An introduction to Direct Methods

17
An Introduction to Direct Methods.
The Most Important Phase
Relationships and their Application in
Solving the Phase Problem
by
H. Schenk
TIO, AL UN,OI ]
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Copyright in this electronic edition (i)2001 International l.Jnion of
Crystallography
Published for the
International Union of Crystallography
by
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© 1984 by the International Union of Crystallography.
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Pub]islaed by the University College Cardiff Press for the
International Union of Crystallography with the
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This pamphlet is one of a series prepared by the
Commission on Crystallographic Teaching of the
International Union of Crystallography, under the
General Editorship of Professor C. A. Taylor.
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the series may be ordered direct from the
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ISBN 0 906449 71 5
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Series Preface
The long-term aim of the Commission on Crystallographic Teactiing in
establishing this pamphlet programme is to produce a large collection of
short statements each dealing with a specific topic at a specific level. The
emphasis is on a particular teaching approach and there may well, in time,
be pamphlets giving alternative teaching approaches to the same topic. It
is not the function of the Commission to decide on the 'best' approach but
to make all available so that teachers can make their own selection. Similarly,
in due course, we hope that the same topics will be covered at more than
one level.
The first set of ten pamphlets, published in 1981, and this second set of
nine represent a sample of the various levels and approaches and it is hoped
that they will stimulate many more people to contribute to this scheme. It
does not take very long to write a short pamphlet, but its value to someone
teaching a topic for the first time can be very great.
Each pamphlet is prefaced by a statement of aims, level, necessary
background, etc.
C. A. Taylor
Editor for the Commission
The financial assistance of UNESCO, ICSU and'of the International Union of Crystallogra-
phy in publishing the pamphlets is gratefully acknowledged.

Teaching Aims
To help students, with some basic knowledge of Crystallography, to
understand the principle of direct methods.
Level
This course is suitable for the first years of undergraduate study in any
direction of science.
Background
Students should have understanding of Fourier Analysis of the electron
density, and some knowledge of the structure factor equation.
Practical Resources
No particular resources are needed.
Time required for Teaching
This course may occupy 2-5 hours of teaching.

An Introduction to Direct Methods.
The Most Important Phase Relationships and their
Application in Solving the Phase Problem,
..<.;c,'-.',L~'.. 4 "
H. Schenk ....~x ~k. ; ~
Laboratory for Crystallography, U n i v e ~ ~ ~ 7
Introduction
The term 'direct methods' is applied to that class of methods which seek
directly to solve the phase problem by the use of phase relationships based
on the observed intensities.
The object of this pamphlet is to familiarize the reader with the phase
relationships used in Direct Methods, and to explain why they work and
how they are used in practice. Some prior knowledge of the phase problem,
the structure-factor equation and the application of Fourier theory in
crystal-structure analysis is assumed.
Notation
H = Laue indices h k I of a reflection
K = Laue indices hj k~ l~ of another reflection
OH= Bragg angle of a reflection H
I. =intensity of a reflection H (I.alFH[ 2)
F. = the structure ractor of H. In this work F. will be
corrected for thermal motion and brought onto on
absolute scale.
EH = normalised structure factor of H
UH = unitary structure factor of H
GH = structure factor of H based on the squared structure
~bn = phase of reflection H
N = number of atoms in the unit cell
= scattering factor of atom j
Zj = atomic number of atom j
rj = position of atom j in vector notation
xi, yj, zj = position of atom j in Cartesian coordinates
e, = N- I e.eke ._ KI
E4= N-'IEHEKEt.E-H-K-t.]

Strong and weak Structure Factor Magnitudes FH
If, in a crystal structure, atoms lie in the neighbourhood of a set of planes
H, as indicated in Fig. l a, then reflection by planes H is strong and hence
the intensity I H is large. Of course, the converse is also true: if one observes
a large intensity In, then the atoms lie near planes as indicated in Fig. la.
This statement follows also from the structure-factor expression:
N
FH = IFHI exp (i4,H) = r£ exp 27ri(hxj+kyj +lzj)i
A large FH will be found if (hx~ + kyj + lz~) rood I is approximately constant
for all j; or, in other words, if all atoms lie near one of the planes H. The
phase 4'n depends on the value of the constant and changes with the origin.
Conversely, a structure-factor magnitude IF.I is small, if the atoms are
randomly distributed with respect to the planes H, as shown in Fig. lb.
Fig. I. A reflected beam H has is strong when the atoms lie in the neighbourhood of the set of
planes H (a) and low when the atoms are spread out with respect to the planes H (b).

