Effect of non-local barrier and relative intensities of alpha-spectra in rare-earth nuclei

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Effect of non-local barrier and relative intensities of alpha-spectra in rare-earth nuclei

M.L. Chaudhury, D.K. Sen

To cite this version:

M.L. Chaudhury, D.K. Sen. Effect of non-local barrier and relative intensities of alpha-spectra in rare-earth nuclei. Journal de Physique, 1981, 42 (1), pp.19-26. �10.1051/jphys:0198100420101900�.

�jpa-00208986�

Effect of non-local barrier and relative intensities of alpha-spectra in rare-earth nuclei

M. L. Chaudhury ^{and D. K. Sen}

Department ^of Physics ^and Mathematics, Indian School of Mines, Dhanbad-826004, Bihar, ^India (Reçu ^{le 17} juin 1980, accepté ^{le 26} septembre 1980)

Résumé.

La méthode pour calculer les ^taux de décroissance 03B1 en prenant pour barrière le potentiel ^coulombien

et un potentiel ^{non local} alpha-noyau ^{avec terme} d’échange ^est appliquée ^aux isotopes 149,151Tb, ^149Tbm et 153Dy.

Ce calcul montre que l’intensité relative mesurée, 03BB1/03BB0, peut ^en grande partie ^être reproduite par les rapports ^des pénétrabilités, ^ce qui permet de calculer les largeurs partielles 03B421. Des calculs similaires utilisant un potentiel alpha-noyau statique ^donnent 03B42l ~ 03B420, ^en contradiction avec l’analyse ^des amplitudes spectroscopiques qui donne 03B421 > 03B420. ^{Il est} remarquable que ^cette méthode corrige ^{dans une} large ^{mesure ce} désaccord, ^{donnant un} rapport 03B421/03B420 de l’ordre de 1,1 ^à 2,2. ^On peut ^en conclure que les valeurs de 03B421 ^{obtenues avec un} potentiel statique

sont trop ^{faibles et} que celles données dans le tableau II peuvent ^être utilisées, dans des limites raisonnables, pour des études de structure nucléaire.

Abstract.

The method of calculating alpha-decay intensities taking the barrier to be the usual Coulomb potential superposed by ^{a non-local} alpha-nucleus potential ^with exchange ^{term is} applied ^{here to the fine} structures of the rare-earth isotopes 149,151Tb, ^149Tbm and 153 Dy. ^The present calculations consistently show that the measured

relative intensities, 03BB1/03BB0, ^are largely accounted for in terms of the non-local penetrability ^{ratios and hence the}

values of the reduced widths 03B421, ^are calculated. Similar calculations using ^static alpha-nucleus potential give

03B421 ~ 03B420 ^{which is contrary to the} finding, viz., 03B421 > 03B420 obtained ^{from other sources, for} example, ^{from the studies}

of alpha-spectroscopic amplitudes. ^{It is} remarkable that the present ^method largely makes up this discrepancy giving 03B421/03B420 in the range of 1.1 ^to 2.2, and it is concluded that the values of 03B421 from static potential ^{are too low}

and those given in table II are within reasonable limits for use in studying ^nuclear structures.

Classification

Physics ^Abstracts

23.60

1. Introduction.

The derivation of a-decay ^inten-

’sities has been a long-standing problem in nuclear

physics. Early ^{theories were based on the} hypothesis

of a pure Coulomb barrier together ^{with a} frequency

factor for the internal a-wave. Lack of agreement

with experiment, except ^{for a} few limited cases, has led to developments ^of a-decay theory along ^the following ^{two lines :} viz., (i) ^{use of a more} realistiç

barrier than the pure Coulomb potential, ^{for cal-} culating penetration ^factor (see below) ^and (ii) repla-

cement of the aforementioned classical frequency

factor by ^an overall internal transition probabilitÿ.

