LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN BIO .^4 vrt\0.5>53-S5?) GOf Z CENTRAL CIRCULATION AND BOOKSTACKS The person borrowing this material is re- sponsible for its renewal or return before the Latest Date stamped below. You may be charged a minimum fee of $75.00 for each non-returned or lost item. Theft, mutilation, or defacement of library materials can be causes for student disciplinary action. All materials owned by the University of Illinois Library are the property of the State of Illinois and are protected by Article 16B of Illinois Criminal law and Procedure. TO RENEW, CALL (217) 333-8400. University of Illinois Library at Urbana-Champaign Jul x ± i\}\}\ JUL i i 2001 When renewing by phone, write new clue date below previous due date. L162 Digitized by the Internet Archive in 2013 http://archive.org/details/partialwaveanaly553merc 5/^ M yu-SS& UIUCDCS-R-72-553 PARTIAL WAVE ANALYSIS OF ELASTIC AND INELASTIC SCATTERING OF DLRAC PARTICLES BY Robert L. Mercer October 1972 OET HE & f~ - i '•*• UIUCDCS-R-72-553 PARTIAL WAVE ANALYSIS OF ELASTIC AND INELASTIC SCATTERING OF DIRAC PARTICLES BY Robert L. Mercer October 1972 Department of Computer Science Department of Physics University of Illinois at Urbana-Champaign Urbana, Illinois 61801 'This work was supported in part by NSF Grant No. GP 25303 held by the University of Illinois, Department of Physics, and was submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree, October 1972. £10,1 u S$~JSS$ 111 ACKNOWLEDGMENTS I express my sincere gratitude to my advisor, Professor D. G. Ravenhall, for his continuing interest and encouragement and for the many invaluable suggestions and insights which he has provided. I am sure that this thesis could not have been completed without his aid. I also thank the National Science Foundation for financial support during the preparation of this thesis. Finally, I would like to thank my wife Diana for her patience and encouragement throughout my graduate studies. This thesis was prepared under the de jure advisorship of Professor David Kuck of the Department of Computer Science. IV TABLE OF CONTENTS CHAPTER Page 1. INTRODUCTION 1 2. PARTIAL WAVE ANALYSIS FOR DIRAC PARTICLES 4 2. 1 Introduction 4 2. 2 Spherically Symmetric Coulomb Well 6 2.3 Coulomb Scattering from a Dynamic Spin Zero Nucleus 28 2.4 Electromagnetic Scattering from a General Nucleus 33 3. THE ELECTROMAGNETIC INTERACTION 49 3. 1 Introduction 49 3. 2 Charge and Current Densities 49 3. 3 Maxwell' s Equations 56 3. 4 Matrix Elements of the Interaction 60 3.5 Specifying the Transition Charge, Current, and Magnetization Densities 67 4. PROGRAMMATIC METHOD 7 2 4. 1 Introduction 7 2 4. 2 Numerical Determination of the Partial Wave Phase Shifts and Amplitudes 7 2 4. 3 Summing the Partial Wave Series 89 4. 4 The Nuclear Potentials 93 4. 5 Organization of the Programs 97 4. 6 CNADIR 100 5. TESTS OF THE PROGRAMS 101 5. 1 Introduction 101 5. 2 Tests of NADIR 101 5. 3 I'ests of ZENITH 109 6. APPLICATIONS OF THE PROGRAMS 124 6.1 Isotopic Variations in the Charge Distribution of Calcium 124 6. 2 Magnetic Dipole Scattering at 180 ° 131 6. 3 Other Possible Applications 132 APPENDIX 1 135 2 140 LIST OF REFERENCES 15 2 VITA 15 4 1. INTRODUCTION There arises, in the course of analyzing high energy electron- nucleus scattering experiments, the problem of determining what the differential cross section is given a particular charge distribution 1/ for the scattering target.^ This problem may be solved using the Born approximation which relates the differential cross section to the Fourier transform of the charge distribution. It is known, however, that the Born approximation is inadequate for scattering situations involving large values of Z, the total nuclear charge. Under these circumstances it is necessary to solve the wave equation for the scattering process properly using the method of partial wave analysis. Computer codes which perform this type of analysis have been available for many years [ 1 ] . These codes, which are able to handle single-channel scattering, are adequate to the extent that the nucleus may be approximated by a static, spherically symmetric charge distribution. Therefore they have found their most fruitful applications in analyses of scattering data from doubly or singly magic nuclei in which there are no low-lying excited states (e. g. Pb , Ca ) . In the last ten years, it has become increasingly desirable to analyze scattering data from more complex nuclei. A step in this direction is provided by the distorted wave Born approximation in which the dis- tortion of the incoming plane wave due to the Coulomb field surrounding 1. A history of electron-nucleus scattering together with an extensive bibliography is provided by Uberall [ 2 ] . the nucleus is correctly accounted for by the partial wave method and nuclear distortions, etc. , are treated as first-order perturbations. It has been suggested that, at least for certain situations, these treat- ments may not be adequate and that the second-order effects of nuclear excitation on elastic scattering may be appreciable. A proper treatment of the higher order effects of excited states and higher multipole fields requires a complete coupled channel partial wave analysis of the Dirac equation. The analogous problem for bound states was first solved by McKinley [ 3 ] in the course of accurately determining the eigenvalues of muonic atoms. In the bound state problem, the wave function is localized in the vicinity of the nucleus. In the scattering problem, this is no longer the case: One must determine the wave function throughout all space. This makes the scattering problem much more difficult, as will become evident in later chapters of this thesis. Although partial wave analysis of coupled-channel problems for the Dirac equation presents no conceptual difficulties, there is a great amount of algebra associated with it which must be carried out if we intend to write a program eventually. Some of this has already been done by McKinley. However, because we consider a more general electron- nucleus interaction, and because our point of view is necessarily some- what different, we have developed everything from the beginning. In Chapter 2, we treat the partial wave method in detail starting with the single channel problem and moving on to the more general coupled channel problem. Our treatment of the single channel problem is more cumbersome than it need be because we have actually developed it as a special case of the coupled channel problem. In Chapter 3, we discuss the electromagnetic interaction deriving there the reduced matrix elements of the potentials which enter into the wave equation. In Chapter 4, we discuss the programs that have been written and the numerical methods used. Chapter 5 is a compendium of tests with which the programs were debugged. Finally in Chapter 6 we discuss applications that we have made of the programs and indicate a number of directions for future applications. Throughout we have made frequent reference to the "Handbook of Mathematical Functions" by Abramowitz and Stegun [ 4 ] referring to it simply as HMF. Detailed references to the parts of the partial wave calculation that have been done earlier may be found in reference 2. 2. PARTIAL WAVE ANALYSIS FOR DIRAC PARTICLES 2. 1 Introduction — » The wave function ^(r) for a free Dirac particle of mass m with total energy E is a four-component spinor satisfying the wave equation H ^(r) s (-ihca- V + ^mc ) ^(r) =E^(r) • The Dirac matrices a and (3 are 4 by 4 matrices operating on the components of \&(r) and are defined by - / ex \ / I , a ; -I ; where I is the 2 by 2 unit matrix and o~ = (a , a 9 /0~ ) are the familiar Pauli spin matrices. /0 1 \ i -i 10 I CT = I I 0" = | J CT = 1 1 / 4 \ i / 0-1 We include here several very useful identities involving the Pauli matrices. Let e be the totally antisymmetric tensor of the third rank equal to + 1 ljk when i,j,k is an even permutation of 1,2,3; equal to - 1 when i,j,k is an odd permutation of 1,2,3; and equal to zero otherwise. Then the following identities are easily verified, e „ (r,\) satisfies the wave equation (H D + V(r,\) +iy ¥(r,v) =E*(r,v) . (4) In terms of the variable r, Equation (4) is a set of coupled linear first- order partial differential equations. The partial wave method exploits the invariance of Equation (4) under spatial rotations and the consequent conservation of total angular momentum to separate the angular dependence of \I/(r, v) from its dependence on r = |r| thus reducing Equation (4) to an infinite collection of sets of coupled ordinary differential equations. The remainder of this chapter is devoted to the detailed application of the partial wave method for several possible choices of V and 1L,. In Section 1.2 wl- treat the case of a simple Coulomb interaction with a static spin zero nucleus; in Section 1.3 we consider the Coulomb interaction with a nucleus that has a finite number of spin zero energy eigenstates, and in Section 1.4 we consider a general electromagnetic interaction with a nucleus that has a finite number of energy eigenstates with arbitrary spin. 2. 2 Spherically Symmetric Coulomb Well In Equation (4), let the nuclear Hamiltonian, H^ ; be a constant and let V(r) be the Coulomb interaction between the electron and a spherically symmetric charge distribution representing the nucleus. Then, since without loss of generality we may assume that H^ = 0, Equation (4) becomes H^(r) = (-ihca-V + Pmc +V(r)) ^(r) = E ^(r) . (5) The total angular momentum operator is -> - h 1° °' 1 \o a where L = -irxvis the orbital angular momentum operator. 2. 2. 1 Spinor Spherical Harmonics Given eigenstates |LM) of (L) and L with eigenvalues L(L + l)fr and Mh respectively together with eigenstates \— m > of (— a) and T C7« with 3 2 " eigenvalues - H and m h respectively, we construct eigenstates, |JLy,) , o f (lT-l-~ o) , (0 , andL +-"G with eigenvalues J(J + l)h, L(L + l)h, and U,h respectively. These are the spinor spherical harmonics. If L 1 L 2 L 3 C M M M are the c l eb sch-Gordan or vector addition coefficients describing 12 3 the coupling of systems with angular momenta L and L to create a system with angular momentum L , then v L * J |jLu> = l C |LM>||m s >. (6) Mm M m \i s s To each value of J, there correspond two possible values of L: L = J - •§■ and L = J+-|. Consequently, J and L may be combined into a single quantum number X as follows: X(J,L) = 2(L-J)(J+|) J(X) = |X| - § L(X) = J(X) + | sgn(X) . Hereafter, X and the pair J, L will be used interchangeably without explicitly illustrating the dependence of one upon the other. Also, we will frequently identify the ket |lm) with the spherical harmonic i Y M (9cr>) = (9cp|LM). The phase factor i is necessary because we choose the phases of the kets |lm> and |-§M ) so that t|lm)= (-1) |L-M) and t|^M > = (-1) |§ - M ), when T is the time reversal operator. It S o follows that t|Xu)= (-1) |X-n). This choice of phases is unimportant for the monopole cases but is a great convenience when considering more general scattering situations. The spinor spherical harmonics form a complete orthonormal set for the space spanned by the kets |lm)|-^M ). — * — * -» a o r — ► A 9 Consider now the operator or = . From Equation (3), (a- r) z = 1 . Therefore, (or) |x.|i) = |X|i) . But or is a scalar operator which, since a is an axial vector while r is a polar vector, changes parity. Hence O r |X[i) = e y I -X[i) and e y e y = 1 . Furthermore, using the time reversal operator T, we have T a- r|Xu> = T e |-X|jl> X 1 = e* t|-Xu> = e*(-D J+ii |-X-n>. On the other hand, T a- r T =-a- r. Therefore, T a. r|Xu> = T a- rT~x|Xu) = - a-r(-l) J ^|x-u> = - e x (-l) J " Hi |-X-^> . Comparing this with the previous result, it follows that e is imaginary. 2/ One can show rather easily that e = - i sgnX.— ' We choose a straight- A. forward but rather tedious way of showing this which previews methods that will be useful in Chapter 3. From the orthonormality of the |Xp.), one has that e y = <-Xu |a* r |Xu) . If a has spherical components G , then m ?-= jf E(-l) m Y lm (9cp)a. m . m Therefore L h J L k J m L-L -1 e L s J L B J r~ 1 = 1 L c M , . „ c Mm ./— (-i) x M 1 m' ii Mm ll V 3 Mm s s s M'm m s x Jdfi Y L M , (Q) Y lm (Q) Y LM (n) x (hm } \o |%m ) ~ s ' -m ' * s where we have used L to denote L(-X) . 2. See, e.g. Rose [5] page 28. His phases differ from ours by i .L Now, / 3 f2L+l) 1LV - 1LL - Jd fi Y L , M , (Q) Y lm (n) Y LM (n) -J ttffafc c mMM , c o and in Section 3.4, it is shown that -i i i 2 J- 2 x s ' -m 1 A s m -mm' s s Also, it is clear that i = sgny- We have therefore 1 LLL I3(2L+1) \ 2L.+1 "0 e =isgnX ,p L. * J LtJ ILL % 1 X 1 , , x m ' _ s " "2 x £, (-1) C M , m , m C Mm m C mMM , C M'n M m Mm U- Mm u. m M M' m -mm Mm s s s s s s m Sums of this nature over several Clebsch-Gordan coefficients are more easily managed when expressed in terms of the more symmetrical Wigner 3-j symbols defined by / J l J 2 J 3i (-l/ 1 "^"" 3 A J 2 J 3 M, MM, M M -M \/2J-+l M l 2 3 12 3 The Wigner 3-j symbols have the following symmetries. i J l J 2 J 3', / J 2 J 3 J H J l +J 2 +J 3 / J l J 3 J 2i / J l J 3 J 2 .M 1 M ? M 3 J ' l^M-M-M.; ( " 1) IM M M / \-VL -H -M (8) 10 Therefore e X = isgnX (6(2L+l)(2L.+l)(2J+l)(2J'+l))^(-li'- +L4 * +J ! .J £ £-) L.+l+l+m+M 1 -m f L_ | J j j § 1 § ,/l L_ L / § L Jj x Zj^" 1 ) M' m' -u/\-m' -m m Am -M* M/ M M -J Mm » S p ' \ s s / 1 / \ s 7 M' m> m s T x i (1 L L_\ ( L. L ll = isgnX (-1) J 2 (6(2L_+1)(2L+1)) 2 [ Q Q Q j ^ i j[ . In making this last step, we have employed Equation (3.34) which defines the Wigner 6-j symbol, s . These remaining coefficients are iJ l J 2 J 3 easily evaluated and one finds indeed that e = -isgnX. We have shown therefore that a-r|Xu.) = -isgnX|-X^> . (9a) Since a* r is hermitean, we have also shown that (Xu|a-r = isgnX <-Xu| . (9b) 2. 2. 2 The Radial Partial Wave Equations Since the spinor spherical harmonics constitute a complete set of functions with respect to the space spanned by the vectors |LM)| 1 rm ) , any solution of Equation (5) may be expressed as g x (r)|Xu> \ (10) *<*> = £%, \ -sgnXf (r)|-Xu> Xu ^ r \ where the partial radial wave functions g v (r) and f v (r) are, as indicated, A. X functions of r = |r| alone. Note that the Wigner-Eckart theorem allows us 11 to place all of the i-i-dependence in the coefficients a^ . Substituting this expression into Equation (5), we arrive at the following equations. £ a. f(mc 2 + v(r) - E) ^ §x (r) |Xu> + i sgn X he a-"v (~ f x (r) | -Xu»] =0 Xu L a, (-ibca- ^(^g v (r)|Xu» - sgnX(-mc 2 + v(r)-E)^f y (r)|-Xu>} =0 Xu (12) Let us consider for a moment the operator a- V . We have (r xL) . = e. r. L. = e. e , „ r . r .V v y i ljk j k i. jr . , fc. . 1. . i. „ V Now, I . ., ?■ = 5. . 5, - 5 . „ 5. ' ljk kj?.m tJL jm j I lm Therefore, i r x £=?(?•?) -r Z v* , or v=r- — -lrx— dr r where we have used the fact that r-v = r — . Multiplying on the left ^r by a and applying Equation (3), we have jW = (a-r) — + r. \ - (a-r)(— -) . Since r • — = 0, we have finally • " = l - 1 ( T " — } - Therefore, for any function h(r), (c-?)(-h(r) |u» =?.r (^ r r r )(-h(r)|x u » . r j — •r jo+i)-l(l+1) - 7 \ (*h(r)|Xu» 12 For X>0 , J(J+1) - L(L+1) - | = (L - |) (L+§) - L(L+1) - £ = - L - 1 = -X - 1 and for X < , J(J+1) - L(L+1) - 7 = (L+|) (L+4 ) - L(L+1) - -r = L = -X -1 . Taken together with Equation (9a), we then have £v (i h(r)|Xu» = -i sgn) c A + 21 h(r)|-Xu> . If we now return to Equations (12), multiplying the first on the left by (X[i | and the second on the left by (-Xu] , we have Mdr" ' r) f X (r) + ( E " mc2 ) § X (r) = V(r) 8 X (r) hc (dT + r 8 X (r) " ( E+mc2 ) f X (r) = V(r) f x (r) . Er In terms of the dimensionless radial variable p = r— , these equations are he dV'pjV^ + g x ( e> = ^H^ dV p) 8 x (p) " < 1 + T > f x ( "> =-v(p)f x (p) (13a) (13b) •*.u n- mc j / n V(r) with T- ■— and v(p) = -*-*■ 13 Physically we know that the probability of the Dirac particle being in any neighborhood of the origin must be finite. This means that H (£y(p) + g^(p)) dp must be convergent. Equations (13) have two independent solutions, the behavior of which near the origin is given by lim(f x (p) p" X ) =1 lim(g x (p) p" X ) =0 (14a) p - p - and lim(f x (p) p ) =0 lim(g x (p) p ) = 1 . p-0 p-0 (14b) For positive X therefore, we must choose solution (14a), while for negative X, (14b) is the choice we must make. 2. 