DEPENDENCE OF AVERAGE MULTIPLICITY ON IMPACT PARAMETER

AND MOMENTUM TRANSFER

G. G. Arushanov, K. G. Gulamov, I. Z. Artykov, and M. M, Muminov

UDC 530.145

Using a previously proposed model as a basis, an investigation is made of the func- tional dependence of the average multiplicity on impact parameter and momentum transfer. The results are compared with those obtained using a probabilistic ap- proach. The advantage of this method is that the information desired is derived from elastic scattering data.

Among the important characteristics of multiple processes arethe dependence of the aver- age multiplicity on the impact parameter of the colliding particles p and the squared momen- tum transfer t. Different models yield various characteristics for these quantities. For example, it is intuitively obvious on the simplest geometrical picture that the average multiplicity n(s, p) (s is the mean-squared energy in the center of mass frame) becomes larger as the impact parameter decreases, while on a naive multiperipheral model, the average multi- plicity falls as p decreases [i]. A paper by Troshin [2] and the references cited therein discuss the dependence of the average multiplicity ~(s, t) on t on the various models. In Troshin's paper [2], this dependence is studied within the framework of a model based on the possibility of a probabilistic interpretation of the unitarity condition [3]. In the present work the behavior of the average multiplicity as a function of the impact parameter and momen- tum transfer is studied on the basis of a previously proposed model [4, 5] and the results compared with those obtained from a probabilistic approach. One of the advantages of this method is that it does not require a knowledge of the differential cross section for the pro- duction of n particles in order to calculate n(s, t) from Eq. (16); it is enough to know the elastic scattering differential cross section, which is measured with much better accuracy than the corresponding cross section for inelastic processes.

We start from the expression for the total cross section of the inelastic processes Oin and the partial cross section for n-particle production On in the impact parameter representa-

tion

~in (s) = ~ ~ ~g (s, p) d~ =, (1) 0

% (s) = ~ ~ ~ (s, p) d~ ~, (2) 0

where ~g(S, p) and ?gn(S, p) are the inelastic total and partial overlap functions, with

*t.(s)= ~ %(s), ~g(s, p ) = ~ wg.(s, v), %~(s, p)~lT~(s,p)l =. (3) n

Here T n is the amplitude for n-particle production and is determined by the mechanism of mul- tiple production. At high energies, the function [6]

(s, n/E(s))=~(s)%(s)/~. (s) (4)

is a function only of z = n/~(s) ("KNO-scaling"), where ~(s) is the average multiplicity

~ i n n 0 0

Samarkand State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 18-22, ~arch, 1979. Original article submitted February 24, 1978.

236 0038-5697/79/2203-0236507.50 �9 1979 Plenum Publishing Corporation

and 5(s, p) is the average multiplicity for a given impact parameter, i.e.,

n(s, p)-----~n~e~(s,p)/~.~,g.(s, p). tz tz

(6)

If the sum over n is replaced by an integration, the normalization of the ~-function can be written in the form

~ ( z ) a z = S z ' r ( z ) d z = l, z =nl~(s), (71 0 0

which is valid at energies high enough that n(s) is no longer small. If we are concerned only with charged particle production (which is in fact the case in experiments), i.e., if we make the replacement n § nch in the preceding equations, the summation is carried out over the even values of nch; this is automatically taken into account by the replacement o n + On/2, i.e., ~(z) § ~(z)12.

In geometrical treatments, it is convenient to introduce the mean-squared inelastic total and partial impact parameters

(8) 0 0 0 0

and also the mean-square elastic impact parameter

0 0

where ~e(S, p) is the elastic overlap function. It is also useful to introduce the mean- square total impact parameter

co oo

0 0

(10)

where i~ (s, p) is the elastic scattering amplitude. From the unitarity condition we have

t t

(11)

The mean-squared impact parameters determine the slope of the corresponding quantities at t = 0; for example,

2 pi. (s) /4- dlna(s, t) I d t f=o '

(12)

where

oo

0

i s t h e i n e l a s t i c o v e r l a p f u n c t i o n i n t h e t - r e p r e s e n t a t i o n .

The mean multiplicity for a given squared four-momentum transfer

= o--P0-- Pi 1=I

(13)

(14)

in exclusive reactions of the type

P~ + Pb "--* P ~ -k Pb + P, + P2 + ... + Pn

s a t i s f i e s t h e r e l a t i o n

(is)

237

(16)

where don/dt is the differential cross section for n-particle production in process (15).

