18
Nuclear Physics A529 (1991) 467-484 North-Holland ACCURATE TREATMENT OF COULOMB CONTRIBUTION IN NUCLEUS-NUCLEUS BREMSSTRAHLUNG D. BAYE, C . SAUWENS, P. DESCOUVEMONT' and S. KELLER2 Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus Daine, C.P . 229, B10S0 Brussels, Belgium Received 2 January 1991 Abstract: Partial-wave expansions of nucleus-nucleus bremsstrahlung cross sections converge very slowly as a function of orbital momentum, especial' y at small deflection angles. While nuclear effects can easily be restricted to a limited number of partial waves, Coulomb effects contribute to much higher partial waves because of the long range of the force. We accurately solve this problem by separating the bremsstrahlung matrix element into a purely Coulomb part and a fast-converging series . The Coulomb contribution is calculated by numerically integrating analytical expressions of the Coulomb bremsstrahlung matrix element . The accuracy of the results and the importance of the corrections are studied in a potential-model description of the a(a, ay)a reaction . 1 . Introduction Electromagnetic transitions in the continuum are the main tool for testing model wave functions of nuclear systems without (or with very few) bound states . Indeed, while scattering data are only sensitive to the asymptotic form of these wave functions, electromagnetic matrix elements also test their inner part. In spite of its interest, nucleus-nucleus bremsstrahlung has received relatively little attention, experimentally or theoretically . The theoretical situation has however changed in recent years . Well-founded models providing various bremsstrahlung cross sections are now emerging . Within the potential model describing the nuclei as two interacting massive points, the first ab initio calculation of nucleon-nucleus bremsstrahlung has been performed by Philpott and Halderson ' ) . In that work, special attention is paid to an accurate treatment of the slowly converging radial integrals . Nucleus-nucleus bremsstrahlung has first been studied by two of us in a microscopic model taking account of all the nucleons involved in the reaction and of Pauli anti symmetrization 2-4 ) . The theoreti- cal cross sections for the a(a, ay)a reaction 2 ) are in good agreement with_ the existing experimental data 5,6) . This agreement is a significant success for a model which does not contain any free parameter . Indeed, the wave functions are calculated from a microscopic hamiltonian involving a standard two-body interaction, whose exchange parameter is adjusted to reproduce a + a elastic scattering data . However, the very peculiar geometry chosen for the experiment does not allow a really severe ' Chercheur qualifié FNRS . ' Present address : Department of Physics, University of Wisconsin, Madison, W153706, USA . 0375-9474/91/$03 .50 c() 1991 - Elsevier Science Publishers B .V. (North-Holland)

Accurate treatment of coulomb contribution in nucleus-nuclues bremsstrahlung

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Nuclear Physics A529 (1991) 467-484North-Holland

ACCURATE TREATMENT OF COULOMB CONTRIBUTION INNUCLEUS-NUCLEUS BREMSSTRAHLUNG

D. BAYE, C. SAUWENS, P. DESCOUVEMONT' and S. KELLER2Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus Daine,

C.P. 229, B10S0 Brussels, Belgium

Received 2 January 1991

Abstract: Partial-wave expansions ofnucleus-nucleus bremsstrahlungcross sections converge very slowlyas a function of orbital momentum, especial'y at small deflection angles. While nuclear effects caneasily be restricted to a limited number of partial waves, Coulomb effects contribute to much higherpartial waves because ofthe long range of the force. We accurately solve this problem by separatingthe bremsstrahlung matrix element into a purely Coulomb part and a fast-converging series. TheCoulomb contribution is calculated by numerically integrating analytical expressions of theCoulomb bremsstrahlung matrix element . The accuracy of the results and the importance of thecorrections are studied in a potential-model description of the a(a, ay)a reaction .

1 . Introduction

Electromagnetic transitions in the continuum are the main tool for testing modelwave functions of nuclear systems without (or with very few) bound states . Indeed,while scattering data are only sensitive to the asymptotic form of these wavefunctions, electromagnetic matrix elements also test their inner part. In spite of itsinterest, nucleus-nucleus bremsstrahlung has received relatively little attention,experimentally or theoretically . The theoretical situation has however changed inrecent years . Well-founded models providing various bremsstrahlung cross sectionsare now emerging .

Within the potential model describing the nuclei as two interacting massive points,the first ab initio calculation of nucleon-nucleus bremsstrahlung has been performedby Philpott and Halderson ' ) . In that work, special attention is paid to an accuratetreatment ofthe slowly converging radial integrals . Nucleus-nucleus bremsstrahlunghas first been studied by two of us in a microscopic model taking account of all thenucleons involved in the reaction and of Pauli antisymmetrization 2-4 ) . The theoreti-cal cross sections for the a(a, ay)a reaction 2) are in good agreement with_ theexisting experimental data 5,6) . This agreement is a significant success for a modelwhich does not contain any free parameter . Indeed, the wave functions are calculatedfrom a microscopic hamiltonian involving a standard two-body interaction, whoseexchange parameter is adjusted to reproduce a + a elastic scattering data . However,the very peculiar geometry chosen for the experiment does not allow a really severe

' Chercheur qualifié FNRS.' Present address : Department of Physics, University of Wisconsin, Madison, W153706, USA.

