M Manser Invicta Grammar School A2 PHYSICS Radioactivity By the end of this presentation you should be able to: Define the terms “activity” and “count rate” Recall and use A = N Recognise, use and represent graphically solutions of the decay law based on x = x 0 e – t for activity, number of undecayed nuclei and corrected count rate. Define “half-life” as the mean time for the number of nuclei of a nuclide to halve Use the relation  t ½ = 0.693

M Manser Sackville School Half life This is the mean time for the number of nuclei of a nuclide to halve. If the number of radioactive particles drop, then the activity of the sample will drop as well. Graphs illustrating half-life can then have “number of radioactive nuclei”, “mass”, percentage of radioactive substance” or activity on the Y axis.

M Manser Sackville School Using half lives, we can predict how long something will be strongly radioactive for, and how much will be left. Half-life is constant, regardless of when you investigate the sample. It takes the same 15 minutes for the activity to fall from 400 to ½ of 400 as it does to fall from 200 to ½ of 200 or from 100 to 50 (etc) Time (minutes)

M Manser Sackville School Decay graphs The three curves shown illustrate the decay of three different radioactive substances. Each has a different rate of decay and a different half life. They all have an EXPONENTIAL shape however. Any decay curve can then be modelled mathematically by an exponential function similar to y = ke -x Amount of radioactive material time Longest half lifeShortest half life

M Manser Sackville School Mathematical decay Consider y = ke -x For the graph below, y is the amount of radioactive substance. Usually we use N to represent the number of radioactive nuclei. N o represents the original number. This is a constant for a particular example (k). The power of e, will determine the curvature of the line, so must be affected by t, as well as a factor particular to the substance. This is the decay constant of the radioisotope,  Amount of radioactive material time N = N o e - t

M Manser Sackville School Variations of the decay equation Corrected count rate time RoRo N = N o e - t R = R o e - t A = A o e - t Amount of radioactive material time NoNo Activity time AoAo N = the number of radioactive atoms. This can be directly linked to mass (in kg) R = The corrected count rate. Corrected for background radiation. Obtained experimentally (in s -1 or m -1 etc) A = The activity of the sample. Higher than the count rate as this is mathematically adjusted to account for some radioactive particles absorbed by the sample itself, and for some particles that escape and are not counted (same unit as R, 1s -1 = 1Bq)

M Manser Sackville School The decay constant = the decay constant, and represents the probability of decay per unit time. It stands to reason that if many nuclei are likely to decay in one second that …. 1. The activity is high and, 2. The half life is short. Also, if the probability of decay is low, but there is a very large number of radioactive nuclei present, that this could cause a high activity i.e. A = N

M Manser Sackville School The decay constant and half-life If N = N 0 e – t, then after a time = t 1/2, the number of radioactive nuclei left = N o /2, so… N 0 /2 = N 0 e – t ½, so ½ = e – t ½, now taking natural logs on both sides gives …. Ln ½ = - t 1/2 lne, so = - t 1/2 and = / t 1/2 or t 1/2 = 0.693