Assuming that the width of the multiplicity distribution is infinitely narrow in the im- pact distance p [i] (henceforth we omit the energy variable),

~gn (P) = ~ (P) ~ (n -- ~(p)) (17)

and using the inelasticity overlap function [4, 5]

?~ (p) --~ 2a exp (-- p2/261) -- ca = exp (-- p2/bl), (18)

we obtain these relationships between the average multiplicity and the impact parameter [7]:

=~+) z c n (p) --.~ 2 ] / - - = - ' ln A (p), z~)(p)---- .2 ] / ~ - ' I n ( l - - a ( ~ ) ,

ca _ ,o \ (19) A (V) = 4 ,naa, _ "7 e ,), i(V) = (v)l (s),

where the plus (minus) sign corresponds to the assumption that n(p) is a monotonically in- creasing (decreasing) function of p. Equation (19) is plotted in Fig. i, in which the param- eters were given the values a = 0.68, b, = 13.1 (GeV/c) -s, Oin = 35.6 mb, c = I.i and 1.38. These choices give a good description of the elastic scattering of protons at laboratory mo- mentum P L = 1500 GeV/c (it should be noted that the solution of the unitarity condition for c > 1 [5] leads to the following behavior of the slope at the diffraction peak: The slope rises initially with increasing ltl, then begins to fall; this pKe~iction was recently veri- fied experimentally [8]). As seen in Fig. i, the increase o~f z~)(p) with p is nearly lin-

ear (as suggested by Moreno [9]), and the curves for c = i.I and 1.38 coincide except at small p. The latter feature is consistent with the fact that the main effect of the param- eter c on the scattering is at high momentum transfer. In the model under discussion we ob- tain

~ 2 2 2 P~n = b~ ( 8 - - ca) ( 4 - - ca) -~, 0 . = Pn, ( 2 0 )

where Pn is found from the equation

n = n (p). (21)

The approximate solution of this equation for the case of n(p) tending to zero* and small mul- tiplicities n < ~(s)~4ab~in is

7~ -) = = p~(-) - - - - 2ba In (n ~ ~i./4 ab, nZ (s)), (22 )

*This is the case which will be considered in what follows, since it leads to agreement with

experiment.

2

i ,

i i

0

%

S Fig.

1

2 g

238

i.e., pn ~-J approaches zero logarithmically at large n . The quantity V ~ i F.

In order to find n(t), p = p(t) must be substituted in n(9), where it is assumed in the geometrical treatments that --t(p) is a single-valued, monotonically vanishing function of 0. We have

co 0 oo 0

r. %(p) dp'-= ~ d t , r. ~g(p) do -~ J dt dr. p (t) t (p) p (t) t (p)

By differentiating these equations, we obtain

( ( d % t = dp~'~ a%, ~ %(P) , at ~ ?' (p) ~ 7 / ~ = n ' a--T = -7/- /~=~,

(23)

(24)

where Pt is the root of the equation

From Eq. (24), we have

"p = ,~ (t). (25)

d%t / d~i.____~n dt dt = ?e (P (t))/?g (p (t)). (26)

Adopting the usual parametrization (we neglect the real part of the amplitude) at small momentum transfer (~0.~ (GeV/c) 2)

we have [ 7 ]

2 3t d%l~__ o~ e~t, ~(?)~.~a~.e_etb" a~--~/4~b, b~.b1~. , (27)

clt t6= I6~. %l

exp ( - - [7 (t)/b) _~ 1 -- exp (bt). (28)

Substituting this pl(t) in fi(p), we obtain for small It[

oz,, ~- (1 - e ~') (29)

If bltl << l , z ~ ) (t) m 4 (a,'=z,,) m O3U r t l TM _~ (o I t I ) ' ( (3o)

Recent experimental data for the elastic scattering of protons at 1500 GeV/c can be also well fitted at high ]t[ by exponentials [i0]

d %/dt ~ A1 exp (bt), (31)

where Ax~ 2 x i0 -s mb/(GeV/c) a, b ~1.8 (GeV/c) -2 for 2 < [t[ ~ 6 (GeV/c) ~, and Ax--I x i0 -5 mb/(GeV/c) 2, b ~I (GeV/c) -2 for 6 ~ [t[ ~ i0 (GeV/c) a. From an equation of type (28), at these values of ]tl we have