2. 3 The Relativistic Coulomb Wave Functions Since v(p) represents the Coulomb interaction between the charge, ze, of the Dirac particle and the charge, Ze, of the nucleus, the latter being contained in a spherically symmetric distribution, we must have, for values of p greater than the nuclear radius, v(p) = - * (15) P where 2 zZe Y :: " he By nuclear radius, we mean that value of p, say p^ such that a sphere of radius p^ centered at the origin contains all but a negligible amount of the nuclear charge. 14 It is instructive to consider the case in which v(p) is given exactly by Equation (15) for all values of p. If we do so, Equations (13) become the relativistic Coulomb wave equations: (£-*) F x (p> + (l-T)G x (p) =-*G x (p) (16a) _d_ X dp p; + -| G Y (p) - (1 + t)F x ( p ) = X F x (p) . (16b) Make the following change of dependent variables, G x (p) = a(G' + F») F x (p) = b(G» -F') . The constants a and b will be chosen later. Equations (16) become 2abf^p- + — ! + (a 2 -b 2 -T(a 2 +b 2 ))G' +(a 2 +b 2 - r(a 2 -b 2 ))F' • dp p > = - 1 ((a 2 -b 2 )G« +(a 2 +b 2 )F' ) P 2abi^-+^M - (a 2 + b 2 -r(a 2 -b 2 ))G' - (a 2 -b 2 - r(a 2 +b 2 ))F' = * ((a 2 +b 2 )G' + (a 2 -b 2 )F') . P We now choose a and b such that 2 2 2 2 2 a -b - T( a Z +b Z ) = 2k 2 2 2 2 a +b - T(a"-b Z ) = , 2 \ 2 where k = (1 - t ) . Provided that k ^ , which is to say that the Dirac particle is not at rest, this system of equations has solutions which, 15 since we are indifferent to the signs of a and b we may choose to be l a = k(l - t)" 2 i b = ik (1 + t)" 2 . Finally, writing x = 2ikp, the differential equations for G 1 (x) and F' (x) are dGj. „ ,'1 + 1Y G , + /X _ ±X\ dx \ 2 kx. x kx' F' - dx 2 kx \x kxy We can solve the first of these for F 1 in terms of G' and use the resulting expression in the second to obtain a single second-order differential equation for G 1 : ik + i if. [i + i i + ix .*£ G , =0 .2 xdx 4x2 k / 2, dx \ ' x l Let H = x 2 G' . Then d^i l + l/l + iv . j-^W) H = (17) dx ' x / which is Whittaker 1 s equation. The solution of Equation (17) that is regular at the origin is H(x) = e 2 x' s + S M(l+x+ 4 1 i l + 2s, x) k 2 2 ! where s ■ (X -y ) . The definition and many properties of the confluent hypergeomctric f unction, M(a,b, z), may be found in Chapter 13 of the Handbook of Mathematical Functions. 16 Ultimately, we want F x (p) and G x (p) to be real when p is real. Therefore, we write iru-ikp G' (p) = e R (2kp) S M(l+S + -^ , l+2s, 2ikp) with rv, yet to be determined. Now, from Equation 13.4.11 of HMF we have (b-a) M(a-l,b,z) = (b-a- z) M(a,b,z)+z ^ (M(a,b,z)) and therefore, using Equation 13. 1. 27 of HMF _Jc e ^- ik P (2kp) S (s- f )M(s+ f ,l+2s, 2ikp). JL. We require that G 1 (p) = (F* (p)) . Therefore we must have (..+ f > e = - ke kX + i Y T and hence 2ir k e = kX - i Y T ks - iy The regular solutions of Equations (16) are thus i -i-n -ikp G X R (p) =A r( 1+t ) ? ( 2k P) S Re ^ e M(s+^, l+2s, 2ikp)} F X. i -iru-ikp R (p) = A r (1-t) ? (2kp) S Im{e M(s+^, l+2s, 2ikp)} We proceed now to determine A_, such that asymptotically G y _ oscillates with unit amplitude. Asymptotically, we have from Equation 13. 5. 1 of HMF, 17 +irra R-l , N ,., , N T-./-LN - _a v-i ( a ) (1+a-b) m/- u \ r(b) e z V n v ' n , _ x -n n=0 fO(i 2 r R ). (18) I S-l (b-a) (1-a) ~] . e z z a - b r(b) y a. s - n + oc|z|- s )l , r < a > n=0 the upper sign being taken if - — tt < arg z < — tt; the lower sign if O 1 ij- - — it < arg z < - — F . In our case, argz = — and so the upper sign is appropriate. By considering Equation (18) with R=l and S = one finds that Ag = (1+T)" S e 2k r(i+s - f ) r(i+2s) Asymptotically, therefore, S.R^ Lf , c sin(kp+£ In 2kp - — + r^ ) (19) F X,R ( P> — cos (kp + £ £n 2kp - -y + t^ ) with k ' iir(L-s) ks + iy r(s+^) (20) The irregular solutions of Equations (16), F v (p) and G v T (p), x,i x,i may be obtained by replacing s with -s throughout the above discussion. Then -r|,-ikp G x x (p) =A i (1+t) 3 (2kp) _, Re{e M(-s +^ , l-2s, 2ik P ) } -s -ir^-ikp F x I (p)-A I (l-T) : (2kp) Im{e M(-s + ^ , l-2s, 2ikp)} -s 18 2lT1 I ks-iy kX + iyT j. 1 r *i " < 1+r > s e r(i-2.) Asymptotically, G X I (p) ~ cos(kp+^jto2kp- y + Tij) F X I (p) ~ - -^ sin(kp+^£n2kp--y+rj) (21) with 2lT ll _ ks + ly r( ' S 'T^ ) iir(W-s) , . " kX + i YT - T< _ t+ kL> • (22) Since Equations (13) behave asymptotically in just the same way as Equations (16), we must have g x (p)\ [ G X,R^ ;G (p)\ ~^ B,, + CL, /\.j J- f x (p>/ p-co X \V VBtf ! 2 ' TF" K > 20 in which j (kp) are the regular spherical Bessel functions. Asymptotically then, the spherically incoming part of ^ r _]_ ane (P) is ^ , , ~l(kp --T-) /. \"2 *. (p)=- l^ir(2l+l) -±r e [±f\ \jt,0)\%m). (24) in &=0 2ik \ / / s For simplicity, we write only the upper components of the Dirac spinor. It is clear from Equations (23), however, that the spherically incoming portion of gy(p) is given asymptotically by (g x (p)). n ~ -i e (25) Because of the logarithmic phase, present in Equation (25) and absent in Equation (24), Equations (24) and (25) are incompatible. This means that there is no solution of Equation (5), the spherically incoming portion of which is asymptotically equal to ^. (p) . The problem illustrated above arises because we have been too narrow in requiring the incident wave to be a plane wave. Physically, the incident Dirac particles are in a narrow beam which we are restricted to investigating only at great distances from the scattering center. As such, we are able to determine two things about the incident beam which, from an experimental point of view, characterize it completely: its wave length \ = 2n/k; and its direction. In particular, we can say nothing about its phase relative to its phase at the scattering center. We are justified therefore in choosing the incident wave from the much broader class of wave forms, 21 *(p)= Z \Att(24+1) ^ sin(kp+h(p) -4f) Uo>||m s >(^ '' 1+T V / where h(p) is any function of p for which Aim h(p)/p = 0. For the P -»oo Coulomb problem, it is clearly convenient to choose h(p) =~- In 2kp. Thus, we replace Equation (24) by " -i(kp+Jj?.n 2kp -^)/ *■' 2im kl < n (P)=-I o ^2m)^ e |l±T y j_^J Uo) | |ms> . (26) 2. 2. 5 The Scattering Amplitudes and Differential Cross Sections From the considerations of Section 2.2.4, it is clear that we must choose the coefficients a^. such that £ An 2kp - — 2ikp *" I 2 £ , , -i £=0 i V -i(kp + ^n2kp-f +6> ,,) Multiplying both sides of this equation on the left by (Xu| we find after some algebraic manipulations, . =^(2LH4J I <* * J A" 1 e 16x (i±tf 6 . (27) x.u k Ommx \ 2 / mu s s s That solution of Equation (5) which satisfies the scattering boundary conditions is given exactly by 22 *(p) =iV47T(2L+l) — C A e I— ■ kp Omm V 2 \-sgnXf v (p)|-Xm > S S \ A, ' s and asymptotically by i(kp+*£n2kp) tf(p) - #. (P) +~ e f (6cp) . in p m P ->03 S Using Equation (23), we find the scattering amplitude f (8cp) to be s given by /. \| L | J \ L ^ J| \ 2 / Ml m' m LM! | 2 / M! m' m L Ml L S S - L where once again L = L(-X) and \- 2i6 x 2ik In terms of the scattering amplitude, the differential cross section is given by (i) - f l ( *> f m < to > • m s s s The spherical harmonics Y„ (Geo) bear the following relationship £m o to the associated Legendre polynomials P (0) : V im (8») . <-»" C Vm Ce»»*= C-l)"^ M" ^ P>>, ->0 . (29) 23 The Clebsch-Gordan coefficients occurring in Equation (28) may be written & I J oil sgnX X 4 I J 24+1 /_ X o -I -I i ^ 24+1 c = A- i I j , l i J 24+1 1 S 2 V 1 1 2 "S sgnxyi X_ 24+1 Therefore, writing all four spinor components explicitly, P?i*|xtf(» fl(9c D ) = Z ^ l sgnX 1±I)' e^pjo) 1-t *|x|p° (e) \ 2 . - and 1-T ,* icp ■sgnX *j*. e iV P L (9) -sgnX f*Cj* e" 1 * Pj(e) f i(ecp) = £ \ A. ^) l"|P°(9) l^,"e-^lt (6) -i^Vtf.Ce) 24 From a computational point of view, it is valuable to write the scattering amplitudes as sums over L rather than X. For example, ,1-H-j P° (LM L + (L+1) M -L-l } fiOco) = £ M e e-p£< v «. t . x ) \* r?J P L < LM -L +(L+1)( W Tf e "'i ( »-rV Now, the Legendre polynomials obey the following well known recursion formulae: P^(cose) = sin & - [cose p^(cose) - p°_ 1 (cose)} (A-mH-l)P I J +1 (cose) = (21+1) cos0P^(cose) - U+m) P™(cos9) and, as a consequence, the additional formula p i (cose) = " ^ine ^ p ° + i( cose ) " cose p°( cos9 )} By using the first and last of these, we can eliminate P (cosG) from the Li second and fourth components of f^ in favor of P (cosG). Doing so we 2 L obtain 25 >] t)* a^+a+DM.^) fi(ecp) = Z^t L=0 e lcD (^) [-cote(Ii! l +(wa)M i _ L . 1 ) + i =5 (LM^+CL+DM^)] 1 icp /1-t 1 l_i, r . cot9(LM _ L+(L+1)ML+i)+ __ (L^+a+DM^.pl Let and f + (9) - Y- p ° (1^+ (L+1)M_ L _ 1 ) L=0 03 f_(e) = I! p^-cotea^+a+DM^^H-^ (LM^+a+D^p). L=0 Then clearly fi(GcD) - 1 1+T * i 1 : f + (9) i icp /1+T * , , Q v e i ~~ 2~~ J 1-T (cose f + (9) +sinB f_(f») ico 1- i (sine f , (e) - cose f (e)) + Similarly, f i(Ocp) can be expressed in terms of the two complex numbers f (6) and f (6). It is easily seen that the lower pair of components 26 may be derived from the upper pair by the application of the operator 0-k. eco where k is a vector of magnitude k pointing along the direction 1+t ecp determined by 9 and cp. When the incident beam comprises an incoherent average over all possible orientations of the initial spin of the projectile, as is frequently the case experimentally, the relevant cross section is d£l _ i V da dQJ " 2 L IdQ/ 1 m m s In terms of f,(Q) and f_(9), this is ;j2j=(|f + (9)i 2 + |f.(9)| 2 ) 2.2.5.1 Polarization It is apparent that while f (9cp) depends upon the azimuthal angle cp , s i-rp: does not. If, however, we consider an incident particle with \ ''m s spin oriented not along the z-axis but in the xy-plane, this is not the case. To describe the variation of the cross section with cp under these circumstances, we introduce the polarization P(9) defined as follows. Let the coordinate axes be as shown below. -$> z 27 Then if (S\ be the differential cross section for the scattering of Hm y s particles incident along the z-axis from negative z with spin projection m ( along the positive y-axis, we have da, ; dcr, 3J5<*,l -l§§(^q)-o) P(0) _2LZ .A* dQ or equivalently, ^ (e '*- 0) ) 4 + (^ (e ^ =0) ) iy p(0) = (g<*,9-0>) £2 if! A ^y_ V do ( ^ fp - ») + Jgc.^ We can construct the scattering amplitude fi (Srp) from the amplitudes that we have previously developed for the case of incident particles with their spins parallel and anti-parallel to the z-axis. Thus f i „ ( cp) ■ -i (fi(^p) + if 1 (Sep)) • Since we understand how to obtain the lower components of a scattering amplitude given its upper components, we may safely proceed using only the upper pair of components. Then 'f + (0) - ie" 1Cp f_(0) f£ y ( cp) = td+T) i(f (9) - ie 1Cp f (9)) 28 Thus i |f,(0)+if (0) i(f + (0)+if_(9)) and the cross sections are !^(^cp = f±f)) = (|f + (9)| 2 +|f.(e)| 2 + 2Im(f*(0) MS))) In this last equation, we have included the lower components and there- in fore lost the factor (1+T) ? . Finally, the polarization is 2Im(f*(0) f (©)) P(0) = ~ 9 : -5 • (30) |f + (0)r+|f_(0)l 2. 3 Coulomb Scattering from a Dynamic Spin Zero Nucleus Let the nucleus be described by a set of N energy eigenstates |n), n = 0, • .., N-l of zero spin for which Hjjn) = ejn> i = 0, ..., N-l . Without loss of generality, we may choose e_ = . Also let V be the Coulomb interaction between the Dirac particle and the nucleus. Then, if V ill «i n (r) |Xn> \ t(rl= Z V7 in h> , (31) X^in ^ r \-sgnXfJ n (r) | -Xp.) J we have, after some manipulation, the following differential equations for the partial radial wave functions: 29 i± " f)*!N> + (t m" '>*£"<*> - 2 V nk (p) ^ k(p) (32a) (£ + ggfto) - (t. + rtf^p) - - E v nk (p)< n ( P ) • (32b) Equations (32) are given in terms of p and T as defined for Equations (13) E-e m l and in addition, t = — - — , v , (p) = — (mlvlk). ' m E ' mk E ' ■ Of the 2N independent solutions of Equations (32), we are interested in the N solutions which are regular at the origin and distinguish among them by means of the index i which ranges from to N-l. For X > , we have jfcim(f* (p)p" ) = 6 ik £im(g^ (p)p~ ) =0 , p ->0 p -*0 while for X < , we have j£im(f£ (p)p )=0 £im(gj (p)p ) - 6 ife . p -0 p -0 As before, we must determine the asymptotic behavior of these regular solutions. Asymptotically, v . (p) ~ - 6 , , with y defined as in p -»» Section 2. 2.3. In fact, the deviation of v , (p) from - -^ 6 , decays mk p mk J exponentially outside the nucleus. Therefore, Equations (32) decouple in the asymptotic region, reducing there to N pairs of relativistic Coulomb wave equations. The discussion of Section 2.2.3 may be applied 30 to these equations by replacing 1 with t throughout as required by i . • « i m i dimensional arguments. In this way, we arrive at constants A^ and . in* . 6 y such that vt T im , . . im . /. . m . 01 Ltt . . im N g x (p) ~ ^ sin(k m p+— Xn2k m p- T +6 X ) p -*co m and (33) k vt T ^ lm / n A im m ,, . m ... Ltf . -im. p ->c° m m We must now choose the constants a^ in Equation (31) such that, asymptotically, \&(p) represent a plane wave approaching the nucleus in its ground state, |0) . Following the discussion of Section 2.2.4, this means that the asymptotically incoming part of "^(p) must be equal to"*-. (p)|0). Therefore, in , — m 2k p - — , o k o^ 2/t+T ■iCk^pH--^- In 2k,p --^-) i_ , _»§ E^lTttML) 2iF-e ° U— U0>|i m >|0> 2=0 o p \ zt o Yt n' , n1 Ltt , .in 1 V v n ,K k ~-— n »" 2 ' W X' "A Av Up" e |XV>|n-> ■K^.p+^T^- £n2k T , l p-^+6 VI ) S'',,' ~o77~ e XV' in' and hence, multiplying on the left by (n|(X|a( ■i6* P , , L I j/t o +r\ £ I4u 4' e " X =^<2L+1) i C .CH -X ■"^" *' k ~ 2t 1 o m m \ o / s s x 6„ 6 . (34) op um s While t Q = 1, we include it explicitly for the sake of symmetry. An expected consequence of the Wigner-Eckart theorem is that a^ is non- 1m zero only for u = m . Since the solutions g (p) are independent, the X 31 . T ip A ip X , . ,ip matrix ft. = Ay e has an inverse, say Yl, , for which I $ hf - I hj» «f . 6. . lm pm Therefore, if we multiply both sides of Equation (34) by tv, and sum over all values of p, we find a£ m - TS^ULrt) A C 1 „ , om ; o ' k ^ ~n * 2t o Omm \ o s s Sub stituting this value for a^, in to Equation (31) we find x.m Yt i(k p +-7 _E - £n 2k p ) vi P k p m *(p) ~ -Z C in (p)\0) + 1 ± e P f o S . D ^ co >|P> P -+oo p ~* P with the scattering amplitudes f (0cp) given by m f S (9cp) = Y, v/477(2i+l) C °^ P Xmm' ic m P x , om mp L -£ J h AT e 2ik Omm o s s s t +T o 2t L \ J Y (Pep) m -m' m' m l m -m' s s s s s s H) t +T P m -m m m s s s s Y (Pcf L m -m 1 s s 32 As in the single state case, the lower pair of components may be derived p 2t \ ° f o n (9) O ->p e ic P f" (9) o ->p \ with O ^p I P^LM^P+CL-fDM^^) and O -*p £ p i (M °. p - m - p l-i> L=0 The cross section when averaged over spins is da) dnA o -*p k _ m m jj I (f s ) f s k *~ o -*p o ->-p o m r r s Vp J / JQ + T j ,,_+ 1.2,1-- ,2 N P / 1 o o / F F (36) 33 2t where we have picked up a factor of - — -^— by including the lower P components. The ratio of the outgoing momentum k to the incoming momentum k , which is identically 1 in the single state case, takes account o of the fact that the scattered particle in general has a different speed from the incoming particle. 2. 4 Electromagnetic Scattering from a General Nucleus 2.4.1 Introduction We turn now to the partial wave analysis of the electromagnetic scattering of a Dirac particle from a nucleus with a finite number of energy eigenstates. In particular, let the nuclear eigenstates be In I p M) , 0 N' n n n ' n n rj|nl p M> = 1(1 +l)|nl pM) N n n n n ' n n P In I p M) = p In I p M) N n n n' n n I In I p M) = Mln I p M> N- ' n n ' n n Furthermore, we choose the phases of these states so that under time reversal, we have I +M T|nM > = (-1) n |n - M) . 