Experiment: Neutron Scattering and the Structure of Solids
- Why the neutron, and not the X-ray
- Shull and Smart: antiferromagnetic order in MnO (1949)
- Brockhouse and Stewart: phonons in aluminium (1955)
- Brockhouse: spin waves in magnetite (1957)
- Brockhouse and Iyengar: the normal modes of germanium (1958)
- The Debye–Waller factor, measured on the same patterns
- What the neutron settled
- Hodgkin, Hauptman and Karle, and Klug: the phase problem (1934–1969)
Phonons and Lattice Dynamics builds the thermal physics of a crystal on a function it never measures. The heat capacity, the thermal expansion and the thermal conductivity all follow from the dispersion relation \(\omega(\vect{k})\) of the lattice waves, and the classical route to that function — fitting a Debye temperature to a measured heat capacity — cannot recover it, because the heat capacity depends only on the density of modes \(g(\omega)\) and many different dispersion relations share one \(g(\omega)\). Magnetism in Matter is in the same position twice over. Néel's antiferromagnet is an arrangement of moments that cancel exactly, so that no magnetometer can see the order at all [Neel:1936]; Bloch's spin waves are collective excitations whose dispersion fixes the exponent of the \(T^{3/2}\) law but which no magnetization curve displays directly [Bloch:1930b]. Three predictions, all of them about the interior of a solid at the scale of the interatomic spacing, and none of them within reach of any instrument before about 1945.
What closed all three was a particle found for quite other reasons [Chadwick:1932a] and made available in quantity, a decade later, by the fission reactors of Nuclear Forces and Nuclear Structure. A neutron that has come to thermal equilibrium with a moderator at room temperature has, by an accident of its mass, a de Broglie wavelength of the order of an interatomic spacing and a kinetic energy of the order of a lattice vibration. No other probe has both at once. This chapter reports four experiments that exploited the coincidence: Shull and Smart's diffraction pattern of manganese oxide, which made a magnetic structure visible for the first time; Brockhouse and Stewart's scattering of neutrons by the phonons of an aluminium crystal, the measurement that turned lattice dynamics from a theory into a data set; Brockhouse's spin-wave scattering in magnetite, which did the same for magnons; and the branch-by-branch dispersion of germanium, taken with the triple-axis spectrometer in its mature form. It then shows that the same patterns deliver the Debye–Waller factor of Phonons and Lattice Dynamics as a by-product, and closes with what the technique does and does not settle. The Nobel prize of 1994 went to Brockhouse [Brockhouse:1995] and to Shull [Shull:1995] for the two halves of this work.
Why the neutron, and not the X-ray
The case for the neutron rests on four of its properties, and the decisive one is arithmetic.
A neutron of kinetic energy \(E\) has, non-relativistically, \(p=\sqrt{2m_{n}E}\) and therefore
Put \(E=25\,\mathrm{meV}\) into Equation (132.1), the mean kinetic energy scale of a gas of neutrons in equilibrium with a moderator near room temperature: the speed is about \(2.2\,\mathrm{km}/\mathrm{s}\) and the wavelength about \(0.18\,\mathrm{nm}\). The wavelength resolves the interatomic spacing, so the neutron diffracts from a crystal exactly as an X-ray does. The energy is of the same order as the quanta the crystal has to offer, which run from about \(1\,\mathrm{meV}\) to about \(100\,\mathrm{meV}\): a phonon or a magnon can take a large fraction of the neutron's whole energy, and the change is measurable with an ordinary crystal analyser.
For any other probe of the same wavelength the second condition fails catastrophically. Table 132.1 makes the comparison. An X-ray photon of wavelength \(0.18\,\mathrm{nm}\) carries about \(6.9\,\mathrm{keV}\), so a \(25\,\mathrm{meV}\) energy transfer is a part in \(3\times 10^{5}\) of it — and it must be resolved twice over, once in the incident beam and once in the scattered one. That measurement is possible, but it needed a synchrotron and a generation of crystal-optics development [Burkel:2000]; it was not available in 1955 and is still not the routine instrument.
| Probe | Energy at $\lambda=0.18\,\mathrm{nm}$ | \(25\,\mathrm{meV}\) as a fraction |
|---|---|---|
| Neutron | \(25\,\mathrm{meV}\) | $1$ |
| Electron | \(46\,\mathrm{eV}\) | \(5\times 10^{-4}\) |
| X-ray photon | \(6.9\,\mathrm{keV}\) | \(4\times 10^{-6}\) |
Three further properties matter, and each of them is used by one of the experiments below.
The neutron is uncharged. It feels no Coulomb barrier and no electron cloud, so it penetrates centimetres of most materials rather than micrometres: the sample can sit inside a cryostat, a furnace or a pressure cell, and what is measured is the bulk rather than the surface. The electron of Table 132.1 fails on this count as well as on the energy count.
The neutron scatters from nuclei, not from electron clouds. A nucleus is of order \(10^{-15}\text{–}10^{-14}\,\mathrm{m}\) across, four to five orders of magnitude smaller than the wavelength, so it is a point scatterer: the nuclear scattering length \(b\) is a single number, independent of the scattering angle. There is no form factor to fall away at large momentum transfer, which is what makes the high-order reflections strong enough for the Debye–Waller measurement of Section 132.6. And \(b\) varies erratically from element to element and even between isotopes of one element, instead of growing smoothly with atomic number as the X-ray amplitude does. Hydrogen is therefore visible next to uranium, and neighbouring elements in the periodic table can be told apart [Squires:1978].
The neutron carries a magnetic moment while carrying no charge. Its dipole couples to the moments of unpaired electrons, with an amplitude of the same order as the nuclear scattering length, so magnetic order produces diffraction peaks of ordinary strength [Halpern:1939]. This is the property that made Section 132.2 possible, and it is the one no other structural probe has.
Shull and Smart: antiferromagnetic order in MnO (1949)
Until 1949, antiferromagnetism was a hypothesis about something no instrument could detect. Néel's two-sublattice molecular field [Neel:1936] predicted a state whose sublattice moments cancel, so that the susceptibility rises on cooling, peaks at the ordering temperature and then falls, and so that the substance is magnetically almost inert below it. Every one of those signatures is indirect: they are consistent with the two-sublattice picture but do not exhibit it. The neutron exhibits it.
Apparatus
A beam from the graphite reactor at Oak Ridge, monochromated by Bragg reflection from a single crystal so that a narrow band of wavelengths near \(0.1\,\mathrm{nm}\) enters the specimen, with Soller collimators and cadmium and boron shielding defining the beam. The specimen is a powder of manganese oxide in a thin-walled container mounted in a cryostat that can be held at liquid-nitrogen temperature or allowed to warm through the transition. The detector is a boron trifluoride proportional counter carried on an arm that rotates about the specimen axis, recording counts against scattering angle [Wollan:1948] [Shull:1949] [Shull:1951].
