Analytical expressions for inelastic scattering cross sections—Part II

ANNALS

OF

PHYSICS:

39,

435452

Analytical

(1966)

Expressions for Inelastic Cross Sections-Part II”

Scattering

S. VARMAt AND A. DAR Department

of

Nuclear

Physics, Weizmann Institute

of Science, Rehovoth, Israel

Closed-form expressions are derived for the differential cross section for inelastic scattering via mutual and double collective excitations and under strong absorption conditions. The expressions result in the generalized “Blair phase Rules.” The predictions are compared with experimental data and good agreement is obtained. I. INTRODUCTION

Austern and Blair (1) have shown that the adiabatic distorted wave theory of inelastic scattering, under suitable approximations, can be expressed in terms of the derivatives of elastic scattering amplitude. Bassichis and Dar (2) demonstrated that the same result may be derived in a simpler way using the W.K.B. approximation (z?), and then may be further reduced to an analytical formula (4) ready for comparison with experiment. This technique is now being extended to mutual and double collective excitations. The expressions obtained for mutual collective excitation are analytic and in this way they present a simple version of the earlier analyses of Hiebert and Garvey (5) and Bassel, Satchler, and Drisko (6). In Section II the expressions for the differential cross sections for mutual and double collective excitations are derived. These expressions are further reduced to analytical form in Section III. A formal discussion of the resulting “generalized Blair phase rules,” which include the phase rules for single, double, and mutual collective excitations, is presented in Section IV. The theoretical predictions are compared with experimental results in Section V. For the sake of complet,eness, the analyses of the elastic scattering and the inelastic scattering via single collective excitation are also included in this section. II. CLOSED-FORM

EXPRESSIONS FOR MUTUAL COLLECTIVE EXCITATIONS

AND

DOUBLE

We consider the inelastic processes a+A+a+A’

I

* The research reported in this document has been sponsored in part by the National Bureau of Standards, Washington, D. C. t Israel Government International Cooperation Scholarship holder, 435

436

VARMA

AND

DAR

a + A + a + A”

II

a+A-+a’+A’

III

via single, double, and mutual collective excitations. In what follows the particles in the incident channel are taken to be in spin zero state. Following ref. 2’ we assume that the effective ineraction between the colliding particles a, A is given by VaaA= vi,

+ CA(5‘%, EA ) r),

where V”, VN are the Coulomb and nuclear potentials, .& represent the internal coordinates of the particles. Using the adiabatic approximation v:A(fa

, tA

, r>

respectively,

(1) and (, ,

z VN(Ra , RA , r),

(2)

the coordinates R, , RA can be expressed in terms of the deformation Ra = RiO -I- at;

parameters

i = a,A,

(3)

where CYi= c Therefore, if the deformations

VN(Ra,

RA,

&‘, rniYTi , rni(i).

(4)

are small

r> = a2v aA + ___

aR, dRn

(5) 0

(Yaa A ’

where 10 = IR, = R,O, Using the Gell-Mann form

Goldberger

Ra = R;.

(6)

theorem one may write the T matrix in the

Tif = (x(-y VLZA- v,, 1PC+‘) + (x(-)1 v*, I&)

(7) where V,, is an optical potential, XP(+’ is the total wave function for the system, with Van replaced by X (-) is a distorted wave solution of the total Hamiltonian V,, and @‘iis a product wave function of the internal wave functions of the colliding particles and of a plane wave for their relative motion. If VO, is chosen to depend only on the relative coordinate of the colliding particles then Tif = (x(-)1 VaA - Vo, l-@(+‘)

(8)

ANALYTICAL

@+’ is approximated

EXPRESSIONS-PART

by a product gc+,

437

II

wave function =

xdAJ/i

(+I

,

(9)

where xa, XA are the internal wave functions of the colliding particles a and A respectively, and +‘+I is the optical wave function for their relative motion. V,, is taken to be v,,

= v 10+ V”.

Clearly only the last term of expression and its amplitude may be written as: Ti,(zlml,

zm2> = (a’ I EI$&, 1 a)@

I tl:A

(10)

(5) contributes

to the mutual excitation

( A)

The overlap integral in the above expression for the T matrix may be further simplified by utilization of the WKB approximation. The method is analogous to the one described in ref. d and results in the following expression, 6) (

I &

o yL,M(e,

4)

I Q>

a

(12) eik.&

Y&n/2,0)

!I$ (

go (21 + 1)1’2t?2”ul3

YI,-M(B, 0).

