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APPENDIX 10. B PATTERSON FEATURES AND PHASE RELATIONSHIPS
Patterson features and phase relationships
which, submitted to a Fourier transform, is related to the triplet invariants. The criterion (10.B.2) is exploited directly by a criterion proposed
by Collins et al. (1996).
(b) Estimate of one-phase structure seminvariants from Harker sections.
(Ardito et al., 1985; Cascarano et al., 1987c). In accordance with
¯ − Rs ) is a s.s.;
Section 4.3, the reflection with vectorial index H = h(I
¯ − Rs )rj = exp 2πihT
fj exp 2π ih(I
¯ − Cs )rj .
fj exp 2π ih(I
According to equation (10.6), the vectors (I − Cs )rj are Harker vectors and
lie on the sth Harker section, or, in other words, define the density on the sth
Harker section. In a more general expression we can rewrite (10.B.3) as
exp(2πi h · Ts )
P(u) exp(2π i h · u)du,
where u varies over the complete sth Harker section, and L is a constant which
takes into account the dimensionality of HS. Relation (10.B.4) provides a phase
estimate for the s.s., FH .
The reader will find more details on all the methods mentioned in this
appendix in Phasing in Crystallography, Chapter 4.
(a) Patterson synthesis of the second kind. Patterson (1949) defined a second
type of Patterson synthesis, say,
P± (u) =
ρ(u + r)ρ(u − r)dr =
|Fh |2 cos(4πh · r − 2φh ).
P± (u) will show a large peak when the product of the densities at u + r and
u − r, integrated over all r values, is large. In this case, u is a pseudo-centre
of symmetry for part (or all) of the electron density; the larger is P± (u), the
larger is the percentage of electron density related by the inversion centre in u.
Since each atom is centrosymmetric in itself, small peaks will be present at the
atomic positions. Small peaks will also be present at midpoints between atoms,
with intensity proportional to the product of the centrosymmetrically shared
part of the two atoms. P± (u) may be trivially used when a centric structure has
been solved in P1 and one wants to recognize the positions of the inversion
centres. It may also be used during a Patterson deconvolution process to check
if the centrosymmetric features of the Patterson function have been eliminated
from the model (Burla et al., 2006b). A large value of P± (u) is some u would
suggest that the enantiomorph has not been well defined.
Phasing via electron
and neutron diffraction
Among the statistics freely available on the webpage of the Cambridge
Structural Database, there is a detail of interest for this chapter: of the
596 910 crystal structures deposited up to 1 January 2012, only 1534 were
solved by neutron data (see Table 1.11). No information is provided on the
number of structures solved by electron data because it is negligible (organic
samples are soon damaged by the electron beams).
A statistical search of the Inorganic Crystal Structure Database (ICSD,
Ver. 2012–1, about 150 000 entries; by courtesy of Thomas Weirich) on structures that have been solved by means of electron diffraction, eventually in
combination with other techniques, indicates a total of about 0.7%.
In spite of limited impact on the databases, electron and neutron diffraction
play a fundamental role in materials science and in crystallography. The main
reason is that they provide alternative techniques to X-rays. Let us first consider
electron diffraction (ED) techniques.
The study of crystalline samples at the nanometer scale is mandatory for
many industrial applications; indeed, physical properties depend on the crystal structure. Unfortunately it is not unusual for compounds to only exist
in the nanocrystalline state; then, traditional X-ray diffraction techniques for
atomic structure determination cannot be applied, because of the weak interactions between X-rays and matter. As a consequence, such structures remain
unknown, in spite of their technological importance. This limits the contribution of X-ray crystallography to nanoscience, a growing scientific area, crucial
to many fields, from semiconductors to pharmaceuticals and proteins. The result is a lack of knowledge on the underlying structure–property relationships,
which often retards further research and development.
