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Sintesi di Fourier - Clegg

Materiale didattico per il corso di Cristallografia della prof.ssa Elena Buonaccorsi. Trattasi del capitolo 6 a cura di W. Clegg del testo dal titolo "Crystal Structure Analysis. Principls and Practice", all'interno del quale è affrontato il tema della sintesi di Fourier.

Esame di Cristallografia docente Prof. E. Bonaccorsi

Anteprima

ESTRATTO DOCUMENTO

Uses of different kinds of Fourier syntheses

8.4 109

F

Here, the structure factor has been separated into its amplitude |F|

and its phase both of which are needed in order to carry out the

φ,

calculation.

Different kinds of Fourier syntheses use different coefficients instead

of the amplitudes and they may also in some cases apply weights

|F|,

to the individual terms in the sum, so that not all reflections contribute

strictly in proportion to these coefficients. These are all attempts to obtain

as much useful information as possible at different stages of the structure

determination, even if the phases are not well known.

8.4.1 Patterson syntheses

These are discussed in detail in the next chapter. The coefficients are

2 instead of and all phases are set equal to zero. In this case

|F | |F |,

o o

all necessary information is known and the synthesis can be readily

performed. The result, of course, is not the electron density distribution

for the structure, but it is related to it in what is often a useful way, as

is explained later. There are some slight variations even within this use,

and these are covered in the Patterson synthesis chapter (Chapter 9).

8.4.2 E-maps

These are an important part of direct methods for solving crystal struc-

tures, and are discussed more fully in Chapter 10. The coefficients are

the so-called normalized observed structure factor amplitudes,

|E |,

o

which represent the diffraction pattern expected for point atoms (with

their electron density concentrated into a single point instead of spread

E-values

out over a finite volume) of equal size, at rest. are calculated,

with a number of assumptions and approximations, from the observed

amplitudes and only the largest values are used, weaker reflections

|F |,

o

being ignored because they contribute less to the Fourier synthesis any-

way. Phases for this selected subset of the full data are estimated by

a range of techniques under the general heading of ‘direct methods’,

and usually a number of different phase sets are produced and used to

E-maps.

calculated These maps tend to contain sharper (stronger and

narrower) maxima than normal Fourier syntheses (F-maps), and this

can help to show up possible atoms, but they also tend to contain more

noise (peaks, usually of smaller size, that do not correspond to genuine

atoms).

8.4.3 Full electron-density maps,

using (8.2) or (8.3) as they stand

These actually tend not to be used very often in chemical crystallog-

raphy, except for demonstration purposes, because the other types of

syntheses have particular advantages at different stages. However, let

us consider how we can carry out such a synthesis without having any

Fourier syntheses

110 experimental phases. Such a procedure can be used when some of the

atoms have been located (perhaps from direct methods or a Patterson

synthesis) and others still remain to be found. Once we have some

atoms, we can use them as a which we know is not

model structure,

complete, but it contains all the information we currently have. From

the model structure we can use (8.1) to calculate what its diffraction

pattern would be. This will not be identical to the observed diffraction

pattern, but it should show some resemblance to it, the more nearly so

as we include more atoms in the correct positions. There are various

measures of agreement between the sets of observed and calculated

amplitudes, and but the important thing is that the calcu-

|F | |F |,

o c

lated diffraction pattern includes phases, as well as amplitudes

φ |F |.

c c

Although these are not the same as the true phases we would really like

to know, they are currently the nearest thing we have to them. A Fourier

synthesis using coefficients with the phases would just repro-

φ

|F |

c c

duce the same model structure and get us nowhere, but combining the

true observed amplitudes with the ‘current-best-estimate’ phases φ

|F |

o c

gives us a new electron-density map. If the calculated phases are not too

far from the correct phases (as is usually the case if the model structure

has atoms in approximately correct places and these are a significant

proportion of the electron density of the structure), then this usually

shows the atoms of the model structure again, together with new fea-

tures not in the model structure but demanded by the diffraction data,

i.e. more genuine atoms. Because of all the approximations involved in

this process, there may also be peaks in the electron-density map that

do not correspond to real atoms, and the results need to be interpreted

in the light of chemical structural sense and what is expected. Addi-

tion of these new genuine atoms gives a better model structure, and

the whole process can be repeated, giving better calculated phases and

yet another new, and clearer, Fourier synthesis. This is done repeatedly

until all the atoms have been found and the model structure essentially

reproduces itself.