The electron density can be thought of as a superposition of density
waves parallel to lattice planes, the amplitudes of which are the [FH [-values,
the relative phases being given by the <bH-values. We will see later that
these density waves afford a physical picture of the phase relationships hsed
in Direct Methods.
Normalized Structure Factors EH
Note: in this text FH designates the structure factor corrected for thermal
motion and brought to an absolute scale; generally this is done using a
Wilson plot. Since the scattering factor of any atom decreases for larger
reflection angle 0, and the expected intensity ([F[2)o of a reflection is given
by
N
(lF[2)o = ~ f~.(8) (1)
j=l
reflections measured at different 8-values can not be compared directly.
Expression (I) can be used to calculate the so called normalized structure
factor
le.12-Z~, f} (2)
It is obvious from a comparison of (1) and (2) that (E~) = 1 for all values
of O.
The structure-factor expression in terms of the normalized structure factor
is then:
1 N
EH - (E ~ J¢~)'J~,=£ exp 2 zri(hxj: + kyj + lzj). (3)
If the form factor fj has the same shape for all atoms (fj = ZJ), expression
(3) can be written as
1 N
EH (~t f~)½j~=Ifi exp 27ri(hxi +kyj +lzJ)" (3)
This is clearly the structure factor formula of a point atom structure, because
no 0-dependent factors are present any more.
In order to find the maximum value of [E I, let us consider an equal atom
structure for which the structure factor (4) further reduces to
I N
Eu - (NZ2) ,_Z ~. exp 2zri(hxs + kys + Izs)
j=l
| N
=--~-~r,Zt~-= exp 27ri(hxj +kyi +lzj).
(5)

The maximum possible value of ]EH] is N/N"-= N".
-The unitary structure factor U was used extensively in the early literature
on Direct Methods:
IF~,I
Iu. I- Zj~, £ (6)
The denominator represents the maximum possible value of FH and thus
Ut.t varies between 0 and 1. In the equal atom case the relation between
U. and IE. l is given by
IE. I"= NIu.I 2. (7)
which can easily be verified by the reader from (6) and (2).
The ]El's of H and 2H: The ~1 Relationship
The Et relation is the first phase relationship which will be considered
here; it estimates in centrosymmetric space groups the phase of reflection
2H on the basis of the magnitudes [EHIand IEzH[.To start with, geometrical
considerations will be applied to reflections with simple indices.
In a centrosymmetric crystal only phases of 0 and ~- Occur; provided that
the phase of the 110 reflection is 0 the maxima of the associated electron
density wave are found at the lines I of Fig. 2 and the minima at the lines
Fig. 2. Lines of equal contribution to the electron density of the reflections hho. E.g. for reflections
110 with d~l~o = 0 the contribution to the electron density is maximum at lines I, minimum at lines
II and zero at lines IlL

II. If the phase of I 10 is rr, the maxima and minima are interchanged. The
lines where the electron density wave has 0-value are marked with III. Thus
in the event ]E~,o[ is large and 05t,o=0, the electron density is mainly
concentrated in the shaded areas of Fig. 3. For the electron density, wave
associated with the 220 reflection the maxima are found at both lines I and
II in Fig. 2 in the case its phase is 0 and the minima at the lines III. Thus,
when IE2,_olis large and 05220= 0 the atoms must lie in shaded areas in Fig. 4.
A similar drawing can be made for 0522o='Tr.
The combination of the tWO electron density waves, associated with the
reflections 110 and 220 leads: to Fig. 5, in which in the areas I "ma.xima are
found of both density waves. In the areas II.the maximum of 220 cdincide
with the minimum of I I0, resulting in a low density. In the event, that both
reflections have a large IE[-value it is likely that the atoms are Corlcentrated
in the double shaded area. : .: "
In case the phase 052,_0= ~r,'the vertically shaded area's shift ro the blank
regions of Fig. 5 and then thei-e is no overlap between the h'orizontally (1 I0)
and verffcally (220) shaded areas; this impl!es that no position for the atoms
can be found in which they contribute strongly to both.structure factors.
As a result for 05220= ~r and 05, ~o= 0 it is not likely that both structure factor
magnitudes I.E,,ol and [E22o1 are large.
In conclusion, for large structure factors ]E,,o[ and IE_,,.ol,it is likely that
052,_0= 0; this relationship is known as the Y.~ relation.
_=-
Fig. 3. In case/Ello/is large and dgtta =0 the atoms are likely to be found in the shaded areas.

