Blake A.J.(ed.) Crystal Structure Analysis

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242 Random and systematic errors

16.6.2 Uncertainty propagation

For a function f(x

, x

, ..., x

)

(f) =



i,j=1

∂f

∂x

∂f

∂x

· cov(x

, x

), (16.32)

where cov(x

, x

) is the same as σ

), as we saw before.

For two functions f

and f

cov(f

, f

) =



i,j=1

∂f

∂x

∂f

∂x

· cov(x

, x

), (16.33)

but this is not often needed.

Note that, if the variables x are all independent, the covariances are

zero except for the variance terms themselves, so in such a simple case

(f) =



i=1



∂f

∂x



), (16.34)

and thus, the variance of f is just a weighted sum of the variances of the

individual independent variables.

The full variance–covariance matrix following the ﬁnal least-squares

reﬁnement must, therefore, be used in calculating molecular geometry

s.u. values. Calculation using the co-ordinate s.u. values alone, with

neglect of correlation effects between atoms, does not give the correct

geometry s.u.s: it is equivalent to using (16.34) instead of (16.32), and

potentially important terms are missing. This is particularly evident

when calculating the bond length between two atoms related by a sym-

metry element, because the co-ordinates of these atoms are completely

correlated.

Such calculations are normally performed automatically together

with the least-squares reﬁnement. Proper calculation in a separate

step later would require that the reﬁnement program output the full

variance–covariance matrix.

16.7 Systematic errors

The preceding sections have largely discussed the effects of random

errors in a data set. Systematic errors usually lead to a reduction in

both precision and accuracy in a structure determination if they are not

corrected.

16.7 Systematic errors 243

16.7.1 Systematic errors in the data

(a) Absorption

Absorption reduces the observed intensities of diffraction, but by dif-

ferent factors for different reﬂections. The effect is greatest at low Bragg

angle, so that there is a systematic error even for a spherical crystal.

Uncorrected signiﬁcant absorption causes atomic displacement param-

etersto be too low, inan attempt tocompensate for theeffect.Anisotropic

absorption (for a non-spherical crystal) affects the apparent atomic

vibration differently in different directions, so that elongated ‘thermal

ellipsoids’ are produced for the atoms in a needle crystal. The atomic

co-ordinates are generally not signiﬁcantly affected, but the s.u.s are

increased because the observed and calculated data do not agree so

well; the atomic displacement parameters can not completely mop up

the absorption errors.

(b) Extinction

This also attenuates the observed intensities, and it is most severe for

low-angle, strong reﬂections. Like absorption, it reduces overall preci-

sion and systematically affects atomic displacement parameters, while

having much less effect on atomic co-ordinate values.

TDS, produced as a result of co-operative lattice vibrations, has the

effect of increasing observed intensities. The effect, however, increases

with sin

θ, so the net effect, if no correction is made, is once again

to reduce atomic displacement parameters from their true values. TDS

effects have received little attention in routine crystal-structure deter-

mination, and they are generally believed to be small. Data collection at

reduced temperature is an advantage here, as well as in other ways.

(d) A poorly aligned diffractometer

Several errors can be introduced into the diffraction data by this fault,

including an improper measurement of intensities if the reﬂections are

not completely received by the counter aperture. The most common

error, however, is probably in unit cell parameters. If a badly aligned

instrument involves systematic errors in the zero points of the circles

(especially 2θ), there will be a corresponding error in reﬁned cell param-

eters, which may well be much greater than their supposed s.u.s. This, in

turn, leads to systematic errors in molecular geometry, and no indication

of these can be seen in the commonly quoted measures of the ‘quality’

of the structure determination (structure factor residuals, goodness of

ﬁt, etc.), which refer only to diffraction intensities and not to diffraction

geometry. Such errors as this, therefore, are the most invidious, because

it is difﬁcult to detect them.

(e) Anomalous dispersion

This may be considered as a systematic effect (and, hence, a potential

source of systematic error) in the data, or as a possible fault in the struc-

tural model if properly corrected atomic scattering factors are not used.

244 Random and systematic errors

Neglectof the correctionin anon-centrosymmetricstructurewitha polar

axis or, even worse, a ‘correction’ with the wrong sign, results in a sys-

tematic shift, along the polar axis, of all atoms displaying signiﬁcant

anomalous scattering effects, because this shift, relative to the rest of the

atoms, mimics the phase shift produced by the anomalous scattering

(Cruickshank and McDonald, 1967). In a centrosymmetric or a non-

polar non-centrosymmetric structure, atomic positions are not affected,

but atomic displacement parameters are.

