a t-matrix analysis for the scattering cross section

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8-1976

A T-Matrix Analysis for the Scattering Cross Section A T-Matrix Analysis for the Scattering Cross Section

Michael J. Linville

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A T-MATRIX ANALYSIS FOR THE SCATTERING CROSS SECTION

byMichael J. Linville

A Thesis Submitted to the

Faculty of The Graduate College in partial fulfillment

of theDegree of Master of Arts

Western Michigan University Kalamazoo, Michigan

August 1976

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

ACKNOWLEDGEMENTS

The author would like to thank Dr. Michitoshi Soga for his patience and guidance in the preparation of this thesis. His advice and explanations have been greatly appreciated. Thanks are also extended to Dr. Allen Dotson and Dr. Haym Kruglak for their constructive criticism in this project, j must also than?< Renee for keeping her promise to Dr. Soga.

Michael J. Linville

ii


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MASTERS THESIS M-9012LINVILLE, Michael J.

A T-MATRIX ANALYSIS FOR THE SCATTERING CROSS SECTION.Western Michigan University, M.A., 1976 Mathematics

Xerox University Microfilms , Ann Arbor, Michigan 48106


TABLE OF CONTENTS

CHAPTER PAGEI INTRODUCTION................ 1

II T H E O R Y .............................. 5A. Potential Scattering ........... 5B. Spinless Projectile, Spin Z„

Target Inelastic Scattering. . 13C. Spin U Projectile, Spin Io

Target Inelastic Scattering. . 18D. Spin lo Isospin to Projectile,

Spin It, Isospin T0 Target Inelastic Scattering ......... 23

III APPLICATIONS....................... 29IV RESULTS.............................. 53

REFERENCES........................ 66

iii


LIST OF FIGURES

FIGURE1 Coordinate System Defining & and <f)2 Schematic Energy Diagram3 Coordinate System Defining Scattering

Plane

PAGE8

34

39


LIST OF TABLES

TABLE PAGE1 Summary of T Matrices 362 Coefficients Au for Positive Parity 45,463 Coefficients kw for Negative Parity 48,49

50,5152

v


I. INTRODUCTION

One of the prominent features of nature is •motion*. If this motion is a relative motion of two objects it is possible to consider a collision occurring between them. It is likely that during the collision process these objects will be deflected from their original path. Through a systematic analysis of the collision it should be possible to determine some information about the forces of interaction between the objects. Possibly information about the internal structure of the objects could be derived from the investigation. If we regress momentarily, the marble ring was where many experimentalists actually did their first scattering experiments. If the concern be with the collisions of objects in the context of the human senses, the concern can also be with the collision of particles or systems with atomic or sub-atomic dimensions.By the nature of atomic particles or systems there is little recourse other than scattering to obtain detailed information. This detail involves very

- 1 Csmall separations of the order 10 meter; the primary probe for such dimensions is a flux of atomic particles with relatively high kinetic energy and

1


2De Broglie wavelength that is small, at least compared to the scatterer. Bohr established the theoretical structure of the nucleus through the analysis of data from the scattering of nucleons by nuclei.

The physical quantity called the differential scattering cross section characterizes the angular distribution of the scattered particles. Physically it is defined as the ratio of the number of particles dNscat scattered per unit time into the solid angle aft. to the flux density of the incident particles.

It is necessary to distinguish between elastic and inelastic scattering. In elastic scattering the internal state of both the scatterer and the scattered system remains unchanged. In inelastic scattering the internal states of one or both are affected. Initially we will discuss elastic scattering because of its relative simplicity since internal structure can be ignored.

The elastic scattering cross section can be expressed in terms of the physical quantity, the so-called "phase shift". We may recall that the only effect a central potential U(r) has on an incident wave is to change its phase with respect to the phase of the same wave if the potential were absent.


Solving the Schrodinger equation for the case of the centrally symmetric potential will eventually yield the expression for the total cross section

. The phase shifts thenlinks the formal theory/ Schrodinger's equation, with the physically measurable total cross section. This bridging between the theory and the physically measurable will be generalized to the inelastic case. Instead of the phase shifts of the elastic case, the analogous quantity will be the Transition "T" matrix in the inelastic case. An expression for the inelastic scattering cross section will be sought that has the geometry of the problem distinguished from the physical factors. These purely physical aspects of the cross section will be represented by the T matrix.

Chapter two will see the analysis of the elastic scattering, of a spinless projectile from a spinless target, by means of Green's Function techniques. The T matrix will be developed for this case.The next step in the progression will be to the theoretical analysis of a spinless projectile from a target with spin £. As a natural extension this will also yield a T matrix by means of the Green's Function technique. The third case to be considered


4will be the scattering of a projectile with spin i0 from a scatterer with spin I» . The most general case to be analyzed in terms of the technique will be the inelastic scattering of a spin L0 , isospin £ projectile from a target with spin I0 , isospin £ .

Chapter three will apply the expression for the scattering amplitude to some practical low-energy nuclear physics; protons with energy less than 12 MeV scattered inelastically by a target nucleus with initial spin-parity 0* and final spin-parity 2*This is the case most encountered with the Western Michigan University Tandem-Vande Graf accelerator.

Chapter four will show some examples of how to extract physical quantities, T matrices, from experimental data. The relations may not give unique solutions, but definitely will provide information about the T matrix combined with additional data, say polarization, asymmetry experiments; these results may yield unique solutions for a group of T matrices.


II. THEORY

A. Potential Scattering

The scattering of spin *■<> isospin I, particles by a spin I0 isospin H target will be investigated. To help clarify the theoretical investigation less complicated scattering problems will be considered first. The least complicated is the potential scattering of a spinless projectile from a spinless target. The two-particle Hamiltonian is

H (i)Ztn, ' Zmi

for a projectile of mass m, incident on a target of mass wvx with the interaction potential Vcr)

potential depends on the relative displacement V,

defined as Y*«((Yi-Vil) t where V7 is the position of the projectile and is the positionof the target in the laboratory frame. Using the commutation properties of the operators for momentum and position and using the definitions /JL & f*Jand NAs rn,*TYit the Hamiltonian takes the form

H = ffcv £ + £ + Vcn (2)where is the center-of-mass momentum operator and


is the relative momentum operator of the system. Since the center-of-mass motion is a plane wave, only the relative motion will be considered. The energy of relative motion is E. = k.* where "Ris the wave number vector of the projectile. Thus the Schrodinger equation in the relative frame is

[T+ Vtr)] ^ C r ) = (3)— i

where the kinetic energy (/zja) Pr*( operator is

T = (4)

noting that L is the orbital angular momentum operatorand -T2. = ( e,4>) are angles defined in Figure 1.

Since Lz commutes with any function of V~ , particularly H*T+Vtr) t it is possible to separate the Schrodinger equation into angular and radial coordinates. All solutions of the Schrodinger equation are obtained as linear combinations of the product of spherical harmonics and radial solutions.

