1

Nuclear Properties of Cobalt Isotopic Chain

Balsam Nader Ata

Faculty of Graduate Studies, An najah national university, Nablus, Palestine

Abstract:

In this paper, ten nuclear properties of cobalt isotopic chain were discussed: binding energy, nuclear

angular momentum, parity, nuclear electromagnetic moments, decay modes, half life time, Q value, one

and two neutron separation energy and proton separation energy. For each property some category were

discussed: the definition, the experimental and theoretical data if it available and Supportive figures with

explanations. HFODD code which is a programming code written in Fortran language used to get a

numerical calculations of the nuclear properties .

1. INTRODUCTION

Cobalt is an atom of symbol Co and atomic

number 27 which classified from transition

metals, cobalt nucleus has 29 isotopes which

have the same number of protons and different

number of neutrons.

Cobalt isotopes mass number range from A=47

up to A=75 which means from 47

Co – 75

Co, all

of isotopic chain of cobalt are radioactive except 59

Co stable which has 100% abundance.

Cobalt isotopes decay in different allowed

decay modes with different half life time and Q

value for each decay to be stable such as

electron capture, negative beta decay β- , β

- with

neutron emission.

The chain of isotopes behave different behavior

from one property to another, the binding

energy values increase with increasing of the

number of nucleons A up to reach the stable one

then decrease, the separation energy of one

neutron make a series of increasing and

decreasing, the two neutron separation energy

decrease smoothly with increasing A, the proton

separation energy increasing linearly with A.

The experimental data of nuclear angular

momentum, parity and electromagnetic

moments were obtained.

2

FIG.1. Nuclear chart [2].

2.THEORY

FIG.2. The binding energy per

nucleuon [2]. The curve reaches a peak

near A = 60, where the nuclei are most

tightly bound [2].

Binding energy:

The binding energy, EB(Z, N), is the

energy required to remove all Z

protons and N neutrons from the

nucleus and is given by the mass

difference between the nucleus and the

sum of its components [1]

EB(Z,N) = {ZMH+ NMn - M(Z,N)}c2 , (1)

The average binding energy of most

nuclei is about 8 Mev per nucleon [2].

3

FIG.3. Binding energy per nucleon for

cobalt isotopes.

The figure shows increasing in binding

energy values with increasing "A" up

to A=59 which 59

Co the stable isotope

then the values decrease after this

point up to the end of the chain . This

is depends on the stability and half life

time , when the half life time increase

the nucleus become more sable so it

requires more energy to split it which

means high binding energy.

Nuclear angular momentum and

parity J∏:

Nucleons have spin quantum number

of S=1/2 and moving in a central

potential with orbital angular

momentum L, the total angular

momentum

J=L+S [2].

FIG.4. the coupling of orbital angular

Table.1: Values of binding energies per

nucleon for the isotopic chain of cobalt

in Mev [3].

The binding energy per nucleon was

plotted with the number of nucleon

"A" in FIG.3,

7.9

8

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

45 55 65 75 85

B/A(Mev)

A

59Co

Isotope B/A(Mev) Isotope B/A(Mev)

50Co 8.004

65Co 8.65688

51Co 8.1933

66Co 8.60594

52Co 8.319

67Co 8.58174

53Co 8.47764

68Co 8.5243

54Co 8.569205

69Co 8.492

55Co 8.669606

70Co 8.440

56Co 8.694825

71Co 8.399

57Co 8.741871

72Co 8.330

58Co 8.738959

73Co 8.287

59Co 8.768025

74Co 8.225

60Co 8.746757

75Co 8.178

61Co 8.756141

62Co 8.7213

63Co 8.7178

64Co 8675.5

4

Table.3: Sn values of cobalt isotopes

(Mev) [3].

