Theories of alpha-decay

ANNALS

OF PHYSIC’S

105. 151-186 (1977)

Theories DAPHNE

of Alpha-Decay

F. JACKSON AND M.

RHOADES-BROWN

The derivation of the width for l-decay is examined with particular emphasis on methods which do not involve arbitrary channel radii. A new method of treating the initial decaying states is introduced and the use of ambiguous phenomenological potentials is avoided. This method yields consistent and acceptable quantitative results for the g.s. to g.s. (ground state) transitions in even polonium isotopes and for the branching ratio for the decay of the isomeric state 212mPo.

CONTENTS I. Introduction 1.1. Statement of the problem 1.2. R-matrix theory 1.3. Sensitivity to ambiguities in the optical potential 1.4. Other formulae for the width 1.5. Definitions and notation 2. Time-Dependent Theory of Decaying States 2.1. Derivation of an expression for the width 2.2. Choice of the Hamiltonians K and K’ 2.2.1. Method of Kadmenskii and Kalechits 2.2.2. Method of Harada and Rauscher 2.2.3. Method of Furman, Holan, Kadmenskii 2.2.4. Creation of an initial bound state 2.2.5. Method of Schlittet

and Stratan

3. Unified Reaction Theory 3.1. Feshbach’s projection operator formalism 3.1.1. Derivation of an expression for the width 3.1.2. Relation to other theories 3.2. Macdonald’s formalism 3.2.1. Derivation of an expression for the width 3.2.2. Relation to other theories 3.3. Excitation of the residual nucleus 3.4. Excitation of the a-particle 4. Calculations for 0’. + O- Transitions 4.1. Cluster formalism 4.2. Construction of the microscopic potentials 4.3. The functions & and xX’4.4. Results for the decay widths 151 Cop>right X11 rights

IQ 1977 by Academic Press, Inc. of reproduction in any form reserved.

TSSN

0003-4916

1.52 5. Calculations for 5.1. Structure 5.2. Decay to 5.3. Decay to

JACKSON

AND

RHOADES-BROWN

the Decay of 311nlPo of the isomeric state the g.s. of ?OaPb the lowest 3- and 5. states in ““*Pb

6. Antisymmetrization 6.1. The effect of the exclusion principle 6.2. Cluster formalism 6.3. Discussion 7. Summary and Conclusions Appendix. Estimates of the cluster parentage coefficients Acknowledgments

1. INTRODUCTION

1.1. Statement oj’ the Problem The study of a-decay in heavy nuclei has a long history. A qualitative explanation of experimental data is readily given in terms of barrier penetration but it has proved more difficult to obtain consistent quantitative results for the absolute values of decay rates in heavy nuclei [l, 21. Interest in a-decay of light nuclei has developed recently in connection with studies of four-particle correlations [3], and of nonconservation of parity [4]. Lifetimes of superheavy nuclei are also of interest. The theoretical formalism for a-decay and, in particular, expressions for the decay width can be developed by means of the perturbation theory of decaying states or in terms of resonance theory for the scattering of the a-particle by the residual nucleus. Several authors have displayed the connection between these methods, usually in terms of R-matrix theory [l, 5, 61. The problems associated with the prediction of absolute decay rates can be stated as follows: (i) in R-matrix theory the penetrability is extremely sensitive to the choice of the arbitrary channel radius [7,8, 9, IO]. (ii) The phenomenological optical potential for a-particle scattering from nuclei at medium energies shows ambiguities. In R-matrix theory [8, I 1] and in other methods [12], these ambiguities causesubstantial variation in the calculated widths. (iii) The initial state is normally treated in the oscillator shell model while the final state is representedby an a-particle moving in the a-nucleus potential. This can lead to problems of nonorthogonality, and it is difficult to show convincingly how the shell model state evolves into the final system of an n-particle and residual nucleus. In this paper we wish particularly to circumvent problems (i) and (ii) and we therefore build our formalism on those methods which do not involve an arbitrary channel radius, and also construct the optical potentials microscopically. We also work as far as possible within the cluster model, and will be able to show that this contributes substantially to the resolution of problem (iii) and yields satisfactory quantitative

THEORIES OF ALPHA-DECAY

153

results. For examples to illustrate our method we use the decays of the polonium isotopes listed in Table I. TABLE

I

Decay Parameters for the Polonium Isotopes

Decay ?I”Po + ?““Pb(g.s.) mpo ““aPb(g.s.) =rP~ --f 210Pb(g.s.) mpo ---f 21ZPb(g.s.) ?mpo + 208Pb(g.s.) + =8Pb(3--) + ?“Ypb(S--)

1.2. R-Matrix

T,/, (set) 1.19 ,’ IO’ 3.04 < IO-’ 1.64: lo4 1.58 .~ IO 1 4.5 \ IO’

rusr, (MeV) 3.82 1.52 2.55 2.89 9.84 1.01 2.02

/I IO-“!’ .‘. lo-” ‘b’\ 10 IX ,’ lo-“’ ‘c IO-” , 10-2;’ :: 10~“”

5.30 8.80 7.10 6.81 1 I .65 9.03 8.45

Theor!,

In R-matrix theory the configuration space is separated into two regions; in the external region defined by R > R,. , the total wavefunction can be expressed as a sum of channel functions of simpleproduct form while the wavefunction in the internal region is replaced by a complete set of eigenfunctions X, with the real eigenvaluesE, . In the single channel approximation, the width of a narrow resonanceis given by r = 2Py,”

(1.1)

where y,,? is the reduced width determined by the overlap of A’, and the channel function at the channel surface. and P is the penetrability (1.2) The functions F and G are those solutions of the one-body Schrodinger equation containing the Coulomb potential and the nuclear optical potential whose asymptotic behavior corresponds to the usual Coulomb functions F and G. The R-matrix is

The one-body widths are related to the many-body widths through the spectroscopic factor S, . i.e.

(1.4)

154

JACKSON AND RHOADES-BROWN

The experimental width r, is related to the mean lifetime 7 : 1;A by the relation r = ” T

0.658 > IO-l5 eV 7(sec) ’

It is well-known that variation of the channel radius R,. can lead to an order of magnitude variation in the penetrability. Calculations by Scherk and Vogt [8] for the decay of z12Poto 2osPbusing the a-particle optical potential for 20EPbdue to lgo [13] show that the penetrability increasesby a factor of h IO as Rc increasesfrom 9,fnz to IOfm. Strictly, R, should be sufficiently large so that there is no coupling between different channels in the external region but values of Rr well inside the inner turning point have been used [7, 91. Values of the penetrability derived from the microscopic optical potentials calculated in this work (see Section 4.3) are plotted in Figure I. The WKB results are also shown, and are plotted at the position of the inner turning point. In agreement with earlier work [l I] we find that the WKB approximation gives an underestimate of the penetrability. Penelrobillty. i_R-F?gtr*)

_ ,.

L

II.0

R,

--A

Fic(fm)

FIG. 1. Variation of the penetrability, defined by Eq. (1.2), as a function of the channel radius . Crosses mark the value obtained using the WKB approximation.

