PHYSICAL REVIEW C 82, 064605 (2010)

α decay of nuclei in the range 67�Z�91 from the ground state and isomeric state

K. P. Santhosh, Jayesh George Joseph, and Sabina SahadevanSchool of Pure and Applied Physics, Kannur University, Payyanur Campus, Payyanur 670 327, India

(Received 19 September 2010; published 17 December 2010)

The Coulomb and proximity potential model for deformed nuclei (CPPMDN) is used to study the favored andunfavored α decay of nuclei in the range 67 � Z � 91 from both the ground state (g.s.) and isomeric state (i.s.).The computed half-lives are in good agreement with experimental data. The standard deviation of half-life isfound to be 0.44. Geiger-Nuttall (GN) plots for various parent isotopes are studied. It is found that all four typesof transitions (g.s.→g.s., g.s.→i.s., i.s.→g.s., i.s.→i.s.) lie on a straight line. The isomeric state α decay showsa behavior similar to that of the ground state and the nuclear structure of the isomeric states imitates that of theground states. Some predictions are done for α transition from both ground and isomeric states, which will beuseful for future experiments.

DOI: 10.1103/PhysRevC.82.064605 PACS number(s): 23.60.+e, 23.35.+g, 27.70.+q, 27.80.+w

I. INTRODUCTION

Alpha decay is one of the most important decay modes forheavy and super-heavy nuclei. It was observed by Rutherford[1,2] a century ago and first interpreted as a consequenceof quantum penetration of α particles by Gamow [3] andindependently by Gurney and Condon [4] in 1928. Alpha decaycan provide some reliable knowledge on nuclear structure[5–11], and it can also be used to identify new isotopesor new elements via the observation of α decay from anunknown parent nucleus to a known daughter nucleus. Thereare many effective theoretical approaches that have been usedto describe α decay, such as GLDM [12], GDDCM [13],UMADAC [14], and CPPM [15], and all of them have beensuccessful in reproducing the experimental data and also inpredicting half-life values. S. Peltonen et al. [16] used thestationary coupled channel approach to study the α decay toexited 2+ and 4+ states of daughters, and the results obtainedare in good agreement with experimental data. D. Ni et al. [17]studied α tunneling through a deformed potential barrier byusing a new version of the generalized density dependentcluster model. The decay width is computed using the coupledchannel Schrodinger equation with outgoing wave boundaryconditions.

Usually α decay takes place between ground states that havethe same angular momentum and parity, and such α decay istermed a favored α decay. If the states of a parent nucleus havedifferent angular momentum or parity from that of a daughternucleus, such decays are probably hindered and are calledunfavored α decays. But now, with the progress in technology,detection of α decay of nuclei in exited states [18–21] havingrelatively large life spans (isomeric states) is experimentallypossible and favored and unfavored decays of such excitedsates are relevant for theoretical studies.

The Coulomb and proximity potential model (CPPM) hasbeen used to study the radioactive decay of parents in thetrans-tin and trans-lead region via cluster emission [22,23],following which there has been an extensive study on α decayof nuclei in heavy and super-heavy regions [24]. It could re-produce the experimental data, even though the deformation ofnuclei has not been taken into account. Different versions of theproximity potential have been used for studying fusion cross

sections [25–27] of different target-projectile combinations.Now the Coulomb and proximity potential model (CPPM) isestablished as one of the effective models to study α and clusterradioactivity. In this study we have improved our formula-tion (CPPM) by incorporating the ground-sate deformations(β2 and β4) of parent and daughter by treating the α particle asspherical. Using this modified version of CPPM for deformednuclei (CPPMDN), we have studied α decay between groundstates and isomeric states of parent and daughter nuclei.Among the various α-decay studies, favored and unfavoreddecay between isomeric states is a rare study and this is thefirst one using the CPPMDN formulation.

The formalism of the CPPMDN is presented in Sec. II. Adiscussion and conclusion on favored and unfavored α decay ofnuclei between ground and isomeric states is given in Sec. III,and, in Sec. IV, we summarize the entire work.

II. THE COULOMB AND PROXIMITY POTENTIALMODEL FOR DEFORMED NUCLEI

In the Coulomb and proximity potential model for deformednuclei (CPPMDN), the potential energy barrier is taken asthe sum of the Coulomb potential, proximity potential, andcentrifugal potential for the touching configuration and forthe separated fragments. For the pre-scission (overlap) region,simple power-law interpolation as done by Shi and Swiatecki[28] is used. The inclusion of the proximity potential reducesthe height of the potential barrier, which closely agrees withthe experimental result. The proximity potential was firstused by Shi and Swiatecki [28] in an empirical manner andhas been quite extensively used over a decade by Guptaet al. [29] in the preformed cluster model (PCM). R. K.Puri et al. [26,27] have been using different versions ofthe proximity potential for studying the fusion cross sectionof different target-projectile combinations. In our model,contributions from both the internal and external parts of thebarrier are considered for the penetrability calculation. In thepresent model, the assault frequency ν is calculated for eachparent-cluster combination and is associated with vibrationalenergy. But Shi and Swiatecki [30] get ν empirically and findthe unrealistic values of 1022 for even-A parents and 1020 forodd-A parents.

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SANTHOSH, JOSEPH, AND SAHADEVAN PHYSICAL REVIEW C 82, 064605 (2010)

The interacting potential barrier for a parent nucleusexhibiting exotic decay is given by

V = Z1Z2e2

r+ Vp(z) + h2�(� + 1)

2µr2for z > 0. (1)

Here Z1 and Z2 are the atomic numbers of the daughter andemitted clusters, z is the distance between the near surfaces ofthe fragments, r is the distance between fragment centers, �

represents the angular momentum, µ is the reduced mass, andVP is the proximity potential given by Blocki et al. [31] as

Vp(z) = 4πγ b

[C1C2

(C1 + C2)

]�

(z

b

). (2)

The nuclear surface tension coefficient γ is

γ = 0.9517[1 − 1.7826(N − Z)2/A2] MeV/fm2, (3)

where N, Z, and A represent the neutron, proton, and massnumber of the parent, respectively, and � represents theuniversal proximity potential [32], which is given as

�(ε) = −4.41e−ε/0.7176, for ε � 1.9475, (4)

�(ε) = −1.7817 + 0.9270ε + 0.0169ε2 − 0.05148ε3,

for 0 � ε � 1.9475. (5)

Here, ε = z/b, where the width (diffuseness) of the nuclearsurface b ≈1 and the Siissmann central radius Ci of fragmentsis related to sharp radius Ri as

Ci = Ri −(

b2

Ri

). (6)

For Ri , we use the semi-empirical formula in terms of massnumber Ai [31]:

Ri = 1.28A1/3i − 0.76 + 0.8A

−1/3i . (7)

The potential for the internal part (overlap region) of the barrieris given as

V = a0(L − L0)n for z < 0, (8)

where L = z + 2C1 + 2C2 and L0 = 2C, the diameter of theparent nuclei. The constants a0 and n are determined by thesmooth matching of the two potentials at the touching point.