50Co 1.55E+01

51Co 1.76E+01

52Co 1.472E+01

53Co 1.674E+01

54Co 1.3422 E+01

55Co 1.40913 E+1

56Co 1.00819 E+1

57Co 1.13765 E+1

58Co 8.573

59Co 1.045 E+1

60Co 7.49192

61Co 9.3192

62Co 6.598

63Co 8.50

64Co 6.01

65Co 7.465

66Co 5.295

67Co 6.985

68Co 4.67

69Co 6.32

70Co 4.8

71Co 5.50

72Co 3.50

73Co 5.20

74Co 3.60

75Co 4.70

momentum L to spin angular

momentum S giving toal angular

momentum J.

Parity is the behavior of the wave

function of the state under a reflection

of coordinate system through the

origin, which equal ∏ = (-1) L

[1].

This table shows the known values of

J∏

of the cobalt isotopes, these values

took from appendix C of introductory

nuclear physics of Krane and from

national nuclear data center NNDC.

Table.2: J∏

of the cobalt isotopes [2,3].

Neutron separation energy Sn:

Is the amount of energy needed to

remove a neutron from a nucleus z AXN

, which is the difference in binding

energies between z AXN and z

A-1XN-1

[2]:

Sn = B (z AXN ) – B(z

A-1XN-1 )

= [m (z A-1

XN-1)-m (z AXN ) + mn] c

2 , (2)

50Co +6

59Co -7/2

68Co -7

51Co -7/2

60Co +5

69Co -7/2

52Co +6

61Co -7/2

70Co -6

53Co -7/2

62Co +2

71Co -7/2

54Co +0

63Co -7/2

72Co -6 , -7

55Co -7/2

64Co +1

56Co +4

65Co -7/2

74Co +0

57Co -7/2

66Co +3

75Co -7/2

58Co +2

67Co -7/2

5

Table.4: Sp values of cobalt isotopes

(Mev) [3].

50Co 1.40E-1

51Co 1.08

52Co 1.615

53Co 4.3516

54Co 5.0644

55Co 5.848

56Co 6.027

57Co 6.954

58Co 7.363

59Co 8.274

60Co 8.774

61Co 9.792

62Co 10.262

63Co 11.445

64Co 11.505

65Co 12.476

66Co 12.543

67Co 1.31E+1

68Co 1.36E+1

69Co 1.52E+1

70Co 1.53E+1

71Co 1.61E+1

72Co 1.61E+1

73Co 1.71E+1

74Co 1.72E+1

FIG.5. Neutron separation energy with

neutron number.

In FIG.5 the neutron separation energy

values were plotted with number of

neutrons of cobalt isotopes, the figure

shows a jump at N=28 which is a

magic number, which means a closed

neutrons shell [4].

The same figure shows also a series of

increasing and decreasing in Sn with

increasing of the number of neutron,

from the figure the increasing occurs at

even N values and the decreasing at

odd values which means that we need

more energy to separate a neutron

when N even than odd.

Proton separation energy Sp:

Is the amount of energy needed to

remove a proton from a nucleus which

equal [2]:

Sp = B (z AXN ) – B(z -1

A-1XN )

= [m (z -1A-1

XN) -m (z AXN ) + mp] c

2, (3)

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

20 30 40 50

Sn (Mev)

N

6

57Co 2.1458E+1

58Co 1.9949E+1

59Co 1.9026E+1

60Co 1.7945 E+1

61Co 1.6811E+1

62Co 1.5917E+1

63Co 1.5096E+1

64Co 1.451E+1

65Co 1.3477E+1

66Co 1.2759E+1

67Co 1.2279E+1

68Co 1.166E+1

69Co 1.099E+1

70Co 1.11E+1

71Co 1.03E+1

72Co 9.00

73Co 8.70

74Co 8.80

75Co 8.30

FIG.7. two neutron separation energy

with number of neutrons.

FIG.6.Proton separation energy with

nucleon number.

FIG.6.shows linearly increasing of proton

separation energy with increasing of the

number of nucleons or increasing of the

number of neutrons because the number of

protons is constant = 27.