Structure analysis by electron diffraction began as early as the 1930s (in
particular, by Rigamonti, in 1936), but the interest of the crystallographic
community in such a technique soon faded, mostly because electron diffraction intensities are not routinely transferable into kinematical |F|2 . In spite
of this limitation, the technique has been used for investigating the structure
of many inorganic, organic, and metallo-organic crystals, biological structures, and various minerals, especially layer silicates. We will therefore, in this
chapter, describe briefly (see Sections 11.2 to 11.7 and Appendix 11.A) the
specific features of electron scattering, in order to allow the reader to understand the special problems one has to face when direct phasing by electrons
data is attempted. For further reading, the reader is addressed to the IUCr
monograph, Electron Diffraction Techniques, volumes I and II (Cowley, 1992),
with special emphasis on Chapter 6 by Vainshtein, Zuyagin, and Avilov,
also, to a monograph by Dorset (1995) and to the International Tables for
Crystallography, Vol. B (1993) and Vol. C (1992). Probably the most updated
and complete presentation of electron crystallography is the monograph by
Zou, Hovmöller, and Oleynikov (2011).
Let us now consider neutron diffraction. This is advised for when crystal structure solution is attempted in order to obtain details which are not
available via X-ray crystallography (e.g. the accurate positions of H atoms
in organic, inorganic, and biological molecules). Phasing via neutron data is
today common practice, mostly when powder diffraction techniques are used
(see Chapter 12); here, only the aspects concerning single crystals will be
described. Since the application of appropriate phasing techniques requires
a prior knowledge of the neutron scattering mechanism, we briefly recall this
topic in Section 11.8. In Section 11.9, possible violation of the positivity
postulate, which may occur when neutron radiation is used, will be discussed.
11.2 Electron scattering
Electrons are produced in an electron gun by a filament a few micrometers in
size and they are accelerated through a potential difference of E volts. Their
divergence is restricted to 10–4 rad or less (smaller than for conventional X-ray
sources) by electromagnetic lenses and the spread of wavelengths is small
(10–5 or less) . The wavelength may be calculated as
(E + 10−6 E2 )1/2
If high energy electron diffraction (HEED) is used, E ≥ 100kV (the range may
be extended to 1 MeV) and λ ≤ 0.05 Å. Since electrons are charged particles,
they are strongly absorbed by matter; therefore, electron diffraction in transmission is applicable only to very thin layers of matter (10–7 to 10–5 cm). While
the electron density distribution is responsible for X-ray scattering, for electron
scattering it is the potential distribution which plays that role. Such a distribution is the sum of the field caused by the nucleus and the field caused by the
electron cloud. Two processes contribute to electron scattering:
1. Elastic scattering: the electrons are scattered by the Coulombic potential
due to the nucleus. Since the proton is much heavier than the electron, no
energy transfer occurs.
2. Inelastic scattering: electrons of the primary beam interact with the atomic
electrons and are scattered after having suffered a loss in energy. In a microscope, such electrons are focused at different positions and produce the
so-called chromatic aberration, which causes a blurring of the image.
Phasing via electron and neutron diffraction data
The strong scattering of electrons by matter implies some advantages but
also serious hindrances to structure analysis work. If we look to the atomic
scattering amplitudes, f e , f x , f n (for electron, X-ray, and neutron scattering,
respectively), on average f e ∼ 10–8 cm, f x ∼ 10–11 cm, and f n ∼ 10–12 cm.
In terms of intensity, I, the ratios will approximately satisfy the relation,
Ix : Ie : In = 1 : 106 : 10−2 .
Minimum specimen thickness for each diffraction technique may be roughly
summarized as follows:
about 0.1 mm for neutron radiation (at SNS, Oak Ridge, USA);
about 0.1 mm for common laboratory X-ray diffractometers, about a few
microns at the Grenoble synchrotron;
about 30 nm for electron diffraction.