155

THEORIES OF ALPHA-DECAY

For light nuclei, the deficiencies of the approximate formula (1.1) have been fully examined by Arima and Yoshida [14]. In more recent work on the polonium isotopes, DeVries et ul. [15] replace the penetrability P by the expression k/Rc 1&(RC)12 where $J(;(R,) is a resonant state wavefunction with a fixed number of nodes in the interior region. They choose Rr to be the radius corresponding to the peak of the last maximum of JjCi. which gives R,. - 8.3-8,s fm. This decreasesthe equivalent penetrability and increasesthe value of yh2derived from Eq. ( I. I). 1.3. SemitiritJ- to Ambiguities in the Optical Poterltial The y-particle optical potential determined by fitting low and medium energy scattering data has ambiguities in the parameters owing to the strong absorption features of the scattering process [ 13, 161. Using optical potentials for “lJ8Pbwhose depths varied between 50 MeV and 150 MeV, Scherk and Vogt [8] found widths for 10-l” MeV. while Bencze and Sandulescu the a12Podecay in the range (I .35-3.43) [l I] found that the penetrability for the 2n8Pudecay increasedby a factor of 5 when a phenomenological potential of depth 74 MeV was replaced by a calculated microscopic potential of depth 231 MeV. In the latter case the shape of the potential was also changed. DeVries et ul. [ 151have removed much of the uncertainty in the phenomenological potential for decay of slaP~ to ?O*Pbby using experimental data for a-particle scattering from ?lZPb and 209Bi in the energy range 17-22 MeV. and requiring a fit to these data and the correct number of nodes for the decaying state (seebelow, Section 4.3). However, their method of calculating the resonant state wavefunction #G is sensitive to assumptions about the imaginary potential for scattering and the upper limit on the decay widths is not well determined. In Section 2, we review methods developed by various authors to avoid the use of R-matrix theory and arbitrary radii. These methods do. nevertheless, involve use of phenomenological optical potentials. Table 11 shows the variation in the results obtained by Harada and Rauscher [I31 using four different setsof parameters (A-D) for the optical potential. These results emphasisethe need to remove the uncertainties arising from potential ambiguities. TABLE

11

Predictions for \-Decay Widths of y1u,z12Po(in MeV)

““PO Reference -~ Kadmenskii

& Kalechits [I61

Harada & Rauscher [IO] A B C D

r 1.43 ,\ 10~ L9 3.5 ,’ IO 32 6.5 I. lo-?’ 2.3 ‘\ IO “R 3.8 10 30

?IPPo l’w,, r,:,hT

r PLIIC ~~~-~.___

- --

rcxp lb:,,, -~-

2.68 1.1 I IO” 5.8 IO 2 I .6 \ lo-:’ I.0 IO’

I .4 1: IO-‘” 3.3 10-15 4.8 .. IO-‘” I.0 ’ IO-‘”

I.1 10’ 4.6 .\j IO 1 ?.I . 101 1.5 lo:’

156

JACKSON

AND

RHOADES-BROWN

I .4. Other Formulas for the Width Within the framework

of the optical model, the resonance width can be expressed as

where u(r)/r is the optical model wavefunction and W(r) is the imaginary part of the optical potential. This formula is not very useful in the present context because W(r) is not well determined for low energy m-particle scattering from heavy nuclei [15, IS]. The resonance width may also be written as [5] ( 1.5b) Here the range of ambiguity excitation

upper limit B of the integral is arbitrary, except that it must be beyond the the nuclear potential. With this expression we still have to deal with the in the real part of the optical potential. and the formalism does not describe of a definite final state in the residual nucleus.

1.5. Dejnitions

and Notation

The total wavefunction

for the A $ 4 system is a solution of the equation (E--H)Y=O.

The Hamiltonian

can be written

(1.6)

as

H = fl, i H4 + V,, + TR -:- Vc

(1.7)

where r, is the kinetic energy operator for the motion of the or-particle relative to the residual nucleus, Vc is the Coulomb potential and V,, is a many-body operator representing the total nuclear interaction between the nucleons in the ‘u-particle and those in the residual nucleus. We choose a set of relative coordinates R -L R, - R,

(1.8a)

q. = ri -~ R,

( I .8b)

El; y= r.4i A - R,

( I .8c)

where RA = d $ ri c-1 The .$,,, vi are not independent

(1.9)

since (1.10)

157

THEORIES OF ALPHA-DECAY

In terms of these coordinates, the interaction C:,, can be written as

The wavefunction for the initial state can be written in a cluster expansion as (1.12) where the symbols 5, 11represent the set of coordinates defined above, En implies summation over all appropriate quantum numbers, and for describes the relative motion of the four-particle system and the residual nucleus. At this stage we omit the antisymmetrization operator, but its role is considered explicitly in Section 6. The symbols n, , lzA represent all relevant quantum numbers for the four-particle system and the A-particle system, respectively, and /I, -= 0, 11,~= 0 implies the g.s. of each system. For simplicity we will frequently neglect terms involving excitation of the a-particle, and whenever n, 1 0, nA =: 0 or 11= 0 is intended the superscript will be omitted. The angular momentum coupling will be omitted except when explicitly needed. The wavefunction in a given exit channel can be written in the samenotation as (1.13)

Y’ - @I;“‘(q) 4,,(E) xS’.+W where x,$ is a scattering state for the emitted cl-particle.

2. TIME DEPENDENT THEORY OF DECAYING

STATES

In this section we follow the description of the theory of decaying states given by Goldberger and Watson [19. Chap. 81. 2. I . Derivation

of an E-xpression,for the Width

The time-dependent wavefunction for the A + 4 system can be written as

Y(t) = & 1 ‘ECE-H

Y,

(2.1)

where H is the total Hamiltonian and y, is the wavefunction of the initial state at t = 0. The time dependent wavefunction can also be written as

where the second integral vanishes for t > 0. Tt can also be expanded in the form Y(t) = Y,J,,(t)

+ 1 Yh’J,“(t)

(2.3

158

JACKSON

AND

RHOADES-BROWN

where U,,’ represents the final states of the A + 4 system and

J&t)

= &

/ G(E) / Y,,;,

(2.4a)

J’ tlE e--iEflfL(Yb’ 1 G(E) I yo::

(2.4b)

i LIE e-i”t:fi(!?‘,,

J*,,(t) = &. with

1 G(E) = E _ H .

(2.5)

We wish to describe the initial state as one of a set of discrete eigenstates of a Hamiltonian K and to treat the difference H-K as the interaction giving rise to the decay process. The final states are taken to be eigenstates of a Hamiltonian K’ and if K Z K’, as is usually the case, some nonorthogonality is introduced into the description of the process. This is discussed in more detail in the following sections. Following Goldberger and Watson, we introduce the operators R=(H-KK)F

(2.6)

FY, = ul, + & where Q is a projection elements of R are

operator

which

QW projects

(2.7)

Kj Fyo

of? the ground state. The matrix


(2.8)

R,,(E) =
(2.9)

R,(E) =

and R*(E) The analysis by Goldberger

and Watson then gives

Joe(t) t-im

0

R&(E,) e--i&t/n JtJ”(t) tto? (E,, - -4, - R,+(E,J~ ’ The probability

for a transition

(2.10)

= Tim R(E + k).

(2.1

la)

(2.1

lb)

to a definite final state at energy Ei, is given by (2.12)

THEORlES

where &I$,)

OF

159

ALPHA-DECAY

is the density of states per unit energy interval and

r,=

2Tr *: R&(Eb)~’ dp(E,). J

The width I’, is to be evaluated at Eb = E, 7. R,(EJ. The expression (2.14) for the width can be rewritten

r,, = 2TrJ* !(lu,’ (H

(2.14)

using equation (2.6) to give

-- K) I I??‘,~” dpp(E,)

(2.15)

which can be iterated using Eq. (2.7) to give

--i- (lv,’ ) (H - K) & The second term represents a process intermediate state and then decays. 2.2. Choice oj’ the Harniltorlians

(H - K) 1 ‘lu,) ... i2,

in which

the nucleus is excited to a virtual

K and K’

2.2.1. Method of Kadmenskii and Kalechits. Kadmenskii and Kaiechits [20] have derived a similar expression for the width following the formalism of Goldberger and Watson [19]. They give as their choice for the unperturbed Hamiltonian K z= K’ = HA $- H, 7 TR 4. 2ZeZ/R and consequently the states Yn , Yb’ (b + 0) form a complete set of eigenstates. It is evident, however, that this Hamiltonian will not give any bound states for the relative motion because there is no potential barrier and the whole of the nuclear interaction is contained in the residual interaction H - K = V,,, -

V, - 2Ze”lR.

Hence, at t = 0 the nucleus is not in a bound state or in a resonant state. In practice, Kadmenskii and Kalechits take the wavefunction Y, to be a shell model state constructed from single-particle wavefunctions of oscillator form. This means that the nonorthogonality of the states Y(, and Y,,’ is reintroduced through the use of a different (shell model) Hamiltonian which is not related to the operators Hand K defined in their paper. They comment that in order to interpret the expression for the width, K must correspond to two spatially separated regions. internal and external, but this separation is not present in their formalism. Their value obtained for the decay width of nlnPo by the method is listed in Table II.