Procedure
The powder pattern is recorded twice, once with the specimen near \(80\,\mathrm{K}\) and once at room temperature, and the two are subtracted. The logic of the comparison is what makes it decisive, and it rests on Section 132.6: the nuclear scattering is temperature-dependent only through the Debye–Waller factor, which weakens existing reflections and creates none. Any line that is present in the cold pattern and absent from the warm one therefore cannot be nuclear. The transition temperature is then located by following the intensity of the new lines as the specimen warms, and compared with the temperature of the susceptibility maximum measured on the same material by ordinary magnetometry.
Observations and data
At room temperature the pattern indexes completely on the face-centred-cubic rock-salt cell of manganese oxide, with the reflection conditions that cell requires. On cooling below the transition, additional reflections appear at scattering angles lying between the nuclear lines. They index on a cubic cell of twice the edge — equivalently, at half-integral indices of the chemical cell — and their intensities fall away with increasing scattering angle markedly faster than the nuclear lines do. They vanish on warming through a transition near \(120\,\mathrm{K}\), where the susceptibility has its maximum [Shull:1949] [Shull:1951].
A substance whose magnetic moments order with zero net moment scatters neutrons into reflections that are absent above the ordering temperature and absent from its X-ray pattern at every temperature. In manganese oxide these extra reflections index on a magnetic unit cell of twice the chemical edge, appear below about \(120\,\mathrm{K}\), and fall off with momentum transfer faster than the nuclear reflections [Shull:1949] [Shull:1951]. Their positions record the periodicity of the magnetic arrangement, their intensities its moment size, and their disappearance the ordering temperature — three quantities no magnetometer can supply, because the net moment they describe is zero.
Derivation. Let the magnetic ions occupy a Bravais lattice with primitive translations \(\vect{a}_{i}\), and let the moment at the site \(\vect{R}\) be \(\sigma(\vect{R})\,\vect{m}\) with \(\sigma=\pm1\), reversing under each primitive translation. The neutron's dipole couples to the moment, so that site scatters with amplitude \(p\,\sigma(\vect{R})\), where \(p\) carries the magnetic form factor and the geometrical factor but not the sign; the amplitude scattered by the crystal into momentum transfer \(\vect{Q}\) is
Any such sign function can be written \(\sigma(\vect{R})=\ee^{\ii\vect{k}_{M}\cdot\vect{R}}\) for a propagation vector \(\vect{k}_{M}\) obeying \(\vect{k}_{M}\cdot\vect{a}_{i}=\pi\) modulo \(2\pi\) for every \(i\); that is, \(2\vect{k}_{M}\) is a reciprocal-lattice vector of the chemical lattice while \(\vect{k}_{M}\) itself is not. Substituting into Equation (132.2),
which is the ordinary lattice sum and vanishes unless \(\vect{Q}+\vect{k}_{M}=\vect{G}\) for some reciprocal-lattice vector \(\vect{G}\) of the chemical lattice. Magnetic intensity therefore appears at \(\vect{Q}=\vect{G}-\vect{k}_{M}\) and nowhere else: displaced from every nuclear reflection by \(\vect{k}_{M}\), hence at half-integral indices, hence on the reciprocal lattice of a real-space cell of twice the edge. The two things the observation reports are one thing: the extra reflections at half-integral indices are the doubling of the magnetic period, and their absence from the nuclear pattern is guaranteed because \(\vect{k}_{M}\) is not itself a reciprocal-lattice vector. In manganese oxide the magnetic ions form a face-centred cubic lattice and \(\vect{k}_{M}=(\pi/a)(1,1,1)\), which reverses the sign across each successive \((111)\) sheet: the moments lie ferromagnetically aligned within a sheet and antiparallel between neighbouring sheets, which is exactly Néel's two sublattices. The faster falloff of the magnetic intensities with angle is the form factor inside \(p\), and is the second, independent signature that the reflections are magnetic: the nuclear amplitude has no form factor at all, as Section 132.1 explains, while the magnetic amplitude is the Fourier transform of an electron distribution of atomic size.
∎Interpretation
The reflections are the two-sublattice structure itself, seen directly. A hypothesis proposed to account for a susceptibility curve [Neel:1936] became a measured atomic arrangement, and the systematic determination of magnetic structures — moment directions, moment magnitudes, propagation vectors — began at once [Shull:1951]. It should be said what the measurement does not fix. A powder pattern from a cubic crystal averages over the orientations of the magnetic domains, and the moment direction within the ordered sheets is not determined uniquely by the intensities alone; the propagation vector and the moment magnitude are. The technique also says nothing about why the manganese ions couple antiparallel through the intervening oxygen, which is the separate question of superexchange treated in Magnetism in Matter.
Primary references
[Shull:1949], with the systematic follow-up [Shull:1951]. The prediction under test is [Neel:1936].
Brockhouse and Stewart: phonons in aluminium (1955)
Diffraction alone locates atoms; it says nothing about how they move. The step from the elastic experiment above to a measurement of \(\omega(\vect{k})\) requires the energy of the scattered neutron to be measured as well as its direction, and it was first taken in a single crystal of aluminium.
Apparatus
A beam from the Chalk River reactor, monochromated by Bragg reflection from a single crystal so that the neutrons entering the specimen have a known energy \(E_{i}\) and wavevector \(\vect{k}_{i}\). The specimen is a single crystal of aluminium — face-centred cubic, one atom in the primitive cell — mounted on a goniometer that sets its orientation relative to the incident beam. The scattered beam is analysed in energy by a second Bragg reflection from a crystal before reaching the counter, so that the instrument records the number of neutrons arriving with a chosen final energy \(E_{f}\) in a chosen direction. This arrangement, monochromator–specimen–analyser, is the triple-axis spectrometer in its first form; Section 132.5 uses it in its mature one.
Procedure
Fix the incident energy and the scattering angle and record the distribution of final energies. Each pair of readings converts, by the kinematics of Equation (132.4) below, into a momentum transfer \(\vect{Q}=\vect{k}_{i}-\vect{k}_{f}\) and an energy transfer \(\hbar\omega=E_{i}-E_{f}\). Repeating over a range of angles and crystal orientations samples a region of the \((\vect{Q},\omega)\) plane. The single crystal is essential: a powder averages over orientations and destroys the direction of \(\vect{Q}\), which is exactly the information a dispersion relation is about.
Observations and data
The distribution of scattered energies at a fixed geometry is not smooth. It carries resolved peaks standing above a background, and the \((\vect{q},\omega)\) pairs deduced from the peak positions do not fill the plane: they fall on curves. In aluminium the curves are three in number, they rise linearly out of the origin with slopes that agree with the sound velocities computed from the elastic constants of the same metal, and they flatten towards the zone boundary, so the group velocity \(\dd\omega/\dd q\) falls to zero there [Brockhouse:1955]. The measured frequencies of the primary paper are not reproduced here; what is quoted is the shape of the result, which is what the theory predicts.