Bohr, ^Frôman and Mottelson [1] assumed this pro-

bability ^{for an} a-particle ^{1-wave in a} deformed nucleus to be proportional ^{to a} squared Clebsch-Gordan coefficient. The effect of possible ^axial asymmetry [2]

of deformed nuclei was taken into account by ^Ros-

tovskii [3] and others. On the other hand, ^{in the}

theories of Thomas [4] ^and Mang [5, 6] ^{the internal}

transition probability ^is expressed ^{as an a-reduced}

width, b2, derived from R-matrix theory using ^the overlap of shell-model wavefunctions. The sensitive

dependence of this method on the choice of an arbi- trary channel radius (which turns out to be less than the inner turning point) ^{and the} ambiguity ^{of the} potential parameters have been discussed [7-10].

Other effects, ^{such as channel} couplings [21a, 12], configuration-mixing [13], ^and complex ^energy eigen-

values [14] ^have also been considered. It has, however,

been found that a common difficulty ^{with these and}

similar other calculations is their inability ^{to repro-}

duce absolute decay-rates; for relative decay-rates, although the trend is reproduced, results deviate by

factors varying ^{from 2 to} 10, ^{as has been noted in the}

literature (see ^for examples, [12, ^{16, 17,} 35]).

From what has been said above, ^the question naturally arises whether ^{one can} expect ^{a solution}

to these difficulties by attempting ^to improve only

the intemal transition probability, ^without adequately refming ^the penetrability ^{which has} long ^{been reco-}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420101900

20

gnized ^as the main factor goveming 0153-decay.’ ^As

mentioned in the beginning, the other approach ^to

the problem ^{has been to} review the barrier hypo-

thesis. From a study ^{of the} systematics ^of a-spectra, it was suggested [18] ^{that the pure} Coulomb barrier in a-decay ^is unrealistic, and should have superposed

on it an appropriate ^a-nucleus potential ^{due to the}

interaction between the nucleons of the just ^emitted a-particle and those of the residual nucleus, operative

until the two fragments ^are separated beyond ^the

range of nuclear force. This point of view is now

well-known (see, ^for example, [19-21]). ^{A static}

a-nucleus optical potential, ^{such as} Igo’s potential [24]

superposed ^on the Coulomb barrier, ^{was used in} the formalisms developed by ^{Rasmussen and other}

authors [22, 23, 37]. ^{It was, however, soon} pointed

out [25] that since the particle-nucleus interaction is

momentum-dependent, ^{as has been known for a} long

time (see [26]), ^{and is now} well-established from

scattering experiments (see, ^{for example,} [27-32]),

then the consequent non-locality ^{of the a-nucleus} potential should be taken into account in treating a-decay problems ^{also. Of course, if non-local effects}

on barrier penetration ^{are not} negligible, ^{it should be} possible ^{in this} way ^to explain ^{the observed} 0153-decay characteristics, particularly ^for large 1, ^{which are not}

otherwise explicable.

In this connection it need be emphasized ^{that the}

situations in a-decay ^and scattering ^{are much} different,

with different boundary conditions, energy-ranges and the added complexity ^{due to} the formation- factor entering only ^{in the} a-decay process. It is also

known that the same optical potential ^{does not hold}

even in different areas of scattering, and there is no

indication _{from any} experiment ^to suggest ^{that if} non-local effects are small in scattering, they ^should

also be small in ce-decay processes. ^It is therefore reasonable to be guided by a-decay ^{data, in our}

search for the appropriate ^non-local part ^{of the said} a-nucleus potential ^for calculating ^the penetrability, ^P.

From this consideration, ^a model of non-local a-nucleus potential ^was suggested, ^{and a} formula for the penetrability ^factor PN,l (where subscript ^{N will be}

used to refer to non-locality) ^{for a} spherically sym- metric barrier was derived. Straightforward ^calcula-

tions have shown [33] ^{that the} ratios, PN z/PN 1" ^largely

take account of the measured intensity ^ratios (mostly

within 50 % ^{of their} values) for almost all ce-spectra of spherical ^nuclei (both ^{even and odd-A} isotopes