160

JACKSON

AND

RHOADES-BROWN

2.2.2. Method of Harada and Rauscher. Harada and Rauscher [12] also take Y,, to be a shell model state and draw attention to the problem of nonorthogonality which arises. Their nucleon single-particle wavefunctions are generated in the potentials given by Blomquist and Wahlborn [21]. They proceed by time-dependent perturbation theory and obtain an expression for the width which in our notation is

r,, =: 27TI%l.Ylh l(H ~ K’)j !Poy

(2.17)

with K’ = HA -L H,, -+ &. $- V, ;

T,

(2.18)

H -- K’ = v,4 - lJv

(2.19)

where U, is the nuclear part of the phenomenological optical potential for the a-particle. In order to examine the physical meaning of the expression (2.17), we consider the transition from the ground state of the decaying nucleus to the ground state of the residual nucleus and assume that the wavefunction for the initial state is represented reasonably well by a real a-particle interacting with the residual nucleus in its ground state, i.e., we take a single term from the expansion ( 1.12). The initial and final wavefunctions can then be written as (2.20a)

y,, - YL@az, u,l -

(2.20b)

4n@AXa+

where $a is the internal wavefunction of the .u-particle, z1 is the relative wavefunction in the initial state, xa+ is the relative wavefunction for the emitted a-particle and 0, is the ground state wavefunction of the residual nucleus. The expression (2.17) for the width reduces to r, cc I<&+. 1 ii,

~- u,

Xx“ 1”

Q‘V = C&@/l I V&/l I +,@A;> c (0 I v,,, Now,

Feshbach’s

formalism

(2.21)

for the generalized optical potential

1 Us = (0 I Vu, I Oi -t ;;O I Vcx,Q ___E _ QHQ = c!, -I- A O.,

(2.22)

0, . [22] gives

QV~A I O>

(2.23a) (2.24b)

and hence the difference 0,” - UN corresponds in principle to the second order term A CT,,,. In practice, Harada and Rauscher take various phenomenological potentials for U;, so that there is no simple connection between c!, and r/, . In this approach, the width is that of a resonance in the system consisting of an a-particle and the residual nucleus. The interaction is the residual nuclear interaction which couples the scattering channel to the resonant state of the decaying parent system. The Coulomb potential and the barrier appear to play no special role in the formalism which could apply equally well to nucleon emission. However, in the

THEORIES

OF

161

ALPHA-DECAY

calculations by Harada and Rauscher on decay of “‘?Po and rr”Po the role of the Coulomb potential is crucial in determinin g the magnitude of x*-m in the interaction region. Values obtained by Harada and Rauscher for the decay widths of 21nPo and ?laPo using four different alpha-nucleus optical potentials are given in Table IT. 2.2.3. Method of’ F’wtzatt, Ho/an, Kahetwkii, at& Stratan. Furman et al. [23] have developed the method of Kadmenskii and Kalechits [20] taking the same unperturbed Hamiltonian to give Y,, and introducing a shell model wavefunction for u/,’ They also replace the nuclear interaction I :,., by the real part of the nucleon~~nucleus optical potential summed over each nucleon in the ,x-particle. Since V,,l is a many-body operator, replacing it by the one-body optical potential excludes the possibility of treating decay processes in which the residual nucleus is left in an excited state. The use of the phenomenological potential presents difficulties for the proper description of the finite size of the ,u-particle, but Furman et N/. [23] have given an approximate method involving expansion of the optical potential in a power series of the ratio of the ,*-particle coordinates to the halfway radius of the potential. 2.2.4. Creation qf 0 hod ittitial state. Even for long-lived naturally radioactive nuclei, the true initial state is not a stationary state which persists until acted on by the interaction H-K. We can make the initial state into a bound state, however. by introducing the approximate Hamiltonian

[I,

=

[i,(R)

=- C’R(rtj),

pi- I’,.(R).

R ..: rt, R

(2.25)

,rtI

where rb is the barrier radius, i.e., the position of the top of the barrier formed by the potential Liv v(. . The required bound state for the x-particle has the singleparticle energy.

The interaction

is now (2.37)

If we take U,, =~=i., , the width for the g.s. to g.s. transition

is now given by

The physical interpretation of this expression is that the transition represents the lowering of the constant potential U,(r,) to allow the n-particle to escape. as shown in Fig. 2.

162

JACKSON

AND

RHOADES-BROWS

u, CR);

i -... ~ ~_ ~. FIG.

2.

Sketch

of the behavior

--1 of the bound

state potential

c/R for L ~- 0

This approach is likely to be a good approximation for the long-lived decays in which penetration through the barrier is the dominant aspect of the process. In this casethe wavefunction for the final state must be obtained from the Hamiltonian K’ =~ H,4 + H, -+ T, 4 U.v 1~ V(.,

(2.29)

This leads to nonorthogonality again but, since K -~ K’ is zero in the region R c:: r-1, where the bound state wavefunction for the lu-particle is large. this should not be a significant effect. This method has been applied to decays of the polonium isotopes in Section 4. 2.2.5. Mefhod qf’ Sclilitter. Schlitter [24] also starts from the concept of decay of a bound state of a model Hamiltonian. This bound state 4 is transformed into the decaying state + according to the equation (2.30)

THEORIES

OF

163

ALPHA-DECAY

He then shows that the total width is given by

where &? is a continuum chosen to be

state of K. The second form for r follows because h’ is h’ =mmPHP

QHQ.

(2.32)

The problem with this method then arises in the calculation of the channei state +fi:cwhich is a solution of (QHQ - E) &'

~~0

12.33)

becausePC&~ : 0 and & is orthogonal to 4. In an example of potential scattering of an l-particle from Y, three ditlcrent approximate methods of choosing the initial bound state are examined. The results for the width vary by a factor of 2.

3. 3.1. Feshbaclt’s

Ptwjectiotl

UNIFIED

Operator

REACTION

THEORY

Fortnalistn

In this section we follow the unified theory of nuclear reactions developed by Feshbach [22] using projection operator techniques, and seek a formula for the width of a scattering resonance for the system consisting of an a-particle and the residual nucleus. 3. I. 1. Derirariott of’ at1 espressiott ,fbr the width. We introduce the pro.jection operators P and Q. suchthat PY representsall the open channels with a free \-particle at infinity and Q mm1 ~~ P. The Schrodinger equation

reducesto

where HpO =mmPHQ. equation

etc. We also introduce the wavefunction $ which satisiiesthe (E

Ht,t>) J,

0

(3.3)

164

JACKSON

AND

RHOADES-BROWN

We neglect excited states of the Cu-particle in order to write the projection in the form

operator

P

(3.4) and define the scattering wavefunction

for the z-particle

?,‘I- ~~ xtt ~~ ’ f$$$“’ The x,

as (see Section I .4)

~J,’

(3.5)

then obey coupled equations of the form (3.6)

where CT,,‘,,,’ = {cja@,;* ’ I/,,4 ; +,@,II;~“’

(3.7)

Neglecting the coupling, we obtain. for the g.s. to g.s. transition. ( T, + cl,,, -b b;) x,, ‘- = E,X, /

(3.8)

where c,,, is the folded potential already defined in Eq. (2.22). Eliminating PPfrom Eqs. (3.2) and using Eq. (3.3) we find iE

-

HQQ

~~

HOP

-pp F _

I

f&o)

H

Qy = Ho++

(3.9)

where .7 ____ I E - i-irG(E ~ Hpp). E -- Hpp E - HP,

(3.10)

For the single channel case it is easy to show that (qht j HP0 1 QY, where

= FL%

(3.11)

\

K’, is the reaction matrix defined as [22] (3.12)

E - cOQ = E - HQo - HOP &-

HP0

(3.13)

PP

Feshbach [22] introduces

a set of eigenfunctions

of the operator

eQQ

so that (3.14)

THEORIES

165

OF ALPHA-DECAY

(3.15) Thus (KJ is related to the R-matrix

The corresponding

transition

Tfi = TfiPot TI e2is

(see Section 1.2)

matrix element is (6

(3.16a)

1 + in(K:

= TTft -1 e2i”(#’

1 HP0

_ T;;” + ,2i”(#+

! HP0

QY, 1 E - HQQ -

(3.16b) -

Hop ) ~‘1,

(3.16~)

wQQ

where we have used Eq. (3.9) and have put W QO =

l H E - Hpp Op ’

HOP

(3.17)

If we assume that the space defined by Q contains only one resonant state, denoted by Y,, , we have Q = ~Y,,, ‘Y. I and