>

This expression is derived under strong absorption conditions and when excitation energy is small in comparison with the incident energy. Its derivation involves the condition

fi2k s -

+O” a2V(b + 1,~) dx m -m aR,dRA o

>> s

+O” dV(b + 1,~) -co aRCI

t-00 a.2 Is aV(b

0

-cc

+ 1,~) 8RA

(13)

dz o

*

This is satisfied only if the condition MdV x +c l

(14)

438

VARMA

AND

DAR

but this is necessary for the applicability of the WKB approximation, hence is not a further restriction. (d is the distance in which the potential V varies, m is the reduced mass of the two incident particles, and k is the average of the initial and final wave number.) The same procedure may be also applied to the double excitation case and it results in the Austern Blair expression for double excitation (1). The closed form expressions for inelastic scattering via collective excitation, as well as for elastic scattering, may be summarized as follows: 1. The differential cross section for elastic scattering of spinless particles is given by g = I f@>l”

(15)

f(0) = & go (21 + 1) (1 - q2eziu’) Pl(cos 0).

(16)

where

2. The differential cross section for inelastic scattering via single collective excitation, a + A 3 a + A’ of the multipolarity L, is given by

(17) where (1)

< A’ I by I A.> YLM(?TI~, 0) (18)

. g (2Z + l)l” 2 e2’OtYI,+(~,

0).

3. The differential cross section for double excitation of the same multipolarity L, a + A ---) a + A” is given by

(19) where (1)

T.21(2L, M) w --ik $’-

($)

( A”

II f:

11 A

> Y2&4/2,0)

’

(20)

*z

(2Z + 1)“2 ‘2

e2i’rYI,-M(f3, 0).

4. The differential cross section for inelastic scattering via mutual collective excitation, a + A --, a’ + A’ is given by

ANALYTICAL

da

m

z=

(42aP

2 (211 + 1)

EXPRESSIONS-PART

(al2

+ 1) c

4n

II

01 0

L,M

0 I L 0)” I Ti,(L, JoI2 (55 + 1) 12

439 (21)

where

(22)

* YL ,(n/2 > 0) 5 (21 +

1p2

I=0

III.

ANALYTICAL

3J e2irz yz,-‘+A@, 0). az2

EXPRESSIONS

The physical assumptions and mathematical approximations that will now be employed are those of the strong absorption model (‘7) and will therefore merely be listed. (a) The strong absorption model approximates 711by a continuous function of 1

rlz = 91(Z)+ i/.l a$

.

gi(Z) approximates unit step function at LO. LOis the orbital angular momentum for a grazing collision E = LOGO + l>fi” + 2, ZA e2 2mR2 ___R > * ( The analytic treatment of the strong absorption model is independent of the specific functional form of gi(Z). The only practical requirement is that dgi/dZ possesses a simple Fourier transform: i(Z--Lg)B

where Ai is the “rounding (b) For Lo >> 1

parameter”

da 2dl LOrz00 where n is the Sommerfeld

parameter. e2ifll ze

dz

(24)

of gi .

= 2arctg

E 0 0 In the vicinity of Lo we may write 2iu~~eiB~(Z-LrJ

(25)

(26)

(c) For 2 >> M and 0 I 19<< ?r - (~Lo)-’

(27)

440

VARMA I

AND DAR

I

I

Elastic p

I

I

Scattering

p.$4 Nl4)

EC,,,,,= 66.5

Cl2 Mev

100sI-

)-

0!I -

0.c 11 0

I

!

IO

20

I

30 0 C.".

I

I

I

40

50

60

7( 1

FIQ. 1. Experimental result and theoretical calculation for C12(N14, N14)C1*,elastic scattering. E,.,. = 68.5MeV. Application of the above approximations yields simple analytical formulas for expressions (lj), (17), (19), and (21). For brevity we have restricted ourselves to the Woods Saxon form for the absorption amplitude: gl = g2 = {l + exp [(LO - Z)/AI)-‘.

(28)

For this special choice F(A0) =

TAO sinh (TAO)

(29)

ANALYTICAL

EXPRESSIONS-PART

441

II

3 cl2

tol6

o16 I CR*

Ec.m. =72.0

MeV

:

0.01I 0

IO

20

30

40

50

@c.m. FIG. 2. Experimental result and theoretical calculation for C”(016, OlG)CB*, inelastic scattering. I%.,. = 72.0 MeV, Q = -4.43 MeV, L = 2 transition.

and expressions (15)) (17), (19)) and (21) reduce then to: 1. Elastic Scattering (7), & < 0 << ?r - (4&-l

where

pi

= [1 + PL(BOf e)]

d(O sinh Me0

f 8) f e)]

(304

442

VARMA

I 5

.Olo

I IO

I 15

AND

DAR

I 25

I 20

I 30

I 35

1 40

5

8. c.m.

