IEEE Transactions on Plasma Science, Vol. PS-6, No. 4, December 1978

INSTABILITY LIMIT ON THROUGHPUT FOR CALUTRONS.(ELECTROMAGNETIC ISOTOPE SEPARATORS)

Igor AlexeffUniversity of TennesseeKnoxville, Tennessee

Received 4/25/78ABSTRACT

Using a model of two interpenetrating ion streams in a background of magnetically immobilized electrons, Iobtain a density threshold for separation disruption that agrees with the values observed. The density thresholdfor disruption is far above that for instability production. The inference is that in ordinary operation a calutronis operating in an unstable mode, but this unstable level is tolerable.

In previous computational and experimental work at theY-12 isotope separation plant at Oak Ridge, evidence wasobtained that the limit on isotope separation throughput wasdue to an instability in the beam.1 Experimental studies of thebeam environment led to improvements in isotopethroughput.' In this work, I accept the postulate that thelimits on throughput occur in the beam. Consequently, Iderive a model that apparently predicts the disruptionthreshold, and also suggests methods of overcoming it.

The basic model has been briefly and incompletelyoutlined in an unpublished but classified report.2 The basicidea is that the isotope separator output consists of twointerpenetrating ion streams as shown in Figure 1. The twostreams are moving axially at different velocities, and alsomove radially as they separate. The relative velocity in theaxial and radial direction is approximately the same. Electronsare present to provide bulk neutralization of the ion streams.However, due to the magnetic field, these electrons are unableto neutralize ion waves propagating across the magnetic field.Quantitatively, the electrons can respond only to frequencies>ce' and the instability occurs at a much lower frequency,-pi' Consequently, the phenomenon is easily described in

terms of a two-ion-stream interaction. These ideas are nextdeveloped quantitatively below.

Note U '"'lons Drift RadiallyInward Relotive To

t34P U I31ons

UU dFocus

.. seams Overlapo Far Much Of Their

Source P odilL

Figure 1. Geometrical diagram of the beams in an isotopeseparator.

A brief review of theory is given for the benefit of isotopeseparator workers.

Consider first the threshold for a 2 stream instability witha dense stream n1 and a rare stream n2. The dispersion relationis found in the center-of-mass of the dense stream nl. We have,

2

1=9 ~+2Cd= 2

2

(w-k.v)2(1)

Here 2pl wpp2 are the plasma frequencies of beams n1,2n2 respectively, where 02 is given by ne- , is the angular

frequency of the unstable wave, k is the wave number of theunstable wave; and v is the small velocity difference betweenthe two ion species.

Plotting the left hand (1) and right hand (f(S)) sides ofEquation 1, we get the graph in Figure 2. Solutions are givenby the circled intersections. In the case of instability onset, itis clear that the unstable solution will occur for w close to kv.Therefore, let w = kV-6w, where 6w is a smaller number. Inow get Equation 2, below,

21 = .W p

(kv-6w)2

2+ p2

(6)2(2)

This can be rewritten as shown below,

(6wd 2[(kv)- 2 ] - 2kv(6w,) 3 + (6W)4=,2 (kv-6w)2p1 ~~~~~P2 (3)

d rop

On the right hand side, the 6w in the parenthesis is a

0093-3813/78/1200-0539$00.75 ( 1978 IEEE

539

correction term to a small number and is dropped. Considerthe above to second order in the small term 6 ,

22 W p2 (4)

(kv) 2 2p1

In this case, if kv < 6w becomes imaginary (growingwaves) and approaches infinity. Hence, gain appears to go toinfinity! Actually, looking back at Equation 3, I find merelythat the 2nd order term has droped out, and one needs to goto third order, which is:

(6w) 32

2< kv2

This is the condition for peak temporal gain.As I have set kv =copl, to obtain Equation 5, I find,