- ~QT
-

Hpp) HpQ 1YJ

(3.18)

:iI’

Finally, we use Eq. (3.5) to expand #+ in a complete set of eigenstates so that the partial width is given by (3.19a) (3.19b) and the resonance energy is Eb --= E, f A. For the g.s. to g.s. transition r,

= 271.ICY0 ; HOP i xZ’+&@A’

we have (3.20)

166

JACKSON

3.1.2. Relation to other theories.

AND

RHOADES-BROWN

We have

(Y,, / QHP 1 I/%+; = (u-‘, j HP - PHP / $+) = <:Y, 1H - E / ++I,

(3.21)

where we have used Eq. (3.3) and the relation Plc,- = $+. Hence the partial width (3.20) becomes r,, z= 2x I,
(3.22)

which is the same as the unantisymmetrized formula given by Mang [I] and by Fliessbach [25]. They note that if #” were an exact description of the A -14 system it would satisfy the equation (E - H) $” 7: 0 but this does not hold for a model wavefunction. They argue that differences between the true wavefunction and the model wavefunction occur in the exterior region R > R,, and hence I’, = 257 i(!Po 1H - E i x,:‘+&@..,‘R>R~, 1’. The radius R, is an arbitrary radius in the theory. We note. however, with our expression (2.28). Using Eq. (3.8) the expression (3.22) can also be written as

(3.23) some similarity

(3.24) Because we have neglected coupling to intermediate states, our optical potential has reduced to ON which is real. It follows from the definition (3.7) that Eq. (3.24) yields I’ = 0 for the g.s. to g.s. transition. Thus, this formalism, in which the width is determined by the residual interaction, is entirely satisfactory for the decay of an excited state by nucleon or a-particle emission but it does not give a simple description, at least in lowest order, or the type of decay process which arises as the result of tunnelling through a barrier without excitation. 3.2. MacDonald’s

Fomalism

3.2.1. Derivation of an expessiorl ,for tile width. MacDonald [26] has developed a unified theory for nucleon-nucleus scattering which pays particular attention to the structure of the compound states and allows these to be constructed in a shell model basis. In our case, these states will be cluster states of the a-particle and the residual nucleus. The transition matrix element is written in the form T,/ -z ,:& 1 U i &‘I\

+ <&-

~ T I & -;

(3.25)

where & is a plane wave state and T =z p f

ji((E - H + ic)-l v

B = VT, - u

(3.26a) (3.26b)

H = H,,, + H, + TR + V,, + V, = K -I- i?

(3.26~)

THEORIES

OF

167

ALPHA-DECAY

Here U is an arbitrary real potential. MacDonald now introduces projection operators P, , P, such that Pd projects on to discrete bound states of K. These discrete states are denoted by I n>. The compound state X/L can be written as

where the A,” are coefficients

for which

(t,, - K - P,$P,)

X,, = 0.

Here ‘i’ is the effective interaction p = v -- v(E - K + k-’ and, since it is complex, the energy E,, = E,‘ [26] proves the relation

&ir,

I’,.?’

(3.27)

is also complex. MacDonald

Im p == -nTTII^(P,. 6(E - K) P,T’ so that the expression for the width becomes

where we have introduced the complete set of continuum states for K. If there is one discrete compound state which we can identify with the initial state Y-‘,,. then using the lowest order approximation ;1’ = v, we obtain for the partial width the expression r, = 2n [ dp(E,,) !yP” / P; z&,-‘,;“.

(3.29)

3.2.2. Relation to other theories. In MacDonald’s theory the potential U is real and arbitrary. It may, therefore, be chosen to give a bound state instead of a resonant state, as we have done in Section 2.2.4 where U = CiR-~ 1’; For the g.s. to g.s. transition we have

where we have usedthe definition of cT,V, Eq. (2.22). Thus, the expressionfor the width does not reduce to zero in this case. Formally, this result resemblesquite closely that of Harada and Rauscher [121,although their treatment of of Y,,l ~ U is quite different in practice. Dumitrescu and Kiimmel [27] were also concerned with the structure of the compound state. They assumethe validity of the shell model with effective residual interactions (SMERT). and obtain an expression for the total width of the form

168 where

JACKSON

&’ is a continuum

AND

RHOADES-BROWN

solution of K = H, + H, + T, + V, .

The many-body

resonance function [Eb

~

H]

For n-decay, Dumitrescu

where

X(E)

X(E) is a solution of Eb :z E + (2i)-1 T;,(E).

=z 0,

and Kiimmel

[27] assume that

Vent is the real part of the nucleon-nucleus

optical potential, and that (3.3lb)

w.

1

z

yP*t 1

--

u!”

.I

(3.31c)

where Up” is the shell model potential. The difference between H and A is included using the exponential operator technique, i.e. i h(E))

= e” 1X(E)),

[Eb - I?] X(E) = 0

(3.32)

so that the resonance character is generated by the operator es. We note that this model still contains uncertainties arising in the choice of the phenomenological optical potential, and that there is no formal connection between V”Dt and VP. Excitation of the residual nucleus is possible only if V”pt is treated as a generalized optical potential containing collective coordinates. Numerical evaluation of the a-decay width by this method would appear to be quite complicated. 3.3. Excitatiorl

qf’ the Residual Nucleus

The expressions (3.19), (3.29) for the decay width allow the possibility of excitation of the residual nucleus. In this section we develop the formalism for this, still neglecting excitation of the n-particle. The wavefunction for the initial state can be written in the form

y$fJ= ,z,,nG,fN~~~~~M IJMJ)@%I)MS>R:~‘(R)

(3.33)

where q, 5, R are the internal coordinates defined in Section 1.4, and C,(I) is the fractional parentage coefficient. The relative wavefunction j& can be expanded as -L,,i X’l = R-*ii,vl,( R) Y,““(lri)

(3.34)

THEORIES

so that the normalization

169

ALPHA-DECAY

is 1’

The wavefunction

OF

,;“’

1’ rl”R = j,;

U.,,,_ I2 dR :

I

(3.35)

for the final state must be coupled to the same channel spin, to give Y;J:Jn!f = 1 (Ivf’M’

i JA4,) @f;““(y)

+,(<) s,:+(R)

(3.36)

with 7u+ XL = A,R-‘u,(R)

Y,v(fi) Y;*(k).

(3.37)

We have to evaluate the matrix element .Y(JMJ - I’M’)

= <‘#‘L”“” / k’,,, ~~- U / ‘?FfJ,z

(3.38)

where the potential U(R) connects only those terms with 1 == I’. Since we have assumed that the n-particle is not excited, the many-body interaction k’,,, can be folded with the +particle density to give

and the matrix element of this interaction

between states of nucleus A is given by

-= ; (47T)‘.S (Iiclh-y ~I’M’) .I

V~‘(R)[PY,.~(R)]*.

Hence the matrix element X becomes

1 L uL( R)[ V;“( R) ~- S,;,U(R)] U,,,(R) t/R.

-0

(3.41)

The effect of the integration over the density of statesindicated in Eq. (3.32) is to give

When I == I’ corresponds to the gs. of the residual nucleus we have

170

JACKSON

AND

RHOADES-BROWN

where we have used the definition (2.22) of the optical potential 0,” and P,.~is the g.s. density distribution for nucleus A. Similarly, when either / or I’, but not both, corresponds to the g.s. of the residual nucleus we have

where py’ is the transition density for nucleus A. The other terms Vi.” are the potentials which would appear in a couple channels description of excitation of several levels in the residual nucleus. The transition density may be constructed microscopically using appropriate wavefunctions for nucleus A, or it may be parametrized in a macroscopic model, or it may be taken from fits to data if the g.s. of A is stable or sufficiently long-lived to allow inelastic nucleon or electron scattering experiments to be carried out. In each case, the optical potential and the interaction potentials VI: may’ be treated consistently with the same effective interaction K,,, using the single-folding technique introduced into the analysis of elastic and inelastic a-particle scattering [28]. Alternatively, the potentials V, may be parametrized directly in terms of the derivatives of the optical potential, i.e., I V,,(R) -= --i”‘[4n(2k A- I)]-“’ (,B&) y; , fi .’ 0, (3.45) where PI;& is the transition strength which may be related to the rotational tional model, as appropriate. Equation (3.41) simplifies considerably in certain special cases. (i)

