A general expression in the form of a multiple integral involving a delta function is derived for the polycrystalline spectra of radicals. The expression is exact when the hyperfine structure is due to the interaction of one proton with the unpaired electron. The extension to a system involving the interaction of a number of protons with the unpaired electron is straightforward but was not done here.Because of the formidable integrations encountered when the complete spin Hamiltonian is used, a simpler case is considered to illustrate the technique developed. It is shown that if one assumes thatIH,the nuclear spin component along the externally applied magnetic field, is a good quantum number, then the peak to peak separation of the broad doublet that is obtained bears a simple relationship to the hyperfine interaction.For a properly chosen Cartesian axis system, the hyperfine spin Hamiltonian can be written asHhfequals;hA SzIzplus;hBSxIxplus;hCSyIy.The peak to peak separation corresponds to the middle value of verbar;Averbar;, verbar;Bverbar;, verbar;Cverbar;.The line shape for a Csngbnd;H fragment with the unpaired electron in a carbon 2pxorbital is worked out in detail assuming thatIHis a good quantum number. Two peaks with very slightly asymmetric shoulders are obtained, the peak to peak separation being 61 Mc.
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