phases by brute force methods, testing a series of phase values until spherical structures are observed in the resultant electron density map. This works because atoms have a characteristic structure when viewed in the sub-Ångström range. The technique is limited by processing power and data quality. For practical purposes, it is limited to "small molecules" and peptides because they consistently provide high-quality diffraction with very few reflections. Molecular Replacement (MR) Phases can also be inferred by using a process called molecular replacement, where a similar molecule's already-known phases are grafted onto the intensities of the molecule at hand, which are observationally determined. These phases can be obtained experimentally from a homologous molecule or if the phases are known for the same molecule but in a different crystal, by simulating the molecule's packing in the crystal and obtaining theoretical phases. Generally, these techniques are less desirable since they can severely bias the solution of the structure. They are useful, however, for ligand binding studies, or between molecules with small differences and relatively rigid structures (for example derivatizing a small molecule). Isomorphous Replacement Multiple Isomorphous Replacement (MIR) Multiple Isomorphous Replacement (MIR), where heavy atoms are inserted into structure (usually by synthesizing proteins with analogs or by soaking) Anomalous Scattering Single-wavelength Anomalous Dispersion (SAD). Multi-wavelength Anomalous Dispersion (MAD) A powerful solution is the Multi-wavelength Anomalous Dispersion (MAD) method. In this technique, atoms' inner electrons absorb X-rays of particular wavelengths, and reemit the X-rays after a delay, inducing a phase shift in all of the reflections, known as the anomalous dispersion effect. Analysis of this phase shift (which may be different for individual reflections) results in a solution for the phases. Since X-ray fluorescence techniques (like this one) require excitation at very specific wavelengths, it is necessary to use synchrotron radiation when using the MAD method. Phase improvement Refining initial phases In many cases, an initial set of phases are determined, and the electron density map for the diffraction pattern is calculated. Then the map is used to determine portions of the structure, which portions are used to simulate a new set of phases. This new set of phases is known as a refinement. These phases are reapplied to the original amplitudes, and an improved electron density map is derived, from which the structure is corrected. This process is repeated until an error term (usually ) has stabilized to a satisfactory value. Because of