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APPENDIX 10. B PATTERSON FEATURES AND PHASE RELATIONSHIPS

APPENDIX 10. B PATTERSON FEATURES AND PHASE RELATIONSHIPS

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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

then,

N

FH =

¯ − Rs )rj = exp 2πihT

¯ s

fj exp 2π ih(I

j=1

N

¯ − Cs )rj .

fj exp 2π ih(I

j=1

(10.B.3)

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

FH =

1

exp(2πi h · Ts )

L

P(u) exp(2π i h · u)du,

(10.B.4)

HS

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 =

V

2

V

h>0

|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.

233

11

Phasing via electron

and neutron diffraction

data

11.1 Introduction

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

Electron scattering

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

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

λ=

12.3

.

(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.

235

236

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:

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

techniques improves.

11.3 Electron diffraction amplitudes

For electron diffraction, the structure factor may be written as

FhB =

N

fB

j=1 j

exp(2πih · rj ),

where

f B (s) = 4π K

ρ(r)r2

sin sr

dr,

sr

K=

2π me

,

h2

0

s = 4π sin θ/λ,

(11.1)

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) =

Kλ2

(Z − fx (s)) / sin2 θ ,

ε0

(11.2)

Non-kinematical character of electron diffraction amplitudes

237

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) =

4π me2

Z r2 ,

3h2

(11.3)

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

(Vainshtein, 1964):

(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

diffraction amplitudes

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

B

9 f

8

7

6

5

4

3

Br–1

2

1

–0.1

0.1

Br

sin ϑ/λ

0.2

0.3

0.4

0.5

–230

Fig. 11.1

Kinematic electron scattering factor f B

for Br and Br–1 .

238

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

λ

total number of electrons

unscattered

inelastic

singly scattered

doubly scattered

t

Fig. 11.2

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.

V

(11.4)

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

smaller wavelengths.

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 + · · · ,

(11.5)

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

kinematic intensities.

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.

for electron diffraction studies

The traditional experimental diffraction procedure may be summarized as

follows:

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,

239

240

Phasing via electron and neutron diffraction data

β*

c*

a*

Tilt

angle Zone

27º (101)

–10º

–16º

–19º

–31º

(201)

(301)

(401)

(501)

(100)

Fig. 11.3

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:

Zone

Tilt angle

Reflection type

(101)

(301)

(401)

(501)

(100)

27◦

–10◦

–16◦

–19◦

–31◦

(hkh)

(hk3h)

(hk4h)

(hk5h)

(0kl)

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

(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 [001], [100], and [110]

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

241

242

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

desired image.

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.,

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APPENDIX 10. B PATTERSON FEATURES AND PHASE RELATIONSHIPS

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