Of course, anomalous dispersion effects can be used for determin-

ing the correct ‘handedness’ of a non-centrosymmetric structure (see

Chapter 18 on twinning), and also for determining phases of reﬂec-

tions in structure solution, but these are really separate subjects and not

directly concerned with errors and results.

16.7.2 Data thresholds

It is fairly common for the weakest reﬂections not to be used in least-

squares reﬁnement, though there is considerable controversy over this.

The inclusion or omission of weak reﬂections usually makes no signif-

icant difference to the derived parameter values. Weak reﬂections tend

to increase the residuals R and (to a lesser extent if they are correctly

weighted) wR, but this is compensated somewhat by the larger excess

of data over parameters (N − P), so it is not clear how ﬁnal s.u.s will

be affected. In fact, it has been demonstrated that they, too, are scarcely

alteredby the inclusionoromissionofweakreﬂections orbythedecision

of just where to set the threshold (Stenkamp and Jensen, 1975). Thus,

the difference in the results is to a large degree just a cosmetic one. On

the other hand, the weak reﬂections can play a crucial role in decid-

ing between centrosymmetric and non-centrosymmetric space groups

in ambiguous cases, because their omission tends to bias statistical tests

towards a decision against centrosymmetry. If no sigma cut-off is to be

used during reﬁnement, it is essential to examine critically the resolu-

tion at which the data vanish into the background. In general, there

seems to be little to recommend the exclusion of weak data in resolution

shells where there are plenty of strong data – their weakness conveys

important information. The inclusion of swathes of weak high-angle

data just because the data reduction software has written them to a ﬁle

only degrades the quality of a structure determination.

16.7.3 Errors and limitations of the model

The parameters reﬁned by least squares are an attempt to describe the

structure we are trying to determine. They represent an approximation

to the actual X-ray scattering power of the structure. No such model

can be a perfect representation, and there are various limitations on the

simple models we use, and various errors that may be made in choosing

the elements of the model.

16.7 Systematic errors 245

(a) Atomic scattering factors

The tabulated scattering factors commonly used are reasonably accurate

representations of the scattering power of individual, isolated atoms at

rest. They have spherical symmetry, and probably their greatest limi-

tation is the lack of allowance for distortion of this spherical electron

density when atoms are placed together and bonded to each other. The

greatest effects of this approximation are seen in the low-angle data,

and an analysis of variance of the observed and calculated data after

reﬁnement commonly shows the worst agreements for such reﬂections.

In careful work, some of the largest peaks in a ﬁnal-difference electron-

density synthesis are found between atoms (bonding electron density)

and in regions where lone pairs of non-bonding electrons are expected

to lie. This is, of course, one reason why bonds to hydrogen atoms are

found to be systematically shortened in X-ray diffraction studies.

An incorrect assignment of atom types, so that a wrong scattering fac-

tor is used foran atom, scarcely affectsatomic co-ordinatesin most cases,

although there are circumstances under which these may be subject to

a systematic error. In an attempt to compensate for the wrong scatter-

ing factor, reﬁnement will adjust the atomic displacement parameters,

often to a very considerable degree; an incorrectly assigned atom may

be recognized in many cases by its anomalous displacement parameter,

especially at the early isotropic stage of reﬁnement when there is a single

parameter for each atom.

(b) Constraints and restraints

Properly used, these are valuable tools in reﬁnement, allowing us to

deal with problems of parameters that are not well determined from the

diffraction data alone. We must, however, be quite sure of the validity

of any constraints or restraints we apply. Any that strongly oppose the

course of unconstrained/unrestrained reﬁnement, rather than gently

guiding it, prevent convergence to the data-determined minimum and

so force a different result. This will, in particular, signiﬁcantly affect the

geometry around the regions in the structure where the constraints are

being applied.

A common example is the use of a constrained C–H bond length of

1.08 Å, chosen because it is the ‘true’ value determined spectroscopically

for simple hydrocarbons. Since C–H bonds are systematically shortened

in X-ray work, the effect of the constraint will be to push both atoms

further apart than the diffraction data alone would indicate. Although

the hydrogen-atom position will be most affected, there will be a small,

but possibly signiﬁcant, effect on the carbon atom. Misplacing the atoms

in this way will also affect their displacement parameters.

Other cases of inappropriate constraints include the imposition of too

high a symmetry on a group of atoms that is genuinely perturbed to a

less regular shape by bonding or packing interactions. Phenyl groups

are systematically distorted from regular hexagonal symmetry, and the

use of such a simple model, frequently used in reﬁnement, may not be

appropriate.