(6w)3 1- 2 p(6w) 2 w p2 wpl

or

To estimate the instability threshold, I note that thewave-length of the instability, 2ir/k, must be smaller than thesmallest dimension a of the beam, or the equations becomeinvalid and growth terminates. Hence, for instability to bepresent, k> 2a . But for large k, Equation 4 shows that the

asystem is stable,( 6w )2 is positive, and 6w cannot be imaginary(wave growth). Thus, the instability threshold is given by,

a k <op1 (8)

Choosing a to be the smaller cross sectional dimension of thebeam (a few cm), and v to be the velocity difference3 betweenspecies (the least unstable case) one finds a value forcpl thatcorresponds to an ion density in the beam several orders ofmagnitude lower than that common in operating practice.Thus, one can say that the standard calutron is operating in anunstable mode, but this instability is not sufficient to disruptthe beam.

We already have the maximum growth rate for theinstability in Equation 7. This value is remarkable, in that it isindependent of the velocity splitting of the beam componentsv. Waves along the beam direction where v is large, and wavesacross the beam where v is small, all grow at the same temporalrate, but with different wavelengths! The most dangerous wavehas the longest wavelength, because for a given amplitude, itproduces the largest disruptive voltage.

To complete the model, one needs to know the time thewave has inwhich to grow, the wave's initial amplitude, and theamplitude required at the collector to cause beam disruption.

The time available for instability growth in an isotopeseparator having 1800 focussing is half the ion cyclotronperiod. Thus,

t = 12

1(n2i 3

(9)

In this time, the electric field must grow from some initialvalue Eo to some final value Ef sufficient to disrupt the beam,or,

(10)(6)

The gain is given by the imaginary component of theabove, so the wave grows as,

wlPI tThe initial value Eo can be computed from shot noise -

the granular nature of the beam itself. Consider a calutron(7) separating uranium,3 with 20 kV on the extractor, 100 mA

540

6w =-1

1

2

+i. r3

_ 1 _-- } 1

21/3

exp . In Il(rY

27T /w

n1/3

.3- 2Li

1 21E = E exp 2 wf 0 24/ 3 nI p c

beam 1, and the beam spread over 40 cm2 (4 X 1 0' m2) A.I find vo (2eV) =2 1 X5 meter/sec. I also find

v02 _m - 800 meter/sec.

Here, m is the mass of a U238 ion (238X1.67X1027kg),and 5m is the mass difference between U238 and U235(3 X 1.67 X 10-27kg). Also, I find

of two beams takes place when instability saturation occurswith other catastrophic effects such as trapping of beam ionsin electrostatic waves. I can estimate saturation as follows: Thewave grows as long as the particle stream is in resonance withthe waves or kv = Wpl , as shown by the resonance conditionof Equation 4. As the wave grows, v is decreased by a velocityperturbation bv. I can estimate bv from the equation ofconservation of energy, assuming the energy goes exclusivelyinto electric field energy,

1 2v 2 =-22in n2 - r~mnn2 (v - by) __

(1 1)

This equation may be approximated as,

1.23 X 1001 /m3c

2

m n v bvy 2Using this value in kv = wp l1 we find

k p1_ 3.00 X 10kmi n : v OiO80J

6I wish to compare this velocity decrease with the

half-width of the resonance peak given by Equation 3. Thisequation to third order is rewritten as follows:

= 1.68 X 10 3m .

Assuming kv - PIp$ obtain,2

kv - p1 = , + p2wplPI 2In a slab 4 X 103 m2 in area A(beam cross section) and half awave-length long, there will be the following number of ions,

nAX

w( )2 k v~5Lo(+ P

2 2

( )

(6 )2

Putting in half the peak value of 6w given by Equation 6, find= 4.13 X 109

kv - pThe rms scatter corresponds to the square root of the abovenumber,

n2 1/3flu

or the velocity decrease is given by

bn = 6.43 X 104

The electric field produced by such a slab of charge is,

E = 6n = .145 volt/metere0 2 0A

Consider disruption. The termination of the interpenetration

k. - :P-l , so 6 vn2 1/3

( )nl

V.

