,; -<yIC/80/160

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

A CORRECTION TO THE WATANABE POTENTIAL

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

A. Rabie

M.A. El-Gazzar

and

A.Y. Abul-Magd

1980 MIRAMARE-TRIESTE

ic/8o/i6o

International Atomic Energy Agency

and

United Hations Educational S c i e n t i f i c and Cultural Organization

IHTESNATIONAL CENTRE FOR THEORETICAL PHYSICS

A CORRECTIOH TO THE WATAHABE POTEHTIAL •

A. Ruble • • M.A. El-Gazzar • • •

International Centre for Theoretical Phys ics , T r i e s t e , I t a l y ,

and

A.I . Abul-MagdPhysics Department, Faculty of Medicine, Abdulaziz Universi ty ,

Jeddah, Saudia Arabia.

ABSTRACT

Using the adiabatie approximation, an analyt ic expression for the

correction t o the Watanabe potent ia l was obtained. In addi t ion , we have

corrected through a proper choice of the energy at which the potent ia l

parameters of the const i tuents o f Li should be taken.

HIRAMARE - TRIESTE

October I960

• To be submitted for publ icat ion.

• • Permanent address; Physie« Department, Faculty of Science , MansouraUnivers i ty , Mansoura, Egypt•

• • • Permanent address: Hattxeutlcs and Theoretical Physics Department,Atomic Energy Establishment, Cairn, Egypt.

I . Introduction

Using trio 'iiatinabo foldir.3 n.odol, ono can't intend to obtain, a good

f i t to t.ia sluBtio scattering au.ta 6inou no account lor absorption

modes iu i.-iuluded in i t 3 formulation. Vi<itson has improvod tiio f i t

the modification of tun iaaginury wall. Usin^ Sliiubar

-aoattcrititi tneory, the correctian to tho 'iVit-in^ba potential

was calculated analytically for deutarons uni tritona ' . A'e exten-

ded thia ctathod to the case of Li with tho result of the necessity

to modify both the real and imaginary wells to improve the f i t t ing .

In addition, an analytic expression relating tha pioter energies of

both the o( - and d- clusters which are no longer 2. E, and JL.£..

respectively was outiiirKid. Taking th« optical parameters of both the

(X- and d- nucleus scattering at the proper energies made some

•improvement in fitt ing*

2. The analytical evaluation of the o rrsotion

In (Hauberk theory, the scattering amplitude for a Li projectile

given by

where tha braoket 4. > denotes the itverage over the internal coordin-

ates of, Li neglecting the internal motion of both the o(- and d-

cluaters, b ia the Li impact parameter and"3 ia the projection of

th» vector r ^ in tha plane perpendicular to the direction of inci-

dsnee. The phase function is given by •

(2.2)

where

and v is the velocity of Li projeotilas. Let us defina the Li

optical potentia.1 V . Aa tue effective potantial yiel

ing amplitude identioal with thutgivan by eq,. (2.1),

i.e.

where

Glaubor iiis used :t similar defini t ion for tns OiitissU

shi f t s i.-. a ia study of hadron-nucleu3 so.i.tt

(2 .5)

Following Glauber, the solution of (£.5) i^ j -v -n by

(2.6)

wfaere

If we ignore the distortion of Li during scattering, we are left with

tha first tarm of the expansion (2.6) a.na.%. will raduoe to

Changing tha variable z .in the first integral inside the square bracket

by a' • 8 + — a and the variable a, in the second, integral by

z1 • a, - I z and comparing with eq,. (2.4), *-s jjetd 3

irhioh i s exaotly eq.util to the "Aatanabe po ten t ia l . If .va express the Li

opt ica l po ten t ia l as -

"> (2.10)

- 3 -

tiian ^V(v)"-iar^ctt;ri^03 the contri'uutiun from the distortion .ind

virtual diamtijjratian of the Li in tins proocaa of scattering.

Numei'ic l studies with the Watanaae model ' ' concluded that

corrections to the real part for deutoron3 aru of thu ordii£ ol' trie

leading term. Since the correction to the iiti^inary part is not

expected tu exceed the real n'at&nabe potential, we can safely assume

the series t,2.6) to ocnver^e so rapidlythat the second term is

sufficient for estimating the correction. In this onus "B have

To obtain an analytic expression for e^. (2 .H), we shall take both

— and d^nucleus optical potentials of tha Gaussian form

Subst i tut ing eqs. (<{.I2) and (2.3) into (2.6), we o«.n, in pr inc ip le ,

calculate "Y flijup to any order. The expression i'ounu in th is way v/ill

consist of a superposition of Gaussian i'unctiona, and the corroapon-

ding Gaussian potent ia l can be imawui i t ° ly foun<i riue to the l inea r i ty

of e<i« ( 2 . 4 ) . If we r e s t r i c t ourselves to the f i r s t t*o terms in o'i.