Two neutron separation energy:

The energy required to remove 2

neutrons from a nucleus with Z protons

and N neutrons. S2n(Z, N), can be

computed from the ground state

nuclear masses M(Z, N) and M(Z, N–2)

and the neutron mass mn with the

relation [5]:

S2n(Z, N) = –M(Z, N) + M(Z, N–2)+2mn ,(4)

Table.5: S2n values in Mev for cobalt

isotopes [3].

50Co 3.48E+1

51Co 3.32E+1

52Co 3.24E+1

53Co 3.146E+1

54Co 3.016E+1

55Co 2.7513 E+1

56Co 2.4173 E+1

0.00E+00

1.00E+01

2.00E+01

3.00E+01

4.00E+01

20 25 30 35 40 45 50

S2n (Mev)

N

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

20 30 40 50

Sp (MeV)

N

7

Q(58

Co) = +0.21 ± 0.03b

μ(60

Co) = +3.790 ± 0.008n.m.

Q (60

Co) = +0.42 ± 0.05b.

μ (57

Co) = +4.722 ± 0.017n.m

Q (57

Co) = +0.49 ± 0.09b

μ(58

Co) = +4.035 ± 0.008n.m

Which are the same values were taken

from the table of nuclear magnetic

dipole and electric quadrupole

moments [7]

The magnetic moment of Co59

= 4.627

from the table of nuclear magnetic

dipole and electric quadrupole

moments [7].

The nuclear quadrupole moment of 59

Co including Sternheimer

corrections up to second order yielded

the value of 0.41b from hyperfine-

structure measurements [8], which

equal 0.380 b from the hyperfine

structure of the seven lowest atomic

levels in has been examined with the

atomic-beam magnetic-resonance

technique [9], and equal 0.40 from the

table of nuclear magnetic dipole and

electric quadrupole moments [7].

FIG.7 shows that S2n values decreases

smoothly as the number of neutron

increases.

Nuclear electromagnetic moments:

Any distribution of electric charges

and currents produces electric and

magnetic fields that vary with distance.

The simplest distributions of charges

and currents give only the lowest order

multipole fields. In the nucleus the first

non- vanishing magnetic moment is the

dipole. [2]

Magnetic dipole moment μ =gsSμN,

where gs is the g factor, S is the

intrinsic spin for proton or neutron, μN

is the bohr magneton [2].

The first non- vanishing electric

moment is the quadrupole.

The quadrupole operator measures the

lowest order departure from a spherical

charge distribution in a nucleus, is

defined Q0=e (3z2 – r

2), where < Q0>

=0 for spherical nuclei [1].

The method of nuclear magnetic

resonance on radioactive nuclei,

oriented in a dilute paramagnetic

crystal, was applied to three Co

isotopes (57

Co, 58

Co, 60

Co).

By using the accurate measures values

of resonance frequencies of the

hyperfine structure parameters and

comparing these with the hfs

parameters of stable 59

Co, accurate

values for the magnetic moments and

electric quadrapole moments were

obtained [6]

8

Table.9: the t1/2 of ε decay for the

ground state cobalt isotopes (from

Co50

- Co58)

[2, 3].

50Co 38.8 ms

51Co >200 ns

52Co 115 ms

53Co 240 ms

54Co 193.28 ms

55Co 17.53 h

56Co 77.236 d

57Co 271.74 d

58Co 70.86 d

And from Co60

to Co73

and Co75

the

decay mode which observed is

negative beta decay [3], in this decay

neutron change into proton,

N and Z change by one unit and A

doesn't change [2].

z AXN z+1

AXN-1 + e

- + ύ

Table.10: the t1/2 of β- decay for ground

statae cobalt isotopes (from Co60

- Co73

and Co75

) [2, 3].

60Co 1925.28d 68

Co 0.199 s

61Co 1.650 h 69

Co 229 ms

62Co 1.50 m 70

Co 108 ms

63Co 27.4 s 71

Co 80 ms

64Co 0.30 s 72

Co 59.9ms

65Co 1.16 s 73

Co 41 ms

66Co 0.20 s 75

Co >150 ns

67Co 0.425 s

Decay modes and the half life time:

The unstable nuclei spontaneously

emitting radiation energy in different

form: alpha, beta and gamma rays to

form stable nuclei.