(14)

Equating the two values of bv. I find the condition of velocitydecrease equal to the half-width of the resonant peak to yield,

2 22n2 l/3-(2):

flu-

0 E2f4 (15)

541

= In - n _ --I1 A v0e

or

max

(12)

_ 271k

(13)

P1

3. ;'4 X 10( 3 tri -1

1 ( k v i. L'i ) ( kv G)PI )2kv PI

which is the same value for saturation given in the literature4.Note that this same mathematical treatment can be used tocompute the new saturation level for the Buneman instability5and indeed this treatment was adapted from Ref. 5. One finds

1 2m e 1/3 1 2

To compute Ef, note that n, = 1.23 X 1015I/M3,n2 = -1- n1, mn = 238 X 1.67 X 10-27 kg, v= 800meters/sec. I obtain for Ef,

1 23X1015 ~27 2 11 1/-2?3XO.41 ) (238 x 1 .67 x 0 ) (800)(li) X 4 X- ~~~~~~~~~~~8.85 X'

Eqfa10s lVowt/m.With the above values, I can rewrite Equation 10 as follows:

EIn ( f)

e

2 4/3

F3

n 1/3

-n) cC =

p1

or

{ E1 ne E

2 4/3

3

n( )n2

1/3 2B1 600= n

1j

(16)

Using the previous values, and B = 0.7 Tesla3, getni = 3.72 X 101 5/m3 or 3.73 X 109 cm-, as shown below;

account for the beanmi's "tailvr'aqrrt prnnt losS 01separation due to a coherent radial oscillation of the beamfocus across the collector slits, as observed by Shipley, Yonts,and others.'

Ways for overcoming the above instabilities to provideincreased ion throughput have been stuidied suiccessfully hoththeoretically and experimentally, arid will be discuLssed ini afuture publication.

The author acknowledges valuable discussions vvith E. D.Shipley and 0. C. Yonts of the Oak Ridge NationalLaboratory, and also excellent suggestions by the arbornyrnousreviewer.

.f%)

I I

W _/IJ

/ I/ ~%II

J/ ~ % II

-- - 1.

-4zr

Figure 2. Graphical derivation of solutionis of EqLuation ?.(Circled)

0

1/3 2 8.85X 2

S 238 X 1.67 X 10

3.72 x olS HI or' 3. 72 x 10cm

Thus, the total beam current would be z n e a v0 = 302 mA.This value approximates the values observed. Note that initialand final assumptions of Eo, Ef are not critical, as they appearin a logarithmic term.

A few final points are of interest. We find °p >> "ICUnder this frequency condition, the ions are capable ofignoring the magnetic field, and the use of the unmagnetizedtwo-beam model (Equation 1) is vindicated. Alsowce > Wpe > "p' so the electrons are immobilized by the

magnetic field.Note that other instabilities may also be present

(density-gradient-across-beam, etc.) .6 However, the instabilitydiscussed here is sufficient to account for the observed effects.Also, it is an instability composed of plane waves, and should

REFERENCES

1. Thermonuclear Division Annual Report ORNL 3652 PP55-56, April 30, '64., IBID, ORNL 3760, PP 45-47, Oct.31, '64, U.S. Patent 3260844, July 12, 66, E.D. Shipley,O.C. Yonts, A.M. Veach Ref 5 A. Hirose, Plasma Physics(Pergamon) 20,481 (1978).

2. I. Alexeff, "Instability Limits on Throughput forCalutrons", OR N L TM-4006, Dec., 1972.

3. I. Alexeff, Journal of Applied Physics, 44,5492, 1973.4. V. D. Shapiro,JETP, 17,416, 1963.5. A. Hi rose, on Amnplituide Saturationr of Bunemtian

Instability and Anomalous Resistivity, to be published itrPlasma Physics.

6. A. Hirose and 1. Alexeff, Nuclear Ftision, 12,315, 1972.

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{ 1lf ( 310 2 4/3]45) 13

IIIII