J = MJ

A’(00 -r I’M’)

(ii)

0 = 1 C,,“(t)(I.1,

l)“iL

( -i)”

AI, Y;;,“‘*(1;)(LOkO

: [(2/i- + 1)/(2/’ -I- 1)2]1/2 1% uI,[V;” .0 1’ : j$f’ == 0

/U(JM, --, 00) : c C,‘(Z)(-LI (iii)

J

.-= M,

:=

0, 1’

i)’ A, Yf’+)(JOZO

/ Z’O) S,.,U] U.v, r/R

(3.46)

1LO) [ ’ u][ V;” ~- 6,,,0’] iiyL r/R ‘0 (3.47)

-M’z:

X(00 --f 00) = 1 C,“(L) L

3.4. Excitation

or vibra-

0

A,Y,“(ir)

f n un[ I’,“” -

S,,U] ii,, t/R

(3.48)

-0

qf the a-Particle

The excited states of the four-particle system should be included in the espansion (3.33) so that the wavefunction becomes

THEORIES OF ALPHA-DECAY

171

When the Eq. (3.49) is inserted into the analysis of the previous section it is evident that the \-nucleon interaction V,,,( R. 7) must be replaced by

Thus the inclusion of excitation of the ,-\-particle in the formalism of the previous section requires a double folding procedure starting with the nucleon-nucleon interaction I’,,., . This procedure has also been introduced into the study of a-particle scattering [29].

4. CALCULATIONS

FOR 0, -0

TRANSITIONS

4.1 . Cluster Fortnalim In this section we calculate the widths for the g.s. to g.s. transitions in the polonium isotopes, whose parameters are listed in Table 1. We use the lowest-order formula for the width developed in Section 2, Eq. (2.16) and in Section 3, Eq. (3.29), i.e.

where H-K is the perturbation and p(&,) is the density of states. We treat the initial system by the cluster expansion, represented by Eqs. (1.12) or (3.33). The transitions studied in this section are all Of ---f O+ transitions and the residual nucleus has 82 protons. In these preliminary calculations we therefore approximate the cluster expansion for the initial nucleus to the single term with L = I P: 0, which representsa real cx-particle moving relative to the residual nucleus in its ground state. Using Eq. (3.48), the expression for the width becomes

r ==[C,,“CO,l” ro,,

(4.2)

where the one-body width is

Thus, [C,,“(O)]’ has the role of the spectroscopic factor defined by equation (1.4) and represents the probability of finding in the initial system a component which resemblesa free -y-particle and the residual nucleus in its ground state.

172

JACKSON

Comparing the definition Eq. (2.24) we see that

AND

RHOADES-BROWN

of the Hamiltonian

where U, is defined by Eq. (2.25) with U,v ~~~ii,, potential, defined by Eq. (2.22). 4.2. Construction

of the Microscopic

K given by Eq. (3.28) and by

and 7?.Yis our calculated optical

Poterttials

We have to construct the microscopic formula for o,v can be reduced to

potential

r’:, defined by Eq. (2.22). The

o,v(R) = .I‘ PA(r) V,,( 1R -- r ~) d3r

(4.5)

where V,, is the a-nucleon interaction and pA is the ground state density for the residual nucleus. Tn order to evaluate (4.5) we use the following forms for I,,, [30, 311: Gaussian

Saxon-Woods

H : V, == 43 MeV,

K

0.526 fin-l

H’ : C; = 53.15 MeV.

K

0.526 fm-l

M : V, = 42.5 MeV,

a m= 0.34 fm

R = (1.43 ~ 0.0009E) fm where E is the lab energy for the nucleon-alpha interaction. By using these forms for va, we take into account, at least approximately, the size and structure of the ,-y-particle in its ground state and exchange in the II ~ 01system. The nuclear density distributions have been constructed from the single-particle model using the Batty Greenlees potential [32] for the protons and the Zaidi-Darmodjo potential, as moditied by Batty [33], for the neutrons. The depths of the potentials were varied to give the correct separation energy for the least bound proton or neutron in each lead isotope. In the caseof zlsPb this procedure cannot be used for the protons: consequently, we have usedthe proton distribution for ao8Pbin this case. The optical potentials derived by this method have been shown by several authors [lg. 30, 311to give good agreement with the elastic scattering data. The values of the barrier height V(rb) and its position rt, for each isotope are given in Table III. The values obtained are in good agreement with those deduced from scattering data [34]. In calculations on the decay “l*Po -, zOsPbwe have also used the phenomenological optical potential of Barnett and Lilley [35], which we denote by BL, and the potential of DeVries et a/. [I 51,which we denote by DLF.

THEORIES

OF

173

ALPHA-DECAY

TABLE

III

Calculated barrier heights and positions TI, (fm)

C’k,,) (MeV)

10.8

20.61

11.0

20.34

10.8 11.0

20.60 20.33 20.56

11.0 10.9

'0.49

10.8 11.0

20.54 20.21

10.9 I I.0

20.21

20.48

4.3. The Functions Xa and ,y$ The quantum numbers NL of the bound state function xu for the a-particle were deduced from the oscillator rule 2(N ~ 1) -+ L = t [2(/l, ~ 1) + /J i-1

(4.6)

where tlili are the quantum numbers for the two protons and two neutrons in the least bound single-particle levels in the A t 4 system. This procedure then yields TABLE Four-Nucleon

IV

Configurations in 21sPo

Coefficient

2(N-

1) +

L

Lowest s-state

0.6682

22

I2

0.3900

22

12

-~0.2386

24

13

0.1067

22

I2

0.2305

22

12

0.1064

22

12

0.1665

24

I3

0.2030

24

13

0.1026

26

I4

0.2385

22

I2

174

JACKSON

AND

RHOADES-BROWN

N ~- 12 for the lowest relative s-state for 212-e16Poand N == 11 for zlOPo. When configuration mixing is taken into account higher values of N are also possible; examples of possible values of N for ?lOPo are given in Table IV, where the admixture coefficients are taken from the work of Glendenning and Harada [36] and terms with coefficients less than i-O.1 have been omitted. In order to obtain a bound state for each isotope at the correct energy E, , defined by Eq. (2.26), we have to vary the potential. The potential U, , defined originally be Eq. (2.29, is therefore modified to be U,(R)

=

=

g[Lj,(R)

uR(rb),

+

V,(R)]

R :- rb

+

(1

-

g)

u&b).

R 5; t-1)

(4.7)

and the parameter g is allowed to vary until a bound s-state with the required N is obtained at the correct value of E, . Figure 3 showsthe potentials for zosPbafter com-

IMeV)

~

20 ‘0 L.~-.-

1

2 -20 -Lob -6OL -80

THEORIES

OF

175

ALPHA-DECAY

pletion of the search procedure. It follows from the definition (4.7) that the barrier heights and positions are unchanged. The parameter g lies in the range 0.4-0.6 for the microscopic potentials but is 1.54 for the phenomenological potential (BL). The phenomenological potential, as we will see later, is quite different in shape from the microscopic potentials and this has a substantial effect on the gound state wavefunction. The scattering wavefunction x,, ’ is generated in the potential R < rb

C/(R) = U,(R), =

o,,(R)

+

V,(R),

R :- rk,

(4.8)

which has the correct behavior at large distances and the barrier height and position consistent with elastic scattering data. The wavefunction can be expanded in the form x,-(R) = (2~--~,‘? k (g)“’

4~ ; i’e’“if;(kR) Y;*(k) Y,“(R)

(4.9a) (4.9b)

where we have put (4.10) The radial function ~1~ obeys the boundary conditions llz = 0

at

R = 0,

(4. I la)

111= (a/2)[(F, -,- X,) -)- PyFl

- iG,)]

at large R.