0375-9474/91/$03.50 c() 1991 - Elsevier Science Publishers B.V. (North-Holland)

468 Bate et al. / Nucleus-nucleus bremsstrahlung

test for the theory. Potential model analyses of the same reaction 7.8) show acomparable agreement with the data so that discriminating between the models isnot possible . Other bremsstrahlung reactions have been studied it the potentialmodel 9`0) . More recently, the microscopic model has been employed to study thea + 1 He and p+a bremsstrahlung reactions ", '2 ) . Again, this parameter-free modelagrees fairly well with the experimental data . However, the scarcity of these dataand their rather large error bars do not really allow serious questioning of the model.The potential model analysis of a + a bremsstrahlung performed in our group 8)

revealed a theoretical problem. Although an excellent convergence is reached inthe calculation ofthe contribution of each partial wave, convergence is not achievedin the partial wave expa "~sion with respect to the orbital momentum. The origin ofthe problem is easy to trace : while nuclear bremsstrahlung is restricted to a fewpartial waves, Coulomb bremsstrahlung remains important for higher waves becauseof the long range of the Coulomb force . The lack of convergence does not cast adoubt on the agreement between theory and experiment since experiments havebeen specially devised to detect 9Q° deflection in the centre-of-mass frame, whereCoulomb effects are weak. For the existing data, the experimental error bars aremuch larger than the theoretical uncertainty . On the contrary, the Coulomb contribu-tion responsible for the slow convergence is important at small deflection angles.The aim_ of the present paper is to provide a tractable solution to the Coulomb

bremsstrahlung problem. This solution is inspired by the analog problem for elasticscattering . Ass®C. ..s

s well known, the series expansion for the scattering amplitude f ofüi .v ...rr

vs

charged particles converges very slowly. Therefore, f is rewritten as

.Î=k+(.f-fc-)1

(1)

where

(- is the Coulomb amplitude for the scattering of two point charges ") .Indeed, fc- is known in a compact analytical form and f -(, is a rapidly convergingseries expansion . In the following, we apply decomposition (1) to the bremsstrahlungmatrix element, i.e . we decompose it in a purely coulombic term and a termcontaining the nuclear effects . This residual term is calculated with a partial waveexpansion and converges rapidly . The method therefore requires a compact formulafor the purely coulombic matrix element. Such a formula can be obtained from abasic integral derived by lüordsieck 'a) .

In sect . 2, general formulas for bremsstrahlung established in ref. 2 ) are rewrittenin a way suitable for introducing the Coulomb correction . Partial wave expansionsand their specialization to the potential model are described in sect . 3 . The conver-gence problem and the principle of its solution are explained in sect . 4 . The compactform of the Coulomb bremsstrahlung matrix element is established in sect . 5 forthe electric multipoles, and commented upon. In sect . 6, computational aspects arediscussed . The a + a bremsstrahlung is revisited in sect . 7, with emphasis onconvergence aspects . Concluding remarks are presented in sect . 8 .

D. Baye et al. / Nucleus-nucleus bremsstrahlung

2. Bremsstrahlung cross sections

In the centre-of-mass (c.m.) system, let us consider two zero-spin nuclei withreduced mass W colliding with relative momentum p;, which we assume to be inthe z direction, and energy E; = p;/2A. The corresponding wave function is denotedas If j(' ) . After emission of a photon with energy E,, and momentum py = hky inthe direction ,fly = (8,,, (p,,), the system is in a final state IP'f ) with momentum pf,in the direction Of = (Of, (Pf ), and energy Ef= pf/2j,c. Up to small recoil corrections,energy conservation requires

Ef=E;_Ey .

(2)

Energy and momentum conservation leave five free parameters which are chosento be Ey, ey , v-1 -5 Of and Vf .The c.m. differential cross sections given by 2)

d30.= Ey f

~!+

'il2P j(*'f)(Df)IHe(ky, EqdEy d1ly dflf (21rh)4 hc q .

In this expression, He is the operator describing the emission of a photon with wavevector ky and circular polarization E q (q = :1: 1) . The multipolar expansion of thisoperator reads

He(k, Eq) =- E

ÀJ~Ag

1£-q(

ÇPy-j -0y-) 0)Agir

4

where a= 0 or E corresponds to an electric multipole and o, =1 or M to a magneticmultipole . The form of the multipole operator S.,, depends on the model: eqs. (8)and (20) of ref. 2) correspond respectively to the microscopic model and to thepotential model [see also eq. (20) below] . The coefficients aA are given by

while the symmetry of T; + ' with respect to a reflection in the xz plane leads to

uÂ-~.(®f~ 0) _ (-)g+Qu0g(®f, 0)

(8)

With eqs . (4) and (6), the cross section (3) can be rewritten as

3

2= 2

Ey

_Pf

(-)A+A'+1

(-)gLIA u''

*FAk"

+Q'(D,) .A sidEy dDf dny(21rh)4 fsc A

~Y

A'a'

aAE = iaM = (iky)A [2 1r(2A + 1)(A + 1)/A]1/2/ (2A + 1)!! (5)

and the Dam,-q functions are rotation matrix elements .Let us define the multipole matrix element

