Dynamic evolution of the averaged distribution function

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Dynamic evolution of the averaged distribution function

H.I. Abdel-Gawad

To cite this version:

H.I. Abdel-Gawad. Dynamic evolution of the averaged distribution function. Journal de Physique,

1982, 43 (6), pp.883-891. �10.1051/jphys:01982004306088300�. �jpa-00209466�

Dynamic evolution of the averaged distribution function

H. I. Abdel-Gawad

Laboratoire de Physique ^{des Gaz et} des Plasmas (*), Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France (Reçu le 8 décembre 1981, accepté ^le 22 fevrier 1982)

Résumé.

Dans cet article nous décrivons une alternative à la méthode quasi ^linéaire classique pour décrire l’évolution de la fonction de distribution moyenne, évitant les difficultés associées à la résolution d’une équation

de diffusion à trois dimensions. Nous considérons explicitement ^{le cas} des électrons en présence de la turbulence

acoustique-ionique générée par ^{un courant} de dérive. Notre formulation apporte ^une confirmation aux arguments développés par Balescu dans ^un travail récent, ^selon lesquels ^{un état} auto-similaire qui ^se comporte ^comme exp 2014 (w5/03BD50) n’est pas compatible ^avec l’hypothèse ^{de stabilité} marginale imposée dans le modèle de Vekstein, Ryutov ^et Sagdeev.

Abstract.

In this article we derive an alternative to the classical quasi-linear ^{method to} describe the evolution of the averaged distribution function avoiding the difficulties associated to the resolution of a three dimensional diffusion equation. ^{We consider} explicitly ^{the case} of the electron species in presence of ^a current-driven ion-acoustic turbulence. Our formulation aims to confirm the arguments developed by Balescu in a recent work, where it has been pointed ^{out that a} self-similar state which behaves as exp 2014 (w5/03BD50) ^{is not} consistent with the marginal ^sta- bility hypothesis imposed ^{in the model of} Vekstein, Ryutov ^and Sagdeev.

Classification Physics ^Abstracts

52.35

1. Introduction.

In the frame of the QL-approxi-

mation of weak turbulence theory, it has been pointed

out that in a one dimensional plasma ^the averaged

distribution function (adf) ^{can have} asymptotically

a self-similar solution (sss). ^{Such a sss was found to}

be of the form of a plateau [1, 2, 3]. ^{In a recent} publi-

cation [4], ^it has been demonstrated that _many other sss rather than the flattened form are possible.

These non-plateau solutions have been observed in

computational experiments [5]. ^{In a} three dimen- sional plasma, ^the situation is more complicated

because a self-similar state of a turbulent plasma requires that all its components ^must behave self-

similarly.

For the problem of simultaneous evolution of the adf and plasma instability, many studies have revealed the important dynamical aspects produced by ^the

reconstitution of the adf. Experimental [6, 7] ^and

numerical [8] ^studies of current-driven ion-acoustic

instability showed that an energetic ion tail is _pro- duced before the instability reaches its maximum.

The reduction of wave energy by ^these energetic particles may constitute an efficient stabilization mechanism.

Many analytical studies have been devoted to the

(*) Laboratoire associ6 au C.N.R.S.

dynamics ^of production ^of energetic ions, ^first by ^Choi

and Horton [9], using ^{the non} local Green _propa- gator ^for deriving ^the solution of the QL-diffusion

like equation for the adf of ions. It has been shown that fi ^behaves asymptotically ^as exp( - V4/3/VÓ/3).

In a recent publication, ^{Tu Khiet} [10] improved

the results of the production ^{rate of} ion-tail, employing

the method which will be developed ^later.

For the evolution of the electron adf and electron

heating, ^numerical simulations, ^{carried out} by ^Bis- kamp et ^al. [11] ^{and Dum,} Chodura and Biskamp [12],

showed that a self-similar state of the adf is attained

rapidly. These numerical results are also corroborated

by analytical ^{studies, due to} Vedenov and Ryutov [3], Vekstein, Ryutov ^and Sagdeev (VRS) [13], ^{and more} recently ^{to Dum} [14] and Galeev and Sagdeev [15].

