Electron cooling

Nuclear Instruments and Methods 215 (1983) 27-54 North-Holland Publishing Company

27

ELECTRON COOLING Allan H. SORENSEN and Ejvind BONDERUP

Institute of Physics, University of Aarhus, DK-8!X10 Aarhus C Denmark Received 6 August 1982

A theoretical analysis is given of the phase space compression of a beam of (anti)protons thermalizing in a cold, dense beam of electrons moving parallel to the protons with the same average velocity. The electrons are guided and confined by a longitudinal magnetic field. In previous discussions, the interaction of a proton with the cooling electron gas has been treated as a series of independent two-body collisions . This leads to divergences at large impact parameters, due to the neglect of polarization forces . and the difficulties are particularly severe because of the presence of a magnetic field . These problems may be avoided through the application of a dielectric description of the gas, and such a treatment is presented here. For a given magnetic field strength collisums with electrons may roughly be divided into two groups, one appearing as for vanishing field, and one corresponding to an infinitely strong field . Since the dielectric description is quite simple in these limits, they are discussed first as a basis for the treatment of the general situation. The presence of the magnetic field gives rise to a significant reduction in cooling time, and to become thermalized a penetrating proton must typically only spend of the order of lt? - 1 s in the cooling gas, as measured in the proton rest system Numerical results are presented for a standard case .

1 . Introduction

This paper contains a theoretical discussion of the possibilities of reducing the phase-space volume occupied by an ion beam through interaction with a cold, dense electron beam, i.e., a dense electron beam. which is strongly focused in momentum. The experimental setup for this electron-cooling effect is an ion-storage ring with a straight section, through which a continuous flow of new cold electrons i maintained at the same average velocity as the ions [ 1,21 since the most rapid cooling is obtained when the relative electron-ion velocity is small . The electrons are kept together and guided through the cooling section by an external longitudinal magnetic field . The electron-cooling process has been analyzed theoretically in many papers, originally for the simplified case, where the magnetic field is neglected [3,4]. Later, also the effects of this field have been considered [5,61, but in general the treatments are rather ;rude and simply consist of combinations of results obtained in the two opposite limits of vanishing and infinitely strong fields. An exception is the paper by Ogino and Ruggiero [7] . However, these authors only carry their results beyond a very general framework in the limit of an infinitely strong field. In all cases, the screening of fields resulting from electron-electron interaction is considered merely in connection with the introduction of a cut-off in otherwise divergent integrals, and the procedures adopted are based upon qualitative arguments only. In the present paper, we shall show how such difficulties may be overcome through the application of a dielectric description of the electron gas. Particular attention will be paid to the simple limits of vanishing and infinite magnetic-field strengths, and we shall elucidate the connection to the binary-encounter model, which has been applied in the past and which shall also serve as a basis for our initial considerations . The cooling process is characterized by a cooling time and a final ion-beam temperature . and these quantities may be obtained when friction (the drag) and diffusion in ion-velocity space are determined . In the system of vanishing average electron velocity, the so-called particle frame. the electrons with a tçpical temperature, T-- 1 eV, tend to stop penetrating charged particles such as the (ant0protons considered here. In practice, the response time of the electron gas is shorter than (or of the order of) the time it takes a proton to traverse the cooling section, and we shall therefore assume the proton to be screened as in an electron gas of infinite extent . The drag force acting is the result of electron-proton collisions. and in the 0167-5087/83/0000-0000/$03.00 C 1983 North-Holland

A . H. Ssrensert. £. . Bottdenap / Elearon eoolinq

28

absence of external fields, it is approximately given by [8] F- 4srae a L n: F ac vo .

K,

d' t~ f(q:) ~ "

a=-2,

10
K'

111

-ur

s.

-

vp >vC , vF,
The quantity r~, is the proton velocity and n the number density of electrons with mass in, charge - e, and xvlocity distribution f(v,,). Here and everywhere in the following, all symbols refer to the particle frame unless otherwise stated. L is an appropriate logarithmic factor. For an isotropic Maxwell distribution, the corresponding slowing-down time for a proton T --- Kl/v,,)dv P/dtj- ' is obtained by standard techniques [8] . If the influence of proton ("betatron" and "synchrotron") oscillations in the storage ring is neglected, the slowing-down time as observed in the laboratory frame is mm l _ _7`_ -x 'ti e4 în- L

I 4 zr vP , 3 (TIM) 442 ir

VP 3/2,

tip

> vC ,

(2a )

<

(2b )

Cie .

Here A!, n L, and rl denote the proton mass, the number density in the laboratory of the electrons moving with average velocity z,, and the fraction of the circumference of the storage ring occupied by the electron beam, respectively. The relativistic factor Y' = I/( 1 - v o/c`) originates partly in time dklation, T,. = yT, and partly in Lorentz contraction, "L = yn . At high velocities, the cooling becomes slow by a factor (rf,/(t' )' "=)z as compared to the low-velocity limit. When electrons from a thermal source are accelerated to high energies, the velocity spread in the direction of motion becomes extremely small, and the resulting velocity distribution is maintained during the passage of the cooling section . The introduction in eq. (1) of a "flattened" distribution, the "electron disc". where the thermal motion survives only in the transverse directions, leads to a reduction of the value of T for vp < v, For a transverse pro ,,un velocity, the reduction factor is 37x/4 = 2.4, and for a longitudinal . the Factor is as large as (3v'-ir /2)(v'/v,,), where v denotes the transverse root-mean-square (rms) velocity electron velocity related to the temperature through the relation T = ' my- . The standard textbook formuk. (?; is based on a binary-encounter model with pure Rutherford scattering above a certain cut-off ang'e and with a velocity-independent logarithm L. In section 2, we shall introduce an improved binary description, which takes the flatness of the electron-velocity distribution into account, and which is based on the scattering cross section corresponding to a properly screened potential. The drag force obtained agrees to within a few percent with the simple result except for very low longitudinal proton velocities . i n addition, in the absence of external fields, the drag derived within a dielectric description agrees with the binary force to an accuracy - l /L. The importance of fluctuations in momentum transfer in the slowing-down process is estimated, and we determine the diffusion tensor, which also agrees with standard results [8] to within terms of relative magnitude - 11L . The final proton-beam temperatures <_ T correspond to extremely small velocity spreads . When a magnetic field is introduced, the cooling process is changed . In the binary-collision treatment, collisions are usually divided into two groups, according to whether they are fast or slow with respect to the Larmor period of the electron, F = Ffa,t ., Fti1". and these groups are treated as if the magnetic field were of vanishing or infinite strength, respectively. At infinite magnetic-field strength, the transverse degrees of freedom of electrons are effectively frozen, and only the longitudinal velocities enter the collision problem. Since these velocities are zero with the flattened distribution, we obtain F""' ac c - ` . r

A. H. Sorertsen, E. Bo»derup / Electron coohiig

At high velocities, vp > v, the introduction of a magnetic field therefore only leads to minor changes in F. In contrast, at low velocities, vp < v, the contribution from slow collisions becomes much larger than the force in eq. (1), which decreases with vn, and cooling times are significantly reduced . Since the binary-encounter model turns out to be somewhat ambiguous with respect to the slow-collision contribution, the dielectric description is needed to determine the drag force in the presence of a magnetic field. In section 3, we shall show how the two pictures may be combined in a useful manner to give simple and accurate results . Acceptable cooling times for stacking purposes are typically a few seconds . With realistic electron-beam performance, these short times are obtainable only if the initial electron-proton velocities are already of the order of the thermal spread of the electron velocities, and therefore we shall mainly consider such cases in the following. In future applications to, for example, antiproton beams, electron cooling may be applied as the second stage in a process, where the initial cooling is the result of so-called stochastic cooling, which is most effective for beams with a large momentum spread [9]. In the following sections, all calculations are performed in the particle frame, and the result of the simple transformation to the laboratory, cf. eq. (2) . is not presented . Electron beam space-charge phenomena, which have been discussed in refs. 4 and 7, as well as "intrabeam scattering" within the proton beam will not be considered. The cooling effects are illustrated by numerical examples, in which we choose as our standard an electron-beam temperature T = 1 OV. a number density n = 3 x 10' cm - 3, and a magnetic field B = 700 G, these values being t ;~pical in an experimental situation .

2. Electron cooling in the absence of external fields 2 .1 . Drug force : Binary-encounter model

Consider a proton of velocity v,, penetrating a spatially homogeneous electron gas ~%ith -elocit\ distribution .f(v,). In the proton rest system, which may be considered an inertial frame because of the smallness of the electron-proton mass ratio, an electron of velocity r, corresponding to a relative electron-proton velocity ~= v, - vp, is scattered by the proton through an angle 0 and thereby acquires a velocity is, ', where ti,' = tit . Within a time dt . the average momentum transfer to the proton is -dp= (dnrv dt)o(9, w') dS2(x,' - %,)in .

where a(0, w) is the differential scattering cross section, df2 the solid angle, into which the electrons are scattered, and do = of(v, )d- v,, the number of electrons per unit volume with velocity in the range [v,,, v, + dvj Integration over scattering angle 0 and relative velocity w yields the drag force F=rnn

J

d v~f(uN)K'atr(tit')x' .

(h)

where the transport cross section is defined by a,=

" (1-cos0)a(0 .K, )2irsin0d0 .

Ju

(?)

In the particle frame, the rms electron velocity in the beam direction is typicalb. - 10 = times less than the orthogonal rms velocity v, cf. ref. 10, and therefore the electron-velocity distribution in the _- direction ma, be represented by a S function. Since the electrons originate from a thermal source . a reasonable chowe k~f the distribution f is f(Ur)

_ 1 _ 2 - 4rv ,- exp(-Lrl1v2)8(vrCl-te,+

30

A . H . SG^msen, E. Bmderup / Electron cooling

Hence, in the particle frame, the total drag force decelerating the proton is given by ntn

d;

t~

exp

( _relv')-,a (-')w,

%, =v~ -%,

where q"-9 = 0. For symmetry reasons, the average force is restricted to the x-z plane, where the x direction is determined by the projection of vp onto the plane of electron velocities . The expression (9) for the drag may be simplified into a one-dimensional integral through a change of variables, We separate into transverse and longitudinal velocities, w = ( wl , w~. ) = (v" -- vp,. , -- vp cos ®), where w_ = w- and where 9 denotes the polar angle of up. Introducing n'1 as the variable of integration, .e d' = d2 w,. = dw 1 w 1d%, w, = w1 cos T, vé = vpsin'O + wl + 2 t)p sin 19w, cos T, we may perform i ., the integration over the azimuthal angle qD analytically with the result F= -

2mn

m- exp(-

t. 2

sin29

Jo

- 1 1,(2vp sin en, l w2 ),VV

dw l is-, COS

exp -

v

;

wo', (W )

91,(2vp stn 9w 1. /e2 ), .