Experimental result and theoretical calculation for CY’(Ol6, O1e*)O*, inelastic scattering. E,.,. = 72.0 MeV, Q = -6.14 MeV, L = 3 transition. FIG.

3.

and Ji = Jit(L0 2. Inelastic 7r - (4LJ1,

scattering

.{(F+” 3. Double

via single excitation

+ F-2)(JyM,

collective

+ ?!m.

+ Jyaq-1)

excitation

of multipolarity

+ 2 F+ F-L&,

of the same multipolarity

Wb) L. For 0 S 0 <

- JyM,-l)}.

(31)

L. For 0 5 fl <<

ANALYTICAL

EXPRESSIONS-PART

443

II

)-\ Ec.m’ 68.5

MeV

0 = -4.4 MA’,

L=2

l-

20

IO

30

40

50

8c.m.

FIG. 4. Experimental result and theoretical calculation for Cu(Nl4, N14)CB*, inelastic scattering. E,,. = 68.5 MeV, Q = -4.4 MeV, L = 2 transition.

7i-- (4LJ1 da _

M p I ( A” II 4: II A >I2(Lo + J4>2 16

dQ . ,g2,

j Y2L,M(7r/2,

0) I2 { (P+2

+ E-vJ;M,

+ JL4

(32)

I-1)

+ ~L++P-(J~,,N,

- J;M,-I> 1,

where (2L, MI cu2 100) = (A”[\ EL’ jlA)Y2~,,(4

41,

(32a)

and &

= (e, f

B)F* .

(32b)

444

VARMA IO-

I

I

AND

I

DAR

I

I

I

I

I

s6-

c12(o16

,o16*)c12”

E c.m = 72.0 Cl = - 10.57

4-

MeV MeV

c .z 3 2 : 0 s-4 2 m c y

2-

l.O-

cl

0

b -

0.8

-

0.6

-

0.4

-

0.2

-

0.1 o

$1.

r

0

-

Lq 0

I 5

I IO

I 15

I 25

I 20 6

I 30

I 35

I 40

0

-

45

c.m.

5. Experimental result and theoretical calculation for CP(O16, O16*)Cu*, inelastic scattering. Eo.m. = 72.0 MeV, Q = -10.57 MeV, Lw = 2, Lolo= 3 transitions. FIG.

4. Mutual ww $ = &

collective excitation of multipolarity

1 ( a’ II Et1 II a ) I2 I ( A’ II fit’

(211+ 1)

Ci% +

ZI + 12. For 0 s 0 << T -

II A > 1’

1) (LO + %)"(fllsinf9> .&

.(I; ; f) I YL,MhP, 0)12{(P+”+ f-“)u:M, + J&) +

2P+

t”--VfM, - &q-l)}.

(33)

ANALYTICAL

EXPRESSIONS-PART

TABLE PARAMETERS Experiment ClZ(N’4

> N14)C’Z

C12(016 cy(y6 C’2& ClZ((y6

Fig.

No.

5 5 8 5

Experiment NiGO(a,a)NiGO Ni60(a,a’)Nie0* Ni60(a,a’)Ni@* PbZ0*(a,a)Pb208 Pb208(a,a’)Pb208* Pb207(a,a’)Pbzo7*

d,(fermi)

1.45 1.41 1.39 1.45 1.41

TABLE PARAMETERS

SCATTERING

v,,(fermi)

1 2 3 4 5

8

@o)c12* @o*)(y2 N’4)C’2* (p*)cn*

I

FOR HEAVY-ION

Ref. No.

Fig. No.

9 9 11 9 10 10

6 7 8 9 10 11

c(fermi)

0.40 0.44 0.4 0.44 0.44

-0.06 -0.06 -0.05 -0.06 -0.06

0.78 0.48 0.32 0.48

II

FOR ~-PARTICLE

Ref. No.

