can provide information about the atomic or molecular orbital containing the unpaired electron. The g factor Knowledge of the g-factor can give information about a paramagnetic center's electronic structure. An unpaired electron responds not only to a spectrometer's applied magnetic field but also to any local magnetic fields of atoms or molecules. The effective field experienced by an electron is thus written where includes the effects of local fields ( can be positive or negative). Therefore, the resonance condition (above) is rewritten as follows: The quantity is denoted and called simply the g-factor, so that the final resonance equation becomes This last equation is used to determine in an EPR experiment by measuring the field and the frequency at which resonance occurs. If does not equal , the implication is that the ratio of the unpaired electron's spin magnetic moment to its angular momentum differs from the free-electron value. Since an electron's spin magnetic moment is constant (approximately the Bohr magneton), then the electron must have gained or lost angular momentum through spin–orbit coupling. Because the mechanisms of spin–orbit coupling are well understood, the magnitude of the change gives information about the nature of the atomic or molecular orbital containing the unpaired electron. In general, the g factor is not a number but a 3×3 matrix. The principal axes of this tensor are determined by the local fields, for example, by the local atomic arrangement around the unpaired spin in a solid or in a molecule. Choosing an appropriate coordinate system (say, x,y,z) allows one to "diagonalize" this tensor, thereby reducing the maximal number of its components from 9 to 3: gxx, gyy and gzz. For a single spin experiencing only Zeeman interaction with an external magnetic field, the position of the EPR resonance is given by the expression gxxBx + gyyBy + gzzBz. Here Bx, By and Bz are the components of the magnetic field vector in the coordinate system (x,y,z); their magnitudes change as the field is rotated, so does the frequency of the resonance. For a large ensemble of randomly oriented spins (as in a fluid solution), the EPR spectrum consists of three peaks of characteristic shape at frequencies gxxB0, gyyB0 and gzzB0. In first-derivative spectrum, the low-frequency peak is positive, the high-frequency peak is negative, and the central peak is bipolar. Such situations are commonly observed in powders, and the spectra are therefore called "powder-pattern spectra".