The spherical harmonics simultaneouslysatisfy

L* = JJUtOte Yifif*’L % YjifW = mH. <5>


Figure (1). Coordinate system defining & and


8

£

X


9where I is a non-negative integer and Vr\ has some integral value

Since the eigenfunctions with angular dependence are completely known, it remains then to determine the eigenfunctions of the radial equation. We write the Schrodinger equation as an inhomogeneous partial differential equation

When IJir)-0 the radial portion of equation 6 has the form of Bessel's differential equation which has as

combination with the spherical harmonics form a

free particle Hamiltonian, the two forms must be equivalent, which results in the useful expansion

of the plane wave. There remains to find the eigenfunctions of the inhomogeneous radial equation. The method of Green's function will be used to that end.

The resulting Green's function method is

(6)

complete set of eigenfunctions for H(\/Cr) = o) e ik-rSince <£• also forms a complete set over the

(7 )

.L'(n.)+fexjGi(rK) = . (8)


10To solve this, expand Gittr') linearly in terms of an as yet unknown radial function and

S i w * f Z Y & ’ Y t i * 1 o)The radial equation

£ t & r * ° ' ^ 7 p ~ do)

results from separating the orthogonal functions of equation 9. The Bessel function is the regularsolution at Y^O and the Bessel function is the regular solution at Y"*= . Matching theseat 'T=Y,/ yields

(11)

After integrating equation 10 from to r+£ withexpressed in terms of the Bessel functions

and using

1. General properties of Green1s function2. General properties of continuous functions3 - C ( V ^ ‘ = - Ty

where 1 could be complex, we allow which results(2)in the determination of C , namely C=ik . Theexplicit form of the Green's function expansion, equation (9) is then


Analogous to the expansion of the plane wave in terms of the Bessel function and spherical harmonics, we expand formally x (r in terms of an as yet indeterminate eigenfunction of the radial problem

^ F £ ^ Y i y Y t£ a) d 3 )

The integral equation

{^cr) = e ‘h - <**■'

can be used to explicitely determine U Cfe*") by substituting the e^qpansions for the plane wave and the outgoing wave. Upon separating and factoring we find that

ULJL(hY'l=(14)K (Jxcr0 U tCkrO r'* 6r'

where we have used the interaction potential, defined as

U ^ > U n » - f V ^ j Y & ■ (15)

A simplification of equation (14) is possible by considering the form that ignores the details near the scattering center, since from physical considerations r is much greater than (distance of detectorfrom the scattering center is very large compared to


scattering center dimensions). We consider the large-V1 form of h* •

, 0) , . i i(kr-Jlir/0f, cfer) — rh e

thenu J / u , i iC f c r - l T / O - r -

u .(h n = itC h r ) + ^ r e T x (16)

where Ti is defined as

1i = - k j j 1l l i r ' jU f k r O U 1( ta ' ' j r ' I d r ' . (17)

The transition matrix Ti describes the scattering of a partial Jl incident wave by the potential.

The explicit radial expression for U.4Ckr) can now be substituted into equation (8) yielding

(i8)

recalling the plane wave expansion and defining the scattering amplitude

$ t*,,*.) s + i r r r T t . d9)

The scattering amplitude fen-) describes the probability amplitude of a particle scattered from an incident direction to the arbitrarydirection SL - . This result is completelyindependent of the coordinate system chosen.


B. Spinless Projectile,Spin Xq Target Inelastic Scattering

The next case to be considered introduces another term in the Hamiltonian, an interaction that depends on the internal variables of the target and the relative distance of the projectile from it. The projectile is spinless, but the target has spin I<> initially and has excited states X* . Schrodinger1s equation for this case is

[T * - £ $ V j ) (20)

The spin dependence of the total wave function is known and satisfies

where is the spin Hamiltonian of the targetand is chosen as an eigenfunction of andLj , the square of the spin operator and its component. The internal spin coordinates of the target are represented by f The energy relation is

= where is the wave numberof the projectile.

The radial eigenfunctions become more complicated with the introduction of spin, but the technique of

13


14finding the radial eigenfunction remains similar to the previous pure potential scattering case. Thus, express equation (20) inhomogeneously and then separate variables. The radial equation has solutions which with the spherical harmonics will generate the Green1s function

Eigenfunctions of the total angular momentum JT=L+T will be generated from a linear combination of the

the expansion will be the Clebsch-Gordon coefficients with values that are dictated by the angular momentum operator. Equation (22) becomes

the total angular momentum by standard techniquesdiscussed in textbooks on quantum mechanicsUsing these same general techniques of adding angularmomenta enables the initial wave

(22)

eigenfunctions of L and X . The coefficients of

(23)

upon forming the set of eigenfunctions of

(3)


to be expanded in terms of eigenfunctions of T

* « •it (iME.n.lJ’n) cU t A p (25)

which can be written in the form

= ««■ f L‘t k _ r o (26)

XvXiU » a . n . i J k ) SiSi

which will be more useful later.At this point assuming there is a complete set

of eigenfunctions of the non-homogeneous radial_ + _

equation formally expand

q j V ? ) = *r Z J L T L Z T L ^ ' u . ^ ^r ££.rt-‘ Jl <M E* T H I' JLUImX*^ /> ^^ ti° Tn) y / £

- ( 2 7 )

The factor (9w.iort.(TK) in the expansion will allow a later simplification. The integral equation with these expansions becomes

*■ f ‘X i k 2= * ■ f £ £ f £ f (28)* ~ P **«xt l ^ T 1 U j£ .i‘ * V(r f ) {4-r Z z .i,z .Z Z -frz* Qxr* { / / Hz^g-'W r


16

*4 t e )n i ***'-> 1

in which the potential is defined as

J yjpk'1 \Z(r‘Y) J-a'Jf'

- s T ^ r f c y § f> jrJ (29)= \/c'r'y C c

v4'l"r l>x ow'd tin*

Factoring the common terms in equation (28) leaves

= §; r ^il'x'&k:L

(30)

As in the spinless case consider the asymptotic region : then

U-MlSi ' v ’ ^ f ‘ ’ * LT (31)x _j_ J f a r - M r . . r)UT(^r) V/CW) r^Y'7

k^v- J ^ * C ^ } X x l. 7/1."

The transition*matrix is defined as

T / = - 2:21 U ^ , (32)1 V i x t - T t 'T x iK r i i x i e

and describes the transition from a state i,loJ to . The asymptotic radial expression

becomes

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission

17

(/u.y-1 s j (fe i-iC c tC r 1 T i-ix.r. • (33)

R R i ^ V . r i y u w )ti*ruR - x x x z

* ‘‘’kif ,fe’r174.i, Y1| )«i«T.rtofjK)

» I t C i V ' x » ^ / T « J Y 4 ^ <X » S>1' k*.

The scattering amplitude is expressed as

T £* ^ JX£-2I.Z.Z.X i z—- ) e "U j- m x' n'

x vSz*<j C i m T o c i o ( J r t ' f f M ' i v t i * - ! 7 f t J Y l t ^

usxngJH _

(34)

— f\ *“z; r.ai.-wulJM') Y*.SN,cX i U =

The scattering amplitude describes the transition from a spinless wave incident with direction -fi-R(V1 on a target in the spin state X. with ^-component M* , to the outgoing wave in direction -fl. leaving the target in the spin state X<* with spin componentM * .