/ - (t) _ 2 V - ~'-I in

and since exp(bt) << i, we finally get

[1 4r'abl((l_eOt)b/2b, ca. l a i n -4 ( _ ebt)o/b,) t (32)

z~-~) (t) - - 2 ] / - - ~:-1 in [A + Bebq,

A =-- I 4~ab, I - - , B--- 1-- . ~ln ain

The behavior of E(h) (t) is very sensitive to the values of the parameters A and B. If h >>

Be bt, Zc(h ) is almost independent of t (it approaches a constant value from below as --t in-

creases), and

(33)

239

B e~t)" (34) z~h' (t) ~-- 2 V / - - ~-I ( ln A + ~ -

If, on the other hand, A<<Be bt, for example A = 0, the increase in average multiplicity is quite rapid and linearity proportional to Itl */~, i.e.,

F~ ) (t) -~'2V - =-' (lnB + bt), (35) which is identical with the result found in [2] obtained from completely different considera- tions.

LITERATURE CITED

i. A. Bialas and E. Bialas, Acta Phys. Pol., BS, 373 (1974). 2. S. M. Troshln, Yad. Fiz., 25, 885 (1977). 3. A. A. Logunov and O. A. Khrustalev, Preprlnts 69-20, 69-21. Institute of High Energy

Physics, Serpukhov (1969); Elem. Chastitsi At. Yadra, i, 71 (1970~. 4. S. A. Azimov and G. G. Arushanov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 5, 84 (1971). 5. S. A. Azlmov and G. G. Arushanov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. i0, 25 (1971). 6. Z. Koba, H. B. Nielsen, and P. Olesen, Phys. Lett., 38B, 25 (1972); Nucl. Phys., B40,

317 (1972); B43, 125 (1972). 7. G. G. Arushanov, K. G. Gulamov, Yu. G. Gulyamov, and E. I. Ismatov, Ukr. Fiz. Zh., 22,

45 (1977). 8. A. B. Kaidalov, Rapporteur's Review, XVIII International Conference on High Energy

Physics, Tbillsi (1976); Joint Institute for Nuclear Research, DI, 2-10400, Dubna (1977). 9. H. Moreno, Phys. Rev., D8, 268 (1973).

i0. H. De Kerret, E. Nagy, R. S. Orr, et al., Phys. Left., 68B, 374 (1977).

ELECTROPHYSICAL PROPERTIES OF IONIC ALLOYS OF GaAs OBTAINED BY

IMPLANTING Zn + (150 keV) WITH SUBSEQUENT ANNEALING AT 500-i000~

B. S. Azikov, V. N. Brudnyi, I. V. Kamenskaya, M. A. Krivov, and L. L. Shirokov

UDC 621.315.592

A study is made of the electrophysical properties (N s, ~eff) of ionic alloys of GaAs obtained by implanting 150-keV Zn ions at 20 and 300"C. The ion dose D = 5.10 *s- 10 *6 ions/cm2; the alloys were subsequently annealed for i0 min in an H2 atmos- phere with temperatures in the range 500-1000~ The optimal parameters of the ionic alloys are obtained for T i = 300~ and T a = 700~ Thermal acceptance of the GaAs under a SiO2 film (d = 0.2-0.3 Bm) is observed for T a > 700=C. The limiting concentration of thermal acceptors Ns(TA) = 3"10 xs cm -2 for T = 1000=C and t = i0

min.

Because of its high solubility and large value of diffusion coefficient, zinc has been widely used as an acceptor impurity in ionic alloys (IA) of GaAs [1-9]. Nevertheless there are a number of problems connected with the behavior of Zn in IA which have not yet been satisfactorily solved. Such a problem is the effect of the implantation temperature T i on the IA parameters and the electrical activity of Zn, for which the information available in the literature is contradictory. Some authors [7, 8] have found no significant difference in annealing behavior of IA electrophysical parameters and Zn + electrical activity between the cases of "cold" (20"C) and "hot" (300-500*C) implantation of the Zn + into the GaAs, while another group [9] notes considerable differences in the IA electrophysical parameters depend- ing on whether the Zn + are implanted at 400~ or at 20~

V. D. Kuznetsov Siberian Physicotechnical Institute,, Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 22-26, March, 1979. Original

article submitted! February 28, 1978.

240 0038-5697/79/2203-0240507.50 �9 1979 Plenum Publishing Corporation