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A crystallographic phase is the angle of a nonzero complex structure factor. Writingseparates amplitude and phase for a reflection indexed by the integer Miller index vector $\mathbf{h}$. Measured diffraction intensities constrain squared amplitudes; reconstructing an electron density map also requires phases. This missing information is the phase problem.^[ref]

Using fractional crystallographic coordinates, one consistent Fourier convention isThe phase controls where a harmonic contributes maxima and minima. For example, the original positive profiles $3+2\cos (2\pi x)$ and $3-2\cos (2\pi x)$ have equal Fourier coefficient magnitudes, but opposite phases for their first harmonics. Their squared amplitudes cannot locate the density maxima.

Phase also depends on the coordinate convention. Translating every atomic position by a fractional vector $\mathbf{t}$ multiplies the structure factor by $e^{2\pi i\mathbf{h}\cdot \mathbf{t}}$. For $\mathbf{h}=(1,0,0)$ and $\mathbf{t}=(1/4,0,0)$, the phase increases by $\pi /2$ while the squared amplitude is unchanged. Shifting the coordinate origin by that vector has the opposite sign. A phase is defined modulo $2\pi$; when $F_{\mathbf{h}}=0$, its phase is undefined. Comparing crystallographic phases therefore requires matching origins and Fourier conventions.

An electron density map is a spatial reconstruction of electron density, typically displayed as a grid or contours. In X-ray crystallography, a beam scattered by a crystal produces a diffraction pattern: measured intensities constrain structure factor magnitudes, while crystallographic phases are also required for reconstruction. Coordinates and experimental data are combined in common model-assessment maps.^[ref]

With fractional crystallographic coordinates $\mathbf{r}$, integer Miller index vector $\mathbf{h}$ and unit cell volume $V_c$, an ideal Fourier reconstruction has the formHere the structure factors are complex Fourier coefficients. Finite data and uncertain phases limit the reconstructed detail; this is part of the phase problem, not simply an issue of drawing contours more finely.

An original one-dimensional example makes the phase information visible:Both profiles have the same Fourier coefficient magnitudes, but their maxima occur half a period apart. Squared magnitudes alone cannot distinguish them.

An electron density map is also distinct from a macromolecular structural model fitted to it. A difference electron density map can highlight disagreement between observed and model-predicted scattering. Common map coefficients already incorporate information from the model, so the resulting map is not wholly independent confirmation of every modeling choice.^[ref]

Heavy-atom labeling introduces atoms of relatively high atomic number at ordered crystal sites, giving a strong X-ray scattering contribution so that their signal can supply information for solving its phase problem. In X-ray crystallography, measured intensities constrain the magnitudes of structure factors, while their phases are also needed to reconstruct electron density. A heavy-atom derivative provides an additional measurement linked to a model of the label sites.

Under an ideal single isomorphous replacement model, the derivative keeps the parent structure unchanged apart from the added label contribution. For one reflection, write the complex amplitudes asThe heavy-atom model supplies $F_H$, while measurements supply $|F_P|$ and $|F_{PH}|$. These are vector constraints in the complex plane; subtracting intensity magnitudes does not directly reveal the protein phase.^[ref]

For an original example, take $|F_P|=5$, $|F_H|=3$ and $|F_{PH}|=7$ in consistently scaled amplitude units. If $\Delta \varphi ={\varphi }_P-{\varphi }_H$, thenso $\cos \Delta \varphi =1/2$ and $\Delta \varphi =\pm \pi /3$. The one-derivative measurement leaves two possible phases. In particular, $7-5=2$ is not the heavy-atom amplitude3: amplitudes must be combined with their phases.

Multiple isomorphous replacement adds measurements from further derivatives with consistently referenced label positions. These can distinguish the candidate phases in favorable circumstances.^[ref] Large changes in the crystal or protein would violate the stated parent-plus-label model, so derivatization must be assessed rather than assumed to be structurally harmless. Ordered occupancy and useful diffraction matter alongside the chemical act of adding a heavy atom.