(4.1lb)

where Fi , GI are the usual regular and irregular Coulomb functions. Hence the wavefunctions are normalized so that /‘X,JE) 1x,(F)

:_ 6(E - E’) &ii - ii’,

(4.12)

and the density of final states is d2,. Comparing Eq. (3.37) with Eq. (4.9b). we find that the coefficient Al is given by A 1 - i’e””

A I 12 --~:1

and hence the one-body width for the L :-= 0 g.s. to g.s. transition reducesto

(4.13)

176

JACKSON AND RHOADES-BROWN

4.4. Resultsfor the Decay Width Results for the decay widths are given in Table V. The values deduced for the spectroscopic factors are consistent with expectations from nuclear structure considerations, in particular the very low value for *l”Po reflects the need to break the closed neutron shells. The results for 116 PO are not included becausethey are subject to some doubt owing to the uncertainty in the proton distribution in 310Pbwhich was discussedin Section 4.2. The results obtained with the microscopic potentials H, H’, and M show that we have succeededto a very large degree in eliminating the uncertainty associatedwith the choice of the optical potential, as can be seenby comparing the results in Table V with those in Table II. The remaining variation in the results is due to the slight differences in the barrier heights (seeTable III) which leadsto differences in the binding energy E, defined by Eq. (2.26). This effect has been verified by fixing the barrier for the 01-C zosPbsystem at 20.4 MeV, which yields the results given in Table VI. The consistency of these results indicates that our method can give very precise and reliable results for the decay widths, but that is extremely sensitive to the barrier height V(rb). TABLE V Calculated Values for Decay Widths and Spectroscopic Factors of L : 0 Decays of PO Isotopes

Isotope

Potential

Zl”Po

H

H' ?'2Po

H H’ M BL D.L.F.

314p.

H H

1.30 1.64 I .02 1.46 1.12 1.45 3.84 5.40 6.80

10mZ7 ‘< 10-Z’ ‘! I o-” ,j_ lO-‘4 _I: IO-” ~, IO-‘” :.’ lo-‘1 ,’ lo-” ‘: lo-”

0.029 0.023 0.149 0.104 0.136 0.010 0.040 0.053 0.042

TABLE VI Decay Widths for the f. = 0 Transition in alzP~ Calculated with a Barrier of 20.4 MeV

H H’ M

1.27 .I’ IO-” 1.37 !< 10 14 1.32 x JO-‘4

0.120 0.111 0.115

THEORIES

OF

177

ALPHA-DECAY

The markedly different result obtained from the phenomenological potential (BL) is due to the need to lower this potential to make it contain a 12sbound state, whereas we have to raise the microscopic potentials. This means that a very substantial difference is introduced between the phenomenological potential and the microscopic potentials in the surface region, as can be seenfrom Fig. 3, and this leads to a substantial difference between the bound state wavefunctions. The DLF potential is also a phenomenological Woods-Saxon potential but it has been chosen to give a 12sbound state at the required energy and consequently gives a value for the width which is closer to the values given by the microscopic potentials. As can be seenfrom Table IV configuration mixing in the shell model corresponds to the introduction of states of relative motion in the cluster model having higher principal quantum numbers. We have investigated the effect of taking different values of N using the potential for z”*Pb - 2 derived from the H parameters. The results are given in Table VII which showsthat uncertainty in N of one unit leadsto about the TABLE One-Body

Width for Various Fixed Binding

Relative Energy

VII s-States in *lsPo, c, in the H Potential

Calculated

fol

-

Principal

quantum

number

N

Scaling

factor

12

0.67

13

0.77

14 IS

,
r I~0 (MeV)

1.02 1.38

IO ‘1 IO-14

0.87

1.87

IO ~‘1

0.98

2.48

IO ‘4

same variation in r as that arising from the different choices for the parameters of the ol-nucleon interaction. The full H potential supports a 15s state at the given m-particle energy.

5. CALCCILATIONS 5.1. Structwe

qf the Isorueric

FOR THE DECAY OF z*Cti)Po

State

The isomeric state 312i’iPolies at 2.85 MeV above the ground state. fn the l-decay of the state 97 ‘<, of the decays go to the g.s. of “““Pb, I UC;to the first 3- state at 2.62 MeV, and 2 y{, to the 5- state at 3.20 MeV. According to the shell model calculations of Glendenning and Harada [36] the spin of the isomeric state is IX ‘~and the principal configurations of the extra core nucleons are ( 1lro&,,=,(2g, ,I i,, y)J,,=l,, and (1h,.,Jfi d,p&g9 ~~1 iI1 An -1o. Using the oscillator rule, Eq. (4.6), these configurations yield N = 3, L =L 18 for the lowest allowed state of relative motion when the core is in its ground state with I ~~ 0.

178

JACKSON

5.2. Decal) to the g.s.

AND

RHOADES-BROW&

qf 20sPb

An expression for the decay width may be obtained from Eq. (3.47). We consider first only the term with I : 0. i.e. the simplest cluster term. so that the width for the decay to the g.s. of the residual nucleus is given by

Now,

using Eqs. (4.41, (4.7), and (4.13) we have

The expression for the one-body width has been evaluated using the microscopic potential H with the correctly calculated barrier height. This yields a value of

giving a spectroscopic

factor s, ::: 0.128,

(5.4)

which is very comparable with the values obtained for the g.s. to g.s. transition. Consequently, the ratio of our calculated one-body widths for the decay of a121i’Po and “‘PO to the gs. of S”sPb is in very close agreement with experiment. In contrast. the calculations of Glendenning and Harada [36] using R-matrix theory and the shell model with configuration mixing gave a ratio greater than experiment by a factor of 45 for a channel radius of 9.5 fm and of I8 for a channel radius of 9.0 fm. 5.3. Drcaj, to the Lowest 3- and 5- States in s”nPb Expressions for the widths of these decays can be obtained from Eq. (3.41). For simplicit),. we include in the expansion for the 18’ state only the lowest 0.. 3 m,and 5m~states of the residual nucleos. This leads to the transitions indicated schematically in Fig. 4. The vertical arrows represent transitions with 1 =~z= I’ which have contributions from I, 0, as well as other values of k if Z m= I’ 2’~ 0. The other transitions have contributions from k s 0 through the excitation potential Y,, . which we treat by means of Eq. (3.45). We lirst neglected the second and third terms in the expansion of the cluster wavefunction for 21zPo and calculated the decay widths to the 3- and 5 states in “nxPb. The ratios of these widths to the width for decay to the O-l-g.s. are then independent of the coekient Cl:(O). These results are given in Table VI11 and are denoted as set A. We next obtained the widths corresponding to the transitions represented by vertical arrows in Fig. 4 using the value of @(Z) C~,“(Z)jC:~(O), calculated by perturbation

THEORIES

Initial

state

of 2’2mP0.

state

of “*Pb.

Final

OF

179

ALPHA-I)ECAY

2oBPb15-) FKr.

4.

Diagram

to

illustrate

the

transitions

contributing

to

the

decay

of

Xi~“‘Po.

theory, as described in the Appendix. These results are given in Table VIII ;I\ set B. Finally, we combined these two sets of transitions so that we have taken into account all the transitions represented in Fig. 4 by double-line arrows and have allowed for interference effects. These results are given in Table Vlll as set C. In each case the quantity denoted by I’,,,, is the width calculated from Eq. (3.41) divided by [Ci$O)12. TABLE Widths

for

Decay

of nllP~

VIII to States

of “‘“Pb

(in

MeV)

Theory Set A

Experiment ?““‘po

~-, zuspb

(0 k,

9.84

,‘. 10

212rnpo

-

(3--)

1 .Ol

::

““‘“p.

+“osPb

(5-j

2.02

.< 10

‘“spb

?’

1.67

10 -2,;

1.00

ci

, (p 2.1 ‘. IO-”

9.81

IO 9;

Ratio

08 3-

97.0

76.7

Ratio

0+.5-

38.5

781.8

Ratio

3

‘5

r;,,,

10.7

0.5

For these calculations, and for the calculations of the coefficients taken the collective strength parameters to be 1371 P3R0

0.81

B?(f)

v+c hac’c

/I,&, =-- 0.48.

The uncertainty on these parameters is about IO y,,. They were obtained from DWBA fits to inelastic 2particle scattering from 2”nPb using the generalized optical model with Saxon--Woods potentials. This means that there may be an additional uncertainty in relating the R, parameter to our microscopic optical potential. The derivative dI?,,,,ldR for the H potential peaks at 6.4 fm.