Neutrons scattered by a single crystal into a fixed momentum transfer \(\vect{Q}\) do not emerge with a continuum of energies. The spectrum carries sharp peaks at a small number of energy transfers, and the pairs \((\vect{q},\omega)\) they define lie on a small number of continuous branches that are periodic in the reciprocal lattice [Brockhouse:1955] [Brockhouse:1958]. The excitations of a crystal are therefore quanta with a definite energy for each wavevector, and the dispersion relation is read off the instrument rather than postulated.
Derivation. A crystal is invariant under the discrete translations of its lattice and not under continuous ones, so what the collision conserves is the neutron momentum modulo \(\hbar\vect{G}\), with \(\vect{G}\) any reciprocal-lattice vector. Writing \(\vect{Q}=\vect{k}_{i}-\vect{k}_{f}\) and \(\hbar\omega=E_{i}-E_{f}\), a process that creates one quantum of the branch \(s\) at wavevector \(\vect{q}\) obeys
The first of these fixes \(\vect{q}\) once \(\vect{Q}\) is chosen, since \(\vect{q}\) is defined only modulo \(\vect{G}\); the second then leaves for the energy transfer only the \(3p\) values \(\hbar\omega_{s}(\vect{q})\), one for each branch. A discrete set of allowed energy transfers at fixed \(\vect{Q}\) is a set of peaks in the spectrum of scattered neutrons, which is the observation. Conversely one peak fixes one pair \((\vect{q},\omega)\), so a scan is a set of points on \(\omega_{s}(\vect{q})\). Two features of the data follow without further work. The branches are periodic in the reciprocal lattice because \(\vect{q}\) enters Equation (132.4) only modulo \(\vect{G}\), so nothing outside the first Brillouin zone is new; and the smooth background beneath the peaks is not instrumental error but multiphonon scattering, in which two or more quanta share the transfer, so that Equation (132.4) constrains only their sum and the allowed \(\omega\) fill an interval.
∎The one-phonon scattering cross section itself: the factor that weights each peak by the square of the projection of the polarization vector on the momentum transfer, by the Bose population of the mode, and by the Debye–Waller factor, so that the measured intensities constrain the eigenvectors of the dynamical matrix and not only its eigenvalues
Interpretation
Before this measurement the vibrational spectrum of a solid was known only through quantities that integrate over it. A heat capacity determines the density of modes \(g(\omega)\), and infinitely many dispersion relations share one \(g(\omega)\); a Debye temperature is a one-parameter caricature of the whole function. Diffuse X-ray scattering carries the information in principle but requires an assumed force model to extract it. The neutron measurement is direct in the strict sense: one peak, one point of \(\omega(\vect{q})\), no model. That is what entitles Phonons and Lattice Dynamics to speak of the phonon as a particle — the sharpness of the peak is the statement that a wavevector has a definite energy — and it is what makes the Born–von Kármán force constants a fitted, falsifiable object rather than a parametrization.
Primary references
[Brockhouse:1955]. The theoretical frame is the lattice dynamics of [Born:1912] [Born:1954].
Brockhouse: spin waves in magnetite (1957)
The same instrument, pointed at a magnetically ordered crystal, does for Magnetism in Matter what it had just done for Phonons and Lattice Dynamics. Bloch's account of the fall of the magnetization with temperature rests on the claim that the cheap excitations of an ordered magnet are long-wavelength precessional waves rather than individually reversed spins [Bloch:1930b]. The two pictures differ qualitatively — a reversed spin costs a finite energy and would give an exponential law, a spin wave costs arbitrarily little and gives a power law — so the claim is testable by the magnetization curve alone. Seeing the excitation itself is another matter.
Apparatus
The triple-axis arrangement of Section 132.3, with a single crystal of magnetite, \(\mathrm{Fe}_{3}\mathrm{O}_{4}\), as the specimen, at room temperature — far below its ordering temperature, so the ordered state is well developed and the magnon population is small enough for the excitations to be sharp.
Procedure
Scans are taken at small wavevector transfer relative to a reciprocal-lattice point, where a magnon branch that starts from zero is well separated from the phonon branches. Three independent signatures distinguish a magnetic peak from a lattice one, and the identification uses all three. The intensity of magnetic scattering falls with \(\abs{\vect{Q}}\) through the magnetic form factor, whereas one-phonon intensity grows as \(\abs{\vect{Q}}^{2}\), so repeating a scan near a high-order reciprocal-lattice point separates them. The dispersion of a ferromagnetic magnon is quadratic at long wavelength while an acoustic phonon's is linear. And the magnon intensity and dispersion respond to temperature and to an applied field as the lattice modes do not.
Observations and data
A branch is resolved whose energy rises from zero quadratically with wavevector at long wavelength, distinct from the phonon branches of the same crystal, and whose intensity falls with momentum transfer in the manner of a magnetic form factor [Brockhouse:1957]. The fitted coefficient of \(q^{2}\) is the spin-wave stiffness \(D\), the quantity that fixes the coefficient of the \(T^{3/2}\) law in Magnetism in Matter and, through the derivation below, the exchange constant. The value obtained is not quoted here.
Magnetite is a ferrimagnet rather than a simple ferromagnet: it has two inequivalent magnetic sublattices, so its spin-wave spectrum has an acoustic branch, which starts at zero and is quadratic at long wavelength like a ferromagnet's, together with optical branches at higher energy. The branch measured at small wavevector is the acoustic one, and it is the acoustic branch alone that the \(T^{3/2}\) law needs.
An ordered magnet carries collective excitations that inelastic neutron scattering resolves as sharp peaks, on a branch whose energy vanishes as the wavevector goes to zero and rises as \(\hbar\omega=Dq^{2}\) at long wavelength [Brockhouse:1957]. The excitation spectrum of a ferromagnet therefore has no gap, which is what supplies the arbitrarily cheap excitations that the \(T^{3/2}\) law of [Bloch:1930b] requires, and the stiffness \(D\) measured here must equal the one fitted to the magnetization curve of the same material.
