In physics, the phase problem is the problem of loss of information concerning the
phase that can occur when making a physical measurement. The name comes from the field of
X-ray crystallography
X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract in specific directions. By measuring th ...
, where the phase problem has to be solved for the determination of a structure from
diffraction
Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
data.
The phase problem is also met in the fields of
imaging
Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of an image).
Imaging technology is the application of materials and methods to create, preserve, or duplicate images.
...
and
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
.
Various approaches of
phase retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum F(k), of amplitude , F (k), , and phase \psi(k):
::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx
where ''x'' is an ''M''-dimensional spat ...
have been developed over the years.
Overview
Light detectors, such as
photographic plates or
CCDs, measure only the intensity of the light that hits them. This measurement is incomplete (even when neglecting other
degrees of freedom
In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
such as
polarization and
angle of incidence) because a light wave has not only an amplitude (related to the intensity), but also a phase (related to the direction), and polarization which are systematically lost in a measurement.
In
diffraction
Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
or
microscopy
Microscopy is the technical field of using microscopes to view subjects too small to be seen with the naked eye (objects that are not within the resolution range of the normal eye). There are three well-known branches of microscopy: optical mic ...
experiments, the phase part of the wave often contains valuable information on the studied specimen. The phase problem constitutes a fundamental limitation ultimately related to the nature of
measurement in quantum mechanics.
In
X-ray crystallography
X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract in specific directions. By measuring th ...
, the diffraction data when properly assembled gives the amplitude of the 3D
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of the molecule's
electron density
Electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typical ...
in the
unit cell.
If the phases are known, the electron density can be simply obtained by
Fourier synthesis. This Fourier transform relation also holds for two-dimensional far-field
diffraction
Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
patterns (also called
Fraunhofer diffraction) giving rise to a similar type of phase problem.
Phase retrieval
There are several ways to
retrieve the lost phases. The phase problem must be solved in
x-ray crystallography
X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract in specific directions. By measuring th ...
,
neutron crystallography, and
electron crystallography.
Not all of the methods of
phase retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum F(k), of amplitude , F (k), , and phase \psi(k):
::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx
where ''x'' is an ''M''-dimensional spat ...
work with every
wavelength
In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats.
In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
(x-ray, neutron, and electron) used in crystallography.
Direct (''ab initio)'' methods
If the crystal diffracts to high resolution (<1.2 Å), the initial phases can be estimated using direct methods.
Direct methods can be used in
x-ray crystallography
X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract in specific directions. By measuring th ...
,
neutron crystallography,
and
electron crystallography.
A number of initial phases are tested and selected by this method. The other is the Patterson method, which directly determines the positions of heavy atoms. The
Patterson function gives a large value in a position which corresponds to interatomic vectors. This method can be applied only when the crystal contains heavy atoms or when a significant fraction of the structure is already known.
For molecules whose crystals provide reflections in the sub-Ångström range, it is possible to determine 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 the phenomenon of
phase bias, it is possible for an incorrect initial assignment to propagate through successive refinements, so satisfactory conditions for a structure assignment are still a matter of debate. Indeed, some spectacular incorrect assignments have been reported, including a protein where the entire sequence was threaded backwards.
Density modification (phase improvement)
Solvent flattening
Histogram matching
Non-crystallographic symmetry averaging
Partial structure
Phase extension
See also
*
Coherent diffraction imaging
*
Ptychography
*
Phase retrieval
Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum F(k), of amplitude , F (k), , and phase \psi(k):
::F(k) = , F(k), e^ =\int_^ f(x)\ e^\,dx
where ''x'' is an ''M''-dimensional spat ...
External links
An example of phase bias
References
{{DEFAULTSORT:Phase Problem
Crystallography
Inverse problems