A microscopic model for the pd → tπ+ reaction

Nuclear Physics A316 (1979) 215-243 ; © North-Hot7axd Prtbliahtng Co., A»rsterdmtr Not to be reproduced by photoprlnt or microSlm wiWout wriüm permiseioa from the publisher

A MICROSCOPIC MODEL FOR THE pd -~ tn+ RÉACTION A . M. GREEN t and E. MAQUEDA tt

School of Mathematical and Physical Sciences, University of Sussex, FaGner, $righton, BN19QH Sussex, UK Received 25 September 1978 Abstrat~t : A microscopic model for the pd ~ trz+ reaction is constructed using two-body isobar wave functions containing the d(1236) . This model results in an analytical form for the reaction cross section and so it is possible to check the validity of those more appróximate models that relate the pd -+ tn+ cross section to the pp -~ drz + cross sectioe. The calculations are performed at the three proton lab energies of 400, 470 and 600 MeV .

1. Introduction An ever increasing . amount of experimental data is now being obtained on the class of reactions p+A -~ (A+1)'-f-ti+ for proton lab energies of less than 1 GeV. This includes not only differential cross sections but also measurements involving . polarised beams t) and targets z.3). However, the theories used to describe these reactions are far from satisfactory ; often requiring sizeable normalisation factors to reproduce the experimental magnitude of the differential cróss sections. The most successful of these theories and the ones often employed by experimentalists, when presenting their data, are based on ,the idea originally suggested by Rudermann a) in 1952 in an attempt to understand the pd -. tn+ reaction and later on developed and discussed amongst others by Ingram et al. 5), Barry 6), Fearing' -9 }, Locher and Weber t ~ and Duck et al. 11). , In particular Fearing shows in rèf. ') how the . pA -.. (A+ 1~+ cróss section can, after a series óf approximations, be expressed in terms of the pp --> dn + cross section in the form

where C is a .kinematical factor and F a form factor depending mainly on the wave functions of the initial and final nuclei . The aim of the present work is to propose a .model that is more microscopic than the ones referred to above. In this way we are ablé to study the validity of those t Address after 1 August 1978 - Research Institute for Theoretical Physics, University of Helsinki, Siltaworcnpenger 20C, SF-00170 Helsinlá .l7, Finland tt pn leave from Departámento de Fisica, CNF,A, Avda Libertador 8250, 1429 Buenos Aires, Argentina . Fellow of CONICET, Argentina.

215

216

A. M: GREEN

AND E . MAQUEDA

earlier models based on the pp -> d~c+ cross section. For simplicity we restrict ourselves to the pd -. tom+ reaction . Furthermore, we use simpl~ed wave functions for the deuteron and triton so that the pd --. tom + cross section can be expressed in an analytical form. This enables us to study in a transparent way, the nature and validity of some of the approximations made in refs . 4-11 ). In sect . 2 we write down the expression for the pd -~ tom+ cross section in terms of a wave function containing the isobar resonance d(1236). In sect. 3 the form of this wave function is discussed and in sect. 4 numerical results for the pd -. tn+ cross section are given. In sect. 5 the expression for the pd -. tn + cross section is approximated so that a comparison can be madewith thosemodels which relate the pd -~ tom+ cross section to that of the pp -> d~c+ reaction . In sect. 6 a discussion is made of possible improvements of the very simple deuteron and triton wave functions used in sect. 2 and in sect. 7 a comparison is made between the piesent theory and experiment . 2. The pd -" tn+ differential cross section In this section we show how the pd -~ tic + cross section can be expressed in terms of a wave functiozx containing the d(1236). The latter is then obtained from coupled channels equations, involving only two baryons. Since throughout this work the aim is to derive an analytical expression for the cross section all radial wave functions will eventually be approximated by Gaussians. In this way we are able to test in detail those models a- i i) for the pd -" t~ + reaction that express the cross section in terms of the pp -~ dn+ cross section. The initial all-nucleon wave function can be written as ~~) _ (2~)_s exp {iko ~ (ro-~rl+rz))}~(d, ri - rz)~

(2.1)

where ko is the momentum in the pd c.m . system and ~(d) is the deuteron wave function, expressed in momentum space as ~G(d, r i -r2) _ (2~)-~

(2.2) ~ exp {ik ~ (rl -rz)} .f(k)~ J Since we eventually choose nucleons 0 and 1 to be the pair that interacts to produce a pion, eqs. (2.1) and (2.2) aré combined and written in such a way as to emphasise the ro ~ rl. dependence. On introducing spin and isospin and restricting nucleons 0 and 1 to be in a relative singlet-even state, the most important for pion production at these energies, we.. get

x (2,~)~ dk exp {~k+ïko) . (~{ro+r1Y-rz)} exP {~ro - rl)' (áko -zk)}I(kk . J

(2.3)

In this equation only the correlation between nucleons 1 and 2 has so farbeen included to give a deuteron. lin order to introduce the correlation between particles 0 and 1 first an expansion in partial waves is made of the plane wave exp (i8 ~ r), . where K = 4ko -ik and r = r° -rl. In analogy with the pp -. d~ + ~ reaction, which is known 1 z- l e) to be dominated by the mechanism .=z,s = °'~ 1 -+ (d~ °,s=z,T=1 --~ (~~.=°,s=l .r=o~+ ~ d~+ PP -' (~~ the assumption is now made that the pd -+ t~+ reaction is mainly due to the chain of processes Pd -' C(NoNI)z°' 1Nz~~ . t -. C(doNl)°z,1Nz7~.~ _.., C~oNI) °l ' °Nz~~ . ~~ -. t~+ . Therefore becomes

r) °z,1 +~ ~Kr {C(doNl) Nz~m%+~+hr, ~+ C(Nod 1)°z ' 1Nz~~%+m+~r, ~Îl where u(K, r) and w(K, r) are respectively the NN( 1 Dz) and Nd( S Sz) radial wave functions obtained by solving coupled differential equations for the two-body : system as described in ref. 13 ). In the present calculation only the isobar component w(K, r) is retained since this is known from the pp -" d~ + reaction to be by far the dominant term. The initial wave function ~, is then an antisymmetric combination of terms of the form (see fig. 1) 3

