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How was the modern day atom discovered?

To answer this question, we must first consider the previous model of the atom.

The atom was initially believed to be a sphere of positive charge with electrons dotted around the inside, this was known as the plum pudding model.

During an experiment, Ernest Rutherford fired alpha radiation -- or signals -- through an object with atoms(in this case a metal foil) and observed the directions in which the radiation was deflected. Most radiation passed straight through but very few bounced back more than 90 degrees (something that was very unexpected, scientists expected all radiation to pass through the atom).

This led them to believe that there is a incredibly dense region of an atom that can reflect Alpha Radiation, this is known as the nucleus which we know now to consist of protons and neutrons. As 1/100 000 were reflected, he concluded the nucleus is 1/100 000 the size of the whole atom.  The rest of the radiation that passed straight through the atom, was identified as the empty electron shells around the nucleus. 

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