aimed toward the axis of the machine. By carefullytiming the confining fields with respect to the injectedbursts, it was possible to capture much of the plasmaat the time the beams intercepted each other in thecenter of the machine. Figure 8 shows two views of amachine used for studies of radial plasma injection.

High-Energy InjectionOther experiments have been carried out, which are

aimed at studying the injection of high energy ionstreams into a Mirror Machine.18 Because of thedesirability of extending the time scale of the injectionphase, close attention had to be paid to the role ofcharge exchange processes between trapped ions andthe background gas, since such processes represent apowerful loss mechanism in this type of experiment.The apparatus developed for this experiment, shownin Fig. 9, was therefore designed with unusual attentionto vacuum problems. A thin-walled stainless steelvacuum chamber was used, with metal gasketsthroughout. The entire chamber was baked out atelevated temperatures. With these precautions it waspossible to reach base pressures below 10~9 mm Hg.

Observations were conducted both with pulsed andwith dc fields. Careful studies of ion orbit trajectorieswere made with collimated sources at ion energies ofabout 10 kilovolts. These studies confirmed the generalpicture of the ion behavior in the machine, anddemonstrated the precession effects alluded to in- thediscussion of injection methods. An attempt was thenmade to increase the current density and to trap ahigh current density beam. It was then found thatserious space-charge "blowup" effects were occurringin the vicinity of the source leading to the loss (bystriking the back of the source) of most of the injectedbeam. Although it was proved that a small fractionof the beam was actually trapped and continued tobe reflected back and forth in the machine for severalthousand reflections (several milliseconds), insufficientbeam was trapped to call the experiment a success.At this time the blowup process is being studied, andmethods are under consideration to improve thedegree of space-charge neutralization.

CONCLUSION

One might characterize the theoretical and experi-mental philosophy of the Mirror Machine program todate by a few words: "An attempt to study pheno-mena in a high temperature plasma by the extrapola-tion of ideas and methods already established in the

field of charged particle dynamics into the realm ofplasma dynamics." Thus the attitude has been to setup situations which were clearly and demonstrablyworkable at sufficiently low particle densities and thenproceed to push the density and temperature as faras possible into a regime where the reactions of theparticles on the confining fields is no longer negligible.Although such an approach necessarily implies theuse of externally generated confining fields, it hasmuch to recommend it. One deliberately picks thosesituations where symmetry and the method of con-finement tend to minimize the possibility of whatmight be called "uncooperative" phenomena in theconfined plasma, and where collision effects play asecondary role throughout the experiment. The ad-vantages gained are: simplicity of understanding, thepossibility of starting with very low densities andworking up, and the likelihood that simple conserva-tion laws will be adequate to define the essentials ofthe behavior of the plasma. The price which one paysfor this is to introduce new technical problems of amore difficult nature than those encountered in someof the other approaches, for example, the simplelinear pinch, and to be confined to working in a regimewhere the basic time scale is essentially set by a singlecollision time of the plasma ions. In the long run,although it clearly represents a compromise, this

-regime might still turn out to be a necessary one forthe attainment of fusion power.

Judged in terms of the eventual goal of fusion power,progress thus far in the Mirror Machine program islimited indeed: where seconds of confinement timewill be needed milliseconds have been achieved; whereion mean energies of a hundred kilovolts or morewould be needed, perhaps a kilo volt has been attained.But judged in terms-of the growth of understandingof the basic processes and problems of injection,heating, and confinement, substantial progress hasbeen made. The goals for the future still lie in thedirection of understanding these processes and their

Figure 9. Mirror machine for injection of high energy ions.Entire vacuum chamber may be baked to reach pressure of

10~9 mm Hg


limitations more fully. From this understanding anassessment can be made of the future of the MirrorMachine in the quest for fusion power.