Using the one dimensional WKB approximation, the barrierpenetrability P is given as

P = exp

{−2

h

∫ b

a

√2µ(V − Q) dz

}. (9)

Here, the mass parameter is replaced byµ = mA1A2/A,where m is the nucleon mass and A1 and A2 are the massnumbers of the daughter and emitted clusters, respectively.The turning points a and b are determined from the equationV (a) = V (b) = Q. The above integral can be evaluatednumerically or analytically, and the half-life time is given by

T1/2 =(

ln 2

λ

)=

(ln 2

υP

), (10)

where υ = ( ω2π

) = ( 2Ev

h) represents the number of assaults on

the barrier per second and λ represents the decay constant.The energy Ev is calculated using the empirical relation [33]

given as

Ev = Q

{0.056 + 0.039 exp

[(4 − A2)

2.5

]}, for A2 � 4.

(11)

For α decay,

Ev = 0.095Q. (12)

In the classical method, the α particle is assumed to moveback and forth in the nucleus and the usual way of determiningthe assault frequency is through the expression given by ν =velocity/(2R), where R is the radius of the parent nuclei.But the α particle has wave properties, therefore a quantummechanical treatment is more accurate. Thus, assuming thatthe α particle vibrates in a harmonic oscillator potential with afrequency ω, which depends on the vibrational energyEv , wecan identify this frequency as the assault frequency ν given inEqs. (10)–(12).

The Coulomb interaction between the two deformed andoriented nuclei, which is taken from [34] and which includeshigher multipole deformation [35,36], is given as

VC = Z1Z2e2

r+ 3Z1Z2e

2∑

λ,i=1,2

1

2λ + 1

Rλ0i

rλ+1Y

(0)λ (αi)

×[βλi + 4

7β2

λiY(0)λ (αi)δλ,2

], (13)

with

Ri(αi) = R0i

[1 +

∑λ

βλiY(0)λ (αi)

](14)

where R0i = 1.28A1/3i − 0.76 + 0.8A

−1/3i . Here, αi is the

angle between the radius vector and the symmetry axis ofthe ith nucleus (see Fig. 1 of Ref. [35]). Note that the

80 90 100 110 120

-2

-1

0

1

2

g.s.---->g.s. g.s.---->i.s. i.s.---->g.s. i.s.---->i.s.

log

10 (

T1/

2,ca

l. / T

1/2,

exp. )

Neutron number of daughter

FIG. 1. Deviations of calculated T1/2 values with the correspond-ing experimental data for different types of favored and unfavored α

decay.

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α DECAY OF NUCLEI IN THE RANGE . . . PHYSICAL REVIEW C 82, 064605 (2010)

quadrupole-interaction term proportional to β21β22 is ne-glected because of its short-range character.

Nuclear interactions [37,38] can be taken in two variants:the proximity potential and the double-folding potential. Thelatter is more effective in the description of interactionsbetween two fragments. The proximity potential of Blockiet al. [31,32] has one term based on the first approximation ofthe folding procedure, which describes the interaction betweentwo pure spherically symmetric fragments. The two-termproximity potential of Baltz et al. (Eq. (11) of Ref. [39])includes the second component as the second approximation ofthe more accurate folding procedure. The authors have shownthat the two-term proximity potential is in excellent agreementwith the folding model for heavy-ion reactions, not only inshape but also in absolute magnitude (see Fig. 3 of Ref. [39]).The two-term proximity potential for interaction between adeformed and spherical nucleus is given by Baltz et al. [39] as

VP 2(R, θ ) = 2π

[R1(α)RC

R1(α) + RC + S

]1/2 [R2(α)RC

R2(α) + RC + S

]1/2

×{[

ε0(S) + R1(α) + RC

2R1(α)RC

ε1(S)

]

×[ε0(S) + R2(α) + RC

2R2(α)RC

ε1(S)

]}1/2

. (15)

Here, R1(α) and R2(α) are the principle radii of curvatureof the daughter nuclei at the point where the polar angle is α,S is the distance between the surfaces along the straight lineconnecting the fragments, RC is the radius of the sphericalcluster, and ε0(S) and ε1(S) are the one-dimensional slab-on-slab functions.

III. RESULTS AND DISCUSSION

Using the Coulomb and proximity potential model fordeformed nuclei (CPPMDN), we have calculated α-decayhalf-life values for favored and unfavored transitions of nucleiin the range 67 � Z � 91. The energy released in α

transitions between the energy level of the parent nucleus withexcitation energy Ejp and the level of the daughter nucleuswith excitation energy Eid is

Qj→i = Qg.s→g.s. + Ejp − Eid . (16)

The Q value corresponding to various exited states iscalculated using Eq. (16) with the corresponding excitationenergy taken from Ref. [40]. Experimental Q values are takenfrom Audi et al. [41] and some of them are taken fromRefs. [12,13,42]. The α-particle emission from a nucleus obeysthe spin-parity selection rule:

|Ij − Ii | � � � Ij + Ii andπi

πj= (−1)�, (17)

where Ij , πj and Ii , πi are the spin and parity of the parentand daughter nuclei, respectively.

Table I encompasses the study on favored α decays fromparent nuclei in the region 67 � Z � 91. In this study we mainlyconcentrate on four types of α-decay transitions: (i) groundstate to ground state (g.s.→g.s.), (ii) ground state to isomeric

state (g.s.→i.s.), (iii) isomeric state to ground state (i.s.→g.s.),and (iv) isomeric state to isomeric state (i.s.→i.s.). The firstcolumn denotes α transitions from parent and daughter nuclei.The superscript m1, m2 and m3 on nuclide names denote thefirst, second and third exited states respectively. The secondand third columns label the spin parity of the parent nucleus(Iπ

i ) and its daughter (Iπf ), respectively. The fourth column

represents the experimental Q values. Since the spin and parityof parent and daughter nuclei considered in this table arealike, the minimum value of angular momentum carried bythe α particle is zero. The barrier penetrability P is givenin column 5. The barrier penetrability of the α particle ina deformed nucleus is different in different directions. Theaveraging of penetrability over different directions is doneusing the equation

P = 1

2

∫ π

0P (Q, θ, �) sin(θ ) dθ, (18)

where P (Q, θ, �) is the penetrability of α particles in thedirection θ from the symmetry axis for axially symmetricdeformed nuclei. The calculated α-decay half-lives usingCPPMDN and corresponding experimental values are givenin columns 6 and 7, respectively. For computing α-decay half-lives, we have used the experimental deformation-parametervalues taken from Ref. [43] and, for the cases in which there areonly theoretical ones, we have taken them from Ref. [44]. Asone can see, although the experimental α-decay half-lives varyin magnitude over a wide range (10−4 to 107), the calculatedα-decay half-lives agree very well with the experimental data.This not only proves the basic validity of our model for groundand isomeric states, but also gives some confidence for thepredictions given in Table V.