445

IL

SCATTERING

ro(fermi)

d(fermi)

1.33 1.36 1.40 1.46 1.46 1.47

0.24 0.28 0.43 0.28 0.30 0.30

d4A

c(fermi1

0.32 0.25 0.34 0.10 0.20 0.20

0.32 0.17 0.31 0.34

Though expressions (31), (32), and (33) were derived for al OO} --p a /&?FL~), A 100) -+ A IZz~mz)collective transitions, they can easily be extended to the general case of collective core excitations. If a cluster cd 1j,,,) is coupled to a core CA 100) which experiences a collective transition to its excited state CA ( LM) to form an excited bound state of il, A 1JfAMJA), (34) Expression (31) should reduced matrix element is coupled to a core C, cited state C, 1I1n~l) to

be multiplied then by ( 2JfA + 1 )/ (2L + 1) and the stands for the core. In the same way if a cluster ca [j,~~) ( 00) which experiences a collective transition to its exform an excited bound state of a, a 1J,“d/!“),

( J,aM,a> = c hdkh1~ PamI

Jf”M/“)

Ijdd

I hl)

(35)

One should replace (211+ 1) (2Z, + 1) in Eq. (33) by (2Jf” + 1)(2JfA + 1). IV.

PHASE

RULES

Different phase rules can easily be obtained from expressions ( 30)) (31) , (32)) and (33). For 0 >> ) M I/(Lo + ,15)

446

VARMA

AND

Ntso lo,a ) Ni” EL,,= 44 Mev

;

00001-~

0

IO

20

30

FIG. 6. Experimental tering. ELA = 44 MeV.

40

50

60

70 60 e c.m

result and theoretical

90

calculation

Jydd, + t&f-I

=

- &+I

=

JLI

DAR

I

I

I

I

I

loo

II0

120

130

140

for NP(a,

a)NP,

elastic scat-

2 av.4 + ?@ (-l)l”’

(36)

sin [(2L0 + l)el a& + $$i/z>e .>

(37)

Also Y,,,(?r/2,0)

= 0 for

L + il4

odd

(38)

and

MgLI YL,Mb-I& w” = 2T.

(39)

Therefore each of the expressions (30), (31)) (32), and (33) consists of a

ANALYTICAL

0.001

L

0

,

IO

EXPRESSIONS-PART

447

II

I

I

/

I

I

I

I

I

I

I

4

I

I

20

30

40

50

60

70 Qcm

so

90

loo

II0

120

I30

14c

I

50

FIG. 7. Experimental result and theoretical calculation for N?o(a, Cy’)NP*, inelastic scattering. ELLS = 44 MeV, Q = -1.332 MeV, I, = 2 transition.

smoothly varying term plus an oscillatory interference term. One can isolate a common factor from the interference term of the form: 1 2F+ F- . ___ .sin [(2L, + 1)0]. a& + pge eo2- e2

(40)

The remaining factor has the form: 1, (-1)“(0: - 0”), (-)“(&” - e’)‘, and ( -1)11+L2(e~ - 0”)” for elastic, single excitation, double excitation, and mutual excitations processes,respectively. One obtains therefore the generalized “Blair Phase Rules,” namely, in the D region the single excitation transitions of even parity are out of phase with elastic scattering while transitions of odd parity are in phase. The double and mutual excitation transitions of even parity are in phase with elastic scattering and are out of phase with single excitations of even parity, while transitions of odd parity are out of phase with elastic scattering and out of phase with single excitation of odd parity. In addition to the generalized “Blair Phase Rules” there is an “intrinsic phase rule”: Since the oscillatory interference term is proportional to the product [i + de0 + e>l

x

[1 + de0 -

e)l = 1 + 53.h +

~“~002 - &t

(41)

448

VARMA

IO.0

I

I

AND DAR I Double Ni’~a,a’l 5,243

Ij) 0

IO

20

30 @cm.

I

I

I

Excitaiion Ni6’ Mev,P=-2.5

40

MEV

50

60

FIG. 8. Experimental result and theoretical calculation for NP(a, scattering. ELnb = 43 MeV, & = -2.5 MeV, L = 4 transition.

70 a’)NP*,

inelastic

it changes sign for e = A 2/l

CL

+ 2/&

+ d f%?,

(42)

which is usually in the physical region, Therefore the differential cross sections for elastic scattering and inelastic scattering via collective excitation experience a phase reversal at 0 = 8. This fact may be used in order to determine uniquely the value of P,

ANALYTICAL

0.0001’

25

35

EXPRESSIONS-PART

,

I

I

I

I

I

45

55

65

75

85

95

449

II

I

I

I

f

I

105

II5

125

135

145

J

155

165

175

8 c.m. FIG. 9. Experimental result and theoretical scattering. EL&~ = 44 MeV.

V. COMPARISON

WITH

calculation

for Pb20*(a, a)Pbeos, elastic

EXPERIMENTS

In order to demonstrate the validity of the theory, the following experimental data were analyzed. (a) Heavy Ions: 1. Elastic scattering (8) : C”( N14,N14)C12,seeFig. 1. 2. Inelastic scattering via single collective excitation (5, 8) : C12(O16,0’“)C’“’ and

See Figs. 2, 3, and 4, respectively. 3. Inelastic scattering via mutual collective excitation (5) : C12(016,O16’)P*, seeFig. 5.