34 Again we assume without loss of generality that e Q = 0. For V(r,v) in Equation (4), we take the electromagnetic interaction which we write in the following form: V(r,v) = S(r, v ) + ^(r, v ) ■ a . (37) 2. 4. 2 Partial Wave Decomposition The total angular momentum operator for Equation (4) is f=3+T. The total parity operator is P = P P where P is the parity operator for the Dirac particle. We wish to expand ^(r^v) '3 in eigenfunctions of f , F_ and P. One such set of eigenfunctions are J I f |fm fP Xn>= /_, C |x^>|nM> (38) (i,M [i M m for which p = p (-1) . In general, to a pair of eigenvalues f and p, there correspond several possible pairs (X,n). Let 7?(f,p) be the set of such pairs and let N(f,p) be its cardinality. It is clear that if (X,n) e7?(f ,p) then (-X,n) e^(f , -p) . As a notational convenience, we introduce a single quantum number, say j, such that as j ranges from 1 to N(f,p), (X.,n.) ranges over all of the pairs in 7?(f,p). Finally, if j is the quantum number corresponding to the pair (X,n) in 7?(f,p), let j be that cor- responding to the pair (-X,n) in 7l(£, -p). We now make the Ansatz 35 ^(r, v )= Li a]. - fm f p f r g^(r) |fm f p j) ■sgnX. f^(r)|fm f -p j_> \ (39) in which i,j are analogous to i,n in Equation (31). Substituting this expression into Equation (4) and making use of the form we have assumed for V(r,v), we have, in terms of the familiar dimensionless quantities defined in Sections 2. 2. 2 and 2. 3, the differential equations £ - X -A f^(p) + (t -T)gjJ(p) = dp p / fp n. fp E{^.(p) ig'w -v^.cp) fJJ'o.)] (40a) (£ + £■) gjJ(P) - (t Bj +T)fjj and f sgnX l V(r,v)•a V j.j•(p) = . (40b) 36 In the next chapter, it will be shown that as p tends to zero. 1 X. - X.' J st\, (p) and v f .,(p) tend to zero at least as fast as p Therefore the solutions of Equations (40) that are regular at the origin have the following behavior near the origin: -X Urn (f^(p)P J ) = S ij 3^0 -X. l±m (g^(p)P J )=0 X.>0 'fp ■ (40 c) X ..X. lim (f" J (p)p J )=0 Mm (Sfp(P)P J ) = 5ij X i <0 p->0 p -»0 It is also shown in the next chapter that asymptotically, I f , , y _.f£ .-i-i s ,(p) £=0 a. ., p JJ and v..,(p) I ',=0 ,f£ -A-l >. -i P JJ .fjfc ,o in which a. , and 8.7. are independent of p. In particular, JJ JJ a f °, =- Y 6.., and B f0 , =0. JJ' JJ' JJ 2.4.3 Determining the Partial Wave Phase Shifts and Amplitudes When some of the constants a... and 8... are non-zero for 1>1, JJ JJ Equations (40) do not decouple asymptotically. Corresponding to the relativistic Coulomb wave equations which were so helpful in the Coulomb scattering problems, we are faced with the differential equations 37 d ip P / r -V i t=o r If - - 1 ; F* j (p) + (t -T) G^(p) = \dp p / f p n . f p * J G^ J (p) - (t +T) F^(p) = fp n. fp P' l ' l l<*?y F f P (p) + ^V 4 P ( ^ )] • (41b) These equations have an awkward singularity at the origin and analytic methods which were successful in the Coulomb case avail us nothing here. However, it is possible to obtain an asymptotic expansion for the solutions of Equations (41) which is valid as p tends to infinity. In obtaining this expansion we drop the indices f and p. We write TO _ F lj (p) = £{sin(k.p+Y ^ ta2k p- -£) s^ m=0 i L. L.7T , v + cos(k p ^r m 2k.| - — r-) t:/j p 1 K. . 1 Z 11 1 38 G 1J (p)= I {sin(k.p + Y ~ to2k.p- -|~)u^ ; m=0 i t. L.7T / x + cos (k . p + y r~ An 2k . p - — - ) v . / } p i k. i 2 ij 2 2 2 where we have written t. = t , . . , k. = (t . - T ) and L. = L(X(i)) . Note i n(i)- 7 ii i that k,t, and L depend on the first superscript associated with F and G which enumerates sets of independent solutions. If we introduce these expansions into Equations (41) and equate the coefficients of sin(...)p -m and cos(...)p in the resulting expressions to zero, we obtain t ._ • c ^t i ( m ) ( m ) (ni) , recursion relations for the unknown constants s. . , t. . , u. . , and ij ' ij ' ij ' (a). Let A. . (t.+T) J (t.+T) t.-T J k. l t.-T J 1 / / y M (m) X H-m-1 J t. l " Y kT i Y t. l Y k7 i X. +m-l J -X. + m -l J t. l ~ y k7 i o t. i Y k7 -X. + m -l 39 JJ = sgn- '-sgnYJj' sgnx.„f>, V ■sgn Y] y (-2) ■sgnX ,a. '., j j j sgnx.a;., j jj sgnX. lP . ., "\ sgnX.a. . . J JJ sgnX.. e «, Then if ,(m) = 'ij we have ("0 .. M (m) c (m-l) , V V c (£ a. . s;t = m: / s; ij ij s,\, ij ij i=x r JJ 1J (42) Let t.-T J IJ -k. (t.-HT) J t.-T J (-t.+T) -k. J 1 2 2 -\ . 1 2 Clearly, B.. A..=k7-k., i.e. B . . = A. . (k . - kT) . Therefore, multiplying ' lj ij i y ij lj i j ' 5 both sides of Equation (42) on the left by B . . , we have 40 K i J ij ij ij X J ^ y X J JJ X J (X) e (m-j&-l) (43) When k ^k , we can divide Equation (43) by (k. -k.) to obtain the desired i J i J recursion relation. However, when k. = k., we can replace m in Equation (43) by m+1, obtaining after some rearranging, X B..M< m+1) S« = IJ IJ IJ jPl j 1 ' ' EE^as" This can be added to Equation (42) to give (A. . + B. . M? m+1) ) S. (m) =M< m) S^~ l) + £ Z (C?S S (m - lj ij ij ij ij ij £=1 j* n ' ' P.) (r -4-1) _ fi c (4) gCm-^v ij jj 1 ij 1 (44) If A. .+B..M. . is nonsingular whenever A., is singular, then ij i] ij ij ; Equation (44) will provide a recursion relation whenever Equation (43) fails. Since we are interested in the case that A. . is singular, ij & ' k. =k. and, of course, t. =t.. Therefore, in considering the matrix A. . +B. . M ij IJ ij (m+1) we wi 11 write k.=k.=k, t.=t.=t, and t 4- j = t. . Then, i J ' i J ' + ' A..+B M< m+1) ij ij ij ■k(X-fm-l) t + (X+m+l) T Y k(X-hn-l) TY ■t (X+m+1) t_ (-X+m+1) t fry T Y -k(-X+m-l) t_ (-X+m+1) k(-X+m-l) T Y 41 and we find after considerable algebra, (A.. + B..« (m+1) )" 1 =^ ? - J J 4k m r ry k(m-l-X) k(m-l-X) Ty t (m+l+X) — ty ^n t + (m+l+X) t_(m+L-X) y T V -— TY -t_(m+l-X) TY k(m-l+X) / - T Y / ■k(m-l+X) -ty / J Thus Equation (44) yields a useable recursion relation whenever A. . is singular and m*0. Finally, when m = 0, Equation (42) gives us the following condition (0) on S ij A.. S< 0) =0 (0) If A., is nonsingular, this simply states that S.. =0, but when A., is singular, there are nontrivial choices of S.. which can be made. In face, there are two linearly independent solutions which may be taken as the first two columns of B... Thus altogether, there are 2N(f,p) sets of independent solutions to Equations (41). We distinguish among these according to their behavior at infinity as follows t L.tf C^ ] (p) ~ sin(k nm P+Y TT^ 1 £n2k p- -*- ) 6.. R p^oo n(l) k n(i) n(l) 1J FJ J (P) n(i) C n(i) + T cos(k p+7 n(i) -niil n(i) in 2k n(i) L.77 42 and GJJ(p) ~ cos(k p+ Y ^to2k p--i-)6 p ->co n(i) n(i) Because these functions play a role in the general theory analogous to that played by the relativistic Coulomb wave functions in the single state monopole theory developed in Section 2. 2, we have used the sub- scripts R and I indicating regular and irregular as was done there. These do not, of course, imply anything about the behaviour of these functions at the origin. The regular solutions correspond to choosing S.. proportional to the second (or third) column of B., while the irregular solutions correspond to the first (or fourth) column. Equations (40) and (41) have the same asymptotic behaviour. There- ii 1 ii' fore, we can find constants B_ , C. such that fp' fp /4>> ! -' i . Z b- N R(P) g fp (p> \ G f P >> or equivalently, + C fp F^' J T (p) fp,I K> \ 8 fp (p) ~ A f P Sin(k n(j) p + Yt -J&ttl , n2k h&ijm +a JJ } k n(j) n( J> 2 f P *> ^-Ai j co S (k ,. n + lisiii jta 2 k t ,. N , "fp n(i) k ,.. p -»» n (j) +t F vjy n(j) L(X(j))77 + ij n(j) P 2 +C V 43 with partial wave amplitudes and phase shifts given by $-[<«# 2 +eg> 2 i / C 1J • 6* j - tan" 1 -ft] • 2.4.4 Cross Sections and Scattering Amplitudes If we consider scattering of Dirac particles from a nucleus originally in its ground state which we take to be |0M), then the spherically incoming portion of the incident wave will be "&. (p) |0M). We must now choose the constants a in Equation (39) so that the fm f p spherically incoming part of S&Cpjv) is asymptotically equal to this. Therefore, we must have ~ , , -I ;t +T\* L^(2*+D ^e ° k ° ° [f—\ |xO>|*m |0M) P=0 Z1 o P o ' S t . L.77 .. ij -i(k ,,-p + v ," llj in 2k p - --j- + 6^ J ) v-i • A r n(j) k ,. N n(j) 2 fp V o 1 _l£ n (j) lr -\ ,> fm.p 2ip ' f r fm p f r ij Multiplying this on the left by r fm f p jk MM MM s s J ( L, * J, \ M S M J W^ |iM )|r V k L. | J r _ k_ - k r+ T M L M S Mj L (to)|iM s >|rM I > "l-f (45) with M fpr I ,v4tt • c ik 1 , Jl .lk fr> ^rrj— h-i A. e rp 6 6 2ik fp fp on. rn. o j k The corresponding differential cross section is m M . , v s k mM . m M dO.)^ r = iT (f o^r (9CD)) f o^r (9 ^ 45 Averaging over the possible initial spin states of the scattering particle and the nucleus, we arrive at the differential cross section for unpolarized scattering: m M idQJ o-*r 1 2(21 +1) o y (is) s s o ->r 2.4.4.1 Simplifying the Scattering Amplitudes As in the monopole cases, it suffices to consider only the upper pair of components of any scattering amplitude. The lower pair then a- £. ( 9co) follows from the application of the operator — — . Let r m MMM_ _ _ _ tIt t t il t l t sIS _ .. J.I f L. * J. J, If L. t? J. f (ft) = Z M i \ \ ; 2L.+1 C c: fpr i m M m,. Omm M T M_ r m r . ft M n M T o-*r fp K J s f ss JlfLSJ jk -iM cp x e Vl^' Then by applying the identities „ J 1 J 2 J 3 , .. J 1 + V J 3 /l J 2 J 3 - M l" M 2- M 3 M l M 2 M 3 and M -2iM cp Vm/** = (_1) 6 L Y LM, (9fD) we find -m -M-M-M„ I -I +L,-L.-M, m MM T M n f s 1 S ( ) = (-1) o r k j T, f s I S (Q) o ->r o ->r 46 L k -L. Because mJ contains the factor 6 nn 6 , (-1) J = pP pP = P P fpr on. rn, r o r o and we have finally -m -M-M -M„ I -I -M m MM M f s 1 S (9)=P r P o (-D ° r X f S 1 S (9) . o -+r o -*-r Therefore, we can express all of the scattering amplitudes in terms of h MM T M n IS the numbers f (9). In particular, the unpolarized cross section o -*r is - — ,t k v ..t +T , „ |MM_M 9 JLjis '— —j, Z |f X S 0)| 2 • (46) dn ! It +r \t k / (21 +1) . \ r The first factor on the right of Equation (46) comes from including the lower components. 2.4.5 The Zero Mass Limit Let f f (p,T) and g f (p,T) be the solutions of Equations (40) and i i i i let f (p,-T) and g f (p, -r) be the solutions of Equations (40) with T replaced by -t . Then, one may easily establish that 8 f J -p (p '~ r) = (s S n V f fp ( P^ T) • If we now use the asymptotic forms derived earlier, we find ij Y t ( .. L(X(j )) ij A f _ p (- T )sin(k n (J) P +T ^ta2k n (J)P-— F^^f-] P ( - T)) n(j) k n(i) ii Yt n(i) L(X(j)) - r^ A fp^> s ^ cos( \ (J) ^i^^ 2k na)^ — r- - +6 fp (T)) - 47 Therefore, A " J - (. T)= AOI^ A j j (T) f-p t n(j) 4T fp S f J -"p ( ^ )= 4p (T) • k nC) In the limit that T tends to zero, the ratio L-- approaches one. i j . . i j We have then that A £ " (0) = A*°(0) and 5 , "* (0) = 6* J (0) . This means f -p fp f-p fp that in the zero mass limit, we need only solve Equation (40) for one value of p, say p = 1. We can then use the result to determine the phase shifts for p = - 1 . ]k From the definition of H. we also have fpr M r " (t=0) =M-] K (t=0) . (47) f -pr fpr Using the relations following Equation (29), we see that L.ij. \J 2L.+1 C J i i = -sgnX #v /|x.| . Therefore, in the zero mass limit, both the first and third components of f*~ (0cp) are given by o -*■ r J.I f J, I f Z cj ° C k r ' sgnX J\X I M jk f f k \ M m f Mj M ] . m f g J V • J ' fpr p-1 X ( ' ^H ^^""X'^J \M L (to)) l rM I> On the other hand the first component is (9) e \ M M \ iM cp 48 and the third component may be expressed in terms of the first and second components yielding i M M _ £ i^cp |MM-| iM r c P cosG f (9) e + sinB f (6) e Equating these two, we find that f (fi,T=0)=tan | f (9,T=0). (48) Thus in the zero mass limit we have only half as many independent amplitudes. The differential cross section is tk. t +r 1 + tan 2 ? „ %MM * (I wj ■ (in?) (iV: tt^ J l f J (» I • («) \ ' n _* r v r'nn n MM I 49 3. THE ELECTROMAGNETIC INTERACTION 3. 1 Introduction Information about the detailed structure of the target nucleus is communicated to the passing Dirac particle by means of the electromagnetic field which surrounds the nucleus as a result of the charges and currents which it comprises. This electromagnetic field is embodied in the matrix elements s . (p) and v..,(p) defined after Equations (2.40) in Section JJ JJ 2.4.2. The purpose of this chapter is to express these matrix elements in terms of matrix elements of the nuclear charge and current density operators. In Section 3. 2, we obtain multipole expansions of the nuclear matrix elements of the charge and current density operators. Section 3.3 displays a similar set of expansions for the electromagnetic scalar and vector potentials and by means of Maxwell's equations relates these to the charge and current densities. Finally, in Section 3.4, we actually determine the matrix elements of the interaction, s . . , (p) and v..,(p). Section 3.5 deals with the actual method used for specifying the multipole charge and current densities. While this section logically belongs in Chapter 4, it is included here so that all of the equations relating to the densities are collected in one place. 3. 2 Charge and Current Densities The electromagnetic structure of the target nucleus is completely determined by the matrix elements of the nuclear charge and current 50 density operators C(r, v) and ?(r,v). We denote these matrix elements by c nMn'M' (?) = (nM | c( - v) | n . M> and f Mn ' M ' (?) = ( nM | ?(?, v ) |n'M' > , respectively. When the possibility of confusion is remote, we omit the superscripts and write simply C(r) and ?(r) . The exact nature of the density operators and their matrix elements is not of immediate interest to us but it is important to point out several properties of a general nature. The first of these is that C(r, v) and ?(r, v) are Hermitean operators from which it follows that C n ' M ' nM (?>= [C nMn ' M '(r>f (U) and J (r) = [J (r)] . (lb) The second is that the matrix elements satisfy a continuity equation expressing the conservation of electromagnetic current in the nucleus. Thus, v* • J(r) -ia)C(r)=0 (2) with to = e , - e n' n Finally under time reversal T, we have TC(r,v) T _:L = C(r,v) (3a) and TJ(r,v) T _1 =-J(r*,v) . ( 3b ) 51 3. 2. 1 Multipole Expansion of the Charge Density Since the matrix element C(r) is a scalar function of r, we may obtain for 1 t an expansion in terms of spherical harmonics. Thus we have , , s V / 1s m J'i I i ,n n' . N „ . „ . C(r) = [_, (-1) C -zrz: C (r) Y (to) . (4) to M'-mM /2I+1 * l The Clebsch-Gord.m coefficient in Equation (3) is present as a result of the Wigner-Eckart theorem. Inserting the time-reversal operator times its inverse before and after C(r,v) in the matrix element de- fining C(r), we have, since T is antiunitary n M n' M' -k i -l -► -l i C (0 = (nM|(T L TC(r,v)T T|n'M' \) = [(\'nM|T" L ) C(r,v)T|n'M' )]* (5) m ( _ ir I-M + I' + M' (c n-Mn>.M' (r1) * But, from Equation (4), we have n-Mn'-M' ft .. V ■ ■ C I- | I J_L n n - * . V -M' -m -M T7TT I _ ™ Cm \. 21+1 I I we now change the sign of the dummy index m and use the identity -M' m -M " ^" i; V -mM we find 52 v 21+1 Comparing this to Equation (5), we find that the reduced matrix elements C nn (r) are real. Hereafter, we refer to these reduced matrix elements I as transition multipole charge densities. Now, from Equation (4), cV)= l(-D m i\ l l 4^T < n ' (r) Y, (Bcp). im M " mM V2I+1 l ^ m Changing the sign of the dummy variable m and noting that V HI . n I' -I+m 21+1 I i I' M' mM K " ' \ 2I'+1 M-mM 1 we have £m \ 21' +1 Comparing this with Equations (la) and (4), we see that C7'(r)= (-l) 1 " 1 ' «c»" n (r). (6) In order that C(r) be well behaved at the origin, we must have that for small r, C (r) goes like r . Since the nucleus presents a localized system of charges, the matrix elements, C(r), as well as the transition multipole charge densities C (r) must decay exponentially for large r. Integrating the matrix element C(r) over all space, we have from the orthonormality of the nuclear eigenstates, T C(r) d r = Ze 6 , 6 MM , •! nn MM' 53 This gives us the following condition on the transition densities when & = , 2 nnV % . /2I+1 P r' C^(r)dr= /^± Ze 8 , . (7) .1 v J V 4tt n n 1 o 3. 