In Table II we have enclosed half-life values of unfavoredα decays. Columns are the same as in Table I, except that anadditional column for angular momentum � is introduced torepresent the minimum value of � transferred by an α particle.In unfavored α decay, the centrifugal term in Eq. (1) comes intoeffect and, as a result, the potential barrier increases. As one cansee from the table, most of the calculated T1/2 values coincidewith experimental values while some show large deviations(and these are the highly hindered ones). The deviations aredue to the fact that, for such decays, microscopic structure maybe different for both initial and final states and the transitionbecomes structurally hindered. From the table, the nuclei inisomeric states 183Pbm and 194Bin show a longer half-life valueeven though their Q values are greater than the correspondingground state to ground state values. This is because, in thesetransitions, the effect of angular momentum dominates thatof the Q value and transition becomes angular-momentumhindered. At a glance, deviation of calculated half-life valuesof favored and unfavored transitions from experimental valuesare given in Fig. 1. The large deviations corresponding tounfavored transitions and most of the favored transitions areof order 1.

Figure 2 gives Geiger-Nuttall (GN) plots of log10(T1/2)against Q−1/2 for favored transitions from various parentisotopes. It is found that all four types of transitions lieon the straight line. These GN plots are found to belinear with different slopes and intercepts and follow the

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SANTHOSH, JOSEPH, AND SAHADEVAN PHYSICAL REVIEW C 82, 064605 (2010)

TABLE I. Comparison of calculated α decay half-lives for ground and isomeric states of all favored transitions of nuclei in the range 67 �Z � 91 with experimental and Geiger-Nuttall-formalism values along with HF values.

Transition Iπi I π

f Q (MeV) P T cal1/2(S) T

expt1/2 (S) T GN

1/2 (S) HF

151Ho →147Tbm1 11/2− 11/2− 4.640 7.82 × 10−24 4.15 × 102 1.60 × 102 4.10 × 102 0.39151Hom1 → 147Tb 1/2+ 1/2+ 4.729 2.40 × 10−23 1.33 × 102 5.90 × 101 1.34 × 102 0.44152Ho →148Tb 2− 2− 4.505 1.43 × 10−24 2.35 × 103 1.35 × 103 2.38 × 103 0.57152Hom1 → 148Tbm1 9+ 9+ 4.574 3.54 × 10−24 9.32 × 102 4.58 × 102 9.60 × 102 0.49153Ho → 149Tbm1 11/2− 11/2− 4.015 1.22 × 10−27 3.07 × 106 2.35 × 105 2.95 × 106 0.08153Hom1 → 149Tb 1/2+ 1/2+ 4.119 6.16 × 10−27 5.94 × 105 3.10 × 105 5.85 × 105 0.52154Ho →150Tb 2− 2− 4.042 1.98 × 10−27 1.88 × 106 3.71 × 106 1.93 × 106 1.97154Hom1 → 150Tbm1 8+ 8+ 3.820 5.24 × 10−29 7.53 × 107 >1.86 × 107 7.29 × 107 0.25153Tm → 149Ho 11/2− 11/2− 5.240 7.70 × 10−22 3.74 1.63 3.92 0.44153Tmm1 → 149Hom1 1/2+ 1/2+ 5.233 7.12 × 10−22 4.05 2.69 4.24 0.66154Tm → 150Ho 2− 2− 5.088 1.28 × 10−22 2.32 × 101 1.50 × 101 2.21 × 101 0.65154Tmm1 → 150Hom1 9+ 9+ 5.165 3.06 × 10−22 9.53 5.69 9.12 0.60155Tm → 151Ho 11/2− 11/2− 4.568 2.14 × 10−25 1.55 × 104 2.43 × 103 1.56 × 104 0.16155Tmm1 → 151Hom1 1/2+ 1/2+ 4.568 2.14 × 10−25 1.55 × 104 >2.25 × 103 1.56 × 104 0.15155Lu → 151Tm 11/2− 11/2− 5.811 1.98 × 10−23 1.31 × 10−1 7.00 × 10−2 1.30 × 10−1 0.53155Lum1 → 151Tmm1 1/2+ 1/2+ 5.734 9.34 × 10−21 2.82 × 10−1 1.82 × 10−1 2.76 × 10−1 0.65156Lu → 152Tm 2− 2− 5.598 2.54 × 10−21 1.06 4.94 × 10−1 1.08 0.47156Lum1 → 152Tmm1 9+ 9+ 5.714 8.18 × 10−21 3.23 × 10−1 2.02 × 10−1 3.37 × 10−1 0.63157Lu → 153Tmm1 11/2− 11/2− 5.054 6.66 × 10−24 4.48 × 102 1.23 × 102 4.41 × 102 0.27157Lum1 → 153Tm 1/2+ 1/2+ 5.129 1.61 × 10−23 1.82 × 102 7.98 × 101 1.82 × 102 0.44156Hf → 152Yb 0+ 0+ 6.208 2.44 × 10−19 9.96 × 10−3 2.34 × 10−2 2.35157Ta → 153Lum1 1/2+ 1/2+ 6.277 2.12 × 10−19 1.13 × 10−2 1.05 × 10−2 1.25 × 10−2 0.93157Tam1 → 153Lu 11/2− 11/2− 6.379 5.23 × 10−19 4.52 × 10−3 4.30 × 10−3 5.12 × 10−3 0.95158Ta → 154Lu 2− 2− 6.123 3.78 × 10−20 6.52 × 10−2 7.20 × 10−2 5.01 × 10−2 1.10158Tam1 → 154Lum1 9+ 9+ 6.205 7.99 × 10−20 3.04 × 10−2 3.77 × 10−2 2.38 × 10−2 1.24159Ta → 155Lu 1/2+ 1/2+ 5.661 8.73 × 10−22 3.05 2.44 4.53 0.80159Tam1 → 155Lu 11/2− 11/2− 5.745 2.06 × 10−21 1.27 9.09 × 10−1 1.92 0.72160Ta → 156Lu 2− 2− 5.449 4.94 × 10−23 5.61 × 101 �1.7 4.32 × 101 0.03160Tam1 → 156Lum1 9+ 9+ 5.552 1.50 × 10−22 1.81 × 101 4.56 1.42 × 101 0.25158W → 154Hf 0+ 0+ 6.613 9.62 × 10−19 2.37 × 10−3 1.37 × 10−3 0.58161Rem1 → 157Tam1 11/2− 11/2− 6.430 1.54 × 10−19 1.53 × 10−2 1.64 × 10−2 1.46 × 10−2 1.07162Re → 158Ta 2− 2− 6.240 1.25 × 10−20 1.94 × 10−1 1.14 × 10−1 7.59 × 10−2 0.59162Rem1 → 158Tam1 9+ 9+ 6.271 9.56 × 10−20 2.51 × 10−2 9.00 × 10−2 5.77 × 10−2 3.59163Re → 159Ta 1/2+ 1/2+ 6.018 1.32 × 10−20 1.54 1.22 5.72 × 10−1 0.79163Rem1 → 159Tam1 11/2− 11/2− 6.069 2.21 × 10−20 1.12 × 10−1 3.24 × 10−1 3.56 × 10−1 2.89165Re → 161Ta 1/2+ 1/2+ 5.629 3.48 × 10−23 7.71 × 101 >5.22 × 101 2.62 × 101 0.68165Rem1 → 161Tam1 11/2− 11/2− 5.657 4.69 × 10−23 5.68 × 101 1.58 × 101 1.97 × 101 0.28165Irm1 → 161Rem1 11/2− 11/2− 6.882 1.47 × 10−18 5.90 × 10−3 2.31 × 10−3 5.68 × 10−3 0.39166Ir → 162Re 2− 2− 6.724 1.07 × 10−19 2.10 × 10−2 1.13 × 10−2 2.17 × 10−2 0.54166Irm1 → 162Rem1 9+ 9+ 6.723 1.06 × 10−19 2.12 × 10−2 1.54 × 10−2 2.19 × 10−2 0.73167Ir → 163Re 1/2+ 1/2+ 6.504 1.69 × 10−20 1.37 × 10−1 7.19 × 10−2 1.52 × 10−1 0.52167Irm1 → 163Rem1 11/2− 11/2− 6.561 2.80 × 10−20 8.20 × 10−2 3.19 × 10−2 9.10 × 10−2 0.39169Ir → 165Re 1/2+ 1/2+ 6.140 6.55 × 10−22 3.75 8.40 × 10−1 4.78 0.22169Irm1 → 165Rem1 11/2− 11/2− 6.267 2.21 × 10−21 1.09 4.75 × 10−1 1.39 0.44171Irm1 → 167Rem1 (11/2−) (11/2−) 6.067 3.64 × 10−22 6.83 2.60 9.94 0.38172Pt → 168Os 0+ 0+ 6.475 5.47 × 10−21 4.26 × 10−1 1.03 × 10−1 0.24170Au → 166Ir 2− 2− 7.174 5.57 × 10−19 3.77 × 10−3 2.00 × 10−3 3.52 × 10−3 0.53170Aum1 → 166Irm1 9+ 9+ 7.277 1.25 × 10−18 1.66 × 10−3 1.48 × 10−3 1.60 × 10−3 0.89171Aum1 → 167Irm1 11/2− 11/2− 7.164 5.52 × 10−19 3.82 × 10−3 1.89 × 10−3 3.80 × 10−3 0.49173Au → 169Ir 1/2+ 1/2+ 6.830 4.06 × 10−20 5.43 × 10−2 2.07 × 10−2 5.48 × 10−2 0.38173Aum1 → 169Irm1 11/2− 11/2− 6.891 6.81 × 10−20 3.22 × 10−2 1.30 × 10−2 3.32 × 10−2 0.40175Aum1 → 171Irm1 11/2− 11/2− 6.589 5.62 × 10−21 4.07 × 10−1 1.70 × 10−1 4.27 × 10−1 0.42177Au → 173Ir (1/2, 3/2)+ (3/2, 5/2)+ 6.303 4.47 × 10−22 5.35 3.66 5.66 0.68177Aum1 → 173Irm1 11/2− 11/2− 6.266 3.85 × 10−21 7.69 1.79 8.01 0.23176Hg → 172Pt 0+ 0+ 6.917 3.24 × 10−20 6.72 × 10−2 2.10 × 10−2 6.72 × 10−2 0.31