C. Spin L„ Projectile,Spin I, Target Inelastic Scattering

The projectile and target both have initial spin U and i0 respectively. After the inelastic collision the projectile is in the excited spin state ,.<n and the target is in the spin state In . The spin wave function is known and satisfies

iS chosen as an eigenfunction of X 1 and as in the last section. The internal spin coordinates

The target spin Hamiltonian from the last section remains intact.

As a direct extension of previous Green's functions we have the expression

(35)

where is the projectile spin Hamiltonian and

of the projectile are represented by 02 . Theenergy relation of the system is E.-E**- tZ*. =(.&>*.) .

ft') ^ i t

«vi

' XhAh Tm K<k

(36)

18


Using the vector addition properties

„ *J1*JY ^ ’Y u ^ = (37a,

■p" *•and the property

(37b)

where i( and ji.* are any two general angular momenta, equation (36) becomes

Gcrr"ir>f'JjO = - ^ tO) A i*™- a h** (38)

x h l c k ^ y c A ' r p 1 I ( ^ V •

Expanding the initial wave with eigenfunctions of L 3 I and applying general vector addition properties the initial wave expansion becomes

<y coT) ( V = £^ A u k u 1 V J V ( I h u l a l ' j ’

2

% L §ULo £'4 h i •x.

f + ____Analogously expand < p < r n v formally in termsEUV»Xo^tof the radial eigenfunctions and the vector spherical harmonics

(40)


20m

® v S l f Vcr'lf ’, 5 vt J^ j r <4i>

Substitute the expansions, equations (38), (39), (40) in the integral equation in which

*JK „ .TH

- V/Tc o ? C

is defined as the interaction potential associated with the temporary state IC to”In M* f J*to/ ) and the final state J J W-j) . Factoring thecommon terms of angle and spin dependence and recalling that the expression will only be used asymptotically, that is for f much larger that Y / , thenthe asymptotically regular Bessel function is removed from the integral, resulting in

lucjjr. = Jo<-fe*r)JrA

- 3 * ik^ m i z . r£- c 'f**r -‘l7r/z) Cj ck ,J"K. niiJL-y x Jx*' 1

x l/(r0 (/(k*r) / /

If the definition

7 ~ t = - % 2: 212:21 ,42)


21

ck%r)rmflutja.

is used then

I'Wj'JI.jtuuJl. (43)

This T matrix describes the transition between the initial and final states. If the explicit radial expression equation (43) is introduced into the formal expansion of the total wave function, the resulting asymptotically valid wave function

results if the initial wave expansion and the properties

n. h, uwhere and J are representative angular momenta,are used.

cp+ (r%f) = e'6vf % cv Y t f )

.4rr znz.nz.7L7LZ.7LZ.a. * 1 * j r n I* t * tr y #*r

* < J Tt^cj'jTnjJLiupU *jJ

X ( J * * T c H ' / m ) U W c n U t ^ / J

(44)


The scattering amplitude is, then, expressed asf = 4-ir 2IZ:Z.ZlZ_Z-Z_Z-2LnI ux.Iort. 4 V J * , x m i J' w n j

* l'*‘t <«>

* 1 % r . M . / j ' M * 'c„n.li‘iiJ'>Ci'Xj%n,Uh) Y{^ .This scattering amplitude describes the transition from a wave characterized with spin £© and direction -fl** incident a target in the spin state T„ ,

to an outgoing wave in the spin state L'* with the residual target in the spin state


D. Spin Isospin to Projectile,Spin I. Isospin To Target Inelastic Scattering

The scattering formalism of this section will be the most general case in this thesis. The isospin states of the projectile with initial spin 2« and the target with initial spin To are respectively to and T. . After scattering, the target is in the spin state X* and isospin state ; theprojectile is in the spin state (.<* and isospin state ^ . The spin wave function remains intact fromthe previous analysis. The isospin wave function satisfies

(46a)

where is the projectile's isospin Hamiltonian.is chosen as an eigenfunction of T’7’ and

T _ . The internal isospin coordinates of theprojectile are represented by ^ . The isospin is represented by and its component by UJn .A similar notation is followed for the target

<4 6 b >The isospin eigenfunctions are formally analogous to

(4)the spin eigenfunctions

23


Using the properties of the Clebsch-Gordon coefficients for addition of angular momenta the Green's function is

G c w ' W H W - - # ik'.to * TIA _

u ^ ' f n (47)- U v T „ AiLMCj)T^

/) 0 i Tlrffiq'V) ii(jv)

and the initial wave is

,;6' f ‘Y c ^ > <Y(?) Y c ? ) Y f w/Viviu® /Vxlt-u Af.ii. / ToIaJ*

* i l 'Jx ^ r) Y x * ^ anuyx.U n ,Y iu ,l^ .lm )

kljTld) <T 7;

Expand the total wave function formally in terms of U. the radial eigenfunctionis

^CrriJ?P) = 4n TLZ.Z.T.tX.Z.Z.TiZ-Z'Z.'Z'L J - e * Miuj *,i* j n v r t wt*Mi 7*10*

x « * • '• * • / * * » ) (48)TVv< ^

uk/rw]

The explicit form of the radial wave function


25will be determined by substitution of equations (47) and (48) into the Integral equation. Using the definition of the interaction equation (41)

where the interaction is between an initial state characterized by the projectiles spin and isospin

and the targets spin and isospin Z’v,7>/ *and a final state characterized by the projectiles spin and isospin and the targets spin andisospin . The initial states refer to avirtual compound state consisting of the target and projectile combined in an excited system that decays into an outgoing wave and target in excited states

L x , j •Upon eliminating the common terms in the ex

panded form of the integral equation, and using the interaction the asymptotic form of the radial wave function is explicitly determined

; X uepi. = (49)t v ; T» °


26

(50)

£ Z.Z z:% Z . it e./ft“r'-hr/SJ*t» r j* & 7i **

^ r"/ ♦» '■*• 4* ; 4k T«The transition matrix is then defined in the

preceding equation as

= * v ‘fe"

w r ' ^ ' til «m / X* 7*and describes the transition from the state

to ) .Rewriting equation (49) using the preceding definition yields the expression

' T m > U

If this explicit expression for the radial eigenfunction is then used in the formal expansion for the total wave function the resulting equation satisfies the asymptotic requirements. After some manipulation parallel to earlier sections, the wave function becomes

« 2 £ > ■ ^ S ‘K a ‘ % £ X ( 2

♦ Vi ^ z z z r z z z z z z z z f ^ ' rI * -5 *J J M (.-* Tv V T T kf +H 7i « r


z L x r z4 * T%,; *» * * * ' • » « * , ' ft* .

x a'« ' ^ £ X0l ( z

and expressing (£ E <fi2£f(J' in terms of eigen-__functions of 3 and T

CD <*njf V) =^ C-loXloIoVC.

^ %ll> <*>

T T T T r z x x z X X - 2L e l ^ ^ T T+ 11 x i i t j m u Xh v 3' "tee tv* fcwt" 1 ' J L w y )l* ; lu c jiro

^ -+•* T ^ ; -u r .

