Half-life period: The half-life period of an element is defined as the time in which the number of radioactive nuclei decay to half of its initial value.
Decay constant: The decay constant of a radioactive element is defined as the reciprocal of time in which the number of undecayed nuclei of that radioactive element fails to $$\dfrac{1}{e}$$ times of its initial time.
Relation between Half-life and Decay constant: The radioactive decay equation is
$$N = N_{0}e^{-\lambda t}$$ ........(i)
when $$t = T, N = \dfrac{N_0}{2}$$
$$\therefore \dfrac{N_0}{2} = N_0 e^{-\lambda T}$$
or $$e^{-\lambda T} = \dfrac{1}{2}$$ .....(ii)
Taking log of both sides
$$-\lambda T log_{e} = log_{e} 1 - log_{e} 2$$
or $$\lambda T = log_{e} 2$$
$$\therefore T = \dfrac{log_e 2}{\lambda} = \dfrac{2.3026 \,log_{10}2}{\lambda} = \dfrac{2.3026 \times 0.3010}{\lambda}$$ ........(iii)
or $$T = \dfrac{0.6931}{\lambda}$$