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TABLE I. (Continued.)

Transition Iπi I π

f Q (MeV) P T cal1/2(S) T

expt1/2 (S) T GN

1/2 (S) HF

177Hg → 173Pt (7/2)− 6.734 7.12 × 10−21 3.15 × 10−1 1.18 × 10−1 3.14 × 10−1 0.37177Tl → 173Au 1/2+ 1/2+ 7.067 3.86 × 10−20 5.53 × 10−2 2.47 × 10−2 4.89 × 10−2 0.45177Tlm1 → 173Aum1 11/2− 11/2− 7.660 3.96 × 10−18 4.97 × 10−4 4.69 × 10−4 4.84 × 10−4 0.94179Tl → 175Au 1/2+ 1/2+ 6.719 2.19 × 10−21 1.03 2.30 × 10−1 9.70 × 10−1 0.22179Tlm1 → 175Aum1 11/2− 11/2− 7.378 5.34 × 10−19 3.83 × 10−3 2.25 × 10−3 4.05 × 10−3 0.59181Tl → 177Au 1/2+ (1/2,3/2)+ 6.321 5.95 × 10−23 4.01 × 101 >3.20 × 101 3.99 × 101 0.80181Tlm1 → 177Aum1 9/2− 9/2− 6.727 2.66 × 10−21 8.44 × 10−1 3.65 × 10−1 9.04 × 10−1 0.43183Tlm1 → 179Aum1 9/2− 9/2− 6.475 2.95 × 10−22 7.9 5.13 9.09 0.65185Tlm1 → 181Aum1 9/2− 9/2− 6.108 6.76 × 10−24 3.65 × 102 ≈1.18 × 102 3.38 × 102 0.32179Pb → 175Hgm1 (9/2−) (9/2−) 7.518 5.23 × 10−19 3.84 × 10−3 3.50 × 10−3 3.68 × 10−3 0.91180Pb → 176Hg 0+ 0+ 7.419 2.59 × 10−19 7.84 × 10−3 4.00 × 10−3 7.72 × 10−3 0.51181Pb → 177Hgm1 (9/2−) (9/2−) 7.175 3.91 × 10−20 5.38 × 10−2 3.60 × 10−2 5.24 × 10−2 0.67183Pb → 179Hgm1 (3/2−) (3/2−) 6.711 8.04 × 10−22 2.80 2.23 2.66 0.77183Pb → 179Hgm1 (13/2−) (13/2−) 6.854 2.89 × 10−21 7.63 × 10−1 4.23 × 10−2 7.63 × 10−1 0.06185Pb → 181Hgm1 3/2− 3/2− 6.631 4.35 × 10−22 4.23 42.00 5.60 9.93185Pb → 181Hgm2 3/2− 3/2− 6.426 6.25 × 10−23 37.56 33.15 3.65 × 101 0.88185Pbm1 → 181Hgm3 13/2+ 13/2+ 6.550 2.04 × 10−22 1.13 × 101 8.14 1.14 × 101 0.72187Pb → 183Hgm1 (3/2−) (3/2−) 6.328 2.68 × 10−23 8.89 × 101 3.62 × 102 9.33 × 101 4.07187Pb → 183Hgm2 (3/2−) (3/2−) 6.120 3.24 × 10−24 7.61 × 102 5.43 × 102 7.39 × 102 0.71187Pbm1 → 183Hgm3 (13/2+) (13/2+) 6.210 8.19 × 10−24 2.97 × 102 1.53 × 102 2.98 × 102 0.52187Bi → 183Tlm1 9/2− 9/2− 7.147 1.44 × 10−20 1.46 × 10−1 4.02 × 10−2 1.33 × 10−1 0.28187Bim1 → 183Tl 1/2+ 1/2+ 7.883 4.41 × 10−18 4.34 × 10−4 3.10 × 10−4 4.48 × 10−4 0.71188Bim1 → 184Tlm1 3+ 3+ 7.144 1.49 × 10−20 1.42 × 10−1 6.00 × 10−2 1.36 × 10−1 0.42188Bim2 → 184Tlm2 10− 10− 6.961 3.12 × 10−21 6.94 × 10−1 2.91 × 10−1 6.46 × 10−1 0.42189Bi → 185Tlm1 9/2− 9/2− 6.811 8.77 × 10−22 2.53 6.17 × 10−1 2.42 0.24189Bim1 → 185Tl 1/2+ 1/2+ 7.453 1.93 × 10−19 1.05 × 10−2 5.54 × 10−3 1.13 × 10−2 0.53190Bim1 → 186Tlm1 3+ 3+ 6.569 9.91 × 10−23 2.32 × 101 7.33 2.24 × 101 0.32190Bim2 → 186Tlm2 10− 10− 6.595 1.27 × 10−22 1.81 × 101 9.25 1.76 × 101 0.51191Bi → 187Tlm1 9/2− 9/2− 6.443 3.11 × 10−23 7.53 × 101 2.43 × 101 7.51 × 101 0.32191Bim1 → 187Tl 1/2+ 1/2+ 7.017 5.98 × 10−21 3.60 × 10−1 1.78 × 10−1 3.99 × 10−1 0.49192Bi → 188Tlm1 3+ 3+ 6.189 2.54 × 10−24 9.58 × 102 2.88 × 102 9.60 × 102 0.30192Bim1 → 188Tlm2 10− 10− 6.181 2.34 × 10−24 1.04 × 103 3.96 × 102 1.04 × 103 0.38193Bi → 189Tlm1 9/2− 9/2− 6.021 4.53 × 10−25 5.53 × 103 2.00 × 103 5.65 × 103 0.36193Bim1 → 189Tl 1/2+ 1/2+ 6.613 1.76 × 10−22 1.30 × 101 3.81 1.48 × 101 0.29194Bi → 190Tlm1 3+ 3+ 5.764 2.71 × 10−26 9.67 × 104 2.07 × 104 9.86 × 104 0.21194Bim1 → 190Tlm2 10− 10− 5.717 1.57 × 10−26 1.68 × 105 5.75 × 104 1.70 × 105 0.34195Bi → 191Tlm1 9/2− 9/2− 5.534 1.85 × 10−27 1.47 × 106 6.10 × 105 1.51 × 106 0.41195Bim1 → 191Tl 1/2+ 1/2+ 6.234 4.71 × 10−24 5.14 × 102 2.64 × 102 6.04 × 102 0.51196Bi → 192Tlm1 3+ 3+ 5.260 5.96 × 10−29 4.81 × 107 2.68 × 107 4.89 × 107 0.56196Bi → 192Tlm2 10− 10− 5.219 3.46 × 10−29 8.36 × 107 8.00 × 107 8.43 × 107 0.96187Po → 183Pbm1 (1/2,5/2)− (1/2,5/2)− 7.693 1.62 × 10−18 1.21 × 10−3 1.40 × 10−3 2.84 × 10−3 1.16188Po → 184Pb 0+ 0+ 8.082 6.81 × 10−18 2.74 × 10−4 2.70 × 10−4 1.66 × 10−4 0.99192Po → 188Pb 0+ 0+ 7.319 2.61 × 10−20 7.90 × 10−2 3.23 × 10−2 5.52 × 10−2 0.41193Po → 189Pb 3/2− 3/2− 7.095 4.22 × 10−21 5.04 × 10−1 4.20 × 10−1 3.69 × 10−1 0.83193Pom1 → 189Pbm1 13/2+ 13/2+ 7.150 6.75 × 10−21 1.27 × 10−1 2.40 × 10−1 2.30 × 10−1 1.89195Po → 191Pb 3/2− 3/2− 6.747 2.24 × 10−22 9.97 6.19 8.35 0.62195Pom1 → 191Pbm1 13/2+ 13/2+ 6.839 5.18 × 10−22 4.26 2.13 3.58 0.50197Po → 193Pb 3/2− 3/2− 6.412 1.00 × 10−23 2.34 × 102 1.27 × 102 2.13 × 102 0.54197Pom1 → 193Pbm1 13/2+ 13/2+ 6.515 2.76 × 10−23 8.40 × 101 3.10 × 101 7.65 × 101 0.37199Po → 195Pb 3/2− 3/2− 6.074 3.79 × 10−25 6.56 × 103 2.74 × 103 7.29 × 103 0.42199Pom1 → 195Pbm1 13/2+ 13/2+ 6.183 1.21 × 10−24 2.02 × 103 6.35 × 102 2.26 × 103 0.31201Po → 197Pb 3/2− 3/2− 5.798 1.58 × 10−26 1.65 × 105 5.74 × 104 1.64 × 105 0.35201Pom1 → 197Pbm1 13/2+ 13/2+ 5.903 5.23 × 10−26 4.88 × 104 1.84 × 104 4.89 × 104 0.38191At → 187Bim1 1/2+ 1/2+ 7.714 1.89 × 10−19 1.03 × 10−2 1.70 × 10−3 1.03 × 10−2 0.17191Atm1 → 187Bim2 7/2− 7/2− 7.817 4.13 × 10−19 4.67 × 10−3 2.14 × 10−3 4.82 × 10−3 0.46192Atm1 → 188Bim1 7.593 7.85 × 10−20 2.53 × 10−2 2.05 × 10−2 2.55 × 10−2 0.81

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TABLE I. (Continued.)