246 Random and systematic errors

Space group determination is based on several experimental measure-

ments and deductions: (i) the metric symmetry of the reciprocal and

direct lattices; (ii) the Laue symmetry of the observed diffraction pat-

tern; (iii) systematically absent reﬂections; (iv) statistical tests for the

presence or absence of symmetry elements, especially an inversion cen-

tre; (v) ultimately, a ‘successful’ reﬁnement. Reports appear relatively

frequently in the literature of space groups that are reputed to have been

incorrectly assigned by previous workers. In many cases, the problem

is not a serious one, in that two molecules, actually equivalent by unno-

ticed symmetry, are reﬁned as independent, and their geometries are

not signiﬁcantly different: the results are reliable, but contain unneces-

sary redundancies. Where the missing symmetry is an inversion centre,

however, there is a real problem, in that reﬁnement is unstable (strictly

speaking, the matrix is singular), but this may be masked by the par-

ticular reﬁnement technique used. The geometrical results in this case

are quite unreliable: parameters that should be equal by symmetry may

be found to differ by a large amount, and the molecular geometry often

displays considerable distortions.

(d) High thermal motion and static disorder

It is not always easy to distinguish these two situations, except by car-

rying out the data collection at a reduced temperature (which reduces

dynamic disorder but not usually static disorder unless there is actu-

ally an order-disorder phase transition at an intermediate temperature).

High thermal motion increases the foreshortening of interatomic dis-

tances generally observed in X-ray diffraction, so there is a considerable

systematic error in bond lengths, which was discussed in Chapter 14.

The usual six-parameter (ellipsoidal) model of thermal motion becomes

increasingly inadequate as the motion increases in amplitude, so the

displacement parameters are of dubious value and their precision is

generally poor.

The presence of disorder in a structure, unless it is very simple and

can be well modelled, reduces to some extent the overall precision of

the whole structure, not just of the particular atoms affected. For this

reason,certain atomic groupingsnotorious for disorderarebest avoided

if possible: these include ClO

−

,BF

−

and PF

−

anions.

High thermal motion and/ordisordercan make the geometrical inter-

pretation of a structure difﬁcult, and may lead to incorrect deductions

about the molecular geometry and conformation. A classic case is that

of ferrocene, (C

)

Fe, which appears to be staggered because of unre-

solved disorder at room temperature, but that (contrary to statements

in some standard inorganic chemistry text-books!) is actually eclipsed.

(e) Wrong structures

Such errors as those just mentioned, with an incorrect molecular geom-

etry, are bad enough, but it is possible, though very uncommon and

unlikely, to ﬁnd a completely incorrect structure, in the sense of identi-

fying the wrongchemical compound.Acase of mistaken identity caused

16.7 Systematic errors 247

by 30-fold disorder involves the supposed structure of dodecahedrane

(Ermer, 1983). Wrongly assigned atom types were suspected in the struc-

ture of ‘[ClF

][CuF

]’, which in reality is probably [Cu(H

][SiF

]

(von Schnering and Vu, 1983); the original workers were misled by the

similarity in scattering powers of Si and Cl, and of O and F, and perhaps

by some wishful thinking!

16.7.4 Assessment of a structure determination

The above discussion should encourage us to take a critical view of

crystal-structure determination in general (Jones, 1984; Ibers, 1974), and

to seek to evaluate carefully any particular reported structure. Several

research journals issue detailed instructions for authors of crystal-

structure reports, and some provide separate checklists for referees.

These can provide a useful framework for assessing a structure, whether

it be one reported in the literature, or one of your own.

Below is a summary of some useful points for checking the quality

of a structure determination. It is derived from a number of sources,

including standard tests applied by Acta Crystallographica Sections B/C/E,

and a list distributed by David Watkin at a British Crystallographic

Association Intensive School.

Check for consistency of the crystal data. If you have a suitable com-

puter program, you can input the cell parameters and chemical formula

and check the volume, Z (number of chemical formula units in the unit

cell), density, absorption coefﬁcient μ, etc. For a quick check by hand

calculation:

(a) count the non-hydrogen atoms (N) in the molecule or formula

unit;

(b) check that abc sin δ ≈ V, where δ is the most removed of α, β, γ

from 90

◦

;

is usually about 18 Å

for organic and many other compounds.

Assess the description of the data collection.

(a) Check that |h|

max

/a ≈|k|

max

/b ≈|l|

max

/c, and that at least the

correct minimum fraction of reciprocal space has been covered.