John Cowdery Kendrew helped establish how X-ray crystallography can reveal the three-dimensional arrangement of a protein. Nobel's primary nomination archive names John Cowdery Kendrew, also identifying J. C. Kendrew in its award-sharing comment.^[ref]

His team's1960 myoglobin paper presented a Fourier analysis at2-angstrom resolution. Its opening recalls how earlier phase determination compared native crystals with heavy-atom derivatives containing localized mercury or gold sites. Such comparisons supply information missing from a simple measurement of diffraction intensity.^[ref]

An original one-dimensional example explains that phase problem. Write a periodic electron density using Fourier series coefficients $F_k$. The two positive densitieshave $F_0=3$, $F_{\pm 1}=1/2$ and $F_{\pm 2}=\pm 1/4$, with the final sign distinguishing them. Their squared coefficient magnitudes are identical:9 for mode zero,1/4 for modes $\pm 1$ and1/16 for modes $\pm 2$. The relative phase of the second harmonic has changed while the first harmonic is fixed. Thus equal intensities need not identify one density.

The myoglobin experiment was more demanding than this toy calculation, but the distinction motivates its chemical strategy: modifying crystals at known heavy-atom sites changes their scattering in a way that constrains phase information. Reconstructing an electron density map therefore requires both measured diffraction and a justified phase-determination method, rather than treating the recorded spots as a direct photograph of the molecule.

Max Perutz, fully Max Ferdinand Perutz, investigated protein structure through X-ray diffraction. The Nobel nomination archive explicitly pairs his full name with the initials M. F. Perutz and records his 1962 Chemistry prize.^[ref] His structural research connected the organization of protein atoms with the behavior of oxygen-binding hemoglobin.

With J. A. Frier, Perutz coauthored the structure of human fetal deoxyhemoglobin. The associated primary record, 1FDH, identifies alpha and gamma chains and a diffraction resolution of $2.50\mathring{\mathrm{A}}$.^[ref] The study relates structural differences to interaction with 2,3-Bisphosphoglycerate. A structural explanation is strongest when its proposed interaction can also be tested through functional measurements.

An original diffraction example shows why obtaining a molecular structure is not equivalent to photographing atoms directly. In a reduced model, a complex scattering amplitude iswhere $x_j$ are positions and $f_j$ scattering strengths. Measured intensity is proportional to $|F(q){|}^2$. Translating every atom by $x_0$ multiplies $F$ by $e^{iq{x}_0}$ without changing that intensity. Intensities therefore do not by themselves fix every phase or coordinate convention. Recovering an electron density model requires appropriate phase information and structural constraints, connecting the measurement with the phase problem.

The fetal structure also illustrates a separate limitation: identifying a deoxygenated conformation does not directly measure the entire oxygen-binding curve. Structural comparisons, ligand-binding experiments and quantitative analysis supply complementary evidence. Perutz's contribution lies in connecting those molecular questions with experimentally determined three-dimensional organization.

To determine a crystal structure, scatter probes whose wavelength is comparable to atomic spacings, typically $0.1\mathrm{nm}$. X-ray diffraction, neutron diffraction and electron diffraction probe periodic order through coherent elastic scattering. With incident and outgoing wavevectors ${\mathbf{k}}_i,{\mathbf{k}}_f$, define $\mathbf{Q}={\mathbf{k}}_i-{\mathbf{k}}_f$. Elastic scattering has $|{\mathbf{k}}_i|=|{\mathbf{k}}_f|$ and constructive interference occurs when $\mathbf{Q}=\mathbf{G}$, a reciprocal lattice vector. The equivalent Bragg's law iswhere the detector scattering angle is $2\theta$, not $\theta$. Peak positions reveal plane spacings and the unit cell; systematic absent reflections constrain centring and the atomic basis.

For X-ray diffraction, a laboratory tube or a synchrotron supplies a collimated beam. In a monochromatic single-crystal experiment a goniometer rotates the specimen and an area detector records reflection positions and integrated wave intensities. A powder experiment uses many randomly oriented crystallites; the reflections form cones recorded as rings or peaks versus $2\theta$, permitting indexing and determination of lattice constants without selecting one crystallite. For a cubic cell, for example, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. The white-beam Laue method is especially useful for crystal orientation.