(2 .6 ) , we find the following expressions for the Vtatana'ue potent ia l

V, ( ' ) and i t s co r r ec t ion o \A ( v j

1 ,! . I3 )

where

- i t -

KV

where

w L

for th« volume Integral and.'Man aquaae radius of the Gtm33ian

potential V2.I2) td« voluM inte*ra.l uia tKe me -in squaie radius of

the

where

ifid

'(2.20)

1 j i l i i t . - ; - . il vijuji of t.n volume

/M tnat liil'l'srent pot9n-

-5--6-

square radii <V* (jlve equivalent degree of fitting. Hsnoe we can use .„

the optical potential parameters to calculate the expressions (2.17) -

(2.50).

3. EBjerioal evaluation of the correotioh

THe optioal modal analysis of proton and neutron scattering indicates

that tiie depth of the real part of tha optioil potantial and hence its

volume intejjral is linearly decreasing with enerjy, i.e.

where C and B are parameters dejerxir.t' on the target mass number.n n

Th» «a..a can be fairly expected for deuteror.s and alpha particles, i.e.

(3-2)

and (3.3)

Aooording to eq. (2.17), the volume integral por nucleon of the real

part of Li optical potential is

S.4)

But the volume integral per r.i>claon of the rfitanabo potsntia.1 is

given by

Comparins e^s- (3>4) and (3.5), one easily obBerveo that i t i s posaibla

to make the voluna intBgral of the tfatj.nibe potontial equal to tlie

- 7 -

volume integral of the potential, ,jivBn by 8^.(3.4), whiob. takes into

aooount tiie disinta(jration of the .Li by taking tha deuteron j,nd tho

(*<•-.particle energies not at i- E . and?: Sj^ respectively, but at

energies related by

•This i s a straightforward extension of a method used by M. H. Siabol

for deuterona and tritona

4. Results and discussions

Expressions (2.17) ar.d (2.13) Gu^gest that wa can improve tha fit t ing

by varrying the depths of the real and imaginary ports of the potential.

Moreover, i t was found that the correction to tha mean square radius,

which contains a l l the ^-eorr.e-trioal parameters, i« negligible. So , i f the

Watanabe potential i s exprosaed as

the potantial expected to iaprova the fittint; will take the form

vco - io ,where a , b , ^nii o are fi t t ing parimBtsra. Figures I and 2 rupror.ont

the results of t:-.a r^tio G~/ai*.n& tho vootor polariaation P induced by th»

the potar'.tiilft.X ), whereas fiijuras 3 and 4 display the results induc»& '

by tho pototitiul C1*.?) v hich show an improvement in fi t t ing especially

for the vector fol-j.i'ii:j.tion F. The optical potential parameter* used «tr«

-8-

listed i n t a b l e

thu suitable vj,l-,;<sa 'of a and b , tha location of

and oiniaa is racijcr.ably fitted for both the ratio<T/ J_a.na P as

in fijjs. 3 and 4. I t userr-a that the paraiaoter b is not sufficient to

predict correctly the amplitude of the oscillation. Por the sake of

compariaoni we hava calculated tha correction factors; a t.nd b .uoord-

Ing to expre&iicnu (2.17) and (2.18) usinj the p:u-J.mjt«rs of table I .

Tha oal?ula.ted and fitting values are liated in t^ble 2.

Chan,ji:i& the depths onl/ and neglecting the geometry otian^e

may l)a a reason for trie misuud optinrum coincidence. Soally, v;e have

not this clioise of varying tha geometrical parameters otherwise we have

to vary 9 parameters for thec<- and d-optical potentials • This i s large

enough to hide any physical meaning evon when an optiauiu f i t 1B attained.

T« uletar.iiine the constants ; e.iB, ( 3 . 2 ) j.tui ( 3 » 3 ) f

the volume intjgral of the real part of both the dsutaron and the alpha

partiole optical potential was calculated according to the approximate

formulae.9)

(4.3)

for •different onurLcioB. Then, by the least square method, ths best values

of the constants that give the best linear dependence of e'»a. (3.2) and

(3*3) £-T© found. The optimum forma are found to be

(4.4)

Substituting thase C'a and B 's values and the values of the correction

parameters a ( a = I —at (£7) in tho expression (3-6)j v/e have found thea*

relations

3 for ^ u = 2

for

for

for the 6Li - I2C scattering

In principle, any two values of fi and E. satiafying any of the

relations (<+»5) are expected to give a fit t ing of the same quality as

that giv«n by correction through the depths. Due to tho lack of data,

we tried one set for each of the four cases. The results are shown in

Fig. 5 for tha two corrections compared with the Crf.se of no correction.