10

Here, 1, and 1, denote modified Bessel functions of the first kind, and the relative velocity iv is given by cos29)'-"'. The two force components refer to the x and z directions, respectively .

w = (%. i + vp

?.?. The scc,-tering cross section Let us consider the problem of obtaining an appropriate electron-proton-scattering potential and the corresponding transport cross section. The proton polarizes the surrounding electron plasma to an extent determined by the dielectric function F of the medium within a linear description, and this results in a screened Coulomb potential . in which the electrons are scattered . Capture of electrons into bound states may be neglected since, first, we know, partly from experiments, that such processes take place at negligible rates 11. 11 ], and . second, our main goal is to cool and stack antiprotons . For a proton of high velocity in the particle frame tip > v, the potential from the perturbing charge is essentially screened out beyond the adiabatic cut-off distance A = v,/wp,, where wp, = (41rne2/m)'i2 denotes the plasma frequency of the gas . This result is well known and will also emerge from the discussion of the dielectric functions to be presented in section 2.4. In the opposite limit of a proton at rest, we may expect a potential similar to the exponentially screened Debye-Hückel potential valid for an isotropic Maxwell gas. In this limit, the screening length to be expected is therefore of the order of the Debye length A u, --- (T14irne 2 )', ` = vjV_2wp, . The effect on the total potential V(r) from anisotropy in the electronvelocity distribution can be evaluated from the collisionless Boltzmann equation for the electron distribution p(r. vc , t) in phase space [S] :

ap at

+v

a+

` ar

e a V ap = 0 . ni ar ave

We are interested in the steady-state solution to this equation corre .ponding to ap/at = 0. The deviation p, from the homogeneous equilibrium distribution po = of (ve) due to the presence of the proton is assumed sufficiently small for the equation to be linearized, i.e., that p may be replaced by po in the term containing induced AV /ar_ For an ex11Onentially poential n" d i taw ..~s 1, r- .----. . . .»""> screened Jr . .. . . ., the .. .e u . .- ..,.v,. spatial .ti. nd at spatial u density u.Sta~e~-a.S r, SIIOrt compared to the screening length A, is n,nd = (4 wrA2 ) ', which, in the Debye-Hückel case equals the unperturbed density at r = do --= e2 /T. Hence, the linearization procedure is adequate for r > d(, or, equivalently, in Fourier space for k < km., -- 1/do. Relating the induced potential V;nd(r) to the induced charge density through the Poisson equation and defining the dielectric function re(k) through the relation V(k) = V,(k)/r(k),

(12)

A . H. Sorensen. E. Bonderup / Electron cooling

31

where VO (k) denotes the Fourier transform of the source potential, we obtain for the dielectric function i e(k)_

1

k2

f du

au~'

u-UC'klk .

(13)

Insertion into eq. (13) of the flattened distribution eq. (8) for fo results in r(k) = 1 + (k 1 A o ) 2, with k1 -- / k x 11 denoting the transverse component of k. A transformation to configuration space of V(k) of eq. (12) with VO(k) = 41re/k 2, corresponding to a proton at rest at r = 0, now yields the potential V(r)=e

f o'> dkl e -k

j'IZIJO(kl r l )/(1

+ (k,AD) -2 ~,

(14)

where JO denotes the zero-order Bessel function of the first kind. As expected, the characteristic length is A n, and the potential is anisotropic . In fig . 1, we show the screening function O(r) --- V(r)/(e/r) both in the transverse plane containing the proton and along the z axis. Electrons cluster around the transverse plane through the proton, and at the expense of a screening weaker than usual in the z direction at distances I > Ao, a repulsive potential is obtained in the transverse direction for r = r l >_ f2 Ao. For an antiproton, the sign of the total field is clearly reversed, and the above clustering of electrons is changed to a corresponding dilution of the electron gas. As a dot-and-dash curve, we furtaei show the screening function averaged over angles, and the dotted curve represents a simple approximation to this average in the form of exponential screening with screening length A = 0.55v/wpt. With the purpose of obtaining an expression for the scattering cross section, we shall in the following neglect the anisotropy and consider the simple potential V= (e/r) exp( - r/A) a valid approximation, since screening lengths A typically appear in huge arguments of logarithms, and since A - A n in all directions . The perturbation approach applied to determine the screening length is valid since A is orders of magnitude larger than do. We may now determine the differential scattering cross section. The screening length A -- max(v, v,)/w,, is normally orders of magnitude larger than the collision diameter d = e 2 / z mw ' , A >> d, and at impact parameters < A, i .e. deflections 0 > d/A, the Rutherford cross section is obtained. With this cross section. the integrand in eq. (7) becomes proportional to 0 - t at small angles . The screening of the Coulomb field .

Fig . l . Screening function O(r) for a proton embedded in the flac.tened electron gas at vanishing magnetic field . Tie full curves show 0 in axial and transverse directions, and the dot-and-dashed one 0 averaged over angles. The dotted curve corresponds to isotropic exponential screening with screening length It = 0.55v/wp, .

A. N. Sorensen, E. Bonderup

f

Electron cooling

however, leads to an effective cut-off in the integral at 0 = d/A, and as a convenient expression for insertion in eq. (7), we may therefore apply e2

0(6, w) =

,,~ sin2

4m n

)21 2 + (d

2

(15)

The corresponding transport cross section is given by a«(w)

= 4vre log{ 1 + ~) ( m`w a 1

_

1/2 1 + (d/2A )

2 .

(16)

By insertion of eq. (16) into eq. (9), we obtain eq. (1) with the logarithmic factor L remaining underneath the integral and given by the term inside the square brackets in eq. (16). For very low relative velocities, the collision diameter becomes larger than X, d > A . In our standard case, this happens for w/v < 10 - ;. Here the screened Coulomb potential has been approximated by power laws, For r - 71, we have approximately V - exp(-1) he/r 2, which gives Q, r = exp( - 1)(ir 2/2) Xd a w - 2, Since this expression agrees fairly well with that in eq. (16) in magnitude as well as in slope at the velocity where d becomes equal to k we have extrapolated eq. (16) as a power a, a w -2 for w-:5 w(d =A). At velocities much lower than w(d = X), the transport cross section actually diverges less rapidly. However, except for protons of very low velocity moving in the longitudinal direction, this is unimportant . Throughout we have assumed that the differential scattering cross section may be determined within a classical treatment, and, according to Bohr (121, this is justified since the wavelength in the relative motion is short compared to both the collision diameter and the screening length. 3. Cooling times In the limit of high proton velocities, up > v, eq. (6) for the drag force reduces to Fv = - mnv pa, . ( vp )v .,

p > v.

(17)

The quantity (d/2,\) 2 appearing in the expression for v, r , eq. (16), is typically as small as 10- 12 or lower . For the cooling time -r --- Mop /F, we therefore obtain the expression in eq. (2a) with the logarithmic factor L 9.iß en by L = log

rrtcp

VP >v .

e lop l

(The plasma frequency appearing in the expression for L corresponds to the electron density in the particle frame.) The limit v p -i 0 may be obtained from eq. (10) with vv - tv,1 - v, and the Bessel functions expanded to the lowest order, which immediately yields F = - Mvp (sin

EQ, , cos OX ), 

vp --> 0,

lq

where the quantities A 1 and A have been introduced through the equations x 2 rnn A~ --_ --J v°exp( Mu 4 n

2 mn

ML` l0

- v, /v2 )a (vr)dv

x ve exp(-vc/v2)o,,(v,)

dv, .

(20

)

These expressions are well defined since véa,,.(v,,) - 0 for v,, -> 0 for any positive .c . The exponential decay rates of the velocity components X 1 _  are in general very different with a ratio of typically X 1 /X  - 10 2 .

A.H . Sorensen, E. Bonderup ,' Electron etahng

The low-velocity limit is re~,ehed when vP is smaller than the relevant electron velocities . As is seen from eq . (20), the velocities determining A are much lower than those responsible for a , . and we therefore expect the limiting behaviour, eq. (19), to be valid at substantially . higher proton velocities for transverse then for longitudinal motion . We note that instead of velocity-proportional drag, eq. (1) in the longitudinal Case predicts a velocity-independent force since L is assumed constant, and this, in turn, implies that A is neither defined nor interesting. Inserting the transport cross section, eq. (16), with a = 0.55 v/wP , into eq. (20), we observe that the functions Gl .,, -= T3; 2\ l .~~ /n depend only on a single variable ~ -_- T/n t ' -; . Fig . 2 shows G, ., versus ~. . 'tie function G,(~) is very closely proportional to log ~, and we obtain ne 7r~7r ~l mM 4

!

2

log[

1 2 3 (?"/e) ] - (TIM) 700n

_

( )i )

with a precision of 0.5% in the range 10 -4 < ~(eV cm) < 10-2 . Hence, for low transverse proton velocities, eq. (2b) is recovered with 2L given by the logarithm in eq. (21), and with the change in front factor corresponding to the transition from a three- to a two-dimensional distribution, L

2

log[

700n (T/e

2)3],

3 _ __ -+ 4 2 7l 7r%/2 7T 4

1

,

VP

=LP1<1, .

