Titanium

Astronomers and astrophysicists study objects great distances from the earth. Often the objects of study are obscured by clouds of galactic dust. Certain forms of electromagnetic radiation can penetrate this dust allowing scientists to "see'' what is inside or behind the clouds. Gamma radiation serves as a good probe for studying supernova remnants in our galaxy. One source of this radiation is Titanium-44(⁴⁴Ti). The amount of radiation depends directly on the amount of Titanium-44 present.

An ongoing problem in physics has been the accurate measurement of how fast Titanium-44 decays. Experiments over the last 30 years have placed the half-life of ⁴⁴Ti anywhere from 46 to 67 years. Two recent experiments^1,2 have determined a more accurate half-life of 59.2 ± 0.6 years.

In this project we wish to study the effects of error on exponential and logarithmic quantities. The half-life (T_½ ) of a radioactive material is the measure of how long it takes for one-half of the material to decay. For example, if 10 grams of a material has a half-life of 4 minutes, then after 4 minutes, 5 g will remain. After 4 more minutes, 2.5 g will remain. Mathematically, this can be written as

½P_o = P_oe ^-rT½

where P₀ is the original amount of material, r is the decay rate, and T_½ is the half-life. Once the rate r is determined the amount of material P(t) can be predicted for any amount of time using the equation

½P_o = P_oe ^-rT½

a. What is the decay rate for a half-life of 46 years? What is the decay rate for a half-life of 67 years?

b. What percentage of a 100 gram sample for each half-life in part a will decay in 6 months? in 1 year? in 2 years? in 5 years?

c. If the mass of a radioactive sample can only be measured to within ±0.5% of its actual value, what range of half-lives does this imply for a 100 gram sample which decays to a 98.7 gram sample after 1 year? (Hint: an error of ±0.5% means the 100 gram sample has an actual mass between 99.5 grams and 100.5 grams.) You will need to find the rate first and then the half-life.

d. If the mass of a radioactive sample can only be measured to a precision of ±0.0005% what range of values for the half-lifes does this imply for a 100 gram sample which decays to a 98.7 gram sample after 1 year?

e. In a more recent and more accurate experiment the half-life of ⁴⁴Ti is reported to be 59.2 ± 0.6 years for a 5 year experiment. Using a half-life of 59.2 years, how much of a 100 gram sample would decay in 5 years?

¹Ahmad, I., G. Bonino, G. Cini Castagnoli, S. M. Fischer, W. Kutschera, and M. Paul. Three-Laboratory Measurement of the ⁴⁴Ti Half-Life, Phys. Rev. Lett. 80, 2550 (March 23, 1998).

²Gorres, J. et al. Half-Life of ⁴⁴Ti as a Probe for Supernova Models, Phys. Rev. Lett. 80, 2554 (March 23, 1998).