~i~) . _ ~(i1~mM) ~ (2izim~(m+~)

x {C(doNl )°z,1Nz~~%+~+ar.~+C~odl)oz,1Nz~~% +~+ai.~k} x (2n) -~ fdk éxp {~k+iko)' (z(ro+rl)-rz)}

~Krr) Yi~"(~.Î(k)"

(2:4),

The moré general result including NN components is ~ given in the appendix. This Jastrow-like wave function is, ofcourse, only an approximation to the true three-body wave function . However, a better approximation would mean solving coupled Faddeev equations, a formidable task, which only now is being attempted for the simpler case of the triton ground state l'). For the three-body system the above Jastrow type of approximation has been used before with.reasonable success 18" '9). The final wave function is taken to be the product of a plane wave pion with a

A. M. GREEN AND E. MAQUEDA

218

pure s-wave triton, i.e.' SYP = (27c) -s exp (iq' r,~ exp ( -iq ~ ~( r° +rl +rz))Nt~ro - ri~(ro - rz~ri - rz)

where the a are the relative radial wave functions within the triton and momentum in the tn+ c.m . system .

q

is the

9~

ko

-ko

Fig. 1. The main mechanism responsible in this model for the pd -~ tom+ reaction.

Knowing the initial and final wave functions, the pd -. tic+ cross section can be obtained from the matrix element ~~P~T-1S' q~~i~~

where T and S are the transition operators transforming a d(1236) into a nucleon wee refs. i z, z°) for more details concerning the properties of these operators. On making the transformation r = r ° - r l , s = 2(r° + r l ) - r z and R = 3(r° + r l + rz) the above matrix element reduces to the form A=

J

~f (k)Y~"(~

J

dra(r) exp (- Ziq ' r)

~Kr

r)

x ds exp {is " (k+Q)}a(s+Zr~c(s-2r~ J

(2.6)

where Q = Zk o -3-q . This triple integral can now be reduced to a double integral if a is assumed to have the simple form of an uncorrelated Gaussian, namely a(x) = exp ( -axz):

(2.~

219

The resulting double integral is

r) x dr exp (-Ziq ' r) exp (-~ar2) ~Kr J where wemust remember that K T 4ko -2k. Ifthe further approximation is now made that the deuteron wave function`i~~also a Gaussian, or sum of Gaussians, then the angular part of the k-integral can be performed to give ~ . B = -128a Z Nd exp ~ -4a 2ká -

8a ~k°

+ Q)Z Yz~"~Î ~

x ~dKK 2 exp ~- Ca2+ 4K2 } jz(iK~F(~ _ -128rsZ Y?~,"(~.M1Vd. (2.8) 8a) J )o where r) ~TZ1o(~) ~P (-i~) (2.9) ~Kr ~ J wheref(k) = Nd exp (- a?k 2) is the Fourier transform ofthe deuteron wave function and 11 1 (2.10) ß = 6aZ+ k°- .q . a 6a F(~ =

The pd -" t7c+ differential cross section in the c.m. system can now be written as ~[E(P)~ ex~ _

*a as 5 ~d)E(t)E(P) q3 _ .Î (1 +3 cost ~~~ ~212M2, ~) ko~2 Ors

(2.11)

where q and k° are as in fig. 1, E(p) = k° +mp, etc. are evaluated in the pd c.m. system, s{pd) is the invariant energy of the pd system, f * z/ors = 0.35 is the. r~Nd coupling constant and b is the angle between the pion direction and the vector ß defined in eq. (2.10). However, since ß is almost in thedirection ofthe incident proton, . S approximately equals B,~ the pion angle with respect to k° in the c.m . system. 3. 1Le~isobar wave fämction w(~ ~)

The expression for the cross sectión, eq. (2.11), still involves a double integral over K and r. Iq order to make this result more transparent and in a form amenable to further approximations, we now express the isobar wave function w(K, r) as a sum of Gaussians in "r". The double integral can then be carried out analytically .

220

-

A. M. GREEN AND E. MAQUEDA

Fig. 2. The real (R) and imaginary (I) components of the isobar wave function w(K, r) at 600 MeV. Dashed line ; ;exact wave functions. Solid line ; approximate wave functions from eqs. (3 .1a) and (3 .1b) .

The isobar wave functions used in this calculation were supplied by Sainio -and correspond closely to those of ref. 1S). A typical wave function is shown in fig. 2. When fitting these wave functions by a sum of Gaussians, care is taken to reproduce as close as possible the magnitude of the first maximum and alsó the position of the first node, The outcome of this are the two approximate wave functions w(real, approx) _ { -12.492 +9.504K -1.728KZ +r2(51.298528 - 84.4252K + 51.677914K2 -13.920906K 3 +1 .389312K4)}G(r,d = 0.3~ +r2(-1.75863+ 1.446021K-0.296548K 2)}G(r, d = 0.2), where

G(r, d) = r exp ( -dr1 )rl -exp (-1.4r2)].

(3.1a) (3.1b)

When using these wave functions it is necessary to retain alI the digits quoted for them to be a good approximation. In fig. 2 the comparison is also shown between these approximate wave functions and the exact ones at a lab energy of 600 MeV in the pp system. It should be added that at this particular energy which corresponds to the maximum in the pp -. dam+ cross sectión, our fit is at its best and as we go away from 600 MeV, down to 500 MeV and up to 700 MeV the fit deteriorates . However, since it is not easy to see by inspection how good or bad à particular wave function is, we also study the matrix element that arises in the pp -> dam+ reaction,

namely ' N(~

= J ~Z

~(~.

(3.2)

r)1o(igar) ~Kr r) '

where ~(d, r) is the deuteron wave function and qa is related tó K by the kinematics of the pp -> dn + reaction . But first in fig. 3 we show the ábsorption cross section for various combinations ofthe >/~(d, r) and w(K, r) appearing in eq. (3 .2). We only calculate the cross section for proton lab energies E(pp) of 300, 425, 510, 600 and 690 MeV joining these results by straight lines. The dotted line is the cross section obtained from the best type of calculation as discussed in refs. ts. ts. t6), namely >/~(d, r) is the Reid soft-core deutéron wave function and the final wave function contains not only w(K, r) in its exact form but also the NN( t DZ ) and Nd(sD 2 ) configurations with their radial forms generated by solving coupled differential equations. The dashed line is the result of using the Reid soft-core deuteron wave function with simply w(K, ~) in its exact form. The difference between these two curves shows that, at resonance; w(K, r) the Nd( s S 2 ) configuration, is dominant, but at lower energies the other two configurations becomeimportant. In this second calculation, ifthe Reid wave function is repláced by the Hulthén wave function then the resulting curve is within 10 ~ of the dashed curve. The solid curve is the result of using the Hulthén wave function for the deuteron and w(K, r) in its approximate form given in eq. (3.1). Since the Is

~o~ (m b.) lo

I s """ v

.0 0

ftes)

1C ~

í~jj ,

`l .