Derivation. Take the Heisenberg model of Magnetism in Matter on a simple cubic lattice of spacing \(a\), written with each bond counted once,
and treat the spins as classical vectors of length \(S\) — legitimate for the long-wavelength modes, in which neighbouring spins are almost parallel and the quantum corrections are of relative order \(1/S\). Each spin precesses in the field of its neighbours,
the sum running over the \(z=6\) nearest neighbours. Linearize about the ordered state by writing \(\vect{S}_{n}=\left(S^{x}_{n},S^{y}_{n},S\right)\) with the transverse components small, and keep terms of first order in them:
The pair decouples in the combination \(S^{+}=S^{x}+\ii S^{y}\), giving \(\hbar\,\dd S^{+}_{n}/\dd t =-\ii JS\left(zS^{+}_{n}-\sum_{\vect{\delta}} S^{+}_{n+\vect{\delta}}\right)\), and the lattice translation invariance of that equation makes plane waves \(S^{+}_{n}\propto\ee^{\ii(\vect{q}\cdot\vect{R}_{n}-\omega t)}\) its normal modes. Substituting them,
Two consequences are the content of the experiment. First, \(\omega\to0\) as \(\vect{q}\to0\): the uniform mode is a rigid rotation of every spin together, which costs nothing because Equation (132.5) depends only on relative orientations. The gaplessness is thus a consequence of the rotational symmetry that the ordered state breaks, not of any detail of \(J\), and it is what makes the low-temperature magnetization deficit a power of \(T\) rather than an exponential. Second, expanding Equation (132.9) for \(\abs{\vect{q}}a\ll1\) on the cubic lattice, where the six neighbours contribute in pairs along the three axes,
which is the measured shape and which converts the measured stiffness into the exchange constant. The numerical factor in \(D\) depends on the convention for Equation (132.5): writing the Hamiltonian as an unrestricted double sum over pairs, as Magnetism in Matter does, counts each bond twice and doubles \(D\) at fixed \(J\). What is convention-independent is the shape \(\hbar\omega\propto q^{2}\) and the vanishing at \(\vect{q}=0\), and those are what the scan measures.
∎Interpretation
Two independent measurements on the same material must now agree: the stiffness \(D\) read off the neutron dispersion and the stiffness implied by the coefficient of \(T^{3/2}\) in the magnetization curve. That is a closed loop of the kind an evidence-based treatise should insist on, because either measurement alone can be accommodated by adjusting a parameter and the pair cannot. The experiment also settles the question Bloch's calculation raised and could not answer from thermodynamics alone: the excitations are collective waves with a gapless spectrum, and not reversed individual spins, which would have shown a finite minimum energy in exactly the scan performed here.
Primary references
[Brockhouse:1957]. The prediction under test is [Bloch:1930b], with the operator formulation in [Holstein:1940].
Brockhouse and Iyengar: the normal modes of germanium (1958)
The aluminium measurement showed that a point of \(\omega(\vect{q})\) can be taken. Germanium showed that a whole dispersion relation can be taken — every branch, along the symmetry directions, in a crystal with more than one atom per cell — and it is the measurement against which lattice-dynamical models have been tested ever since.
Apparatus
The triple-axis crystal spectrometer at Chalk River, in its mature form: three axes of rotation, one at the monochromating crystal, which selects \(\vect{k}_{i}\) out of the reactor's broad spectrum; one at the specimen, which sets its orientation and hence the direction of \(\vect{Q}\) in the crystal; and one at the analysing crystal, which selects \(\vect{k}_{f}\) before the counter. The specimen is a single crystal of germanium, whose diamond structure has \(p=2\) atoms in the primitive cell.
Procedure
The instrument's three angles and two crystal settings give more freedom than the measurement needs, and Brockhouse's use of that freedom is the methodological content of the experiment. In the constant-\(\vect{Q}\) method the incident and final energies are stepped together, with the arm angles moved to compensate, so that the momentum transfer stays fixed while the energy transfer is scanned. Each scan is then a cut through the \((\vect{Q},\omega)\) plane along one axis, and a branch appears as a peak at a definite energy rather than as something to be reconstructed from a locus of intersections. Scans are repeated along the \([100]\), \([110]\) and \([111]\) directions, and the peak positions are collected into \(\omega_{s}(\vect{q})\) branch by branch.
Observations and data
Germanium shows six branches, as \(3p=6\) requires. Three are acoustic: they vanish at the zone centre, rise linearly there with slopes that agree with the sound velocities computed from the elastic constants of the same crystal, and flatten towards the zone boundary. Three are optical, with a finite frequency at \(\vect{q}=0\); branches degenerate along a symmetry direction separate away from it, as the symmetry of the direction requires. The complete set was compared with Born–von Kármán force-constant fits, and the fits require interactions extending beyond nearest neighbours [Brockhouse:1958].
The measured dispersion relation of a crystal consists of \(3p\) branches \(\omega_{s}(\vect{q})\), where \(p\) is the number of atoms in the primitive cell; three of them vanish as \(\vect{q}\to0\) and the remaining \(3p-3\) do not. Germanium, with \(p=2\), shows six branches, three acoustic and three optical [Brockhouse:1958]; aluminium, with \(p=1\), shows three and they are all acoustic [Brockhouse:1955].
Derivation. Let \(u_{\alpha}(\vect{R}\kappa)\) be the Cartesian displacement of the atom \(\kappa\) of mass \(M_{\kappa}\) in the cell at \(\vect{R}\). In the harmonic approximation the equations of motion are linear with coefficients depending on \(\vect{R}-\vect{R}'\) alone, so they are diagonalized by the Bloch form
and reduce, for each \(\vect{q}\) separately, to the eigenvalue problem
for the dynamical matrix \(D\), which is Hermitian because the force constants are symmetric, and which is \(3p\times3p\) because the indices run over three Cartesian directions and \(p\) atoms. A Hermitian \(3p\times3p\) matrix has \(3p\) real eigenvalues, so there are \(3p\) branches; and \(D\) is built from \(\ee^{\ii\vect{q}\cdot\vect{R}}\) with \(\vect{R}\) on the lattice, so the branches are periodic in the reciprocal lattice.
Three of them vanish at \(\vect{q}=0\). A rigid translation moves every atom of every cell by the same vector \(\vect{c}\); it leaves every interatomic distance unchanged, so it costs no potential energy. It is of the form Equation (132.11) with \(\vect{q}=0\) and \(e_{\alpha}(\kappa)=\sqrt{M_{\kappa}}\,c_{\alpha}\), hence an eigenvector of \(D(\vect{0})\) with eigenvalue zero; there are three independent choices of \(\vect{c}\), so three branches with \(\omega\to0\) as \(\vect{q}\to0\). The remaining \(3p-3\) modes move atoms within a cell against one another and have no reason to be free, so they start at a finite frequency. Nothing in the argument uses the values of the force constants: the count is a consequence of translational invariance and of the size of the cell, which is why it is a test of the framework rather than a fit to the data. Germanium must show six branches with three acoustic, and does.
∎Interpretation
Two things are established at once, and they are of different kinds. The first is structural and survives any change of model: the number of branches and the three zero modes at the zone centre, which is the content of Phenomenon 132.4. The second is quantitative and model-dependent: the shapes of the branches determine the force constants, and in germanium they cannot be fitted with nearest-neighbour forces alone. That is a genuine result about the bonding of a covalent semiconductor, obtained from a mechanical measurement, and it is the sort of thing that only a full dispersion relation can deliver — a density of states, and therefore a heat capacity, is insensitive to it. With this experiment the function on which Phonons and Lattice Dynamics rests stopped being an assumption.
Primary references
[Brockhouse:1958], following [Brockhouse:1955]; the model it is fitted against is that of [Born:1912] [Born:1954].