2. 2 Multipole Expansion of the Current Density The vector spherical harmonics, defined by ^i<* )= I c nf- ii. V (W % (8) m' u in which the spherical unit vectors e are given in terms of the cartesian unit vectors e ,e , and e by x' y' z 1 — /s /. e . i = 7^ ( + e - i e ) +1/2 x y e = e , o z ' form a complete orthonormal set for vector functions on the surface of a sphere. A number of useful properties of the vector spherical harmonics are collected in Appendix . If we take the complex conjugate of both sides of Equation (8) we have (Y.Vecp))*- I cj X J Y* ,(9cp) e* . J £1 ^ m V m' u m jgm 1 ^ u But Y , (8co) = (-l) m Y _ m ,(6cp) and e" = (-1)^ e_ . Therefore, by changing the signs of the dummy indices m 1 and u, we have, using the identity 54 A J 2 J 3 ri /l + V J 3 C J 1 J 2 J 3 C M 1 M 2 M 3 = ( - 1} C -M r M 2 -M 3 > (^ 1 (ecp))*=(-i)" +1 ' j4m Y-^ 1 (ecp) . (9) — > ~* m It is customary to write X„ (9cp) for Y -(9cp) when the other harmonics j£m HI axe not being used. The current density T(r) may be expressed as the sum of a transverse T — ► T part J (r) and a longitudinal part J (r). Thus, J(r) = J T (?)+J L (r) (10) with V- J T (r)=0 (11a) and 7xJ L (r) =0 . (lib) The continuity equation becomes V- J L (r5 + icu C(r) =0 . (12) Equations (lib) and (12) together essentially determine the longitudinal current density completely. In fact, taking the divergence of Equation (12) and using the identity V(V- V) = V V+ V x (V x V) , we have v 2 j L (?) = + iU)VC(r) . Since the longitudinal current must vanish at infinity, this determines it completely. 55 rr We may now expand ? (r) in a series of vector spherical harmonics, obtaining j An j j where the phase factor i ensures that the matrix elements J. „ (r) be real. This is easily verified using Equations (3b) and (9). From the — * — ► fact that J(r,v) is Hermitean we have, in analogy to the case of the charge densities, r nn* . , ..vl-l'+l-j T n'n. . .... J \£ (r) = ( "*' J \£ (r) * (14) 3.2.3 An Alternative Expansion of the Transverse Current Density The expansion of the transverse current density presented in the previous section is especially useful when one is concerned with the coupling of the current to various angular momentum eigenstates. It is cumbersome, however, for some of the manipulations that we shall have occasion to undertake presently. For example the simple fact that J (r) is transverse is not at all clear from Equation (13), but must be injected through an additional relation that necessarily obtains among the reduced matrix elements. Therefore, in this section we develop another expansion for 7 (r) which retains much of the form of Equation (13) and yet from which the transverse nature of 7 (r) is immediately evident. For any function F(r), V. (F(r)X, (9cp))= . Also, it is clear that '&m ■=t zt -* -*r -* ~*c -* -«m -»• v. (V X (F(r)X. (9cp)))=0. Therefore, we write J (r) = J (r)+J (r) with am 56 jjH(x) =^xjT(r). We then expand both J (r) and iT(r) in terms of X„ (0cp). We have ^(a- £<-»■£ ii ^jT'wVm (i5) and (?) - 2 (-D m £ i I 7*& C' M x e (ecp) . (16) £va M 1 -mM s/21+1 -2 An Because of the way in which |i(r) contributes to J (r) } we think of it as a magnetization density. Hereafter, we refer to the reduced matrix elements J» (r) and \x„ (r) as the transition multipole current and magnetization densities respectively. Using the properties of the vector spherical harmonics from Appendix 1, we find that Jj5' (r) = J7' <*) (17a) T nn' , s £ [ d £\ nn' , . ,,_.. J ii+l (r) = V2mldr " r) ^ (r) (17b) T nn' t . l+l / d , ^t-l\ nn' , x ,„„ . J £ i-l (r) = • V2il idr" + ~ ) ^ (r) " (17c) 3. 3 Maxwell's Equations We now wish to consider the scalar and vector electromagnetic potentials cp(r) and A*(r) which are generated classically by the charge and current densities C(r) and J (r) . We arrive at these through Maxwell's equations which are V 2 cp(?) + iau 7 • A(r) = - 4tt C (r) V 2 A(?) + u 2 l(r) - t . (7* . t(v) + icu cp(r) ) = - 4tt ?(r) where again, ou = e , - e , n n 57 3.3.1 The Coulomb Gauge From a given charge density, C(r), and current density, J(r), Maxwell's equations do not uniquely determine the electromagnetic potentials cp(r) and A(r). In fact, for any function F(r), the potentials cp' (r) = cp(r) - icuF(r) and A' (?) = A(?) + V.F(r) satisfy Maxwell' s equations whenever cp(r) and A*(r) do. Transformations of this type are referred to as gauge transformations. When one chooses a gauge, one specifies additional constraints which the potentials must satisfy. It is convenient here to choose the Coulomb gauge for which the additional constraint is that the vector potential, A(r), be transverse. Thus we have V-A(r) = . (18) It can be shown that for the Coulomb gauge, Maxwell's equations reduce 3/ to V 2 cp(?) = - 4tt C(?) (19a) V 2 A(r) +w 2 A(r) =- 477 J 1 (r) (19b) J T (r) = J(r) - J L (r) . (19c) 3. Jackson [ 6 ], page 182. 58 To make the scattering problem computationally tractable, we must neglect ou in Equation (19b). This is equivalent to replacing the four- momentum in the tranverse photon propagator by the three -momentum. We shall also neglect J_ (r) in Equation (19c). Both of these approximations are expected, by Born-approximation arguments, to affect scattering cross sections only in the forward direction, where jq| >-^kt(*-f1 ^ n ' (r) A nn' , s J+l/d ^+1 .Jin' N ,.. N A ^-i (r) = -vIm^d7 + — rf i (r) • (24c) 3.3.3 The Radial Maxwell Equations Substituting Equations (4), (15), (16), (20), (22), and (23) into Equations (19a) and (19b) we have by orthogonality of the various spherical harmonics involved the radial equations ~2 dT r dr" V r) " ""2" V r) = " 4ff C £ (r) (25) r r 1 d 2 d . , % l(H-l) . . . /„ t / n ,o^ - dT r dT V r) " 2~ V r) = " 477 V r) (26) r r 1 d 2 d x, t n -g(^-l) UM /„ , V "2 dT r dT V r) " ~2 V r) = " 4ff ^ (r) ' r r (27) In deriving Equations (26) and (27), we have used the fact that w=0. Actually, Equation (27) need only be the third order differential equation resulting from the statement that Vx (V^r) + 4ff£(r>) = . Any solution of Equation (27), however, will certainly be a solution of this less restrictive third-order equation. We are justified therefore in simply using Equation (27). 60 Equations (25), (26) and (27) all have the same form. Since we wish each of the potentials to vanish at infinity, we have the following solutions 1 »,w --sSki-k f c /*> «** dx+r ' ! c / x) xl "' dx } < 28) J 1 V r) " 1*1 1~m I J i (x) x dx+r J J / x) x dx r (29) r r I 477 fx £+2 1-J.l V r > =firal V x > x dx+r I ^ (x) x dx r (30) j Thus each of the multipole potentials, cp»(r) A„(r), and M.(r) behaves at the origin like and (24c), we have H - £-\ at the origin like r and at infinity like r . Using Equations (24b) nn* . / 1 A ££+V V) '"'' ' * ff v ; 2^-l £+2 J A *£ r Ki J M x) x dx and nn £ £ _ x (r) = - 4tt J ■£+l £-1 2M-1 r l-£ J M-^(x) x dx . (31a) (31b) 3.4 Matrix Elements of the Interaction The matrix elements S nMn M (r) = -> I V (r) = (nM|V(r,v) |n' M' > of the interaction given in Equation (2.37) are related to the potentials by S(r) = ze cp(r) V(r) = ze A(r) 61 in which ze is the charge of the scattering Dirac particles. In this section, we will use this relationship to express the matrix elements s£ feI (r) = (31) and f v kk , (r) = (f m f p k jsgn)^, \P(r, v) • ?| f m f -p k^ > (32) .nn in terms of the matrix elements cp„ (r) and A. * (r) defined in the Section 3. 3. Using Equation (2.38), which defines the kets |fm pk), we have kk &M I MjM L M s Wj "jVf Wi M jVf M 1 M A M S XC «i-»i J ^T tP " n ' J' Y LM L (n)Y * (n)Y L'^ (r;)dn The identities Jl J2 J 1 - H)^^ £ ^ J 3 "l M 2- M 3 v2T^ M 1 M 2 M 3 and (2i 1 +l)(2£_+l)(2X_+l)r/i. ^M/^i i,i, 11 22 33 \ / 10 0/lm m m (35) al low us to write 62 3 £ k , (r) = ze ^ (-1) L+J+L,+J ' " i " I " 1 (2f+l) ((2L+1) (2L'+1) (2J+1) <2J»+1) (2H-1))* i^ L '" L /L i L'\ , n nn' , x — =" l0 <"•> *J (r) n/4tt with m+M +M +M'+tL.+2m M jWs MLM1M! m x /l 4 J \/ J Iflfl'l l\/ L IV \JV Ij'Wj'I 1 t i\ M s - M j/\ M A*'»f/\ M i-'» M i/K mM L/\ M L M s M y\ M ? i i- ffl f With the aid of two more identities, the first of which may be considered as a definition of the Wigner 6-j symbol { J l J 2 J 3, \ J 2 J 3 I ( _ 1) J 1 +J 2 +J 3 + V M 2 +M 3 M 1 M 2 M 3 m l m 2 J l J 2 j 3! M, -M„m J 2 J 3 j l J 3 J l h\h j 2 j 3 1 - M 2 m 3/ \ M 2 " M 3 m l/ \ M 3 " M l m 2/ \ m l m 2 m 3 (36) 1 J J 1 J 2 J 3 Jo Jo ™*K 2j_+l [J, J. J '3 J 3 3 3 2 J 3j and I ( . 1) J l +J 2 +J 3 +M l 4M 2 HM 3 M M M 12 3 J l J 2 j 3 J 2 J 3 j l \ M 1- M 2 m 3 /\ M 2- M 3 m l J l J 2 J 3 j l j 2 j 3 \ m l m 2 m 3 / \ J l J 2 J 3; J 3 J l h V M l m 2 (37) 63 we are able to carry out the indicated sum. Briefly, we use Equation (37) to perform the sum over IL , M|, and M , after which we use Equation (36) to perform the sum over the remaining indices. Introducing the result into the equation for s,,, (r), we have sf k ,(r) =ze(-l) f+I "2 ((2L+l)(2L'+l)(2J+l)(2J , +l)r ' L I L'l I J J' x (38) Y~ l) loo oJ|l-l mJi'i fjvhr^ (r) _. nn' , . , , N I-I'+i n'n, N . Since cp (r) * (-1) cp. (r), we have ■£,■<*> - s* = sgnx sgnx' (fk_|s(r, v )|fk^> or 64 s£ k , (r) = sgnX sgnX' s£ fcI (r) . (39b) We turn now to the vector portion of the interaction. Using Equation (21), we have £,(.)-.« sgnX- I I c-i) m tf* J c M J J S"J "J"l m f linr MLM'X M 1 V 171 S 1/ t J' J' I 1 f I 1 j I i 1 j C* C* C* f 1 X MlM'M' M j M i m f Mj-mMj m 1 |i m -0+U-L 2 ^=f A j" W<*" S l°| i l* M S> J"*!^/® V (n) V^«» dfi The Wigner-Eckart theorem tells us that But -l-.«|JJ- % Therefore (40) <|M s |a^| 4 M-> = /3C* J* . o S Using this together with Equations (34) and (35), we have v kk . JJ-J 1 j = | |X. | ~ |X, , | |. The behaviour of cp„ (r) near the origin then implies that s, , , (r) tends to 1.1 Vi - IV' 1 1 zero as r tends to zero at least as fast as r A similar result follows for v., , (r) . 67 3. 5 Specifying the Transition Charge, Current, and Magnetization Densities In order to make calculations of the scattering of Dirac particles from a nucleus we need a general method of describing the charge, current, and magnetization densities that characterize the nucleus. In this section we present a method which while allowing the specification of virtually any functional form for the densities also makes it possible to perform the Integrals indicated in Equations (28) through (31) analytically. This is of great importance in the general scattering problem since it obviates the need for storing the potentials at a large number of radii. We define the following matrix elements. nn' li+l nn* . 5 2 (r) = e "~UF *£ (r) (44) and vff(,).e(-i)', (ziyi) A „„. _ (45) Then, using Equations (28), (29) and (31), we have (omitting the super- scripts where convenient) S £ M' e J?M '-fe J' C i< x > « i+2 dx + r i f Ci (x) * 1 '*4m \ (46) v |(j (r)-e(-l)V4irJ-j^ i f J p (x) x £+2 dx + r £ J J £ (x)x l ' £ dx> (47) f-fl 1 « 9A-0 v i+i (r)=e(-l) ^ 4^(2^+3) • -j~ J U; (x)x^dx. (48) r Since these three are quite similar, we deal in detail only with the case of the charge multipoles. The results for all three cases however, will be presented. 68 We define the strength s - of the transition potential s^(r) to be the coefficient of r 'in the asymptotic expansion of s^(r) for large r. Thus ■i-1 V r ->°° a/r) ~ s^r Since s ,(r) has the units of an energy (which since we choose h=c=l JU -1 & is (length)" ), the strength s^ has units (length) . In the special case 2 that i = and n=n' , and if furthermore 1=0, we have simply s^=ze =y . We express C„(r) as the product of s and a shape function which is suitably normalized. Thus, we put _nn' , N nn 1 -norm, . ,, Q . eC Z (r)= s i F w (r) (49) with «°° ^norm. . Z+2 , J2M-1 , Kn . ! F w < r > r dr = 24+1 -x _ /x\ . /x\ -% - c x e f - f . it x dxa5 , . •J m \2) m 1 \2) mm' Now the polynomials which are orthogonal on the positive real line with 2,0+i _ x (X weight x e are the generalized Laguerre polynomials L (x) with 2 a =24+? (22.2.12 HMF). Accordingly, we set f ((r/a) ) equal to 2 j? 1 1 2 L ^(2(r/a) ). Equation (52) becomes m F.(r)-rV< r /*> I«A(r/,) ! ,. (53) * _ m m m=0 For a given charge density C.(r), we have 00 ; 4rr r 4+2 s - = e £ m V 2^+1 J r C^(r) dr (54) m m! P 34+2 -(r/a) T 24+? , 0/ / . 2. _ , N , ,--,. 1 m = T(2^+3/2) I r e L m (2(r/a) )C^(r)dr. (55) m The shape parameters a clearly depend on the length parameter a. In practice one chooses a so as to minimize the number of coefficients a necessary to achieve an adequate representation of C„(r). Thus for £ -r 2 example if C (r) =r e we need only one coefficient a if we make the wise choice of a=l, whereas choosing a=1000 generates the need for many coefficients. It is clear from Equation (7), that in the special case £=0 and n^n' , the normalization indicated in Equation (50) is impossible. We replace Equation (50) then by 70 it 4 ^norm, N , n 4 norm. N , , IN J r F 0nn' (r)dr = I r F 000 (r) <^ n > (56) where F _ (r) is the shape function for the ground -ground monopole charge density. While Equation (55) still applies in this case, Equation (54) must be replaced by p 4 ^nn 1 . x , J r C (r) dr 00 U nn s n = s °° 4 ,.00 (n^n') . J r c < r > dr u (57) The corresponding results for the current and magnetization densities are V r) .~„ V j^ r " ' ' r -*°° t /„\ _,norm . . e J / r) = V U G i < r > ,"1 m=0 m m v ii = e £+2 (-1) \ 47T J r ^ J,(r)dr (58) (59) (60) (61) a = m: .3^-2 -(r/a)%24+| m r(2^fm+372T I r " L^ (2(r/a) ^) J/r (62) norm. e ^ (r) - v i m H i < r > (63) norm., _ ( i+1 / 1 H * - X. > J a.<°> - 1J h (0) - * b ij " 6 ij x. < J The series in Equations (2) are then used to generate values of f..(p) and g..(p) to be used in starting a numerical integration of Equations (2.40). The particular numerical integration scheme used in both NADIR and ZENITH for the bulk of the integration inside the nucleus is Hamming's fourth-order predictor - corrector method. Each step of this method requires one derivative evaluation plus knowledge of the functions and their derivatives at the four previous integration steps. The initial values for the functions and their derivatives are created by making three fourth-order Runga-Kutta steps starting from values generated by the starting series. The question arises of where to evaluate the starting series in order to achieve a desired accuracy at the end of the integration. Equations (2.40) have 2N(f,p) independent solutions of which half are regular and half are irregular. Although Equations (2.40c) specify a particular set of regular solutions, this set has been chosen in a purely arbitrary manner. Any other set would serve as well. We can see this as follows. Let f' (p), g' (p) be any set of N(f,p) independent regular solutions fo Equations (2.40). Then we must have 76 f (p)-fij(P)- Z a iihi (p)=af (P) g' (P)=gij(p)= Ea i£ g £j (p)=ag(p), the matrix, a, being nonsingular. Refering back to Section 2.4.4, we see that the cross sections do not depend directly on the partial wave phase shifts and amplitudes but only on that combination of them which makes up Mi defined after Equation (2.44). But if we go through the algebra v fpr of that section using the new set of solutions we find that ik (M J ) = /, 477 ~rr. — h (a ) „ a . A e 6 6 *—• 2ik i£m o 2ik_ in mi on. rn. V / 1 , J 1 A ik x 6 . c = / / 477' -rr; — h A e 6 6 V 2ik on . rn. l o j k = M jk Therefore, the cross sections do not depend on the particular set of regular solutions that we happen to choose. Our only problems then arise from mixing in some of the irregular solutions. To simplify the discussion let us consider the case in which N(f,p) = 1. Equations (2.40) will then have one regular solution g (p), f R (p) and one irregular solution g T (p), f (p) . We choose the irregular solution so that asymptotically, the phase difference between the regular and irregular solutions is tt/2. We wish to evaluate the 77 starting series at a radius p such that the solutions g(p) and f(p) which we obtain numerically are given by g(p) = c g R (p) +d gjCp) f(p) =cf R (p) +df ] .