Phase Problem

The phase problem is the difficulty of reconstructing an object from measured wave intensities when the corresponding phases are unknown. It is central to X-ray crystallography, where a diffraction experiment records the strengths of scattered waves but does not directly record their relative phases. Both quantities are needed to calculate the distribution of electron density within a crystal. Solving the problem therefore requires additional measurements, mathematical constraints, or prior structural information. Closely related reconstruction problems arise in lensless imaging and optics. (journals.iucr.org)

Physical and mathematical basis

X-rays undergo scattering by the electrons in a crystal. Waves scattered from different positions combine through interference, producing reflections whose amplitudes and phases depend on the arrangement of matter. Each reflection is described by a structure factor, a complex number conventionally written

where $\mathbf h=(h,k,l)$ denotes its Miller indices, $|F_{\mathbf h}|$ is its amplitude, and $\phi_{\mathbf h}$ its phase. After appropriate experimental corrections, the measured intensity is proportional to $|F_{\mathbf h}|^2$. The phase does not appear separately in this measurement. (iucr.org)

Electron density and structure factors are related by a Fourier transform. Using fractional coordinates $\mathbf x$, the inverse relation is

where $V$ is the volume of the unit cell. The sum combines waves associated with points in the reciprocal lattice. Their amplitudes determine their strengths, while their phases determine how they align. Incorrect phases can therefore yield an uninterpretable density map even when the amplitudes are accurately measured. (journals.iucr.org)

The problem is an inverse problem, not merely an instrumental precision limit. Better intensity measurements improve the available constraints but do not themselves supply the missing phases. Nevertheless, phases are not wholly independent: the requirement that they describe a physically plausible crystal imposes relationships that reconstruction methods can exploit. (journals.iucr.org)

Information available without phases

A useful phase-independent construction is the Patterson function, obtained by Fourier transformation of squared structure-factor amplitudes:

It represents the autocorrelation of the electron density. Its peaks describe displacement vectors between atoms, rather than their absolute positions. A Patterson map can thus reveal structural relationships without providing an ordinary electron-density image. Overlapping vectors make interpretation difficult for large structures, but prominent contributions from strongly scattering atoms can be especially informative. (iucr.org)

Intensity data also retain some intrinsic ambiguities. Translating an isolated object changes Fourier phases without changing Fourier magnitudes. For a real density in the absence of anomalous-scattering effects, inversion likewise preserves the intensities. Consequently, reconstruction requires an origin convention and may need additional information to distinguish alternative orientations or handedness. (journals.iucr.org)

Principal crystallographic solutions

Direct methods. Direct methods infer phases through mathematical relationships among measured reflections. They exploit properties such as the nonnegative, atom-like character of ordinary electron density and use probabilistic relationships between phase combinations. They are particularly effective for small-molecule crystals with sufficiently high-resolution data; conventional applications to proteins are more demanding. Herbert Hauptman and Jerome Karle received the 1985 Nobel Prize in Chemistry for developing these methods. (iucr.org)

Isomorphous replacement. In isomorphous replacement, diffraction from a native crystal is compared with diffraction from derivatives containing additional heavy atoms while retaining approximately the same underlying structure. Intensity differences help locate those atoms. Their calculated scattering contributions then constrain the phases of the original structure. A single derivative generally leaves a phase ambiguity; multiple derivatives or complementary information can help resolve it. Structural changes between crystals are a major limitation. (journals.iucr.org)

Anomalous diffraction. Anomalous scattering introduces wavelength-dependent contributions that can make opposite-index reflections differ in intensity. Single-wavelength anomalous diffraction, or SAD, uses these differences at one wavelength; multiwavelength anomalous diffraction, or MAD, combines measurements at several wavelengths. Once the anomalously scattering atoms are located, their contributions provide phase constraints. These methods require accurate measurement of relatively small intensity differences and careful treatment of experimental errors. (journals.iucr.org)

Molecular replacement. Molecular replacement uses a sufficiently similar structural model. An algorithm searches for its orientation and position in the target crystal, and the placed model supplies initial calculated phases. Similarity need not imply identity, but substantial errors or missing components can prevent success. Because the initial phases come from a model, subsequent maps can inherit model bias and require critical assessment during structure determination. (journals.iucr.org)

Phase improvement and structural interpretation

Initial phasing normally produces uncertain estimates rather than exact phase values. These uncertainties can be represented by a probability distribution and incorporated into weighted map coefficients. Density modification improves estimates by enforcing expected real-space properties, including relatively uniform solvent regions or similarity between repeated molecular copies. Model building and refinement then alternate between interpreting density and comparing calculated scattering with observations. Agreement with measured amplitudes remains essential, but it does not by itself establish that every structural detail is correct. (journals.iucr.org)

Phase retrieval beyond periodic crystals

In coherent diffraction imaging, an isolated specimen produces a diffraction pattern sampled more densely than ordinary crystal reflections. This oversampling, together with constraints on the specimen’s finite spatial extent, can supply additional reconstruction information. Iterative phase-retrieval methods alternate between Fourier-space amplitude constraints and real-space constraints, retaining or updating estimated phases at each step. (journals.iucr.org)

Successful reconstruction depends on the available constraints and data quality. Noise, missing measurements, inaccurate support estimates, and stagnation in iterative searches can produce artifacts or competing solutions. Oversampling therefore supports phase recovery but is not an unconditional guarantee of a unique, accurate image. (journals.iucr.org)