8.4.4 Difference syntheses

These are widely used in preference to full electron-density syntheses

for expanding partial structures. The coefficients are and the

|F | − |F |

o c

phases are obtained from a model structure as described above. The

result is effectively an electron-density map from which the features

already in the model structure are removed, so that new features stand

out more clearly, and it usually makes it easier to find new atoms. This

is rather like saying that, if the tallest peaks in a range of mountains

were somehow taken away, the foothills would appear to be much more

impressive! Peaks lying at the positions of atoms in the model structure,

or negative difference electron density there, indicate that the model

has either too little or too much electron density in those places, and can

indicate a wrongly assigned atom type, e.g. N instead of O or N instead of

Uses of different kinds of Fourier syntheses

8.4 111

C N

C C

C C C

C

C N

C Et

C C

C

Fig. 8.3 A section through a difference synthesis showing the effect of wrongly assigned

atom types and missing hydrogen atoms; the assumed model structure is shown, together

with the positions of its atoms and bonds in the map.

C for these respective effects. An example is shown in Fig. 8.3. There are

potentially some considerable problems with difference syntheses when

the proportion of known atoms is quite small, because the calculated

phases can have large errors. Also, weak reflections with relatively large

uncertainties in their intensities can cause disproportionate errors, and

it may be best not to use the weakest reflections; alternatively they can

be given reduced weights, as discussed below. It is important to ensure

that the observed and calculated data are on the same scale. Another

F

reason why difference syntheses can be better than full syntheses is

o

that series termination errors (small ripple effects due to the lack of data

beyond the measured cancel out through use of the differences

θ )

max

instead of full amplitudes.

8.4.5 2F syntheses

F

o c

The use of coefficients 2|F with phases calculated from a model

| − |F |

o c F

structure combines the advantages of standard and difference synthe-

o

ses. The resulting map shows both the known and the as-yet unknown

features of the structures, with the new atoms emphasized, and it is

less subject to some of the errors of the simple difference synthesis.

It is more widely used in protein crystallography than by chemical

crystallographers.

Fourier syntheses

112 8.4.6 Other uses of difference syntheses

Towards the end of structure determination, difference maps are often

used to locate hydrogen atoms. These can not usually be found until

all other atoms are present and have been refined with anisotropic

displacement parameters, so that their contributions are correctly rep-

resented in the model structure. This is because hydrogen atoms have

very little electron density, and even that is significantly involved in

bonding, so the positions found in difference maps are usually closer to

the nearest atom than are the actual centres (the nuclei) of the hydrogen

atoms. Unless data are of good quality, and particularly when heavy

atoms are present in the structure, hydrogen atoms can easily be lost in

the noise of an electron-density map. This is particularly true for non-

centrosymmetric structures, where the absence of hydrogen atoms in a

model structure is partially compensated by shifts in the phases from

their correct values; any Fourier synthesis using calculated phases will

always have a bias towards the model structure from which they were

obtained. Hydrogen atoms contribute relatively more to low-angle and

less to high-angle reflections, because their atomic scattering factor falls

off more quickly with than those of other atoms, so it may help to

θ

leave out the high-angle data, or use weights that reduce their contri-

bution to the sums. Right at the end of a structure determination, when

refinement is complete, a final difference synthesis must be generated

in order to see if there is any remaining electron density unexplained

by the refined model. This must include all data and use no weights.

Residual electron density may be an artefact of inadequate data correc-

tions (usually absorption), or may indicate poorly modelled disorder or

other problems and imperfections in the model. The sizes of the largest

maxima and minima in this final difference map, together with their

positions if they are of significant size, are important indicators of the

quality of a structure determination, and should always be included in

any summary of the results.

