Beta -ray spectra

\(\beta\) -ray spectra:

A \(\beta\) -spectra is an electron emitted from nucleus of a radio-active substance. The penetrating power of\(\beta\) -particle is greater than that of \(\alpha\)- particles. the speed of\(\beta\) -particle is approximately \(1.96\times 10^6m/s\) to \(2.46\times 10^7\)m/s.

There are two types of\(\beta\) -particles

1. \(\beta^-\) particle (electron emitted from nucleus = -1\(e^\circ\)
2. \(\beta^+\) particle (positron emitted from nucleus = +1\(e^\circ\)

1. \(\beta^-\) emission \((_{-1}e^\circ\) emission)

During\(\beta\) emission a neutron is converted into proton is atomic number is increased by 1 unit.

i.e \(_0n^1\longrightarrow _{+1}p^0+_{-1}e^\circ+\bar\nu\) (Anti-nutron)

\(_zX^A\longrightarrow _{z+1}X^A+_{-1}e^0+\bar\nu+Q\)

According to conservation of energy.

\((M_p-Zm_e)c^2=[m_d-(z_1)m_e]c^2+m_ec^2+Q\)

where, \(m_p\)= mass of parent atom

Z= atomic number of atom

\(m_e\)= mass of electron (\(\beta\)-particle)

\(m_d\)= mass of daughter atom

(z+1)= atomic number of daughter atom

q= energy emitted during radioactive disintegration

\(M_p-zm_e^2\)+ mass of parent nucleus

\(m_d-(z+1)me= mass of daughter nucleus

\(M_p-c^2-Zm_ec^2=m_dc^2-Zm_ec^2-m_ec^2+m_ec^2+Q\)

Q=\((M_p-M_d)C^2\dotsm(1)\)

Equation (1) gives the energy of emitted \(\beta-particle\) for the disintegration the value of Q must be positive

i.e \(Q>0\Rightarrow (M_p-M_d)C^2>0\)

\(\Rightarrow M_p>M_d\)

\(\Rightarrow M(z)> M(Z+1)\)

2. \(\beta^+- emission\)

During \(\beta^+\) -emission (positron-emission) the atomic number of parent atom decreases by 1 unit and mass number remains same.

A proton inside nucleus is converted into neutron and\(\beta^+\) -particle along with neutrino.

i.e \(_{+1}P^1\longrightarrow _0n^1+_1e^0+\nu\)

\(_ZX^A\longrightarrow _{Z-1}X^A+_1e^0+\nu\)

According to conservation of energy

\((M_p-Zm_e)C^2=M_d-[(Z-1)m_e]c^2+m_eC^2+Q\)

Where, \(M_p\)= mass of parent atom having atomic number (Z)

\(m_d\)= mass of daughter atom having atomic number (Z-1)

\(M_p c^2 - Zm_e c^2= M_d c^2- Zm_e c^2+ m_e c^2+m_e c^2+Q(M_p-M_d)C^2-2mec^2=Q\)

For the\(\beta^+\)- emission, Q>0=(\(M_p-M_d)>2m_e\)

The graph of relative of\(\beta^+\)- particle along y-axis and energy of\(\beta^+\)-particles along x-axis represents\(\beta^+\) -particle spectra. The maximum energy of\(\beta^+\)- particle is known as end point enegry (Q- value disintegration)

Experimental determination of\(\beta^+\) -particle spectra:

The motion of\(\beta^+\) -particle is deflected applied magnetic field.

The experimental setup for the determination of end-point energy consists of a source of\(\beta^+\) -particle placed in lead box. The\(\beta^+\) -particle emitted from source travels with speed V inside the magnetic field perpendicular to he motion of particle. The centripetal force experience by\(\beta^+\) -particle is

$$\frac{mv^2}{r}=B\;\;\epsilon\;\; V\dotsm(1)$$

where, m-mass of particle

r= radius of circular path

B= strength of applied magnetic field.

or, mv=Bev \(\Rightarrow p\propto r\)

The linear momentum of\(\beta^+\) -particle is proportional to radius of curvature of circular path.

the kinetic energy of\(\beta^+\) - particle

K.E=\(\frac12 mv^2=\frac12\frac{m^2V^2}{m}\)

K.E=\(\frac{B^2e^2r^2}{2m}\dotsm(2)\)

By determining the value of r from the photographic plate and using values of B,e and m

The kinetic energy of\(\beta^+\) -particle can be determined.If speed of\(\beta^+\) -particle is to high then the kinetic energy of\(\beta^+\) -particle

K.E=\(mc^2-m_\circ c^2\)

=\(m_\circ \cdot c^2\biggl[\frac{1}{{\sqrt{1-\frac[v^2}{c^2}}}-1\biggr]\dotsm(3)\)

Problems Associated with \(\beta\) -particle spectrum

1. Conservation of energy:

The end point of energy of \(\beta\) -particle is \(Q=(M_p-M_d)C^2\)

Where \(M_p\)= Mass of parent atom.

\(M_d\)= Mass of daughter atom.

Q= end point energy

The K.E of\(\beta\) -particle is constant (Q is constant). Experimentally, it's found that all\(\beta\) -particles do not have same kinetic energy. There is disapearence energy of K.E of\(\beta\) -particle. It couldn't explain continuous\(\beta\) -ray spectrum.

2. Conservation of Angular momentum:

Electron proton and neutrons are fermion. Each fermion have spin angular momentum \(\frac{\hbar}{2}\). If total angular momentum of nucleus is integral multiple of \(\hbar\).If the total number of nucleus is odd than the angular momentum of nucleus is odd integral multiple of \(\frac{\hbar}{2}\). During\(\beta\) decay mass number remain same, but angular momentum is not conserved due to emission of\(\beta\) -particle.

3. Conservation of linear momentum:

During\(\beta\) -decay\(\beta\) -particle & daughter nucleus does not move in opposite direction in all cases. If the parent atom is at rest the daughter nucleus and\(\beta\) -particle must move in opposite direction to conserve linear momentum.

There is apparent violation of conservation of linear momentum.