from Bi to U). ^A logical extension of this formalisa

was to define a spheroidal ^{non-local a-nucleus poten-}

tial superposed ^{on a} similarly deformed electrostatic

potential, ^and this has been successfully applied ^to

the problem of angular distributions of alpha-particles

from oriented 237 Np ^nuclei [35]. ^{This method has}

also been applied ^{to the} alpha-spectra ^{of the} deformed

even-even nuclei from U to Fm [36]. ^{It is} significant ^that

the penetrability ^{ratios are} consistently ^{found to be} quite ^{close to} the observed relative intensities. On

the other hand, the values of PIIPI, using ^static alpha-

nucleus potentials have been found to be larger (by

factors varying ^{from 2 to} 10) ^{than the observed’ rela-}

tive intensities in the trans-lead nuclei, ^thus rendering tJi’ /tJi: incompatibly small. This finding ^{has been}

confirmed by other authors. For example, Toth, Bingham ^and Schmidt-Ott [38] ^have recently ^calcu-

lated tJZ2 ^{for the} alpha-spectra of the rare-earth nuclides

using ^static alpha-nucleus potential ^{in the} penetra- bility factors. Their results also show that for the

alpha-spectra ^of 149,1 s iTb and 149Tbm, à/ z (1/10) bd

and for those in 153Dy, à/ z ^0.005 bô. ^{This is contra-} dictory ^{to the} finding, viz., bi > bô ^{obtained from}

other sources and we shall show that the present method is in conformity with this result (see ^{section 5}

for discussion).

_

In the present work, ^our purpose is :

(i) ^To apply ^{the non-local} barrier method to the a-spectra ^{of the} rare-earth isotopes for which satis-

factory theoretical results have not yet ^been reported.

The values of spins ^{used in our} calculations are based

on what is known about their assignments ^{for the}

concerned states connecting a-decays. We have pre- ferred to follow the usual _sequence of spins ^{in the}

shell-model with changes ^which appear very probable

from other sources. On this basis, calculated values of PN,IIPNI, ^{are given} in table I.

(ii) ^{The measured} a-decay intensity ^is given by

where we use the symbol F, ^to represent ^{total internal} transition probability. ^{Various other} symbols ^are

used to represent ^this quantity, ^{such as b2 or} y2 (referred ^{to as} the a-reduced width), ^{or 92 or} A2

(called a-spectroscopic amplitudes). ^{We shall} prefer ^to

use here a simpler notation and write

In table 1 are also given ^{values of the} experimental

reduced width ratios, Fi/Fi,, ^{obtained from} equation (l.l) ^and (13) below, ^{where l’ refers to} decay ^{to the} ground ^state.

2. Wave équation. ^{- In a} previous paper [33] (which

will be referred to as 1 below), ^{the wave} equation ^for

the non-local barrier was obtained, ^{with some res-} trictions. In this paper, ^{a more} general form of the

equation ^is given.

The Hamiltonian of the two-body system consisting

of the alpha-particle ^{and the} daughter ^{nucleus is}

where y ^{is the reduced mass of the} alpha-particle,

V r 2 ^{is the Laplacian with respect to the vector r}

(= r, 0, lp) which is the position ^{of the} alpha-particle

relative to the centre of mass coordinate system ^with

axes fixed in space, Hf denotes the excitation _energy of the fmal nucleus due to collective rotation, particle-

motions and nuclear vibrations. The term HP, ⁱⁿ equation (2) represents ^the interaction energy between the alpha-particle ^{and the fmal} nucleus for a spheri- cally symmetric barrier. As already ^{mentioned in the} introduction, H n, ^{cannot be} purely electrostatic and is given by :

Z being ^the charge number of the parent nucleus and

e is the electronic charge. ^{The term} V,(r) ⁱⁿ equation (3)

denotes the isotropic ^non-local potential. ^{Now, for}

deformed nuclei terms must be added to 2(Z - 2) e2/r

and to V,(r) ^{to represent} deviations from spherical symmetry of the barrier. This is denoted by Hin, ⁱⁿ equation (2), wherein the electrostatic part ^{will be}

and V,(r) ^{is to be} replaced by V,(r) ^with corresponding

terms due to deformation, ^and R; ^is equal ^{to the}

inner turning point ^of the barrier.