J4cA IV (6)

Rotation invariance of VY;' around the z axis provides the property

u A' (Of, (Pf) = e''w f uAw (Of, 0) (7)

TO

The

and

and

with

hoton-angle functions F"'r are defined as

lim r ( y ) = (47r)ß/" Y_'(2j+1) - ' I-'(AA'Iu,- lz 'BjM- - iu.')(AA'1 - 1L10)Yi

(10)

where the prime on the summation symbol indicates that the sum is restricted to jvalues for which j + A + A'+ e is even . Useful properties of these functions are

F~,~(®y,0)=(-)~+"F~,F(0 0)-

(12)

Other differential cross sections are interesting. Integrating (9) over ,fl,, leads to

d2o-

E,, p`'=87T-

fIdE,, dOf

(2Trh )4 hC AMrr

e cross section dcr/dE,, is then obtained by integrating (13) over 12f . Because ofthe cylindrical symmetry property (7), this double integral reduces to an integralover 8f only.

3 .

et a9 / Nuelea:s-nucleus bremssirahlung

artial-wave expansions

l; m;

(2A+1)-'1u;,,,12 .

(13)

Practical calculations require an expansion cf W ;+ ', 'hf- ' and u,,, in partial-waveseries . Let 41

r"' be a partial wave of a unit-flux scattering wave function . For arbitrary

directions ,fl; and ,f2 f, IP +' and 1f-' can be expanded as 2)

+'=(47r)'/21 (21;+1) - ' / 2 y~mj(, Lj~f,Ijmj(14)

f-' = (47r)'/`1 (21f+ 1)-' f`'

Yin, (nf) exp [-2i(o-,r+ S,r )]qi j r'M ',

(15)Ifmr

where o-, and S, are respectively the Coulomb and quasinuclear phase shifts . Forsymmetric systems, the sums over l; and if are restricted to even partial waves. Takinginto account the particular choice for fl;, we obtain the expansion of LIÄ,, as

uA _ (41r)1/2

(21f+ 1) - ' Y;(~f)(1~AOM- lflu)Xä~;rr

(16)�

Xä,;r r = a,(21f+ 1) 1/2 exp [2 i(o -,r +S,r )1WfIrj~~~ä (~

f)

(17)

Introducing eqs . (16) and (10) in eq . (9) allows one to recover the result (13) of ref. 2 ) .Until now, all the equations displayed are valid for the microscopic case as well

as for the potential model . Differences arise in the expressions of IRA, and of

~'n .

Now, we specialize to the local model. Let us denote by Rl(r) the real radial solutionsof the local Schrödinger equation at energy E = h2k2/21L with the asymptoticnormalization

where F, nd G, are the regular and irregular Coulomb wave functions and where312 is equal to unity if the system is symmetric and to zero otherwise. The factorv - '/2 corresponds to the fact that si p" is assumed to be a partial wave of a unit-fluxscattering wave function . The partial waves 41'" can be rewritten as

or by

where

RI (r)

= (1+5,2)'/2v-'

/2(kr)-'(Fj(kr) cos &,+G,(kr) sin 8,],

(1S)r-oo

D. Baye et al. / Nucleus-nucleus bremsstrahlung

471

,PIm = (4r) 1 /2(21 + 1)1/2il exp [i(Q, +81 )]YÎ (12)RI(r) .

In the potential model, the spin-independent part of the electric multipoleoperators reads

.~«Â~ _ (2A + 1)!!(A + 1) 'ky"-'(eh/»cl

(19)

x [Z1XAp.(A2kY/A, r) +( - )AZ2XA~.(Atky /A, r)] ' ®

(20)

[see eq. (20) of ref. 2)] . In eq. (20), Z, e, Zee, A, and A2 are respectively the chargesand masses of the colliding nuclei, and A = A,+A2 . The vector functions X,, aregiven by

X EN.(k, r) =

k2r+® 1+ra

ja(kr)Y,(0)

(21)ar

XE,,,(k, r) _ (47riA ) -' I

Y~ ( 1lk)[2ik+ k2r-k(k

r)] exp (ik

:) ûd2 k .

(22)

Notice that we do not make use of the long wavelength approximation in order toensure the convergence of the integrals calculated below in eq. (24) . Here and inthe following, we only consider the electric multipole operators. Similar expressionscan be derived in a parallel way for the magnetic multipoles .With eqs . (19) and (20), the reduced matrix element appearing in eq. (17) can

be written for electric multipoles as

(

'`iI E 110 1

') =exp[i(~,,+5,,-af-S,f)]

x (2A + 1)!!(A + 1)-'ky " - '(eh/tLc)[4,7r(2A + 1)]'/2(21;+ 1)

x (I;A00 1 If0)[Z,IÂlf(A,k,,/A) +( -)AZ2IAilf(A,k,,/A)] ,

(23)

I' -'r(k) =A(A+1)

rR,,(r)j,,(kr)d

[Rl;(r)]drJ()

dr

+;[I;(1;+1)+A(A+1)-Ir(Ir+1)]

axf(

R1, (r)

[rj,(kr)]Rl;(r) d r .

