Intensities of alpha spectra of radioactive nuclei

Intensities of alpha spectra of radioactive nuclei

B. Pal

Department ^of Physics ^and Meteorology, Indian Institute of Technology, Kharagpur ^{721302, India} (Requ ^{le 9} juillet ^1985, révisé le 13 mai 1986, accepté ^{le 11} juillet 1986)

Résumé.

Le modèle d’exciton pour les réactions nucléaires

été appliqué

spectres alpha des noyaux radioactifs au-delà du Pb. Dans le cadre général donné par le théorème de Wigner-Eckart,

^loi empirique

pour la valeur du ^moment angulaire orbital de la particule alpha ^{émise par}

^noyau pair-impair

impair- impair ^est proposée, ainsi que cela existe pour ^les noyaux pairs-pairs. ^On

^utilisé

approximation ^linéaire

pour ^la relation entre le logarithme de l’intensité alpha ^et celui du facteur de réduction afin d’identifier les structures 1à où elles

sont pas aisément identifiables. Les résultats ^sont satisfaisants pour les petites ^valeurs

de J0.

Abstract.

The exciton model of nuclear reactions has been applied ^to obtain the intensities of alpha spectra of the nuclei in the post ^lead region. ^An empirical ^selection rule, within the general ^rule given by ^the Wigner-

Eckart theorem, for the unique value of the orbital angular ^momentum of the emitted 03B1-particle ^{from odd-A}

and odd-odd nuclei has been suggested

it is obtained for the 03B1-particles ^from

nuclei. An

approximate ^linear relationship between the logarithm ^of alpha intensity ^{and the} logarithm of hindrance factor has been used to identify ^band-like structures particularly where it is not easily discernible. The results

attractive for low J0 ^values.

Classification

Physics ^Abstracts

21.60

23.60

1. Introduction.

Precompound emission of a-particles ^{in the reactions}

of the type (p, a) ^and (n, a) [1-2] ^is explained ^{on the} assumption that there is a finite probability ^{of an}

particle pre-existing ^{in the} ground ^{state of the} parent nucleus. Bonetti et al. [3] ^have calculated the alpha preformation ^{factor for the} naturally occurring

active nuclei and compared with the values given by (p, a) ^and (n, a) reactions. They assumed that the

decaying nucleus is found in « one exciton

states, the exciton being the a-cluster in its single-particle

levels. It has been found by ^{Basu and Sen} [4] ^{that the} a-decay ^rates (g.s. ^to g.s. transitions) ^{in even-even}

nuclei give straight ^{lines when} plotted against ^the

square ^root of the absolute values of the simulta-

neous separation energies ^of two neutrons and two

protons ^{of the} parent nucleus for all nuclei having

constant Z. This was sought ^{to be} explained ^{on the}

basis of existence of a-clusters. Further, ^{a rather}

indirect connection of a-decay probability ^{with the} single particle ^{effects has been found} from the fact

(unpublished) that when the shell-corrections obtai- ned either by using the Weizsacker [5] ^or Myers-

Swiatecki [6] mass-relations are plotted against ^the logarithm ^of half life for different a-decaying ^chains (N-Z

constant), ^each decaying chain for even- even (and odd-A ) nuclei follow approximately ^a pair

of straight lines with a break at about Z

90 or 92.

It is ^now well-recognized ^{that at} about this proton number, ^{there is a} shape change in the nuclei and the shell corrections have their origin ^{in the} fluctuations in the density ^of single-particle ^states of the nucleus.

These facts about the existence of the a-clusters in the nuclei and the simple connection of the a-decay

half-lives with the single particle effects made us

examine the a-intensities in the light of the exciton model as proposed by ^Griffin [7] ^{and Blann} [8]. ^It

must be mentioned here that in order to perform ^a split ^{of a nucleus either} by way of alpha-decay ^or by

way of fission, the nucleus must pass through

intermediate states where energy is much higher

than the energy of the initial and final states [9] ^and

the exciton model deals with the intermediate pro-

cesses in nuclear reactions. One interesting point

about the exciton model is that it provides ^{a unified} description ^{of the} preequilibrium processes in nuclear reactions and the decay ^{of the} equilibrium compound ^nucleus [10].

