Broadly speaking, radioactivity is not something one should mess with just as a pastime. Indeed, ionizing radiation has the potential of causing carcinogenic mutations in your cells DNA, as well as produce damage to cell tissue. Indeed, it makes me chuckle that until 50 years ago or so kids could play with it by purchasing stuff like that shown below...

If you know what you are dealing with and take the necessary precautions, however, radiation _can_ be fun to study at home. The tools and the primary matter are not found at the corner grocery, though, so you need to have a specific interest in it before you get ready to start. 

Since I studied nuclear physics some time ago, and I even took radiation worker training in my early days at Fermilab, I guess I do qualify to have some fun with radioactive elements. As for the tools, I own a nice "Radex" radiation meter from Quarta, which can detect beta and gamma rays (and is also sensitive to X-rays). However, until a few days ago I did not own any radioactive substance in significant amounts. Or so I thought.

The matter changed last week, when I received a box of old minerals I had stored eons ago in my parents' house. The house is being sold so I brought the lost sheep to my home, to re-join a mineral collection that counts over 500 specimens. Among the old stuff I brought home was a small piece of uranium ore. It might be carnotite, an uranium vanadate, or some other compound. It had been given to me by Giulia Giudice, a friend of my mother who had a large minerals collection, some 40 years ago. Shortly after my acquisition the piece had broken in smaller bits, and since then sat in a small plastic box, waiting to be rediscovered. 

When I tested the sample for radioactivity with my Radex meter, I was surprised to see that the material shows a detectable amount of activity. My surprise stems from the fact that the sample -or better, what remains of it - is rather small; its weight is of just 11.13 grams. If we assume that the sample contains a 10% concentration of undecayed carnotite, for example (just an educated guess), that weight would correspond to a number of uranium atoms which can be extracted by knowing the formula of the substance (it is  K2(UO2)2(VO4)2•3(H2O) so there are two uranium atoms per molecule). One gets that in 902.18 grams of the substance (one molecular mass) there is a number of uranium atoms equal to twice the Avogadro number 6.022*10^23. My sample could thus contain of the order of 2*6.E23*0.1*11.13/902.18 = 10^21 uranium atoms, more or less.

Uranium in the ore is mostly the less dangerous, long-lifetime 238U variety (as opposed to the bomb-grade 235U), which has a half-life of 4.46 billion years. That means that in one second one might expect the following number of decays from 238U in my sample:

N/1s = 1.E21 * [1- exp (-1/(86400*365.26*4.46E9/0.693))] = about 4500 decays per second.

Above, I used the fact that the number of decays in a given time interval (1 second) is the original number times (one minus the survived fraction), and the survived fraction in a time interval Δt is exp(Δt/lifetime). Finally, half life and lifetime are related by the 0.693 ratio above.
We might feel proud of having extracted a rate of decays from scratch, but unfortunately, computing in a careful way the number of detectable decays from my sample is not as simple as that - the devil is in the details! To start with, we must acknowledge that the decays of 238U produce a chain of other decays of lighter elements. The main chain of processes is the following:

238U --> 234Th (Thorium) --> 234Pa (Protactinium) --> 234U --> 230Th --> 226Ra (Radium) --> 222Rd (Radon) --> 218Po (Polonium) --> 214Pb (Lead) --> 214Bi (Bismuth) --> 214Po --> 210Pb --> 210Bi --> 210Po --> 206Pb.

The last element, lead-206, is stable, but that's quite a journey to get there! The decays that change by four the atomic mass emit alpha particles (helium nuclei, that is), a form of short-range radiation that is hard to detect with commercial counters such as the one I own; while those which change the substance without modifying the atomic number are beta decays, which cause the emission of electrons - to which my Radex is sensitive. 

Since I am at it, I can list the half-lives of each element in the above chain. These are tabulated somewhere, and amount to:

- HL(238U) = 4.46E9y
- HL(234Th) = 24.1d
- HL(234Pa) = 1.17'
- HL(234U) = 245E3y
- HL(230Th) = 75.4E3y
- HL(226Ra) = 1.6E3y
- HL(222Rd) = 3.82d
- HL(218Po) = 3.11'
- HL(214Pb) = 26.8'
- HL(214Bi) = 19.9'
- HL(214Po) = 1.63E-4"
- HL(210Pb) = 22.3y
- HL(210Bi) = 5.01d
- HL(210Po) = 138d

Above, y=years, d=days, ' indicates minutes, and " indicates seconds. Without computing expected rates, we may observe that the beta decays that my instrument is sensitive to arise from combinations of chains from 234Th, 214Pb, and 210Pb. While the first only has one fast decay in the sub-chain (234Pa->234U), the second has two consecutive beta decays with comparable lifetimes (27' and 20'). The third also gives two beta electrons. In total, every 238U decay eventually produces 5 beta electrons, randomly distributed with a complex function of the listed lifetimes. 
All of the above is to say that the rate of betas from my source should equal 5 times 4500 Hz, or something in the range of 20 kHz. But wait - we know nothing of the energy range of these beta electrons, nor do we know much about the detection efficiency of the instrument I have. Furthermore, my calculation was very roughly approximating several of the ingredients, as e.g. I have only very little knowledge of the fraction of 238U actually making up the source. See, quantitative physics is hard!

Despite all these caveats, it is fun to just get the counter close to the uranium mineral and see the counter ticking. When I place the sensitive area of the counter directly on top of the specimen, radiation level overshoot the counter's range, with values above 999 microRem per hour. Check it out for yourself in the video at this link (20Mb) (NB, the video shows the rate up to 444 microRem/hr - I did not center the source well when I took it).

Thinking about it, I am startled by that reading - I was not aware of the significant activity of this source. Luckily, the inverse-squared-radius law saved me from getting a dose by sleeping for some ten years a few meters away from the cabinet where that piece of rock sat: levels of radiation drop below the background of 20 microRem per hour if you get the counter farther than about 4 inches from it.