„ v /^ a 4lj(:i.vio>uu(^liijXdlu3r*ho(JK) LZLZIZ.* Tim Mv.

* (4ouo0To IK). It v j ) 3 ^ ( K ^ - T vs U)vt (Tvo) ^ *

The scattering amplitude is then expressed as

C --h.vs **"*-) _ A—* V 7“ "S” T- 5~ T T JL ' ~- /T S S & S & 2 S 3 : * * i * « 1

* Tx* 5Tw>JUotj)r« ^ ' va^- t ^ T V ; -fc. T . ( 5 3 )

xUwL#ua,(JtaJy j « JT;fto)JK)^J[,«u./ u K l X ^ 'J

x (yiw/ XviMvi ('T'KX' a ^ o ^ ofTw}(4».U)v. Ixk^w 111\)) .

This scattering amplitude describes the transition from a wave characterized with spin and isospin__


Lo,^o and direction - ^ a (&k^Aky.) incidenta target in the spin and isospin states ~Lo(To toan outgoing wave in the spin and isospin states with the residual target in the spin and isospin states

T m .> •


III. APPLICATIONS

The series of the partial wave expansion of the preceding scattering amplitude will converge to a

in application to low energies. A classical justification is based on the short-range nature of the nuclear forces.

Consider the target to be a hard massive sphere

mass number, and the projectile, a proton, to be a

projectile's impact parameter must be less than R-r or no nuclear interaction will occur.

The impact parameter of a 12 MeV proton (which is available at Western Michigan University tandem Van de Graff accelerator) is found from the expression

practically acceptable value for small £ if limited

point mass with The

v = 4 .9 • rw«A<-r

and when 4-p = S. ? • lO

29


if A s (p4 the nuclear radius is

l^T - ^ ^ *uM*~

The £=4- proton's impact parameter is greater than the radius of the ^ = (*4 nucleus. Therefore for these circumstances the partial waves can bedropped from the expansion.

The total angular momentum and parity of the projectile-target system is conserved. Consider the scattering process:

Where ) represents the target initially withspin, parity (To/IT;) , representsthe partial wave A, = of the incidentprotony the intermediate state is represented byftf/K K with spin, parity (j,T) ; the

A ^ H iresidual state of the target is represented as with spin, parity (x^TCf) , representsthe partial wave of the outgoing protons. The spin angular momentum of the compound state must equal the spin orbital angular momentum of the projectile combined with the spin of the target

J t I. = y <54>


31The total angular momentum of the projectile is

J = 1 * L (55)

where U is the spin of the proton (magnitude %)y thus

( i t u ) t i . * 5 “ (56)

Since parity is conserved the parity of the projectile- target system must equal that of the compound state

TT;. < - 0 a = TT (57)

Similarly, the spin angular momentum of the intermediate compound state must equal the total angular momentum of the final system

. (58)

This results in the expression

X = C l ' + r O , (59)

upon separating the total angular momentum of the outgoing proton into its components of orbital and spin angular momentum I' and Lx (magnitude %). Theexpression for conservation of parity in the transition from the compound state to the final system is

-n- = (-i)* irf (60)


Equations (57) and (60) express the equality of parity in the initial and final systems

-TClC-O* - T f . (61)

This leads to the conclusion that only odd-numbered outgoing partial waves are associated with odd-numbered initial partial waves, if 7 a similarrelation holds for the even-numbered partial waves.

These conservation relations are applied to the system described in figure 2. Table 1 summarizes the allowable partial waves, thus the corresponding T matrices that contribute in the partial wave expansion of the scattering amplitude, for a transition from 0^ to 2 + of the target nucleus.


Figure (2). Schematic energy diagram corresponding to + *1>' through acompound state with spin-parity T"* .


34


Table (1). Summary of T matrices, incomingpartial wave, all possible outgoing waves, that contribute in the partial wave expansion of the scattering amplitude.


36

Po.n't-y OdA Parity

'Tz'Jt'-Vi.tJ Tl'x'+ytJi?

T2.9a.oyz

Tnt(Vz

T>*.|Vx.

Tmu’A. T ’/zOt

TaSfct**. Tim ?*.

"Wz2. K

Tz^zfk

T ^ z \

loy*z9t

Tzftzft.

'Tih.zi'i.T(/t39t TiK39x

Tm?,** T3K39t

*rvn3?t TW^St

T|J4.3K. T|*3?i

73%3% ~l3%.3*i

T V *3* “l?«fc.33z


37Consider next the previously mentioned example where

Zj - 0* and 1 ^ = 2? . Figure (3) defines thecoordinate system where the xy plane is the scattering plane. If we use the relations established

the last section we find that the scattering amplitude has the form

r i f f i c i ' " • ' w

* <'■'«* M °!d (62)

« (t'-m-mk-niM-HM. >k iu*(j

Heft*.Define the constant

nuli

x (j?' ft* ?z HA*1*o-tiw)

which is the coefficient of 'Wl-rHt-l* n'Mnin equation (62).

f6)Bohr1s Theorem is

P « e iT’,’ = P) V e.:i r S '6 3 >where and , "Wj are respectivelyparity, spin projection of the system in its initial and final states. Since tL* - it follows that


Figure (3). The coordinate system used for this calculation. The x axis is the incident beam direction and the xy plane is the scattering plane.


39


40

/>n] - >n/ = 0,±Z,±4 , ■ ■ ■

yrt0- Ow*-rAf*J= 0, ± 2, ± 4 , . .. .

When Him , are parallel and 4k* - 4**,

M . -- o , ± Z , * 4 t • • •

which is the spin—nonflip case. The spin-flip case is represented when , to* are anti-paralleland "to* = - TTI'k t then

M * « . . .

From these relations the scattering amplitudes representing spin-nonflip cares are

t-ji 2.; -*

v7+"> 0 1 (64)

~yz

and the spin-flip amplitudes are

(65)


Equation (62) evaluated for the spin-nonflip case,

say. • is= - T z % . o ^ Y ^ ’ *«** - I \PS* H x x K I 1 %

+ V x ( . Y i S - ' H W - Y i - V f W )

*

+ ■£***. f ?+ - YSf° i fW )- W ^ Y S T ’lfSOt Tttz?.(\t?ii'f¥-Y;.?'IS'Y4a,i:'f¥)

a^- •, i;WfcY4“+'HF(Y > 1 < * - Y £ ? > $ ’I ¥ )

►iT»..*.(\£?>*<K - N £ T » f ( ? ) ___

-‘" W > * “>|/F- V S * 0 » / I *- Y i r ^ ( ¥ )

*;YwR(Y£?'ff -N£raf‘¥ )-' -Ys-f Y ^ - V k f ^ r )-*T*.ws.(YS*>,t',Sf-Yf*15t*5 )-; TTsiiKCyi? 7 <« - Y y > + Yi"-’ )- '■ " W ( Y ^ Y ^ - Y Y H 3? +'r£*’l'l%t )

■ r H * & ( Y & > K m - YY°^^- Y~r5 )- i Y v ^ (YY'f’Y ' P I r - Y ^ ^ ' T w

+ Y y ° £ : «■ * Y c" ’ & ’fiF ) •

permission of the copyright owner. Further reproduction prohibited without permission.