(24)dr

72 ye et aL / Nucleus-nucleus 6remsstrahlung

ese equations are similar to those providing the external correction described inf. ®) . Here also, the Siegelt theorem is not employed for convergence reasons. The

integrals appearing in eq. (24) converge very slowly but the convergence can beaccelerated by performing a contour transformation beyond some r value') .

In the to

model, the radial Schr6dinger equation is solved numerically at theenergies E; and Effor different partial waves. Then the integrals Iâ

rf(k) are calculated.

They allow the calculation of the uE [eq . (16)] and of the cross sections . However,the number of calculated partial waves being finite, a truncation error appears . Thiserror and its correction will be the main topics of the following sections .

4. Coulomb convergence problem

The series (16) giving the matrix elements u", L must be truncated in some way.At a even energy E® , an orbital momentum I�,, exists beyond which the quasinuclearphase shifts S, and the nuclear component of the partial wave functions 0" becomenegligible. Beyond It,,, the partial waves are purely coulombic. However, the integrals

1~1¬ (k) [eq. (24)] do not vanish for purely coulombic wave functions . They onlydecrease (rather slowly in some cases) as a function of l; and If . One easily guessesthat the decrease is especially slow when Coulomb elastic scattering is important,i.e . at small ®f angles.

Except for small values of Of, convergence can be reached in practice by addinga sufficient number of partial waves to IM . However, this technique is very time-consuming, in particular because high-partial-wave matrix elements are longer tocompute. We propose here a much cheaper technique similar to the one used forthe solution of the Coulomb problem for elastic scattering .

Let us denote as uâ, and X',�f the quantities uA~, and X,,,,f obtained in the purelyCoulomb case. Then eq. (16) can be rewritten as

uA~ = uâ,A +(47r)'/' ~ (2If+ 1) - ' Y;(12f)(hADlu, Iflu)(X-i ; l, -

X âi f )

(25)

The series in (25) is now converging fast since the difference X',,,r -Xo f becomesnegligible beyond 1�,, . The use of (25) requires two additional calculations : XA�f.and u;~ . The coefficient XA�` is just a particular case of X .,;, f and does not leadto any additional complication . The time lost in computing this coefficient is largelycompensated by the fact that the sums over l; and 1f can be truncated near 1�,, . Theinterest of (25) therefore relies on the existence of a compact expression for u_ Â~ ,and on fast algorithms for computing it . The expression of uâ~ is derived in sect . 5.

5. Coulomb bremsstrahlung matrix element

For the sake of simplicity, we restrict the presentation to a non-symmetric system .Symmetrization requires calculating the matrix element u,,,L twice. With the present

D. Baye et al. / Nucleus-nucleus bremsstrahlung

473

With (5), (20) and (22), the Coulomb matrix element for an electric multipolebecomes

uâw- Q'a

(fc )(af) I

Âjz l

oc

=[(2A + 1)/81rA(A + 1) ]'/2(eh/uck,,)

is the orthogonal component of a vector X, with respect to k. The matrix element(31) can be expressed in a (rather) compact form.

Let us define the auxiliary integral

J(A, q)=(W (fc ) (flf)I r-'exp[i(q-kj+kf) - r - Ar]jV1(jc'(ni)),

(33)

where A is a real parameter. Other variables (such as k; and kf) on which J dependsare not displayed explicitly . With this definition, Ic can be expressed as

Ic(k) = (2k - k2®k) ' [kiJ '(0, q) - ik;®kJ(0, q)]q=k;-kf+k

In (34), J'(0, q) is the derivative of J(A, q) with respect to A, calculated for A = 0.The differentiation with respect to k ; is performed with v ; and 'R ; considered asconstants. The replacement of q by k ; - kf+ k takes place after the calculation of®k;J, i.e . q is also a constant during the differentiation process.The expression of Ic is then obtained after a heavy but straightforward calculation,

from J(A, q) whose expression is derived by Ncrdsieck in ref. '4) (see appendix A).

In order to display the final result, we separate Ic into two parts

(34)

(35)

x[Z,I;,,(A2k,,/A) +( - )"Z2I �;,(A,k,,/A) ], (29)

where

I (k) = YZ(nk)I c(k) dd2k . (30)

In (30), Ic is given by

Ic(k) = (1Pfc)(flf)I(2ik + k2r1 ) exp (ik - r) - Q ~c'(fli)) (31)

where

X1 =X -k-2k(k- X) (32)

conventions and notations, the Coulomb wave functions read ' 3 )

W;c "(fl;) = Ci exp (ik; - r),F,[-irii ; 1 ; i(kir-ki - r)] (26)

Wfc , (0f) = Cf exp (ikf - r),F,[inf; 1 ; -i(kfr+kf - r)] (27)

where

C = v - 'i2F(1 + ij) exp (- 6rl) . (28)

474

D Bave e' al. / Nucleus-nucleus bremssirahlung

nd respectively to the terms 2k and -k'®k in the first operator ofeq. (34), after simplifications .

e first display and comment on I;

which corres

Ii = 2

- [U1.Î(x) - ik(1-x)f'(x)] ,

(

wheref(x) is a shorthand notation for the hypergeometric function

.f(x) = 2F, (I + i7lè, -i7lf ° 1 ; x)

(37)

with

with

1-x=(2k r

e multiplicative factor F is given by

+ 92)(2kà - q - q2)/ q,[(kà+ kf)`' - k2] ,(38)

q=k® -k r+k .