8.5 Weights in Fourier syntheses

It was noted above that the calculated phases, derived from the current

model structure, are only an estimate of the true phases. Clearly the

approximation improves as the model structure becomes more com-

plete. In any given set of calculated phases, some will be more in error

than others. For a reflection with large and almost equal and

|F | |F |

o c

there is greater confidence in the reliability of the phase than there is

when is small. This variation in reliability of the phases can be

|F |

c

incorporated into the calculations by multiplying each contribution by

a weight, which increases with expected reliability. Various weight-

ing schemes have been developed and used, with weights calculated

from the values of the observed and calculated amplitudes and the pro-

portion of unknown electron density in the structure. Appropriately

Illustration in one dimension

8.6 113

chosen weights can help to enhance the genuine new features of Table 8.1. One-dimensional Fourier

contributions.

Fourier syntheses and reduce noise. Weights that are can

θ-dependent

be used to aid the search for hydrogen atoms in the later stages, by true sign model sign

l |F | |F |

o c

down-weighting the higher-angle data containing less information from

these atoms. No weights may be used in the final difference synthe- 3 8 17 + −

sis for checking the completeness of a refined structure; by this stage 4 64 51 − −

the calculated phases will be as close to the correct values as they 5 56 64 − −

6 74 55

can be. − −

7 15 26 − −

8 5 9 + +

9 46 39 + +

8.6 Illustration in one dimension 10 45 53 + +

11 43 47 + +

z-axis)

For a one-dimensional structure (this direction taken as the with 12 17 26 + +

13 9 3 − −

inversion symmetry, (8.3) simplifies considerably: 14 26 28 − −

15 31 41 − −

1 16 23 39 − −

s(l) cos[2π(lz)] (8.4)

|F(l)|

ρ(z) = 17 12 23 − −

c l 18 14 1 + −

19 20 19 + +

and the Fourier summation can easily be demonstrated pictorially. Only 20 33 31 + +

21 63 30 + +

l

positive values of the index need to be considered, each giving a double

F(l) F(−l),

contribution to the sum, since in addition to the single con-

=

F(0).

tribution of The phase of each reflection is now just the (unknown)

s(l)

positive or negative sign, or We use some data measured

= +1 −1.

a number of years ago for a compound containing a long alkyl chain

and a bromine atom (the detailed molecular structure is not important

here); this crystallizes in a unit cell with one long axis the molecule

(c),

being stretched out so that its projection along this axis gives resolved

atoms, Br and several C. There are two molecules per unit cell, appear-

ing as inversions of each other along the two halves of the cell axis. This

projection can be investigated with just the reflections, with the

(00l)

irrelevant zero indices ignored here.

Table 8.1 lists the observed amplitudes of the measured reflections

l

with between 3 and 21 (|F and the amplitudes calculated from

|),

o

a model structure consisting only of the two symmetry-equivalent Br

atoms (|F via the one-dimensional equivalent of (8.1)); how these Br

|,

c

atoms can be found from the data is considered in the next chapter, on

Patterson syntheses. Also given are two sets of signs (reflection phases):

the correct signs obtained by calculation from the complete structure

once it is known (true signs), and the signs obtained from the model

containing only the Br atoms (model signs). Below Table 8.1 are shown

in Fig. 8.4 the individual terms that contribute to the

(l)|cos[2π(lz)]

|F o

s(l).

sum in (8.4), ignoring the signs Carrying out a Fourier synthesis 0 1

z

to obtain the one-dimensional electron density just means adding up

these ‘electron-wave’ contributions with the correct signs. Since there Fig. 8.4 The contributions of each of

the reflections in Table 8.1 to the one-

are 19 terms to add together, the number of possible sign combina- dimensional Fourier synthesis, all shown

19

tions is 2 , which is over half a million: not a good case for trial here with positive sign (zero phase angle);

and error! Several different variants on (8.4) are shown graphically in l

the reflections are in order, with 3 at the

=

Figs. 8.5 to 8.8. l

top and 21 at the bottom.

=


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DETTAGLI
Corso di laurea: Corso di laurea in scienze geologiche
SSD:
Università: Pisa - Unipi
A.A.: 2011-2012

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Atreyu di informazioni apprese con la frequenza delle lezioni di Cristallografia e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università Pisa - Unipi o del prof Bonaccorsi Elena.

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