(22)

(Again, the density underneath the logarithm refers to the particle frame.) As indicated in fig . 2, the function G (~) depends more strongly on ~ than G 1 (s ). In the region (T, n) - (1 eV, 3 x 10' cm we obtain for the longitudinal damping rate X, , approximately . Ä  = 400(n/10' em-3) . (T/CV) -3i2 . (~/ 10 - ; eV cm)o ."' s - t . (23)

In contrast to the transverse rate, eq. (21), X is sensitive to the low-velocity part of the transport cress section, where Qtr(vc) a v,- " with a <_ 2, as well as to the exact choice of screening length. Consequent]" . the overall accuracy of the result, eq. (23), is typically - 50°x . However, since velocity-proportion al stepping . eq. (19), turns out to apply only at extremely low velocities for longitudinal protons . the accuracy obtained in the expression (23) is sufficient . In the general case, the drag force is obtained numerically from eq. (10). The transport cross section t " evaluated as described in the previous section with the screening distance chosen its ,\ = [, 111 . 55 t )= -

TO ` 10 °

,- ~,-T,T,--- _----~T 10 10 ° (evcm)

Fig. 2. Scaling of the damping rates X 1 ,,, . Shown are the funct.ons 7" 2 A, , /n vs j = T,ln' ~.

0 Fig. ?. Drag force FI up ) for tanishtng magnettr field %it'iin the hinarv-collision model for a transverse and a lonaitudiral Pr, , it),, velor`tv . The dashed cures gi%e the a.smptotic h~ haoi~ ur of Fexcept for low longitudinal velocities . %here the ass mptote «ould he nearly vertical in the figure . The insert sh ,%-s the local poser a of F as determined by the relation F cx r~ .

34

A. H. Sorensen. E. Bonderap / Electron cooling

to fit the asymptotic screening lengths. Other combinations of v an,i up have been attempted, but the numerical results are insensitive to the exact choice within a few percen". The force is shown in fig. 3 as a function of up together with the asymptotic expressions for both a purely longitudinal and a purely transverse proton velocity. The maximum drag force appears at a much lower velocity for longitudinal than for transverse motion. For small longitudinal proton velocities, eq. (1) leads to a constant drag, F= --4wne 4LIT, which with L - 10 is comparable to the maximum value in fig . 3. The insert shows the local power a of va as obtained from the relation Fa up. In the limit of low and high proton velocities a = I and a = - 2, respectively, in agreement with eqs. (19) and (18). To get an over-all view of the slowing-down process, the time evolution of vp as obtained in the continuous approximation from the relation dvpjdt = FIM is shown in fig. 4 for an initial velocity vp = v and for different polar angles . A proton with a purely longitudinal velocity, ® = 0, is in our standard case brought to rest within 0.15 s, whereas in all other cases, the time is close to - 0.3 s valid for a proton with transverse velocity 6 = -a/2. The reason for this is revealed by the insert in the figure, which shows that the velocity of an off-axial proton becomes parallel to the electron disc relatively quickly during the slowing-down process. vR }t' - j/'~apt

2.4. Dielectric description

Above, the drag force was evaluated within a binary-collision model, where collective effects were taken into consideration only as a means of obtaining screening of interaction potentials. The validity of such a binary-collision model might be questioned since collective plasma oscillations may contribute significantly to the energy loss. However, although the single-particle picture sometimes fails in a detailed description of the spectrum of energy quanta transferred to the medium, the average energy loss Fop usually comes out correctly 1121 . As an example, in the Born limit reached at sufficiently high proton velocities, an exponentially screened Coulomb potential leads to the scattering cross section, eq. (15), with the collision diameter d replaced by the wavelength for the relative motion. Inserting the corresponding transport cross section (16) with the adiabatic screening length Aa = vpfcop, into eq. (6), we obtain the correct Bethe

y C

ó .~

00

I

01

\

t

(sec)

oz

4 13

-_

Fig. 4. The time evolution of vp for initial speed vp _ v and varying initial polar angle 9 for vanishing magnetic field . The insert shows the time evolution of the polar angle.

A. H. Sorensen. E. Bonderup / Electron cooling

35

formula for the specific energy loss, although in this case half of the energy loss is due to excitation,. (if plasmons . As a counterpart to the binary-collision picture, we shall now within our classical framework develop a dielectric description, which automatically contains all collective phenomena, including the proper screening of electric fields . We describe the electron gas a continuous charged fluid, the motion of which is governed by the collisionless Boltzmann equation ('.1). Since the number of particles within the range of fields is large, nXn > 104, a continuum description should be valid. The drag force opposing the motion of a penetrating proton of velocity vp is due to the dielectric response as the medium is polarized, F(rp(t))=eEi d(rp(t))=e[E,,,t(rp(t))-Eo(rp(t)) ] -

(24)

Here, E,o, , Eind, and EO denote the total, the induced, and the source field, respectively . The equation is conveniently expressed in terms of Fourier transforms of the potentials, and with rp(t) = vpt, we obtain F= -e - -

r 2n1 4,rd3kdwe'("~[Vp,(k,w)-VO(k,w)~ ei

(21r)

4

~

rero(t)

w) 1 _ 1 d3k dwk ei(ko~,+Wo 4~g',(k' [ e(k, w) k2 11

(25)

where e(k, w) is the dielectric function and qo(k , w) the Fourier transform of the charge density of the source. Since we have go(rp(t))= eS(r- vpt), implying qo(k, w) = 21reS(kvp + w), the equation for F takes the form 2

2

F= - e i d3k k 1 - 1 = e d3 kk Im 1 e(k, -kvp ) k 2-7r 2 J 2 2~r 2 J k 2 e(k, - kvp )

(26)

The last step follows since the force is a real quantity . As in section 2.2, the dielectric function entering these equations is determined from the lineari .zed Boltzmann equation, but now the time derivative ap/at is retained, and a Fourier transformation is introduced with respect to both time and spatial coordinates . The result is 2 r-oo

:(ku+w)+1/T

k2 ƒ

where the quantity 1/T appearing in the denominator derives from a collisional damping term introduced on the right-hand side of eq. (11), Op/at )«,11 = - p,/T. An infinitesimal damping is necessary to ensure the proper interpretation of integrals like the one in eq. (27), which otherwise would contain poles on the path of integration. In the steady-state limit w = 0, the perturbation approach used in solving the Boltzmann equation breaks down at kR, .x - 1 /d o = T/e 2, as discussed previously . For high proton velocities rF > t* . the induced density equals the unperturbed one at a distance r = d/2 --- e 2/mr-, corresponding to k ,ux - 1/d. In general, we may therefore expect the dielectric description to be valid for k max(v2, vP) m/e 2. As a first example, we shall determine the drag force at high proton velt-ittes or. equivalently, in the simple case of an electron gas at rest. Insertingfo = S(ve ) into eq. (27), we obtain for the dielectric function WP1

e(k, w) = 1 - lim T- x (w -

, .

1/T)~

(ZK )

With w = - kvp = - kvP cos 0, this equation reduces to e = 1 - (k,\,, cos 9) - 2. and this shows that the field around a proton of high velocity vp > v is screened out at distances of the order of the adiabatic cut-off length A u . This result was already applied in our discussion of the binary-collision model. For svmmetr`

A. H. Snrricen. E. Bonderup / Veerrnn roatinq

1+

reasons, the drag force is directed along t~, F= Fîß,, and eq. (2á) reads e" k~ ~ ' 1 e .i: _ = _ dww Ini[ dklk 1 d'k F', U kL~, 2-.r ƒ 7TUp

~

1

W

(29)

The integral receives contributions only from regions where Re e = 0, corresponding to w'` = w;,, and therefore eq. (29) may be rewritten as (30)

F= - e-, 1J1 ü dk/k ƒx dww Im x 4t` ~ ~( w )

Since the function r t (w) is analytical below the real axis in the complex w plane, the w integration in eq. (30) may be performed along a large semi-circle in the lower-half plane . From the asymptotic behaviour 1 - " (w~,,/w) = for jwj --+ oo, one then immediately obtains _

F -'

-

As is typical for this type of consideration, a logarithmic divergence appears at large k since the dielectric description becomes inadequate for k > max{v`, v~; )m/e 2 . Such k values correspond to very close collisions, and the force may safely be determined from a binary-encounter treatment, which leads to an effective minimum impact parameter equal to d/2 --- e `/mv,, as is apparent from eq. (16). For insertion into eq (31), we shall therefore use k ., - 11d, the exact choice being immaterial since the divergence appears underneath a large logarithm. The expression (18) from the binary-collision calculation is now retrieved when we introduce the following substitution into eq. (31): log(k m  r a ;, ) -+ log( mv,;/e'w,, ) - ; _ log(1 .2A,/d ). We shad next consider the drag force at low proton velocities vp < v, and here the details of the electron velocity distribution become important . Inserting the flattened distribution into eq. (27), we obtain for the dielectric function after integration over electron velocities perpendicular to k, e(k . w )=F(k :,c,,)=1+(k -,X

t))_

2

1-

lim

-

_w

1

Vqr

K 1

A ku

!-Lo

~d x

A

u

.-

e

l'

(32)

u + wlk

The principal value may be expressed in terms of the integral D(q) = e -'' Dawson's integral (14j, and one ends up with F(k

l

,

, c~ )=

1 +(k 1 X a) `[1 - 2r1D( ij ) - i~~~1~ 1 e - "' ] . -q _-kW 1

V

~ e' ~ d (ref. 13) known as Jc, t

*

(33)

(The same expression for is found for a Maxwellian distribution provided we replace k 1 by k and prodded is still interpreted as the two-dimensional rms velocity.) We now limit ourselves to the simple case of transverse velocities vp = v,,, « v. Again the force is directed along vp , and we obtain by performing the k: integration F

Lv

F=

11

r, J()

dkik

i

J

a

d T.

cos qp

imF-

'( k1 ,

-

k _,

r~, cos q) ) .

Since the proton velocity is low, we may expand the quantity Im F _ I to first order in 71, and we find x ~,! . e` ki ~ 2 ir ne`' '/1 I t7

(34)

A . H . Sorensen, E. Bonderup / Electron cooling

37

where L is given by L- log(k,,a,X1,)- 1/2 . Once again agreement is obtained with the result of the .p )nt/e` = 1 binary-collision calculation, eqs . (19) and (21), when we choose k , :,x = max(c2 . (The binary-encounter model and the dielectric description supplement each other in the sense that they contain divergencies in opposite extremes, namely at small and large k values, respectivcl) . The dielectric formulation is by far the more complete of the two since it takes proper account of electron-electron interactions. The apparent difficulty at large k values is easily removed since it originates in close encounters, which are correctly described as two-body collisions within a pure Coulomb field . 2.5. Diffusion processes

So far, only the average slowing-down of a proton has been considered. The neglect of fluctuations in velocity changes requires some justification . Consider therefore the mean value of the square of the fluctuation in velocity decrements 12' experienced by a monochromatic beam of protons, 92 --- ~(dVP -

(

,IVP

)

(36)

)Z> .