The spherical harmonics for are

2 ; ( 6 7 )

for Wv<0 , we may use?

VC6,+)=(-0"Y,.if*'vn **“

and the Legendre polynomials are given by

p ccoseo* (cos’fe-O*'W 2»SL( (!-»-) > S ' «j («««)»-

Equation (67) has the form

Y (\e>p)- c!^ <2_1,ITl f (68)

k.&-H ot-»oG _ n\.defining y A c U**. J = if thescattering angle -TL-a (e ©p) , as defined infigure (3), is used.

Substitute for the spherical harmonics in equation (66) using equation (68) and take the product of the scattering amplitude and its complex conjugate.A term of the product has the general form

•nMx M - m ' - j - _____ * i(*i-W)Qp

which can be written

(69)


This form can be expressed in terms of the Legendre polynomials, as

- IQtu. ( “T&’I'jLJ U l. R (-CoSeP

noting the change of dummy variable L(7)

A. computer program calculates the coefficient

Table 2 and table 3 list the numerical values of the coefficient for the I© ® 0 to ' 2.excitation by Cn. ~ 2-. projectile.

The scattering matrix j has theform

/ p ii ( pcuen f' «»<** I *•I = ) T Cep) + f C6» |

, /■'eotntz j r o&A / *The | rC&p) j and j j- C6p) ( terms contribute tocoefficients of while theterms contribute to coefficients of P ®p)U-Oe*<iIf we are interested in only even terms of p ccos^p)L»«Ajeviwhich is the case in this paper, we need only |9(e )(z

and |


Table (2). Coefficients A-u for positive parity- partial waves. Inelastic coefficients in column 1, 5 and 9? spin-flip coefficients in column 2, 4, 6, 8, 10, and 12? spin-nonflip coefficients in column 3, 5, 7, 9, and 11.


45

mm— i 2 A xM«*6 1 t

AtoM*«d • 2 T r j ' i j T j/n^ iri" TT«+

, Xto i-------

IXi k */*.! 1

0 •rit 4ft0 iff ffi 4 * Iff0 .{(I f<* 4 K Iff 4ff I P Ta***.Ti*JV

0 ■41? Iff iff 414 Iff -Iff

o Hfi -HI i t -f* -|ff -Z&

o iff iff -vfi -W* ■ w -*ff TVvt^ V*.TdYt<l V*

0 •*ff I R m»/T Iff -111 ’44 X)4i>y\ 1

1 2. XT 22.io Itlfc,*!’

Q 111 -a I ® 4 A *11 1 X'Ri>»"Ts4'«.«A'Vzo ♦4? -414 |<S -441 H I -Mi Taft.*** (d-KftTz0 41? -Iff ■ w iff - I f f la s/t. lament

0 - H I -iff -4Ji ill

o -Jfc ♦ a ?JS -sR U S -*6 -IS XiVt.i i T a vta^

0 *77 -4J5S17 * S -41? iff -ffX J.* r k«■ \ T ^ ro 2. fL<r -i. -is -itr _*t*r T^aii. Idi-zdVto i e H i 1 -|C -?€ - ? £

o **5- _fcr iT Trtav*. 1

0 'S<7 H ? - H f - I f f I f f s'? I VtdTi leJVtift.o • H ? - I f f ■ H i f f 2 J i

T17 T V ^ o t-

£ as-

4.r

2.r

_fcr _ t .r

- 2r

o s £ * e s in w lijl; n i •« s ~laita1ft.TJV2.ivt1 s 4 r 7 ft » IO It 12


46

A*. A*

o

JUL.*

1

- f

ft

f Ukr* * n->

lS.r -1

_ Nm O"

<...

o fff iff tit M '«4

0 w » HI f # W f f a

1 £ St9f

is.3T •jr - 4 ' K it7

o -MS W U f £ tfi

o Sff e f f iJt M M All iff 4T1* lift-«

o 1 C * « S « i QCzrf g c * « K iSfk * « &.£

3 1so ]Xa* 12to

0 u flff w s ? 4 iff *ff

o iff iff ■ * # f f f -iff - M ■•ff

3 U7« StTP

SfeTO -itw -3LIf - aV

0 -AK 21K - asrS741??

301 205s*r -it7 -Jtfr*r

-/££W

3 HiIHO eft.i**o BLi*»o iiIf <t80 &LMS' £ - * -_1♦T -Jt49

< ft 3 4 r 6 7 9 IO u tz

E

\4*h**H\dVx!ih

I T & dYz I Z

Ujwy*"^ry*a^ T^*wWTj%j9i. U^Jh. [<&***%.

( 1

~Uvxd'?rr<±vi<i9>.

(Td%<l9i[Z


47Table (3). Coefficients A.u. for negative parity partial waves. Inelastic coefficients in column 1, 5, and 9; spin-flip coefficients in column 2, 4, 6, 8, 10, and 12; spin-nonflip coefficients in column 3, 5, 7, 9, and 11.


43

t i 2 Aiv i 2 A*M«iO 1 2 TT«-

1 1i£ %

2T l^pHpyJ 1

0 i f f it? I f f Tp*.Pfc

£> *6 & i f f "W p/x ip*.?**.

a 1<S |C f « "Wpr*. lpnP»A

0 "ft? f t? " I f f i f f I f f i f f TpHP*Ti%.p%.

0 'ft? I f f ff ■Mp517 ' f f i f f ' f i 1pHP*n?JkPfc

o •fJ? i f f f t I f t f ’I * -ZyR

0 f j r i f ? & • i f f i f f "Tpli-Pti.

o I f f i f f i f f I f f I f f

0 ?\liu (S Jn i f f • c TpTi.pjt

0 & i f f - i f f 0 - f f W f 1 PV*PJi.lpit$K

o i f f s f f 0 I f f - f f I f f * f f I f f w * l?%.f fc.

0 i f f i f f ff**« 0 - f f ? ff I f f I f f 7~ph.pyiTf.h-fkI 3

zo iu> aZjO0 K i f f - f f I f f 4\£ If^PA. IPTiPVt

o i f f ft? i f f I f f i f f If^pyJpStPfc.