The parameters used in fitting the data are given in table 3» From thea«

parameters we have calculated a and b ;iuoordintj to expressions (2.17)

and (2.IC). The calculated and the fitting valuoa aro liatad in tabl*>4»

The fitting parameters used in the case of correction by energy

ohoioe are liated in table 5*

for this data.The ratio /£"-, and Ej/f, were plotted against

Figure 6 shows tnat both ratios begin with a large value at umall &,.

-10-

then docreise . t i l l i t rot.chas the xt z,.^, fr 220 Jlev for

and the value i- at 2. . ~ 124 Hev for the ratio £./ i...

•-.. Fron t':.a expressions (^.17) and (2.18) it is clear that the correc-

tions to the voluce integral per nuoleon of the real and icao-inai-y parts

of the Viatanibe potential are inversely proportional to E and to theiji

target raasa number A . These syatematics are seen in table A for t'.:a

'parameter b. Yiita respect to the mass nurr.ber varriation thesa syatenutics

are missed both with respect to energy and ciaaa nun.bar varriation. ?ha

brave adoption that expression (2.17) and (2.18) ai-a independent of the

'form of the potential needs justification. The ainbijuitius in the c,tic^l

potential parameters used in calculations ;:.'ty ba a reaaon. T:ia r.Ji;-e i.u.-.li-

ty Of the quality of fitting Tor different t^r^-eta and different endi-lus

and also tha ambiguity in the parameter Eay be a reaaon for the missed

systematic?;.

Ooncernin;j tha calculated values of a , i t its found to bs small

compared with b . This means that the contribution of the distortioia of

Li is irapar-tint only in tlia calculation of the iir^^inary part of tlio Li

optical pot^nti^l for li^ht nuclei aa seen from tab la 2 tir.d for low-

as seen froiu table 4.

As for the fitting values, this i3 not the oaso. This lead.3 to the

Important result that modification of the roal depth Must not be neglected.

The calculations of the corrections were baaed on t:ia adiabatic approxi-

mation method which was not intended to ;iva nora than qualitative values.

The au'.horo woiil'i l ike to thar.k F r c : ' - r ~ r [.. Forida for rendi:;.'; tin;

manuscript. TVo of th'-.-. ( I .R . ) ;-id (M.A. Y,-). ) vould l i k e to th;mk

Frofessor AbduG Enlam, '-.he In te rna t iona l Atc~i,- linerrj Agency arid UNJ-XCO

for h o s p i t a l i t y at the In te rna t iona l Centre for Theoret ical Fhyaics , T r i e s t e .

REFERENCES

I.Vatson, J . V;., "uci. i:ij-ji. A

2.01<iubar, R. J . , Lscturea in theoretical ;:iy3ics, Vol. I

(Interscier.ca, Ne.v iurk I^j?; p. 3I <

3 # 3 l ^ i i ^ ^ l | u . . 1 . a n a ^ D i l — U - J j l j ^ V . I * , - u c l . r n y 3 . h . 1 (f\ ±}j [ x , ) $ £ C i

4cSi:!-1ael, 11. H., Phys. -iev. C IO(lS74)lC1-J.

5.Sii;ibdl, a. H., Pays. rtev. J J^ Ii< 73)35-

7.Coffou, £ . and GolcUarb, L. J . B., Suol. Fhys. A94{ 1967)241.

B.Oreenlaes, 3. W. et 3.1.» Phy3. Eev. I71(x^S8)lI15.

y.Bindal, P. K., e^ ^ 1 . Pays- Kev. CS(I57-'l)2l5'i.

iO. Persy, G. LI. and Paroy, F. G.| Atou4.o uat-i and Hqul. d^ta tables

17(1576)1.IJ.V.'eiaa, \i. et 3.1. Fhys. Le t t . 6IB{ly7B)237.