For the sake of argument, we may group the velocity changes in the scattering events into small intervals around values 4vp , (ref. 12). If n, denotes the fluctuating number of events of type i, we have =F_(n,>avp . ,Q2=

(Y_(n, - )VvP .,)

2

=F, «n,2 >

-<

n, >2)wp ,

(37)

With n, given by the Poisson distribution, we obtain in analogy to eq. (5) in the binary-collision model 522

=

jt

f dnw f

Q

dSl( w' -

w)2m2/M` .

(38)

Integration over scattering angle yields E22

=Jt2n(m/M) 2

f dvJ(ve)'t''Q<<('t'),

(

19)

and by introducing the distribution (8) and changing variables. we may . i n analogy to ey . ( 10) . etipre"" !'= as l; x l .QI = At 42 (n: 'M )2 exp - t sin2 9~ 1t' 3cTt r ( w ) tu (2 r. sen ç) tit' ` ,' u-' ) . (4() ) divlwl exp( V L' ,

2

f

In the limiting cases, this may be simplified to mvp - ~ 81r2e 4 log for vP > v, S22 =,A, M vP e w P, ~ for 2 -v2 ~ 1 v. < i, .

(41)

where X l was introduced in eq. (20). corresponding To estimate the importance of fluctuations, we shall compare Sl to ~-AvP to a typical time cßí, which we choose as the slowing-down time, T = i 'v do %dr ' . introducing -it = T into e 4i 1. tie obtain by means of (18) and (19) for vP >v, t Àl //X il

for vP = vp .r < v, for vP = vP . « v .

(42)

A . H. S~sen, E. Bonderap / Eket- rnolinq

With it -r, the average velocity decrement is of order vt , ((Avt,)l - vp, and from eq. (42), we obtain for vp > v, for up_

1 l

VP

/A .

vp ., < v ,

(43)

for vp = vp.
As is observed, fluctuations are most important at low proton velocities, and the quantity 92 remains small as compared to j(Avp>) for proton velocities above the very small values v[2m/M]ti2 and v[2(m/M)A,/\ jt/= for vp = vp1 and vp = vp=, respectively. Thus diffusion processes are unimportant unless the protons are extremely close to thermalization, and the slowing-down times based on a continuous drag force are indeed relevant. In fig. 5, wee show the quantity Q2 /Bt for ® = 0 and ir/2 as obtained numerically from (40) and from the asymptotic e.pressions in eq. (41), and obviously the asymptotic forms are adequate for estimating the relative fluctuations. 2.6. 77w equitibritun proton distribution

The trend towards the final equilibrium for a proton beam exposed to electron cooling may be determined from the Fokker-Planck equation [8,15] " ( F (gD, , at avp, g M, + ` a vp,avpl which governs the time evolution of the proton-velocity distribution g(vp, t). Here, the diffusion coefficients are defined as the averages D,l --- d(Avp;Avp.,)/dt, where Avp, is the i th component of the velocity change for a proton during a single collision with an electron . In terms of the transport cross section (16) and the quantity Qo --- 4sre ° /m e w ° , w --- vC - vp , the diffusion coefficients may, according to appendix A, be written as D =n(m/M)'J d 3Vef(t%)

'[(1v`-K~`)a<<-

z(w` -3 u'~

d?vef($)wn, %, [a, - iQO ] ,

- rt(nt/M) J

)QO ~ .

(45)

i e- j.

I f the variation of the logarithm L is neglected and the a terms of relative order I /L - 0.1 are omitted, the expressions agree with the standard results [4,8] 4~ne s % 3 j M2-LJ d 3$f(v~)(w 2s,. - wrw )/w .

D =

(46)

This result may also be derived in a simple manner through application of the impulse approximation . As in the previous section, only low proton velocities are of interest, and in this case, the diffusion tensor for the flattened distribution f becomes 0 'Mn v ' Xy) 2 '0

i

0 0

0

1

vp - 0,

(47)

et,- e>, -4 -.1 vent  refers to the longitudinal direction z. Tennis resulting from neglected since they lead to only a 5% correction in our standard case. With the force from eq. (19), the diffusion equation assumes the form uh

a

-

allpp .:

( g~ ~~ vp : )

+

8~

a va'

( ga L Vp,)

+

A1

82g M

8

vp~

+ i i-

a2a a vp,

CIO have been

(48)

A. H. Serensen, E. Bonderup / Electron cooling

39 i I

0

12

0.0

c .0

2.0

3 .0

P

V /V

4 .0

5.0

Fig . 5 . Rate of average mean square fluctuation in velocity change for a transverse and a longitudinal proton velocity at vanishing magnetic field. The dashed curves show the asymptotic expressions.

I I

Fig. 6 . Geometry for a distant binary encounter in a magnetic field .

where we have introduced the electron temperature T = v22z ML 2 . To determine, e.g., (r2, ), we multiply this equation by vPX and integrate over all velocities, assuming g to vanish rapidly as v tends to infinity. This p leads to at(

P

V ,)_

-2 A1(vP,)+XlT/M,

vp,) =-2 A~,

at

(vP,)

+ 2X 1 T/M.

i=x,y, (49)

These equations describe an exponential "decay" of the proton velocity distribution. TI. .- damping rates are twice the corresponding rates calculated from eq. (19) since eq. (49) describes the decay in mean energy in each direction rather than the decay in velocity . The final velocity distribution of the proton corresponds to the temperatures TP

1

=

i

j-,

7'r, II

=

i

T.

(50)

For T = 1 eV, the final longitudinal temperature is 1/400 eV. The actual electron temperature in the beam direction [10], which here has been assumed to be equal to zero, is far below this value, and therefore our treatment is adequate. Final proton temperatures have been presented in ref. 5, but only the value for TP ; agrees with eq. (50). The diffusion coefficients were evaluated within the binary-collision model. As demonstrated by Thompson and Hubbard [16], a dielectric description may be applied for the determination of D,,, as we did for the drag force. Close to thermal equilibrium, their procedure: results in diffusion coefficients essentially identical to those obtained within the binary-encounter model, and therefore we shall not discuss the dielectric treatment further in this connection.

.4 . H. Sorensen. E Bedenep / Electron rooláng

40

3, The effect of a longitudinal magnetic field If a longitudinal magnetic field B is applied to guide and confine the electrons within the cooling swkm, and the magnetic field is sufficiently strong for the Larmor frequency w,. of the electrons to exceed the plasma frequency, w, > wp,,, the cooling rate may be changed significantly . Qualitatively the influence of the magnetic field may be explained as follows: The momentum transfer to an electron from a proton of velocity t:~ is essentially perpendicular to v., and this momentum may conveniently be resolved into components parallel and perpendicular to B. If the collision is slow, i.e., if the impact parameter p is large cx-wmpared to the Larmor radius rL , the electron is oscillating rapidly in a direction perpendicular to B within distances smaller than p, and therefore only the momentum transfer in the longitudinal direction is accompanied by an energy transfer. Since the longitudinal electron velocity is zero for the flattened distribution, the corresponding drag force becomes essentially proportional to v,- ', cf. eq. (4). As discussed in the introduction, a significant increase in damping rate may then be expected at sufficiently low proton velocities. In the binary-collision model, the collisions are divided into two limiting groups [5,6], i.e., fast collisions, which are treated as for vanishing magnetic field (wc ' = cc) and slow ones, for which B is considered infinitely strong (w, ' = 0). Such a division is somewhat uncertain, in particular because the slow collisions, which dominate at low proton velocities, occur within a fairly narrow region of impact parameters, corresponding to a logarithm L''' of only - 2. In addition, a sharp division at a definite impact parameter may result in mysterious correction terms to the drag force [6] . In the presence of a magnetic field, also the question of screening of electric fields is less transparent . These difficulties lead us to introduce a dielectric description . At this point it should be mentioned that with the electron number densities of practical interest, the magnetic field needed for giving w,: = wp, is relatively weak. In our standard case, only 56 G is required. In cowling experiments, the field strengths applied have been 500-1000 G 11,21. and as our standard magnetic field strength we shall therefore choose B = 700 G. .'. fnflnitely strong magnetic field As a guide to the general case it is instructive first to compare the binary collision treatment in the impulse approximation with the dielectric description in the simple limiting case B -+ x . i .e., for a maenetic field sufficiently strong for all physically relevant times and lengths to be large compared to wl.- ' and r, , respectively (but sufficiently weak as not to influence the motion of the heavy proton) . In this c.xtreínc situation an electron can only move in the direction parallel to B, and the energy transfer resulting from a momentum transfer in this direction may easily be evaluated . Consider in the particle frame a proton with impact parameter p and velocity up penetrating an electron gas with- 'a velocity distribution flattened along B, and choose a coordinate system with x, axis parallel to up and x ., axis in the B-z, plane (fig. 6). If ~ denotes the angle between the impact parameter and the x 2 axis and a the angle between B and the t, axis. we obtain for the momentum transfer in the longitudinal direction

:Jp s =f7 FB (t)di= x

2e P vP

sin ~cosa.

(51)

The corresponding charge in electron energy averaged over impact parameters smaller than an appropriate screening length is given by 2 `- r k

t1 E > = .., *a , J dp d~p 2 m

t n ` ' - logl mvP a 2 Ri .

2va 1

Jos -a.