The Debye–Waller factor, measured on the same patterns
The diffraction patterns of Section 132.2 carry, without any extra apparatus, the measurement that Phonons and Lattice Dynamics needs to close its account of thermal motion. It is set down here because the procedure of that experiment leans on it, and because the neutron measures it more cleanly than the X-ray does: the nuclear scattering length has no form factor, so the whole falloff of intensity with momentum transfer at fixed temperature is thermal, with nothing to disentangle.
The intensity of a Bragg reflection falls as the crystal is heated, by a factor that grows with the momentum transfer; the reflection stays at the same angle and keeps its width. The intensity lost from the peaks reappears as a diffuse background rather than disappearing. And the attenuation does not extrapolate to unity at zero temperature: a harmonic crystal at absolute zero still attenuates its own reflections [Debye:1913] [Waller:1923]. The same factor governs the recoilless fraction in nuclear resonance absorption [Moessbauer:1958].
Derivation. Let the nucleus \(n\) sit at \(\vect{R}_{n}+\vect{u}_{n}\), with \(\vect{R}_{n}\) its equilibrium position and \(\vect{u}_{n}\) its thermal displacement, and let \(b_{n}\) be its scattering length. The Bragg peaks come from the part of the scattering that is coherent with the average lattice, that is from the thermal average of the amplitude:
In a harmonic crystal \(\vect{u}_{n}\) is a linear combination of the normal-mode coordinates, each of which is Gaussian in thermal equilibrium, so \(x=\vect{Q}\cdot\vect{u}_{n}\) is a Gaussian variable of zero mean. For such a variable \(\avg{\ee^{\ii x}}=\ee^{-\avg{x^{2}}/2}\), whence
and the measured intensity, being the squared modulus of the amplitude, carries the factor \(\ee^{-2W}\).
Each of the three observed features is now a line of the calculation. The factor \(\ee^{-W}\) multiplies the lattice sum in Equation (132.14) without altering it, so the peak positions and widths are untouched: the average lattice is still perfectly periodic, and it is the average lattice that the Bragg peaks see. Nothing is destroyed, because the total scattering from a set of nuclei is fixed by their scattering lengths and cannot depend on temperature; what leaves the peaks reappears in the one- and multi-phonon scattering that fills the space between them, which is the same background beneath the peaks of Phenomenon 132.2. And \(\avg{u^{2}}\) does not vanish at \(T=0\): each normal mode retains its zero-point amplitude, so \(W>0\) at absolute zero. That last term is Waller's correction [Waller:1923] to Debye's classical calculation [Debye:1913], and it is a direct consequence of the quantization that Phonons and Lattice Dynamics imposes on the modes. At high temperature, by contrast, equipartition makes \(\avg{u^{2}}\) proportional to \(T\), so \(\ln I\) is linear in temperature with a slope proportional to \(\abs{\vect{Q}}^{2}\) — the form in which the factor is fitted, and from which the mean square displacement of an atom in a crystal is extracted.
∎The mode sum for the mean square displacement, giving \(2W\) in terms of the measured dispersion relation, its evaluation in the Debye model with the zero-point term separated from the thermal one, and the resulting temperature dependence of the attenuation at fixed momentum transfer
What the neutron settled
| Experiment | Tests | Result |
|---|---|---|
| Shull–Smart 1949 | Magnetism in Matter | extra Bragg reflections below \(120\,\mathrm{K}\) on a cell of twice the chemical edge: Néel's two sublattices seen directly |
| Brockhouse–Stewart 1955 | Phonons and Lattice Dynamics | sharp energy transfers at fixed momentum transfer; three branches in aluminium, linear at long wavelength and flat at the zone boundary |
| Brockhouse 1957 | Magnetism in Matter | a gapless branch with $\hbar\omega=Dq^{2}$ in magnetite: magnons resolved, and $D$ measured independently of the magnetization curve |
| Brockhouse–Iyengar 1958 | Phonons and Lattice Dynamics | all six branches of germanium along three symmetry directions; force constants beyond nearest neighbours required |
Table 132.2 closes four gaps at once, and the 1994 Nobel prize recognized the two halves of the technique: the diffraction method that shows where the atoms and the moments are, and the spectrometry that shows how they move. It is worth being explicit about what the closure does and does not amount to.
What is measured directly is a scattering law [vanHove:1954], and its interpretation as a dispersion relation uses the one-phonon — or one-magnon — approximation, whose corrections are the multiphonon background of Phenomenon 132.2. Where the excitations are not sharp, the peaks broaden and the notion of a dispersion curve itself weakens; the measurement then reports a lifetime as well as an energy, which is more information rather than less, but it is no longer the clean reading of a function that Phonons and Lattice Dynamics assumes. The neutron is also a flux-limited probe: sources are few, samples must be large by the standards of modern materials work, and weakly scattering or strongly absorbing elements are difficult. Inelastic X-ray scattering at synchrotron sources now reaches phonons in specimens far too small for a neutron beam [Burkel:2000], at the price of the extreme energy resolution that Table 132.1 shows it needs.
Neither reservation touches the conclusions of this chapter, because all four are qualitative statements that a quantitative correction cannot reverse: the magnetic cell of manganese oxide is twice the chemical one; the energy transfer at fixed momentum transfer is discrete; germanium has six branches and three of them are acoustic; the magnon spectrum has no gap. Each is what its theory chapter predicted, and each was, before 1949, a claim about something nobody had seen.
Hodgkin, Hauptman and Karle, and Klug: the phase problem (1934–1969)
One step has been taken for granted in every diffraction measurement of this chapter, and it is not innocent. A detector counts quanta, and its reading at a reciprocal-lattice point is proportional to \(\abs{F(\vect{G})}^{2}\), the squared modulus of the scattered amplitude. A modulus is half of a complex number: the phase \(\arg F(\vect{G})\) leaves no mark on any counter, and the Fourier synthesis that turns a diffraction pattern into a picture of the scatterers needs both halves. Bragg's law [Bragg:1913a], in the form of Equation (125.1), fixes where the peaks are and therefore the lattice; what sits inside the cell is carried by the intensities, and the intensities do not contain enough to recover it. This is the phase problem, named as an open question in Section 125.2.3 and taken up here.
It did not obstruct the four measurements above, and the reason is worth stating, because it is the exception and not the rule: each of them had few enough unknowns to be fitted rather than inverted in the first place. The chemical structure of manganese oxide was already known from X-rays when Section 132.2 began, so the unknowns there were a propagation vector, a moment magnitude and a direction — a handful of numbers, which a comparison of computed with observed intensities settles without any Fourier synthesis at all. The inelastic measurements use peak positions and barely use intensities. A molecule of a hundred atoms in the cell offers three hundred coordinates, and there fitting has nothing to start from: a model must be guessed before it can be compared, and nobody can guess three hundred numbers.
