atoms of hydrogen with similar results. Later scientists conducted experiments using other atoms that have only one electron in their valence shell: (copper, gold, sodium, potassium). Every time there were two lines formed on the metallic plate. The atomic nucleus also may have spin, but protons and neutrons are much heavier than electrons (about 1836 times), and the magnetic dipole moment is inversely proportional to the mass. So the nuclear magnetic dipole momentum is much smaller than that of the whole atom. This small magnetic dipole was later measured by Stern, Frisch and Easterman. Electron paramagnetic resonance For atoms or molecules with an unpaired electron, transitions in a magnetic field can also be observed in which only the spin quantum number changes, without change in the electron orbital or the other quantum numbers. This is the method of electron paramagnetic resonance (EPR) or electron spin resonance (ESR), used to study free radicals. Since only the magnetic interaction of the spin changes, the energy change is much smaller than for transitions between orbitals, and the spectra are observed in the microwave region. Derivation For a solution of either the nonrelativistic Pauli equation or the relativistic Dirac equation, the quantized angular momentum (see angular momentum quantum number) can be written as: where is the quantized spin vector or spinor is the norm of the spin vector is the spin quantum number associated with the spin angular momentum is the reduced Planck constant. Given an arbitrary direction z (usually determined by an external magnetic field) the spin z-projection is given by where is the secondary spin quantum number, ranging from − to + in steps of one. This generates different values of . The allowed values for s are non-negative integers or half-integers. Fermions have half-integer values, including the electron, proton and neutron which all have s = 1/2. Bosons such as the photon and all mesons) have integer spin values. Algebra The algebraic theory of spin is a carbon copy of the angular momentum in quantum mechanics theory. First of all, spin satisfies the fundamental commutation relation: where is the (antisymmetric) Levi-Civita symbol. This means that it is impossible to know two coordinates of the spin at the same time because of the restriction of the uncertainty principle. Next, the eigenvectors of and satisfy: where are the creation and annihilation (or "raising" and "lowering" or "up" and "down") operators. Energy levels from