The Radioactive Disintegration Constant, Lambda

Abstract

To Dr. Howard P. Doub Editor of Radiology Dear Dr. Doub There is a misconception regarding the disintegration constant, lambda (λ) which seems wide-Spread among radiologists, radiological residents, and not a few radiation physicists. In most texts the disintegration constant is defined as “the fractional change in number of atoms which occurs in unit time.” This definition is correct only in the limiting case where the unit time interval is infinitely short. In all other cases, lambda may be a fraction or it may be a number greater than one; it is never the fractional change in number of atoms as given in the above definition. Consider radon, whose half-life is 3.825 days. Its disintegration constant is 0.693/3.825 = 0.18 per day. In this case lambda is a fraction. However, it is equally justifiable to consider the disintegration constant per month. In this case the half-life of radon is 0.1275 months, and the disintegration constant is 0.693/0.1275 = 5.43 per month, a number which is certainly not a fraction. Lambda will thus be either a fraction or a whole number depending on the half-life and the units in which it is expressed. I have asked many radiologists, and a few radiation physicists, this question: “The decay constant of radon is 0.18 day−1. What per cent of the atoms disintegrate per day?” The answer is invariably “18 per cent.” This is not correct. The per cent of radon atoms disintegrating per day must be obtained from the equation N = N0e-lambda t If N0 is taken as 100, lambda as 0.18 day−1, T as 1 day, then N equals 84, and 16 atoms out of 100 have decayed per day, or 16 per cent per day, not 18 per cent, as would be concluded from the common definition of the constant. It is possible that this misconception has gone along so many years because in the above classical example, using radon, the correct and incorrect answers are so nearly alike that most persons have assumed a small arithmetic error somewhere along the line. It is not an arithmetic error at all; it is a philosophical error. When the disintegration constant of radon is taken per day, the per cent disintegrating per day is about 11 per cent less than the constant. As the unit of time decreases, so does the error. The per cent of atoms decaying per hour is only about 0.5 per cent less than the constant. However, it must be realized that there will always be a difference until the unit of time becomes zero. This misconception, which occurs even in the early radium literature, probably stems from a misuse of the basic differential equation dN/N = -lambda dT. The left-hand term of this equation cannot be treated as an ordinary arithmetic fraction, which is done when lambda is defined as the fractional change per unit time.