28 of cobalt isotopes are radioactive

which decay in different principal

decay modes: negative beta decay or

electron capture or negative bata decay

with neutron emission.

From Co50

to Co58

the decay mode

which observed is electron capture ε

[3], in which a proton captures an

electron from its orbit and converts

into a neutron + neutrino [10].

p+e-

n + ν

z AXN + e

- z-1

AXN+1 + ν

Half-life time t1/2 of the decay is the

time needed for the activity to be

reduced by half, where activity is the

rate at which radioactive nuclei decay

which is the number of decays per

second [10].

The half-life time related with stability,

the shorter half-life the faster of decay

so the less stability [10].

The t1/2 for a decay different from one

isotope to another for example the half

life time for Co67

= 0.425 s for

negative beta decay and for Co57

=

271.74 d for electron capture but Co59

is stable [2,3].

Table.9 shows the half life time for

electron capture decay mode.

9

In the case of Co60

Ni60

the

Co60

decay by emitting electron into an

exited state of Ni60

, then decays to the

ground state of Ni60

via two gamma

decays [12], the mass table give

QB- = [M(Co

60) – M(Ni

60)]c

2

=[59.933820-59.930788]931.5MeV/u

QB- = 2.824308 5MeV, in a good

agreement of QB- from the

experimental data from Brookhaven

National Laboratory =2.82281 MeV.

FIG.8. negative beta decay of cobalt.

Table.11: Beta negative Q value in

Mev [3].

60Co 2.82281

61Co 1.3237

62Co 5.322

63Co 3.661

64Co 7.307

65Co 5.940

66Co 9.598

67Co 8.421

68Co 11.54

69Co 9.81

70Co 12.3

71Co 11.0

72Co 14.4

73Co 13.2

74Co 15.1

Co74

decay by negative beta decay with

neutron emission wit by 18% with t1/2

= 25 ms and in the form of β- [3].

Q value:

The Q value is the kinetic

energy released in the decay of the

particle at rest [11], and it is the

difference between the initial and final

nuclear mass energies [2].

Beta negative Q value Which is the

energy released in beta decay.

For a typical negative beta decay

process

z AXN z+1

AXN-1 + e

- + ύ

The Q value is

QB =[MN(z AXN )-MN(z+1

AXN-1)-Me-]c

2 (5)

Where MN is the nuclear mass, the

mass of neutrino equal to zero. To

convert the nuclear masses into neutral

atomic masses use

M( AX)c

2=MN(

AX) c

2 –Z Me c

2 - i=1∑

zBi (6)

Where B is the binding energy of i th

electron. Then QB in terms of atomic

masses with neglecting the difference

in electron binding energy be

QB-= [M (

AX) – M (

AX

' )]c

2 (7)

The QB value is the energy shared by

the electron and anti-neutrino QB = Te

+ Eύ

The neutrino is massless and moves in

speed of light so its total relativistic

energy is equal to its kinetic energy Eύ

= Tύ.[2].

10

z-1 AXN+1 is in an atomic exited state

,so the calculation of the Q value must

take this into account.

Qε = [M(AX) – M(

AX

')]c

2 -Bn

Bn is the binding energy of the captured

n-shell electron (n= K,L,..) [2].

Table.12: Electron capture Q values in

Mev [3].

50Co

1.67E+01

51Co 1.286E+01

52Co 1.434E+01

53Co 8.288

54Co 8.244

55Co 3.451

56Co 4.566

57Co .836

58Co

2.307

.

FIG.10. Electron capture Q values of

the chain. Where the maximum values

appear in odd odd nuclei.

The QB value of 66

Co was measured

yielding 9.7 MeV by multi-nucleon

transfer-reactions [13].

FIG.9. Q value for negative beta decay.