iil' ,lil IllI ,:!1 Ill" ,Ill IIJ"
ii~,, I1" ~ I1~'~
Fig. 4. For/Eeeo/large and cb,,o = 0 the electron density is more likely to be present in the uerticaUy
shaded areas.
rl '
Fig. 5. Superposition of Figs. 3 and 4. In the area's I the shaded areas from the 110 and 220
reflections coincide. In case both reflections are large this is a rather likely situation.

H '/":il.........
Fig,. 6. The drawn line H gives the electron density wave for (b~s= O, and its dotted mirror image
the wave for c~H = rr. The maximum of the dashed line 2H coincides with the maxima of the drawn
!irte H in P and with the maxima of the dotted one in Q. Thus if /EH/ and /E2, / are large, it is
likely that d~eH= 0 whatever the phase of H.
Up to now no attention is;paid to the situation qS~o= rr; the reader is
invited to show that this gives no change in the formulation of the ~.~ relation.
The comparison of H and 2H can be considered as a one-dimensional
problem which can beunderstood by looking along line A in Fig. 2. In Fig.
6 the situation along this line is sketched with ~bH= ~b2H= 0 while in Fig. 7.
~bH= 0 and ~bzH= ~r.Areas labelled P in Fig. 6 denote regions of consider-
able positive overlap, whereas in Fig. 7 only regions of minor positive
overlap are seen. The implication is that for large IE. Iand IE2HIthe situation
depicted in Fig. 6 is more prgbably true and thus ~b2, = 0. When ffH = ~',
as denoted by the dotted line in Fig. 6 the overlag areas marked Q show
that 052H is still zero.
H 2H
Fig. 7. Here the unlikely situation is depicted that for strong reflections H and 2H dpH= 0 and
.~,-H= rr. There is no positive overlap and therefore if /Eu/ and /E,H/ are both large this situation
is much more unlikely to exist that the situation of Fig. Z

The El-Relation from a Harker-Kasper Inequality
In 1948 Iqarker and Kasper published their paper on inequality relation-
ships, which actually opened the field of direct methods. They applied the
Cauchy inequality:
a~ ~ E lajl~ 2 Ibjl'- (8)
j=I j=l j=l
to the structure factor equation. For instance the partitioning of the unitary
structure-factor equation in P1 into:
Uu = Z nj cos 2wH" r= ~2 ajbs (9)
j=l j=l
such that as = n)/'- and b~= nJ/'-cos 27rH" r leads to
U~ (j~, nj)(j_~_, njcos 22~rH" r). (10)
From the definition of the unitary structure factor it follows that
N (11)
, ~nj=l
j=l
and the second factor can be reduced as follows
n~cos'- 2~rH" r = ~ ½n~(1+ cos 27r2H" r)
j=l j=l (12)
=~(l + u,,).
These results used in (10) give
u,, -<-',(1+ u,_,). (13)
In case U~>~ then U2H/>0 or in other words the sign of reflection 2H is
positive whatsoever its IU2HI-value is. Note that the sign of H may have
") I
both values. In practice Ui~ > ~does not often occur. However, when IU2HI
is large, expression (13) requires the sign of 2H to be positive even if UH
is somewhat smaller than ½.Moreover, when [UHI and IU2Hl are reasonably
large, but at the same time (13) is fulfilled for both signs of 2H, it is still
more likely that S2H = + than that S_,H= -. For example, for IUHI=0.4
and IU2H[=0.3, Sz, = + leads in 13 to 0.16<~0.5 +0-3 which is certainly
true, and S2H = - to 0.16~<0.5 -0.3 which is also true. Then probability
arguments indicate that still SzH = + is the more likely sign. The probability
is a function of the magnitudes IUHI and IU,_H[ and in this example the
probability of SzH = + being correct is > 90%. In conclusion the mathemati-
cal treatment leads to the same result as the graphic explanation from the
preceding paragraph: the ~'~ relationship.
8

Large IE.I, IEKIand IE-H-KI: The Triplet Relationship
If two reflections H and K are both strolag then the electron density is
likely to be found in the neighbourhood of the two sets of equidistant planes
defined by H and K. That is to say the electron density will be found near
the lines of intersection of the planes H and K as indicated in projection
in Fig. 9. A large IEI for reflection -H-K as well implies that the electron
Fig. 8. A few large terms (I:Ft<FH_K: H:FK,FH_K,: etc) from the right hand side of expression
(27) in a phase diagram. It can be seen that their phases (l:dpK "I-chH_K:2:CkK.+~bH-K,: etc) are
approximately equal to 4~H.