180

JACKSON AND RHOADES-BROWN

We seefrom Table VIII that for set A the relative width for the 3 state is about right but the relative width for the 5 state is too small by a factor of ~16. To correct this it would be necessaryto increaseP,R, by a factor of 4. For set B the relative widths for both the 3 and the 5 states are too small by a factor of 2 but the ratio is in excellent agreement with experiment. Thus, in this case, relatively small changes in the coefficients By(Z) would yield exact agreement with the data. This implies that what is important in the decal, C$ the isotneric state is tlot the collective excitation qf the residual nucleus Z”nPbbut rather the cluster structure of’ the deca~~irg state Z1siliPo.

Further support for this view comesfrom the result that the value of [C$O)]” deduced from the cluster perturbation calculation differs by only a factor of 2 from the value of $, obtained for the transition to the g.s., as can be seenby comparing Eqs. (A.10) and (5.4), despite the simplicity of our calculation of the coefficients in the cluster expansion. The difference could arise from small errors in the coefficients By(Z) due to the uncertainties in P,(R,, and from neglect of terms in the cluster expansion involving the lowest 2. and 4’ states in ZOxPb,or from the neglect of the contributions to the g.s. transition from terms with I + 0, or from neglect of Coulomb excitation. Contributions from additional terms in the cluster expansion will reduce the value deduced for [C$O)]p. In set B the most important contribution to each transition comes from the term with the lowest value of / due mainly to the effect of the angular momentum barrier. When the two modes of transition are put together to give set C we find that the interference changesthe widths so that the relative width of the 3~ state is too large by a factor of 7 while the 5- width is too small by a factor of 2. This cancellation depends sensitively on the value of /$R,, and on the values of the coefficients By(I) which also depend on PkR,, . For comparison we note that the calculation of Rauscher et al. [9] using the shell-model with configuration mixing and R-matrix theory gave a relative width for the 3 state which was too small by a factor of 250 and a relative width for the 5 state which was too small by a factor of 14. Both results were obtained with channel radius of 9 fm; it was found that the width for the 5 state could be brought into agreement with experiment by reducing the channel radius to 775 fm but this still left the width for the 3~ state too small by a factor of 100.

6. ANTISYMMETRIZATION 6. I. iQj%ct qf the Exclusion

Principle

So far we have taken the exclusion principle explicitly into account only in the way we have determined the lowest allowed state of relative motion in the initial A -1~ 4 system, using Eq. (4.6), although the use of effective interactions which tit elastic and inelastic scattering data should incorporate someexchange effects. However, a correct treatment of antisymmetrization leads to modification of the wavefunction describing the relative motion. Arima and Yoshida [14] take this into account approximately in the framework of R-matrix theory by renormalizing the interior wavefunction and

THEORIES

181

OF ALPHA-DECAY

hence the reduced width but, because R-matrix theory is used, the normalization coefficient is a function of the channel radius. Fliessbach [20] has developed the reaction theory due to Feshbach [22] to give a normalization coefficient [N(R)le2 which is very small inside the nucleus. 6.2. Cluster Formalism Ln this section we introduce antisymmetrization into the cluster formalism we have previously used, following resonating-group techniques. We rewrite Eq. (1.12) introducing the operator which exchanges nucleons between the four-particle system and the A-particle system, and take & and @.4 to be fully antisymmetrized wavefunctions for these systems. Thus

Following the method of Park, Schied and Greiner [38] we separate this wavefunction into direct terms of the form

and exchange terms @ix,Jn) which correspond

to exchange of IJ particles,

so that (6.3a)

= 1 [@u(n)+ @es(n~l

(6.3b)

,i

where the sum over S denotes all the different functions arising v-exchange. By inserting Eq. (6.3b)for Y,, into the Hill-Wheeler equation

from

a given

(6.4) we obtain a set of coupled integrodifferential

equations for 2,

[T(R) + &CR) -t- VAR) -- ~1 xx(R)

(6.5)

182

JACKSON

AND

RHOADES-BROWN

where we have used our previous definition, Eq. (2.22), of the optical potential i;, and have put E,, = E - EA ~ E, , as before. The direct coupling potential is defined as Lln.4nfi Oil = (q&Q,, ~ I’,, j ~J@~A,z

(6.6)

which is identical with Eq. (3.7) when II, L_:0. Hence, if we omit the exchange terms and put II, = 0, Eq. (6.5) reduces to Eq. (3.6). The exchange integrals can be represented as f di&) j” Q,,(v)

$,((I

%M

dy & -:- -

j K,(R,

R’) za”(R’)

d”R’

(6.7)

C’,.@&)

do c/f == --

f G,(R,

R’) flyn(R’) d3R’

(6.8)

&@)

where the K,( R, R’) are matrix elements of the exchange operator K introduced by Feshbach [22]. By neglecting the coupling to excited states we obtain the formal equation given by Arima and Yoshida [ 141

[T(R) + Vc(Rj + ~.dR)l X,? %(1 ~ n ,y*L k[T(R’) -J. V,(R’)] j& --- Gg,, . (6.9) The sameset of equations apply to the scattering function x, . The correct normalization of these functions is now [22,25,39] “x1(E)! I - IC i x,(E’)l

: S(E -~ E’)

(6.10)

instead of Eq. (4.12). In order to take account of this change in normalization a new scattering function Q(E) has been introduced such that [22, 251 Q,J E) = (I ~~ Ky

xu-( E)

(6.1 I)

and a new bound state function is defined as [25, lo] G,v z: (1 - @‘I’ za .

(6.12)

However, the operator K is nonzero only in the interior region which does not contribute to the g.s. to g.s. transitions in our formalism. 6.3. Discussion The normal procedure in resonating group calculations is to take the wavefunction to consist of only one antisymmetrized term from Eq. (6.1), namely the term with n, -= n.4 0. This is a valid approximation for many problems such as nucleus nucleus scattering at low energies but is less convincing for a-decay from or to an excited state. A study of the role of exchange in a-decay is now in progress, although the evaluation of the functions K, and G,Lis extremely tedious for heavy nuclei. We note. however, that the low-lying states of relative motion excluded by the oscillator rule (4.6) have radial wavefunctions which fall rapidly at large distancesand will therefore

183

THEORIES OF ALPHA-DECAY

have little influence on the u-decay widths for g.s. to gs. transitions which, in our formalism, have no contribution from inside rh . On the other hand there is an overall renormalization arising from antisymmetrization and this may modify our values derived for the spectroscopic factors S, .

7. SUMMARY AND CONCLWONS We have developed a method of calculating the widths for (r-decay of heavy nuclei which does not contain any arbitrary radii or phenomenological potentials. Indeed, we have only one adjustable parameter. namely the scaling factor g which changes the depth of the bound state potential to give a bound state with the correct number of nodes. We have shown that this method leadsto consistent and acceptable results for the one-body widths of g.s. transitions in the even polonium isotopesand for the branching ratio for the decay of the isomeric state 212’J’Po. The values derived for spectroscopic factors are in the range 0. IO-O.15 and our calculation of the cluster parentage coefficients for the isomeric state supports this magnitude. The shell model yields much smaller values than this. For example, a calculation for ?l”Po with admixture of 89 proton levels and 83 neutron levels yields 1.20 ~8’lo-” while the lowest configuration (Ih, Z)Z(3gS, J” alone yields I .98 ‘j 10~” [40]. DeVries et a/. [15] obtain results for a-decay widths using R-matrix theory with R, w 8.3-8.5 fm which are IO’-IO3 times larger than theoretical shell model values. Thus, it appears that jbr these /rear)>mclei the shell model, el’erl itlit substantial cotljiguration mixit7g, is jar ,fronl adequate and any attempt to conlpemate it? R-matrix theory leads to abnormall~~Ion2values of‘ the channel radii. There are many ways in which our calculation can be made more rigorous. A study of the effects of antisymmetrization is in progress but, as already pointed out. we do not expect the exchange kernel to have a significant effect on the one-body width for the g.s. to g.s. transitions although it could have a greater effect on the cluster parentage coefficients. In any case it would be more satisfying not to depend on the oscillator rule, Eq. (4.6). to determine the quantum numbers of the relative motion. More terms can be included in the cluster expansion for the initial state, the excitation of the residual nucleus for k .’ 0 can be treated microscopically but still within the cluster model. and the excitation of the n-particle can be included. All these modifications are feasible within the framework of our formalism.