(52)

The divergence at p -- 0 appears since the impulse approximation breaks down in this limit, and as usual the remedy is the introduction of a cut-off distance p,, = d/2, cf. eq. (16). Within a time dt the proton traverses a layer of thickness vpdt and therefore interacts with dv = vpdtira 2 n electrons. By energy

A. ƒ/. Sorensen, E. Bonderup / Electron coohng

41

conservation the change in proton energy then is dE

2 I rne 4 l2a _ = log+ m

Url

(53)

where vp ,_ is the component of vp in the perpendicular direction. This equation also immediately determines the drag for vp = vp,, or vp = vp , whereas the general case is more complicated, since the force is no longer parallel to vp, cf. ref. 5 . The remaining problem in the binary-collision treatment is as always the determination of the screening length A . In analogy to the considerations in section 2.2 we consider the linearized Boltzmann equation, where the unperturbed density po(r, ve) now represents a homogeneous electron gas in the presence of an infinitely strong magnetic field . Since the field is infinitely strong, each electron is pinned to a fixed position in the transverse plane, and thus the polarization induced by an external charge cannot in any way depend on transverse velocities. We may therefore disregard transverse velocities altogether, and in its linearized form the Boltzmann equation (11) assumes the form ap t ap 1 e av apo . °e~8 : +mozav =o at +

(54)

Fourier transforming and solving as in sections 2.2 and 2.4 we obtain corresponding to eq. (27) for the dielectric function e(k, w) = 1 - lim WP f J

)/d v, dv 1 k. df0( v,

(55)

The static screening is determined in the limit w = 0 and for a Gaussian distribution f,(ce-) of longitudinal ) -2 corresponding to an isotropic. exponentially screened  electron velocities, one obtains r = 1 + (OD Coulomb potential with screening length equal :o the longitudinal Debve length. A = X 1, = v /W r V~ ~ v,',,) Hence for B - oc the static field around a proton is screened out at a distance ~~ hich for the flattened distribution under consideration is exceedingly short as compared to the screening distance A pl obtained for B -* 0. This indicates that the question of screening must be treated carefully . Proton velocities are, however, generally large compared to c  for the flattened electron velocity distribution. and to determine the dielectric function e(k, w) for this case, we insert a 3-function for the distribution l;,( r, ; 1 in eq. (55) and obtain I

---

e(k, w) = 1 - Tlim

k

w-

i/T

This may be compared to the corresponding expression in eq. (28) for the case B = 0. Inserting !a: = - kr, we observe that the screening length to be applied in the energy loss formula (53) is the adiabatic cut-off distance A and neither A D, nor the very much shorter longitudinal Debye length X,, . Let us next attempt to evaluate the proton energy loss directly from the dielectric description of the flattened electron gas with r given by eq. (56), and let us first consider transverse protons. ri, = vpz . Inserting eq. (56) into (26) and changing to polar coordinates with z as the polar axis we obtain dt

lim

-oo

21r 2

vp Jp

k J0

dq)

x

dww 1 -

, k"

(te

+1/T)

_

( 57 )

The integral receives contributions only for Re F = 0 equivalent to w -' ~ wp,( k :,'k )- . and therefore the k integration has been extended to infinity. In this integration the poles are clearly in the lower half plane . and thus the integration path may be replaced by a large semi-circle in the upper half plane . For fired w, lw) -+ oo, the integral over q) may be evaluated to yield

A . H . Sorencen, E. Bonderup / Electron cooling

21r ~1 +(k~~,)

(58)

2

`- Wpl/w ~

within the curly brackets does not contribute to the w integration, which is now trivial, and performing the k integration we obtain the number I

d

t

2

_ -

P

2

[Iog( kn,a,Aa) - I],

(59)

., = d', where the length d was In the limit of large logarithmic factors, Nve have retrieved eq. (53) with k,,, with previous discussion we conclude impact parameters. In agreement our introduced at small the cut-off that the screening length is the adiabatic cut-off distance A a . Finally we consider longitudinal proton velocities, u = v !. As before the integral in (26) only receives contributions for 12e E = 0., which now corresponds to k _ A a ' . We first perform the k: integration for fixed k, _< X; ', In the complex k . plane the quantity 1 /k 2 e has poles below the real axis only, and therefore the inte~ along the real k: axis may be replaced by an integral along a large semi-circle in the upper half plane. Since 1/k -e -+ k= 2 for Ik, I -+ oc we obtain z e2 2~rne° e Im(-iir)= vP = VPZ . dB-FvP= d'k 1 (6 kimax = mv vPJ 27r 2 P P

p

2P

With the resonance in r - ' obtained at k = Aa' the formula corresponds to c :ntributions from distances p_
a

.i.2.

L

Finite magnetic field

The determination of the dielectric function for finite magnetic field strength 0 < B < oo is somewhat more complicated than in the limiting cases B = 0 and B = oo, and for a derivation we refer to Appendix B. In the limit B - oo the general result obtained in eq. (B.7) agrees with eq. (55). Furthermore, the formulas (B-8k (B.9) reproduce the simple result from statistical mechanics that a constant magnetic field does not alter the dielectric properties of an isotropic Maxwellian gas in equilibrium (17]. For the flattened electron velocity distribution, the dielectric function r is obtained through insertion of f,(v, :) = S(c ,) into eq. (B.7). Integration over electron velocities then leads to the expression e

T -. x

k`

Lri W. /t~2 w+nw .-ilT

k

2

(61)

A. H. Sorensen, E. Bonderup / Electron cooling

43

where we have added an infinitesimal term -i/T in the denominator as in eq. (27). In the limit B --' x. we retrieve the result (56) through application of the relation 1jlu)

- (fL/2)"/nl

for ju - 0.

(62)

In the binary-collision model, it was argued that collisions with impact parameter larger or smaller than -- rt, _ 01W, could be treated as if the magnetic field were infinitely strong or weak, respectively . For the dielectric description to be consistent with this picture, the contributions to the drag force, given as an integral over k of a function determined by -(k, -- kvp ), should show similar limiting features for k << r, ' and k >> r L ' . For this to be so, the magnetic field should appear only in the combination kr i , such that, e.g., the limit kr L << 1, when replaced by krL = 0, may be interpreted as corresponding to B = oo. From eq. (61) we do, in fact, obtain 1 k up v 2 f kr L , -, - , Xo = , (63) e(k, -kvp) = 1 + i k v ) f2wi ( pi (k D) where f is a function of the three parameters indicated . gas at rest It is instructive first to study an electron gas at rest, v - 0, in which case we may reduce eq. (61) to

4.2 .1 . Electron

r(k, w) = 1 --

lim

- ,c

k2

Wp1

k2

(W

_ 1/T

)2

+

k2

(W

- 1/T )2 _

(64)

Wc

by means of eq. (62). For B = 0, eq. (64) is identical to the expression (28). Initially we consider a proton with transverse velocity. For symmetry reasons the force in eq. (26) is in the direction of the proton velocity, which we take as the direction from which azimuthal angles 4 are measured . As in the absence of a magnetic field, the contributions to the integral in eq. (26) come from poles in Im corresponding to vanishing Re r. With the dielectric function (64) the quantity Re F becomes zero at azimuthal angles 92, and 'T2 determined by the equations k1 vp cos 92, k1

L

p

cos

422

=

± wp,kz/k,

= ± ( w + wF2, k 2 /k 2

iz

valid to second order in the small quantity wp,/w,:. Apart from the simple trigonometric factor k _, k . the upper poles are situated at f wp,, "plasma poles", whereas the lower ones are very close to + w, . "cyclotron poles" . Introducing the variable w = kvpcos T, we may perform the integration over T. Opposite poles ± w give the same contribution, and we obtain

r" o2 -f d(p92

k'

cos 99 k2c

a_1

( k 2 u2 1

p

2 _w + )

aw

'

p

27rA  2 k`/k ; (kik2 _ k?/X2 +

27r11d `k /k 3

(kik2-k`/r,2-kï/X,)~ X,, - up /wp , . r, = vp/w, .

(66)

where (9(x) denotes the Heaviside step function . To perform the remaining integrations over k ~ and k ; we change to variables k and 6, where (k_, k 1 )=k(cos 0, sin 0), and for the plasma-pole term in (66) we

.4 . N. Sorensen, E. Bonderup !' Elect-It c-finR

obtain 2e, X a-

= 2srAn=

2 3 k; k 4. 2 k 2 k :/~

dkl_

dk ;

dk j~ d8

t `o

°

k

cos28sin6

® k X -k2 k(kX )_

` ,/`

(sin2 6-cos`9/(kX N ) )

~

- cos 29

1 +(kX  )`

(67)

2 _2 ./(1+k,n  x~u)~/ %' l)I/Y,-kmi~xX + _ - il a ilog[kma,A . + ( k m

As usual, it is necessary to introduce a cut-off kn ,x at large values of k. In the special case of an infinitely strong magnetic field the length r, tends to zero, and therefore the second term in (66) vanishes . Up to terms of order (kmpx~a) `, which typically are as small as 10 - "' we have therefore rederved the expression (59). For finite magnetic fields like our standard field of 700 G, we have r, >> k n; ; x , and we obtain an additional contribution to the force from the cyclotron resonance, 2r:X;

;~x

rx

dk_J o

-Q2>-Jlo a g

ï

dk

1 ki

kmax r l +

hl/k

: 2 X.2. ~ .;2®(ki k 21 k - -k` /rI -k 1/ u) ?

9

(kmax 1 -

(rl/~a

)2)1/2

2 I/2

-

Áa r,

- rl

)2

I

)2 )

~ma~x~u ni .

2 

I /`

-

( Îl + rl

.

A (68)

Adding the two contributions in (67) and (08), we finally obtain the total force 2Ane

niv2

4

[2 log( kmaxr,)+log(2h /r,) - ; +O,+O.,J,

vP =v. . .

(69)

here O, - O((k,r,)-') and O_ - O(w 2 ,/cap ). For longitudinal protons vp II ;; a similar analysis ray be carried through . The 9) integration is now trivial and having determined the position of the poles we may perform the integration over k .. The calculation is very similar to, and even simpler than that presented in Appendix C for a protortof velocity vP < u` penetrating an electron gas with a flattened velocity distribution, and we therefore only give th,~ result,

v.

F. - -

2~ne4 r12Lp

[lo8(klmaxrl +1)+(kimaxrl +1)-

+0,1,

vP llv,

(70)

the other force components being zero. In the limit B -- oc, eq. (60) is retrieved, whereas under standard conditions, we obtain in analogy to eq. (69) F_= -

2-,r ne° mt= P

[2(log(klm,,xrl)-i)+1+O,+OZ],

vP

ll~ .

(71)

The final integration, which was performed to obtain F l and F_ was over the variables log k and log k : , respectively, and in fig. 7, we show the integrands 1, and 1  , 4Tne4

MVP

20

_ ~ d(log

#c)

1 :. .  .