-11-

-12-

Table I

Optical potential parameters for<^ (d)-nuclou3 soa ^ ^

Target

20.0

I2C

22.8 22.8 22.8

28.,

22.8

C 0 0 31 J H

23.0 6.67 16.0 7.0 17.2 ' 7-8 15-7 8.0 21.0

185 103-2 125

1.4 1.05 I.S7

0.52 0.8 0.5

•' 25.0 9.58 0.0

0.0 0.0 1.5

1.4 1.28 I.97

0.52 0.755 0.5

130.1 150 lie 121 117.4 180.7

0.9 1.93 1.0.1.55 1.05 1.37

0.6 0.52 0.SI2 O.56

0.0 17.5 0.0 21.9

4.9 0.0 19.7 0.0

1.9 1-55 1.6? 1.37

0.9 0.5

II .0 0.0

0.0 7.5

2.X 1.93

0.6 0.5 0.6 O.52 • O.46 O.56 0,629

Tabl» 2

Pitting parameters

Target

20.0

I2C

22.8

12-

22.8 22.8 22.8

Calculated

fitting

a

0.98

1.4

b

1.47

I . I

a

I.OI

0.3

b

1.32

1 . 0

a

0

0

.98

.6

. I

I

b

.44

. 0

I

I

a

.0

• 3

I

0

b

.03

-52

a

1.0

1 . 0

b

1.03

1.0

-13-

ae j I 2 _Op-tioal potantial parameter sots foro<(d)— 0 scattering

II.8

106.1

I.07

0.807

0.0

16.6

1.35

.629

V

roao

16.0

125.0

1.97

0.5

0 ,0

X.5

1.97

a.5

24.5d

132.0

0.9

0.9

0 . 0

7.73

1.665

0.569

3d. 6

Of

27^2

52.5

1.9

0.6

6.0

0 .0

1.9

0.6

a12.1

118. 0

0.97

0.93

0 . 0

9.44

I.83

0.47

Ov

32.0

71.87

I. Of

0.45

12.23

0.0

1.09

0.446

50.fi

d

15-9

HI. 7

0.9

0.9

0 . 0

11.2

1.92

0.449

63.0

42.0 20,5

24.0

1.99

0.43

13.0

0 .0

1..99

O.42

103.4

0V9

0.84

0 . 0

7.46

1.74

0.624.

Table 4

Pitting parameters

2 * 5 36.6 5a 6 63.0

Calculated

Sitting

1.0

0.4

X.53

1.5

0.99

0.52

1,06

1.2

1.0

0.6

Table 5 I 2

Optical potential parameter seta for°((d)- 0

i^Olev)

Ew(d)(ilev)

ro*0

V

V

166.0

85.0

1.34

0.7

17.70 . 0

1.77

0.52

d

34.4

92.41

1.038

0.783

0.0

9.75

1.426

0.693

36.6

139-0

103.1

1.22

0.76

16.90 . 0

1.85

0.47

d

52.0

71.8

1.25

0.7

0 .0

I I . 0

1.25

0.7

50.6

139.0

106.1

1.22

O.76

16.90 . 0

1.85

0.47

1.04

0.9

0.98

0.94

scattering

d

23.2

IO9.O

0.9

0.822

0.0

8.39

1.545

O.729

63.

5t1.0

113*8

1.5

0.555

24.0

0 .0

1.5

0.4

1.03

0.87

0

&

60.6f

r.230.666

10. <X

I.X5?

0.8O8

- 1 U -

Pigui« Captions

Fig. I. The angular distributions of the ratio <5"/ccalculated accordingto the c< — d aluatar model Ot Li. The experimental data aretaken from re.?. I I ) .

—5^

Pig. 2. The angular distributions oi' the vector polarization V calculatedaccording to the X, -d oluster ^odel of Li.

fig. 3. The anGular distributions oi' the rotioc>^r calculated through the

corrected tfatanabe potential.

Jig. 4, The angular distributiona of the vector polarization P calculated

through the corrected Watanabe potential.

Pig. 5« The angular diatributicma of the ratio ^"^ror Li- C scatteringat different ST. calculated thxou.jh the depth corrected and the

til

energy corrected '.fatanata potential conpared with the nonoorrec—

ted 'Tatanabe potential. The e:cpe;'iaer.tal data are token from ref.

9).

ooEo

CDoCM

Fig.6 The ratios E /E. .Cl L i

and against E

CMCC

to

o6'©0

3-

CMK

1IK

daUlo

CM CM

%5o «?

S2 CMCM

8

-15-

a

I . ICMo'

O

la.

CMd o

" • i

(OO

[

COO1

CNi

CO

2 ?

" CM

X

b , •-."•-.

- C Oc o W CM"« CM

CO

~ CM

<D

88(

s

' CO

CO LL.

8

*

-18-

CO

a*5cqciCM

111

C/CO

CMCMII..

iif

Q .X

111

8

SQ e

CD

102

oCMbo

8O

oCM

oCO

oCM

10'

CD

101

162

— Without— Energy corr

Depth corr.o Expt

12C(b) (c)36.6MeV =5tt6MeV E, =63MeV

Li Li

(d)

&

20 60 106 40 80 20 60 20 60

8cmFIG 5

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A.R. HASSAN and F. Abu-ALALLA: Exciton-polarizations by two-photon absorptionin semiconductors.