} ..

i . OS (326)

.

.

.

~ . L0 (425)

.

~ IK lE

.

~

1 . 15 (5611

1ppIMeV )

.

2A f12í1

Fig. 3. Then+d -. pp crosssectionas a function ofg/k, the pion momentum, [and app) the corresponding pp lab energy forthe inverse reaction]. Dotted line ; >/r(d) _ ~(Reid) and the final wave fimction contains the components NN(ID~, Nd(3S2) and Nd(3D~ from ref. Is) . Dashed line ; ~r(d) _ ~(Reid) and the final wave function contains only the component Nd(SSz) from ref. 1'). Solid line ; yG(d) _ ~(Hulthén) and the final wave function contains only the Nd('Ss) component but expressed in the approximate form of eqs. (3 .1). Crosses; ~i(d) given by eq . (3 .3) and the final wave function as in dashed line . The experimental points are the samcas those quested in refs. "-IS).

222

A. M. GREEN AND E. MAQUEDA

Reid and Hulthén wave functions are essentially equivalent in the present problem, the difference between the solid and dashed curves is mainly due to differences between the exact form of w(K, r) and its approximation in eqs. (3.1) and can be understood later from fig. 4. In fig. 3 the differenceß between the dashed.curve and the crosses arises when the Reid soft~ore deuteron wave function is replaced by the Gaussian wave function >G(d, r) = 0.362 exp (-0.075r2),

(3.3)

which enters in the next section. As. can be. seen, this very simple deuteron wave functión appears to be a good approximation in this type of problem. In fig. 3 we also show the experimental situation and note that even the so~alled best calculation, the dotted curve, gives an underestimate. There are several possible . reasons for this : (i) Uncertainty'in such parameters as the Ndp coupling constant and the ranges of the form factors needed in the transition potential generating isobars from nucleons 13,14) . 16). (ü) The neglect of s-wave rescattering (iü) The neglèct of other Nd and dd configurations. Later, when we make a comparison between our theory and experiment for the pd -. tom + reaction, this underestimatè in the pp -+ dam+ case will be kept in mind.

0.0

0.05 F

Fig. 4. The matrix element N(1~ of aq . (3 .2) as a function of K the c.m . nucleon momentum ( ms s CECPP) _ /~N'7 ~ (a) for w(Real), (b) for w(Ima~. The dashed and solid lines are for the same wave function combinations as is fig. 3. The dot~ash line ie for t4(d) = y~(Hulthén) in e4 . (3 .4) and the exapt form of w(K, r). The chained line is for ~(d) _ ~(Gaussian) of eq . (3 .3) and the approximate fprm of w(K, r).

pd -" ta +

.

223

In order to get a better understanding of the effeçts of various forms of ~(d, r) and

w(K, r) we show in fig. 4 N(K) ofeq. (3.2) for the wave function combinations discussed

above. Again the dashed line corresponds to ~r(d, r) being the Reid soft-core deuteron wave function and w(K, r), the Nd(SSZ) wave function in its exact form. So that we can perform analytically the integration of eq . (3.2), a Hulthén type of function is next chosen for the deuteron : ~(d, r, Hulthén) = 0.875r -t{exp (-0.228r)-exp (-1.37r)} .

(3.4)

This choice of ~r(d, r) is checked by integrating numerically eq. (3.2) with w(K, r) again in its exact form : The corresponding results are_joined by the dot-dash line and are hardly distinguished from the dashed line in which ~(d, r) is the Reid deuteron wave function. Finally eq. (3.2) is analytically integrated using the ~y(d, r) of eq. (3.4) and. the approximate w(K, r) of eq. (3.1). These results are joined by the full line in fig. 4, the same notation as in fig. .~3. In addition, we calculate N(K) analytically when ~~(d, r) is the Gaussian of eq. (3.3) and w(K, r) is in its approximate form . This result is shown by the chained line and will be of interest in the next section when we discuss approximate ways of evaluating matrix elements for the pd --~ tn+ reaction. The conclusions to be drawn from the prévious paragraph are as follows. The diûerence between the solid and dashed curves of fig. 4 shows that when the approximate form of w(K, r) is used in integrals of the form shown in eq. (3.2) it performs reasonably well in the region 2.5 5 K 5 3 fin- ' [ie., 500 5 E(pp) 5 750 MeV] but fails badly for smaller values of K = as seen earlier in fig. 3. However, in the next section when we discuss the pd -. tom+ reaction the functions arising similar to N(K) are weighted by K-dependent factors that greatly cut down the contributions from the smaller values of K appearing in eq. (2.8). 4. N®erkal results for the pd -. ta + differential crues section Once w(K, r) has been approximated as in eqs. (3.1a) and (3.1b) the double integral ~ JdkJdr of sect. 2 can be performed analytically. With the notation of eq. (2.8),

x [ ~ A(n, R or l7K"I(q, a, d(R or n)+ ~ B(n, R or 1N(q, a, d(R or n)], .=o .=o,

(4.1)

where A(0, R) _ -12.492, E(4, R) --' 1 .389312, etc. as given by the parametrization

224

A: M . GREEN AND E. MAQUEDA

of eqs. (3.1a) and (3.1b), and

n) = f ~z ~P f -i~ Z)Jofz4r) ~P f -dr2)[1- ~P f - 1:4r~)]

I(q, a, d(R or

J0

-fia+d+1 .4)- exp {- i~4zfia+d+1 .4)-1 }], J(q, a, d(R or

n) _ ~ drr4 exp ( - ~Z )Îof~z4r) exP f -dr2)[1-~P f - 1 .4rZ)] 0

_ ~ä~[fi-ísg2f~+~- lx~+~-~ exP {-s42fia+~-1}

-(i-3sg2(~a+d+1.4)-lxza+d+1 .4)-~ exp { -í-6g2(Za+d+1 .4)-1}].