K=

kr,

k, r,

for vp 1 f, for VP 1 I ~.

(72)

A. H. Sorensen. E. Bonderup / Electron cooling

15

h

T~0 ß = 700 Gauss n = 3x10

45

vp =vp i

8cm 3

1 .01-- ---______-- -1_WC

Wpl

Fig. 7. Drag force as determined from the dielectric treatment before the final integration over log u = log(kr, ) or log x log(k, r,) for protons with transverse and longitudinal veloctties, penetrating an electron gas at rest, exposed to a longitudinal magnetic field. The figure shows the quantities I, . defined by eq . (72), vs . log sc . and the plasma and cyclotron contributions are identified by the symbols wr,, and w, . The position of the reciprocal of the adiabatic screening distance. ~~ 1 , is marked in the figure, znd the dashed line I - 1 indicates the level obtained in the binary treatment for B = 0.

An abrupt change in 1 appears around is = 1 corresponding to distances r = r, = ct,/'w. . Apart from the adiabatic screening distance A., the reciprocal of which is marked in the figure, r, is the only characteristic length in our problem . In the binary-collision treatment, it enters as the impact parameter, at which the separation between fast and slow collisions is introduced. For ti; > 1, the dashed level 1 = 1 valid for scattering in a Coulomb field for B = 0 is quickly reached . For h < 1, only the plasma pole contributes . However, the plasma-pole contribution is in general not restricted to K < 1, as is observed from the figure . For transverse velocities, the plasma pole contributes half of the force at large k . It may be noted that in the longitudinal case, the small, well-isolated plasma-pole contribution accidentally cancels against a term of relative magnitude - 1 /log(Ktt,ux) deriving from the cyclotron resonance, as indicated in eq. (71). Since the change in 1 around K - 1 appears quite abruptly, a simple prescription for obtaining the force with the usual accuracy of - 1/Lf">t is to use the pure Coulomb term for h > 1 together with that part of the plasma-pole contribution, which is found for a < l . For the electron gas at rest considered here, the plasma-pole contribution is identical to the one obtained for B - oc since the limit c -" 0 corresponds to r,_ - 0. In this case, the simple combination of results obtained in the opposite limits B = 0 and B = x is therefore adequate when the contribution at infinite fields calculated within the dielectric description . The division is introduced at r = r,, as suggested within the binary treatment . As we shall demonstrate in the following section, for a thermal electron gas exposed to a finite magnetic field, we can no longer adequately determine the contribution from distant collisions by considering the limit B -- oo. However, when the finite value of the field is retained, the drag force obtained within the dielectric picture should be very accurate in the important case of low velocities cr since slow collisions when the dominate, and the only remaining üconnww uncertainty connected with the choice of 4, ,__ disappears , , contribution from fast collisions becomes small . 3.?.2. Thermal electron gas In the general case of a flattened electron gas of finite (transverse) temperature in a finite magnetic field . the dielectric function r is given by eq. (61) . In the special situation v - 0 considered above, only the terms with n = 0, 1 survived to yield the function in eq. (64), and with these two terms, the poles in Im F

A. H. Sorensen, E. Bonderup ,/ Etectn)n cooling

46

determining the drag force were situated at contribute, the poles are located at w _ fwp, ;

w= fnw,

w=

f wp, and w - t

w,

Similarly now, where all values of n

a= 1 .2,3 . . . . .

As in the special case, the contr, ibution from the distant, slow collisions is part of the plasma pole, and the cyclotron resonances cre limited to the close, fast collisions. While the binary treatment is ambiguous for the slow collisions, it does lead to a fairly accurate result for the fast ones since the Coulomb logarithm appearing in the expression for the drag force is large in this case. This is fortunate since a very large number of cyclotron resonances contribute within the alternative picture, i.e., the dielectric description . We should therefore be able to obtain a simple and fairly accurate expression for the drag force by treating the close collisions as binary ones and by identifying a suitable part of the plasma-pole term from the dielectric description with the contribution from distant collisions . The accuracy of this procedure will now be investigated for longitudinal and transverse proton velocities, and as for an electron gas at rest, we shall show that the simple Coulomb expression is applicable for k > rt ', whereas the contribution from slow collisions is given by the part of the plasma-pole term corresponding to k < r; '. Initially, we shall consider a low-velocity proton moving in the longitudinal direction and evaluate the drag force opposing its motion within the dielectric descriptiton. As shown in Appendix C, the calculations may in this case be performed analytically except for the final integration over IL = ; (k 1 rL )2 for the cyclotron resonances. For the drag force, we find the expression ,

e2

= - , . go+ (rLIX.)~J0 re Po

=

:(rllÄa) 2

e -Pol0(tt0) ,

m~ .

d!~

x n-t

e _ ~In(F~)

lun - 2(»U/Vp)2,

"

(ia+lun)

2

lUp 2 4~ 3 + 2I -) - +" g gn v

.'~,nax - 2(klmax r L) 2 ~

(74)

vp - up-

where the term proportional to go derives from the plasma pole, which gives rise to the total force for B = oc. In fig. 8 we present the drag Fd versus Lp as obtained from (74) under standard conditions . Since the cut-off at large k values appearing in the quantity IL,,, ., is determined by the fast collisions, corresptinding to vanishing magnetic field, we have chosen expressions obtained from a comparison of (31)

Fig. S. Drag force for protons with longitudinal velocities penetrating a flattened electron gas in a longitudinal magnetic field of 700 G . The full curves and F~ are from a binary

P'

V

P

V

collision and a dielectric calculation, respectively, and the dotted curve is the binary-collision result at vanishing field strength.

A.H. Sorensen, E. Bonderup / Electron cooling

47

with (18) and from (35) with (22), valid also for not too low longitudinal velocities, 2 +

i . i, 2 kmnx - (kmax

i ~, inax

~~

~. kn,áx

i. _

-

2 exp( -

-2 )

T(

2 Vp )~ L

k ,,-,. max

ir exp(1) _T 175 e2 *

(75)

For comparison, we also show a simple estimate Fb obtained within the binary-collision model. Here the slow collisions do not contribute at all for vp 11 B, and the close, fast collisions extend out to impact parameters, where the collision time becomes of order we 1 . This corresponds roughly to a replacement of wPl by w,, in the screening length A appearing in the expression for the transport cross section, eq. (16), and the resulting reduction of the drag force is only of the order of 20`x. At high velocities vP > v, the difference between F~ and Fb does not exceed the uncertainty already introduced through the choice of k,,,., . but at low values of vP , a very significant discrepancy is found due to the neglect of the contribution from the slow encounters within the binary-collision model. The drag force in eq. (74) is given as an integral over k l , and in analogy to eq. (72), we write it in the form

F=

-

r

21rne°

7,

J

ao

d(log(klrL))11 i *

(76)

in fig. 9, we show the function 1  for the .,pecial case of a proton velocity of vp = v P : = 0.2v . As for longitudinal protons in an electron gas at rest, the plasma-pole term is well separated from the w, contributions, and it extends only up to Ago /rL = 0.96Aß' << r, 1 . For larger k l , the cyclotron contribution starts to emerge, and for log(k 1 r L ) >_ 2-3, it reaches a constant level. Since large k values correspond to close collisions, the constant level may be reproduced by results for binary encounters with a pure

7F

T= lev

r. = 700 Gauss In-300e cm 3

2327

6~

vP= "PZ= 0.2 v

rk 3

1=

Ó~rL

::096

Xo

-4

1 i

Fig . 9 . Dielectric drag force for a proton of longitudinal velocity v p = vt, . = 0 .2v in a thermal electron gas exposed to a longitudinal magnetic field . The quantity 1  defined through eq . (76) is shown vs . log(k 1 r L ).

Fig . 10 . Dielectric drag force for proton: with transver .e %elocities v p = up , = 0 .2 c, 0 .5v in a thermal electron gas exposed to a longitudinal magnetic field. The quantity 1 . is shown -s . log(kr t ) .

.A . H . Serensen. E. Bonderup / Electron cooling

Coulomb interaction. The force in eq. (Il`) may be expressed in the form (76) with log k, replaced by log, p ', where p denotes the impact parameter, and inserting the transport cross section corresponding to Rutherford scattering, we obtain for p ~s- d (see ref. 20), ( v2 v v (77) 1, = 2 -- 2vp F exp - erf( v [l where erf(x)= 21r - ~ , 2Jó e-`` dt denotes the error function [141. This value agrees with that obtained from a numerical evaluation based on eqs. (74) and (76). In fig. 9, we have indicated the position of k, = r; ', and it is observed that by approximating the cyclotron contribution by the expression 1,i - ®(k,. r, - 1), we introduce errors only of the order of 1/L - 1 /log(k 1.a,r r, ) in the part of the drag force originating in The extensive sum close collisions, and such uncertainties are already present through the choice of k 1 in eq. (74) from the cyclotron resonances may therefore be replaced by the expression in eq. (10) when the distance r, is introduced into the transport cross section instead of the screening length A. This simple scheme for calculating the we contribution becomes less accurate with decreasing vr , but here the plasma term, which contains no uncertainty, dominates strongly, and for t',< v, the overall error is typically well below 10%. A similar analysis may be carried out for transverse proton velocities. As in eq. (66), valid for an electron gas at rest, the q) integration may be performed analytically, but the 9 integration must now be done numerically . Once again the w, poles are cumbersome, and it is necessary to perform the integration for each n separately. As for longitudinal velocities discussed in Appendix C, the contributions from the two poles bracketing any multiple of wc nearly cancel . Since the upper limits on the integration over 6 are slightly different for the two poles, the numerical evaluation was performed in two steps. First, the sum of the two integrands was integrated up to the smaller of the upper limits (small integrand, large interval), and then the remaining function was integrated between the two upper limits (large integrand, small interval) . The result for the quantity I1 defined as in eq. (76) with the replacement k 1 k is shown in fig . 10 as solid curves for the two velocities v~, = 0.2v, vp = 0.5v. At small values of k, only the plasma pole contributes to the drag force. For k = r,` , the first w,: poles set in and cause a significant reduction with respect to the plasma-pole term, which, as shown by the dotted curve, extends far beyond this value of k . A dip is observed ,%-henever new resonances occur at k = nr,-- ' . The excursions made by the function I 1 are quickly dainned . and at large k, the total contribution approaches a constant . which as above may be determined ,,~iuiin a binare calculation or, equivalently, from the dielectric function for 6 = 0 and kA  >> l,

2

)

v)1,

-~

vr,.

exp - 1 2 c~ '

Up v2

I

1 °p - 1 1 UP 2 v2 2 v2 ~~~ `~ 1 ~ 0

)]

(78)

This level is shown by a dashed horizontal line in fig . 10. The error in the drag force introduced via a replacement of the correct function 11 foi k L > r, ' by the constant corresponding to pure Rutherford scattering is again small. One might have expected the separation between close and distant collisions to occur at impact parameter rL rather than r, for protons of low velocities. The result obtained must be connected to the fact that the dominating distant collisions correspond to a strong magnetic field, for which the transverse motion is effectively frozen out . Generalizing to a proton moving in an arbitrary direction in the presence of a magnetic field, we may calculate the drag force as a sum of two terms, the plasma-pole contribution from the dielectric description, truncated at k = r,- , and the contribution from pure Coulomb scattering for impact parameters p :5 r,, _

F=(FI,F) = - 2Te

a

('iog(r,

,

) IWi,

1(k)d(logk)+F''.