(39)

The expression (38) defines a variable x which is non-negative and strictly smallerthan unity .

e computation of the hypergeometr c function (37) is thereforerelatively simple .

e variable x vanishes when kà - kf= kàkf and kà - k = k àk, i.e. atforward angles for both the final state and the emitted photon. The vectors U, and

are defined as

F=4-gCàCfexp ( ,T'gà)q -211+'nd (2k' - q_q')id,,~-nd[(ki+kf)2 - k=]44

e quantities appearing with imaginary exponents in (44) are positive (see appen-dix),

e expression a 4l is unambiguously defined by exp (iq In a).e matrix element I(' defined in eq. (31) changes sign when interchanging

respectively à and n® with -kf and i7 f . The same property is verified by the matrixelement appearing in the calculation of the total Coulomb bremsstrahlung crosssection between two charged particles, with all multipoles - electric and magnetic- included . The operator relevant to this matrix element is Ey exp (ik - r) - V whereE q - k = . This matrix element is very similar to I; except for the replacement of°2ik by E4 . Therefore, one expects I,- to display the above-mentioned property.owever, except for x which is symmetric when exchanging kà and rl à with -kf andf, this =vmmetry is not apparent in eq. (36) . In order to reveal this hidden property

and to lin :- eq . (36) with previous works, we define a new variable y by

1-y=(1-x)-'

(45)

=(-lf- gi)kikr, (40)

U,= k®k®'- kfk® , (41)

k;= kà/(kr - q+2-g2), (42)

k f= kf (kà - q - q2) . (43)

and the "symmetric" hypergeometric function

with

D. Baye et al. / Nucleus-nucleus bremsstrahlung 475

g(Y) = 2FI(- i71i, -i7lf ; 1 ; Y) -(46)

The variable y is negative and jyj may be larger than unity . Using simple propertiesof the hypergeometric function 'S ) and ki 7l i = kfqf, Ic, can be rewritten as

I, =81rCiCf exp (7r71i)q-2(1+"ni"nd(2ki , q-g2)'ni(2kf . q+g2)'n6

xk- [ki77i(ki-kf)g(Y)+i(kikf-kfki)(1 -Y)g`(Y)] .

(47)

This form of I; possesses the expected symmetry, if one understands the ambiguousproduct of phase factors (-1)'ni(-1)'nr as exp [ir(rl f-71 i )] . In fact, an expressionsimilar to (47) has first been derived by Sommerfeld for the calculation of the totalCoulomb bremsstrahlung cross section [see e.g. eqs . (4), (l0a), (11), (11b) and (13)in sect . VI I.6 of ref. '6)] . However, the factor exp (a7li) is not apparent in thatreference . The advantage of the form (36) is that it is derived from the Nordsieckexpression for the matrix element (33) in which the phase factors are unambiguous.An additional advantage is that the argument off(x) is always smaller than unity,so thatf(x) is easier to compute than g(y) .

Now, let us display the more complicated second term 12 of eq. (35):

I2 =I'K.2{U ; - (VI +kf)f(x)-i[U2 - (VI +q')-Ui - (iV2+71f'ki)]

x (1-x)f'(x)+iU- . V,(1-x)2f"(x)}

(48)

q'= 2q/q2,(49)

VI = (1+t71i)q ' +l(1if

71i)kf,

50

V2=g~_kf_ki .

(51)

In eq. (48), the second derivative off(x) can be eliminated with the differentialequation of the hypergeometric function. Therefore, expression (48) is not muchmore complicated to program than eq. (36) . Notice that the orthogonal componentsof the vectors [eq . (32)] which are employed in (48) can be partly modified withthe relation X - Y1 = X1 - Y.The complexity of (36) and (48) indicates that the integral (30) cannot be

performed in compact form. However, we display here the dipole integral (A =1)in the long wavelength approximation. In this case, I2 can be neglected and (30)provides

with

Iig(k)=34?TZCiCfeXP (~71i)[(ki-kr)2]- -n1i(ki_]Cr)`( ni-n~)~

x (ki+ kf) icn i+n,i-' kikrk[(71f - 71i) Yi(Df)f(xo)

-i[ Yi (12r) - (3/47r) 1i2SMo](1 -xo)f (xo)] ,

xo=2(kikr- ki - kr)/(ki-kr)2 .

(52)

(53)

76 'are et al. / Nucleus-nucleus bremsstrahlung

is expression, as well as I

in the general case is larger at forward angles thanA juat backward angles because of the factor 1 /( i - r)`.

