proportionality between the size of a population of radioactive atoms and the rate at which the population decreases because of radioactive decay. Suppose N is the size of a population of radioactive atoms at a given time t, and dN is the amount by which the population decreases in time dt; then the rate of change is given by the equation dN/dt=−λN, where λ is the decay constant. integration of this equation yields N=N₀e^−λt, where N₀ is the size of an initial population of radioactive atoms at time t=0. This shows that the population decays exponentially (exponential function) at a rate that depends on the decay constant. The time required for half of the original population of radioactive atoms to decay is called the half-life. The relationship between the half-life, T_1/2, and the decay constant is given by T_1/2=0.693/λ.