2. Formulation.

The emission of the a-particles from the radioactive nuclei can be considered as a compound ^emission problem in. the exciton model of Griffin [7] ^{with the}

a-clusters assumed to be an exciton in its single particle ^states. Now the basic assumption ^{of Griffin’s}

model is that a set of eigenstates for the nuclear

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470110189300

single particle Hamiltonian exists which can be subdivided into classes characterized by ^various permissible ^numbers called the exciton numbers. All the energy conserving transitions are assumed to

proceed ^{via weak} two-body interactions with the restriction that the exciton number changes by ^{0, or}

± 2 for which the transition matrix element M I ^is

constant and otherwise I M I = O. With these

assumptions, ^the density ^of single particle ^{states at}

the excitation E is

where g is the average single particle ^state density.

The total density ^of states at the excitation E is

We like to calculate the probability ^{for an exciton}

state with an exciton in the continuum having ^the

channel energy between Eo ^and Eo ⁺ dEo. ^{This can}

be worked out as the probability relating ^{to a finite}

stochastic process the study of which is facilitated by

means of the tree diagram (Fig. 1) ^{as follows.}

Let a set p in the uniform probability space be

partitioned in subsets p 1, p 2"’p x of ^{the set} having ^the property ^that

where _p is the number of states of the whole system per unit energy interval ^at the excitation energy E ^as

given by equation (2). ^Let _p, ( E ) be the number of states per unit energy interval with the exciton

Fig. ^1.

^Tree diagram. ^Three partitions consistent with the restriction, ^An

^{0 or ±} 2,

the transition matrix have been taken.

number j

( n - 2), ^n, ( n ⁺ 2) corresponding ^to

the branches Al, A2, A3 respectively ^{of the tree} diagram (Fig. 1).

Then

the probability ^for

the conditional probability of , given

and

the conditional probability ^of

exciton states (which decay instantly by emitting ⁱⁿ alpha-particle with channel energy Eo), ^where

E

U + Eo ⁺ Eb, Eb being ^the binding energy of the a-cluster to the _core, U the excitation _{energy of} the residual nucleus and

By ^the partition theorem of statistics [11], ^the probability for emission into states available to the

alpha-particle ^{as an exciton}

The above sum is taken over j = ( n - 2 ) ,

n, (n + 2). Therefore, ^total compound ^emission probability

If it is assumed that the a-particle ^{as an exciton}

exists with its orbital angular ^{momentum I} already decoupled in the virtual daughter ^{state, the} equation (5) ^{is to be} multiplied by

the probability ^of decay through the channel of spin Jo. Thus the total probability

It requires ^some justification ^{for the} assumption ^of

a unique ^orbital angular ^{momentum of the emitted} a-particle. ^{For the even-even nuclei there is no need}

to have any justification, ^{for the} decays ^{in these} type

of nuclei take place ^almost invariably ^{from a state}

having ^the spin ^and parity ^{0+ .} Following points ^may

be considered in support ^{of the} justification ^{for the}

odd-A ^or odd-odd nuclei: (i) alpha-gamma angular

correlation offers a way ^to know the angular

momenta of the emitted alpha-particles. Although

there are some experiments ^{to show} that there is a

mixing ^{of the} alpha of different angular ^{momenta, it}

has been found [12] ^{in the case of} a-decay ^{from the} ground ^{state of} Bi212, having spin ^and parity 1- , ^to

the 40 keV state of Tl208 with spin ^and parity ^{4+ that}

the experimental anisotropy agrees with pure 1

3

or nearly pure 1

5. But the pure 1

3 is the better choice on decay ^rate grounds. (ii) It has been shown

[13] ^{that a} number of levels of ^some light nuclei,

which can be approximated ^{as a} two-body system ⁱⁿ its predominant channel, fall into reasonably ^well-