The result of the double integration is IM~ z =

~cZ

( aZQ2 4Z) exp{ } . a . ßa 2 Cfn)Af~ R) ( If4~ a, dfR)) ~~~~ ß ~] (4.2)

where Z = ßs/[16fx 2 + 1/8a)], .ß is defined in eq. (2.10) and the polynomials G(n) are those following from the asymptotic expansion ofconfluent hypergeometric functions given in eq. (13.5.1) of ref.?1). The use .óf this asymptotic fonm is .adequate since . Z ,: 100. The precise form of the G is as follows: Gf0) 1-~Z_ i, G(2) - 1-Z-i+áZ_z~ G(3) = 1, Only G(1) is an infinite series . However, it is found that it is su~cient to kcep terms up to Z_s .to ensure an accurate result: Before the pd -> tn+ cross section of eq. (2.11) can be evaluated it only remains to specify the parameters a2 and a of the deuteron and triton wave functions respectively. For the deuteron aZ = 3.33 fm2 is used, which corresponds to the wave function ofeq. (3.3) and as can be seen from fig. 4 gives a ~+ d -. pp cross section that is. in reasonable agreement with the one obtained with the Reid .soft=core deuteron

EtPI = 400MeV

xl0~

~o . oo

t+o

~ao

Ey1-400 MeV

eo ro'

oó ~ow mo

tso iw"

E y1 ~ 470 MeV

~o

~o

~o

ao

iso md

E t~1= 600 MeV

Fig: 5. Fondions M(R), M(I) and 1MJ for proton lab ~ergies of 400, 470 and 600 MeV.

226

A. M. GREEN AND E. MAQUEDA -

wave function . However no attempt was made to optimize this agreement by making fine variations in aZ. Later, in sect. 6, the effect of using- the Hulthén wave function for >/r(d) is estimated. But this can only be done in an approximate manner, since the

30

W

90 I20 ~(OO)

á0

IM

Fig. 6. The pd -. trz+ differential cross section for the proton enérgies (a) 400 MeV, (b) 470 MeV and (c) 600 MeV.

pd --> tn +

227

double integrals of eq. (4.1) can then no longer be performed analytically . For the triton radial form of eq. (2.7) a(z) = exp (-O.lrZ) is used as in ref. 9). The relative . ranges of the deuteron and triton wave functions are as expected, namely the deuteron . i s more diffuse than the triton . In fig. 5 are shown M(R), M(n and also ~M~ as a function of BR(c.m .) for the three proton energies 400, 470 and 600 MeV. In fig. 6 the differential cross sections at the same energies are shown. Before these results are discussed the question should be asked about their dependency on the choice of w(K, r). In fig. 7 the K-integrand of eq. (2.8) is shown for the three proton energies of 400, 470 and 600 MeV. The solid (dotted) curves correspond to the real (imaginary) part of the integrand calculated using w(K, r) [as in eq. (4.1)]. The crosses (circles) stand for the corresponding values of the same iíitegrand .but calculated using' S) the "exact" w(K, r) at the relative moments K of 1.90, 2.26, 2.48, 2.69, 2.88 and 3.09 fin-1 . Therefore in this latter case the radial integrals have to be calculated numerically. Fig. 7 shows thatfor the values ofK which contribute most to the integrals, the approximate curves are in good agreement with the "exact" values. It also shows that the results are expected to be more accurate where the integrands peak around K = 2.69 fm- t (corresponding to a proton energy in the pp -. dam+ case of 600. MeV). It . is at this point. where the approximation to .w(K, r) is at its best . When going away from this value of K, the

K (f~')

Fig. 7. The Kintegrad of eq. (2.8) for the proton energies (a) 400 MeV, (b) 470 McV'and (c) 600 MeV. Solid (dotted) line ; w(K, r) as in eq. (3.1a) (or eq. (3.1b)). Crosses (circles) - real (imaginary) part of w(K, r) from ref. `s) .

228

A . M . GREEN AND E. MAQUEDA

largest differences arise for the imaginary (real) part of the integrand at E(p) = 400 MeV (600 MeV). However, in those cases it is the other part of the integrand, i.e. the real .(imaginary) part, which contributes more to ~M~ 2 as can be seen from fig. 5. The error introduced by the use of the . approximate w(K, r) can now be estimated numerically by performing the integral in eq. (2.8) using the w(K, r) from eq. (3.1) but . between definite values of K chosen to compensate some of the erroneous featiues ofthe approximate integrands in fig. 7. The dashed curves in fig. 5 correspond to these modified values of M. It is therefore to be expected that for the three proton energies under consideration the pd -> tn+ cross sections are affected, when using the approximate w(K, r) by at most 27 ~ át E(p) = 400 MeV and at forward angles. The differences for the proton energies of 470 and 600 MeV are hardly significant. In fig. 6 the differential cross sections - labelled by G - are shown for the three proton energies 400, 470 and 600 MeV along with the current experimental information i. as-za). The latter have been corrected from the energies used in the experiments to the nearest of the above three energies of interest by méans of the multiplicative factor E(d)E(t)E(p) qs

in eq: (2.11). The curve labelled H is obtained when the Gaussian form ofthedeuteron wave function . in eq. (2.8) is replaced by the .more realistic Hulthén wave function . In this case the dil%rential cross section cannot be derived analytically but requires an ádditional: approximation. However, as will be discussed in sect. 6, this approximation seems to be ~ reasonable, so that the curve H should be a better estimate for the theoretical differential cross section. A detailed comparison between theory and experiment will be postponed until sect. 7, since the main object until now has been to derive an analytical form for the differential cross section that can be approximated in various ways as will now be discussed. 5. Approximations for the pd -i m+ cross section

One of the motivations for beginning this study was an attempt to understand a popular approximation 4-11) to the matrix element M of eq. (2.8), in which F(K) is replaced by F(Ka) on the assumption that F(K) is a slowly varying function of K. By means of this the differential cross section for pA -. (A+ 1~+ reactions; and in particular pd -> tx +, can be expressed in terms of the pp -. dx+ differential cross section at some average pp energy corresponding-to Ko. In the following we shall attempt to show whether this replacement F(K) -. F(K o) is meaningful or not, and also which, if any, is the optimum value of Ko. Using the approximate forms of w(K, r) given in eqs.. .(3.1a) and (3.16) we get

229

that from eqs. (2.8) and (4.1) M(KQ)