In the above publications, pertaining ^{to the ion-}

acoustic instability, the electron adf behaves self-

similarly ^as exp( - w’lv’) ^{where the} system ^{is basi-} cally ^{assumed to be at} marginal stability. ^{But this} hypothesis ^{is so} restrictive that it makes the des-

cription ^of many physical aspects ^of plasma ^turbu-

lence far from the dynamical formulation of the

problem. For, ^it may be expected that, ^as pointed

out by ^Balescu [16], the obtained sss for the adf is not consistent with the starting hypothesis ^{of the}

literature referred to above.

On the other hand, Horton, Choi and Koch [17]

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004306088300

884

claimed that the time scale required by the adf of the electrons to behave as exp(- w’lv’) ^{is not} fully

reached.

To clarify ^this question, ^we develop ^{in section 2}

a method describing the evolution of the adf of electrons starting from ^the Liouville theorem. We shall _prove (in the frame of the QL-approximation)

that the proposed ^{sss of the} isotropic part ^{of the} adf is not attainable. Further, ^an asymptotic ^state

will be found. In section 3 we describe the evolution of the anisotropic part of the adf and show how our

adf tends to be isotropized.

2. Evolution of the averaged distribution function. - The distribution function for ^a plasma ^without

external field satisfies the Vlasov equation :

Introducing ^the characteristic orbit functions of

plasma particle :

with initial conditions x(T=t)=x ^and v(T=t)=v.

In equation (2.2) the statistical averaged fluctuating

electric field is assumed to be zero I.e. ( E(x, t) > = 0.

Equation (2.1) ^can be written as df /dT

^{0, and} integrated ^to give

or

where x°(t) ⁼ x(0) ^and v°(t) ⁼ v(0).

Averaging (2.4). ^{one obtains}

Hereafter, ^for simplicity, ^we shall write x(t) ^for x°(t)

and v(t) ^for v°(t). The distribution function fs may be

decomposed ^{into a} homogeneous part ^{and a small} inhomogeneous part :

where bf,, Fos. Owing ^to (2.6), equation (2.5)

reduces to

Now we prove that (2.7) insures the conservation of the number of particles. ^{For this we} rewrite it in the form :

An integration of both sides of (2.7a) ^over v-space

yields ^the required conservation law.

Hereafter we shall be interested in studying ^the

evolution of the adf of electrons which is initially ^a drifting Maxwellian :

where ve ^{and u} = I u are the thermal and drift speed respectively. ^{For u} > c.

(T e/mi) ; ^{T e} is ^{the elec-}

tron temperature and m; is the ion mass ; ^{the ion}

acoustic modes are generated ^{with an} eigen ^fre-

quency : ^: Wk

kc,(l ^{+ k2} A2 ) - D 112, ^{where k -1} repre- sents the wavelength and AD ^{is the} Debye length.

The study is restricted to the dynamics ^{of the ion-}

acoustic instability ^over the electrons and we drop

the subscript ^{o s ».}

For convenience, ^we represent the adf of electrons in the frame where the electrons are at rest. In this

frame, the initial distribution function is isotropic

with respect ^{to the variable w}

^{v - u, we then} have, owing ^to (2.7)

Here, ^the fluctuating electric field is responsible ^for

the change ^{in the} particle velocity.

Transforming Fo(w, 0) ⁱⁿ Laplace space, then

reporting ^{the time} dependence, equation (2.8) ^beco-

mes (dropping ^the subscript O)))

where F(wo, 0) ^is given by (2.8).

Referring ^{to the} property ^{that the} scattering ^of

electrons by ion acoustic waves is a quasi-electric

process [18]; i.e. I Aw I ^{w; we then set}

where A is purely ^{due to} the turbulent field. Inserting

this into (2.11) yields

where we have singled ^{out the} stochasticity ^{of the}

turbulent field which is represented by A(t).

For evaluating exp i&A(t) >, ^we employ ^{the cumu-}

lant expansion, ^where up ^to second order terms (2.13)

is represented by :

where C1 ^and C2 ^are the first and second cumulants

respectively. Bearing in mind that a cumulant _expan- sion can be truncated if :

i) the first cumulant (as ^a parameter of the expan-

sion) ^{does not} exceed the thermal width of the initial distribution function (the function about which we

expand) ;

ii) the correlation time of the stochastic electric field is much smaller than its characteristic evolution time (as A(t) depends ^on E) ^{which is a basic} assump- tion of the QL-theory.