behaved Regge-trajectories. ^For example, ^the 5/2+

level at 6.38 Mev in N13 is considered as resulting

from the coupling ^of C12 (in ²⁺ state) ^{to a} proton with 1

0. There are other levels which arise due to the coupling ^{of the} proton ^with 1 = 1 with C12 (2+)

state. The concept ^{of a} Regge trajectory ^{is also} intimately connected with that of a rotational band structure [14] ^obtained through ^the a-decay ^process.

(iii) ^{In the case of even-even} nuclei, it is found that the a-decay ^{is more and more hindered or the}

intensity ^decreases gradually ^{as one} goes up the sequence of levels in ^a band to which the transitions

are taking place. ^Both favoured and many unfavou- red transitions in the odd-A nuclei behave very much in the same way ^as in the even-even nuclei. (iv) ^An analysis ^{of the} intensity pattern [15] ^{of the} a-decay

from Cm243 having ground ^level spin ^and parity 5/2+

to the 3/2+ ^{of Pu239} shows that the strength ^{of the} partial ^{wave with 1}

^{4 is} stronger ^{than 1}

^{2 wave} the reason for which is not known with certainty. ^All

these made us suspect that there might ^{be a} unique

orbital angular ^momentum 1 for the emitted

particle although ^the arguments put forward above

are not sufficiently convincing. ^{However, the} justifi-

cation can be had a posteriori from the final result.

The quantity ⁱⁿ (6) is the number of states per unit energy interval available in the continuum ^to the

particle ^{as an} exciton. Its reciprocal is, therefore, ^the

average distance of ^states into which the decay ^is taking place ^and qualifies [9] (with ^the decoupling assumption) ^{to be used as} D> which is related with the decay probability leading ^{to a definite state of}

the residual nucleus. The intensity ^{I of the} alpha- spectra will, therefore, ^{be related} with the above

expression for the average distance of ^states.

Thus,

with

or

assuming ^{the level} density parameter « g » to ^{be the}

same for both the parent ^{and the} daughter. (8) ^can

further be simplified by noting ^the following points : (i) ^E

^{U +} Eb ⁺ Eo, U is the excitation energy of the residual nucleus, Eb ^and Eo ^{are the} binding

energy of the alpha ^{cluster to the core and its kinetic}

energy respectively, (ii) ^{there is} always ^{a constant}

shift d in energy whenever the single particle spec-

trum is bunched into ^a periodic system ^{which has}

actually been done in equations (1) ^and (2) ^or

whenever pairing interaction is introduced into the level density ^formula [16].

The final result after simplification ^is

both are assumed to be constants when g ^{and A are}

constants. Thus, ^{if the left} hand side of (9) loglO I + 1 ₂ lo glo Eo - log10 (21 +1) (which ^{is deno-}

ted by y) ^is plotted against Eo, ^{it will} give ^a straight

line provided ^{all the} assumptions ^{involved are reaso-} nably ^correct.

3.1 EVEN-EVEN ^NUCLEI.

The equation (9) ^invol-

ves the orbital angular momentum 1 and Eo ^{which is}

the alpha kinetic energy including the recoil of the residual nucleus. It is easy ^to find I for the ^even-even

nuclei, for all transitions are taking place ^almost invariably ^{from J"}

^{0+ to} any level Jo ^{when the}

selection rule for 1 is given by J - J 0 I "’" 1 "’" J ⁺ Jo

consistent with the conservation of parity

7r i 1T f

( -1 ) l. ^The plot of y against Eo (Fig. 2)

shows quite good straight lines for all the rotational bands of the nuclei considered up ^to 1

4 after which there is a change ^{in the} slope. For the nucleus with Z

92, ^A

232, ^a very good straight ^{line is}

obtained up ^to 1

6. Landau [17] has shown that

highly prolate ^even-even nuclei which emit practi- cally ^all a-particles ^{from the} region surrounding ^the pole, ^the decay probabilities ^{to the} different levels of

a rotational band can be obtained from where

Jo ^{is the} spin ^of the rotational level which the

particles populate ^on decay, ^{C and K are theoreti-} cally ^{related to} the dimension and eccentricity ^{of the}

nucleus and do not depend ^on the energy levels.