= exp ~- 4a Z kó -

1 8a (~k° +Q)Z~

x ~dKK 2Îz(iKß) ~P ~J0 4ask°

CJ

where

with

y

F(Ko) _ ~ -o l' =

J' =

CaZ+

8a)

4K 2~ F(Ko)

1

8a(ak +Q)s.~F(K °) °

~KZ1z(iK~ ~P ~- Caz+ 8a) 4KZ ~J'

A(n)Kó l l'(go, ao, ~+

4

~n~ó 1 J'(go=ao, ~, ~ -°

(5.1)

(5 .2)

~drr a exP ( - aor2)1o(igor) ~P ( -dr~)[1-~P (-1 .4r2)],

J0

~drra ~P ( -aor2)Jo(i~or) ~P ( -~z )~1- ~P (-1 .4r Z )]~ J0

Writing M(Ko) in this form is convenient since F(K°) is now independent of the , geometry of the pd -~ t~+ reaction. On the other hand after performing the K-integral in eq. (5.1) the factor multiplyingF(K°) becomes, in thènotation ofeq. (4.3), - ~ 2Z G(1) . ~- 2oczZ P exp (5 .3) QZ/

which, through the exponential factor, is strongly dependent on the pd -. tn+ geometry . It should be noted that, in anticipation ofattempting to relate the pd -~ tom+ cross section to the pp ~-. dam+ cross section, we have modified I and J so that they are in the form appropriate for a pp -~ dam+ matrix element. This requires three modifications: (a) The factor exp (-?arZ) is removed, since this arose from using e4. (2.7) in the product a(s+Zr)oc(s-~r) as can be seen in the expression following eq. (2.7). (b) The value of the Gaussian,range is changed from,a =~0.1 fm-a which arises . from the .triton wave function, to ao = 0.075 fm z which is appropriate for the deuteron [see eq. (3.3)]. (c) The value of.q is no longer the one corresponding to the incident proton energy in the pd-. t~+ . reaction . Instead it is the value qo which is related to Ko by the kinematics of the pp -. d~ + reaction.,

230

-

A. M. GREEN AND E. MAQUEDA

0.2

K Ifs~)

Fig. 8. (a) F(R, Ko), (b) FCI, Ko), and (c) IF(%~I as functions of xo . . We therefore see that F(Ko) is nothing more than N(Ko)~10 .362, where N(Ko) . is given. by eq. (3.2) with ~(d) being the simple Gaussian of eq. (3 .3) and w(K, r) the approximate wave functions of eq. (3 .1) - the chained line of fig. 4. In spite of these three differènces between F(Ko) and F(Ko), thé proportionality between F(Ko) and. N(Ko) indicates that F(K) may not be a sufficiently slowly varying function of K to justify the replacement of K in F(K) by any physically meaningful value of Ko.

This can be seen either' from fig. 4 showing N(K) or fig. 8, in which F(R and I, Ko ) and ~F(Ko)~ are plotted as functions of Ko . As a measure of the importance of each of the three modifications mentioned above we did some test calculations. At Ko = 2.26, qo = 0.704, when exp (-a or2~) is replaced by exp (- i~Z~ F(R, Ko) drops from 0.113 to 0.085 showing that this is a non-negligible effect . However, at Ko = 2.26 when qo is replaced by 0.75 or 0.65 the value of F(R, Ko) only changes to 0.112 and 0.115 respectively . To give the reader a feeling of the values of q, qo and Ko that arise in the pd -> tn+ and pp -. dn+ reactions at energies in the resonance region, we show in fig. 9 the relationship between q and the proton lab energy in the pd -. tit. reaction, and in fig.10 the relationship between qo and Ko and the proton lab energy in the pp -~ dam+ reaction.

~00 N.111

!00 a.W

~00 a .W

TP (Lab) lMeVl (k,/~.)

T00 ts .s~l

~00 tsn~

s

Fig. 9. The pion momentum q, in the pd c .m . system, as a fimction of the proton lab energy . and ko of fig. l.

From figs. 5 and 8a it is now possible at a given energy (F"(p)) and angle (B,~ to e~itract that value of Ko(R) which satisfies the equation M(R, Ko~)) = PF(R, Ko(R}) = M(It~

(5 .4a)

where F and P.are defined by eqs. (5.2) and(5.3~ Similarlywe can find a Ko (n such that (5.4b)

When attempting to express t]re pd -" .tn+ cross section in terms of the pp -. ~drt+ cross section, the most desirable situation, would be if Ko (n and Ko(R) were equal. However, as seen in table 1 this is not.the cáse, the Ko(n are 0.1-0.21m - 1 larger than

232

A. M. GREEN AND E. MAOZIEDA

o.s

LO Is3zl

Z2 110U

2A N7q

K (foi')IElpp)MeV)

2.i IsWI

2J IfaU

3.0 (7~7)

Fig. 10. Thépion momentum qo, in the pp c.m . system, as afimction of Ko, the relative momentum, and also the proton lab energy.

the K o (R). This may seem a small effect, but a glance at fig. 8 shows that such differences can change the M considerably . Since it is not possible to fit M(R) and M(l7 simultaneously with the same Ko, the next step is to try and choose, with the aid of figs . Sc and Sq a Kó so that M~(~ Kó)+M Z(I, Kó) = pZ~F2~~ Kó)+FZ(I, Kó)Î (s .s)

= M2c~)+MZ(n.

The values of Kó are given in table 2. ïn most cases there is more than one value of Kó which satisfies this equation . However, usually one of these solutions seems to be somewhat more reasonable than the others in the sense that for this solution Kó increases as B~ increases. More disturbing is the fact that for some energies and T~iB 1 The values of Ko(R) and Ko (n which ensure M(R, Ko (R)) and M(I, Ko(n) equal the exact matrix elements M(R) and M(I) from figs . Sa and b. E(p) (Me~ 400 470 600 ~

Ko(R)

Ko~ 180°

í 2.25 2.71 2.83

2.27 2.72 . 2.86.