Three Nobel prizes in Chemistry mark the three ways out. Dorothy Crowfoot Hodgkin's, in 1964, was for carrying X-ray analysis to molecules whose constitution chemistry had been unable to settle, using a heavy atom that the molecule contains, or can be made to contain, as the source of the phases [Hodgkin:1965]. Herbert Hauptman and Jerome Karle's, in 1985, was for showing that the phases are not in fact independent of the amplitudes: a scattering density is non-negative and is made of atoms, and those two facts alone constrain the phases enough to determine them [Hauptman:1986] [Karle:1986]. Aaron Klug's, in 1982, was for taking the phases from a different instrument altogether — an electron microscope forms an image, and an image has its phases in it [Klug:1983].
Apparatus
For the X-ray work: a sealed tube with a copper or molybdenum target, whose characteristic \(K\alpha\) line supplies a wavelength near \(0.1\,\mathrm{nm}\), filtered or monochromated by a crystal; a single crystal of the substance, a few tenths of a millimetre across, on a goniometer head that can be set to any orientation; and a recorder — photographic film in a Weissenberg or precession camera in the early work, a counter diffractometer later — that measures the integrated intensity of each reflection. A protein crystal is largely solvent and must stay wet, so it is sealed in a capillary with its mother liquor. Where heavy-atom phasing is used the apparatus includes the chemistry: crystals soaked in solutions of mercury, platinum or uranium salts, which have to bind at a few definite sites without disturbing the lattice.
The second instrument is arithmetic, and here it is apparatus in the strict sense. A three-dimensional Fourier synthesis over several thousand reflections is not a calculation anyone finishes by hand in a working lifetime, and the vitamin \(\mathrm{B}_{12}\) analysis was among the first structural problems carried out on the electronic computers then becoming available. What can be solved is set by what can be computed, which is why the reach of the subject tracks the history of computing so closely.
For the electron work: a transmission electron microscope with magnetic lenses, at an accelerating voltage of order \(100\,\mathrm{kV}\); a specimen thin enough to be transparent to the beam, supported on a film and imaged either negatively stained or unstained; a tilting stage; a photographic plate, and a densitometer to turn the developed plate into numbers a computer can transform.
Procedure
The X-ray procedure has three variants and one goal: obtain a first estimate of the phases good enough that a Fourier synthesis computed with the measured moduli and the estimated phases shows recognizable atoms. Once it does, the atoms found give better phases, a better map follows, and the cycle converges — the moduli being right throughout, the phases are the only thing being improved.
The first variant computes the synthesis whose coefficients are the measured intensities themselves, all phases set to zero, and reads the position of a heavy atom out of the resulting vector map. Phases computed from that one atom are then good enough to start the cycle, because a heavy atom contributes a large and known term to every structure factor. The molecule may supply the heavy atom itself — the cobalt at the centre of vitamin \(\mathrm{B}_{12}\) — or one may be introduced, by crystallizing a salt of the substance with a heavy cation or by attaching a heavy group chemically.
The second variant is for proteins, where no atom of the molecule dominates. Crystals are soaked with a heavy-metal salt so as to add a few atoms per molecule without changing the cell, the intensities of native and derivative are measured separately, and their differences locate the added atoms; the derivative then plays the part of the known term. In the general case two independent derivatives are needed, for the reason given in the interpretation below.
The third variant uses no heavy atom and no chemistry. The measured moduli are scaled and normalized to what point atoms at rest would give; the strongest few hundred are selected; an origin is fixed by assigning arbitrary phases to a small permitted set of reflections; and phases are propagated through the relations of Equation (132.20) below, many trial sets being carried in parallel and ranked by internal consistency. The map is computed from the surviving set and inspected for a chemically sensible arrangement of atoms.
The electron procedure records micrographs of the specimen at a series of orientations — or exploits the specimen's own symmetry, which presents several orientations at once — digitises each plate, transforms it, places the result as a central plane of the three-dimensional transform, interpolates onto a grid and transforms back. The dose that would give one good image destroys the specimen, so each micrograph is recorded with far too few electrons and many images of identical particles are averaged.
Observations and data
The synthesis computed with intensities as coefficients and every phase set to zero is not noise. It is a map with a peak at the end of every interatomic vector of the structure, weighted by the product of the two scattering powers, and with a large peak at the origin [Patterson:1934]. That map is centrosymmetric whether the crystal is or not, and the measured intensities themselves obey \(I(\vect{G})=I(-\vect{G})\) to the precision of the measurement — with one reproducible exception, in crystals containing an element whose absorption edge lies near the wavelength in use, where the two members of the pair differ by a small but definite amount [Bijvoet:1951].
Penicillin was determined while its constitution was still disputed, and the analysis showed the strained four-membered \(\beta\)-lactam ring fused to a five-membered ring, which the chemistry had been unable to establish [Crowfoot:1949]. Vitamin \(\mathrm{B}_{12}\), a molecule with of order a hundred atoms heavier than hydrogen and no known constitution at all, was determined from the crystals with the cobalt atom at its centre as the heavy atom [Hodgkin:1956]; insulin, a protein, followed by isomorphous replacement and a far larger computation [Adams:1969].
In structures already known, the phase sums built from triples of strong reflections are found not to be spread uniformly round the circle: they cluster about zero, and the clustering tightens as the product of the three normalized amplitudes grows. For a centrosymmetric structure, where a phase can only be \(0\) or \(\pi\), the same statement reads that the product of the three signs is \(+1\) more often than not, the fraction rising towards unity with that same product [Karle:1950] [Cochran:1955a]. Structures with of order a hundred independent non-hydrogen atoms are now solved from the measured intensities alone, with no chemical input beyond the composition [Hauptman:1986] [Karle:1986].
An electron micrograph behaves differently from a diffraction pattern in exactly the way that matters. Its two-dimensional Fourier transform, computed from the digitised plate, reproduces the three-dimensional transform of the specimen on the plane through the origin perpendicular to the beam — with phases. A set of micrographs at different orientations therefore fills the transform, and back transforming yields a density in three dimensions. This was carried out for the tail of bacteriophage T4, whose helical symmetry presents every azimuth in a single image [DeRosier:1968], and set out in general form immediately afterwards [Crowther:1970].
The Fourier synthesis whose coefficients are the measured intensities, with all phases set to zero, is the autocorrelation of the scattering density: a map carrying a peak at the end of every interatomic vector [Patterson:1934]. It is centrosymmetric even when the structure is not, and correspondingly the intensities obey Friedel's law \(I(\vect{G})=I(-\vect{G})\) wherever the scattering is non-absorbing, so that no diffraction pattern can distinguish a structure from its mirror image; near an absorption edge the law fails measurably, and the failure fixes the absolute configuration [Bijvoet:1951]. What the experiment delivers is thus exact and is not the structure.