6. Numerical aspects

Since the integrals (30) are computed numerically, fast algorithms have to beemployed for the calculation of the hypergeometric function and for the integration .Fast algorithms providing the hypergeometric function with complex parametersare described by Luke ") . They are based on expansions in Chebyshev polynomialswhich are particularly efficient if x < 2- . The coefficients ofthe expansion only dependon the constants qi and of in eq. (37) so that they can he stored for given initialand final energies. The variable x is comprised between 0 and 1 but we have chosento use transformation formulas to the variable 1- x when x> 2. Two hypergeometricfunctions then need be calculated . The derivative of the hypergeometric functionis simply given by

F(x) - r%f(7%i - i)2Fl(2+ i71i, 1 - t gr ; 2 ; x)

(54)

and can therefore be calculated along the same lines.The determination of Ic,,,, requires a double integration with respect to ek and

q)k . T'-p matrix element Ic(k) is invariant for a reflection of k with respect to theplane containing ki and kf . With the choices for ki and kf indicated in sect . 2, theintegral reduces to

1i,(k) = 2

sin ®k dek

d4Pk 1Ze [ Y£ (nk)]Ic(k) .

(55)Jo

0

In fact, the function I'(k), in spite of its complexity, presents relatively smoothbehaviou s wail fespect to Ok and lPk . An excellent accuracy can be achieved witha Gauss-Legendre quadrature for 8k and a Gauss-Fourier quadrature 's) for ~Pk (i.e .

constant spacings between the mesh points and equal weights) . For A =1 or 2,typical numbers of points are respectively 16 or 24 for ®k and 8 or 16 for (P k . Thesesmall numbers of points keep the computing times below one second for mostpresent-day computers .The final expression for Ic,,,, needs to be tcsted against programming errors . A

rather severe test is thatICo(k) = 0

(56)

since there is no monopole term in the electromagnetic interaction . Under theconditions described above, we have found that this test is satisfied with an accuracybetter than 10- ' °.The behaviour of the matrix element uAc ®f, 0) as a function of ®f is exemplified

for A = 2 by fig . 1 . The conditions of the calculation are those of the a + a casedescribed in sect . 7 for Ei = 8 MeV and E,, = 1 1VIeV, but UÂ

is not symmetrized .

quantity

N

D. Baye et al. / Nucleus-nucleus bremsstrahlung

477

»=O0.5

0.4

0.3

0.2

30 60 90 120 150 180

(degrees)

Fig . 1 . Moduli of the quadrupolar Coulomb matrix elements U2C multiplied by (k ; - kf)2 (in units ofh'/2c-1/2 fm) as a function of the final angle Of . The (unsymmetrized) matrix element is calculated under

the conditions of a + a scattering (see sect. 7) for E ; = 8 MeV and E,, =1 MeV.

The modulus Su2,û is multiplied in fig . 1 by (ki - kf)2 in order to eliminate the strong

forward peaking due to the g-2 factor in (44) . After this multiplication, the matrixelement behaves rather smoothly with respect to Of . At 0° and 180°, the 1L 00 matrixelements vanish because the initial and final wave functions both possess a cylindricalsymmetry along the z axis . In addition fig. 1 presents an approximate symmetrywith respect to 90°. This effect is typical of small photon energies and disappearsfor larger energies .The slowness of the Coulomb convergence is illustrated by table 1 ; under the

conditions of a + a scattering for E ; = 8 MeV and E,, =1 or 5 MeV. In table 1, the

_ ~ 1û2M - (41r)'i2- (21f+ 1) - ' Y;(f2f)(le20M- lfl

,)X2i f )~ 2~~u2,u ~2

(57)Il

i i i,

TABLE 1

Parameter s measuring the convergence of the series expansion of theCoulomb matrix element for different values of the truncation orbital

momentum IM for E; = 8 MeV [see eq . (57)]

I A,,Ey = 1

O f = 5°

MeV

Of = 30'

Ey =

Of = 5°

5 MeV

Of = 30'

6 0.82 0.178 6.5 x 10 --2 1 .4 x 10-28 0.71 0.151 1 .5 x 10-2 3 .1 x 10-310 0.60 0.072 3 .2 x 10-3 5.3 x 10-412 0.49 0.096 6.5 x 10-4 7.6 x 10-514 0.39 0.08516 0.31 0.05218 0.24 0.058

78 Bale et al. / Nucleus-nucleus bremsstrahlung

is tabulated as a function of Im in different cases [see eq. (61) below for the definitionof 2~] . This ratio vanishes if l�,, tends to infinity . For E,, =1 MeV, the o,:crease isvery slow for ®f= 5° and slow for ®,-= 30° . Moreover, it is not regular. On the contrary,for large photon energies, the convergence is fast.

7 . Application to a + a bremsstrahlung

e present model is applied to the a(a, a y) a reaction . As in other works'"'),we employ the deep gaussian potential of Buck et al. `9) to describe the a + a system .This potential nicely reproduces the phase shifts of the I = 0, 2 and 4 partial wavesup to 20 MeV, with only two parameters . The relation bctween this deep potentialand shallow ones is clarified in ref. 20 ) .

In a symmetric system such as a+ a, parity and angular momentum conservationforbids odd-parity multipoles. The M 1 contribution vanishes in the long-wavelengthapproximation and is therefore expected to be negligible at the energies consideredin the following . To a very good approximation, the bremsstrahlung cross sectionis given by its E2 contribution .

First, we consider the so-called Harvard geometry for which experimental dataexist `,6 ) . In this geometry, for a symmetric system, the nuclei are detected in thecoplanar laboratory directions fl, = (01 , 0) and 12, = (01 , 7r), and the photon remainsundetected. This geometry corresponds in the c.m. frame to

ef= 27r,

(Pf= 0 .