With the value of K given by ^the observed À2/ Ào.

the value of À4/ Ào ^agrees reasonably ^{well but there}

is serious error

in À6/ Ào. ^{The present formula (9)}

has some similarity with the relation of Landau as our 1 is nothing ^but Jo ^{for even-even} nuclei and in

place ^of Jo Jo ⁺ 1 (characteristics ^{of rotational}

Fig. ^2.

^Plot of y ^for

^nuclei against Eo, ^the alpha kinetic energy including the recoil energy of the residual nucleus. Numbers in each graph indicate successi-

vely ^the proton ^and

numbers and those shown

against ^the graphs for each of the nuclei (90, 226), (92, 232) ^and (96, 242) represent ^{the orbital} angular ^{momenta 1}

of the emitted alpha particles

illustration.

levels) occurring ^{in the} exponential factor, ^{there is}

the KE equal ^to Eo ^{which is} intimately ^connected

with the rotational levels. Varshney ^{et al.} [181 ^have

studied the effect of spin ^{on nuclear} shape ⁱⁿ

deformed ^even nuclei in the framework of asymme- tric rotor model by calculating ^the quadrupole

transition probabilities ^BE (2) ^for almost all the medium mass nuclei and some heavy ^{mass nuclei,}

and noted that there is a shape transition at

Jo

⁶ suggesting ^{a limit at which the} rigidity ^{of the}

nucleus is unacceptable. In any ^case, the departure

from the approximate ^linear relationship (9) ^for higher ^orbital angular ^momentum values may be due to the dependence ^{of 4} the energy ^and therefore of

C, on ^the angular ^{momentum. It will be seen in the}

next section that ^some of the odd-4 nuclei also show

a break of linear relationship ^{at about I}

^{6 for the}

emitted alpha-particle.

3.2 ODD-A NUCLEI.

In order to apply ^{the above} analysis ^{to the odd A nuclei, the main} problem ^{is to}

know the unique ^{value of the orbital} angular ^momen-

tum I of the emitted alpha particle. ^{The selection}

rule IJ - Jol :E: ^{1 J + Jo} consistent with the conser-

vation of parity gives ^a range ^of values of I which was

used by ^{workers to sum over the different} partial

waves to get ^the alpha intensities. Chaudhury [19] ⁱⁿ

the study of the intensities of alpha spectra ^{for the} parity unfavoured transitions in spherical ^{nuclei used}

the minimum 1-values consistent with the principle ^of parity conservation. The need to have a particular

selection rule was felt by Poggenburg ^{et al.} [20] ^who suggested ⁱⁿ analogy ^{with the} electromagnetic ^transi-

tions the rule that which electric transition multipola-

rities would be strong ^or weak between the single particle ^states of the odd nucleon in parent ^and daughter ^{is to be} determined ; ^the strong ^{and weak} values for a-decay ^will generally correspond ^{to the} strong ^{and weak} 21-pole transitions. We have, ^howe-

ver, used the following simple empirical ^{rules for}

different situations consistent with the conservation

principles ^{and in} conformity with what is used with respect ^{to the even-even nuclei.}

Selection Rules

(1) ^For transitions with ^no change ^of parity

1ri = 1T f.

up ^to first ( 2 J, + 1 ) ^{number of levels in a particular}

band of the daughter. Thereafter, ¹

for the next number of successive levels of the ^same band.

upto ^first 2 J1 number of levels in the particular

band. Thereafter for the next (2/i+l) ^levels,

(J2 - Jl) ^{+ n,} n = 3, 4,... except when J2 ^starts

with 1/2 in which _case, of all possible ^{values of l, the} highest value consistent with the parity conservation has to be taken.