Transition Iπi I π

f Q (MeV) P T cal1/2(S) T

expt1/2 (S) T GN

1/2 (S) HF

192Atm2 → 188Bim2 (9,10)− (9,10)− 7.378 1.40 × 10−20 1.46 × 10−1 1.07 × 10−1 1.35 × 10−1 0.73193At → 189Bim1 1/2+ 1/2+ 7.388 1.61 × 10−20 1.27 × 10−1 2.80 × 10−2 1.25 × 10−1 0.22193Atm1 → 189Bim2 7/2− 7/2− 7.480 3.39 × 10−20 5.94 × 10−2 2.14 × 10−2 6.06 × 10−2 0.36193Atm1 → 189Bim3 13/2+ 13/2+ 7.256 5.41 × 10−21 3.84 × 10−1 1.13 × 10−1 3.59 × 10−1 0.29194Atm1 → 190Bim1 7.341 1.16 × 10−20 1.77 × 10−1 3.05 × 10−1 1.81 × 10−1 1.72194Atm2 → 190Bim2 (9,10)− 10− 7.329 1.05 × 10−20 1.96 × 10−1 3.97 × 10−1 1.99 × 10−1 2.03195At → 191Bim1 1/2+ 1/2+ 7.099 1.58 × 10−21 1.34 3.28 × 10−1 1.31 0.24195Atm1 → 191Bim2 7/2− 7/2− 7.223 4.57 × 10−21 4.57 × 10−1 1.54 × 10−1 4.70 × 10−1 0.34197At → 193Bi 9/2− 9/2− 7.101 1.79 × 10−21 1.19 3.50 × 10−1 1.29 0.29197Atm1 → 193Bim1 1/2+ 1/2+ 6.846 1.85 × 10−22 1.20 × 101 3.7 1.16 × 101 0.31198At → 194Bi 3+ 3+ 6.894 3.02 × 10−22 7.26 4.67 7.62 0.64198Atm1 → 194Bim1 10− 10− 6.997 7.58 × 10−22 2.84 1.19 3.12 0.42200At → 196Bi 3+ 3+ 6.599 2.11 × 10−23 1.08 × 102 8.27 × 101 1.10 × 102 0.77200Atm1 → 196Bim1 7+ 7+ 6.544 1.24 × 10−23 1.86 × 102 1.09 × 102 1.85 × 102 0.59200Atm2 → 196Bim2 10− 10− 6.677 4.46 × 10−23 5.07 × 101 6.95 × 101 5.35 × 101 1.37202At → 198Bi (2,3)+ (2,3)+ 6.354 2.04 × 10−24 1.16 × 103 4.97 × 102 1.17 × 103 0.43202Atm1 → 198Bim1 7+ 7+ 6.259 7.65 × 10−25 3.15 × 103 2.09 × 103 3.03 × 103 0.66202Atm2 → 198Bim2 10− 10− 6.353 2.02 × 10−24 1.18 × 103 4.79 × 102 1.18 × 103 0.41195Rn → 191Po 3/2− 3/2− 7.694 6.97 × 10−20 2.81 × 10−2 6.00 × 10−3 2.02 × 10−2 0.21195Rnm1 → 191Pom1 13/2+ 13/2+ 7.713 8.08 × 10−20 2.42 × 10−2 5.00 × 10−3 1.75 × 10−2 0.21197Rn → 193Po 3/2− 3/2− 7.410 8.03 × 10−21 2.54 × 10−1 6.50 × 10−2 1.84 × 10−1 0.26197Rnm1 → 193Pom1 13/2+ 13/2+ 7.508 1.78 × 10−20 1.13 × 10−1 1.90 × 10−2 8.46 × 10−2 0.17199Rn → 195Po 3/2− 3/2− 7.132 3.99 × 10−21 5.30 × 10−1 6.60 × 10−1 1.81 1.25199Rnm1 → 195Pom1 13/2+ 13/2+ 7.204 6.94 × 10−21 3.02 × 10−1 3.30 × 10−1 9.90 × 10−1 1.09201Rn → 197Po 3/2− 3/2− 6.862 8.37 × 10−23 2.63 × 101 1.37 × 101 1.9 × 101 0.52201Rnm1 → 197Pom1 13/2+ 13/2+ 6.911 1.31 × 10−22 1.66 × 101 1.12 × 101 1.23 × 101 0.67203Rn → 199Po 3/2− 3/2− 6.630 1.03 × 10−23 2.21 × 102 6.67 × 101 1.62 × 102 0.30203Rnm1 → 199Pom1 13/2+ 13/2+ 6.682 1.70 × 10−23 1.33 × 102 3.59 × 101 9.93 × 101 0.27201Fr → 197At 9/2− 9/2− 7.519 8.15 × 10−21 2.46 × 10−1 6.20 × 10−2 1.59 × 10−1 0.25201Frm1 → 197Atm1 1/2+ 1/2+ 7.605 1.63 × 10−20 1.22 × 10−1 1.90 × 10−2 9.23 × 10−2 0.16202Fr → 198At 3+ 3+ 7.387 2.89 × 10−21 7.07 × 10−1 3.00 × 10−1 3.72 × 10−1 0.42202Frm1 → 198Atm1 10− 10− 7.381 2.75 × 10−21 7.44 × 10−1 2.90 × 10−1 3.87 × 10−1 0.39203Fr → 199At 9/2− 9/2− 7.276 6.38 × 10−21 3.25 × 10−1 5.79 × 10−1 7.76 × 10−1 1.78204Fr → 200At 3+ 3+ 7.172 2.67 × 10−21 0.79 1.77 1.57 2.24204Frm1 → 200Atm1 7+ 7+ 7.108 1.52 × 10−21 1.4 2.89 2.43 2.06204Frm2 → 200Atm2 10− 10− 7.153 2.26 × 10−21 0.93 2.3 1.78 2.47206Fr → 202At (2,3)+ (2,3)+ 6.926 1.75 × 10−22 1.25 × 101 1.90 × 101 8.80 1.52206Frm1 → 202Atm1 7+ 7+ 6.926 1.75 × 10−22 1.25 × 101 1.90 × 101 8.80 1.52206Frm2 → 202Atm2 10− 10− 7.067 2.21 × 10−22 9.64 1.40 × 101 3.24 1.45201Ram1 → 197Rnm1 13/2+ 13/2+ 8.066 1.98 × 10−19 9.43 × 10−3 1.60 × 10−3 9.49 × 10−3 0.17203Ra → 199Rn 3/2− 3/2− 7.742 1.85 × 10−20 1.05 × 10−1 3.10 × 10−2 1.03 × 10−1 0.30203Ram1 → 199Rnm1 13/2+ 13/2+ 7.765 2.22 × 10−20 8.77 × 10−2 2.40 × 10−2 8.66 × 10−2 0.27205Ra → 201Rn 3/2− 3/2− 7.486 2.58 × 10−21 7.80 × 10−1 � 2.10 × 10−1 7.56 × 10−1 0.27205Ram1 → 201Rnm1 13/2+ 13/2+ 7.517 3.34 × 10−21 6.01 × 10−1 � 1.70 × 10−1 5.91 × 10−1 0.28207Ra → 203Rn (5/2,3/2)− (5/2,3/2)− 7.325 7.34 × 10−22 2.81 1.44 2.79 0.51207Ram1 → 203Rnm1 13/2+ 13/2+ 7.525 3.93 × 10−21 5.10 × 10−1 3.67 × 10−1 5.55 × 10−1 0.72206Ac → 202Fr 3+ 3+ 7.944 3.62 × 10−20 5.24 × 10−2 2.20 × 10−2 5.03 × 10−2 0.42206Acm1 → 202Frm1 10− 10− 7.903 2.64 × 10−20 7.23 × 10−2 3.30 × 10−2 6.77 × 10−2 0.46208Ac → 204Fr 3+ 3+ 7.720 7.42 × 10−21 2.63 × 10−1 9.60 × 10−2 2.62 × 10−1 0.37208Acm1 → 204Frm1 10− 10− 7.910 3.29 × 10−20 5.80 × 10−2 2.78 × 10−2 6.43 × 10−2 0.48217Pa → 213Ac 9/2− 9/2− 8.489 4.52 × 10−19 3.93 × 10−3 3.48 × 10−3 0.89

Geiger-Nuttall law

log10(T1/2) = A√Q

+ B. (19)

The coefficients A and B are shown in Table III. Withthese values of coefficients, we have calculated half-livesfor all transitions using Eq. (19), and they are shown in

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α DECAY OF NUCLEI IN THE RANGE . . . PHYSICAL REVIEW C 82, 064605 (2010)

TABLE II. Comparison of calculated α decay half-lives for ground and isomeric states of all unfavored transitions of nuclei in the range 71� Z � 91 with experimental values along with HF values.