The force F' is determined from eq. (10) wit"i A

= r,

inserted into the transport cross section, and

(79)

Vo

A. H. Sorensen. E. Bonderup / Electron cooling

49

corresponding to the plasma resonance is given by the expressions 2

?t

( -1) r w

t t

- 1)'w,

v(t)

=

wpitV`~~lo(01

ktvp..)Iv., , - kt],

2 +tvp~} ,

Y(t)=(1 -t2)Up1 wr

+

Fu

=

z(krL) - ( 1 -t2) ;

l'r'=2ADk 2©(2po -(krL )2 )[0, 1], 2 =0 . vpl 2po=(rt ./\ .) e -u  lo(lao) ;

VP1

0 (80)

The error introduced through the expressions (79) and (80) is typically less than 10% and much less at small velocities . 4. Concluding remarks As a means to obtain phase-space compression of particle beams, electron cooling is of practical interest for beams with a velocity spread of less than or on the order of thermal electron velocities in the particle frame, vp <_ v. We have discussed cooling of a beam of (anti)protons . From fig . 4 it is observed that for vp <_ v and vanishing magnetic field, the damping of the transverse velocity component is significantly slower than that of the longitudinal one . This effect, which is due to the flatness of the electron velocit\ distribution, is maintained for not too low velocities when a longitudinal magnetic field is applied . According to fig. 4, the cooling time To for a transverse proton with initial velocity u p = c equals 0.33 s in our standard case without a magnetic field . This time is reduced by a factor of -- 3 when a magnetic field of 700 G is applied, and already at an initial velocity vp = r, where T = 0.25 s. the reduction amounts to a factor as large as - 7. This is the so-called supercooling effect, which is due to the slow collisions in the flattened electron velocity distribution. For longitudinal proton velocities, cooling times are essentially unaffected by the magnetic field unless the initial proton velocity is below - r . Ho\\ever. for r. = ur :, the r . respecti\ el\ To are already as low as 0.15 s and 0.05 s for initial yeiocities rp = c and rp cooling times the beam-transsituation, it is essential that For the application of our considerations to an experimental electron velocity distribution . port system does not destroy the flatness of the In section 2, fluctuations in momentum transfers during the slowing-down process were found insignificant for B = 0 except for nearly thermalized protons. With the magnetic field, fluctuations are even less important for the slow collisions, and continuous slowing down is indeed a valid approximation . Correspondingly, since slow collisions act as in an electron gas at rest, the final proton-beam temperatures are expected to be even lower than those given in section 2. At the present stage of electron-cooling performance, a discussion of reduction of proton velocities beyond - v/60 is hardly relevant. We are grateful to Karl Ove Nielsen for introducing us to the topic of electron cooling and for hill constant encouragement throughout the work. Also, he made it possible for us to keep in contact with the exrneri-mentaall group at CERN. We wish to thank Jens Ulrik Andersen for numerous enlightening discussions . Appendix A To calculate the diffusion coefficients in formula (45) of section 2.6 consider e.g. the v component of the velocity decrement Av, of an electron colliding with a proton. Let r denote a unit vector in the direction of

A.H . Sorencen, E Bonderup / Electron cooling

the x axis and write .r = a., W

bl _,,

+

b 1. , -%,

=

O.

W2

In terms of the velocities K", w' and the scattering angle 0 introduced in section 2 .1, the velocity decrement is A.2 Jc, = ( M,' - %, )" x = a,w`(cos 0 - 1) + (w' - %,) -b, , and hence (Av, ) = a %r,(cos 0 - 1)` + « w' - w) - b, )2 - 2a,w 2 ( 1 - cos 8)(w' - w)-b, (A .3) When averaged over the azimuthai angle between b 1 , and the projection of (N, ' - w) onto the plane perpendicular to w the last term in (A.3) vanishes . The second term equals 1 ( w' - W12 lb1 _,1 2 cos 2~, where %P is the angle between (x,' - x") and the plane perpendicular to w. Using (A.1) for 6 1., and a.,. we obtain with 4 = 8/2 '_[(1-cos8)2+wJr,

2

- cos8)(1 -=; ) i(1+cos8) . w`

111

(A.4)

ihus we have for the square of the ith component of the decrement in momentum (Ap,) ° =m2(2v+,2(1-cos0) + ; sin29(w 2

-3w,2)) .

(A .5)

Consider next Av, Av, Using expressions analogous to (A.2) we obtain

av,av,

-= a,a,w°(1 - cos 8)2

+(

;t,'- w)"b1 ,a,.w 2(cos 0 - 1)

+ (%,'- w)"b1 ,.a,i+- 2 (cos 9 - 1) + «m,'- w)"b  )((w~ - %, )-b.) . (A .6) The second 4nd third terms vanish after an average over azimuthal angle has been performed . The fourth term equals et--: -!~Jb y , ii b 1 , x"1 2 cos`( 8/2), where denotes the angle between b l , and b l , . Eq. (A.6) is then rewritten as + ti." (1 - cos 8) (b l ,- b 1 , ) 8) = iv.iv, ( 1 - cos - 2 sin`8 i v,3v, (A

3c , . à r,

_ ~,-, ~., (1 - cos 0) 2

cos

= (8/2 ) .7)

or equivalently ;jp,jp,=n2`'»;w, [2(1-cos0)- 1 sin2 6 ],

i _j .

(A.8)

The next step is to integrate (A.5) and (A.8) together with the scattering cross section . Terms with (1 -- cos 8) yield the transport cross section a, [cf. eq. (7)]. As to terms with sin20 = (1 - cos 6)(1 + cos 8 ) we note that except for large angles 8 - v we have 1 + cos 0 - 1 and hence f sin 29a2w sin 8 d0 - a, (A .9) u Furthermore. for Jp,vp, (i and j different or equal) all integrals involve factors tv w,w, and these integrals are of the same type as the expression for X j in eq. (20) (v'a,, ). Such integrals receive virtually no contributions from the region of e+-cessive screening and we need only consider the case where 8 = d/a 1 . We obtain j._ sin28 a 2- sin 8 d9 --- fe°"° + f (A.10) O O em.n . . --2j em. n (1-cos8)a27rsin0d8+j (l+cos8)(1-cos8)a2rsin0d8 . 0Q fo.,1.

A . H. Sorensen, E. Bonderup / Electron cooling

51

the cross section a is proportional to sin °(8/2) . The last integral is then rewritten as For angles 0 > a .)(1 - g) «,, _ 2~rea 2Ire (1 + p &""" dju m 1+1 (1 -p,) m w The first term on the right-hand side of (A.10) together with the last term in (A.11) yield 2a, r, and with Qo = 4 ,ffe 4 /rn, 2 w4, we have f~sin20u21rsin0d0=2a, r -a, .

(A .12)

0

From eqs . (A.5), (A.8) and (A.12) the diffusion coefficients are obtained to be

f

D =n(m/M) 2

1 d3Uef(ve)~'[(w2-w,2)atr-i(~t'2-3w?)Q  ,

f

~ =n(m/M)2 d- vel(Le)ww,W,(301 D

-

atr) ,

j.

;

(A.13)

Appendix B We shall determine the dielectric function E(k, w) for a system consisting of an electron gas in a longitudinal magnetic field of finite strength. As in section 2 .4, the description is based on the collisionless Boltzmann equation governing the time evolution of the electron distribution in position and velocity p(r, ve , t). Letting po denote the distribution for the undisturbed gas (in the magnetic field) and p, the corresponding change induced by an external electric field, e.g., from a charged particle . we easily obtain dpi ~ dt perturbed

orbit

I11

avapo, _ ar aLe

where V(r, t) is the total electric potential resulting from the perturbation . As indicated. the substantial derivative dp,/dt should be recorded along the perturbed electron orbit. but Nti-e no%% assume the perturbation to be sufficiently weak for this orbit to be replaced by the unperturbed one in the evaluation of the time derivative. With a spatiallti homogeneous equilibrium distribution p = nf( v,) . see therefore obtain _ en aV af. Pi(r(t),v,(t),t)-Pi(r(0),v(0),0) =- mJ0 dT ar_ (r(T) .T) ai.~ .(Lr(T))

(B ._)

with ve (T)=(v l cos(w,_T+T), v j sin(wjr+T), r,) , c cos(w~.T+T.), v_T~ +r, . r(T)= ~-1 sln(w~T+q)), -

B 3)

where r is a constant and qo an azimuthal angle in the transverse plane . In terms of the Fourier transform with respect to space and time, eq. (B.2) may be written as

Pi(k ve (t), w) =- en V(k,w)ik m

r

- P,(k , LM), w) dTe~kï~rl+iwr af, ~t(r,(,r a Le

.

where r- - r - r(0). Expressing the induced charge density p,(k. w) = (d?i,.,p,(k . t w) in terms of the induced potential V(k, w)(1 - E(k, w )j by means of the Poisson equation, we obtain

52

A. H. Sorensen. E. Bondernp / Electron cooling

Lctk w)=

2

1

k -'

d,re 'krZrl+iwTk

d~

-i wp1 ƒ

fo Jo

l ( Ve (,.