o i f f I f f ' i f f •? ff I f f • I f f -1$ "foiPJi Iw K

0 i f f - iJ iTlx* rz*42 I f f 5f f i • f i (4%.P^?x.P\

o i * i f f ' i f f i f f I f f * f f lp*.f?i

0 i f f US i f f t f I f f - I f f ■ « U w jnpw %

0 i f f If? • I f f i f f I f f I f f

0 i f f - J i l l1517I f f £ f f I f f w ff i f f

0 i f f I f f o - « h%?h I pTh$\.1 i » 4 5 6 7 8 9 10 7/ /z


49

kMk'O .01 1 AM»*0 U.1 1 A«IWO i “ISeyk-ni-j-x-J* r»—o iff o -*S d£ Ti^pikTRwn0 MF IsV? iff 0 «ff tff iff -ff -ff? *<*1 J.5 is kS l"fp)kp l2‘0 £5 if -* .1sr * .1y ^fePiCTpHPK0 iff - a ■fSfff 'iff-iff TpAPiili^PK0 ♦ft iff o ff ■Wf ff ^PKFiili\Pho i .k5 1yo 'if f IV? 'IV?1 W iff iffo w tff - « iffiff iff -Jff lpw»d?^f%.o H f f f f £ 6> ■ e -ff fit «lff *Co iff w -V? fff iff 7pV>k 1p* l?**o ft# -2E/T/Fl? ?«? iff 5?ffiff Ipw*ClffitK0 IV? ft£ a ♦£ ■"iS'‘iffIff TpJiPiiJfhfkz 7r r j.r -4.r _£r 5" ( Tr'h.pyJ ^o M ¥ £ iffiff TpM xlfltfKo -m W iff W •iffIff-nf 'TpnrhlfkPK0 JLr _*#r X.r -/z.r _ c. f it? -2 ' Phfb. (pa.0 i f f Wf W iff •Iff- * (p&Fh (pM%0 i f f M rl/fHff « W iff *ff rff0 H »e Iff A asyiS -iff ff ?ff iff *<r, TpHP Tyfi-fSLo Iff Iff HF iff •Iff•iff »ff T w T p w i0 'Iff♦ft SSfjEior Sff jiff■iffiff iff 1 * l-f %.0 -iff Iff iff “ff£ff 7ST "fp»kpyzlf?,.??2.\ X. 3 ♦ y (r 7 • 0 IO l« 11


50

i.(fa-*

k .• X iw *o c I

A irw>o « i T jrt'A iT rr-** ’^

5> -

z £ 8 t i -& » t I l f H f O k l *

0 i * H Hi * * i * M I f f T f fc r t t fo k r ik

0 I f f tit ' i f t i t * £ « «

0 * * £ 6 t i i a n k * m ♦ff H M ^pOpikFi

0 *t f * 2 c a * % * S « f t f f * f f

0 - jl I i|0«

-jtM9 IS

zkLMCf t -

-2iXI * *-ilk49 “T«LP\T?X?5t

o « S » f f £§ »« *s56 ’U

0 SS* * w 2e sy* a» Offas « * S* w

0 -i«ft i i

109A-XI w *

-St147

1C.XI ff * 5 1 ft7

I 3Liio 2LTO Ift7

Xz * ♦ f i .

7S h f t p i t l 1

o « T Iff o H S l J ¥ - M O f f ^ v x T T p j f c ^

o i rAV -Jfc

05mm t fc S f * 22.

412 17 " ' ? \ p k T p k ^%.

o 31TO ■» * * m i3 IP ■ f f 12391

-J6.70

W •*£ OS%€-a?J541 M Off t*K?h. I^\f iL

c Off off iff ?W¥>iE fejy»islpii.f3s.

0 a * •** »«-A/Z41 tff

0 * * a * *« 0* S« -m ”5fcpn

3 iZjO

xyr azo I W '

o ¥£ w t i f

a m fff " f t ? fff “Iff -Wl ipk faTm z0 Iff o i f f ff Off■*<? -?6fC Tpm^ x^ z

0 t f i Iff ff

0 i f fff i f SW* *

i ** 3 4 sr <0 * • « lO u IX


51

A*

o

&£0

W

1

-J61

- 6 0

H»o 1

* £ - * £'

- * *

lfc»o i

* «

t.

3 ftm10 1 -fc -11

K

‘A1 139

Jl39 S feM S

<D2u tt £ - * -214-

<**-i£& _12*

49

0 I d'3l{S 35* *• & a * 9 * M «

o a {£K u # e f - c 3 *

0 ^ 6» S c « C • * # * s * f * m

"S-S

0 I <T* r - # £ * * S * i t s 9 # -X J£57**

!*3<5MS'»- i± J i ■ I t f

3 32.•20

2 2 AL1*10

I tI f

- « aia»

fiL*49

212S

_ a.7

-3 . -X549

0 # « V & « a * 4 * o -**J i471*

0 U-S JJ5T«- i f & £ M fH & & - « T?C * *

o * # * e * i * ? c -iv jz o a «

0 f e # * 4 *- X j f/57lb ♦7 # *

aa»53n*m

3 XVO j&i70

12.39

va.7

a .49

ik49 t

_3.7 -H£

H6-1259®

2.7

o' * 6 S t *

B tS § v r § f f 48 O’41 W

o -$ G -2*J?7JS-»S - M ! ! i l « « /«7 “ u

0■xS' -J 2

/o rif i■3.1 -SB3.1 -224Mf7 - m

2 *Zl-1*477 • 3 8

-2»730KX7

li 22 11 no 3 - 4- J2.T 70 •> r lO 7 w- 7 14

o - Ik■as »

ffia.H 9=3. SSflL

147a iZ \

2.38•«

u i ,HS

S t7

0 2 iJ ias113* «

-A 6 57 * V 5 r , # ■24/32»*f

-1<S

4 z.S

3J tIOS

£2=105

32Zt Ik

«wk i-147

2 .7

-12.7

-2 2 -82. ik7

O - f £ S-15t i € * £ l<*7 *3 af< 94 JT77 >3

S S fiwts” 2»£(}BB85 =*•{331 '

4 a3

ik2-t3 -7

227 -J2Q44i

jfeo.147 12-

1■JflS77

- & SS39

-?¥«/T39

c7

i z 3 4 9 o 7 e 9 IO it IZ

r m n w - i T

(pK?9n?^9t

I p M f k T w i ^

IpKffJfhfK.

"Tfitf*. lpK9Jt"fe%l$%S2L^fsCT^/*.

[Tn.fr. Iz TkfsCTwiSfc Tffcffclfitf*

iTpr^l"

l?%.f* If

E


IV. RESULTS

The differential scattering cross section can be expressed in a Legendre polynomial expansion as

Cii- L

The experimentally determined differential cross section is best fit to determine the requiredLegendre polynomial coefficients.

A set of linear equations relating the coefficients Au and the transition matrix elements can be established from table (1). In some cases these* linear equations can be used to establish a definite relation between the coefficients A u . Data extracted from phase shifts of elastic scattering experiments can be used to determine the amplitude of the various background terms. The experimental evidence might indicate a negligible amplitude for a particular background element in which case it could be eliminated from the set of equations.

The following transition matrix elements are expected for an To^ = O to Jlx - 2* process through an intermediate Y-l states

,~Tsrid'Vi ,"~UrrA9,.

53


It is the background terms and ~T"v *.<L9lthat are susceptible to elimination from the experimental evidence. Table (1) yields the following equations if the process is a pure "TV**.*/*, resonance.

A© = iTa^-syt. I'

(Ta>iSYt. ( + "Ta^y^TTsYt.ci'Vt

- ■§='3. TaKsVx

Al - lf' '~Q%.-!>vJTsy*.«i'yt - "Qyr-syZTsyici j.

The set of equations for a pure Idyi y*. resonance is

A* = I T«j^yz\

kz. - § ’Tj^yfUvz.aTz- + |-*3 T n ^ Y C Y ^ d ^ x

Avl 2 -4 iTcL^yxl^y H i ~ Q s?x > v^TUy^ v*.