Derivation. Write the scattering density of one cell as \(\rho(\vect{r})\) — the electron density for X-rays, the nuclear scattering-length density for the neutrons of Section 132.1 — and keep the sign convention of Equation (132.13),
The second of these is the synthesis that would produce the structure, and it needs \(F\) and not \(\abs{F}\). Form instead the synthesis whose coefficients are the numbers a counter delivers,
Substituting Equation (132.15) into the autocorrelation \(\int_{V}\rho(\vect{r})\rho(\vect{r}+\vect{u})\,\dd^{3}r\) gives a double sum over \(\vect{G}\) and \(\vect{G}'\) whose \(\vect{r}\) integral is \(V\delta_{\vect{G}',-\vect{G}}\); the surviving terms carry \(F(\vect{G})F(-\vect{G})=\abs{F(\vect{G})}^{2}\), since \(\rho\) is real, and what is left is exactly Equation (132.16). The intensities therefore determine the autocorrelation completely and without approximation.
Three properties of that object are the three parts of the observation. Relabelling \(\vect{G}\to-\vect{G}\) in Equation (132.16) leaves \(\abs{F}^{2}\) alone and flips the exponent, so \(P(-\vect{u})= P(\vect{u})\): the map is centrosymmetric whatever \(\rho\) may be. For a structure of well-separated atoms of strengths \(Z_{j}\) at positions \(\vect{r}_{j}\) the autocorrelation is \(\sum_{j,k}Z_{j}Z_{k}\,\Gamma(\vect{u}-(\vect{r}_{k}-\vect{r}_{j}))\) with \(\Gamma\) the autocorrelation of one atom, that is a peak of weight \(Z_{j}Z_{k}\) at the end of each interatomic vector, with the \(N\) diagonal terms piled at the origin. And \(\rho\) real gives \(F(-\vect{G})=F(\vect{G})^{*}\), hence Friedel's law; a structure and its inversion \(\rho(-\vect{r})\) have structure factors that are complex conjugates of one another and so identical intensities, which is the statement that handedness is invisible. Absorption is what breaks it: near an edge the effective scattering density acquires an imaginary part, \(\rho\) is no longer real, and the Friedel pair separates.
The same lines say why the vector map is not a solution. It carries \(N(N-1)\) off-origin peaks in a cell that holds \(N\) atoms, so the peak density grows as \(N^{2}\) while the space available does not: at \(N\sim10\) the map can be read, at \(N\sim100\) it is a continuum. The loss is not only of resolution — distinct structures can share an autocorrelation exactly — but overcrowding is what defeats the method in practice. One case escapes, and it is the case Hodgkin exploited: if one atom is much heavier than the rest, the peaks involving it carry weight \(Z_{\text{heavy}}^{2}\) or \(Z_{\text{heavy}}Z_{j}\) and stand above the rest, so the heavy atom alone can be located in a map that shows nothing else.
∎For a structure of resolved atoms the phases are not free given the moduli. The sum \(\varphi(\vect{H})+\varphi(\vect{K})-\varphi(\vect{H}+\vect{K})\) over a triple of strong reflections is observed to cluster about zero rather than to be distributed uniformly, and the clustering tightens as the product of the three normalized amplitudes grows; in a centrosymmetric structure the corresponding product of signs is \(+1\) with a probability rising towards unity with the same product [Sayre:1952] [Karle:1950] [Cochran:1955a]. Structures of the size of a small organic molecule are therefore solved from the intensities alone, with no heavy atom and no chemical assumption beyond the composition [Hauptman:1953] [Karle:1956] [Hauptman:1986] [Karle:1986].
Derivation. Take \(N\) identical atoms, resolved from one another, at positions \(\vect{r}_{j}\), each of the same shape \(s(\vect{r})\), so that \(\rho(\vect{r})=\sum_{j}s(\vect{r}-\vect{r}_{j})\) and, by Equation (132.15),
with \(f\) the transform of \(s\). Now square the density. The atoms do not overlap, so the cross terms vanish identically and \(\rho^{2}(\vect{r})=\sum_{j}s^{2}(\vect{r}-\vect{r}_{j})\): squaring a set of separated equal blobs returns a set of separated equal blobs at the same places. Its structure factor is therefore \(g(\vect{G})S(\vect{G})\), with \(g\) the transform of \(s^{2}\), whence
On the other hand the transform of a product is the convolution of the transforms: substituting the synthesis of Equation (132.15) twice into \(\int_{V}\rho^{2}\ee^{\ii\vect{G}\cdot\vect{r}}\dd^{3}r\) and doing the \(\vect{r}\) integral gives \(F^{\text{sq}}(\vect{G})=V^{-1} \sum_{\vect{H}}F(\vect{H})F(\vect{G}-\vect{H})\). Equating the two expressions,
which is Sayre's equation [Sayre:1952]. It is exact, it contains no unknown but the structure factors themselves, and it is nonlinear — which is what lets it constrain phases, as no linear relation among the \(F\) could.
The phase relation is its leading term. For a real, spherically symmetric atomic shape both \(f\) and \(g\) are real, and over the range of momentum transfer in use both are positive, so \(\theta\) is a positive number and Equation (132.19) makes the phase of \(F(\vect{G})\) the phase of the sum on the right. A sum of many complex terms reaches a large modulus only if the terms dominating it are nearly in phase with one another; so wherever \(\abs{F(\vect{G})}\) is large, the dominant products \(F(\vect{H})F(\vect{G}-\vect{H})\) — those whose two factors are both large — must carry, near enough, the phase of \(F(\vect{G})\). Writing \(F=\abs{F}\ee^{\ii\varphi}\) and \(\vect{K}=\vect{G}-\vect{H}\),
whenever all three reflections are strong.
That this particular combination is the right unknown is forced, and the argument is two lines. An intensity is unchanged by a shift of origin, so nothing an intensity measurement can ever determine may depend on where the origin was put. Moving it by \(\vect{t}\) replaces \(\rho(\vect{r})\) by \(\rho(\vect{r}-\vect{t})\) and multiplies \(F(\vect{G})\) by \(\ee^{\ii\vect{G}\cdot\vect{t}}\), so every individual phase moves by \(\vect{G}\cdot\vect{t}\); the combination in Equation (132.20) moves by \(\vect{H}\cdot\vect{t}+\vect{K}\cdot\vect{t} -(\vect{H}+\vect{K})\cdot\vect{t}=0\) and does not move at all. Individual phases are thus not determinable even in principle — part of each is a choice — while triplet sums are. That is why a structure determination begins by fixing an origin, assigning arbitrary phases to a permitted handful of reflections, and then propagates through relations of the form Equation (132.20).
In a centrosymmetric structure with the origin at a centre, \(\rho(-\vect{r})=\rho(\vect{r})\) makes every \(F(\vect{G})\) real, so \(\varphi\in\{0,\pi\}\) and \(F=\pm\abs{F}\); Equation (132.20) collapses to a statement about signs,
probably rather than certainly. How probably is the quantitative content of the method. With the moduli normalized to what point atoms at rest would give,
— up to a factor accounting for the symmetry of the reflection, and dividing out both the form factor and the Debye–Waller factor of Section 132.6, so that the \(E\) are the structure factors of the fictitious structure Sayre's argument assumed — the probability that Equation (132.21) holds is
for \(N\) equal atoms in the cell [Cochran:1955a], the non-centrosymmetric analogue distributing \(\Phi\) about zero with a concentration \(2\abs{EEE}/\sqrt{N}\) [Cochran:1955b]. Two features of Equation (132.23) fix what the method can do. Only the strongest reflections are useful, since \(\abs{E}\) of order unity leaves \(P_{+}\) barely above one half; and the argument falls as \(N^{-1/2}\), so the relations weaken as the structure grows. Direct methods therefore took the small molecule and left the protein to the heavy atom.
