Radioactivity is the behavior where certain atom nuclei spontaneously disintegrate while emitting energy or particles. 

Shortly after the discovery of X-rays by Wilhelm Röngten , Henri Becquerel experimented with different fluorescencing materials, trying to produce radiation that would show on photographic plates. He did this by exposing the specimens to ultraviolet light, but doing this he found that a certain uranium salt had the desired effect even without having exposed it to the ultraviolet light. This was in 1896, and soon thereafter he started working together with Marie Curie and Pierre Curie, researching this new form of natural radiation. The theory was that the salt emitted radiation that was independent of chemical bindings and temperature. By working on the mineral uranite, pechblände, which interesting enough was discovered by Klaproth, the Curie's discovered two new radioactive elements, Polonium and Radium.

The radiation from these new elements appeared to be of different kinds, called α-, β- and γ-rays. By applying magnetic fields to the radiation, they concluded that the alpha rays were positively charged, beta rays were negative and gamma were neutral. Ernest Rutherford concluded experiments that showed that the α-rays  were emitted from the nucleus itself, and also the relationship between mass and charge. Soon, the alpha decay, beta decay and gamma decay had been completely discovered.

In 1902, Rutherford and Frederick Soddy presented the laws governing decay of the new radioactive materials. Also see radioactive decay. They assumed that the probability of decay is constant over time, regardless of age of the material and also of the amount of material. The law they described said that the number of decays per unit time for a specimen of N atoms is

(1)  -- = - λN            (2)   N = N0 e-λt

This makes sense, since it means that in twice as much specimen material, you'll see twice as many decays. The second equation describes how much material you have after time t, if you started out with the amount N0. The probability constant is called λ, and from this you can derive the half-life T1/2 and mean lifetime τ.

             ln 2                      1
(3)  T1/2 =  ----          (4)    τ = ---- = 1.44 T1/2    
              λ                        λ

The term λN is called activity, and is assigned the letter A, and it can be calculated with

(5)  A = A0e-λt   where A0 = λN0

Activity is an important measurement of radioactive materials, and it is measured in units of Becquerel, Bq. 1 Bq is defined as 1 decay per second. It replaces the older unit Curie, Ci. 1 Ci = 3,7·1010 Bq . It is common to plot activity equation (5) over time, as below

|. <-- A = A0
| .
| .
| .
|  .    
|   .
|    . <-- A = A0/2
|     .            Area below line = N0
|       .        /
|         .     /
|            . /
|             /  .
|                     .
|                          .
|                               .   .  .
     T1/2                               time 

(Pardon my bad ascii art).

Anyhow, it's common that nuclei decay in different ways. For instance, copper-64 will decay either through β-, β+ or electron capture. Each of these have a different λ, which makes calculations a bit more complicated. Each of these decays will have a certain probability.

Artificial radioactivity
All since the beginning of 20th century, radioactivity has been used for treatment of tumors. At first radium was used, which was quite dangerous, but in the 1920s the first artificially manufactured radioactive materials were created. First one used natural alpha particles to initiate nuclear reactions in other materials, but soon the scientists had learned how to use protons or alpha particles from accelerators. Later on, fission was discovered and one started to use neutrons to initiate radioactivity. The process of creating radioactive materials follow the same laws as the decay. The difference is that the activity increases as more materials gets radioactive.

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