We show in Appendix ^{A that} CB ^and C2 ^are given by :

and

where

where Ik(t) ^{is the} spectral distribution function in k-space.

Introducing (2.12) ^into (2.14), ^as C2 > 0 the repeated integral ^is absolutely convergent, ^{and we} may invert the 8-integration, ^{then it can be carried, after} replacing a by ^{zero, and} gives

r

It is important ^to notice that when comparing the last result with the method of resolution of QL-diffusion

like equation by introducing the Green function formalism : Dum [14],-

we see that the Green function is

For t

0, r¡(t

0)

^{0 and whence} C1 ^and C2 ^are zero, G(w, wo,breduces ^to 6(w - wo). Beyond t

0, the Green function is quasi-Gaussian in the variables w and 14,0.

Carrying ^{out the} wo-integration ⁱⁿ (2.18), ^with F(wo, 0) given by (2.8), ^{we obtain :}

where

886

In the regime -u ^{1 we have} C2 ^{1 and the} argument ^{of the error} function, appearing in the second ^term in the curly ^{bracket, can be} approximated by ^{I then} (2 . 21 ) ^{reduces to}

The last formula allows one to study the evolution of the adf. However, ^the complex ^{form of the} first and second cumulants makes the description of evolution of the adf difficult. One may decompose ^the later, given by (2.22),

into an isotropic part (independent of p) ^{and an} anisotropic part ^which depends ^on Jl.

The isotropic part ^{can be} readily ^obtained by ^{setting p = 0 in} (2.15), (2.16) ^and substituting ^into (2.22) yields :

where w > (u - cs/ve).

One establishes that, ^from (2.23), Fiso depends ^{on the time} through ^the dynamic quantity q(t), ^defined by (2.17), ^which contains the dynamic evolution of _energy in the ion acoustic modes. We shall show, ⁱⁿ Appen-

dix B, ^that q(t) ^increases monotonically ^{from zero to an} asymptotic ^value q z (U)-I. This will be done by solving the kinetic equation ^{of waves.}

It becomes clear that the isotropic part ^{of the} adf, ^as given by (2 . 23) ^{does not evolve to a sss as} exp( - W5/vg).

It relaxes to an asymptotic steady ^{state which} corresponds ^to q £ (U) - 1 ^{and is} given by :

From the last equation ^{one finds that}

in the region w (2 u-)’Il ^{which means} that the distribution function is flattened over a small region ^about

its initial maximum. Beyond ^this region ^{it is} approximately Maxwellian.

In the next section we shall investigate the modifi- cation of the anisotropic part and the mechanism of

isotropization.

3. Evaluation of the anisotropic part ^{of the adf.}

We derive here an approximate formula for the ani-

sotropic part by converting equation (2.22), ^des- cribing ^{the time} dependent ^{adf, into the} Spitzer-

Harm form [19] :

where Fi.. ^is given by (2.23). The function #,(W-, t)

can be derived by expanding ^the exponential in (2. 22)

up ^to first order in _p. For this, ^we represent Ci ^and C2 ^{in a more} convenient form :

After substituting ^for C1 ^and C2 ^from (3.2) ^and (3.3)

into (2.22) ^and expanding, ^{one obtains}

From (2.23) ^and (3.4) ^one may establish the fol-

lowing arguments :

i) The second cumulant (C2) plays ^{an efficient}

role in changing ^the isotropic part ^{of the adf} through enlarging the thermal width.

ii) ^{The term} Cl contains the effect of the turbu- lence on the isotropization process of the adf.

The above arguments become clear when we eva-

luate the first and second velocity ^moments of the adf.

The drift velocity of electrons evolves as :

Substituting ^for F(w, t) by ^its approximated ^formula (3.1), equation (3.5) ^becomes

Up ^to lowest order in (ü)2 ’1 (3.6) gives

From the last equation, ^one finds that the drift velo-

city ^decreases during the evolution of ion acoustic

instability. ^{One will} expect, then, that when the

condition I u(t) I > Vph ^Cs breaks down at certain

time, the ion acoustic instability ^{will be} quenched.

This becomes evident from (3.7) and will be shown,

in Appendix ^{« B} >>,’ through ^a systematic ^derivation

to the solution of kinetic ^wave equation.