(p) with | — J " (t n +T) F x„ * p F x„ ■ (5b) 79 These equations may be solved analytically in terms of confluent hyper- gymetric functions and, as was shown in Section 2. 2. 3, the behaviour of these solutions at infinity may be deduced analytically from their behaviour at the origin. Therefore, if we possess a complete set of independent solutions of Equations (5) at the nuclear radius, we can determine the asymptotic behaviour of f (p) and g (p) immediately. The problem of integrating Equations (2.40) from the edge of the nucleus to infinity then, is equivalent to that of determining a complete set of independent solutions of Equations (5) at the nuclear radius. When N=l, and when the mass of the scattering Dirac particle may be neglected in comparison to its energy (i.e. T«t =1) the only parameter present in Equations (5) that depends upon the target nucleus is y- Since there is only a small number of nuclei for which scattering cal- culations need be made, it then becomes possible to compute the relativistic Coulomb wave functions at some convenient radius once, for each of the corresponding values of y, and store them in a dictionary of some sort to be used when required. Each set of Coulomb wave functions will be used in a number of calculations in which details of the nuclear charge distribution are changed each time but y remains fixed. Therefore, the time consumed in computing the Coulomb wave functions is averaged over the total number of calculations in which these functions are used. This means that when N=l and r=0 the calculation of the Coulomb wave functions is not critical. Typically it is done by integrating Equations (5) from the origin to the desired radius in much the same manner as the integration of Equations (2.40) described in Section 4.2.2. 80 However, when N>1, and/or T/^0, Equations (5) involve two or more parameters, several of which one may wish to alter from run to run. The calculation time for the Coulomb wave functions then becomes an important consideration and it is desirable that it be reduced to a fraction of the time required to integrate Equations (2.40) to the edge of the nucleus. To achieve this reduction in computation time we have taken advantage of the simple form of the right-hand sides of Equations (5) (as compared with the right-hand sides of Equations (2.40)) by using a first-order numerical integration scheme of high degree. Specifically, we expand the functions G(p) and F(p) in powers series about points p. succes- sively farther away from the origin until the radius at which the Coulomb wave functions are desired is reached. The details are as follows. Omiting the subscripts, we let F (p) = p }j a „ P n n=0 n 00 G(p)=p X £ b p n n=0 n which we substitute into Equations (5). Equating the coefficients of the lowest powers of p to zero, we obtain the indicial equation s x - X - y 1 1 2_ 2 S S , ,,,2 2. 2 the two solutions of which, s v = (X - v ) and s v _ = - (X - v ) X-, K X, I correspond to the regular and irregular Coulomb wave functions respectively. Equating the coefficients of the remaining powers of p 81 to zero yields the recursion relations a n = " ^M^ (Y(t+T) Vl + (S X^ +X > t'-D b „-l) b n " ^V ««X H - X ) (t+T) a n-l " Y(t-T) Vl> " Y a Finally, b Q = ^- . Although these series converge for all values of p, the limited precision of the arithmetic performed by digital computers makes them unusable outside a small neighborhood of the origin. We can, however, obtain from them F(p) and G(p) for p equal to, say, p . Let x = p-p - Then Equations (5) become (X + P y \te " x^T/ F W + ( x+ P )( t+T ) G W = -YG(x) (6a) (x + p o } id^ + ^~, G(x) " (x+p o )(t+T) F(x) = yF(x) ' (6b) Equations (6) are not singular at the origin (of x) and therefore we may write (x) = l c x n n n=0 BO G(x) = V d _ x n n=0 82 Substituting these expressions into Equations (5), and equating the coefficients of the various powers of x to zero, we find that c n+l " l^kr n f< x -n)c n - (v+P (t + r))d n - (WDd^} and d j.1 = i iK {-(X-hi)d +(Y+P n (t-T))c +(t-T)c J . n+1 (n+l)p n n n-1 Since the only singularities of Equations (6) are at x = - p n and x = ro , these series converge whenever |x[

\-)p5" 8 jj- 84 problem of integrating Equations (2.41) from the edge of the nucleus to infinity. In Section 2.4.3 we derived asymptotic series for the solutions of Equations (2.41) which converge to the desired accuracy whenever p is greater than, say, p . Unfortunately, one cannot relate these 6 ' ■' asm ■" solutions to the solutions of Equations (2.41) in the neighborhood of the origin by analytical means. It is necessary to continue the integration of Equations (2.40) or equivalently Equations (2.41) from the edge of the nucleus (p = p ) to p by numerical means. Since p is max asm asm frequently several times as large as p this region of the integration calls for some close attention. The simple form of Equations (2.41) as compared to Equations (2.40) suggests that we might be able to use the power series method again. £+1 If we multiply Equations (2.41) by p " and let x = p - p n , we have, writing L for £+1, L I d (x+p o> idx - o^y.i plJ(x > + (x+ v = -Z Z(x + P ) X - £ {af.,F i i'(x)-p«,G iJ '(x)} 1=0 2 u JJ JJ (7b) 85 These series are used to perform the integration of Equations (2.41) from p = p to p = p in several large steps in a much shorter time max asm than is possible with a simple extension of the methods that must be used to integrate Equations (2.40) from the origin to p = p ° *> or- 'max 4. 2. 3. 3 Convergence of the Asymptotic Series For a given value of p, successive terms in the asymptotic series defined in Section 2.4.3 pass through a minimum and then grow without limit as m tends to infinity. We must determine a value of p such that the minimum term in the series is sufficiently small in absolute value that the sum of the series to that point has an acceptable accuracy. To be explicit let us assume that we wish to have the minimum term in the series less than or equal to e in absolute value. From the form of the recursion relations given in Equations (2.42) and (2.44), we see that for large m, ,(m), 2 Is:" 1 = ar(mb + c) 1 JJ Then if the asymptotic series is evaluated at p = p , the minimum term asm occurs for m such that (mb + c) = p and its value is r asm a T(m b + c) / , n bm (mb+c) We desire that this be less than e . Therefore we solve the equation a T(m b + c) (mb+c) bm for m and choose p > (mb+c) . It is necessary, of course, to know asm * ' ' the values of a, b, and c. We estimate these as described below. 86 Although Equations (2.41) in general represent a set of coupled pairs of equations, we neglect the coupling in determining p . We then use the corresponding recursion relations to determine three successive values „ (^(nOi . (m) 4 (m-l) . (m) , .(m+1) „ . . , , of |S;/|=A , say A , A v ', and A . Writing z = mb+c, we have, using Stirling 1 s approximation for the logarithm of the gamma function, . (m+1) (z + b - §) £n(z+b) -z -b (z - ■§) in z + z = in r-r- A^ m; Expanding the logarithms in powers of b yields (z-*)£ - -—+••• } + b{inz+~ - \- + I}*- in ^t-t- z ^ 2 z 2z . (m) 2z' A or b(j>nz - -f.) + b 2 -i + 2z 2z = in (m+1) 7(mT" Changing the sign of b, we have b(inz - ~) - b 2 ±- + 2z 2z (m) in (m-1) Therefore, neglecting terms of order l/z , 1 A (m+1) bUnz- ~) = | in A , - 2z *( m "l) = c. u 2 1 b - = in z A (m+1) A (m-1) A (m) A (m) (c 2 ) and finally y z in z 2/¥ 87 This equation may be solved for z from which a, b and c follow easily. We use essentially the same method to estimate the step size which may be taken in one iteration of the power series method. Here the problem is the reverse of that with the asymptotic series. For a given value of x ; successive terms of the series pass through a maximum and then tend to zero. If the computer arithmetic is performed with D significant digits, and we wish to compute the series sums to E significant digits, then the maximum term in the series must not exceed D-E 10 in absolute value. 4.2.3.4 Stability of the Power Series Method The power series method is an integration scheme which replaces a differential equation of order n by a nonlinear finite difference equation of the same order. Since the order of the equations is un- changed by the transformation from differential to difference equations, this transformation introduces no extraneous solutions which might lead to instabilities in the method. Although the power series method itself is a first-order method, the difference equations used to generate successive terms in the series are not first order. We must consider the possibility that these difference equations have solutions which grow with respect to the desired solution thus Leading to instabilities. These difference equations, however, have the general form c = [n c , + terms whose coefficients are independent of nl n p_n n-1 88 where p_ is the radius at which the expansion is made. For large values of n, this is simply c = — c ... Thus all solutions grow at the same ' r J n p n-1 ° rate for large n. 4. 2. 4 The Residual Partial Wave Phase Shifts and Amplitudes The result of the integration described in the previous three sections is the set of numbers A,, and 6,- defined in Section 2.4.3. fp fp These are the partial wave amplitudes and phase shifts respectively. jk As we have seen, the matrix M which is formed from them is independent of the particular set of regular solutions of Equations (2.40) that we jk fpr ik choose to solve. It is convenient here to consider not M^ but the matrix B^Ih^e 1 ^ (8) which is also independent of the set of regular solutions chosen. jk For large values of f , B reduces to fp jk c where TL(X.,t.) is the phase shift of the regular Coulomb wave function associated with X. and t.. If the diagonal elements of the array B J are J J fp 2 i ni B J , J = Bj e ^ fp fp Then the residual phase shifts and amplitudes are defined to be 89 fp fp and 6 £ P ■ 4 P - ^vv These tend to zero as f tends to infinity. In the output of ZENITH and i i i 1c NADIR, we actually quote tan 6_ rather than bi • In terms of B^ , we ' fp fp fp have R fp "fp RJ = |B jk -2iT^(X t) ImCB^ e K J J ) tan6 J f = & (r] H-l+Re(BJ k e R J J }) fp fp When the residual phase shifts fall below a cutoff value specified by the user, they are taken to be identically zero. Thus, the integration may be dispensed with by choosing A^ j = 6.. 6^ j - ru(X.,t.) 6. . . fp R J j iJ 4. 3 Summing the Partial Wave Series From the partial wave phase shifts, it is a short step to the constants Ik M which we require in order to compute the cross sections. These are combined to form the coefficients of the spherical harmonics in Equation (2.45). Unfortunately, the series as it is given there does not converge. 90 Nonrelativistically, the amplitude for point Coulomb scattering is—' f c^ 9; 2k rd+iY) \ 2 i Y -l (9) The divergence of the sum in Equation (2.45) is the legacy of the singularity present in Equation (9) for 9=0. To simplify this discussion, we replace Equation (2.45) by f(9) - E*, <(e) (10) which with regard to matters of interest to us here is equivalent to it. Acting on the assumption that the relativistic scattering amplitude has a singularity similar to the nonrelativistic amplitude, Yennie, Ravenhall and Wilson [ 1 ] suggested that Equation (10) be replaced by d-cose) n f(e) = £a< n) p^(e) in which (n) (n-1) i I l+l+m (n-1) 2 £+3 a i+l l-m (n-1) a 2Z-1 l-l (11a) and (0) 4. Gottfried [ 7 ], page 153. 91 Equation (11a) is easily derived from the recursion formula for associated Legendre polynomials of order m. Although this is the form in which we actually use the reduction in NADIR and ZENITH, we give here a derivation of a more general reduction scheme of which Equations (11) form a special case. Given the Legendre polynomial series S(9) = Z s, ^(9) 1=0 Z Z TO) =Tj t P™' (9) we can determine coefficients u such that i 00 u,(e) = S(8) T(Q)= Yj u. P^' (9) . In terms of s and t , we have from the orthogonality of the associated Jo *> Legendre polynomials IT tn , n(^-m-iti')! V f« _m4m' , . _m , . ^m 1 . . ,. . u r (i+ ^te?)! A s t, t £ n I ^ (e) V e) p i (e)d(cose). *. i! 1 2 1 2 We may evaluate the integral occurring here using Equations (3. 35) and (2.29). Thus, If i i r> „m+m . . m ,.. JBV, » P, (9) P. (9) P. (9) d(cos9) . nrim 1 (i.+m)! . (i„+m')' N (-1) 477 (l+m4m' ) .' 47T 1 4ff 2 2TT V2-WL (0-m-m 1 )! 2^+1 (4 -m)! 2-0+1 (i -m')! f dQ Y (Q) Y (fi) Y, , (Q) « I , -m-m *„ m *„m 92 i 1 )! x 2 /i £. £. i/ I £. £ Therefore, m-tm' V /(l-m-m f ).' 1 __2 u f -(-l) (2 1} ^ Si i ^2 U i4m+m ') ! i+2n -(r/a) 2 2H-2n+l I n = "— F" r e + — I" Vl Now, '? I = -i- r" r 2 ^ 2 e " (r / a)2 dr r Let this be K^ . Then integrating by parts again, we have K --r"" 1 V e~ (r/a) " + (^i) *- K , . m 2 \ 2 / r m-1 Finally, r Similarly, if we let P -(r/a) a/ff _ , , . . -2n £ p 2n+l -(r/a) , J n - a r J r e ^ ' > dr r (13) 95 we have -2(n-l) ^. 2n _ (r/a) 2 J = r e ' + n J .. . n 2 n-1 and j _ J i. e -(r/a) 2 J 2 N Clearly, v,(r) = /_j b (I +J ). Thus the computation of v.(r) may be n=0 organized as follows: V r) = S N s = b (I +J ) + s . n n n n n-1 s -i = ° 2£+2n+l , I = I , - E n 2 n-1 n J = n J , + E n n-1 n , '2 E = ,£ ) E 1 n a / n-1 l ~ K ui K = (2tei) a! K F n I 2 / r n-1 n-1 F = r F . n n-1 F - &- e" (r / a) 2 E = J = F £ K Q - ^f erf(r/a) . 96 4.4.2 Computing the Derivatives of the Nuclear Potentials at the Origin For small values of r, we see from Equation (13) that v i (r) N z n=0 b a •2n - 2+2n+2 i+2n+4 r r r i+2n+6 24+2n+3 a 2 (2 ^-2n+5) 2a 4 (2i^-2n+7) + r . 2nH-2 £4-2n+2 ^+2n44 ^+2n+6 1 & a r r r n: 2n+2 2 " 4 r • • ■ a (2n44) 2a (2n+6) ^ We can also expand v*(r) about the origin using Taylor's theorem. We have dv v,(r) = v/0) + r — 2 d v + r=0 2! .. 2 dr + r=0 By comparing this with the expansion given previously, we can therefore determine the derivatives of v„(r) evaluated at r=0. We require v„(0) & and the first four derivatives. These are explicitly 2 J b n! 6 m n £0 £ v/0) . t. n=0 dv^(r) dr d 2 v„(r) dr' 2 N = § 2 ^ b n nl 6 il r=0 " n=0 N = a 2 I K nl b 90 -\ b, r=0 n=0 n * £2 3 £0 d 3 v f (r) dr 3 97 N = 3a 2 I b n n! 6^ - f b Q 6^ r=0 n= ° d v,(r) N — T-| =12a 2 ib a n!« -f b « J2 +-S 2 (b o -b l )8 J0 dr _ n=0 5a r=0 4. 5 Organization of the Programs 4. 5. 1 Storage Allocation Both NADIR and ZENITH have been written in FORTRAN for use on an IBM 360 computer. It is necessary for many routines to have access to the same data. At the same time, because the computations may be divided into relatively independent segments it is possible for logically distinct arrays to share the same physical space. It is the purpose of COMMON and EQUIVALENCE statements in FORTRAN to allow goals of this nature to be achieved. However, when one has a large number of routines in which COMMON and EQUIVALENCE statements must appear, making changes to them becomes a tedious and error-prone procedure. Since it is not practical to have each array declared with the largest dimensions it will ever require, such changes loom as a recurrent possibility. Therefore, we have adopted an alternative approach which avoids this problem while fulfilling the functions of the COMMON and EQUIVALENCE statements. The main routine of each program has one large array (GM in ZENITH and CMAT in NADIR) from which storage for virtually all the arrays used by other routines is assigned. Che amount of storage allocated to 98 each array is determined based on the input parameters for a particular run. In this way storage can be assigned in an optimal manner. The storage is passed to the subroutines through an initializing call from the main routine. One exception to this rule is FININT in ZENITH which is called from NACSET because it uses arrays, the dimensions of which must change during the computation. 