ACKNOWLEDGEMENTS

I wish to acknowledge especially the work of severalmembers of the Mirror Machine experimental group:

in the high compression experiments, Drs. Fordand Ellis; in the multi-stage compression work, Dr.Coensgen and William Cummins; in the high energyinjection studies, Drs. Damm and Eby; and in thedevelopment of microwave measurements, CharlesWharton. Throughout the work Dr. Herbert Yorkand Dr. Edward Teller have maintained a constantinterest and encouragement.

REFERENCES

1. R. F. Post, Sixteen Lectures on Controlled ThermonuclearReactions, UCRL-4231 (February 1954). Some of thismaterial was presented at the American Physical Societymeeting held at Washington, D.C. (May 1958).

2. H. Alfvén, Cosmical Electrodynamics, Oxford (1950).3. R. F. Post, Controlled Fusion Research—An Application

of the Physics of High Temperature Plasmas, Revs.Modern Phys. 28, 338 (cf. p. 355) (1956).

4. E. Fermi, Astrophys. J. 119, 1 (1954).5. L. Spitzer, Physics of Fully Ionized Gases, Interscience

tracts on physics and astronomy No. 3, Interscience-Pub.,N.Y. (1956).

6. J. Berkowitz, H. Grad and H. Rubin, Problems inMagnetohydrodynamic Stability, unpublished report.

7. M. Kruskal, On the Asymptotic Motion of a SpiralingParticle, Nuovo cimento (to be published).

8. A. A. Garren, R. J. Riddell, L. Smith, G. Bing, J. E.Roberts, T. G. Northrup and L. R. Henrich, IndividualParticle Motion and the Effect of Scattering in an AxiallySymmetric Magnetic Field, UCRL-8076, P/383, Vol. 31,these Proceedings.

9. D. Judd, W. McDonald and M. Rosenbluth, End LeakageLosses from the Mirror Machine, AEC report WASH-289,p. 158, Conference on Controlled Thermonuclear Reactions,Berkeley, California (February 1955).

10. A. Kaufman, Ambipolar Effects in Mirror Losses, AECreport TID-7520 (Part 2), p. 387, Conference on Con-trolled Thermonuclear Reactions, Gatlinburg, Tenn.

• (June 1956).11. R. F. Post, Revs. Modern Phys. 28, 354, 355 (1956).12. R. F. Post, A Theory of High Compression Magnetic

Mirror Experiments, AEC report TID-7503, Conferenceon Controlled Thermonuclear Reactions, Princeton(October 1955).

13. E. Teller, private communication.14. R. F. Pjost, Radiation Processes in a High Temperature

Plasma (to be published).15. R. F. Post, Survey of Injection and Other Problems in the

Mirror Machine, AEC report WASH-289, p. 198, Con-

ference on Controlled Thermonuclear Reactions, Berkeley,California (February 1955).

16. F. С Ford, An Enhanced Magnetic Mirror Machine,UCRL-4363 (July 1954).

17. F. H. Coensgen and F. С Ford, Pyrotron Plasma HeatingExperiments, UCRL-5045, P/378, Vol. 32, these Pro-ceedings.

18. С. С. Damm and F. S. Eby, Pyrotron High Energy Experi-ments, UCRL-5046, P/379, Vol. 32, these Proceedings.

19. F. H. Coensgen, Energetic Plasma Injection and TransferExperiment (to be published).

20. M. N. Rosenbluth and С L. Longmire, Annals of Physics,1, 120 (1957).

21. W. H. Bostick, Experimental Study of Ionized MatterProjected across a Magnetic Field, Phys. Rev. 104, 292(1956).

22. G. Gibson, W. A. S. Lamb and E. J. Lauer, Injection inThermonuclear Machines Using Beams of Neutral Deuter-ium Atoms in Range 100 kev to 1 Mev, UCRL-5047, P/380,Vol. 32, these Proceedings.

23. P. R. Bell, J. S. Luce, E. D. Shipley and A. Simon, TheOak Ridge Thermonuclear Experiment, P/344, Vol. 31,these Proceedings.

24. R. F. Post, AEC report WASH-115, p. 81, Conference onThermonuclear Reactors held at Denver on June 28,1952, J. S. Steller collaborated on making the measure-ments reported here.