The final c.m . energy of the symmetric system depends on ®, according to

Ef = E; tant 0, .

(59)

In the Harvard geometry, the photon energy E,, therefore varies with E; .After an integration over ®,, of eq. (3) in ref. 2 ), the E2 cross section reads

d2a/df2, U22 =7r -'k4kfii - ' sin4 e, sin -5 20,

x [3(u2E0)2 +4(u2,)2+(u2)2+~ Re (u ôu22)] ,

(60)where the amplitudes u~, are defined in eq. (6) . In practice, these amplitudes arecalculated with eq. (25) in which the sums over Ii and If are restricted to even orbitalmomenta and u2C is replaced by the symmetrized expression

7EC EC

EC(u212 - u2w(k,ki,kf)+u2,,k,ki, -kf) .

(58)

(61)The integrals I'* << [eq . (24)] are computed numerically, from radial functions R,obtained with a Numerov algo i~hm 2' ) . They are sensitive to small variations ofthe wave functions . Therefore, since we want high accuracy on the cross sections(at least six digits), the numerical calculations must be performed very carefully .Typical mesh sizes for the resolution of the radial equations are of the order of0.05 fm . In addition, we employ exact Coulomb functions instead of these numerical

solutions when the nuclear effect is negligible . The contour transformation takesplace at rather large distances (>20 fm), depending on the truncation orbital momen-tum IM .The cross sections obtained with eq. (60) are compared in fig . 2 with experimental

data at 8, = 35° [ref. 5)] and 0, = 37° [ref. 6)] . The dashed curves correspond to acalculation without Coulomb correction, in which the sums in (16) are truncatedat 1M = 8. These are the conditions employed in the microscopic calculation of ref. 2) .The full curves include the Coulomb correction. The results are almost insensitiveto the truncation value Im provided that it is at least 6. As expected from the largec.m. angle e f [eq . (58)], the corrections are small . They can be considered as negligiblewith respect to the model accuracy and to the experimental error bars. The smallnessof these corrections explains the good convergence obtained by Liu et al. "''2) .

In order to have a better insight into the importance of the corrections, let usnow consider differential cross sections d 2Q/dE,, dil f [eq . (13)] as a function of 8 f .The corrected results are compared in fig . 3 to uncorrected ones obtained withdifferent truncation values 1M for a rather large photon energy (with respect to theinitial energy) . The error due to the Coulomb force is at most of the order of 7%and it does not appear at angles larger than 30° . The situation is completely differentat small angles for the smaller photon energy 1 MeV (see fig. 4) . The results obtainedfor 1M =6 or 8 are two orders of magnitude too small at very small angles. Even atlarge angles, the different curves still differ by an observable amount.The slowness of the convergence and the accuracy of the correction are better

illustrated by table 2, for E; = 8 MeV and E,, =1 MeV. Here, we perform a moreaccurate numerical resolution of the radial Schr6dinger equation, with the smallermesh size 0.02 fm. The accuracy of the results has been tested by varying the mesh

Nb =pN_p C

D. Baye et al. / Nucleus-nucleus bremsstrahlung

479

Fig . 2 . Laboratory differential cross sections in the Harvard geometry at ®, - 35° and 37°, as a functionof the initial c.m . energy B; . The full lines represent the present corrected results. Dotted lines correspondto a truncation orbital momentum IM = 8, without Coulomb corrections . Experimental data are from

refs . s .') .

480

15

10

100

80

60

40

20

ye et al. / Nucleus-nucleus bremsstrahlung

86

I_1

E, =8 MeV , Ey=5 MeV

16f (degrees)

Fig. 3. Cross section d`'o-/dEy dOf as a function ofthe deflection angle of at E; = 8 MeV and E, = 5 MeV.Comparison of the Coulomb corrected results (full lines) with uncorrected results obtained for different

values of the truncation orbital momentum IM (dotted lines).

parameter and the location of the contour transformation . The digits displayed intable 2 give an idea of this accuracy . At O f= 30°, the truncated results are close to8.5 nb/ eV - sr for Im = 6 and 8. For increasing Im values, they start oscillatingaround this value with an uncertainty larger than 10% . The situation is clearly worseat 5° : the cross section increases steadily with 1M without giving any indication ofconvergence . With the correction, the situation is completely different . The correctedresults displayed in table 2 show that convergence is reached with at least sixsignificant digits ~t Im =10. In fact, lm = 8 and even 6 already give excellent results .

'Of (degrees)

Fig. 4 . Same as in fig . 3 for Ey = 1 MeV .

Corrected

and

uncorrected

cross

sections

d 2Q/dE,, d!1f

(innb/MeV - sr) for different values ofthe truncation orbital momentum

I�,, (E; = 8 MeV, E,, =1 MeV)

vrbl'v w

1 : Uncorrected results truncated at /, .2 : Corrected results truncated at IM .

In practice, the results presented in the figures have a sufficient accuracy with I�,, = 8and a mesh size of 0.05 fm.The Coulomb effect on the total cross sections is displayed in fig. 5 . As expected

from the previous figures, the do,/dE,, cross section is mainly affected at smallphoton energies . Because of the sin Of weight in the 12 f integration, the error at smallangles does not affect too much the total cross sections. Anyway, truncating at 1�,, = 8as was done in ref. 2 ) introduces an underestimation of about 10% on the crosssection for 1 MeV photons.