The above data suggest that electron diffraction may reduce the most severe
limitations of today’s crystallographic research: crystal sample size and the
necessity for a single chemical phase. For example, the chemical synthesis of
thin films and coatings, superconductors, or improved materials for long-life
batteries typically do not yield large single crystals, but rather produce small
grain size multiphase powders. From a report based on data extracted from the
JCPDS-ICDD database, 1997, for a small subset of technologically relevant
substances (by courtesy of Thomas Weirich), it may be estimated that the fraction of unknown crystal structures is about 81% for pharmaceuticals, 65% for
pigments, 67% for general organic compounds, and 33% for zeolites. It can be
envisioned that these large fractions of materials with unknown crystal structure will decrease considerably if the phasing capacity of electron diffraction
11.3 Electron diffraction amplitudes
For electron diffraction, the structure factor may be written as
exp(2πih · rj ),
f B (s) = 4π K
s = 4π sin θ/λ,
h is the Planck constant
and ρ(r) is the atomic potential distribution. f B (s) is related to the atomic
scattering factor for X-rays, fx (s), by the Mott–Bethe formula,
f B (s) =
(Z − fx (s)) / sin2 θ ,
Non-kinematical character of electron diffraction amplitudes
where ε0 is the permittivity of the vacuum. If λ is in Å, f B (s) in Å, and fx in
electron units, (11.2) reduces to
f B (s) = 0.023934λ2 (Z − fx (s)) / sin2 θ .
At low values of s, the Mott–Bethe formula is less accurate; indeed, (Z − fx (s))
vanishes for neutral atoms. In this case, the formula given by Ibers (1958),
f B (0) =
Z r2 ,
may be used, where r2 is the mean square atomic radius.
The f B (s) values (in Å) are listed in the International Tables for
Crystallography (1992), Vol. C, Table 4.3.11 for all neutral atoms and
most chemically significant ions. Most of the values were derived by Doyle
and Turner (1968) using the relativistic Hartree–Fock atomic potential; for
some atoms and ions, f B (s) has been derived using the Mott–Bethe formula
(11.2) integrated with (11.3). Relativistic effects can be taken into account by
multiplying the tabulated f B (s) by m/m0 = (1 − β 2 )−1/2 , where β = υ/c and
υ is the velocity of the electron. In order to obtain the Fourier coefficients of the
potential distribution in volts, f B (s) values (and therefore FhB values) are usually
multiplied by the ratio 47.87/V, where V is the volume of the unit cell in Å3 .
The difference between fx (s) and f B (s) can be schematized as:
1. With increasing s value, f B (s) decreases more rapidly than fx (s).
2. While fx (0) = Z coincides with the electron shell charge, f B (0) is the ‘full
potential’ of the atom. On average, f B (0) Z 1/3 , but for small atomic
numbers, f B (0) decreases with increasing Z.
3. The scattering factor of ions may be markedly different from a neutral atom;
for small sinθ/λ ranges, f B may also be negative (see Fig. 11.1).
The above features of f B reflect the peculiarity of the potential distribution
(a) The peaks of the atomic potential (being related by Fourier transform to
f B ) are more blurred than electron density peaks.
(b) The peak height (that is, the potential in the maximum) is not strongly
dependent on the atomic number. Therefore, light atoms (hydrogen
included) can be revealed in an easier way than via X-ray data. Typical
peak height ratios are
H : C : O : Al : Cu = 35 : 165 : 215 : 330 : 750.
11.4 Non-kinematical character of electron
Crystal structure analysis via electron diffraction was initiated during the years
1937–8 by a group of crystallographers in the Soviet Union, led by Pinsker
and Vainshtein. Ten years later, Vainshtein and Pinsker (1949) published the
Fourier map of Ba Cl2· · H2 O. The same kinematic approach was used by
Kinematic electron scattering factor f B
for Br and Br–1 .