2.28 2.74 2.87

2.33 2.41 3.05

2.355 2.425 3.07

2.38 2.435 3.11

pd -> trz +

233

angles (e.g. 400 MeV, 6x = 0°) no realistic value of Kó can satisfy eq. (5.5) in a reasonable way. Furthermore, on comparing the values of Kó with K°(R) and K°(n . of table 1 we see that they bear little relationship to each other in the sense that Kó is in no way an average of K°(R) and K°(n . The reason for this is the one suspected earlier in this section, namely, F(K) is not varying sufficiently slowly as a function of K to justify the replacement of F(K) in eq. (2.8) by F(K°) to obtain eq. (5 .1). Tesu?2

The values of Kó which ensure e4. (5 .5) is satisfied E(p) (MeV)

B,

400 470 600

0°

90°

180°

NS 2.89 3.215

NS 2.89 3.23

2.28 3.07 3.24

The entry NS means that no reasonable value of Ko existé .

The conclusion from the previous paragraph means that the use of Kó is no more than a way of simulating the pd -" tn + reaction in terms of pp -. d~c+: However, this would still. be very useful if we can determine Kó by other means involving only the kinematics and the structure of the deuteron and triton wave functions. This is the aim in the work of refs. a -11). In refs .'' a) Fearing makes the assumption that K°(F') is determined from the condition that the struck nucleon in the target has its minimum value. Later in ref. 9), Fearing imposes a more accurate and complicated prescription, but he writes that this amendment has little practical ef%ct. In this article we shall only discuss the values K°(F) resulting from his simpler prescription. These are given in table 3 along with the values of K°(LW) used by Locher and Weber t°): The latter are"obtained through the prescription ofneglecting the Fermi motion in the triton and is considerably simpler to get than Fearing's. Tesla 3

The values of Ko (F~ as given by Fearing's prescription ~ E(p) (Me 400 470 600

0°

90°

180°

xo~~

2.50 2.72 3.04

2.52 2.74 3.06

2.67 2.92 3.31

2.54 2.75 3.11

') Ref.').. Fearing in deriving K °(F~ relies purely .on kinematical constraints and does not use any information about the initial and final nuclear states. From eq. (2.8) we see that- an alternative prescription which depends also on the wave functions presents

234

A. M . GREEN AND E . MAQUEDA

itself. The function aa+ 4K2 }jz(iKß) . . (5.6) {- C 8a) acts as a weighting factor for F(K, R and I) and it is the latter that contains the bulk of the : reaction mechanism and is different for the real (R) and imaginary (I) components . On the other hand W(K) depends only on the deuteron wave function (through a2), the triton wave function (through a) and the initial (ko) and final (q) momenta of the pd -~ tic+ reaction [through ß in eq. (2 .10)]. It is easily shown that W(K) is a strongly peaked function with its maximum at W(K) =

Kz exp

where y = 4(a2 + 1/8a). In table 4 the values of Ko( W) are shown for various energies andin fig. 11 there is plotted, for 9x = 90°, the values of Ká, Ko(F~, Ko (LW) and Ko(W) as a function of ko 'the initial momentum. This figure shows by the crossed curves the multiple solutions for Kó. Tes~.a 4 The values of Ko(iT~ the .maximum of the weighting factor W(1~ in al . (5 .6) E(p) (Me~ 400 470 600

Bs 2 .38 2 .58 2 .90

90°

lso°

2.43 . ~.64 2.98

2 .48 2 .70 3 .06

By their definition each of these values of Kó is able to reproduce the exact cross sections of fig. 6. However, as pointed out earlier, none of these games of Kó bears a sufficiently close resemblance to. the various approximations Ko(F, LW or W) for any of the latter to give a good description of the exact cross section. To get a feeling ofhow important are these differences between Kó and Ko (F, LW, W) in fig.12 shows, for the three proton energies 400, 470 and 600 MeV, the function T(F, LW, W), defined by the equation . (F, LW, W) = T(F, LW, W) ~(exact),

(5.8)

where dQ/dál (exact) is from fig. 6. In most cases T is greater than unity. The reason for this follows immediately from figs. 8 and 11 where it is seen that Ko(F, LW, W) take on values near 2.7 fm- t . For this value of Ko there is a maximum in ~F~ corresponding to the maximum in the pp -. dam+ cross section in fig. 3. The function T(F~ has more structure than the others because Ko(F) varies much more with B~, as can . be seen from tables 3 and 4. It is also seen that T(F), at any given valve of9~, decreases as E(p) increases. This trend is itt 'agreement with what is found in ref. 9), but there

pd.-. tR +

235

Lw

F(9d')

w (907

3 .0

z.o

~ t ~ tt .t ? 3.0 ~ Noo1t 451{410)

H1

35

"t . (6001

k (fm")(E(p1hkV)

4.0

Fig. 11 . The relative moments Ká, Ko(F), Ko(LR~ and Ko( ü) as a fimdion of ko for B~ = 90° . For Kó all of the possible solutions to «l. (5 .5) are shown .

the actual values of. T(F) are much smaller. Fearing finds that the ratio theory/ experiment dróps from 1 .14 at 377 MeV to 0.53 at 462 MeV to 0.154 at 590 MeV. However, it should be pointed out that Fearing includes distortion factors that dccrease the simplest form of his cross section results by about a factor of 5. One of these distortion factors, tflat due to ~-t restettering in the final state, is certainly not included in our model. Hówever, the distortion due to p-d restettering in the initial channel is to a great extent taken into account by our .use of the two-body wave function w(K, r) derived from coupled two-body Schràíinger equations and also ensuring that ~~ in fig. 1 is correctly antisymmetrized. 6. Improvelo~ts of the deuteron sad triton wave f®ctia~ns

In. the model described in sect. 2 both the deuteron [eq: (3 .3)] and triton [eq. (2.7)] wave functions are assumed to be unoorrelated Gaussians and this resulted in the pd -. tom+ differential cross section háving a simple algebraic form. Tn this section we now äiscuss to what extent our wave functions can be made more realistic. For the triton, any attempt to go beyond the simple Gaussian of eq. (3.3) leads to expressions that are much more Complicated than the double integral of eq. (2 .8). This is simply because the product oo(s+~r~c(s- Zr) no longer separates into a product of a function of s and a function of r as seen in eq. (A .3). We, therefore, stay with the above unoorrelated Gaussisn . However, as can be seen from refs.'" 9) this` wave. function gives for the model discussed theré results that are. qualitatively similar to those obtained using a .correlated Gaussian. For improving the deuteroon wave fimction more progress can be made. As seen