(d) J1 -+J2 = J1 - 2, J, - 1, ..., only ^{case obser-}

ved for this type of transition is with the nucleus

(Z

^{96, A}

243) ^{for which} 5/2+ -+ 1/2+ 3/2+...,

the highest ^{1 value} consistent with the parity ^conser-

vation is to be taken.

(e) J1 -+ J2

J, ^{+ 2,} J, + 3,

only ^{case observed}

for this type ^of transition is with the nucleus

(Z

^{94, A}

239) ^{for which} 1/2+ -+ 5/2+,... ^and

the values of I ^are unique.

(2) ^For transitions with the change ^of parity

first 2 Jl ^number of levels of the daugliter ^{in a} particular ^band. Thereafter for next (2/i+l)

number of levels in the ^same band

for first levels of the

daughter ^{in a} particular ^{band. For next}

levels in the same band,

in a particular ^band.

successively. ^This occurs ⁱⁿ the nuclei with (Z

98, ^A

249) ^and (Z = 100, ^A

251) ^{where the} transitions are of the The application of these rules in the case of some

of the nuclei is illustrated in the table I. The figure ³

shows the plot of y against Eo ^{for the} a-particles populating the levels of the different bands of the

daughter. ^The highest point ^{in a curve} corresponds

to the spin ^{of a} particular band-head of the daughter

with minimum value of orbital angular ^momentum

of the emitted alpha-particle. ^{The next lower} point

in the same curve represents ^the decay ^of a-particle

to the next higher level of the same band. For the

subsequent higher levels the orbital angular ^momen-

tum of the alpha ^increases generally ⁱⁿ steps ^{of two}

units in accordance with the parity conservation rules. However, it will be observed from the curves

that with the exception ^{of a few} intermediate points

in some curves, ^most of the points corresponding ^to

l = 6 or 8 fall out of the lines as it happens ^{in the}

case of even-even nuclei. This is very much conspi-

cuous in the case of the nucleus (Z

95, ^A

243)

in which the points corresponding ^{to 1}

^{8 and 10} depart very much from the straight line. It may be

pointed ^out that for each of the three bands of the nucleus (Z

98, A

249), almost all the points ^fall

on quite good straight ^lines. Further, ^{the curves} corresponding ^to the nuclei like (Z

94, ^A

239,

Z

96, ^A

²⁴¹ etc.) for which the decay ^is taking place ^{from states with} spin ^and parity 1/2+ ^{show the}

characteristics of rotation-particle coupling. ^The

same characteristic is also seen in the case of transitions to levels having the band-head spin ^and parity 1/2+. We have ventured to have similar plot

for the nucleus (Z

86, ^A

211) ^{in the harmonic} region. ^{In this case there is no immediate} guiding principle by which it may be discerned whether there is any band like ^{structure as} in the rotational region.

Fig. ^3.

^Same plot

ⁱⁿ figure ² for odd-A and odd-odd nuclei. Letters I, ^{II etc.} against

^{of the} graphs indicate the

plots ^for different bands in

nucleus and «

and « b » show the rotation-partition decoupling ^{effect in}

band. Circles with dot and solid circles have been used to indicate which point belongs ^{to which} graph. ^The nuclei (95, 242m ) ^and (83, 210m) show transitions from isomeric states. Two bands of the nucleus (86, 211) ^{have been} identified by ^{the linear} relationship ^between loglo ^{I and} loglo ^{HF together} with their spin sequences which

shown against each of the bands.

The spin ^{of the level at 393 keV} (shown ^{with ?} mark) ^{has been} suggested ^{to be} 5/2 instead of the tentative assignment

(3/2). ^Lower right hand scale is for (83, 212) ^and (86, 211).

Table I.