(b) 2θ

max

should be at least 45

◦

(better, 50

◦

) for Mo radiation, 110

◦

(better, 130

◦

) for Cu radiation.

and the threshhold [which may be expressed in terms of σ(I) or

σ(F): I ≥ 2σ(I) corresponds to F ≥ 4σ(F)]; check for a low value

of R

int

if equivalent reﬂections are merged.

(d) Calculate and compare μt

min

and μt

max

(dimensionless!) for

the minimum and maximum crystal dimensions. If μt

max

< 2,

absorption is probably no problem. If μt

max

> 5or(μt

max

−

μt

min

)>2, an absorption correction is necessary (or there will

be signiﬁcant effects on the U

values). More detailed tests

248 Random and systematic errors

are described in the Notes for Authors of Acta Crystallographica

Section C.

Assess the reﬁnement and results.

(a) The number of observed data should be greater than the num-

ber of reﬁned parameters by a factor of at least 5 (and preferably

10).Anisotropic reﬁnement gives 9 parameters per atom, isotropic

gives 4. Constrained H atoms do not count unless U is reﬁned for

them.

(b) Examine carefully the description of any constraints/restraints,

the treatment of H atoms, and any disorder.

values; high values may indicate dis-

order, low values may indicate uncorrected absorption (unless

low temperature was used); either of these could be due to

misassigned atom types.

(d) Check for convergence (shift/s.u. values preferably < 0.01).

(e) Examine the ﬁt of observed and calculated data, if possible, not

only by the value of R.

(f) Difference electron density outside about ±1eÅ

−3

may be due

to missing, misplaced, or misassigned atoms, systematic errors

such as absorption (especially if large peaks appear close to heavy

atoms), or unmodelled disorder.

(g) Check for ‘absolute structure’ determination if the space group

does not have a centre of symmetry.

(h) Assess the s.u.s (i) of the reﬁned parameters; (ii) of the molec-

ular geometry parameters. Watch out for low s.u.s ignoring cell

parameter uncertainties. Check for s.u.s on values that should be

constrained, symmetry-equivalent, etc.

(i) Check for strange results: unusual geometry, impossibly short

intermolecular contacts, etc.

Many of these tests have been incorporated into the CHECKCIF proce-

dure in PLATON (Spek, 2003), and all structures should be validated with

this program as a matter of routine.

References

Abrahams, S. C. and Keve, E. T. (1971) Acta Crystallogr. A27, 157–165.

Erratum: (1972) A28, 215.

Barlow, R. J. (1997). Statistics. John Wiley: Chichester.

Betteridge, P. W., Carruthers, J. R., Cooper, R. I., Prout, K. and Watkin,

D. J. (2003). J. Appl. Crystallogr. 36, 1487.

Bevington, P. R. and Robinson, D. K. (2003). Data reduction and error

analysis for the physical sciences, 3rd edn. McGraw Hill, New York.

Blessing, R. H. (1997). J. Appl. Crystallogr. 30, 421–426.

Carruthers, J. R. and Watkin D. J. (1979). Acta Crystallogr. A35, 698–699.

References 249

Cruickshank, D. W. J. and McDonald, W. S. (1967) Acta Crystallogr.

23, 9–11.

Ermer, O. (1983). Angew. Chem. Int. Ed. Engl., 22, 251–252.

Hamilton, W. C. (1964). Statistics in physical science. Ronald Press, New

York. [This is out of print, and can be difﬁcult to ﬁnd].

Hamilton, W. C. (1965). Acta Crystallogr. 18, 502–510. See also Pawley, G.

S. (1970) Acta Crystallogr. A26, 691–692.

Ibers, J. A. (1974). Problem crystal structures and Donohue, J. Incorrect

crystal structures: can they be avoided?InCritical evaluation of chemical

and physical structural information, (eds) D. R. Lide Jr. and M. A. Paul.

Nat. Acad. Sci: Washington D.C.

Jones, P. G. (1984). Chem. Soc. Rev. 13, 157–172.

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P.

(1991). Numerical recipes in Fortran. Cambridge University Press,

Cambridge, UK.

Prince, E. (1994). Mathematical techniques in crystallography and materials

science, 2nd edn. Springer: New York.

Prince, E. and Nicolson, W. L. (1983). Acta Crystallogr. A39, 407–410.

Sheldrick, G. M. (2008). Acta Crystallogr. A64, 112–122.

Spek, A. L. (2003). J. Appl. Crystallogr. 36, 7–13.

Stenkamp, R. E. and Jensen, L. H. (1975) Acta Crystallogr. B31, 1507–1509.

Taylor, R. and Kennard, O. (1983). Acta Crystallogr. B39, 517–525.

von Schnering, H. G. and Dong Vu (1983). Angew. Chem. Int. Ed. Engl.