236

A. M. GREEN AND $. MAQUEDA 6

5 4

' ~,

Lw

Elv) ~400 MeY

w

2

°ó

~

~

90

120 Iso t~lcm) (deq)

Iw

EIv1 ~470 MeY

3 T

Lw

0

30

60

90

120 ISO 61~ (c m) (deq)

Ie0

Elvl ~ 600MeV w Lw

T 2

~'~F 9o I2o 8,~(cm) (deq)

Iso

leo

Fig. 12 . The normali7stion factor Ti(F, LW, fil defined by e9. (5.8) as a fimáion of B~ for the . throe proton energies (a) 400, (b) 470 and (c) 600 MeV. The notation F, LW aad W refers to the throe approximations due to Fearing'), Locker i~ and eq. (5 .7).

pd =-" tx

+

237

in eq. (4.2). there is an overall factor exp:[ T (az4Z/aß~Q2], the origin of which can be understood from e4 . (2.~ ~as follows. Ifin this latter equation the "r" integral is taken to be independent of K, as in the approaches of refs. a - 11), then the k-integration can be performed to give 1 ('

A = Yim(Ko) ~~r) ~P ( - ii4 ' r~Ko~ r) Kor J X

~~~ s) ~P (~' Q~s+k~a--isr~

where ~(d, s) is the deuteron wave function in coordinate space. However, because of the simplified foim of a(x) in e4. (2 .ß,e4 . (6.1) can now be separated into a product of two integrals - one over r and one over s. The latter is simply uQ) _ { ~~ s) exP (i8

.

(6.2)

Q) ~P ( -~)~ .

.Replacing ~(d, s) by the uncorrelated Gaussian

leads to the result uQ, Gaussian)

= 2}ns~ 4a-~ C2a+

1 ~2l

-;

exp ~-

s

.

Q , 4(2a l/4as)~

(6.4)

which is closely related to the factors F(d) and F(Q~) that appear in refs.' 1~. The following approximation now suggests itself namely, replace i/r(d, s) in eq. (6.2) by á more realistic deuteron wave function, e.g. we shall use the Hulthén wave function in what follows, and then simply multiply the expression for ~M~Z in e4. (4.2) by the factor

22(Q, Hulthén) R(Q) = L Q, Gaussian) .

The effect of this oorroc~ion can be seen in fig. 6. It should be added that the above approximation of replacing K by Ko in e4. (6.1) can be done. separately for the . real 'and imaginary components of M. Therefore, this is similar to the.first approximation discussed in sect. 5 and is less of a restriction than the approximation of refs. a.'~ also discussed in sect. 5 [eq. (5.3)]. 7. ~ Cus~rleo~ wlth ex~erimeat Even though the main object of this paper has been .to,óonstruct for the pd -. tn+ reaction a simple analytical model capable of being approximated in various ways,

238

A. M . GREEN AND E. MAQUEDA

it is also of some interest to compare the.b~est. theoretical results (curves H in fig. 6) with experiment. .. From fig. 6 it is clear that the model in its present form does not givé a good account of the observed cross sections. At the three energies considered the theory overestimates the cross section by up to an order of magnitude at some angles. For 400 MeV the overestimate is on the average about a factor oftwo, rising to a factor of X10 at 470 MeV. For 600 MeV theoverestimate on the averagedrops again. However, it must be remembered that the theory developed in sects. 2, 3 and 6 involves several major simplifications of which the most important áre: (a) The model neglects the ~+ t final state interaction. As seen from figs. 5-7 in ref.'s this final state interaction combined with the effect of the initial p~ interaction reduces the theoretical cross sections at 340, . 470 and 590 MeV- by factors of about 1.5, 3 and 2 respectively - the trend needed to impróveagreement in thepresent model. Unfortunately the author of ref.') .does not quote the reduction factors arising from the final and initial state interactions separately. Even so, the reduction factor due to the final state interaction alone will certainly.not be sufficient to bring the present model into agreement with experiment . Furthermore, it is stated in ref.') that this latter reduction factor is to á very good approximation independent of angle. Therefore, the shapes ofthe theoretical cross sections will be. unchanged.'For example, . at 600 MeV if the final state interaction improved the agreement at small B it would do the opposite for large 8. (b) The model only involves the chain of events

anc~ neglects pion production from other intermediate configurations suçh as (dl~aa . i and (1VI~Z°~ 1. However, as seen from fig.. 3 for the pp -> .d~ + cross section this leáds to an underestimate and there is no reason to expect this to .change dramatically in the pd ~ tom+ reaction . (c) The model neglects D-wave components in the initial deuteron and final triton. However, from ref. s) the indications are that the effect of the deuteron D-wave component is rather small giving rise at 470 MeV to a 12 ~ reduction at 0°. and a 15 ~ increase at 180° . At 590 MeV the increase at 180° rises to 40-~, which when combinedwith the angle-independent effect of final state interactions could possibly improve the agreement at small angles without effecting the fit at larger angles. Even though no one has yet estimated the effect of D-state configurations in~the final triton, there does not seem to be any reason why such configurations with their 8 probability should lead to significant reduction factors. (d) The model uses an oversimplified triton wave function . Apart from the negléct ofthe D-state configurations just mentioned theassumption in eq. (2.7) ofuncorrelated Gaussians for the relative wave functions between. the three pairs of nucleons in the

triton is very crude. However, às stated in sect. 6, the introduction of correlated Gaussians can only be àchieved by sacrificing the simplicity of the present model - see eq. .(A.8). . Having listed the above simplifications present in the theory; it is not surprising that the agreement with experiment is rather poor . Of these four simplifications only the first, the neglect of the ~+ t final state intéraction, can be removed in an approximate but simple manner as describëd in ref.'). Any attempt to remove the other three simplications leads to a much more complicated model in which the effect of the ápproximations discussed in' sect. 5 would presumably be less~'transpa "rent and so defeat the main purpose of this article: Of course, if this model is ever to be taken seriously in a comparison with experiment all of the above simplifications must to some extent be reYnoved or justified 8. Conclusion