Illustrates the orbital angular ^{momentum I} of

even-even, odd-A and odd-odd nuclei. Values for

other nuclei have not been tabulated for economy of space. ^Data

^taken from reference [21].

(*) ^{Number in brackets in column} (6)

powers of 10.

t

Indicate relative intensities while other entries in column (6)

absolute intensities (transitions per 100 decays ^of

the parent substance).

Table I (continued).

On the basis of the approximate ^linear relationship

between loglo ^{I and} loglo (hindrance factor) ^{in a} band, ^{which we} shall discuss in the next section, ^and

the spin sequence of the levels, ^{we could} _get ^two

bands in the daughter

^one beginning ^with 1/2- ^at

the level 68.7 keV followed by 3/2- ^{at 236.6 keV}

and the level 393 keV with tentative spin (3/2- )

which we consider to be 5/2- and the second band is the ground ^{state band with} spin 5 2- ^{followed by}

the levels at 588.4 keV and 814.5 keV ^with spins ( 7/2 ) - ^and 9/2- respectively. ^We get ^two straight

lines (see Fig. 3) ^{for these two bands} using (9) ^and

the selection rule discussed earlier. The transition from 1/2-, ^{however, to} any of the levels in ^a band of the daughter ^{does not} show any rotation-particle coupling ^effect. In any ^case, it is clear from the

figure ^{3 that} quite ^a good number of odd-A nuclei satisfies the approximate linear relation (9) ^for intensity reasonably ^well.

3.3 ODD-ODD NUCLEI.

The number of odd-odd nuclei having well-formed bands with at least three levels for being populated by ^the alpha-particles ^on decay ^{from the} parent ^is quite few. We have studied

(see Fig. 3) four nuclei in the actinide region ^and

three in the harmonic region. The bands of the three nuclei (83, 210m; 83, 213; ^and 85, 216) in the harmo- nic region ^{were identifed on} the basis of a-intensity

and hindrance factor (HF) relationship (to ^{be discus-}

sed in the next section) ^{and the} spin sequences. The

plot ^of equation (9) for these nuclei, with three

points ^{in each case, is} quite good. ^For (85, 216) although the a-kinetic energy is clearly given ^for

each of the levels, ^the intensity ^{0.2 % for} decay ^to

the levels 210 keV and 238.63 keV is given together.

Further the latter level i. e. 238.63 keV level has a

tentative spin assignment (0- ) whereas the level at 210 keV does not have any assignment. ^{We have}

assumed the spin ^and parity ^{to be 3-} for this level and plotted equation (9) ^to get ^the straight ^{line as}

shown in figure 3. The transition from the isomeric states of (83, 210m) ^and ( 95, 242m) ^{are seen to}

follow the linear relationship ^of equation (9). ^{It is} interesting ^{to note} that in the decay ^{of both}

(83, 210m) ^and (83, 212), ^the magic proton ^num- ber 82 is crossed. Selection rule followed is ^{more or} less in the same way ^as that given ⁱⁿ previous ^sub-

section.

4. Intensities and hindrance factors.

Hindrance factor (HF) ^{is defined as the ratio}

(A-I/AO) calculated/ (AJ/A 0 ) ^observed where A., and A 0

are the probabilities ^of decay ^to the excited level with spin ^{J and} ground ^{state of the} daughter ^nucleus respectively. ^The intensity comes as a factor in the ratio. Therefore it is expected ^{that a} linear relations-

hip ^{exists between} loglo (HF) ^and loglo ^{I. We have} plotted ^{these two} quantities ⁱⁿ figure 4(a) ^{for the}

even-even and in figure 4(b) ^for odd-A, ^odd-odd

Fig. ^4a.

^{Plot of} loglo ^{I against} loglo ^{HF for}

nuclei for different bands which

indicated by ^{I, II etc.}

« a »

and « b », ^circles with dot and solid circles have the

same

meaning

ⁱⁿ figure ^{3. In}

^two graphs

with circle with dot and the other with solid circles coalesce in which

single line has been drawn.

