--

Indian Journal of Pure & Applied Physics Vol. 37, June 1999, pp. 441-448

Deuteron-deuteron collision at high energy A Y Ghaly

Mathematics Department, Faculty of Education, Ain Shams university, Roxy, Helioplis, Cairo. Egypt

Received J February 1998; accepted 4 February 1999

D-D elastic scattering has been studied within the framework of the Glauber high energy approximation as extended by Franco. The elastic scattering differential cross section and the total cross section of D-D have been calculated and comparison has been made with the experimental data at 1 GeV of energy. Some computer programs are given as used for the needed calculations

1 Introduction

To calculate deuteron-deuteron elastic scattering differential cross-section and total cross-section of deuteron-deuteron collision at high energy, a deuteron ground state wave function has been used . In this wave function both the short-range correlations and quadrupole effects have been taken into account.

For nucleon-nucleon amplitude the famous simple Gaussian form has been used. The calculations are at ISR energy -Is = 53 GeV. The effect of short-range correlation and non-vanishing quadrupole moment are discussed . The sensitivity of the deuteron-deuteron elastic scattering differential cross section with respect to some used parameters has been examined. The expectation value for the total cross section has been calculated. The formalism was extended to the study of nucleus-nucleus scattering by Franco', who applied it in particular to deuteron-deuteron scattering generalizations and limiting expressions of the model have also been obtained by Kofoed-Hansen2

as well as by Czyz and MaximonJ for nucleus-nucleus collisions.

In this paper, a theoretical study of available D-D elastic scattering data· has been presented. In our calculation, the model of scattering of high energy particles from nuclei developed by Glauber' has been used. The effect of mUltiple scattering on high energy elastic scattering cross-section has been discussed . We examined the sensitivity of cross

section to the nuclear structure parameter of the ground state wave function of deuteron.

2 Nucleus-Nucleus Collisions

We consider collisions in which the incident beam as well as the target, consists of extended systems with internal structure and generalize the results to obtain scattering amplitudes for such coil isions'.

The elastic scattering amplitude F(q.k) takes the

form"

F(q,k) = ~: fe iijh d 2b

fl\lf(r; ,L,rA )1 2 r,o, (b, SI ,Ls A )8(~ t ~) n d 3 rj AJ=I J=I ... (I)

where \If ( rj ) is the_ gro~nd stat~ wave function of the nucleus, r tot (b, s 1, ® sA) is the -'profile function for the nucleus-nucleus interaction, b is the impact parameter vectors, the s-function in the integrand is Dirac 0- function, ~,j = 1,2, ® A are the nucleon position vectors where A is the target nucleus mass number. In this expression rtot is given by.

rtot< b. S\, ®, ~A) = - - -

1 - eiX tot ( b, s l' ® sA) ... (2)

where Xtot represents the resultant phase shift which take the form " :

442 INDIAN J PURE APPL PHYS. VOL 37, JUNE 1999

Xtot< b, SI , 0 SA) = LJXj( b- s7) where ~ are the projections of the position vectors rj on the Eerpendicular plane to the incident m~mentum k. The scattering amplitude operator apt ( q. k) for collisions of incident nucleon p having momentum hk with target nucleon t, in which a momentum is transferred to the target nucleon, IS related to the profile function r pt by the relation;

( - k) = ~ f iij';; r (b)d 2b ap l q , 27r e 1'1 . • . (3) and the inverse Fourier transform for this relation is;

r (b) =_1_ fe- iQ ';; a (- k)d 2-pI 21rik 1'1 q, q

3 Deuteron-Deuteron Elastic Scattering Amplitude

.. . (4)

The scattering amplitude operators apt ( q) which describe collisions between nucleons may be dealt with most compactly by writing them in the fonn given in Eq. (1):

- - - -apt = I{ q) + tp. t~ ( p) ... (5)

where ~p and ~t are the isotopic spin operators for a nucleon in the target respectively. If we assume charge symmetry of nuclear forces, so thatlpp = Inn and Ipn = fnp , then we may express all nucleon-nucleon elastic amplitudes f and g which we have used in constructing the expression for the scattering amplitude operator, a pt are related to the directly observable amplitudesfpp and fnp via Eq. (I).