- Ta^VzT^xciv-z.^ T^l'Qft.svJXi^y*.

Afr” yz. 'Vz. “■ Wy».c1%- .


The transition matrices expected for a resonance in the to ~ SL* process are

The first three elements are the possible resonant terms. The set of equations for the pure ~\*yidVz

resonance is

Pô= I I2"

/\-t =■ % TsKui>iTd.‘.5y-t. + % 'T5

- r 11 $ *41*1 +

/\t_ = 1 lci%.syz /

for a pure IdYz.iYz ^

,\0 = 5. I'QUnttiJ2'

Avi. “ “ ^ 1aTtinTsJi.ivt - ^ IdVzdyt \sh.<Wz

+• (£TdVt<i>2."laT*.sy*. - "TdîV*.

A j = t [Td.yt.d.'vS^ ? T ^ y T T s ^ d K ~

- y - §" fi Td"»i.<4Y^Td^*.syv

A.1 — ~ * I "T Yl^Yi. i - TIîxiTiTj,*IdTtirns^ ?z.

+ “IdvrQ-s-svj_t "ldy*.«jyi"Td%-s^ /


56

for a pure ,

/^0 = Z I 1

y'*1 ' T a 9 ^ i T s y tdVt+|^'Q9fiJL&i

■+■ 5 ^ 7' T o l ■ ’/JTd"VuSyi- “ "r^vF"~l~d¥t.<lYl- ( elVx. sf*.

/& = JS \Tj^d3/,. T- f </£ TjtShAVxTi^V-L + Z'Il U*jPfllQ\hSh

+ 7 la!UK \ J%.S‘/t- +“

K t = - S + s“ \f£T^=iyJUfe J %- '1 1 jf~j-y. t-i v.~~ (zfiAyi.

The transition matrices expected for a resonancer-ir -rTT ,-+•in the ± 0 =* O to - n ^ process are

i )Ta',/ii9t/Tav*<i%- , lc*‘%.syi-JTi%5},2.llsh4'V*-y *

The set of equations for a pure T*ft.<A3z. resonance is

f\o = ^ | T^ci?*. |

A*«. =

l\o — io lTsft.d%-l ~ S' U>*-d?i-($&.«) Sr^T^dS^Q:}^^

K.t. = ~ ~JhYi~<L¥*- ~ I Jicl^tTId^sKz. /


for a pure resonance,

[\0 —

A t = “ 5 r l" n ifc *s J - f { f l^ jT ts V id V z

- 7 ^ Yz.- I^Ta^^JTd^iSJfc

Ao = 70 -'- rlfl"QVtjeLStUyidK-

+ 7 I t I d f e S f e . + ^ S . T d ’J i . d y z I T a ^ s / z .

A t = ~ It

- ‘ ■'R" Id’VtdVt - Js'j ' "U-Kci^ru^sy-t.

for a pure "Td^.^%- resonance,

Avo = 3 | T ^ u ^ f 2-

A t = -*- fiTA^jsiUteYz. - - s-

Vl Tz. - Tj^*.d*?rnVid-Fz

A a " ~ t" I ’T^%<l9a. I *

A o “ \HC ”Q.9-tcl?zl ■+ ~U^’jA^Qd‘Vt5yt+ y^ldSfciS’t.ldft.sJ'i.


N •

- w ’Qz.jpJlfr.dyu *- iff T s ?t

- m 1 iTd^dJ^l2-- f ^(V TdfcJ&Utoft.

~~ “T 7 ^ ( yj.d9z.

Avh - ~ § ■ (

The set of equations for a pure T r* P Yv , Yl- (8/9) '

resonance is

P \o = ( T p ’APVz .!1

£ v l = T \T F " W p fc . U 9 tP ^ ~ ^ ] / f Ip fipyC U hP *. Ip h fH lph.f?z

A.© = fti”fp\pv\ ■*■ l'Jl”Ip%.pxIi.%.?>i.+ f(^ Tp pj'ZIiistpVz.

- s ^ ’ p- p>«ir f T ^ pxTTp^ ^

- 7pfcPiTip>tf + f ff Tpv*.py*.) py*.? y*

K l = ~ f^TpTtPiC^ %.?*.“ ■§■ f?Tp»/4.pvt1iy*p*.

- I 1 ^ "Tp?t?y*TpyJt 7 * .- % ^ T p x p y J lp y ,.p % .+ z ^ l p y x P v Z ^ x f jj.


59

The set of equations for a pure resonanceisK o = (Tpg.pyJ*

A z - - ¥ Ji v f \IJ T^pyfli?'rP%.

+ T^pyJTpy*.? *L - 7f fuPlfe I P*Af

N > r T f s t P H J p ^ n

A * » - - ^ ~R% .?yZT? - f j g T ^ % . p y l l ? 9 a .

- + ^ l ^ ^ p y r T p ^ T z

A i = r I T?^.p^. % jf Tyftpfc^f'~ sfe 1?5%.pkT^^p Hz

- l ^ ^ p y Z T p A ^ t s^'5^.pyZipMPt+ t£,'5%p^lp?fc¥?l .The set of equations for a pure "TpvtP%.,%. resonance

isfc\a -=*■ 2*. I Tpy*_?%-(7'

Avo r Tp^pWIf^p-*,. - lil-lpAP^'VaPYx' f p ^ PvTTp*?z

+ l{i l?ya.p%Jp^^^-+ Ifi Tpap*Jp h9 % + > t p ^ ^ . p V z

Av2. = " £]/% Jpf^pl^TpyxfVi.

+ I M F T p ^ P h T p te e ^ + f - j l i 7f%_p*z.~Ipy1.p ? i

= - ¥ \f u lp & p ?C fc% p]± - 2 i/fT p JL P vT fc& P h L

~ Q ] / i T p ^ p y jp v if Vx+ T p h .P h fc tf & * fffT tZ P & Ip fiP * .


60The set of equations for a pure *pViP%.j resonanceis

A© - 2- I T p h jp h I *

t& = 9 lTr»*f- r H WnTwK+fif Tekn&hfiK- f TpKPWTpKf* + £ lf f Ir^PKlrKifL * ZtfTr&Kfai X

- 7pHP&7?&.P>'a.^ 'IhnJttti’K -ffy? Ip^rhJfKPh- r 7ph.Pf i7p* t1r* rKTph£&~ $7&r*Jpkf%.

- f i f TpnntffVLPK Al= -rlTpKr&F*-fil£(TpKpyt.1f%.ph -f|/t TpKP*M\py2

+ $ & 1pyj>’A7f%-?&

The set of equations for a pure resonanceis

Ac. = 2'\~^-9LPli.|i

A o = I r I ■»• f r l ? T f S L P & T / ^ ^ i - t(£i Ifp^phlp^-f^-1-3 /f Tf9LPhJp-h£'%--&if 1£&ph1F’AFh*r$7p%pj'zlf&P%i

Az. = -js-l7f£Ph.r+ TfKPhHtkr*. T^phJph.^- x&ff~U%Pn7r-*zf*. -0ffiKP*lphfo-J$Tf%pJilf%r7/2.