For the spherically symmetric ^non-local potentia4

the interaction kernel ^as defmed in 1 is

where the vector r ( = ^r, 0, ç) represents ^the position

of the a-particle ^{in the} body ^fixed system ^{of coordir}

nates with the same origin ^{as for the} space-fixed

system of coordinates, ^{and the vector r’} (= r’, 0’, qJ’) represents ^some position ^{of the} a-particle ^{other than}

that given by ^r.

J(r, r’) suggested ⁱⁿ equation (5) is different from that in effective-mass approximations [46], given by

It may be seen from equation (5) ^that non-locality

is included in the function 8b, ^b being ^its range. Fur-

thermore, ^from analogy and also from observed odd-

parity transitions, ^{it was} found in 1 that a part ^{of the} potential ^{should be of} space-exchange character, ^due

to the Majorama interaction between the nucleons of the a-particle ^and surface nucleons of the daughter

nucleus. Hence ^we take a mixed kemel of the form :

where the subscripts ^{o and e refer} respectively ^{to the} ordinary ^{and the} exchange part ^{and S denotes the} proportion ^of Jo. ^{Hence the} integral operator ^{in the} integro-differential equation ^{for the} alpha-wave ^would

be :

where W, is the radial part ^{of the a-wave function} t/J ri’

Y, ^{is the} spherical ^{harmonics and we assume} bb ⁱⁿ equation (5) represented by

Hence for a spherically symmetric barrier, ^{the radial}

function U,(r) ( = rW 1(r)) ^{satisfies the} Schrôdinger equation

where _x is the radial part ^{of the} expansion ^{of J in} Legendre polynomials ^{and the lower} limit of the integral ^{is the}

minimum of r’ which depends ^{on the} region ^{to which the} equation (8) ^is applicable and Ei ^{is the} energy including

the recoil energy of the daughter nucleus. The ordinary ^{and the} exchange potential ^{terms in the} right ^{hand side}

of equation (8) ^differ only by ^a weighting ^factor. Therefore, taking ^{first the} ordinary ^{term and} dropping ^{the subr} script ^{o and} by ^a Taylor expansion ^of U,(r’) ^around r, integrating ^and retaining ^{the terms} up ^to b2, equation (8)

becomes :

22

where Ul ^and Ui’ ^are first and second derivatives of

UI(r) ^with respect ^{to r and :}

and

It is interesting ^{to note that the non-local} potential

terms of different orders exist, ^{which we} may denote in general by V o(r) g(n)(z) ^{for 1 =} 0, where the super-

script n ^{indicates the order of} p (cf. equation (10) below). ^{We have retained the terms in} equation (9)

up ^to order n

2. It _may also be noticed that for

x-angular ^{momenta 1 #} 0, ^an additional factor 1

(1(1 ⁺ 1) b2/4 r2) ^x (1 --: g(2)(Z)/g(O)(z»)

occurs. Therefore, it would be convenient if we denote

the spherically symmetric ^non-local part ^{of the barrier} by ^the symbol vl(n) (instead of Vl ^{in equation} (3)),

and referring ^to equation (9) ^{we have :}

where the purely ^non-local part

Equation (9) ^{can now} be written as :

The terms g(1) ^and g(2) ^are small and for the present

we neglect ^the higher ^{order terms} involving ^b2.

To this approximation ^and adding ^the exchange ^term,

which gives ^{a factor} (- 1)’, equation (11) ^{reduces to :}

3. Penetrability ^{factors and} intensity ^ratios.

Now, using the WKB method of approximation ⁱⁿ equa- tion (12) between the limits R; ^{and the outer} turning point Ro, ^we get ^the expression ^{for the} penetration ^{factor as}

where

and

The factor (YI ⁺ erf(z))) is the correction term due to the effective non-locality. ^The subscript ^{± in the} equation (13) corresponds ^{to 1 even or odd} respecti- vely. V o and the form function f (r) ^are taken from the

optical ^model potential [24] and S, already stated,

is the mixture proportion ^{of the} ordinary part ^of the potential. ^The integral ⁱⁿ equation (13) ^{is calculat-}

ed on an IBM 1620 computer by using ^{a modified} Simpson’s ^{rule as} given ^{in I, because the} integrand ^is exactly ^{zero for the outer} turning point Ro but approa- ches zero within the limit allowed by ^the computer for the inner root Ri.