.. . (6)

and

form f eiij·' I\{'; (i:i dr, where \vi( ~ ) is the configuration-space wave function of the deuteron ground state. The integral is equivalent to the expression:

seq) = f eiij·' I\{'; (r)1 2 dr = s( -q) ... (9) which is the form factor of the deuteron ground state

It is of important to notice that, using the same consideration and simplifying.!:esult, we have obtained a simple form for f( q, k) compared to the result of Franco'. We finally obtain:

F(q, k) =8s2 (q / 2)f"" (q)+~ 7rk

fs(~)fpp (li)fpp (V){ seq /2) +is2 (~) }d 2~_

- 7r 28e fs(~)s(~')fpp (m )fpl' (li - w)fpp (V)d2~d2~'_

- ;i J fs(~)s(~')fpp (x- m »fpp 7rk

(li - x)fl'p(v+m - x)fpp(x)d 2~d 2~d2t.."

where

A -, 1 - - 1 - A U = q -"2 q , u = "2 q + u,

m = 11 + 11'

and

x = 11 + 11' + 11"

... ( I 0)

... (7) 4 Elastic Scattering Differential Cross Section

Furthermore, g( q ) is si~ply related to the charge-exchange amplitudelc ( q) by

1 g(q) = "2~' (q) .. . (8)

To evaluate the elastic scattering differential cross-section we must make use of integral s of the

The differential cross section of deuteron-deuteron elastic scattering is given by :

do2

= ~ IF(G)12 d 2 k l 1 q

... ( II )

In order to determine some qualitative aspec~ of the elastic scattering processes we sha ll use fpp( q ) in the simple form'

--+

GHAL Y: DEUTERON-DEUTERON COLLISION 443

term when s in the same direction as q. In this ... (12) case, the form factor take the form

where k is the momentum of the incident proton, (J is the proton-proton total cross section, a o is the real part to imaginary part ratio in the forward direction, and A is the slope parameter.

For the deuteron ground state wave function we shall use the form"

where a is the parameter which is related to the radius of the deuteron, "( is the deformation parameter of the deuteron, r represents the short range correlation [I - 2"(3 cos'S -I] "' represents the quadrupole effect, S is the angle between the vector r

and the axis of summetry of the deuteron matter and N is the normalization constant. The constant N can be determined using the normalized condition:

In which, we have"

N = ... ( 14)

The parameter a can be determined by equating root mean square distance between the nucleons in the deuteron

444 INDIAN J PURE APPL PHYS. VOL 37. JUNE 1999

J 2 A( 128,j'a' -12Aa - l) , (J (a O - I) 8( 1.8AaX I, 12Aa) q + . 2 e

16lT\ 1 + SAa)(1 + 12 Aa)

-(\ + S A a) - ------'----'-'-----{

9a2 24(1- 4y)a 2

A (I+SAa)(1+12Aa)(1+4Aa)

(9 2 1 3 ) 3Aa)(1-4y)

4A+ 40a + 20SAa + 256A a - - 32(1 + SA a)(1 + 12 Aa)2

1 1 1 16(1-4y)la 3

(3 1+744Aa+3440A a)q + (I+SAa)(1+12Aa)(1+4Aa)

(1 + 20Aa 2Aa( 11 +SAa) SAa(1 + 16Aa)( 1 + SAa) ---+ + +

4Aa (I + 4Aa) ( I + 12 Aa)( 1 + 4Aa)

12SA 3 a 3 +( 1 + SAa)( 1 + 12 Aa)1 +SAa(1 + 12Aa)1)

(l + 12 Aa)(1 + 4Aa)

+~ 3-~( 1 -. 4y) e-\-;;;;;- q + cr(a" - ) 1 [ 1 ] 1 (1. 'Au 'J ' 1 I

S I7I 16a 471

3-!L..(1-4) 4a(I+12Aa+Sy) e- ~" + { [ '][ ] ( I' U' ),

16a Y (I+SAa)l

A ,

} r

+ ae-'Y (9(1+4Aa)1- 6(1+4 Aa)(I -4AY )+2(1- 4Ad)} (1+4Aa)

cr (Jail -I) e 4(I.KA,,)( I. IU,,) " 1 2 {A(I'llAU'(" "'u' ,

n1(1 + 12Aa)(1 + 4Aa)

{.az(9-. S(1-4y)

l ( I + SAa) 96A 1 a 1(1- 4y)

(! + SAa)\1 + 12Aa)(1 + 4Aa)

6(1 -4y) ) 3A1a 1(1_4y)

(l +12Aa )(I +4Aa) (I_SAa1)(1+12Aa1)

(1 + 4Aa)(1 + 12Aa) + --+ (1 + SAa)q1 + r 16A1a

2) }

\ I+SAa

1 - 4 (1'~A")" ' {2a\I+16Aa+ SOA1a1) +2(1- 4y' e +

) (I + 12Aa)1(1 + 4Aa)1

2A1a

3(1 + 2Aa+ 24A1a 1) 1 A' a' , }

( I + SAa)(1 + 12Aa)l(l + 4Aa )' q + 2(1 + Sa) l (1 + 4Aa)' q - -

)" 2 A(l2KA ~! , , ~ - 1 2 A ' I - I ) ! cr (a" - 6a" + I) - ' (I. ' AuKI.11 Au ) "