APPENDIX:

ESTIMATES

OF THE CLUSTER PARENTAGE

COEFFICIENTS

The general expression (1.12) for the wavefunction of the initial state can be written as

(A.11

184

JACKSON

where

AND

RHOADES-BROWN

B, = C,L,‘CO . In first-order perturbation

theory, the coefficients

B, are given by

and the coefficients of terms with the a-particle in its ground state are

= (@AnAXmn I J’,A I @~x;oI‘ .

B&7,4)

Inserting

the angular momentum

(E, - Er) B;,(r)

(A.3)

coupling using Eq. (3.33) we find (-i)”

=

(-1)”

(LOkO 1 L’O)(Onk

-_ q 1 L’m’)

MmM’,,,

x

w C

where cases.

+ 1Wk + 1) l/2 ccllN,L,vII’ti n* A NL . (2L’ + 1) ‘0

I I

Vf’ is defined as in Eq. (3.40). Equation (i)

for certain special

Z--M--O

+ I)]'/* [? iiN~L+'$ivI. t/R

(E, - E,,) B,J(Z’) = i’\JOZ’O 1 L’O)[(21’ (ii)

(A.4) simplifies

(A.4)

‘0

(AS)

J = MJ = 0

(EI - El,) B;,(Z') = c (-i)" (-l)'+" k

(ZOkO Z'O)[(2k + 1)]1/2lx ii,vy'I,V;"iiAr,dR '0 (A.@

(iii)

J = 0, I = 0 (&,

If we now parametrize

EI,) @(I’)

= i”[(2Z’

+ I)]'/" [“ ii,v&f'fi,o ‘0

c/R.

(A.7)

V,, using equation (3.45) we have I

i”(2Z’

+ l)l/* ,$,' ii,,I>,I';$,,

dR = (-1)"+1(B,.Ro)(4,rr)-112 Jo=fi,,,,L,$

ii,, c/R. (A.81

We have calculated the BLJ(Z') for the 18-k state in 2VzPo taking I’ =z 3, 5 corresponding to the 3~ and 5- states of the zosPb core. In using the oscillator rule to determine the principal quantum number N’ we have added one quantum of excitation of the core to the right-hand side of Eq. (4.6). The values obtained for B:?(Z) are given in Table IX. From Eq. (A. I). the normalization condition becomes Co” = l:/(I

+ c

,I

B,?j.

(A.91

185

THEORIES OF ALPHA-DECAY TABLE

1X

Values of the Leading B Coetficicnts Coefficient

Value 0.97 m-1.14 0.54 0.22 -0.31 0.43 0.20

The values of the coefficients given in Table IX yield

Note ad&d i~r proof. Since we completed this work our attention has been drawn to the work of Tobocman [41] on exchange effects in particle decay.

ACKNOWLEDGMENTS We are indebted to Professor A. Arima. Professor 5. S. Blair, Dr. A. M. Lane and Professor G. H. Rawitscher for valuable comments and correspondence at various stages of this work, and to Dr. Won Sin Pong many helpful discussions and cricitism of the manuscript. We are grateful to Mrs. Joan Hilton for help and advice in computing. Mark Rhoades-Brown is supported by an SRC Research Studentship.

REFERENCES 1. H. J. MAN(;, AIIIL Rev. NW/. Sci. 14 (1964), I, J. 0. RASMUSSEN, “Alpha-, Beta-, and Gamma-Ray Spectroscopy” (K. Siegbahn, Ed.), NorthHolland, Amsterdam, 1965, p, 701. 3. M. IC.HIMURA. A. AKIMA, E. C. HALBERT. AND T. TERASAWA, Nucl. Ph~ss. A204 (1973). 225. 4. M. MFLJBE~K. H. SCHOBER, AND H. WAFFLER, Ph.vs. Rev. Cl0 (1974). 320. D. H. WILKINSOV.

2.

Phys. Rec.. 109 (I 958), 1603. 5. G. BREIT, “Handbuch der Physik” (S. Flugge, Ed..) Vol. 6. R. G. THOMAS, Prog. Theor. Phys. 12 (1954), 253. 7. H. J. MANG AND J. 0. RASMUSSEN, Mat. Evs. S/if. J. 0. RASMUSSEN, Nucl. Phw. 44 (1963). 93.

XLI/I. Dan.

Vicl.

SelsX.

Z

(1962)

No.

3.

186 8.

L.

JACKSON SCHERK

AND

E. W.

V~GT,

Ca/z&.

AND

RHOADES-BROWN

/. fhlx

46 (1968),

1 I 19.

9. E. A. RAUSCHER. J. 0. RASMUSSEN, AND K. HARADA, Nucl. I%.v.v. IO. T. FLIESSBACH AND H. J. MANY;, Nucl. Phys. A236 (1976), 75.

A94

(1967).

33.

11. C. BENCZE AND A. SANDULESCLJ, Phys. Left. 22 (l966), 473. 12. K. HARADA AND E. A. RAUSCHER. f/gx. &I,. 169 (1968), 81X. 13. G. Tco, P/7ys. Rep. 115 (1959), 1665. 14. A. ARTMA AND S. YOSHIDA, Nurl. Ph.vs. A219 (1974), 475.

15. R. M. DEVRIES. 16.

R. M.

DRISKO,

17.

C. E. PORTER,

A. FRANEY, P/ryx Rcr. Lctt. 37 (1976), 481. R. H. BASSEL, Phys. Lrtt. 5 (1963). 347.

J. S. LIL.LEY, AND M. G. R. SATCHLER, AND P/7.1x

Rec.

100 (1955).

935.

18. D. F. 19. M. L. 20.

JACKSON 4ho M. RHOADES-BKOWIU, GOLDBERGER AND K. M. WATSON, S. G. KADMENSKII AND V. E. KALETHITS.

J. BLOMQUIST 12. H. FESHBACH, 33. V. I. FURMAN,

21.

AND

S. WAHLBORN,

Nucl.

P/~Jx

“Collision Yud.

Arkir.

tj~sik.

“Reaction

Dynamics,”

S. HOLAK,

S. G. KADMENSKII.

f?=.

A266

( 1976),

61.

Theory,” Wiley, New York, 1964, Chap. X. 12 (1970).

70.

16 (1960).

Gordon and Breach, New York-London, AND

G.

STRATAN.

,~uc/.

P/73x

A226

1973. p. 169. 0974). 13 I

Nud. Phys. A211 (1973), 96. Zeit. Ph3vih-. A272 (1975). 39. 26. W. M. MACDONALD, Nucl. Pl7ys. 54 (1964), 393. 27. 0. DUMITRFSCU AND H. KijMMEL, A1717. qfPl7ys. 71 (1972), 556. 28. C. G. MORGAN AND D. F. JACKSOC. Phys. Reo. 188 (1969), 175X. 29. A. BUDZANOWSKI. A. DLIDEK, K. GROTOWSKI. 2. MAJKA. AND A. STRLALKOWSKI, Pnrticla Nuclei 5 (1974). 30. C. J. BATTY, E. FRIEDMAN AND D. F. JA~SSOS, Nucl. Phys. Al75 (I 975). 1. 31. P. MAILANDT, J. S. LILLEY, AUD G. W. GREENLEES. P/7ys. Rer. C8 (1973), 2189. 32. C. J. BATTY AYD G. W. GREENLEES. Nucl. Phw. .4133 (1969). 673. 33. C. J. BATTY, P/735. Lett. 31B (1970), 496. 34. G. GOLDRIYG, M. SAMUEL, B. A. WATSON. M. C. BERTIN, AND S. L. TABOR, PI73s. Lctt. (1970). 465. 35. A. R. BARNETT AND J. S. LILLEY. PI7p. Rev. C9 (1974), 2010. 36. N. K. GLENDENNINC~ AFL’D K. HARAIIA, N&. Phys. 72 (1965). 4x1. 37. J. ALSTER, P/73x Lat. 25B (1967). 459. 38. J. Y. PARS, W. S~HEID, AND W. GREIN~R, P/r],s. Rcr. C6 (1972), 1565. 24.

J. SCHLITTER,

35.

T. FLIESSBACH.

39. 40.

B. BUCK, A. ARIMA

41,

W.

1976, preprint. private communicalion. P/73x Rw. Cl3 (1976), 790.

H. FRIEDRICH, AND AND I. TONOZUI(A,

TOBOCMAN,

C. WHEATL~Y,

otd

32B