450

VARMA

lO(

)T

I

I

AND I

DAR I

Inelastic Pb*”

I

Scotiering (Ct,Cfl

E Lob=42

I

rev

Pb*“* a=-

2.615

Mev

70

FIG.

scattering.

10. Experimental result and theoretical EL,~L, = 42 MeV, & = -2.615 MeV,

calculation for L = 3 transition.

PbzoS(,,

a’)PbZ08*,

Table I shows the parameters used to obtain the fits. The quantities ro, d, ~c1/4A, c tabulated are defined in the following In Eq. (28), the quantities Lo , A are calculated by 112 Lo=&R 1-g

[

and A=l+-&][1-$$z respectively.

1

inelastic

way:

(43)

(44)

ANALYTICAL

EXPRESSIONS-PART

‘OO.OI---

lnelostic Pbzo7

4.51

II

Scattering (a@

1 Pbzo7*

ELob = 42 Mev, 0 =- 2.6 MW

O.1 I-

0.0'0

I IO

I 20

I 30

I 40

I 50

I 60

1 70

I 80

I 90

FIG. 11. Experimental result and theoretical calculation for Pbzo’(ol, a’)Pbzo7*, inelastic scattering. EI,&~ = 42 MeV, & = -2.6 MeV, L = 3 transition.

The bars on the Coulomb parameter n, the diffusenessd, the sum of the radii of the intracting particles R, and the c.m. momentum k indicate an average over initial and final states. We define 1’0by the equat’ion R = l.o(a1’3+ A”“)

i4.5)

where a and A are the mass numbers of intracting particles. The reduced matrix element defines the deformation parameter ,B (1) :

c = (A’ II h II A) = (aL’~1)l,2.

452

VARMA

AND

DAR

In the case of mutual excitation, c represents the product of the reduced matrix elements. The parameters obtained are reasonable and seem to compare well with the optical model parameters. Another interesting result obtained is that the product of the reduced matrix elements for the two single excitation cases is approximately equal to the product of the reduced matrix elements for the mutual excitation in the case of C’2-016 experiments. (b) a! particle scattering: A detailed analysis of a! particle scattering with medium and heavy nuclei has been done using the same expressions. Two representative nuclei Ni and Pb were chosen (9-11) . Table II shows the cases analyzed and the parameters used. Angular distribution for elastic scattering is shown in Figs. 6 and 9, for inelastic scattering via single collective excitation in Figs. 7, 10, and 11, and for inelastic scattering via double collective excitation in Fig. 8. ACKNOWLEDGMENT We are grateful to Professor Amos de-Shalit RECEIVED:

for many helpful discussions.

March 4, 1966 REFERENCES

1.

2. 3. 4. 6. 6. 7. 8. 9. IO. 11.

N. AUSTERN AND J. S. BLAIR, Ann. Phys. (N.Y.) 33, 15 (1965); N. AUSTERN, “Selected Topics in Nuclear Theory.” International Atomic Energy Agency, Vienna, 1963. W. H. BASSICHIS AND A. DAR, Analytical expressions for inelastic scattering crossSections, Ann. Phys. (N.Y.), referred to as Part I. R. J. GLAUBER, High energy collision theory. “Boulder Lectures in Theoretical Physics,” Vol. 1. Interscience, New York, 1958. P.J. POTGIETERAND W.E. FRAHN, Preprint. J.C. HIEBERTAND G.T. GARVEY,P~~~.R~U. 136,B 346 (1964);G.T. GARVEY AND J.C. HIEBERT, Proc. Asilomar Conj., 1963. R. H. BASSEL, G. R. SATCHLER, AND R. M. DRISKO, Proc. Asilomar Conj., 1963. W. E. FRAHN AND R. H. VENTER, Ann. Phys. (N.Y.) 24,243 (1963); R. H. VENTER, Ann. Phys. (N.Y.) 26, 405 (1963). JOEL BIRNBAUM, “Studies of Nuclear Transfer Reactions at High Energy,” Ph.D. Thesis, Yale University, 1965. G. BRUGE et al, “Diffusions Elastique et Inelastique de Particules OLde 44 Mev par NiSO Nia, Sn”e et Pbm* entre 30 et 170” c.m.” Preprint. J. ALSTER, “Collective Excitations in Pb207, Pb208and Bim.” Preprint. H. W. BROEK et al, Phys. Rev. 126, 1514 (1962).