-'--''----''--'----=- e 64713 (1 + SAa)(1 + 4Aa)1

!_9a_1

(I +SAa)- 24(1-4y)a1

A (1+SAa )(1+12Aa)(1+4Aa)

- 32(J+ SAa )( I + 12Aa /

(31 + 744Aa + 3440A 2a 2 )q 2

+ 16( 1-4 y)2 aJ

( 1+20Aa

( I + SAa)( 1 + 12Aa )(1 + 4Aa) 4Aa

2Aa(II+SAa) SAa(I+16Aa)(I+SAa) + +

(I + 4Aa) (I + 12Aa)(1 + 4 Aa)

+ 128A 3 a 3 +(I+SAa)(1+12Aa)2 +SAa(I+12Aa)1 )}]2

(I + 12Aa)(1 + 4Aa)

... ( IS)

For comparison we consider also the deuteron wave function of the form :

... (19)

(2/J) 3/4 •

where Ni = ---;; and 13 = 0.0962 fm- 2 Ref. 1.

The corresponding form factor 1 is:

_i.. Seq) =e sp ... (20)

Also the differential cross-section of deuteron-deuteron elastic scattering in the form 1;

r -~( At/J ),, ' (Y 2 { - ~'" -( A+B)l } e - + -(aD - I) e 4 e 4

+ 8Jr +--- -4/3+2A /3+ A

A: P+A ) , - - - (I

2( .... 2 I) 4fJ+6A (J -,a ~ - e

5 Total C ross Section

... (2 1 )

The deuteron-de uteron total cross section 0dd(k) at momentum hk can be written as:

+-

GHALY: DEUTERON-DEUTERON COLLISION 445

... (22)

where ocr is the total cross section defect and the subscripts nand p refer to neutron and proton respectively.

The deuteron-deuteron total cross-section can be obtained using Eqs (10), (12) and (17), or directly from the form of F( q, k) corresponds to the deuteron from factor given in Eq. (17). Using optical theorem 1, in any case we have;

0 2(a~- I) {12a(1 + 24Aa+ 8y) 0,=40+ +

9n (I + 8Aa)2 )

a , (9(1 + 4Aa)2- 6(1 + 4Aa)(I - 4y) + 2(1- 4y)2)) (I + 4Aa )'

_ 4a ' (3a,; -I ) { a 2(9 _ 8( 1 - 4y) _ ) 9n 2 (1 + 12a)(1 + 4 Aa) ( I + 8 Aa )

96 .-1 2a 2 (1- 4y) 6( 1 - 4y)

( I + 8 A a) 2 ( I + 12 A a )( I + 4 A a) ( I + 12 A a )( I + 4 A a )

+ 2 (I _ 4 y I ) ( 2 a I (I + 16 A a + 110 A 2 a, 1

}' ) _

( I + 12 Aa)I (1 + 4.4a)"

a' (a,~-6a,; +I) {9a 1(l+8A a) 1 144 n J ( I + 8.4 a )( I + 4 .·1 a ) ) A

24( 1 - 4y)a 1

(I + 8Aa )( 1 + 12Aa)( 1 + 4Aa)

( 49A + 40 a + 208 A a I + 256 A 1 a ' )

( I + 8r/a)(1 + 12Aa)( 1 + 4.'1a)

( I + 20 A a 2 ,1 a ( I I + R A a ) R rl a ( I + 16 .·1 a )( I + Il A a) ---+ + +

\ -1 /1 u ( I + -1 .. 1 (1) ( I + 12 .. I u l( I + -1 .·1 a)

+ 12R , I 'a'+( 1+ 8Aa}(I + 12 .. la) ' +ILla( I + 12Aa )1 )}

( I + I 2 .. I a )( I + .L-j n )

.,,(23 )

III the case of th e form fac tor give n in Eq . ( 19) we ha ve

a =40' " 2a + +-- -'{ a(a~- I ) r I I ] , 0 87r L 4fJ +2A fJ+A

a2(3a~ -I ) a )(a~- 6a~ +I )

64 7r 2 (fJ+~AXfJ+~A) 4096 7r J A (fJ+~Ay ".(24)

6 Results and Discussion

Deuteron-deuteron elastic scattering differential cross section at the centre of mass energy "S = 53GeV (=the incident momentum in the laboratory system =klab = 1500 Ge Vi c) is calculated using Eq (18). The nuclcon-necleon total cross section and the ratio of the real to imaginary parts of nucleon-nucleon amplitude in the forward direction ' at klab = 750 GeV/c are taken to be crt = 41.32 mb and a o = 0.081 . To obtain the value of ao at the required energy. \ve used the curve, which represents the data of a o at different values of klab (F ig. I).