Transition Iπi I π

f � Q (MeV) P T cal1/2(S) T

expt1/2 (S) HF

155Lum2 →151Tm 25/2− 11/2− 8 7.584 8.19 × 10−17 2.43 × 10−5 2.70 × 10−3 111.11156Hfm1 →152Yb 8+ 0+ 8 7.987 3.88 × 10−16 4.86 × 10−6 4.80 × 10−4 98.77158Wm1 →154Hf 8+ 0+ 8 8.502 1.42 × 10−15 1.25 × 10−6 1.43 × 10−4 114.40174Irm1 →170Re 7+ 5+ 2 5.817 2.12 × 10−23 1.22 × 102 1.96 × 102 1.61174Ir →170Re 3+ 5+ 2 5.624 2.64 × 10−24 1.02 × 103 1.58 × 103 1.55183Pb →179Hg (3/2−) (7/2−) 2 6.928 3.47 × 10−21 6.27 × 10−1 8.11 × 10−1 1.29183Pbm1 →179Hg (13/2−) (7/2−) 4 7.025 2.76 × 10−21 7.78 × 10−1 1.66 × 101 21.34185Pb →181Hg 3/2− 1/2− 2 6.695 4.96 × 10−22 4.55 3.15 × 103 692.31187Pb →183Hg 3/2− 1/2− 2 6.395 3.30 × 10−23 7.15 × 101 2.17 × 102 3.03186Bi →182Tl 3+ 2− 1 7.757 1.42 × 10−18 1.37 × 10−3 1.48 × 10−2 10.80191Bi →187Tl 9/2− 1/2+ 5 6.778 8.28 × 10−23 2.69 × 101 2.05 × 101 0.76193Bi →189Tl 9/2− 1/2+ 5 6.305 1.04 × 10−24 2.30 × 103 1.91 × 103 0.83194Bim2 →190 Tlm1 10− 7+ 3 6.016 1.89 × 10−25 5.10 × 104 5.76 × 104 1.13194Bi →190Tl 3+ 2− 1 5.918 1.33 × 10−25 1.91 × 104 2.05 × 104 1.07195Bi →191Tl 9/2− 1/2+ 5 5.833 7.71 × 10−27 3.36 × 105 5.72 × 105 1.70216Acm1 →212Fr 9− 5+ 5 9.279 6.43 × 10−17 2.53 × 10−5 4.43 × 10−4 17.51216Ac →212Fr 1− 5+ 5 9.235 4.93 × 10−17 3.32 × 10−5 4.40 × 10−4 13.25217Pam1 → 213Ac 29/2+ 9/2− 11 10.349 1.23 × 10−17 1.19 × 10−4 1.48 × 10−3 12.44

column 8 of Table I. From this it is clear that, by usingthe Geiger-Nuttall law, we can reproduce the half-life valuesaccurately. From this, we can also conclude that deformationsof nuclei in isomeric states remain the same as that of groundstates, and T1/2 values mainly depend on Q values. The lastcolumn of Table I gives the hindrance factor (HF). The HFfor the transitions to the different states is simply the ratiobetween calculated (theoretical) and measured (experimental)decay constants or the ratio between experimental half-life andtheoretical half-life and is given by

HF = λcal

λexpt= T

expt1/2

T cal1/2

. (20)

0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52-4

-2

0

2

4

6

8

Pb

Ra

Rn

At

Po

Tl

Au Ir

ReTa

Lu

Tm

Ho

log

10 (

T1/

2 )

Q-1/2

FIG. 2. Geiger-Nuttall plot for favored α-decay transitions be-tween ground and isomeric states of various isotopes.

The HF values obtained are close to unity, so we can say thatmost of the transitions are unhindered. If we take into accountlarger possible angular momentum values instead of zero, T1/2

will increase as angular momentum increases. The variationsof log10(T1/2) with respect to different possible angular mo-mentum values for various transitions are elaborately shownin Fig. 3. It is clear that, as angular momentum increases, thecentrifugal barrier plays a prominent role and the decay willbecome highly hindered. So, naturally, α decay occurs withthe minimum possible value of angular momentum transfer.

TABLE III. Coefficients A and B of GN law for various isotopes.The # symbol denotes unfavored α transitions.

Sl. No. Parent A (slope) B (intercept)

1 Ho 110.74588 −48.799862 Tm 115.95634 −50.062003 Lu 117.72772 −49.723234 Ta 120.96293 −50.184345 Ir 130.71328 −52.072126 Au 128.46967 −50.418417 Hg 128.88462 −50.174538 Tl 134.94359 −52.072599 Pb 134.17575 −51.36844

10 Pb# 125.72231 −47.8895511 Bi 138.22841 −52.5811312 Bi# 145.41519 −54.8027013 Po 141.63467 −53.6058914 At 137.91453 −51.6441715 Rn 140.16525 −52.2264616 Ra 142.03689 −52.0343117 Re 119.94899 −49.1379318 Fr 114.01640 −42.3793119 Ac 140.30503 −51.07829

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SANTHOSH, JOSEPH, AND SAHADEVAN PHYSICAL REVIEW C 82, 064605 (2010)

0 2 4 6 8 10-101234567 (a)153Tm

155Tm155Lu

log

10 (

T 1/

2 )

0 2 4 6 8-2

-1

0

1

2

3

4 (b)187Bi 189Bi 191Bi

0 2 4 6 8 10-3

-2

-1

0

1

2

3

4

Angular Momentum

(c)157Ta 159Ta

0 2 4 6 8 10 12-1

0

1

2

3

4 (d)193Po206Fr160Ta

FIG. 3. Variation of half-life values as a function of possibleangular momentum of transferred α particle for various types oftransitions, (a) for ground state to ground state transitions, (b) forground state to isomeric state transitions, (c) isomeric state to groundstate transitions, and (d) for isomeric state to isomeric state transitions.