Fig. ^4b.

^Same plot

figure ^4a for odd-A nuclei.

Fig. ^4c.

^Same plot

figure ^4a for odd nuclei and the odd-A nucleus (86, 211).

nuclei. It is found in these figures that for different bands in a particular ^{nucleus different} straight ^lines

are obtained. Here also the effect of the rotation-

particle coupling is observed in the case of transitions from 1/2’ ^{states. Thus we} find that the hindrance factors are not only dependent ^{on the} spin change

but also on the energies through ^the expression ^for intensity given by equation (9). ^{But the most} impor-

tant point ^{in the} present study is that the approxi-

mate linear relationship ^between 10glO (HF) ^and loglo ^I helps ⁱⁿ identifying the bands where rotational band-like structure is not immediately evident. This is further examplified ⁱⁿ figure 4(c) where the bands of the levels of nuclei (86, 211) ; (85, 216), (83, 212),

(83, 210m) ^{etc. have been} identified together ^with

the help ^of spin sequences of the levels forming ^a

band which the a-particles ^are populating ^on decay.

All the data used in the analysis of this section and the previous ^{section were} taken from the Table of

Isotopes compiled by Lederer and Shirley [21].

5. Discussion and conclusion.

The equation (9), which has been obtained by ^the

consideration of the exciton model of Griffin [7], gives the intensities of alpha transitions in a particu-

lar band of a nucleus, which may be even-even, odd

or odd-odd, ^{with the} empirical selection rules for orbital angular ^momentum of the emitted alpha particles up ^{to at} least a first few levels. Figure ³

shows quite good ^{results for some odd-A nuclei like} (92, 233), (95, 241), (98, 249) etc.., ^{while for} the odd- odd nuclei as well as those lying ⁱⁿ the harmonic

region the result is also reasonably good ^{for the few}

transitions (mostly three), ^for which the Eo ^and

other data are available, ^{in a} band. Thus the

assumptions of the exciton model together ^with

other assumptions ^work reasonably well in delinea-

ting ^the intensity pattern ^{in a band} provided ^{it is} agreed ^{that the} empirical selection rules can be

accepted. The breaks in the straight ^{lines at about}

l = 6 ^or 8 for even-even and odd-A nuclei may be due to shape transition as it has been surmised by Varshney ^{et al.} [18]. ^The deformation in shape ^itself

is a shell effect and therefore is expected ^to change

both eland C2 ^{which are} dependent on g the average single particle ^state density and d the energy shift parameter. The effects of angular ^momentum

and energy ^on these two parameters ^{are to be}

investigated. Further the level structures in the nuclei for which the transitions are taking place ^from

states with spin ^and partly 1/2+ ^{are to be enquired}

into. It may be argued that various assumptions

involved in the present ^work might have compensa- ted all the errors and therefore the exciton model,

which is used to explain the intermediate processes in nuclear reactions, ^{does not} really give ^{the correct} picture. ^{In that case, the} equation (9) ^{is to be taken}

as an empirical relation the parameters ^{of which}

need explanation. Assumption ^of unique ^orbital angular ^{momentum, which is not} in violation of the selection rule provided by ^the Wigner-Eckert ^theo-

rem, for the alpha-particles emitted from all types ^of

nuclei even-even, odd and odd-odd, put ^them somewhat on the similar footing. However, ^{in the} present study ^{we have not} considered the intensities of those decays in which the alpha particles populate

the levels with inverted spin sequence. In these ^cases the orbital angular ^{momentum 1 is to be} adjusted

such that the linear relation given by equation (9) ^is

satisfied.

Acknowledgments.

The author is thankful to Prof. R. K. Bhaduri of McMaster University, Canada for suggesting ^the problem ^{and to} Prof. D. Sen of Burdwan University,

W. Bengal, India, ^for helpful discussions.

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