A.R. HASSAN; Phonon-polarltonB by two-photon absorption in s o l i d s .

L. LOKGA and J. SKJHIOR: Exact results for some SD-Ising models withperiodical ly distributed Impurities.

ABDUS SALAM and J. STRATHDEE: On Witten's charge formula.

ABDUS SALAM and J. STRATHDEE: Dynamical mass generation in (UT(l) x U_( l } ) 2 .Li I*

K.S. CRAIGIE, P. BALDRACCHINI, V. ROBERTO and M. SOCOLOVSKY: Study offactorization in SJCD with polarized beams and A production at large p_. '

K.K. SINGH: Renormalization group and the ideal Bose gas.

NARKSH DADHICH: The Fenrose process of energy extraction in electrodynamics,

6. KUKHOPADHYAir and S. LUNDQVIST: Dynanical polar l i a b i l i t y of atdma. "

A. (JADIR and A.A. MUFTI: Do neutron stars disprove multiplicative creationin Dirac's large number hypothesis?

V.V. MOLOTKOV and I .T. TODOROV: Gauge dependence of world l ines and invarlane«of the S-matrix in r e l a t i v i s t i c c l a s s i c a l mechanics.

Kh.I. PUSHKAROV: Solitary excitations in one-dimensional ferromagneta atT l < 0 .

Kh.I. PUSKHAROV and D.I. PUSHKAROV: Solitary clusters in one-dimensionalferromagnet.

V.V. MOLOTKOV: Equivalence between representations of conformal superalgebr*.from different subgroups. Invariant subspaces.

MEHDEL SACHS: On proton mass doublet from general r e l a t i v i t y .

L.C.PAPALOUCAS: Uncertainty relations and semi-groups in B-algebraa.

IC/80/85 T»o RUIBAO and Pu FUCHO: A proof of the absence of nodulated phase transitionand'epln density wave phase transition in one- and two-dimensional systems.

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IC/80/109 P. 7UHLAH and R. RACZKA: On higher dynamical symmetries in models ofIHT-REP.* relativistic field theories.

IC/80/111UTT.REP.*

ic/80/iia

IC/80/113IHT.REP.*

A. BREZIHI and 0. OLIVIER: Localization on weakly disordered Cayley tree.

S.K. OH: The BJorken-Faschos relation in the unified gauge theory.

A.A. FADLALLA: On a boundary value problem in a strongly pseudo-convexdomain.

IC/80/117 P. FAZEKAS: . Laser induced switching phenomena in amorphous GeSegS Aphase transition model.

IC/80/118 P. BUDINI: On spinor geometry: A genesis of extended supersymmetry.

IC/8O/II9 0. CAHPAGHOLI and E. TOSATTI: AsF.-lnterealated graphite: Self-consi«t«ntbond structure, optical properties and structural energy.

IC/80/120 Soe Yin and B. TOSATTI: Core level shifts in group IV semiconductors andBemimetala.

IC/80/121 D.K. CHATUEVEDI, G. BEHATORE and M.P. T03I: Structure of liquid aliali.IHT.KEP.* metals as electron-ion plasma.

IC/80/126 Th» Seventh Trieste Conference on Particle Physics, 30 June - U July I960IHT.REP.* (Contributions) - Part I.

IC/8Q/12T The Seventh Trieste Conference on Particle Physics, 30 June - k July I960UJT.BEP.* (Contributions)- part II.

IC/80/128 FAHID A. HWAJA: Temperature dependence of the short-range order parameterand the concentration dependence of the order-disorder temperature for Hl-Ptand Ni-Fe systems in the improved statistical pseudopotential approximation.

IC/80/130 A. 4ADIE: Dirac'a large number hypothesis and the red shifts of distantUtT.REP.* galaxies.

IC/80/131 S. ROBASZKIiVICZ, R. MICIfAS and K.A. CRAO: Ground-state phase diagram ofINT.HEP.* extended attractive Hutbard model.

IC/80/133 K.K. SINGH: Landau theory and GInzburg criterion for interacting boson*.

IC/80/13l» R.P. HAZOUME: Reconstruction of the molecular distribution functions from theIHT.REP.* site-site distribution functions In classical molecular fluids st eo.ui.libri\Hn.

IC/80/135 S.P. HAZOUME! A theory of the nem&tle liquid crystals.IHT.REP.*

IC/80/136 K.G. AKDENIZ and M. HORTACSU: Functional determinant for the Thlrring modelINT.RKP.* vith Instanton