β-decay

There are three different type of β decay:
β-, β+, and electron capture.


The tables give the masses of atoms 
(including their electrons), that is the masses of 
neutral atoms, rather than masses of bare 
atoms, because it is much more diffucult to measure 
with high precision the mass of these "no-normal" 
elements.

1. Beta-minus particle: β^-

A β- is an electron. This process involves 
transformation of a  neutron into a proton, an electron, and 
an antineutrino, in about 15 minutes. β- decay 
occurs with nuclides with too large ration N/Z.

The speeds of β-particles range from 0.9995 c to c. their motion 
is then relativistic. If the recoiling nucleus and the β-particle 
are alone in the products of the decay, the speed of the β-particle 
would be definite. The β-particle are emitted with a continuous 
spectrum of energies. According, however. Therefore, there is a third 
particle which is the antineutrino  in the products of the 
β-deacy. From the conservation of charge, it nust be neutral, 
and from the concervation of momentum, it must be of spin 1/2.

The antineutrino is the antiparticle of the neutrino, without mass 
as a photon and of symbol is &vu;. Both of them have no charge 
and no mass. There are three varieties of neutrinos. the first 
provides from β-decay, the second form the tau-particle decay 
and the third from the muon-particle decay, each of them has its 
proper antineutrino. We write then:

n → p + β- + νe

Here is the β- process:

X(A,Z) = X(A,Z+1) + β- + νe

n → p + β- + νe


β- binding energy

X(A,Z) → X(A,Z+1) + β-
n → p + e-
X(A,Z+1) is not neutral because it laks an electron, 
that is it contains (Z+1) protons and Z electrons.
But with β- which is an electron
e-, gives a neutral atom that exist 
and we know its atomic mass. We can write: 

Mass of neutral X(A,Z+1) = mass of [non neutral X(A,Z+1) + β-]
Therefore, the mass defect is:
Δm = mass[neutral X(A,Z)] - mass[ neutral X(A,Z+1)]


Example:

Co(60,27) → X(A, Z+1) + β-
X(A, Z+1) = Ni(60,28)

Mass of Co(60,27) = 59.933822 u
Mass of Ni(60,28) = 59.930791 u

The process β- is possible 
because the mass defect is positive, that is: 
Δm = 59.933822 u - 59.930791 u = 0.003031u

2. Beta-plus particle: β⁺

β+ is a positron e+, the 
electron's antiparticle.

β+ decay occurs with nuclides 
with too small ration N/Z. 

In the process of β+ decay, we have:

p → n + β+ + νe
&vu;e is the electron neutrino.
 
Here is the β+ process:

X(A,Z) = X(A,Z-1) + β+

p → n + β+ + νe



β+ binding energy

X(A,Z) → X(A,Z-1) + β+ + νe
p → n + e+ + &vu;e

X(A,Z-1) is not neutral because it contains one more 
electron, that is it contains (Z - 1) protons and Z 
electrons. We subtract un electron to obtain a neutral 
atom X(A,Z-1).
The mass of  β+ = mass of  electrom,
then:
Mass [non neutral X(A,Z-1)] = mass [neutral X(A,Z-1)] + 
mass [substracted electron] + mass [β+]= 
mass [neutral X(A,Z-1)] + 2 mass [electron] 

Therefore, the mass defect is:
Δm = mass[neutral X(A,Z)] - mass[ neutral X(A,Z-1)] - 2 mass [electron] 

Example:

Co(57,27) → X(A, Z-1) + β+
X(A, Z-1) = Fe(57,26)

Mass of Co(57,27) = 59.936296 u
Mass of Fe(57,26) = 59.935399 u

The process β- is possible 
because the mass defect is positive, but the process β+ 
is not possible because the mass of the original nuclide 
Co(57,27) is less that the mass of its daughter Ni(57,26) plus 
the mass of two electrons.
Δm =  59.936296 u - 59.935399 u - 2. me = 0.003031 u < 0.

me = 9.11 x 10-31  kg
u = 1/Avogadro's number = 12 g (C)/ 12 x 6.023 x 1023  
= 1/ 6.023 x 1026  kg
me/u = 9.11 x 10-31  x  6.023 x 1026  = 0.0005487 u
2 me = 0.00109739
59.935399 u + 2. me = 0.003031 = 59.9364929
Δm < 0.

3. Electron capture process:

There are a few nuclides for wich the process β+ 
is not energetically possible, but they involve the electron 
capture process in which un orbital electron (from K-shell generally) 
is captured by the nucleus and combines with a proton to form a neutron 
that remains in the nucleus and a neutrino which is emitted.

β- + p → p + νe


X(A,Z) → X(A,Z-1) 

p + e-  → n  + νe


Electron capture binding energy

The mass defect is:
Δm = mass[neutral X(A,Z)] - mass[ neutral X(A,Z-1)] 
so, the binding energy is:
Δmc2  = mass[neutral X(A,Z)]c2  - mass[ neutral X(A,Z-1)]c2  

Example:

The nuclide Co(57,27) decays by the electron-capture 
process:
X(A,Z) → X(A,Z-1) 
Co(57,27) → Fe(57,26)

Mass of Co(57,27) = 59.936296 u
Mass of Fe(57,26) = 59.935399 u
Δm =  59.936296 u - 59.935399 u > 0.