4.5.2 History Files If desired, a record may be kept of runs as they are made. The results may then be analyzed later in ways not directly available through NADIR and ZENITH themselves. In NADIR, two sequential files are used for this purpose. They are referred to internally as I/O units 9 and 10. Unit 9 contains an index to the runs which are stored on unit 10. In addition to the index, it contains the number to be assigned to the next run for which records are to be kept as well as the record number on unit 10 at which the storage will begin. Each pair of files can record the data from up to 57 runs which are numbered consecutively beginning with a number which must be stored in unit 9 before any runs are made. In ZENITH, the index and data recording functions have been combined in a single direct access data set referred to internally as I/O unit 10. The index has room for 193 runs but the storage capacity of the file will probably be exhausted long before the index is full. In addition to providing a permanent record of important runs that are made, the ZENITH history file makes it possible to make an unusually long run in small segments. A run which has been begun with a history file present may 99 be restarted at any time and continued to completion. ZENITH runs of this nature are protected against premature abortion by ENSURE which is called at the beginning of the main routine whenever a history file is present. ENSURE sets a time interval which is slightly shorter than the time limit specified by the user. When this interval expires it causes an interrupt which allows ENSURE to return control to the main routine so that the history file may be properly closed. ENSURE also calls the STAE macro which allows it to return control to the main routine when- ever any non-fatal system error occurs other than an excess time error. 4.5.3 Input and Output All input except that pertaining to the charge and current dis- tributions of the nucleus is processed in the main routine. The remainder is read in UPDINT. The exact format of the various possible input cards is given in Appendix which also explains the significance of the input parameters. All output with the exception of that pertaining to the charge and current distributions of the nucleus is created by WRTINT; the latter is created by UPDINT. Output is primarily in the form of English sentences describing the run. Input parameters, when given numerically, are given with trailing zeroes suppressed. Each line of output is potentially three lines: a superscript line, the line itself, and a subscript line. The lines are collected by OUTSM until they are 117 characters in length. Then, beginning at the 105th position, the lines are searched for a position at which all three lines have a blank. If such a position is found between the 90th and 117th positions the 100 line is broken there and printed. If not, the lines are arbitrarily broken after the 105th position and printed with a hyphen in the 106th position of the central line. 4. 6 CNADIR In addition to the NADIR and ZENITH programs, there exists CNADIR which is the same as NADIR except that it permits one to use complex as well as real potentials. All of the equations given in this and preceding chapters may be used directly in the case of complex potentials. Con- sequently CNADIR differs from NADIR largely in that the variables have been declared COMPLEX- 16 rather than REAL-8. In addition, because CNADIR was created with optical model analysis in mind, it has been provided with special potentials of the type frequently used in such work. These are described in Appendix 2. 101 5. TESTS OF THE PROGRAMS 5. 1 Introduction A complex program matures slowly. Before we can follow it with confidence through unexplored territory, we must guide it with care through familiar countryside. In this chapter we describe the battery of tests which NADIR and ZENITH have successfully faced. These tests are largely comparisons with the Born approximation in one form or another. Accordingly, we present, in Born approximation, the differential cross sections for electric monopole and quadrupole and magnetic dipole distributions. Of course large computer programs are never completely free of small errors but we feel that the tests described here adequately exercise those portions of the program that contribute to the numerical results. 5. 2 Tests of NADIR 5. 2. 1 Scattering of Massless Electrons from Nuclei with a Single Spinless Eigenstate At the inception of work on NADIR, we possessed a very reliable program for computing, by means of the partial wave method, the scattering of electrons from nuclei with a single spinless eigenstate provided that the rest mass of the electron be neglected. The results of NADIR runs on data acceptable to the older program have been found to be in agreement with results from that program to all significant digits. 102 5. 2. 2 Variation of the Scattering with the Charges of the Scattering Particles In Born approximation, the differential cross section for scattering of Dirac particles of charge ze from a spherically symmetric charge distribution p(r) with total charge Ze, is given by to (j^efE (l .4| |F(7)] 2_ (1) dfl \ q 2 J \ « 2 ' In Equation (1), E is the total energy of the incident particle and q = p. -p is the momentum transfer. The form factor, F(q), is defined by F(q) e q p(x) d : J p(x) d 3 x We see then that, ceteris paribus, the cross section is proportional to the square of the charge on the scattering particles. To verify that NADIR does indeed behave thus when operating on input data for which the Born approximation is valid, we have made four runs in which massless particles of charge -2, -1, +1, and +2 respectively are scattered from hydrogen nuclei at an energy of 1108.41 MeV. Table 5.1 displays the results for two values of the scattering angle. The tabulated functions are T^/z and its first two differences. Clearly "jo/z is very nearly a constant. The additional differences show that even in this energy regime, unfavorable as it is to the partial wave approach, the cross sections are behaving in a very regular fashion. 103 Table 5.1 Run # z e do /2 dfi/ z = 11° -2/ do / 2\ 5 idfl/ 5 J 28 i i -2 ! 1 5.61502335(-3) i I i j .00516678(-3) I 31 j -1 1 5.62019013(-3) .000570925 (-3) i ! i i .005737705 (-3) j i 32 ! 1 1 5.63166554(-3) .000573680(-3) | i .006311385(-3) i 27 i i ; 2 { 1 5.63797692(-3) e = 23 Run # z do / 2 dfi/ z »(Sg/- 2 R 2/do 2 6 dfl Z ! 28 -2 1.54136499(-4) i , i .00113666(-4) 31 -1 1.54022833 (-4) .00026402(-4) i .00087264(-4) 32 1 1.53848304(-4) ; .00026581 (-4) .00060683 (-4) 27 2 1.5378762K-4) 1 104 5.2.3 Scattering of Particles with Non-Zero Mass If one performs two independent scattering experiments, one with particles of mass m- and incident momentum p, the other with particles of mass m 9 but the same incident momentum p, then from Equation (1), we see that in Born approximation, the differential cross sections are in the ratio i — \ ,x,2 2 ^2 2.29 \dSlL 4E - q E - p sin - ida \ /r? 2 2 v 2 2.26 - 4E 2 -q E 2 - P sin - 2 2 2 in which E. = m. +p . As a check of NADIR with massive projectiles, we 11 have computed the scattering of massless electrons and also of muons from 12 C . The mass of the muon is taken as 105. 66 MeV. The energy of the muons was taken as 200 MeV. Thus we have E.. = 200 MeV and since the electron ma ss was neglected, p = E = 169. 35 MeV. We see from Figure 5. 1 that the ratio of these cross sections as computed by NADIR agrees with that predicted by Born approximation for small angles but falls steadily below the predicted value as the angle increases. In fact, the Born approximation systematically overestimates the muon cross section. From a classical point of view, the energy of the scattering particle, be it muon or electron increases as it enters the nucleus. This gives rise to an increase in the momentum of the particle which is slightly greater for the muon than for the electron. As a con- sequence, the muon cross section is shifted to smaller angles, or equivalently, decreased at a particular angle. As an estimate of the magnitude of this effect, we mention that in these calculations the energy gained in the nucleus by the scattering 105 Figure 5. 1 The ratio of the differential cross section for muon scattering to the differential cross section for electron scattering at the same momentum is plotted as a function of scattering angle. The dotted line is the ratio as computed in Born approximation. The solid line is the NADIR result. 25 3*0 b b .Tf. IS 106 / / / / Born Awwximat/om / / / / X. / / A/Aj>m SO 6 <0° > + (t n.- T) «0 J(P) = ° < 5a > ii + ^U iJ(p) " (t n. + T > f 3(p) = ° • < 5b > A complete set of independent regular solutions for Equations (5) is 8^ R (P) = k.p J L _ (k.p) 6.. ^ sin (k.p- -f-) 5 ij (6a > ii L i 7T f J (p) = (sgnX ) kp j (k p) 5 ~ cos'k.p- -J- 8... (6b) > K J J L j_ J lj p^co J 2 / lj If we now multiply Equations (2.40a), (2.40b), (5a) and (5b) by 8 R^' ~ f R^> " 8 ^ and f J ^ p ^ we have u P on addi -ng all four equations together, --^ p > t <^ +£lJ < p > i\i = ZC«jji (P) (g^R (p) S ij '(p) +f^ R (p) f ij '(p)) (7) - v jj' (p)(g O (p)fij,(p)+f oli (p) s ij, (p))^ • 113 The left-hand side of Equation (7) is an exact derivative. Therefore we obtain upon integrating Equation (7) from p=0 to p=°= and using the asymptotic forms given in Equations (6), -A.^sina 11 ' = J Z {s. j ,(p)(gJ^(p)g iJ '(p)+fo^(p)f iJ, (p)) -v jjt (p)(g^ R (p) f ij '(p) +f^ R (p) g ij '(p))} . If we now approximate f^p) and g J (p) in the integrand by f J R (p) and g Q J R , we arrive at the Born approximation to the partial wave phase shifts, 00 A. .sin6 lj = f dpk, k. p 2 {-s..(p)(j T (p)j_ (p) + sgnX. sgnX j (p) j (p)) ij u q i J Ji l>. ti^ j j- i- + sgnX v (p) (J L (p) j" (p) +sgnX iS gnX j (p) j (p))}. (8) j i- J i j- We have applied this equation to the pure magnetic dipole scattering discussed in Section 5.3.6. Because of the small size of the nucleus used there, v..(p) may be adequately approximated by its strength divided 2 by p . Using Equation (7) then becomes a matter of evaluating integrals of the form 00 i - r j m (p> j n (p)d P . mn 'i_ m n This, however, is rather easy for we have, applying in succession 10. 1. 1, 11.4.33, 15.1.20, and 6.1.17 from HMF, IT I « 4 nm 2(2n+l) ' ii— ui o , n-m even, n/m n-m+1 1 n 2 - (m+n+1) (n-m) , n-m odd, n>m 114 Table 5.4 Magnetic Dipole Coupling max asm Phase Shift -1 3.0 10.391219 -2. 91614989080 (-1) 11.848131 11.848131 -2. 91614989084(-1) 15.0 15.0 -2. 91614989141 (-1) 3.0 11.848131 11.848131 11.848131 15.0 15.0 3.0 11.848131 11.848131 11.848131 15.0 15.0 7.72085022299(-2) 7. 72085021943 (-2) 7.72085021966(-2) -9.79638914222(-2) -9. 79638914107 (-2) -9. 79638914638(-2) Electric Quadrupole Coupling 1/2 -1 1/2 3/2 -3 3.0 11.848131 11.848131 11.848131 15.0 15.0 3.0 11.848131 11.848131 11.848131 15.0 15.0 3.0 19.321106 11.848131 19.321106 15.0 19.321106 •3.49226752110(-3) ■3.49226750964(-3) •3.4922675374K-3) 1.52234438106(-2) 1.52234438082(-2) 1.52234438054(-2) 4.7406669448K-3) 4.74066697409(-3) 4.74066695366(-3) 115 Table 5. 5 shows the phase shifts in Born approximation and according to ZENITH for several values of f and X. The ZENITH values were taken from run 16. 5. 3. 5 Monopole Scattering from a Spin Nucleus vs. Monopole Scattering from a Spin ■c Nucleus Since a monopole charge distribution cannot couple different magnetic substates of a particular eigenstate the scattering from a spin zero nucleus should be identical to that from a spin one-half nucleus provided each is described by the same charge distribution. Therefore, we are able to test the handling of eigenstates with non-zero spin. ZENITH runs 2 and 5 investigate this property. The agreement here is to 6 or 7 decimal places. The discrepancy is accounted for by the fact that the cutoff value for the phase shifts has a slightly different meaning for spin zero and spin one-half nuclei. Table 5.5 f . X Phase Shift Born Approximation ZENITH 3 -3 1. 206675(-3) 1. 206751(-3) 4 -1. 206676(-3) -1. 206547 (-3) 10 10 4.022255(-4) 4.022321(-4) -11 -4.022254(-4) -4. 022199(-4) 20 20 2.060179(-4) 2.060188(-4) -21 2.060179(-4) -2. 060171(-4) 116 5.3.6 Born Approximation to Magnetic Dlpole Scattering We wish to consider the scattering of Dirac particles from a nucleus of spin I which is completely described electromagnetically by a dipole current distribution. The electromagnetic field of the nucleus is A (r,v) = (0,l(r,v)) with a*/-» x f ^(x* , v) ,3 , /nN A(x,v) = , t" d x • (9) jx - x I If the initial state of the system is — -ip . • x *. m.) = - u(p.,s.) e m. ) x ' l \E x x ' x and the final state of the system is — -ip -x * f l m f > = v| u( Pf ,s f ) e |m f > , then in Born approximation the differential cross section is given by da _ ( 2 e m j \— . -> t r« 3 -iq- x, |7,-* N i \ dQ " \W) l u( Pf> s f) YuCp^s^J dxe M (m f |A(x,v) |m i > (10) In Equation (10) , z and m are the charge and mass of the scattering particle respectively while q = p f -p. is the momentum transfer suffered during the collision. Using Equation (9), we are able to evaluate the integral in Equation (10) as follows: 117 |d 3 xe- iq, % f |A(?,v)|n..) _ _ -lq- (x-x 1 ) .->-», = Jd 3 xd 3 x' * „^ t e _lq ' X (m^x^m.) x-x 4ir f> 3 -iq-x J* x e (m f |?(x,v) Inu) But since x^e are considering a pure magnetic dipole, ( m JS(r-,v)\ m .) = Z(-D M C 1 .^ 1 -=!= J(r) jf (n ) M m f nm , in x Therefore we have, using the expansion •"*•"-£ I * iL V< r > Y LM ( V L=0 M=-L and the definition of ?, W (Q ), the following expression for the differential 1M x cross section, g- (^) 2 Iec-i) 11 -^^?-^©,).^,^) -^- — ^- C 1 l T l(n) ,-,2 v2I+I V-Mrn Uc » ) q f i in which I(q) - f r 2 J (qr) J(r) dr. We must now sum over the final spins s , m and average over the initial spins s ,m.. We deal with m and m first. We need only evaluate T T c 1 l T c 1 l l nyn f MM< m f Mm i m f M ' \ 118 which we can do immrdiately obtaining as a result, 21+1 3 MM This leaves the sum and average over s. and s to be performed, we must evaluate Thus E |«