( k,r,+w) : !w, -

aV

1),

(B.5)

where wv have made the special choice of time t = 2v/w,,,, which implies q(t) = q,(0). With k = k 1 + k.. .,O., k s w = 0, and the x axis chosen along k 1 , we have kf(-r) =(k,v 1 /w,)(sin(wT+f)-sin9» +k .vj and

kafo/a q =ik l (e "w.+P" +e -w-") )afo/av 1 +k afa/aU: .

Applying the identity 1141 e i n si n e = En - .J"(a) e '"O, where J denotes the n th-order Bessel function of the first kind, we may perform the integration over z: e -in sin 4, i~2ar a,ikr~sl+iwT afo d-re k -(ve ( .r » (e - 1) E J"(a) e inQ, £ av~ iw~ n®-00 x

k 1 afo afo k: av: ~+n +2 ~` 1 óv 1

e1 Q ~+1+n+5-l+n

îv:

'

+w '

WC

_ k 1 y,. a= WC

(B .6)

With the identity [14] fä dqD e -in "in T +i" T = 2=Jn(a), we may now integrate over yr in (B.5), and changing the summation index in two of the resulting terms, we obtain wpi

00

k

-x

00

dv F(k, w)= i dv, J l 2zrv 1 J o

00

n __ x

J,?

k 1 v1

(

~

(n/yl)afo/av1 + (k ./w )ó.%/óy:

wc

(k :U: +w)%wc+n

. (B .7)

This result may also be found in refs. 18 and 19. In the special case of an infinitely strong magnetic field, only the zero-order Bessel function survives. J(k 1 v ., /w,) -+ 6,, . , and we are back at our expression in eq. (55). a n Also in the presence of a longitudinal magnetic field, the anisotropic Maxwellian distribution po = n,, exp( - v ~'v = - v:/2e.., ] satisfies the Boltzmann equation for the undisturbed gas . With this expression for !1: inserted in eq. (B.7), the integration over transverse velocities may be performed analytically to yield [20] (x °° Xv+J1-2 Dij k:_ . _ F(k .w)=1+k -2e-" ~ In(p.) dv:fo(v:) J k :v. + w+nw, 00 n s-x 2 !2(k,rc ) , ~p1 = v/Fwp1, ÁDu - v ii /wp,,

, (B .8)

where rr = v/w, is the Larmor radius corresponding to the transverse rms velocity v. For an isotropic gas in the static limit w = 0, eq. (B.8) reduces to 2 F(k, 0) = 1 +(ka0 since we have e-"E"1"(FL)= 1 . This is consistent with the well-known result that a static magnetic field does not affect the properties of a gas in thermal equilibrium 117). Appendix C Let us determine the drag (26) for a proton penetrating the flattened electron gas in the longitudinal direction with moderate velocity in the presence of a longitudinal magnetic field, vp 11 B 11 ü . The dielectric function -(k . w) is given by eq. (61), where we assume wi, >t- wp, . We shall first perform the k, integration . The contributions come from regions around points where Re r = 0, and we obtain

A . H. Sorensen, E. Bonderup / Electron cooling

F. _ -

e2

, 2zr`

f

d 2 k,

f

00 0o

dk,k,

a-- Re(k 2 e(k, - k,vp )),

~

e~ , _ 2ir a2 + #` #

ß --- Im(k 2 e(k,

f

d 2k 1

F_

sign(p)

da

i

.

a(k 2 ))

- kyp )),

with the summation extending over positive k.. values, for which a(k..) = 0. These values correspond to multiples of the cyclotron frequency, i.e., k, = nw,,/vp. With frequencies and lengths in units of ca,, and rL --- v/w, respectively, we have

E e-"In(g)(2-k.o) n-U 00

a=k2-wpl

kZ

w2 +n2 2n2 -n 2 ) 2 + w2 -n 2 (w 2

F~ = zkl ,

(C .2)

where w = -kvp = -k,vp and where vp is measured in units of v. The poles in the k . integration are connected with the various values of n, and to determine the nth poles we may exclude all other values of it in the summation, since wp, << 1 . For n = 0 and fixed k 1 the poles are obtained at the values of k . which satisfy 0=a=k? +k,2.

-Aö 2 C - k2,/2 4( -ik 2

),

X'j

(C.3)

= vp/Wpl .

With k. > 0 one solution exists for each value of k,. less than k 1 ma x, where k',,,,, m .,, = 2p, is determined from the equation 2tu.o = e-"°lo(luo)/A,, . For the force contribution F, from this pole we obtain . from (C.1), inserting # > 0 and as/a(k?) = 1,

e 2

Fo - - 2 p,o, rL

2p0 = e-w,1o(jA,)/Aa .

(C .4)

It is somewhat more complicated to determine the force contributions for n >_ 1 . The equation determining the poles for non-zero n is a cubic equation in the small quantity x --- (k :v,,/n ) 2 - 1 . The cubic term and one of the quadratic ones may be treated as perturbations, and the cubic equation may therefore be approximated by a quadratic one by insertion of the lowest order solutions for x2 in perturbing term!, . For vp < 1, the poles in the k, integration are then given by (

vpk '

)2=

n

1 t2d +d2 3+2vp2 -2 nnnI

d =w',e - "I (lu)/ni,

ni=n 2-- (k l vp ) 2 ,

n>_1,

1~ ~!

valid to second order in d , which is small since we assume wp, << I and since e - I'/,(,u) < 1 _< n ; . The mo we fin,a at the poles solutions corresponding to t 2d  lead to sign(#) = t l, respectively . For (aa/a(k (C.1) by expansion in d , _I 2 )j -',

aa

a(k2)

= " d  1 td  3+2v~2-4n --` ni ni

Insertion into eq. (C.1) finally leads to the total force, 2' 00 ,~ 9 e .. 2 -91 .1 4

(C .6)

4ju 

where juO is determined in eq. (C.4). In principle the summation over n in eq. (C.7) should be truncated at n ,a,, - cp/d since for larger n the poles in (C.5) correspond to k values exceeding the usual limit k,,,,,, - 11d. Except for extremcly low proton velocities, v, :5 lOd- 10-3, where also the finite longitudinal electron vel(wities come into . pla,,

A . H. Sorensen, E. Bondcrup / Electropt cooling

such corrections are immaterial. As usual, the ju integration in eq. (C .7) diverges logarithmically at large lu , and it should be truncated at IA,,., -- 1/d 2 , In the limit B eq. (C.7) ihen agrees with eq. (60), since Xmas -= :(kmexrt .)2 tends to zero in this case. The result (C.7) was derived for vI, < 1 . As is apparent from eqs. (C.5), (C.6) it suffices to require jdj" < l . Since we have d. < wrt M /n, where w,,, = 0.08 in our standard case and where M --max(e-"I.(p))(x 1/n with Mt = 0.2, the inequality dr ; < 1 is fulfilled for up somewhat larger than 1 for all but a few very low n values in the summation in eq. ~C .7), and for these n the inequality is only violated in a narrow range of p values around the one lea dine to e - "1 (lu) = M, We may therefore apply the expression (C.7) also for vp somewhat larger than 1 . Under standard conditions higher order correctionq to eq. (C.7) are extremely small. For example, the asymptotic level to be presented in eq. (77) is typically reproduced to better than 10 -5 . References III (21 131

141 151 161 171 18i

191

101 [III (121

11?1 1141 151

it,!

(171 (181

1191 1=01

M . Bell, J . Chancy. S. Cittolin, H . Herr . H . Koziol. F. KrX , . i, 1.i . Lebée, P . Moller Petersen. G . Petrucci . H . Poth, T. Sherwood . G . Stefanini~ C. Taylor, L . Tecchio, C . Rubbia, S . van d -r % °rcr and T. Wikberg, Phys . Lett . 87B (1979) 275, M. Bell, J . Chancy, H . Herr, F . Krienen, P. Moller Petersen and G . Petruo .i, N-,c- '.^str . and Meth. 190 (1981) 237 . G .1 . Budker, N .S . Dikansky, V.1. Kudelainen, i.v. rleshko , V .V. Parchomchuk, D .V. Pestrikov, A .N . Skrinsky and B .N . Sukhi, a Part . Acc . 7 (1976)+ 197. G .I . u, ker, Sov . J. Atom. Energy 22 (1967) 43R . Y.S. Da "nev and A.N . Skrinsky, Part . Acc. 8 (1977) 1 . Y.S. Derbc- :v and A.N. Skrinsky, Part . Acc. 8 (1978) 23 i . M. Bell, Part. Acc. 10 (1930) 101. T. ogino and A.G . Ruggiero, Part . Acc . 10 (1980) 197 . E .g . . T .J .M . Bovd and J .J. Sanderson, Plasma dynamics (Nelson . London, 1969) . F .T . Cole and F.E. Mills, Ann . Rev . Nucl . Part . !~--; 31 ~ 1981) 295 (Review of cooling techniques) . l' . Derhenev and 1 . Meshkov, Studies on electron _:,üng of heavy particle beams, CERN 77-08 (1977) . L! Bell and J .S . Bell, Part . Acc . 12 (1982) 49. N . Bohr. K . Dan . Vidensk . Selsk . Mat . Fys. Medd . 18, No 8 (1948) . W B. Thompson, An introduction to plasma physics (Pergamon, New York, 1964). \1 Abramowitz and 1 .A . Stegun, Handbook of mathematical functions (Dover, New York, 1972) . Ni N Rosenbluth . W .M . MacDonald and D .L . Judd. Phys. Rev . 107 (1957'. 1 . % B Thompson and J . Hubbard . Rei . Mod . Phvs . 3 2 (1960) 714 . See also J . Hubbard . Proc . R o . . Soc . A260 (1960) 114 and A261 (1961) 371 . E-g., N .W. Ashcroft and N .D. Mermin, Solid state physics (Holt, Rinehart and Winston, New York, 1967) . S . Ichimaru and M.N . Rosenbluth, Phys . Fluids 13 (1970) 2778 . N . Rostoker . Phvs. Fluids 3 (1960) 922 . I .S. Gradshtevn and I.M. Ryzhik, Table of integrals, series, and products (Academic Press, New York, 1965).