The aim of this paper has bcen to wnstruct for the pd -> tn+ reaction a microscopic model that is sufficiently simple to be derived analytically - eqs. (2.11)' and (4.2). By considering this analytical form for the cross section to be the exactsolution to the problem; the effect of various approximations to these exact solutions have then been studied These approximations are the ones that express the pd -~ tom+ cross section in terms of the pp -. d~+ cross section at some average momentum Ko in the pp system. In sect. 5, table 2 and fig. 11 show that indeed usually such a Ko exists . . However, the values (Kó) bear little relationship to what is expected [Ko(F, LW, Tom] îrom physical considerations based on kinematics and the structure of the deuteron and triton wave functions. These differences between Kó and Ko(F, LW, i~ are sufficiently large that the àpproximate cross sections derived from Ko(F, LW, W) cán lead to overestimates of up to a factor of 6 at 400 MeV and small angles, fig.12. This indicates that great caution is necessary when applying these same approximations in models thát are more realistic than the one describèd here ánd alsó whén such models are extended to heavier nuclei. From fig. 12 it appears that the approximate cross sections could be incorrect by up to an' order of magnitude and also distorted as a function of 8,~. Having thrown doubt on the accuracy of those models based on the pp -..dn+ cross section, the question then is - what can be done that is better? In sect. 7 and fig. 6. it is séen that the simple model..of eqs. (2.11) and (4.2) certainly does not give a good account of the observed cross sections. However, as also discussed in sect. 7 this model could be improved in a number of ways. Unfortunately most of these impróvements complicate enormously the expression for the cross section. But this is probably the price that must be paid before a serious comparison with experimental data is possible, especially if any attempt is made to understand cross section asymmetries using polarised protons.

A. M. GREEN AND E. MAQUEDA

The authors wish to thank Professors R. J. Blin-Stoyle and J. P. Elliott for their hospitality, the British Science Research Council for financiál aid and M. Sainio for supplying isobar radial wave functions. . Appeaalg . In this appendix the expression for the transition amplitude corresponding to the pd -> ta+ reaction, given in sect. 2 is generalised for the case of: - (A) A deuteron wave fimction ~d(n - r2) whose Fourier transform is with cx(ko, K) then defined as c2(ko, K) =

J0

dB sin BPx(cos e).i(~kq-2~

(A.1) .

where 8 is the angle between ko and %. For the particular case treated in sect. 2 of a Gaussian deuteron wave fimction, cx(ko, K) =

ac"t exp { -a2~ó+4K2)}( -iiilz(6~2ko~~ 2'CnJ (B) A triton wave function where the expression ofeq. (2.~ is generalised to include correlation, Then in terms of s and a, the triton . orbital wave funbtion is written as 6

(A .3) . = Nc ~ ~~~1)+2B(r~) cosh (br ~ s)] exp { - Y(~1~ - z(n~ 2 h w=i and B(~), D(~), . Y(r~) and Z(~~ are given in table 3. For the case of the unoorrelated Gaussians of sect. 2, C = 0 and only the first rbw of table 5 is operative. ' ~ _ the coupled channel solution of the (C) Both the NN and Nd components of ~c

T~ 5

The parameter's of the triton wave function D(q) 1 2 3 4

1 -C 0 0

G

-C3

Y (n) >

B(~) 0 0 -C C2 ' 0 0

~

~a ~a+b }a+}b ~a+ü8 ~a+~}b ~(a+b)

Z(p)

-

2a 2a 1a+b 2a-~b ~fa+b) 2(a+b)

nucleon-nucleon interaction in the 1 Dz partial wave are considered . The radial solutions u,,(K, r) correspond to Nd(SSz) :

w(K, r) _- uoz(K, r)

(A .4b)

of sect. 2. (D) The one-body transitión operator ® _ ~z= °®(n) is

®(n) _ ~- l ~.f(Q ' 9xtin' ~)+ .Î *(Sn . 4xT~' ~)~S(rA-r~),

(A.5)

where Q (s) is the usual Pauli spin (isospin) operator and S (TA) is the operator acting between a nucleon and a d . For later use a spin-isospin transition operator is defined as with spin-isospin reduced matrix elements
The antisymmetry of the initial and final wave functions allows writing the expression

Te = <`y~(ro~ r i ~ rz~ rx)I®~~j(ro~ rv rz)i = 4V J<~`c(ro, rv rz,

rx)I®(~)I~i(P(ro) ; .d(r rz))),

(A.7)

where ¢;(p(r°) ; d(rt, rz)) is the straightforward generalization of eq. (2.4) including NN components and `Pr(ro, r l , rz, r,~) is given in eq. (2.5). Finally, the transition matrix element for going from the initial colliding proton and deuteron with spin projections m and M respectively to produce a pion and a triton with spin projection m" is given by

x (Z1ZImMX21~(m+MXm" -m-M)) x~

~

x

~ S(1s21m;(m'-m,)xlsll(m+M-m'~(m'-m;))

1

1

~Z(s+1)}
~ LZ(n)7- ~ exP {- ~Zl4Z(rl)} n

A . M. GREEN AND E . MAQUEDA

242

x (~,li2JOm'x~,1 i2~00)I(r1,1, s, ~, li ; ko, B~) ~~~~

zi,i,i,

x rira (1IilzJmimix~islaJa~~-m~-ml)) x ( 12 1a 2~(ml +m2xrn'-ml -ml)1111112J~) x (~,1314JOOx12142J00)Y3. ~ r+ ~,-~,'(B', 0) {Y~~,i(9'+, 0)I+(~Îi 1,

X

S, ~,11,12~

+(-)i2y~m1(B,r, 0)I_(n

;

13i k0, Bx)

h s, ~ 1 1~ la+ ls+ ko~ Bx)}]~

(A.8)

where I and I t are. the double integrals ~.. . I(rl ; h S, ~ 1 1 + ko~ ex) dKK exp { -KZ lZ(n)}ez(ko, ~.li~(`pKlZ(~1)) (A.9a) I t (n,1, s,. ~,11,12,13 ~ k0, ex)

dKK exp { -KZ/Z(n)}ez(ko~ ~1tg(`pK/Z(n))

x

~drr exp { -[Y(~1)-(b2/4Z(n))]rZ}u,,(K, r)

J0

x1<<(xt(n1~)~~2(bxr/z(nN~

(A.9b)

The vector p is defined as p = 3q-2ko and B' is the angle giving its direction with respeçt to ko. The vectors 7Ct(n) are related to the triton wave function and given by 7Ct(n) = z[lf(b/3Z{n))]~I~Lb/Z(~1)]ko . with 9't indicating their direçtion with respect to ko.

pd ~. t~+

References

243

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