Before concluding the discussion on favored α decay, inorder to measure quality of agreement between calculated andexperimental data, the standard deviation of the calculatedvalues over the experimental data has been evaluated using theequation

σ ={

1

(n − 1)

n∑i=1

[log

(T cal

i

T expt

)]2}1/2

. (21)

In Table IV, the first column represents different modesof transition, the third and fourth columns give standarddeviations of T1/2 calculated by the CPPMDN and the Geiger-Nuttall formalism, respectively. From the table, as one cansee, the standard deviation values for ground state to groundstate and exited state to exited state are very close to thestandard deviation of all transitions. The two remaining typesof transitions show small deviations, but we cannot make anygeneral conclusion on that because, in comparison, they arefewer in number.

In addition to all this, we have performed a study onunfavored α decay by using Geiger-Nuttall plots, as done forfavored decay, and the corresponding slopes and intercepts are

TABLE IV. Standard deviation of calculated half-lives usingCPPMDN and Geiger-Nuttall formalism with the experimental datafor favored transitions.

Types of Number of Standard deviationtransitions transitions CPPMDN Geiger-Nuttall

g.s.→ g.s. 55 0.43 0.42g.s.→ i.s. 23 0.55 0.54i.s. → g.r. 10 0.32 0.36i.s.→ i.s. 74 0.43 0.41All transitions 162 0.44 0.43

0.34 0.36 0.38 0.40 0.42 0.44

-4

-2

0

2

4

6

8

194Bi ---> α + 190Tl

Bi

Favored cal.Favored exp.Unfavored cal.Unfavored exp.

log

10 (

T1/

2)

Q-1/2

FIG. 4. Geiger-Nuttall plots of favored and unfavored transitionsof Bi isotopes.

given in Table III. The GN plot for favored and unfavoredα emission for various Bi isotopes are given in Fig. 4. It isclear from the figure that, for both favored and unfavoreddecay, points occur on two straight lines satisfying Eq. (19),and, for most of the transitions, the calculated values are inclose agreement with the experimental data. In the case of186Bi, the experimental half-life lies on the GN curve for theunfavored one but the computed value (T1/2 = 1.37 × 10−3s,� = 1) lies on the GN plot for the favored one. When wecalculate T1/2 with � = 5, the computed value coincides withthe corresponding experimental data. Thus, in our study, it isclear that the angular momentum carried by the α particleis 5 instead of 1. In the case of the unfavored transitionfrom 194Bi, theoretical and experimental values are in closeagreement and both lie on GN plot for favored transitions.This observation implies that α decay from 194Bi is favoredinstead of unfavored. We would like to draw the attention ofboth experimentalists and theoreticians regarding the decayof 186Bi and 194Bi isotopes. We would like to mention thatDelion et al. [45], in their letter, first pointed out the lineardependence of the logarithm of reduced half-life log10 Tred onthe Coulomb parameter χ in the case of proton emitters similarto the Geiger Nuttall plots given in Fig. 4. It is interesting tosee that the data points for prolate and oblate proton emitterslie on two straight lines with different slopes and intercepts(see Fig. 1(b) of Ref. [45]). The authors have shown that animproper choice of � values will spoil the straight-line pattern,and this enables the precise assignment of spin and parity forproton decaying states.

On the basis of the fact that CPPMDN formalism couldreproduce successfully the experimental values for favoredand unfavored transitions, we have extended our study topredict half-lives of a few α-decay transitions from ground andisomeric states. Details of predicted half-lives are tabulated inTable V. The transitions and corresponding Q values are takenfrom Refs. [12,13]. Calculated T1/2 using the GLDM [12]and GDDCM [13] formalisms are given in columns 8 and 9,respectively. In comparison, one can see that our calculated

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α DECAY OF NUCLEI IN THE RANGE . . . PHYSICAL REVIEW C 82, 064605 (2010)

TABLE V. Comparison of predicted half-life values for favored and unfavored transitions with that of GLDM and GDDCM formalism.

Transition Iπi I π

f � Q (MeV) P T cal1/2(S) T cal

1/2,GLDM T cal1/2,GDDCM

161Tam1 → 157Lum1 11/2− 11/2− 0 5.353 1.81 × 10−23 1.56 × 102 2.12 × 102

161Re → 157Ta 1/2+ 1/2+ 0 6.328 6.20 × 10−20 3.84 × 10−2 3.44 × 10−2

164Rem1 → 160Ta 2− 2− 0 5.930 7.33 × 10−22 3.47 4.88165Ir → 161Re 1/2+ 1/2+ 0 6.830 9.57 × 10−19 2.31 × 10−3 3.98 × 10−3

171Au → 167Ir 1/2+ 1/2+ 0 7.089 3.03 × 10−19 7.02 × 10−3 3.34 × 10−3

175Au → 171Ir 1/2+ 1/2+ 0 6.562 4.41 × 10−21 5.22 × 10−1 2.09 × 10−1

176Aum1 → 171Irm1 7+ 7+ 0 6.433 1.44 × 10−21 1.63 2.04187Hgm1 → 183Pt 13/2+ 1/2− 7 5.289 6.73 × 10−29 4.24 × 107 7.69 × 107

212Atm1 → 208Bi 9− 5+ 5 8.050 7.41 × 10−19 2.53 × 10−3 5.16 × 10−3

214Atm2 → 210Bi 9− 1+ 9 9.221 3.32 × 10−17 4.93 × 10−5 3.50 × 10−4

203Frm1 → 199Atm1 1/2+ 1/2+ 0 7.372 1.44 × 10−20 1.42 × 10−1 3.79 × 10−1

216Thm1 → 212Ra 8+ 0+ 8 10.113 2.05 × 10−16 7.29 × 10−6 2.96 × 10−7

245Mdm1 → 241Esm3 7/2+ 7/2+ 0 8.824 9.69 × 10−22 1.76 4.90 × 10−1

257Dbm1 → 253Lrm1 1/2− 1/2− 0 9.308 1.20 × 10−21 1.35 3.00 × 10−1

263Sgm1 → 259Rfm3 3/2+ 3/2+ 0 9.391 1.08 × 10−21 1.48 3.00 × 10−1

values support the predicted T1/2 vales and, we hope, providesnew insights to experimentalists.

IV. SUMMARY

In conclusion, we present in this article a new formalismCPPMDN by incorporating deformations (β2 and β4) into theearlier CPPM formalism. We have performed α-decay half-lifecalculations for favored and unfavored transitions for nucleiin the range 67 � Z � 91. From the comparison with theexperimental values one can see that our formalism could

very well reproduce the experimental values. By plotting theGeiger-Nuttall law for various isotopes, we could concludethat isomeric-state α decay shows a behavior similar to thatof ground states and the nuclear structure of isomeric statesimitates that of ground states. In addition, we have done aninvestigation on favored and unfavored α decays of Bi isotopes.After ensuring the validity of CPPMDN on a wide range ofnuclei for favored and unfavored decays, we have made somepredictions which will serve to guide future experiments. Itis evident from the study that our ground-state decay modelis apt for describing α-decay transitions of both ground andisomeric states.

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