1600

1400

1200

1000

.D

cs? 80 0 I-----------f 600

400

- 0.02 0 0.0 8 0.10 0.12

Fig. I - Dale of Ct a at different valucs of klab (Rcf. 7) . The solid li ne r

In Fig. 2, fo r . a = 0.0035 (GeV /c)' wh ich corre ponds to the root mean sq uare distance

446 INDIAN J PURE APPL PHYS. VOL 37, JUNE 1999

between neutron and proton in the deuteron ,n = --J5 /4u = 3.73 fm and A=10(GeV/c)2. the elastic scattering differential cross section for y = 0 (spherical symmetry case) and y = 0.132 (when the quadrupole effect is taken into account) is represented by curves I and 2 respectively. The results of Eq. (21) of Franco' are presented in curve 3. The value of parameter 13 in the deuteron form factor which is used in Eq. (21) are taken to be 59.55 GeV" which corresponds to the value of

GHAL Y: DEUTERON-DEUTERON COLLISION 447

~-->'---4

~"""""'--5

Fig. 4 - Dependence of deuteron-deuteron elastic scattering differential cross section at .Js = 53 GeY on the parameter u with A =10 (Ge Y/c)-2 and y=0.132 . Curve 1,2,3 ,4 and 5 represent the results with a A = .01109, .00624 .. 0037, .0035 and .002759 (GeY/c)2. The experimental data are taken from Eq. 4

However, even when the quadrupole effect is taken into ' account, the dip is present and the posi~ions of the first minimum and second maximum are displaced to left from the correct place. This may be due to : (i) the fact that the quadrupole contribution is not sufficiently large at the value where the dip occurs (10), (ii) the D-state component is Rot complete in the used wave function , (iii) the simplifying assumption which is used to calculate the deuteron form factor, (iv) the effect of new type of the double scattering (abnormal double scattering), since the dip is onl y due to the abnorm al double scattering whereas the normal one shifts the dip to the smaller values and raises the value of the cross-secti on''', or (v) the va lue of some used parameters.

To examine the sensiti vity of the d-d e lastic scattering differential cross-section at the used

energy with respect to the slope parameter A of nucleon-nucleon amplitude, the d-d elastic scattering differential cross section is calculated for the two values' 8.385 (GeV/ctl and 10 (GeV/ctl Ref. 8 of A, curve I and 2 respectively in Fig. 3. It is expected that the decreasing of the slope parameter of nucleon-nucleon amplitude leads to increase its effect for large angles (in the double scattering region). Also if the difference between the slope values is not large, its effect for the small values of the momentum transfers is neglected . This feature of slope parameter effects is clear in Fig. 3. We also note that the value A=10 (Ge V/C)-l gives a better agreement with the deuteron-deuteron experimental data. But, no considerable effects on the positions of the first minimum and second maximum are noted .

Fig. 5 - Dependence of deute ron-deu teron c lasti c scattering differenti al cross section at --Is = 53 GeY on the parameter a wit h II =8.3 85 (Ge Y/ct 2 and y=0. 13 2. Curve 1.2.3 .4 and 5 represent the resu lts with a A = .011 09 . . 00624 .. 0037 .. 0035 and .002759 (GeY/c)2 Thc ex perimenta l data are taken from Eq. 4

The e lastic scatterin g diffe rentia l cross-section fo r the values 0.01109, 0.00624, 0.003 7, 0 .0035 and 0 .002759 (GeV/ct' of a are represented by curves

448 INDIAN J PURE APPL PHYS. VOL 37, JUNE 1999

1,2,3,4 and 5 respectively in Fig. 4 with A= I 0 (Ge V/ct' and Fig. 5 with A = 8.385 (Ge V/ct' . In both figures y = 0.132. These values of a correspond to the values 1.368, 1.651, 2.016, 2.062 and 2.277 fm +"' = 0.88 fm. The sensitivity of the differential cross-section with respect to the parameter a is clear for all values of t except at I = O. The correct positions of second maximum and first minimum are obtained for a = 0.00624 (Ge Vlct' and A=8.385 (Ge V/ct'. Fig. 5. But, the curve corresponds to values of these two parameters a, A have no agreement with the experimental data .

At the energy -Is = 53GeV. With the parameter a = 0.0035 (Ge V/c)-' and y = 0.132, for the d-d total cross section, Eq. (23), we have