Further, ^the decreasing ^drift velocity ^{shows that}

the adf tends to be isotropized ^when u(t) ^-+ 0, ^and the rate of isotropization ^is

where v(t) ^is the turbulent collision frequency (as

introduced in the literature of Choi and Horton [20]).

Now, ^we evaluate the time dependent ^electron temperature by calculating the kinetic _energy of the

particles ^{relative to the mean :}

By ^{the same} method, ^as above, (3.8) ^becomes (up ^to

lowest order in (u-)’ q) :

From (3.9) ^{the rate of} change ^{of the} particle energy is :

From (3. 8) ^and (3. 5), ^one concludes that the turbu- lent rate of isotropization ^is higher ^{than the rate of} change ^{of the} particle energy following ^the square of the thermal speed ^as compared ^{to the wave} phase

speed.

4. Conclusions.

We have developed ^a statistical

theory describing ^the dynamic evolution of the adf of electrons, ^{in the} presence of ion acoustic instability, avoiding the difficulties associated with the method of resolution of the QL-equation.

It has been shown that the isotropic part ^{of the} adf does not admit, ^at all, ^a self-similar solution of

the form exp( - w’lv’). However, ^an asymptotic steady ^state of the adf (corresponding ^{to a} steady

state of wave energy) is attainable. This asymptotic form is flattened over a small region, ^{in the} velocity space, w (UV;)1/3. Beyond ^this region ^{it is} approxi- mately Maxwellian.

The above conclusions confirm the arguments

developed in the work of Balescu. Our formulation is based on studying ^the dynamical evolution of the

instability ^{as well as} of the adf. Contrarily, ^the pre- vious work assumes a self-similar property ^{of a} turbulent plasma ^{at a} marginal stability ^{and then}

proves the inconsistency.

Our method can be used to describe the evolution of the adf of ions. Further, knowing ^{the time} depen-

dent adf allows us to study the anomalous transport coefficients. This will be our object ^{in a} subsequent publication.

Acknowledgments.

The author would like ^to thank Dr. Tu Khiet for valuable suggestions ^and

discussions.

Appendix ^A.

^For evaluating the first and second cumulants (C1 ^and C2), ^{we have to derive an} explicit

form of the stochastic quantity A(t). Starting ^from w(t) = I w(t) I, ^{writing w(t)}

^{w + Aw(t),}

and developing ^the expansion ^{about w} = I w I (as

I Aw I w) ^{up to} second order in I Aw I (or ⁱⁿ E)

one obtains for A(t) :

The problem ^of calculating C1 ^and C2 ^{is reduced}

to the calculation of the statistical average of the deviation and _square deviation in the particle ^velo- city, taking the lead of (A.1). ^One has, then,

The function 0(t’, t") represents the correlation func- tion of the electric field, ^{which is} symmetric ^{in the}

variables t’ and t". By ^{the last} property (A. 3) ^{can be}

rewritten

888

whence.

_ n At

Now, assuming that the correlation time is small

compared ^to the characteristic time of evolution of the electric field, ^one may represent (A. 4’) (in ^the

frame of a QL-theory) by :

Converting the summation into integration ^and assuming ^a nearly isotropic turbulence (spectrum ⁱⁿ k-space), ^{one obtains :}

and I is the unit dyadic. ^{One finds}

or

where il(t) is defined by (2.17).

By ^{the same method we have}

In (A. 8) ^{the terms of order} (-U)3 ^and higher ^{will be} neglected, ^{on then finds :}

we rewrite (A. 8) ^and (A. 9) ^{in terms of w :}

One derives further :

The direct substitution from ^(A. 10) (A. ^{12) into the} appropriate formulae of C1 ^and C2 (C2 ^is approxima- tely (W.W)2 )/W2 ve) yields (2.15) (2.16).

We have to notice that, rewriting (A. 12) for Av )

in terms of v :

or

where v(t) ^is the effective (turbulent) ^{collision fre-}

quency. The last equation ^{shows that the} velocity

of the electrons decreases due to their interactions with the ion acoustic _waves, whence the electrons will be transported ^{from the} region ^of high velocity

to that of low velocity ^{in the} v-space. This _process

causes the modification of the distribution function.

Appendix ^B.