The crystal basis structure factor isafter geometric and thermal corrections, including the Debye–Waller factor. For X-rays, $f_j$ describes scattering from the atomic electron density. Thus positions provide lattice geometry while relative wave intensities constrain atomic positions and species. Wave intensities do not directly give the phases of $F_{\mathbf{G}}$: structure refinement or other constraints must address the phase problem.

Neutron diffraction uses a reactor or spallation source, wavelength selection and a detector bank or time-of-flight detection. Neutrons interact with nuclei through nuclear scattering lengths that vary between elements and isotopes rather than simply with atomic number. This makes them useful for light atoms, isotope contrast and distinguishing some neighbours in the periodic table; their magnetic interaction also probes magnetic order. Electron diffraction in a transmission electron microscope gives strong signals from very small thin crystallites, although strong multiple scattering often complicates quantitative wave intensity analysis. These techniques complement one another.

A phonon dispersion relation requires both momentum and energy transfer, so elastic diffraction alone is insufficient. In coherent inelastic neutron scattering, the measured transfers for phonon creation areHere $\mathbf{q}$ is the phonon wavevector reduced into a Brillouin zone, and $s$ labels an acoustic phonon branch or optical phonon branch. Negative energy transfer corresponds to annihilating an already occupied phonon. For neutrons $E={\hslash }^2{k}^2/(2m_n)$, so wavelength and energy selection are linked. Typical thermal-neutron energies of a few to tens of millielectronvolts ($1\mathrm{meV}=10^{-3}$ electronvolts) are well matched to many lattice vibrations.

A triple-axis spectrometer uses a monochromator crystal to select ${\mathbf{k}}_i$, the sample as the second scattering axis, and an analyser crystal to select ${\mathbf{k}}_f$ before detection. Rotate these elements and the specimen to hold $\mathbf{Q}$ fixed while scanning $E_i-E_f$; inelastic peaks locate ${\omega }_s(\mathbf{q})$. Repeat at different $\mathbf{q}$ along symmetry directions to map the phonon dispersion relation. The dynamic structure factor governs peak weights, including the selection of phonon polarization by the scattering geometry. A time-of-flight spectrometer instead infers outgoing neutron energy from travel time over a known flight path and records many directions simultaneously.

Inelastic X-ray scattering can also resolve phonons when very high energy resolution is available. Optical Raman scattering and Brillouin scattering measure vibrations near the zone centre because photon momentum transfers are small compared with a typical reciprocal lattice spacing; they complement, rather than generally map the entire zone like, momentum-resolved neutron measurements. Ultrasound measurements give the long-wavelength sound speed, the initial slope $d\omega /dq$ of an acoustic phonon branch.

The phase problem is the loss of the wave phases of the crystal basis structure factors when an experiment records their squared magnitudes. Write $F_{hkl}=|F_{hkl}|e^{i{\varphi }_{hkl}}$. A Fourier series reconstruction of a periodic electron density uses both $|F_{hkl}|$ and ${\varphi }_{hkl}$, whereas an ordinary wave intensity measurement supplies only $|F_{hkl}{|}^2$. Estimating the missing phases therefore needs additional constraints or measurements.^[ref]

There is also an unavoidable choice of origin. Translating all motif positions by the same fractional crystal coordinate vector $t$ multiplies $F_{hkl}$ by $e^{2\pi i(h,k,l)\cdot t}$ without changing $|F_{hkl}{|}^2$. The physical arrangement is unchanged. A reconstructed coordinate list must therefore distinguish a coordinate convention from information actually recovered from the measurements.

X-ray crystallographic structure inference is an inverse problem: infer a crystal structure from measured X-ray diffraction. A proposed unit cell and atomic basis of a crystal predict a pattern; structure determination asks which proposals survive comparison with that pattern. The useful separation is between directions and strengths. Peak positions constrain the reciprocal lattice and hence interplanar spacings; relative wave intensities constrain the motif through the crystal basis structure factor. Knowing a repeat distance does not locate all the chemical atoms within that repeat.

The lattice constant from two indexed reflections example isolates the geometric inference. Basis intensities and phase ambiguity then holds the translations fixed while changing the motif. Even complete, noiseless wave intensity data need not determine all wave phases: this is the phase problem. Chemical composition, symmetry, and additional measurements can supply constraints beyond the original pattern.^[ref]