25. F. H. Coensgen, Progress Report on Toy Top Experiments,AEC report WASH-184, Conference on ControlledThermonuclear Reactions, Princeton (October 1954).

26. W. С Elmore, E. M. Little and W. E. Quinn, Neutronsfrom a Plasma Compressed by an Axial Magnetic Field,P/356, Vol. 32, these Proceedings.

27. A. C. Kolb, High Temperature Plasmas Produced by theMagnetic Compression of Shock Preheated Deuterium,P/345, Vol. 31, these Proceedings.

28. С. С. Damm, J. С Howard and С. В. Wharton, PlasmaDiagnostic Developments in the UCRL Pyrotron Program,UCRL-5048, P/381, Vol. 32, these Proceedings.

Mr. Post presented Paper P/377, above, at the Con-ference and added the following remarks on recentmeasurements made with a multi-stage mirror machine.

In order more effectively to use the magnetic com-pression technique we have constructed a machinewhich employs three successive stages of magneticcompression. This is accomplished by using magneticmirrors at each stage both to compress the plasma andto transfer it rapidly along the machine. In this wayit is possible to take maximum advantage of the com-pression process without requiring very large con-denser banks. What is more important is that it has

been shown experimentally that it is possible toarrange for such a rapid sequence of transfer opera-tions as to leave behind most of the neutral gas andthe impurities which may be produced in the injectionregion. In this way a nearly pure hot hydrogen plasmacan be obtained, minimizing charge exchange losses.

Figure 10 shows an outline drawing of the multi-stage machine and also shows the sequence of mag-netic field changes which are used in the successivecompression and transfer operations. Figure 11 showsa view of the entire machine.

By means of microwave interferometer stations


Axial position - feet

V 9" pulse

coil

Ly pulsecoil coil • \J/

18 pulse coil

Figure 10. Equipment schematic and magnetic field sequences inmulti-stage experiment

Figure 11. Multi-stage compression and transfer apparatus

which are placed at several points along the machine,it is possible to follow the plasma during its successivecompressions, except in the case of the final and mostintense compression. It is found that the densityincreases at each stage in relatively close agreementwith theory. This encourages us to believe that thesecompression and transfer operations do not, as yet,give rise to plasma instability.

Electron heating and confinement effects have alsobeen measured in this machine and have been foundto be similar to those observed in the single stagemachine. Also, under some circumstances, it has beenpossible for us to measure the type and the energydistribution of ions escaping from this machine.Figure 12 shows measured energy distributionsintegrated over the time of the compression pulse. Incase {a) the plasma burst was injected parallel to themachine axis, so that only a small fraction of theinitial ion kinetic energy appeared initially in rota-tional motion. Even so, the heating produced by two

stages of magnetic compression can clearly be seen.Figure 12(6) shows a similar analysis taken withsources directed nearly across the magnetic field. Inthis case, even with a single compression and transfera pronounced ion heating was observed.

These compression experiments have shown us thatwe can inject and heat a plasma by carrying it throughmanipulations by means of magnetic mirrors. We feelthat these results are not only helpful to us in learningabout plasma heating and confinement, but also havea bearing on the possible future practical use of amirror machine. This is because the operations oftransfer and compression, which are carried out adia-batically, should also in principle be reversible, so thatenergy could possibly be extracted directly from theplasma in electrical form by performing controlledtransfers and radial and longitudinal expansionswhich are just the inverse of the operations we havedemonstrated.

In all these compression experiments, although theion energies reached were substantial, the plasmavolumes were too small to yield any appreciablenumbers of neutrons. The equipment is now beingmodified to permit the confinement of larger plasmavolumes at higher final ion energies.

A - 515 ev. averoge for plosma

В • 310 ev. average for source

600 800 1000

D+energy- electron volts1400 1600

-Source and 18" pulsed field

500 1000 1500 2000 2500 3000 3500 4000

Figure 12. Energy distribution for escaping D+ ions: (a) for longi-tudinal injection, (b) for radial injection

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