In the calculations the number of strips ^{is taken to}

be 100 for the entire range of the integral, ^whereas

erf values involved are found sufficiently ^{correct with}

60 strips, ^{since smaller} strip-widths in numerical

integration ^{do not} make any appreciable change ⁱⁿ

the calculated values of the penetrability ^{factors. The}

values of Ri ^{are obtained} by ^{an iterative} method. The mixture proportion ^{S is} fairly well-known for the nucleon potential. ^S

⁶⁰ % ^was found suitable in the trans-lead region, ^and non-locality ^{is found valid}

within 0.5 b ^{0.9 fm} (the penetration ^{factor is}

not sensitive to the value of b within this range). ^We

shall therefore take the _average value of b

0.7 fm.

In table 1 the Ri values for a non-local barrier are

shown in column (6).

It is to be noted that the values of Ri ^{for a static}

and for a non-local barrier are considerably ^different

in rare-earth alpha-spectra ^{due to somewhat} larger

a-energy differences than in heavy ^nuclei. So, ^unlike

the case of heavy nuclei, ^{here we have to} take appro-

Table I.

Intensity ^ratios.

(*) Numbers in braces in columns (4), (7) ^and (8) ^are the powers of 10.

(**) Exceptional ^case.

(°) ^{Chu et al.} [11], (6) Golovkov et al. [49], e) Bingham et ^al. [42], (d) ^Ref. [38], (e) Macfarlane et al. [39], (f ) Baronovskii et al. [15].

priate ^{values of} Ri ^{in the two cases.} The calculated values of penetrability ratios for non-local barriers and the measured value of Âl/Â,, ^are given respectively

in columns (7) ^and (8) ^{of table I.} Hence the values of

bf /bÕ ^are calculated and given ^{in column} (9).

We present ^{in table} II, the values of penetration

factors and reduced widths calculated from _equa- tion (13) and the relation (1).

4. Discussion.

Information on a-decay ^half-

lives and branching ^{ratios are} taken from recent works. Most of these data have also been confirmed

by ^{other authors within} the limits of _error, except

for 153Dy, ^{which case} is discussed below. Regarding

the spins of the relevant states connecting a-decays,

we have preferred ^to follow the usual _sequence in the shell model except where different assignments

are known from actual measurements.

149m . ₆₅

The decay scheme for this case is shown in figure ^1.

The presence of isomerism in this case was suggested by Macfarlane [39] ^{to be due to the 65th} proton being

in the hl 1,2 orbital. That this high-spin ^{state is} depress-

ed below d3/2 ^and Sl/2 in the usual _sequence in shell model, has also been suggested by Brussard et al. [40].

On the other hand, ^the ground-state spin of the pro-

duct 14’Eu ^{has been measured} by Ekstroem et al. [41]

to be dS/2 +. ^{This means that the} g7/2 ^{level is} depressed

below 5/2 +. Whether the isomeric state h11/2 - in 165Tbm decays ^{to the} ground-state dS/2 + ^{or to the next} higher level, ^was considered by Bingham ^{et al.} [42].

From a consideration of E3 life-times, it is found that if isomeric transition ^{occurs to} the next 330.1 keV

level, the life-time turns out to be 0.1 _s, whereas the total half-life is 4.2 min. Hence, the isomeric transi- tion is now accepted ^as leading ^{to the} ds/2+ ^{ground- ,}

state of 14 s Eu. This gives ^an a-angular ^momentum

1 = 3 which is quite consistent with our result, viz., the non-local penetrability ^{ratio comes within 15} %

of the observed relative intensity, giving ô2/ô2 ;:t ^0.9,

where the value for static potential (putting ^non- locality range b

0 in equation (13)) gives P3/Po larger ^than the measured À3/ Ào by ^{about 700} % ^thus renderin b2 ô2

ren erlng 1 10 ⁰

14.9T .