Phasing via electron and neutron diffraction data
Cowley (1953a,b,c; 1955) for solving a small number of structures. Some
years later, Cowley and Moodie (1957, 1959) described the n-beam dynamical diffraction theory which, through multislice calculations, more closely
describes the physical phenomena involved in electron diffraction. Such models are highly successful in explaining the details of scattering but are, in part,
structure and crystal shape dependent (the observed diffraction pattern does
not contain direct information on the crystal shape); therefore, corrections
eliminating the dynamical effects from the intensities are still not trivial.
The significance of dynamical effects, often predominant with respect to
kinematical scattering, has reduced the interest in structure analysis via electron data. In the following, we summarize qualitatively the components of
electron diffraction amplitudes.
Dynamic scattering. The transition from kinematic to dynamic scattering
occurs when the thickness t of the crystal reaches a critical value for which
total number of electrons
Proportion of electrons which remain
unscattered, singly scattered, and doubly
scattered as a function of the thickness t.
The proportion of inelastically scattered
electrons is also shown.
t ≥ 1.
FhB is the structure factor amplitude (first Born approximation, see
Appendix 12.A) and V is the volume of the unit cell. Since <|FhB |> is proportional to Z 0.8 , condition (11.4) is soon violated for heavy atoms. Even a 50 Å
thickness may be enough to produce dynamic diffraction effects for heavy
atom materials. Condition (11.4) may be easily understood from Fig. 11.2,
where the type of scattering is monitored as a function of thickness t. The number of unscattered electrons rapidly decreases with t; with increasing values of
t, singly scattered electrons are scattered again.
The effects of dynamic scattering on the success of direct methods was evaluated by Dorset et al. (1979) (see also Tivol et al., 1993) by calculating n-beam
diffraction data from crystals with increasing thickness and using different
wavelengths. The tangent formula was then applied to the dynamic structure
factors. Failures occurred for accelerating voltages too low with respect to
crystal thickness. However, even if higher voltages are used, dynamic effects
are always present in electron diffraction data. Formula (11.4) suggests that
one should use very thin crystals, and also higher voltages since they generate
Secondary scattering. A perturbation of the diffracted intensities occurs
in thick layered crystals when strongly diffracted beams from upper layers,
uncoupled from the lower crystalline regions owing to defects, act as the
primary beam for the lower layers. The intensities can then deviate remarkably
from the kinematic value. It is worthwhile stressing that the secondary scattering only involves superposition of the intensities of diffracted beams without
any interference between coincident beams (such interference is present in
dynamic scattering). Accordingly, one measures
Ih = Ih + m1 Ih ⊗ Ih + m2 Ih ⊗ Ih ⊗ Ih + · · · ,
where ‘⊗’ denotes the convolution operation.
Among other consequences, owing to (11.5), space group forbidden reflections (because of screw or glide planes) could not remain extinct under the
A traditional experimental procedure for electron diffraction studies
convolution operation. Dorset (1995) applied a secondary scattering model to
correct data for copper perchlorophthalocyanine, obtaining a good fit with the
Diffraction incoherence. An additional source of incoherent scattering is
crystal bending. Even if its occurrence is easily recognizable, the influence on
electron diffraction is often not taken into account. Cowley (1961) and Cowley
and Goswami (1961) noted that bending is a non-negligible source of incoherent scattering; however, it is not easy to model its effects on diffraction
intensities (Turner and Cowley, 1969; Moss and Dorset, 1983).
Radiation damage. Inelastic scattering often damages the crystal specimen
and, generally, the resulting damage is of a chemical nature and it is different for different types of materials. The damage may influence the various
diffraction intensities by different amounts.
The above considerations suggest that, in spite of great experimental and
theoretical advances, the problem of deriving kinematic intensities from
observed data is still not completely solved. Thus, one should be prepared
to apply phasing methods to scrambled diffracted intensities, and to suffer an
unavoidable loss of efficiency. In the case of success, phasing methods should
provide phases which, coupled with the diffraction magnitudes, should provide
approximated potential maps.