Figure 9 shows a series of increasing

and decreasing in Q values with

increasing A, the increasing occurs in

even A nuclei where the decreasing in

odd A nuclei, because in even A cobalt

nuclei N odd and Z odd so the change

of neutron to proton is easily because

the shells not closed so it give higher Q

value than odd A (odd Z, even N)

because the shell of neutron is closed

so much energy used to break the shell

so it gives less Q value.

Electron capture Q value

Which is the energy released in

electron capture decay.

For electron capture process

z AXN + e

-

z-1

AXN+1 + ν

0.00E+00

5.00E+00

1.00E+01

1.50E+01

2.00E+01

45 50 55 60

Q EC

A

0

2000

4000

6000

8000

10000

12000

14000

16000

45 65 85

QB

A

11

one neutron separation energy have a

sequence of increasing and decreasing

depend on even and odd neutron

number , and has a jump in energy at

magic numbers of neutrons. The values

of two neutron separation energy

shows smoothly decreasing with

increasing N, the values of Sp shows

increasing linearly with N.

The principal decay mode of each

isotope where obtained with the half

life time and Q value .

The electromagnetic moments values

of four isotopes taken from different

experiments to compare between them

.

HFODD code: is a programming code

written in Fortran language used to get

a numerical calculations of the nuclear

properties, applied to description of

high-spin chiral bands, magnetic

rotors, superdeformed states, triaxial

octupoles. It used to get solution for

many problems as Solution of the

Skyrme-Hartree-Fock-Bogolyubov

equations in the Cartesian deformed

harmonic-oscillator basis, Nuclear many-

body problem, Superdeformation,

Quadrupole deformation, Octupole

deformation, Pairing, Nuclear radii, Single-

particle spectra [14].

3.CONCLUSION

Ten nuclear properties of the cobalt

isotopic chain were discussed ,the

binding energy values shows

increasing up to reach the stable one

Co59

which has the long life time in

the chain then become to decrease,

the values of nuclear angular

momentum and parity of the chain

were taken from experiments , there is

no relation between one isotope to

another in these values , one and two

neutron separation energy and proton

separation energy concepts where

discussed in details , the behavior of

them with increasing of neutron

number obtained where the values of

one neutron separation energy have a

sequence of increasing and decreasing

depend on even and odd neutron

number , and has a jump in energy at

12

[12] K.H.Lieser Einführung in die

Kernchemie S.223, Abb. (7-22); ISBN

3-527-28329-3, (1991)

[13] U. Bosch, W.-D. Schmidt-Ott, E.

Runte, P. Tidemand-Petersson, P.

Koschel, F. Meissner, R. Kirchner, O.

Klepper, E. Roeckl, K. Rykaczewski,

and D. Schardt, Nuclear Physics A

477,90362, (1988)

[14] N. Schunck, J. Dobaczewski, J.

McDonnell, W. Satula, J.A. Sheikh, A.

Staszczak, M. Stoitsov, P. Toivanen ,

Comput. Phys. Commun. 183, 166

,(2012).

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[1] Samuel S.M. Wong, Introductory

Nuclear Physics, second edition, wily-

vch

[2] Kenneth S.Krane, Introductory

Nuclear Physics, John Wiley & Sons

,ISBN 978-0-471-80553-3 (1988).

[3] National Nuclear Data Center

NBDC, http://www.nndc.bnl.gov

[4] Seven Gosta Nilsson and Ingemar

Rangnarsson, Shapes and Shells in

Nuclear Structure, Campridge

University Press.

[5] Sabina Anghel, Gheorghe Cata-

Danil and Nicolae Vector Zamfir,

Nuclear Physics, (2008).

[6] L. Niesen, W.J. Huiskamp, Physica

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[7] Pramila Raghavan, Elsever, 42,

2,189–291, (1989).

[8] J. Dembczyński, G. H.

Guthöhrlein, and E. Stachowska, Phys.

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[9] W. J. Childs and L. S.

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[10] Murugeshan,R ,Modern physics,

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[11] B.R. Martin and G. Shaw Particle

Physics, John Wiley & Sons.

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