165Tb .

Figure ¹ also shows the a-decay ^{of the} ground-

state of 149Tb, ^{to the first two low} lying ^{levels in}

145Eu. ^The ground-state spins ⁱⁿ 165Tb ^{as well as}

in 163Eu ^{are known to be} dS/2 +. ^{The fact that the} ground and lst excited state in 149Tb are d,,2 ^and hllj2-, ^{strongly suggests from} similarity ^{that the first}

two levels in 163Eu ^{are also the same, because the}

decrease in proton-numbers ⁱⁿ a-decay ^{occurs in}

pairs and, ^{if the} _g7/2 ^{level is} depressed ^below dsj2

24

Table Il.

Penetrability factors ^and experimental ^reduced widths, for ^{non-local barrier.}

See footnote in table I.

Fig. 1. - a-decay schemes for "9Tb and 149Tb"’.

in 149Tb, ^{there is} apparently ^{no reason for this} sequence to be upset ⁱⁿ l4sEu. Besides consistency of our result also provide support ^{to this} spin assignment ^shown

in figure ^1. Calculated values of PN,IIPN,O ^{are then}

found to be within twenty % ^{of the} observed relative

intensity ^{and hence} 1 0

1.25. On the other hand, calculations by ^{Toth et al.} [38] ^{for this case} using ^a

static a-nucleus potential gives ô’ approximately

1 ô2 10 ^Uo.

i s ir .

15’Tb

In this case the ground-state spin ^is reported ^{to be} 1/2+ (Ref. [43]). ^{However, from} électron-capture decay ^of 151Tb, ^{it is} reported [44, 48] that for this

ground-state ^{either of the} spins 1/2, 3/2, 5/2 ^is possible.

It is also worth mentioning ^{that the} ground-state spin ^{of 149Tb is known to be} 5/2. Again ⁱⁿ 153Tb (Ref. [43]), ^{as well as in 149Eu, which is} the isotone of

151Tb, ^the ground-state spins ^{are known to be} 5/2 (Ref. [41]). ^But from the result of reference [43], ^there

arises some uncertainty ^{in the} assignment. ^{So we} give

calculation for two alternative decay ^{schemes with} spins 1/2 ^and d5,2 ^{for the} ground-state ^{of the} parent nuclide 1 s 1 Tb, ^{as shown in} figures ^{2 and 3} respectively.

Now, regarding ^the product 147 Eu, ^the ground-state spin ^{is measured} [41] ^{and found as} 5/2+. ^{The next}

two higher ^{levels are however somewhat uncertain.}

If the spin ^{of the} ground-state ^{of 151Tb is taken as} 1/2,

Fig. ^2.

a-decay schemes for 1 s 1 Tb.

this would mean that the level Sl/2 ^is unusually depress-

ed below the set (hlll2, g7/2, ds/2) ^{of levels in the usual}

shell model sequence of levels. Consequently, ^the spins ^{of the} product ^{147 Eu are then} uncertain, ^{and we}

have ^no alternative but to choose an assignment ^that gives ô2/ô2 consistent with the values obtained for other cases. On this basis the decay ^{scheme as shown}

in figure ² gives the value of ô2/ô2

^1.06.

However, the alternative assignment, ^as given ⁱⁿ figure 3, appears ^{to us to} be more plausible. ^{This is} because, in both the isotopes ^{14SEu and 147 Eu the}

level ^concems the 63rd proton. ^From similarity,

it, is probable ^{that the next} higher ^{levels in both of}

them are the same. Hence, ^to be consistent with the level scheme in ’ 4 s Eu shown in figure ^{1, in the} present

case also (i.e., 147 Eu) ^{we take the first two levels as} dsj2 ^and h11/2 ^{as shown in} figure 3. It is interesting

that in this case ô 2/ô2

2.2, ^{as shown in table I.}

It may be mentioned that the 3rd excited states in

figures ^{2 and 3 are somewhat} arbitrary, ^{and not} being

Fig. ^3.

Alternative decay schemes for 151Tb.

connected by a-decay, ^{do not come into our calcula-}

tions.