∎The probability distribution of the triplet phase sum: the random-atom model in which the atomic positions are independent and uniform over the cell, the central-limit argument that makes the joint distribution of three normalized structure factors Gaussian, and the conditional distribution of the phase sum that follows from it, giving the hyperbolic tangent in the centrosymmetric case and the von Mises form in general
The Fourier transform of a micrograph of a thin specimen reproduces the three-dimensional transform of that specimen on the central plane perpendicular to the beam, phases included, because a lens performs the back-transform physically. Micrographs at a sufficient set of orientations therefore determine the density in three dimensions with no phase problem to solve, as was first done for the tail of bacteriophage T4 [DeRosier:1968] [Crowther:1970]. The number of distinct views required grows as the ratio of the particle diameter to the resolution sought, which is why symmetric particles were done first.
Derivation. Let \(\rho(\vect{r})\) be the density and let the beam run along \(z\). For a specimen thin enough that an electron scatters at most once, and a lens that collects the scattered and unscattered waves and brings them back together, the image contrast is a linear functional of the projection
Compare the two-dimensional transform of the projection with the three-dimensional transform \(\tilde{\rho}\) of the density:
the last step because the exponent is \(\vect{q}\cdot\vect{r}\) with \(q_{z}=0\). The integration over \(z\) that forms a projection is the evaluation of the transform on \(q_{z}=0\). So one micrograph gives a central plane section of the object's transform, amplitude and phase together — the contrast with a diffraction pattern being that a micrograph is an image, formed in the plane conjugate to the object, and not a squared modulus recorded in the far field. The lens has already performed the back-transform the crystallographer cannot perform.
How many sections are needed follows from two facts. The object is bounded, of diameter \(D\), so its transform is determined by samples at intervals \(1/D\); and resolving detail of size \(d\) requires the transform out to radius \(1/d\). Sections through the origin at angular spacing \(\Delta\theta\) are separated at that radius by \(\Delta\theta/d\), which must not exceed \(1/D\), so \(\Delta\theta\leq d/D\) and the number of views over the available range \(\pi\) is
This count is why symmetry mattered so much in the early work. A helical particle presents every azimuth at once, so a single micrograph of one particle already carries the whole set of sections, which is how the T4 tail was reconstructed from one image [DeRosier:1968].
∎Interpretation
The three routes share one feature: none of them extracts a phase from the intensities of a single diffraction pattern, because the phase is not in them. Each brings information from outside — a known sub-structure, a general property of scattering densities, or a second instrument.
The contrast that makes this sharpest is holography, which belongs with the interference experiments of Experiment: Wave Optics. Let \(O\) be the wave scattered by the object, and superpose on it, before the detector, a reference wave \(R\) of known amplitude and phase and coherent with it. What the detector records is
and the last two terms are linear in \(O\): they carry its phase, written into the positions of the interference fringes, which a detector of intensity reads as easily as it reads anything else. Illuminating the developed record with \(R\) again returns a transmitted field containing the term \(\abs{R}^{2}O\), which is the object wave reconstructed, alongside a conjugate image and a background [Gabor:1948]. Nothing has been recovered that was lost; the phase was never lost, because interference with a known wave converts a phase difference into an intensity difference. The crystallographer has no such wave — there was no coherent, splittable source at \(0.1\,\mathrm{nm}\) in 1950, and a reference beam cannot be introduced inside a crystal — so direct methods are what one does instead: a computational substitute for a reference wave, paid for with an assumption about the object rather than an addition to the apparatus.
Heavy-atom phasing sits between the two, and reading it as a reference wave explains its one awkward feature. The heavy atom contributes to \(F(\vect{G})\) a term known in amplitude and phase once its position is known, exactly as \(R\) is known; but it travels with the object rather than beside it, so what the measurement gives is \(\abs{F}\) for the native crystal, \(\abs{F}\) for the derivative, and the known added term, and those three fix the phase of the native structure factor only up to a reflection about the phase of the added term. In the general case the ambiguity is twofold and irreducible from one derivative; a second, independent derivative gives a second pair, and the member the two pairs share is the answer. Where the projection is centrosymmetric the phase can only be \(0\) or \(\pi\), the two branches coincide, and a single derivative settles the sign outright — which is the form in which the method entered protein crystallography [Green:1954b].
What none of this settles should be said plainly. The triplet relations are probabilistic and have no convergence proof; their reliability falls as \(N^{-1/2}\) by Equation (132.23), and the atomicity that Sayre's equation assumes requires data extending to a resolution at which individual atoms are separated — which most protein crystals do not give, so proteins are still phased experimentally rather than directly. The electron route has its own reservation. An unstained thin biological specimen barely absorbs; it is a phase object, and an in-focus image of it has almost no contrast, so contrast is created by defocusing, which converts phase to amplitude through a transfer function that oscillates in spatial frequency and vanishes at a series of values. The phases obtained are the object's phases modulated by that instrument function, which has to be measured and inverted; and since the dose that would give one good image destroys the specimen, what is transformed is an average over many copies assumed identical.
Finally, the problem is not peculiar to X-rays, and this chapter has been living with it throughout. The magnetic structures that followed Section 132.2 are determined by refining a model against the observed intensities and not by inverting them, for the same reason the original measurement could be: a magnetic structure carries few parameters. Where that ceases to be true — incommensurate order, large magnetic cells, disordered moments — neutron diffraction meets the phase problem in the form the crystallographers met it, and borrows their machinery to deal with it.
Primary references
[Patterson:1934] for the vector map and [Bijvoet:1951] for the breakdown of Friedel's law. Direct methods: [Sayre:1952], [Karle:1950], [Hauptman:1953], [Karle:1956] and [Cochran:1955a] [Cochran:1955b], with the Nobel lectures [Hauptman:1986] [Karle:1986]. Hodgkin's structures: [Crowfoot:1949] for penicillin, [Hodgkin:1956] for vitamin \(\mathrm{B}_{12}\) and [Adams:1969] for insulin, surveyed in [Hodgkin:1965]; the isomorphous-replacement method is [Green:1954b]. Three-dimensional reconstruction from micrographs: [DeRosier:1968] and [Crowther:1970], surveyed in [Klug:1983]. Holography: [Gabor:1948].
The contrast transfer function of a defocused electron microscope: the weak-phase-object approximation, the phase shift introduced by defocus and by spherical aberration, and the resulting oscillating transfer function whose zeros must be filled in by combining images taken at different defocus