This situation has a counterpart in the final stages of crystal structure analysis, which usually end with a value of the crystallographic residual, Rcryst ,
larger than for X-ray single crystal data. Rcryst values close to 0.25–0.35 may be
obtained, which reduces to 0.15–0.20 for data with larger kinematical nature.
11.5 A traditional experimental procedure
for electron diffraction studies
The traditional experimental diffraction procedure may be summarized as
1. The crystals are transferred to electron microscope grids; a relatively large
but thin (see below) defect-free region of the sample is selected for which
almost no bend contour is observed.
2. Based on the diffraction pattern of the initial zone, appropriate axes are
chosen for tilting, to provide different zonal projections. Tilting is performed using the tilt holder in the goniometric stage of the electron
microscope. Since the sample is extremely thin, the Fourier transform of
the lattice function in the beam direction is not a delta function. In such
conditions, tilt angles are not well defined and a tilt series is used to better
approximate the tilt angle.
Electron diffraction patterns usually provide a subset of the reflections
within reciprocal space. This weakens the efficiency of direct methods,
since a large percentage of strong reflections, and consequently of strong
phase relationships, would be lost. A good example is the structure, CBNA
(Voigt-Martin et al., 1995; see also Chapter 18), space group P21 /c, for
which the b∗ axis was chosen as the tilt axis (see Fig. 11.3). Firstly,
Phasing via electron and neutron diffraction data
CNBA: reciprocal space view down the
tilt axis b*. Diffraction data have been
collected for the emphasized zones (courtesy of I. G. Voigt-Martin and U. Kolb).
zone (201) was recorded (0◦ tilt angle; (hk2h) reflections recorded). Then
measurements were made for the following zones:
Tilting can be performed only over a range of 31◦ ; therefore, the data so
collected represent only part of the information within the reciprocal space.
It may be worthwhile mentioning two additional reasons for the lack of
popularity of traditional ED techniques.
(i) Organic crystals are often subject to severe electron beam damage, and
recording diffraction data from different zones of the same crystal is
nearly impossible. The problem is made more acute by the low speed of
the manual tilting and alignment of the crystal. As a consequence, data
acquisition may be an extremely time-consuming task, inappropriate
for routine investigations.
(ii) Geometrical restrictions on the specimen holder do not allow the
use of high tilt angles with conventional transmission microscopes.
This causes further severe limitations in data collection (missing cone
problem) and therefore additional difficulties in the phasing step.
3. High accuracy in estimated cell constants is difficult to achieve for ED techniques, because data are confined to very small angles. Consequently, the
accuracy of the unit cell parameters is usually low, and cells with higher
Electron microscopy, image processing, and phasing methods
symmetry may frequently be simulated. A synergy with powder diffraction
data is often practised; indeed, unit cell constants obtained by ED data are
used as a starting point for indexing powder patterns with strong peak overlapping. Conversely, powder diffraction is used to improve the accuracy of
unit cell parameters obtained via ED.
4. Electron diffraction intensities are quantified by application of specifically
designed programs (Zou et al., 1993).
5. The space group is identified. This may be performed in three basic ways:
(a) by exploiting the dynamical effects present in the convergent beam patterns (CBED). The resulting diffraction pattern consists of disks (rather
than sharp spots) of diameter proportional to the chosen convergence
angle. Interplanar spacing and angular information are obtained from
the centres of the disks. The point group symmetry and often the space
group symmetry may be derived from the fine structure of intensity
variations within the disks. Not necessarily high quality CBED patterns may be obtained, and in any case, this technique requires further
supplementary experimental work.
(b) from a few zone axis microdiffraction patterns, i.e. at least the zero and
first-order Laue zones should be recorded and analysed (Morniroli and
Steeds, 1992; Morniroli et al., 2007). This approach implies a four-step
procedure: identification of the crystal system, of the Bravais lattice, of
the glide planes, and of the screw axes.