^To discuss the evolution of the

dynamic quantity 11(t), ^{we have to} solve the kinetic

equation ^{of the} spectral distribution of _energy density ak(t). ^{In the WKB} approximation ^this equation ^is represented by :

where r k denotes the wave damping (or growth) ^rate.

As we are interested in studying the effects of modifi- cation of the electron distribution function on the

wave dynamics, ^we shall restrict our calculations to

r ke(t) :

where Re 6(k, w) ^is given by the classical result :

equation (B . 3) takes the form

In terms of w (B. 4) ^becomes

where Qk

k. u - Wk. For carrying ^{out the} integra-

tion in (B. 5) ^{we have to} substitute for Fe(w, ^t) by ^its explicit ^form, given by (2.21) ^or (2 . 22), bearing ^{in mind}

that (2.22) characterizing ^a plasma system ^where u 1. This will be our object ^for carrying ^{the calcu-}

lation of the growth ^rate.

An approximate ^{form for} Fe(w, t) (which ^is given by (3. 1)) ^{can be used, but one must account for that} (3.1) ^{is a truncated} expansion ^{of the} adf abouts its

isotropic part. ^{Such an} expansion ^{is valid over a} region ^w > (u ’I)

Returning ^to (B. 5), ^after carrying ^{out the} integral

over the angle between p ^and w, the lower bound of the w-integration ^{will be} wo = ilk ⁼ Qk/kve. ^This

leads one to distinguish ^{between two limits :}

i) Qk > (U- 1) 1,4 , Fe(w, t), ^{under the} w-integration,

can be replaced by (3. 1), ^further Fi.o ^{will be} approxi- mately Maxwellian and (B. 5) ^becomes

We may split rke(t) ^into rke(O) ⁺ F ke (’)(t), ^where

and

from (B. 6a) ^and (B. 6b) and the above condition,

one finds ^that F(’)(t) ke ^{cannot balance} F k,,(O) ^because

?I(t) Ek’l-u. ^{This leads one to} estimate that the modification of the adf _may produce ^a significant

reduction to the growth ^{rate when} q(t) >, dk’/-u.

The level of _energy required by the last condition

can then be estimated. As, approximately,

r nv, kyL 1 a ; ^{a is the total} energy; ^one finds that

This will be the case :

ii) (ü fl)1/4 Qk; ^we employ the formula (2.22)

for Fe(w, t).

After operating O/N ^onto (2.22), (B. 5) ^becomes

The last equation ^can be rewritten (up ^to lowest order in S2k (or ü))

where p

^{xl X2 + Yl Y2 COS 0, Y1,2 = (1 -} xI 2)1/2 ;

x2

k. u and F,,(-w, p, t) ^is given by (2. 22), (2J5)-

(2.17). One finds that the dominant term in Cip

is (U’1,u/(W)4), ^{cf (2. 1 5),} then for -W > (u- Ci p « ¹

and C2 ^{1 so that} Fe ^{can be} approximated by ^a

Maxwellian. The 0-integral may be carried out when the w-integral is cut-off at w ⁼ (u- ^{Then we have}

The wave energy reaches its maximum value when the growth ^{rate becomes zero; that is, when}

We notice that i) r ke(t) is, approximately, linear in k

as (Ok - kcg.

ii) ^The growth ^rate (Tke) ^{can be} represented ^{in a}

more convenient form :

890

where ^{we have} approximated ’4 exp - (U-?1)112 by Qk, ⁱⁿ (B. 9), in a consonance with the assumption

that (-U) 2(or (!5k )2) ^1.

The effect of the modification of the adf of electrons onto the growth ^rate is manifested by the function

K( ). ^{It becomes} clear that the adf, during ^{the evolu-}

tion of ion-acoustic instability, changes ^{in such a} way that the net drift velocity of electrons decreases down to zero i.e. the adf tends to be isotropized. ^{This means}

that the source of _energy, furnished by the electrons to the _waves, is exhausted, whence the _energy in the

waves is reduced so that the instability ^{will be} quench-

ed. The above arguments explain the mechanism of stabilization _process of ion-acoustic instabilities

accounting for the modification of the adf of electrons.

