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Radioactivity

Field: Nuclear Physics

The process by which an unstable atomic nucleus loses energy by radiation.

Sequence of Expressions

Definition

Half-Life

Let N(t)N(t) be the number of radioactive nuclei at time tt, and λ\lambda be the decay constant. The decay follows the differential equation dNdt=−λN\frac{dN}{dt} = -\lambda N. The half-life T1/2T_{1/2} is defined implicitly by the condition N(T1/2)=12N(0)N(T_{1/2}) = \frac{1}{2} N(0), yielding:\nT1/2=ln⁡(2)λT_{1/2} = \frac{\ln(2)}{\lambda}
Define the activity AA of a radioactive sample as the rate of decay, A=∣dNdt∣A = \left| \frac{dN}{dt} \right|. If λ\lambda is the decay constant and NN is the number of radioactive nuclei, the activity is defined by the relation: A=λN A = \lambda N
The equivalence between mass mm and energy EE is given by the relativistic formula: E=mc2 E = mc^2 where cc is the speed of light in a vacuum.
Consider a parent nucleus XX with mass number AXA_X and atomic number ZXZ_X. Alpha decay is represented by the reaction: ZXAXX→ZX−2AX−4Y+24α {}_{Z_X}^{A_X}X \rightarrow {}_{Z_X-2}^{A_X-4}Y + {}_{2}^{4}\alpha where YY is the daughter nucleus, and the conservation laws dictate that the total mass number and atomic number must be conserved: AX=(AX−4)+4andZX=(ZX−2)+2 A_X = (A_X-4) + 4 \quad \text{and} \newline Z_X = (Z_X-2) + 2
Theorem

Beta Decay

Consider a parent nucleus XX with mass number AXA_X and atomic number ZXZ_X. Beta decay (specifically β−\beta^- decay) is represented by the reaction: ZXAXX→ZX+1AXY+e−+νˉe {}_{Z_X}^{A_X}X \rightarrow {}_{Z_X+1}^{A_X}Y + e^- + \bar{\nu}_e where YY is the daughter nucleus, e−e^- is the beta particle, and νˉe\bar{\nu}_e is the electron antineutrino. The conservation laws require: AX=AX+0andZX=(ZX+1)+(−1) A_X = A_X + 0 \quad \text{and} \newline Z_X = (Z_X+1) + (-1)
Let EiE_i and EfE_f be the energy eigenvalues of the initial and final excited states of the nucleus, respectively. The transition probability rate Γi→f\Gamma_{i\to f} for the emission of a photon γ\gamma with energy EγE_{\gamma} is given by the Fermi's Golden Rule approximation, leading to the energy conservation relation:\nEγ=Ei−EfE_{\gamma} = E_i - E_f
Define the decay constant λ\lambda (units: T−1T^{-1}) as the proportionality factor relating the instantaneous decay rate R(t)R(t) to the number of nuclei N(t)N(t): \nR(t)=−dNdt=λN(t)R(t) = -\frac{dN}{dt} = \lambda N(t)
Let N(t)N(t) be the number of radioactive nuclei at time tt, and N0=N(0)N_0 = N(0). The decay rate is governed by the differential equation: dNdt=−λN(t) \frac{dN}{dt} = -\lambda N(t) where λ\lambda is the decay constant. The solution is given by: N(t)=N0e−λt N(t) = N_0 e^{-\lambda t} Furthermore, the decay constant λ\lambda is related to the half-life t1/2t_{1/2} by: λ=ln⁡(2)t1/2 \lambda = \frac{\ln(2)}{t_{1/2}}
For a decay process transforming initial mass MiM_i into final mass MfM_f and emitting particles with total rest mass memittedm_{emitted}, the total energy EE is conserved. The energy released, QQ, is given by the mass defect Δm=Mi−Mf−memitted\Delta m = M_i - M_f - m_{emitted}: \nQ=(Mi−Mf−memitted)c2=Δmc2Q = (M_i - M_f - m_{emitted}) c^2 = \Delta m c^2
For a nucleus with ZZ protons and NN neutrons, let mpm_p and mnm_n be the masses of the proton and neutron, respectively, and MM be the measured mass of the nucleus. The Nuclear Binding Energy BB is defined as the positive energy required to disassemble the nucleus into its constituent nucleons:\nB=[Zmp+Nmn−M]c2B = [Z m_p + N m_n - M] c^2