166Dy .

The ground-state spins ^of 153 Dy ^{as well as} 149Gd85

have been measured to be 7/2- (Refs. [41, 45]). ^These

states are thus represented by ^the f7/2 - ^{odd neutron}

orbital, ^{the first one available} beyond ^N

^{82 closed}

shell. This means that the f7/2 ^{level is} depressed ^below d3/2 ^{as shown in} figure ^{4. However, in this case, we}

fmd that the value of PN,I/PN,O ^is higher ^than expected, giving & 2/b2

^{0.1. In fact,} this is the best fit values of 1 and of course would be obtained if the il 3/2 ^level

is also taken to be depressed, ^{to become} the lst excit- ed state in 149Gd.

Fig. ^4. - a-decay schemes for 153 Dy.

For a-decay ^{to the next} higher ^{level in} Gd, ^{we have}

no other alternative but to be guided by the best fit values of 1. With this criteria, ^the sequence of levels

are f,/2-, d3,2’ ^(or 113/2+) and g9/2 + ^{as shown in}

figure 4. The value of bf /1JÕ for the transition to the

9/2+ ^{level is} again ^1.6, consistent with the results obtained in other cases.

5. Conclusion.

The present investigation ^shows

that the non-local barrier method developed ^and

used earlier in understanding the relevant experi-

mental data on trans-lead and trans-uranium nuclei,

is equally ^well applicable ^{to the} alpha-spectra ^{of the} rare-earth isotopes.

In some papers it is mentioned that non-local effect is small in scattering. This, however, ^is not contra-

dicted by ^our results in alpha-decay, because, ^as already ^{discussed in the} introduction, scattering ^and alpha-decay ^{are not} identical situations. Our results

convincingly show that the non-local effects are

significant ⁱⁿ alpha-decay. ^The results obtained in the

previous papers and the present ^one may be summariz-

ed as follows : (i) ^for odd-parity transitions, ^the

26

exchange ^{effect is} quite predominant ^{even for small} alpha-angular ^momenta, 1 ; (ii) ^for even-parity ^tran-

sitions and for small 1, the results from non-local barriers are not much different, ^as expected, ^{from those}

obtained by using ^{a static} alpha-nucleus potential;

(iii) however, ^for alpha angular momenta 1 > 3, ^the

values of PN,,/PN,,’ nearly accommodate the observ- ed intensity-ratios giving ^the expected values for the ratios ô2/ô2

Finally, it may be mentioned that, from table I,

we fmd that for 1 = 1, F,/F 0 ^{= 1.6 and for 1 = 3,}

our results _vary from 1.25 to 2.2, giving ^an average

F3/Fo (i.e., b3/bô)

^{1.72. In} taking ^this average ^we have not included the value of F3/Fo ^{for 149Tbm}

because the situation ^is exactly reversed, ^{the a-tran-}

sition being ^{from an excited state of the} parent ^{to the} ground-state ^{of the} product ^{nuclei. It should be men-}

tioned that _{the range} of F,/F 0 ^obtained by ^us (viz.,

1.1 to 2.2) is lower than that of 3 to 7 expected ^from

the formula of Ichimura et al. [47], viz., FI = (21 ⁺ 1)

Fo obtained from the SU(3) ^model applied ^{to 160-} decay. ^It may however, ^{be seen} that this formula is based ^on the configuration assumed for 160. There-

fore, ^a possible ^{reason for the} difference between the range obtained here and that expected from the above formula _may be due to the différent configurations

for the nuclei of the rare-earths and 160. We conclude that the values of reduced width _given in table II are

within reasonable limits for the purpose of compari-

son with the theoretical reduced width, ^{which is} potentially ^{one of the most} informative factors about nuclear structure.

z

Acknowledgments.

One of the authors (DKS) gratefully acknowledges ^the grant of an I.S.M. Senior Research Fellowship. Calculations were done on an

IBM 1620 Computer of the Central Mechanical

Engineering ^Research Institute, Durgapur. ^{The faci-}

lities are thankfully acknowledged.

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