We rewrite the kinetic ^wave equation

Integrating ^{over the whole} k-space, employing ^the

fact that the wave spectral distribution is peaked ^at

k = k* = À-D/,J2 ^{and then one derives an approxi-}

mate relation between the total _energy a and q, by referring ^to the relation (2.17) :

and we obtain :

Writing ^{T =} 2j; ^{kC,, t and} integrating (B. .15) yields :

Owing ^to (B. 14), ^{the total} energy is represented by :

For obtaining ^the asymptotic value ^{of q we repre-}

sent K( r) ^{in the} limit q Z (iT4)’/-u. Furthermore

K(q) ^is approximately represented by

Introducing (B. 18) ^into (B. 16) yields

where A = (Ut)1/4. ^{The last} equation gives

This yields ^that q,,,, --- 0(1 I-u) ^which is consistent with the value of q derived from the condition ^of validity of the cumulant expansion; namely C1 ^{1 gg 1}

or u r,u ^{1 which} implies that rJ 1/ J.lu ^{and the}

maximum value of q ^is (1-i)-’ (for J1

1). ^{The above} asymptotic ^value of y corresponds ^{indeed to a} steady

state of wave energy (a - ^0).

For q . 1> ?I 1> (Q4)4(U, the total wave energy is

represented by :

References

[1] DAVIDSON, R., Methods in Non-Linear Plasma Physics (Academic Press) 1972, ^{ch. 9.}

[2] IVANOV, ^{A. A. and} RUDAKOV, ^L. L., Sov. JETP 24

(1967) ^1027.

[3] ^{VEDENOV, A. A. and RYUTOV, D.} D., ^Review of Plasma Phys. (1975) ^{vol. 6} (ed. ^{M. A.} Leontovich). ^Consul-

tant Bureau.

[4] TU KHIET and ABDEL-GAWAD, H., ^J. Physique-Lett.

41 (1980) ^L-189.

[5] ^{DAWSON, J. M. and} SHANNY, R., Phys. ^{Fluids 11} (1968)

1506.

[6] ^{BENGTSON, R. D.,} GENTLE, ^K. W., JANCARIK, MEDLEY, S. S., NIELSEN, ^{P. and} PHILLIPS, P., Phys. ^Fluids

18 (1975) ^710.

[7] ^WHARTON, C., KORN, P., PRONO, D., ROBERTSON, S., AUER, ^{P. and} DUM, ^C. T., ^{in Plasma} Physics ^and

Controlled Nuclear Fusion Research (International

Atomic Energy Agency, Vienna) 1971, ^vol. II,

p. 25.

[8] BISKAMP, D., CHODURA, R., ^{in Plasma} Physics ^and

Controlled Nuclear Fusion Research (International

Atomic Energy Agency, Vienna) 1971, ^vol. II, p. 265.

[9] CHOI, D. I. and HORTON, ^W. Jr., Phys. ^{Fluids 18} (1975) ^858.

[10] ^{Tu KHIET, J.} Physique Colloq. ⁴⁰ (1979) ^C7-605.

[11] ^{BISKAMP, D. and} CHODURA, R., Phys. ^{Fluids 16} (1973)

888.

[12] ^{DUM, C.} T., CHODURA, ^{R. and} BISKAMP, D., Phys. ^Rev.

Lett. 32 (1974) ^1231.

[13] ^{VEKSTEIN, G.} E., RYUTOV, ^{D. D. and} SAGDEEV, ^R. Z., JETP 12 (1970) ^291.

[14] ^{DUM, C. T.,} Phys. ^{Fluids 21} (1978) 945, 956.

[15] GALEEV, ^{A. A. and} SAGDEEV, ^R. Z., ^Review of ^Plasma Physics, ^{vol. 7} (1979) (ed. ^{M. A.} Leontovich).

Consultant Bureau.

[16] BALESCU, R., ^{J. Plasma} Phys. ²⁴ (1980) ^551.

[17] HORTON, ^W. Jr., CHOI, ^{D. I. and} KOCH, ^R. A., Phys.

Rev. A 14 (1976) ^424.

[18] CHOI, ^{D. I. and} HORTON, ^W. Jr., Phys. ^{Fluids 17} (1974)

2048.

[19] SPITZER, ^{L. and} HARM, R., Phys. ^{Rev. 89} (1953) ^1977.

[20] HORTON, ^{W. Jr. and} CHOI, ^D. I., Phys. Rep. ⁴⁹ (1979)

273.