-5r 7 'W)- Using well known properties of the y matrices this becomes -2 ?t2- CvVWlX" m M ,2 2 - + (e -m"-P f -P i )x 1M (n q ).x 1M (n q )} . y -* " 3 -+ -> 2 Now ' X (Si ) X 1M (G! ) = 7— and p e -p. =p cosi ^ 1M q 1M q 4tt f K i Therefore, carrying out the algebra involved in the remaining dot products, we have 1 3 r, 2 ( 2 8 , .2 9, ~2 ' 47 P (cOS 2 +2sin 2 } * m Collecting these expressions, the spin-averaged cross section is Ma \ = 8jt dQ j 21+1 / _..\ ze p I (q) 2 q , 2 9,. . 29, (cos - +2 sm -) . 119 Finally, we turn our attention to I(q). We consider a current density of the form J(r) = cr e " (r / a)2 with strength v. Referring to Equation (3. 61) we see that c = ev47T T ,*.-<'/*) 2 4r T 1 4v 3ire a Therefore we may write t/ \ 4v r 3 • < \ "( r /a) . el(q) = - r J r j,(qr) e ' dr 3tt a J Using 10.1.1 and 11.4.29 from HMF, we have el(q) 2 2 // !l(q) = _ ja_ e -q a/4 6\/7T from which it follows that da\ 2 z 2 p 2 v 2 -q 2 a 2 /2 , 28 ±1 . 2 8, do = Im'-7l- e (cos ^ +2sln 2 } • 9q 2 2 2 9 Finally, using the fact that q = 4 p sin — , we have dg \ z dQ/ 18 2 2 2 2 ! v . e " q a / 2 (2 + cot 2 -^) (21+1) u + cot 2 ; . (11) As a check of the ZENITH program, we have used it to compute the scattering of 100 MeV electrons from a spin one-half nucleus with a pure magnetic dipole current distribution. The strength used was 120 Table 5.11 Differential Cross Sections for Magnetic Dipole Scattering Angle 1 V V 111' 178' 179' do an Born Approximation 9. 1194757 (-1) 2. 2802958 (-1) 1.0137809(-1) 8. 3151847 (-5) 8.3119756(-5) 8.3100508(-5) ZENITH (run 16) 9. 1194918 (-1) 2. 2802922(-l) 1.0137762(-1) 8. 2332845(-5) 8. 2300793 (-5) 8.2281569(-5) v = . 05 fermis and the length parameter, a, was taken as one fermi. The results are shown in Table 5. 11. In fact we also checked the differential cross sections obtained by summing and averaging over s. and s only, but we tabulate only the complete, averaged, differential section. 5.3.7 Born Approximation to Electric Quadrupole Coupling In this section, we consider the scattering of Dirac particles from a nucleus of spin I which is completely described electromagnetically by a quadrupole charge distribution. The electromagnetic field of the nucleus is A (r,v) = (cp(r,v),0) with /-"* n P c (x' ,v) ,3 , cp(x,v) = J — b — *— - d x 1 . (12) x - X' 121 If the initial and final states of the nucleus are as in Section 5.3.6, then in Born approximation, dQ = l^lrJ l u 2 . (13) As in Section 5.3.6, we can use Equation (12) to evaluate the integral in Equation (13). We have - . -* -» Jdxe q ' X (m f |cp(x,v) |m i > 471 o.3-* -iq-x - / \ n ,~* .1 s = « d x e n (m C(r,v) m > . |q| Since we are considering a pure electric quadrupole = £ <-l) M P.* £,* -= C(r) Y (C^) . M f i \ 21+1 r Therefore, once again following Section 5.3.6, we have Hfj IZ (" 1 ) M u (Pp s f > Y u(p i ,s.) v 2 da z e m \ dn M x (frir) 2 i c i 2 I , ,2 with 00 I(q) = f r 2 j (qr) C(r) dr. ' Averaging over s., m. and summing over s f , m , we obtain ~gT m i6TL zeE-I(q)\ 2 _ pi 2 dQ; 21+1 ! 2 U „2 2 ; q e 122 2 -(r/a) Finally, if the shape of C(r) is r e ' and its strength is s, have we -1 — / 2 2 ^ 2 d<3 \ 4 z s E \dnj = 45(21+1) 2 2 /o e -q a /2 (1 P_ 4 2 0v o sin o) • (14) We have also checked ZENITH against Born approximation for pure electric quadrupole scattering. The results are shown in Table 5. 2 In this test, s = . 5 (f ermis) , 1=1, E=p=100 MeV, and a=. 25 fermis. Table 5. Differential Cross Sections for Electric Quadrupole Scattering ZENITH (Run 17) 1.9009250 (-3) 1.7732203(-3) 1.4223516 (-3) 9.4727502(-4) 4. 7730938(-4) 1.3648416 (-4) 1.2543268 (-5) Although the Born approximation cross section is going to zero at 180°, the ZENITH cross section is not. In fact this is due to higher order effects which ZENITH computes properly. To see this, we made one additional run with a quadrupole strength only one-half as large. At 150°, the Born approximation then predicts a value of 3.092212(-5) while Angle Born Approximation da 1° 1.9024454 (-3) 30° 1.7713315(-3) 60° 1. 4155386 (-3) 90° 9. 3614807 (-4) 120° 4. 6433204 (-4) 150° 1.2368848 (-4) 179° 1.4030964(-7) 123 the ZENITH calculation gives as a result 3. 195030(-5). At 179° these become 3. 507741 (-8) and 9. 933771 (-7). Although the difference is still present it has been greatly reduced. 124 6. APPLICATIONS OF THE PROGRAMS 6. 1 Isotopic Variations in the Charge Distribution of Calcium Experimental results on the scattering of 250 MeV electrons from Ca , Ca , Ca , and Ca have been analyzed by Frosch et al. [ 9 ] in terms of isotopic differences in ground state monopole charge 40 distributions. However, because Ca , being doubly magic, has no low- 42 44 + lying excited states, while Ca " and Ca both have a low-lying 2 excited state, it has been suggested [ 10 ] that the observed isotopic differences in differential cross sections may be due in large part to dispersive effects rather than to actual differences in the ground state monopole charge distributions. Rawitscher [10, 11] has estimated the magnitude of dispersive effects using a model in which the inelastic excitation takes place by a single monopole transition to state degenerate with the ground state. The strength of this transition is adjusted to give a reasonable value for the total inelastic cross section. Applying his model to the calcium experiments discussed earlier, Rawitscher finds that dispersive effects may account for as much as a 5% difference in the differential cross sections in the first diffraction minimum. The quantity quoted is actually the difference in the cross sections divided by the sum, in agreement with the usage of Frosch et al. Inasmuch as the measured 40 44 difference between Ca and Ca is on the order of 12% there, this would appear to be an important effect. This view has been emphasized by Wall [12]. 125 To discover if this is actually the case, we have computed the 40 44 dispersion corrections in the Ca - Ca case. Although the charge dis- tributions used by Frosch e_t al . are not immediately acceptable to ZENITH we have made fits to them that are. The resulting parameters are shown in Table 6. 1. Using these fits, we computed the elastic differential 40 44 cross sections for Ca and Ca . The ratio of the difference in cross sections to the sum is plotted as a function of angle in Figure 6. 1. It is in excellent agreement both with the experimental data and with the 40 44 Ca - Ca curve given in Figure 8 of reference [ 9 J, indicating that the fitting process leading to the parameters in Table 6. 1 is adequate. 40 Then using the Ca ground state charge distribution, we added a 2 excited state coupled to the ground state by a quadrupole transition charge density. Heisenberg, McCarthy, and Sick [ 13] have fitted this 44 transition charge density for Ca . In doing so however, they treated B(E2) as a variable parameter which resulted in a value for it that does not correspond to that determined by Coulomb excitation experiments [14]. We have therefore used the Coulomb excitation value for the B(E2) and chosen the shape of the transition distribution to be that given in Equation (1) by Heisenberg, McCarthy and Sick. The parameters and strength for this shape are also given in Table 6. 1. Finally, the 2 excited state was assumed to have a charge distribution identical to that of the ground state. The effect of this excited state on the differential cross section is shown in Figure 6. 2. The vertical scale there is 40 times larger than that in Figure 6. 1. Although the effect grows with angle 126 Figure 6. 1 Comparison of the experimental and computed values of < a 40 " CT 44 } 7—^ -— at 250 MeV. (a 40 " a 44 } 127 Scattering Angle 128 40 44 44 Figure 6. 2 Dispersion effects in Ca and Ca . The Ca cross 40 section was computed using the Ca ground state charge distribution and one 2 excited state. 129 to o o © I 75-* %* Scatter //vg Fig. 6.2 130 Table 6. 1 We tabulate here the ZENITH parameters for the charge distributions in calcium. The monopole distributions were obtained by fitting to the 44 + results given in Table III of reference [ 9 ]. The B(E2) for the Ca 2 44 44 d 40 state is 350 [14]. For Ca , P ex (r) = r ^ P ground (p)- For a11 dis- tributions, a =1.99 fermis. 40 (r) ground 44 (r) ground P (r) ex i = & = Q i= 2 s°°= . 145944 2.3092235 1 -1.3886713 2 .46526361 3 -.02099411 4 -.00570991 5 -.00210360 6 .000093475 7 .002158589 8 .002788878 s°°= . 145944 n a n 2.3546959 1 -1.4694359 2 .52464586 3 -.03025631 4 -.01337605 5 -.02261365 6 -.001853795 7 .003543130 8 .003182487 01 S 2 2 = . 22 Fermi n a n -.93126934 1 .6907 2861 2 -. 23785941 3 -.00002116 4 .002930919 5 .011714143 6 .001361289 7 -.00109017 8 -.00140849 131 relative to the experimentally measured difference in Figure 6. 1, it remains negligible throughout the region investigated experimentally. Analyses of experiments in which the ratio is sampled for larger values of the momentum transfer may have to take proper account of these dis- persive corrections. Rawitscher has also suggested [11] that the magnitude of the dis- persive corrections may be energy-dependent, growing appreciably for lower energies. To check this we repeated the runs described above at 50 MeV. The effects are extremely small and to within the accuracy of the com- putations appear to depend only on q, the momentum transfer. 6. 2 Magnetic Dipole Scattering at 180° In nuclei which have magnetic dipole moments, the scattering due to this is in general completely masked by the scattering due to the nuclear charge. At 180°, however, the scattering due to a monopole charge dis- tribution is rigorously zero to all orders for massless Dirac particles. One therefore expects that the differential cross section at 180° reflects only the scattering due to the magnetic dipole. De Vries, Van Niftrik and Lapikas [15] have used this fact to study the magnetic form factor in the nuclei Sc, V and Co. Although the Coulomb scattering is zero at 180°, the presence of the Coulomb potential causes distortions in the electron plane waves which in turn modify the magnetic scattering even at 180°. This is accounted for by applying to the cross sections correction factors which are computed using the distorted wave Born approximation. By varying the energy at which the scattering is 132 done one changes the value of the momentum transfer to which 180° corresponds and thereby samples the magnetic form factor at different values of q. We have made several runs in an attempt to perform by calculation the experiments reported by de Vries, Van Niftrik and Lapikas for cobalt. In these runs we have used a monopole charge distribution obtained by fitting 59 the distribution given by Hahn, Ravenhall, and Hofstadter [ 16 ] for Co . The dipole transition current density was taken to have a shape determined by multiplying the monopole distribution by r. The strength has been chosen to correspond to a magnetic moment of 4. 62 nuclear magnetons [ 17 ]. We expect the cross sections to be independent of spin. To verify this we made one run using the physical spin of 7/2 and the remainder using a spin of l/2. The run with spin 7/2 agreed with its spin 1/2 counterpart.-^ Using Equation (6) of reference [ 18 ] we have computed the distorted wave Born approximation correction factors. These are considerably larger than those determined by ZENITH and although we do not understand the reason for this discrepancy, we do believe that the ZENITH results are correct. The results are presented in Table 6. 2 6. 3 Other Possible Applications ZENITH allows the investigation of a host of scattering problems hitherto inaccessible to exact calculation. The applications presented in the preceding sections are only a small portion of what might be done. Here we briefly indicate other areas in which ZENITH may be useful. 5. The strength of the potential must vary inversely as / 21+1 in order to keep the cross section constant. 133 Table 6. 2 Cross sections are given in square fermis per steradian. The Born approximation results were obtained using ZENITH with the charge set to 0. Energy jjg(BA) ^(ZENITH) ^(Experiment) [15] ^(DWBA)/^(BA) [18] 50 MeV 3.510(-6) 3.799(-6) 3.3(-6) 53% 65 MeV 1.533(-6) 1.141(-6) 9.6(-7) 53% The next generation of high energy electron scattering experiments will be performed on machines capable of considerable increased resolution and precision (for example the 400 MeV accelerator at MIT and the 600 MeV accelerator at Saclay). These may be expected to yield much more accurate information concerning Inelastic electron scattering. It should be possible to analyze such data with the accuracy they deserve using ZENITH. In heavy deformed nuclei the matrix elements linking ground to excited state are quite large. Therefore the effects investigated in Section 6. 1 may become appreciable. If this is so, then an accurate analysis may require ZENITH. Another realm in which accurate theoretical results are not avail- able is low energy electron scattering. The distorted wave Born approximation can only be used when the energy is sufficiently high. Thus when effects due to higher multipole moments are important, the capabilities provided by ZENITH are required. Because NADIR and ZENITH account properly for the mass of the scattering particle they may be used to compute proton scattering as 134 well as electron scattering. Thus it is possible to investigate strong interactions with these codes. Some work has already been done along these lines with NADIR [19]. One might expect that with strong inter- actions the multipole couplings between states will be much stronger. If this is the case the full capabilities of ZENITH may be called for. 135 APPENDIX 1 Al. 1 Introduction In this appendix, we have collected a number of identities that are useful in the manipulations of Chapters 2 and 3. These are taken primarily from Messiah [20], Appendix C, Uberall [2], Appendix A, and Edmonds [21] . Al. 2 Spherical Harmonics For m>0 , / a \ / i \ m !2M. (I-mTl imcp _m , m V 4tt (Mm) I Y,. m (ecp) = (-!)" Y^W* . The associated Legendre polynomials P.(cos0) are defined by v m , m (-D m+£ (Mm) I , i x-m / d ^" m , . Q .U P A cos 6) = ■>-— « -*- — ^- (sin0) 5 (sin©) ? i' (*-m)! \dcos9y and obey the following recursion relations, P^ 1 (cos0) =--~s {(i-m) cos0P™(cos0) - (£tm) P™ . (cos©)} * sinf x> Jo- 1 P^CcosS) = -g-i-- {(2i!+l) cos© P"j(cos0) - (4+tn) P™^ (cos0)} In addition, F J. 1«1L (slne /. 2 « 136 We have /(2i +l)(2i.+l)(2i,+l)x* [ Y t> m (Q) Y P n, (Q) Y f m (Q) dQ = l 'i, i„ i, \ / i, i- £. \ 12 3 / \m. m nu y ' 12 3 x Setting i=m_=0 leads to the orthogonality relation f Y* (Q) Y, (0) dQ = 5. - 6 m m . J ^ i 2 m 2 ^ mi m 2 In terms of the spherical harmonics, Y, (&p), we define vector An m spherical harmonics Y. p . (0cp) by 3 ^* Y™„(ftp)- Z C i X i Y, (ftp) e m |i with 1 - - e = — : ( + e - i e ) ± l /J x y z Under complex conjugation, we have 8J n »i»*- (-D^-^vT^W 'm When j = i, we write Y^CQcp) = X^(0cp). In the following formulae, F(r) is any function of r. 137 *Cr(r>.Y fc W»~ (^j f-|]«r)'?; wi W + (ira) (i^l'W-^iiM v x (F(r) ?» w lW ) - i(A + ^2j r W .(^) ?;, .(ftp) Vx (F(r) ?».<*,)) = l(£ - i) F(r)- (j£jj f^ .(ftp) 1 + 1 d7 + — ; F(r),i 2W Y iMiW i VX (F(r) ?;„ .(ftp))- i(£ - ijl) F(r ).(^| f-^ftp) V. (F(r) ?/ m l(eTO - - (^jj (A + )) s 1 i ^ / d £-1 '■ ■ (isi; ! '£ - i i ) F(r >- Y A n<^> Al. 3 Wlgner Coefficients A more extensive collection of identities is available in Messiah [20], Appendix C. 138 a W3 V 2 M 3 V J 2 + M 3 / (-1) L i v&T+l J 1 J 2 J 3 M lM2 -M 3 J 1 J 2 J 3 [Wl ,./l +J 2 +J 3 J 1 J 3 J 2 K M 2 M 3 = Wl \ M 1 M 3 M 2, J l J 3 J 2 -M -M -M 1 3 2 j I. (-D J 1+ J 2+ J 3 +M 1+ M 2+ M 3 M M M T21 m l m 2 J l J 2 j 3' lJ l" M 2 m 3 fJ 2 J 3 j l 1 M 2 -M 3 m 1 J 3 J 1 J 2 M 3 -M im j l J 2 J 3 m l m 2 m 3 I = 6 J3J3 m 3 m 3 2j 3+ l \hhh\ |J 1 J 2 J 3 I (-D 12 3 12 3 M M M 123 fj 1 j 2 j 3 \ \ m l m 2 m 3 1 ,J l J 2 j 3 M r M 2 m 3. f j l j 2 j : J 1 J 2 J 3J J 2 J 3 j l M 2 -M 3 m r J 3 J l j 2 M_-M.m 2 i j l J 2 J 3 , J 1 J 2 J 3 j 1 j 3 j 2 | J 1 J 3 J 2 > = J 2 J 3 J 1 J 2 J 3 J 1 j l J 2 J 3 J l j 2 j 3 ,J 13 J 24 J ^ M 13 M 24 M / h j 2 J 12 j 3 j 4 J 34 J 13 J 24 J I m 1 m num. M M M 12 34 j l j 2 J 12 im l m 2 M 12 j 3 j 4 J 34 m 3 m 4 M 3 j l j 3 J 13 T3 M 13 j 2 j 4 J 24| / J 12 J 34 J ,m 2 m 4 M 2 M M M W 12 34 W 139 ZENITH makes considerable use of Wigner 3-j, 6-j and 9-j coefficients and must compute them rapidly and accurately. For this purpose, we use the excellent programs of Caswell and Maximon [22] . 140 APPENDIX 2 A2. 1 Introduction We collect in this appendix instructions for preparing input decks to be used with the programs described in the body of the thesis. In addition we briefly describe two auxiliary programs, ZENREQ and NADREQ, which process information generated by the primary programs ZENITH, NADIR, and CNADIR. A2. 2 ZENITH Input Input decks for ZENITH have the following format. Card 1 Variable Format Description IPECM II, 4X IPECM must have an integer value between and 3. It determines the significance of ENOMOM according to the following table: IPECM ENOMOM Total Lab Energy 1 Lab Momentum 2 Total CM. Energy 3 C. M. Momentum ENOMOM D10.3 NS 15 15 15 ENOMOM determines the momentum or the total energy of the scattering particles in either the laboratory or center of momentum reference frames according to the value of IPECM as indicated in the table above. Nucleon number of the target nucleus (note that this is an integer). Proton number of the target nucleus (note that this is an integer). The number of energy eigenstates of the target nucleus. 141 Variable AF Format D10.3 ZF D10.3 Description AF determines the mass of the target nucleus in atomic mass units (one atomic mass unit equals 931.441 MeV). If these columns are blank, AF is taken equal to A. ZF determines the charge of the target nucleus. If these columns are blank, ZF is taken equal to Z. Card 1 + 1. 