22, 408.

Wilson, A. J. C. (1949). Acta Crystallogr. 2, 315–321.

Wilson, A. J. C. (1976). Acta Crystallogr. A32, 994–996.

250 Random and systematic errors

Exercises

1. Show that (16.1) and (16.2) can be derived from (16.3)

and (16.5) if unit weights are used.

2. The data in Table 16.4 are H…O distances taken from

structuresdetermined with neutrondiffraction, contain-

ing a certain type of hydrogen bond.

(a) Calculate the weighted value of χ

and χ

red

using

= 1/σ

(b) Is calculation of a mean justiﬁed for these data?

Discuss your answer in terms of the likely effects

of environmental factors on hydrogen bonds.

about χ

, and says that you must calculate an

average. What standard deviation should you

quote?

Table 16.4 H…O distances from

neutron-diffraction data.

σ(x

)

1.814 0.0015

1.844 0.003

1.728 0.003

1.832 0.003

2.121 0.003

1.997 0.0075

1.808 0.0075

1.833 0.009

1.739 0.009

1.772 0.009

1.742 0.0105

1.877 0.012

1.948 0.012

3. The data in Table 16.5 were measured at points x giving

measured values y.

Table 16.5 Data points

for Exercise 3.

1 7.1

2 34.9

3 111.2

4 258.7

(a) Fit these data to an equation of the form y = a +

, ﬁnding the values of a and b by least-squares.

(b) Work out an R factor.

Hint: The crystallographic R factor is R =



−F



(d) For a particular application the quantity

c = a + b

is important. Compare the standard uncertainties

in c obtained if covariance terms are included or

excluded.

Note: For a function f(x

, x

, ..., x

) the full

propagation of error formula is

(f) =



i,j=1

∂f

∂x

∂f

∂x

· cov(x

, x

where cov(x

, x

) are variances [σ

)], and

cov(x

, x

) are covariances.

4. The following ALERT was issued by CHECKCIF after a

reﬁnement where restraints had been applied:

732_ALERT_1_B Angle Calc 105(4), Rep

104.9(8) 5.00 su-Rat

N2 -O1 -H1 1.555 1.555 1.555

What response might be given?

5. In a particular structure determination the bond angles

in a nitrate anion were found to be

120.1(2), 119.4(2), 119.5(2)

◦

What is the sum of the angles and its s.u.?

6. Bond angles in a substituted cyclopropane ring are

reported as:

59.3(2), 59.6(2), 61.0(2)

◦

What is the sum of the angles and its s.u.?

Powder diffraction

John Evans

17.1 Introduction to powder diffraction

X-ray and neutron powder diffraction are extremely powerful tools for

probing the structural chemistry of materials in the solid state. Both

techniques can be used to gain information about the composition of a

bulk material, the degree of crystallinity of its components, information

about its unit cell size and symmetry and, in favourable cases, full 3-

dimensional structural information comparable to that obtained from

single-crystal methods. Powder diffraction experiments can be readily

performed under the inﬂuence of external factors such as temperature,

pressure or applied magnetic ﬁeld, under laser illumination of a sample,

and even as a function of time or chemical environment during the

synthesis of materials, giving valuable kinetic and mechanistic insight

into their formation.

The primary focus of this book and the BCA crystallography school

on which it is based is on single-crystal methods as applied to ‘small

molecules’. There is no doubt that if one’s primary goal is the elucidation

of high-quality structural data, single-crystal diffraction will always be

themethod of choice. Nevertheless, the opportunities offeredby powder

diffraction methods should not be forgotten. In many cases single crys-

tals of interesting materials of sufﬁcient size/quality for single-crystal

methods simply can not be prepared (though increasingly small micro-

crystals can now be studied at a synchrotron source; see Chapter 22).

This is particularly true for the extended materials discussed in Chapter

14, where low solubility, phase transitions leading to multiple twinning,

or the speciﬁc synthetic conditions required make single crystals hard or

impossible to obtain. For many categories of technologically exploited

materials (zeolites, high-T

superconductors, structural materials, mag-

netic materials, conducting oxides, multiferroics, ionically conducting

polymers, etc.) the key structural insights came from powder diffraction

studies since single crystals were not available.

In other applications powder diffraction may provide complementary

information (such as bulk sample composition) to single-crystal tech-

niques. Non-ambient studies (under extremes of temperature, pressure,

magnetic ﬁeld or optical irradiation) are often simpler to perform on

powdered samples than single crystals, allowing the potential to study

functional materials under ‘real’ operating conditions; powder studies

251