Aa = ~f? J Tf .p . /\ 7? % PrJbrf *4 IfiZPTiJpKfft.

'SffTJKPxJrhf1*--£ril£7fB.rhlphf*~£/f %%P?Jf%P?zA * - - '#• rfTf%.pvCTpy±fy2_

£^ = $$T f% p irf?% ffi-ffl3T iy t.PKTpK'f7/L


61

The set of equations for a pure resonanceis

k o * a

kl ^ 1 Tpxpyj-I **■ f

~ "If &?yrifWH+

Ay.i = + 4 ^ ” ?%P?CTp%.-?%.+ P.Y5- ~Tf.T pfa \ph$\

+ i lS 7f$y>*TfeP% + ^

A?. = ?■ + IpTlfcPft.

- z g f i l t K P h l p h . f K . l * v j > 9 J i w * r j£j?IfZp^farn

The set of equations for a pure "Tp&f?*. ,%. resonance is

A 0 = 3 1 "Tpy-».?7*-'l Ai = (TpvtfS/^ - s

” rj!F TpJ±ftCIp%.P%- " f it \ Tps^?CTfVJ>\ i 't ( ^ IphPpCJfKpyi

A,. = f Tp3t5sCTp«a.+ ^ F

AJ * Ip L m T r to f* ~ £ ft 7 p *P iilp iL fK -$ fi^ r* fo fo

+ ~if*?z.py^Tpyifsk- + Tr^-fft-Tf**p3/zf a * / a f i i i p H t f i G h r * *


62

The set of equations for a pure i.f&; %. resonanceis

kl ” ' S I 7- ic^ TpK?%npK-fK-*-t,IiTp^^T^s?rpi'l + + "Tp’w^xTf%.PK

A t’ " ft |Tp%$fcl*- -Jf-'fg

^1= " lr T"TV^%JPTt K- y-SvTp-v zT p

- 1 ^ 1 ? Tp?L/5i77^p%. + ^ = TpKfs?zTfvxpy*/\‘f~ IP’Yi.-f'f/z. If-KPh.

(\l = ^ / F 7r*.fnlfgPh. + -fr Tpn-fvJli\PVL .The set of equations for a pure 7J%f%) ^ resonance is

3 3 (

w t * i'Jf T&Sltrf *f %.Kxa T%. 2Tp^Vz.-f

sCTf p-fc. '*& ^TfwTpJkfy*.

a5_= £ t hsttfcf-

- iff ^ ^Ah ’ ~qj ( 7Iw&zl*+ 7f*Lfc*.7pK-f%.-‘-ff-f* TfK V kTfftnL *


63The set of equations for a pure , 9x resonance

A*> =

+ Tf*fftlpfcf?z. ~ Tflf&Tphf?z

Ax = Sfe'fs 9*Ph.

+ 5*c ~If M ?x~U %.PK" T*h? ?i.?*2. “ T*Kf *£Ipm7z

+ ?§• - Mai? T m f- 5iAn. •=• ~ 7 ("^Tts-Vxi + ^

- 1^^9JT5Vi.?-»/x T|^^Tpi«fe-5<S1fK<CTfe^.

aI = I -g £ ~l*yl*sLT$y2_?yx

<& T ? y zjps?xT?‘f e ^ -<J,^I "^**1*%.T ?vt^%.The set of equations for a pure T p ^ ^ j %. resonance is

A<> = 4 (Tpii<fyzrAo = f'S~fpyxfKTwy*. - “ g 'W skV? W i

" 9 \R T p -¥^Tpyvf?x ~ T ^ W T ? * ^

Ax- T ^ ^ T ^ t5/z

■*■ "fpvxV/J pfc.*?!. ~T?y>.+hTpyxfr2.

A z = 7 - A)£lpiiSWTi%PV!t TvyzfTtT 92.?%.~ §? & "^Vx xT?tx?'Vx ~ f T?ifcf WlwkS%.- § "im^npjt-?^

^ ” ?x yrrs-«7k?yx- .


64The set of equations for a pure \ resonanceis

fa = 4(T*iUKl1a£ = TifLfhlfzrrx IfiKfhlht?*.

+ £ & Tfof **£ fcf fcAi= I? /7f^k/

M = £n ( TfSLfhfcz?* '£ & ** Ffc-fVfffcP*

" 73r^~ff%f1tTfJtP'¥i~ 7 ^ l/f +

A* a - W Tf9JK (^ * ! / & 7%Pb - 7f\fklf%?K - 3 f ^ TfZ.fPz'Uh.pv-L - R TfhflkTryJVi

Al = - I f - /lf% fh .r + f ] / I If-W^lfs^pii ' # &'$gfkJzpn

' $l/£ “G%.f\V\nk + ?f l/iTfSkthfoL**-The set of equations for a pure resonanceis

ho = 4 Ih i = £ l"If%.f?i.lz- S:lf?'IfTtfvJF^T^ +

hi.- ^ 5 ? - S^Tffc^Mipyi

- IfxfitlpKWi" '^h.^ViTp-h.fVz.A* ~ I'W*-!2'" l4%.f

3tif\PK',>'S Tf%f*3pfc.f9i+ 1 fM * fa & + & 9~fatfbs9i

fas. -'£*

- “’f jr f iS w v l® Iffc W ftr& G T K to tf * -


65

'i« W m > S w V »£« TfWkWS^iJyi* Tfy^Wrt

/ W = - ^ \ T , „ K r

£ =

The preceding equations give information about the transition matrices in terms of the experimentally determined parameters bi. . The coefficients are, as mentioned earlier, determinedfrom spin-flip and spin-nonflip elastic scattering differential cross sections. These relations alone are not sufficient to explicitly determine the transition matrices. Additional elastic scattering cross sections and polarization measurements would provide the means to explicitly determine the transition matrices. For example, these relations with

(3,9)elastic scattering data determine uniquelywhich partial waves predominantly contribute to the % resonances of C .


REFERENCES

1. E. Merzbacher, Quantum Mechanics, (J. Wiley, Inc., New York, 1970)J p. 194.

2. J. D. Jackson, Classical Electrodynamics, (J. Wiley, Inc., New York, 1967), p. 77.

3. L. S. Rodberg and R. M. Thaler, Introduction to the Quantum Theory of Scattering, (Academic Press, New York, 1967), p. 286.

4. Merzbacher, op. cit., p. 508.5. R. M. Eisberg, Fundamentals of Modern Physics,

(J. Wiley, Inc., New York, 1961), p. 566.6. E. M. Bernstein and G. E. Terrell, Physics Review

173. 937 (1968).7. A. Bohr, Nuclear Physics 10, 486 (1959).8. H. T. Chen, Master's thesis, Western Michigan

University, Kalamazoo, Michigan.9. E. M. Bernstein, J. J. Ramirez, R. E. Shamu, Pui-

Wah Cheng, and M. Soga, Physical Review Letters 28, 923 (1972).

10. E. M. Bernstein, J. J. Ramirez, L. Baumann, and M. Soga, Physical Review C8, 1162 (1973).


a t-matrix analysis for the scattering cross section

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