1 = 1, NS Variable Format Description EPS (I) D10.3 EPS (I) is the energy eigenvalue of the I-th nuclear eigenstate in MeV. For 1 = 1, this must be zero. SPN(I) P(D 15 4X,A1 SPN(I) is the spin of the I-th nuclear eigen- state in units of fr/2 (note that this is an integer). P(I) is the parity of the I-th nuclear eigen- state. A plus sign (+) denotes positive parity, and a minus sign (-) denotes negative parity. The next card describes the projectiles . Variable Format Description JORP 15 JORP determines the type of projectiles being scattered according to the following table: JORP Projectiles Electrons 1 Positrons 2 Negative Muons 3 Positive Muons 4 Antiprotons 5 Protons 6 Special (see PM and PZ below) NOMAS S L5 If NOMASS is TRUE (specified by a T right justi- fied in the input field), then the mass of the scattering projectile is neglected. If NOMASS is FALSE (specified by an F right justified in the input field), then the mass of the scattering projectile is not neglected. 142 Variable Format PM D10.3 PZ D10.3 Description If JORP is equal to 6, then PM is the mass of the projectile in MeV. If JORP is equal to 6, then PZ is the charge of the projectile (note that the charge of an electron is -1). The next card specifies several computation parameters. Variable Format Description DR D10. 3 The step size for the numerical integration from the center of the nucleus to its edge. RMAX D10. 3 The nominal radius of the nucleus. PHSCHK D10. 3 The cutoff for the partial wave residual phase shifts. SDEG D10. 3 The least angle in degrees at which cross sections will be printed. DANGLE D10. 3 The angular increment in degrees at which cross sections will be printed. LDEG D10. 3 The greatest angle in degrees at which cross sections will be printed (note that this variable is REAL*8). FTMAX 15 The maximum angular momentum in units of h for which partial wave phase shifts will be computed (note that this is an integer). NPOT L5 NPOT is a LOGICAL variable which if TRUE causes the nuclear potentials to be printed out and which if FALSE inhibits this printing. LMAX 15 The maximum multipolarity used in the nuclear potentials. The remaining cards may be divided into groups, each of which describes a single charge or current density. The format of one of these groups is as follows. 143 Card 1 Variable Format DTYP A2,3X LMPL NUC1 NUC2 ADIM ALNNP NP 15 15 15 D10.3 D10.3 15 Description DTYP must be either ' CD 1 or ' JD T . The former indicates that the transition density described is a charge density. The latter indicates a current density. The multipolarity of the transition. The initial eigenstate coupled by the transition distribution. For this purpose, the eigenstates are numbered from through NS-1. The final eigenstate coupled by the transition distribution. For this purpose, the eigenstates are numbered from through NS-1. The scale parameter for this distribution in fermis. The strength parameter for this distribution in fermis raised to the LMPL power. The number of shape parameters for this distribution. These are specified on the remaining cards of this group as described below. Cards 2 through (NP + 15)/8 Variable Format A(I),I=1,NP 8D10.3 Description The dimensionless shape parameters for this distribution. In addition to these cards, the input deck may begin with one or more comment cards describing the purpose of the run. These cards are dis- tinguished by a ' C in column 1. Remaining columns comprise the body of the comment which is included as part of the output. The results of ZENITH runs may be stored on a disk file for further analysis if desired. This desire is communicated to ZENITH by providing a DD statement for FT10F001, which is the DD name of the history file, 144 in the JCL for the run. ZENITH makes a search to determine if this DD statement is present and if it is, proceeds with the recording process. Runs which have been begun with a history file present may be continued at a later time by submitting an input deck with a single card containing an ' R 1 in column 1 and the number of the run to be restarted, right justified, in columns 13-15. A DD statement for FT10F001 must also be provided and must agree with that used when the run was begun. Two sample input decks are shown below. The first is for computing the scattering of electrons with an energy of 200 MeV in the laboratory 12 frame from a two-state C nucleus. No history file is provided. The second deck is for the same problem with negatively charged muons. Here a history file has been provided. Example 1: /*ID . . . /*ID REGION=260K // EXEC G0F0RT, G0FILE= ! USER. P1450. RLM. FORTLIB (ZENITH) ' , REGION=260K //GO. SYSIN DD * 200. 12 6 2 0. + 0. + T .02 12. 8. D-ll 5. CD 1.705 -.47 1.77 .0437832 2 CD .5 1 .33333333 1.77 .0065 2 CD /* 1.705 1 1 -.47 1.77 .0437832 2 1. 150. 59 145 Example 2: /*ID . . . /*ID REGION=260K // EXEC G0F0RT,G0FILE=' USER. P1450. RLM. FORTLIB (ZENITH) ' ,REGION=260K //GO. SYSIN DD * 200. 12 6 2 0. + 0. + 2 F .02 12. 8 D-ll 5. 1 CD 1.77 .0437832 2 1.705 -.47 CD 1 1.77 .0065 2 .5 .33333333 CD 1 1 1.77 .0437832 2 150. 59 /* //GO. FT10F001 DD DSN=USER. P1450. RLM. ZHIST1,DISP=0LD A copy of this output may be obtained by running ZENITH with an input deck consisting of a single blank card. A2. 3 NADIR Input Input decks for NADIR have the following format. Variable Format Description EL D10. 3 EL is the total energy of the incident particles in the lab frame. Z 15 Proton number of the target nucleus (note that this is an integer). A 15 Nucleon number of the target nucleus (note that this is an integer). NS 15 The number of energy eigenstates of the target nucleus. Card 2 through (NS + 15)/8 Variable Format Description EPS (I), 1=1, NS 8D10.3 EPS (I) is the energy eigenvalue of the I-th nuclear eigenstate in MeV. For 1=1, this must be zero. 146 The next card describes the projectiles and gives a number of computational parameters. Variable Format D10.3 SDEG DANGLE LDEG DR RMAX PHSCHK CHMAX NOMASS D10.3 D10.3 D10.3 D10.3 D10.3 15 L5 Description The least angle in degrees at which cross sections will be printed. The angular increment in degrees at which cross sections will be printed. The greatest angle in degrees at which cross sections will be printed (note that this variable is REAL*8) . The step size for the numerical integration from the center of the nucleus to its edge. The nominal radius of the nucleus. The cutoff for the partial wave residual phase shifts. The maximum angular momentum in units of h for which partial wave phase shifts will be computed (note that this is an integer). NOMASS is a LOGICAL variable which if TRUE (specified by a T right justified in the input field) causes the mass of the scattering projectile to be neglected and which if FALSE (specified by an F right justified in the input field) causes the mass to be included in the calculation. NPOT MUOREL L5 15 NPOT is a LOGICAL variable which if TRUE causes the nuclear potentials to be printed out and which if FALSE inhibits this printing. MUOREL determines the type of scattering projectile used according to the following table: MUOREL 1 2 3 Projectile Electrons Negatively charged muons Antiprotons Protons 147 The remaining cards may be divided into groups, each of which describes the distribution for one transition. There must be NS*(NS+l)/2 such groups describing all possible transitions. If the states are numbered from through NS-1, then the transitions must be described in the order (0-0), (0-1),---, (0-NS-1), (1-1), (1-2),-.., (NS-1 -NS-1). The format of one such group is described below. Card 1 Variable Format NP ADIM RN 15 D10.3 D10.3 Description The number of shape parameters for this dis- tribution. These are specified on the remaining cards of this group as described below. The scale parameter for this distribution in fermis. The strength parameter for this distribution. Variable Cards 2 through (NP+15)/8 Format Description A(I),I=1,NP 8D10.3 The dimensionless shape parameters for this distribution. The results of NADIR runs may be stored on a disk file for further analysis if desired. This desire is communicated to NADIR by providing DD statements for FT09F001 and FT10F001 in the JCL for the run. A2.4 CNADIR Input Input for CNADIR is identical to that for NADIR except that following the last transition distribution, there must be a set of cards which specifies an optical model potential describing the 148 interaction of the scattering projectile with the ground state of the target nucleus. This interaction may be either of two types. Type 1 is a parabolic modified Fermi distribution of the form v(r) . v ( i+ *j r / c >; l+e (r ~ c) / z / Type 2 is a harmonic well distribution of the form V(r)=vi + ^j 2 )«- (t / ,,! . In both of these, V is a complex constant determining the strength of the interaction. The potential may be taken as the fourth component of a four-vector or as a Lorentz scalar. It may also be apportioned between the two. This is governed by the parameter X according to the following scheme. V s (r) = X V(r) V 4 (r) = (l-\) V(r) . The format according to which these variables must be presented in the input is shown below. Card 1 Variable Format Description ITP 15 ITP determines the type of optical potential used and therefore the meaning of the parameters on the next card. 149 Card 2, ITP = 1 (i.e. Parabolic Modified Fermi) Variable Format Description VPWR 2D10. 3 VPWR is the complex constant determining the strength of the potential in MeV. Note that it requires two input fields for its specification. CR D10.3 c in fermis. ZR D10.3 z in fermis. WLG D10.3 w . DLAM D10.3 X . Card 2, ITP = 2 (i.e. Harmonic Well) Variable Format Description VPWR 2D10. 3 VPWR is the complex constant determining the strength of the potential in MeV. Note that it requires two input fields for its specification. ALPHR D10. 3 a . AR D10.3 R . DLAM D10. 3 X . A2. 5 NADREQ and ZENREQ These two programs have been written for analyzing NADIR and ZENITH runs which have been recorded on the history files. They both use the same simple language for specifying their input. The grammar for this language is given below. In this grammar words, enclosed in quotes are words that may actually appear in the input. Words enclosed in angle brackets are merely names used to describe certain input sequences. The grammar describes the rules for generating acceptable input. Thus if :: = (A) 'BY' 150 (REQUEST INPUT) :: = (REQUEST) | (REQUEST INPUT) ';' (REQUEST) (REQUEST) : : = (CHI SQUARE REQUEST) | (PERCENTAGE DIFFERENCE REQUEST) | (CROSS SECTION REQUEST) (CHI SQUARE REQUEST) :: = 'COMPUTE' ? 'CHI SQUARE' [' INFORMATION FOR'] ? (CROSS SECTION) ['AS A FIT TO 1 | 'WITH RESPECT TO']? (EXPERIMENTAL CROSS SECTION) [['FOR ANGLES BETWEEN* ] ? <*N) 'AND' ? <*N)] ? is a rule and 'TWO' is a valid example of (A) while 'FOUR' is a valid example of (B) the 'TWO BY FOUR' is a valid example of (C). If (C) may be generated by more than one rule, then the alternatives may be separated by a vertical bar |. Thus, for example, we might have (C> :: = (A) 'BY' (B>|(A) 'BY' (D) which is equivalent to the pair of rules (C) :: = (A) 'BY' (C) :: = (A) 'BY' (D) . We may further condense this with the square bracket notation as follows. (C) :: = (A) 'BY' [(B)|(D)] . Finally, a question mark following any item indicates that the rule is valid with or without that item. Thus the rule (C) :: = (A) 'BY' ? (B) is equivalent to the two rules (C) :: = (A) 'BY' stands for any number. 151 (PERCENTAGE DIFFERENCE REQUEST) :: = ['COMPUTE 1 | 'PLOT'] 'THE' ? 'PERCENTAGE DIFFERENCE' 'BETWEEN' ? (CROSS SECTION) 'AND' (CROSS SECTION) (ABSCISSA PART) ? (CROSS SECTION REQUEST) :: = ['COMPUTE' | 'PLOT'] 'THE' ? (CROSS SECTION) (ABSCISSA PART) ? (CROSS SECTION) :: = (CROSS SECTION TYPE)? 'CROSS SECTION' (STATE PART)? 'FROM' ? 'RUN' <*N) (FOLDING PART) (CROSS SECTION TYPE) :: = ' FOLDED' ? ['ELASTIC' | 'INELASTIC'] (STATE PART) :: = ' TO EXCITED STATE ' (*N) (FOLDING PART) :: = 'FOLDED OVER AN ANGULAR UNCERTAINTY OF' (*N) 'DEGREES' (ABSCISSA PART) :: = ['VERSUS' | ' AS A FUNCTION OF 1 ] ['MOMENTUM TRANSFER' | ' Q 1 | 'SCATTERING' ? 'ANGLE* ] ['IN THE' [' CENTER OF MOMENTUM' | ' LABORATORY' ] 'FRAME'] ['('<*N) V <*N) ' ,' <*N) ')']? (EXPERIMENTAL CROSS SECTION) :: = 'EXPERIMENTAL CROSS SECTION' <*N) 152 LIST OF REFERENCES 1. D. R. Yennie, D. G. Ravenhall, and R. N. Wilson, Phys. Rev., 95, 500 (1954). 2. Herbert Uberall, Electron Scattering from Complex Nuclei (Academic Press, New York and London, 1971). 3. J. M. McKinley, Phys. Rev. 183 , 106 (1969). 4. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , M. Abramowitz and I. A. Stegun, Eds. (National Bureau of Standards, Washington, D. C. , 1964), Series 55. 5. M. E. Rose, Relativistic Electron Theory (John Wiley & Sons, Inc. , New York, 1961) . 6. John David Jackson, Classical Electrodynamics (John Wiley & Sons, Inc. , New York, 1962). 7. Kurt Gottfried, Quantum Mechanics Volume I: Fundamentals (W. A. Benjamin, Inc., New York, 1966). 8. D. G. Ravenhall, D. R. Yennie, B. C. Clark, and R. Herman, Nucl. Phys. 7_2, 177 (1965). 9. R. F. Frosch et al. Phys. Rev. 174 , 1380 (1968). 10. G. H. Rawitscher, Phys. Rev. 151 , 846 (1966). 11. G. H. Rawitscher, Phys. Letters 33B , 445 (1970). 12. N. S. Wall, Ann. Phys. 66, 790 (1971). 13. J. H. Heisenberjf, J. S. McCarthy, and I. Sick, Nucl. Phys. A164 , 353 (1971). 14. F. R. Metzger and G. K. Tandon, Phys. Rev. 148 , 1133 (1966). 15. H. de Vries, G. J. C. van Niftrik and L. Lapikas, Phys. Letters 33B, 403 (1970). 16. B. Hahn, D. G. Ravenhall, and R. Hofstadter, Phys. Rev. 101, 1131 (1956). 17. G. H. Fuller and V. W. Cohen, Nucl. Data, Tables A5, 433 (1969). 153 18. B. T. Chertok, Phys. Rev. 187 , 1340 (1969). 19. B. C. Clark, R. L. Mercer, D. G. Ravenhall, and A. M. Saperstein, Phys. Rev. (to be published). 20. Albert Messiah, Quantum Mechanics, (North Holland Publishing Co. , Amsterdam, 1966), Vol. II. 21. A. R. Edmonds, Angular Momentum in Quantum Mechanics, (Princeton University Press, Princeton, New Jersey, 1957) . 22. R. S. Caswell and L. C. Maximon, FORTRAN Programs for the Calculation of Wigner 3j, 6j, and 9j Coefficients for Angular Momenta < 80, NBS Technical Note 409, (National Bureau of Standards, Washington, D. C. 1966). 154 VITA Robert Leroy Mercer was born on July 11, 1946 in San Jose, California. He attended the University of New Mexico where he received the Bachelor of Science degree in Mathematics and Physics in June, 1968, graduating summa cum laude with distinction in mathematics. In June, 1970 he received the Master of Science degree in Computer Science at the University of Illinois. He has coauthored the following publications: "Diffusional Deposition From a Fluid Flowing Radially Between Concentric, Parallel, Circular Plates," Aerosol Science, 1970, Vol. 1, pp. 279-285, "Optical Model Partial Wave Analysis of 1 GeV Proton Nucleus Elastic Scattering," i Physical Review (to be published). BLIOGRAPHIC DATA IEET 1. Report Nc. UIUCDCS-R-72-553 Title and Subtitle Partial Wave Analysis of Elastic and Inelastic Scattering of Dirac Particles Author(s) Robert L. Mercer 3. Recipient's Accession No. 5. Report Date October 1Q72 8- Performing Organization Rept. °- UIUCDCS-R-72-553 Performing Organization Name and Address University of Illinois at Urb ana -Champaign Department of Computer Science Urbana, Illinois 6l801 10. Project/Task/Work Unit Nc 11. Contract/Grant No. NSF Grant GP 25303 Sponsoring Organization Name and Address National Science Foundation Washington, D.C. 13. Type of Report & Period Covered Doctoral Dissertation 14. Supplementary Notes Abstracts The problem of elastic and inelastic scattering of particles from a quantum mechanical center of force has been analyzed theoretically and computationally. Programs have been written in which effect dues to excited states are treated correctly. NADIR handles the case of Coulomb monopole elastic and inelastic scattering with spin zero targets. ZENITH handles the general problem. A number of checks of the programs are described. The programs are then applied to estimate the dispersion corrections in elastic scattering of electrons from isotopes of calcium, and to investigate the effect of Coulomb distortion in magnetic scattering at 180°. /Key Words and Document Analysis. 17a. Descriptors 7 Identifiers/Open-Ended Tc 7 < "'.AH Field/Group B R availability Statement - Unlimited '"iivis (10.70) 19. Se< tint y ( lass (This Report ) UNCLASSIFIED 20. Security (lass (Thi: 1 '..*<• UN( LASSIF1ED 21. No. of Pages 157 22. Price USCOMM-DC 40329-P7 OCT 24 19/ J