150

100

50

D. Baye et al. / Nucleus-nucleus bremsstrahlung

TABLE 2

®f = 5°

Of=30°

E; (MeV)

481

Fig . 5 . Integrated cross section do,/dE, . for E.,, =1, 3 and 5 MeV as a function of E; . Comparison of theCoulomb corrected results (full lines) with uncorrected results obtained with the truncation orbital

momentum 1M = 8 (dotted lines) .

1 2 1 2

6 2.70 57.39988 8.48 8.615588 4.50 57.36104 8.54 8.6154910 7.19 57.36050 8.99 8.6154612 10.83 57.36050 9.64 8.6154614 15.32 57.36050 9.51 8.6154616 20.49 57.36050 8.56 8.6154618 26.10 57.36050 7.74 8.6154620 31 .90 57.36050 7.82 8.61546

482 D Raye et al. / Nucleus-nucleus bremsstrahlung

TABLE 3

Correctedand uncorrected crosssections dtr/dE.,(in nb/ MeV) for different values ofthe truncationorbital momentum lM (E; = 8 MeV, E., =1 MeV)

1 : Uncorrected results truncated at !�,, .2 : Corrected results truncated at lM .

More precise information about the convergence of the cross section for E; =8 MeV and E, =1 MeV is given in table 3 (again with 0.02 fm as mesh size) . Althoughthe uncorrected results slowly converge towards the correct value 50.07 nb/ MeV,the error is still of the order of 3% for Im = 20. On the contrary, the corrected resultsare stable beyond IM =10; they are already excellent for IM = 6 and 8.

. Conclusion

Taking the Coulomb term correctly into account has been traditional for a longtime in the calculation of elastic scattering. The correction is simple and is notconsidered as an additional complication . A similar correction for the less-studiedbremsstrahlung process has until now been overlooked . In the present paper, wedetermine this correction for electric multipoles and show that it can easily becomputed numerically . Of course, this correction is far less simple than in the elasticcase but introducing it in a bremsstrahlung code is not a major work and thecomputer time increase remains moderate in any case . In fact, being able to truncatethe series at smaller angular momenta often represents a gain in computer time (andsometimes a very significant gain) .

Bremsstrahlung reactions are a good tool for studying scattering states since thecross sections are sensitive to the inner part of the wave functions . They shouldhelp discriminating between models which give similar descriptions of elastic scatter-ing . However, for this purpose, new experimental data are necessary . Cross sectionshould not be measured only in the Harvard geometry . Such experimental datashould provide constraints on nucleus-nucleus potentials at low energies . In par-ticular, they might allow testing the nature - deep or shallow - of these potentialsby comparing bremsstrahlung cross sections obtained with potentials which areequivalent for elastic scattering . In the present paper, we have studied the Coulomb

lm 1 2

6 44.83 50.082918 45.61 50.07296

10 46.29 50.0728712 46.86 50.0728714 47.36 50.0728716 47.78 50.0728718 48.15 50.0728720 48.46 50.07287

correction in the simple potential model. The next step is implementing it in themicroscopic model 2), which should provide a very accurate description of nucleus-nucleus bremsstrahlung for several light systems .

D. Baye et al. / Nucleus-nucleus bremsstrahlung

Appendix A

The proof of (A.1) in ref. 24 ), initially intended for electron-nucleus bremsstrahlung,relies on the validity of the conditions

k i - q-zq2>0,

kikf+ki - kf+ki - q> kf - q+iq2>0 .

(A.7)

With the physical definition (39) of q, the following relations hold for A = 0,

483

y(o) =1[k2 - (kr-k) 2] ,

(A.8)

a(0)+ß(0)=2[(ki+k)2-k2J,

(A.9)

y(0)+B(0) = ![(ki+kf)2-k2] .

(A.10)

The quantities y(0), a(0)+,S (0) amd y(0) + B(0) are posit="re in the non-relativisticregime so that conditions (A.7) are satisfied .The variable x defined in (38) is

x = z(0) .(A .11)

From (A.6) and (A.8) to (A.10), x is smaller than unity . In addition, one may write

a(0)B(0) -ß(0)y(0) = 1 g2(kikr+ki - kr-2kilIkf ll ),

(A.12)

where kill and kfll are the components of k i and k f parallel to q. This guanrir; isnon-negative so that x is also non-negative .

From ref. '4), we deduce the value of the integral (33)

.j A, q)=27rCi Cf exp('lrni)a -(1+ins) yi(rji-''` )( y+S) i" f 2F,(I+ini, -igf s l;z),

here

a (A) =2(q2+A2) ,

(A.1

(A.2)

,B(A) =kf - q-iAkf , (A.3)

-y(A) =ki - q+iAki-a, (A.4)

S(A) = kikf+ki - kf-ß, (A.5)

z(A) =(aS-By)/a (y+B)=1-y(a+ß)/a(y+B) . (A.6)

484

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iye et al. / Nucleus-nucleus bremsstrahlung

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