(c) using a fully automatic approach (Camalli et al., 2012) based on analysis of the diffraction intensities; similarly to that used for X-ray data
(see Section 2.6). This task is not trivial. Indeed, dynamical effects
introduce discrepancies among expected symmetry equivalent reflections, and Laue groups belonging to the same crystal system can
frequently not be clearly distinguished by checking equivalent intensities. Furthermore, symmetry forbidden reflections (i.e. reflections
expected to be systematically absent) show non-vanishing intensity, so
making it difficult to identify the diffraction symbol.
(d) by direct inspection of the projected potential maps, obtained straight
from the experiment (see Section 11.5). Indeed, each recorded planar
image would show a specific planar symmetry compatible with the
three-dimensional space group. For example, if the space group is
¯ images should show projected
P63 /mcm, then , , and 
p6mm, pmm, and pgg symmetry, respectively.
11.6 Electron microscopy, image processing,
and phasing methods
There is an important supplementary advantage which may be exploited
by electron crystallography. Electron diffracted beams can be focused by
electromagnetic lenses (equivalent to inverse Fourier transform).
These images, however, are sensitive to focus, crystal thickness, orientation,
and astigmatism. Indeed, only very thin and well-aligned crystals can provide
Phasing via electron and neutron diffraction data
images interpretable in terms of crystal structure projection, provided that the
effects of other parameters have been corrected. Point-to-point resolution is
between 1.5 and 2 Å for a conventional TEM operating at about 200–400kV.
In complex structures, atoms overlap (nearly or exactly) in any projection, and
therefore cannot be resolved in a single projection image. The way to overcome this problem is to collect several images from different directions and
to then combine the images to provide a model structure. This was the basic
contribution of de Rosier and Klug (1968), who described a method for the
reconstruction of three-dimensional structures from a set of two-dimensional
microscope images. Their work allowed the solution of hundreds of molecular
structures, including membrane proteins and viruses. Later on, it was found
that the contrast of a high-resolution electron microscopy (HREM) image
changes with optical conditions and crystal thickness. The so-called contrast
transfer function (CTF) plays a fundamental role in contrast changes. The
interpretation of contrast became simpler when simulation computer programs
(O’Keefe, 1973) using multislice methods, became available. It was then the
custom to interpret experimental images via image simulation. This method
was essentially a trial and error technique; a structural model is assumed, various optical parameters (thickness, defocus, etc.) are varied, and calculated
images are compared with the experimental image. The structural model is
modified, then simulation is started again.
The method is time consuming. Klug and his group revived the technique
by application of the crystallographic image processing method, which proved
capable of recovering the correct structure projection from each individual
image. The method has been further improved by Hovmöller and his group in
Stockolm (Hovmöller et al., 1984; Wang et al., 1988; Li and Hovmöller, 1988).
What is the accuracy and resolution with which the projection of a structure can be deduced from an image obtained via the back Fourier transform
performed by electromagnetic lenses? There are two main factors limiting the
immediate use of the electron micrograph:
1. The image does not represent the projection of the crystal potential, but
instead, its convolution with the Fourier transform of the contrast transfer
function. Therefore, a deconvolution operation is necessary to restore the
2. Widely scattered electrons are focused at positions other than those to
which electrons travelling close to the lens axis (spherical aberration) are
focused. As a consequence a point object is spread over length r in the
image plane, so that the real resolution of an electron microscope is no
less than about 100 times the electron wavelength. In practice, 1 Å resolution images can seldom be obtained by high-resolution microscopes; more
often the image resolution for organic crystals is 2−3 Å or lower, and from
4 to 15 Å for a two-dimensional protein crystal.
Images with a resolution of 1 Å are only obtained for special inorganic
structures. Then, atoms are resolved; since images are projections of the threedimensional structure, peak overlapping could, even in this case, hinder correct
three-dimensional location of the atoms. Several projections are therefore
needed for a three-dimensional reconstruction of the structure (Wenk et al.,