Discrepancies observed in $\beta$ decay
(a) Continuous nature of $\beta$ ray spectrum is unexpected since, $\beta$ transition connects two states of definite energy.
(b) Non–conservation of energy : Most electrons were emitted with only about one–third of maximum energy $E_{\max}$ (potential energy).
(c) Non–conservation of angular momentum

$_{1}^{3}H \rightarrow _{2}^{3}He + _{-1}^{0} e$

spin of $_{1}^{3}H$ is 1/2

Both $_{2}^{3}He$ and $_{-1}^{1}e$ are fermions

So, spin of $_{2}^{3}He + {0}^{-1} e$ is either zero or 1.

$n \rightarrow p + e^{-} + \overline{v} \ \ \ \ \ \beta^{-}$ emission

$p \rightarrow n + e^{+} + v \ \ \ \ \ \beta^{+}$ emission

$p + e^{-} \rightarrow n + v \ \ \ \ \ \mathrm{electron \ capture}$

The electron, neutrino and product nucleus share the energy, momentum and angular momentum among got them available from the nuclear transition. The $\beta$ – particle gets maximum-energy when the neutrino is emitted with zero momentum.

(d) Non conservation of linear momentum : Electron seldom moves opposite to the nucleus as required for conservation of linear momentum. For example : $C^{14} \rightarrow N^{14} + \beta^{-}, N^{14}$ and $\beta^{-}$ seldom moves opposite to each other.
These difficulties were overcome by the introduction of the neutrino hypothesis.
Pauli postulated the existence of a new particle, neutrino in 1930, having following properties.
(a) zero charge – it is denoted by the symbol $\nu$
(b) zero rest mass or (atmost few eVs)
(c) intrinsic spin = ½
The introduction of this particle corrected the discrepancies.
(a) The half spin of neutrino allow the angular momentum to be conserved in $\beta$ –decay process.
(b) Second property of zero rest mass allows the conservation of energy.

The decay energy is shared between $\mathbf{\beta}$ particle and antinutrino

$E_{\max} = E_{\beta} + E_\nu$

(c) This sharing of energy is not in fixed proportion. So, curve depicting $\beta$ particle is continuous. It show the way energy is shared.

Properties of antinutrino $\mathbf{\overline{(\nu)}}$

(a) zero charge
(b) zero rest mass
(c) intrinsic spin=½

Difference lies in helicity $\nu$ is left handed particle with H = –1 and $\overline{\nu}$ is right handed particle with H = +1

Helicity

Helicity of elementary particle can be defined as

$H = \dfrac{\vec p \cdot \vec s}{|\vec p| |\vec s|}$

where $\vec p$ is the momentum and $\vec s$ is the spin of particle

H = –1 for neutrino

H = +1 for anti-neutrino

If H = 0, system has definite parity

If $H \neq 0$ , parity is violated.

Properties of Helicity

(a) It has a fixed value for neutrino and antineutrino
(b) Eigenvalues of helicity operation is $\pm 1$
(c) Its eigen values can be used to discriminate between two linearly independent eigen vectors of energy and momentum.
The fixed value of helicity for neutrino and antineutrino leads to implication that they move with the velocity of light c. Otherwise v will have opposite, helicity in the frame of reference moving faster or slower compared to velocity of v. So, it is not possible to overtake v or $\overline{\nu}$ by a faster frame and so, these particles moves with velocity of light c.

Violation of parity in $\mathbf{\beta}$ -decay

In 1957, Lee and yang questioned the conservation of parity in $\beta$ -decay. Let us consider mirror image of $\beta$ + decay. Let the neutrino travel towards a mirror with momentum $\vec p_v$

Its spin $\vec S_v$ is oppositely directed to $\vec p_v$ . The mirror image of this neutrino is different because the direction of $\vec S_v$ and $\vec p_v$ are same in mirror image so, it is like antineutrino.

So, the mirror image of the $\beta^+$ decay is not a possible process. This termed as violation of parity by nutrino

This asymmetry implies that interactions in which v and $\overline{v}$ participate the weak interactions need not violate parity and parity conservation, is found to hold true only in strong & electromagnetic interactions.

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