Fig. 9. If the reflections H and K are both strong, then the electron density will likely lie in the
neighbourhood of the intersecting lines of the two sets of equidistant planes defined by H and K.

density will also peak in planes lying d-H-K apart. It is therefore most
likely that these planes run through the lines of intersection of the planes
H and K, in other words that the three sets of planes have their lines of
intersection in common (see Fig. lOa) Then by choosing an origin at an
H
-K
Fig. 10. When H and K are strong and -H-K is strong as well it is more likely that the planes
of high density of-H-K run through the lines of intersection (a) than just in between (b).
I0

arbitrary point the triplet phase relationship can be found from a planimetric
theorem, proved in Fig. l I:
AO/AD+BO/BE+CO/CF=2 ' (14)
which is equivalent to
~b-H-K + &H + ~b~:+2 • 21r = 0 (modulo 2rr). (15)
Because the choice of the origin is arbitrary it is obvious that expression (15)
is independent of the position of the origin: relations of this type are usually
called 'structure invariants', although a more logical name would be 'origin
invariants'.
In Fig. 10a the ideal situation is sketched and of course a small shift of
the planes of largest density of -H-K does not affect the reasoning given
above. However, the most unlikely position for these planes is the one
indicated in Fig. 10b; here the planes -H-K of largest electron density
keep clear of the lines of intersection of H and K. The triplet relationship
therefore has a probability character and this is emphasised by formulating
it as
~'~H "~ ~)K "I-~--H--K "~"0 (16)
for large values of E3 = N-I/21EHEKE_H_KI. The ~-sign means that the
most probable value of the triplet phase sum is O. Clearly, the triplet product
E3 is large when all three reflections H, K and -H-K have large ]El-values.
A
B D Q C
Fig. I1. In an arbitra~ triangle ABC an origin 0 has been arbitrarily. Theorem: AO/AD+
BO/BE +CO/CF=2. Proof." AO/AD=AP/AC;CO/CF=CR/AC; BO/BE=BQ/BC=
AS/AC; because RP=SC, AP+CR +AS=2AC.
II

The Triplet Relation from Sayre's Equation
The earliest formulation of the triplet-relation (10) for the centro-
symmetric case was via Sayre's equation (Sayre, 1952). This equation can
be derived from Fourier theory as follows.
The electron density can be written as
1
p(r)---~Z p. e×p (-2~i.. r) (17)
and upon squaring this function becomes
1 (18)
(18) is rewritten by setting H = L+L' and K = L' to
1
P2(r)=-~2~ ~ FKFH-K exp (-2~iH. r). (19)
Because p2(r) is also a periodic function it can be written, by analogy with
(17), as
1
p2(r)=~ Z GH exp (-27rill. r) (20)
in which G. is the structure factor of the squared structure. Comparing
(19) and (20) it follows that
1
c.
The structure factor GH is:
N
GH = Z gj exp 2Tri(H. r2) (22)
j=l
in which gj is the form factor of the squared atoms. For equal atoms (22)
reduces to
N
GH = g 2 exp 27ri(H. 5)" (23)
j=l
The normal structure factor for equal atoms is
N
Fn =f ~ exp 2~ri(g. rj). (24)
j=l
12

Thus from 23 and 24 we obtain
g F.. (25)
G. =f
Finally from 21 and 25 it follows that
flV.=gV~ FKF,-K (26)
which is known as Sayre's Equation. It is emphasised that, given an equal-
atom structure, Sayre's equation is exact. The summation (26) contains a
large number of terms; however, in general it will be dominated by a smaller
number of large IFKF,-K[.Rewriting (26) to
IV.lexp,e,.= ZlF.V.-.lexP'( . (27)
and considering a reflection with large IFHI it can therefore be assumed
that the terms with large IFKFH-KIhave their angular part approximately
equal to the angular part of IFHI itself, illustrated in Fig. 8. For one strong
IFKFHi_KI this leads to:
exp i~6H~ exp i (~K + q~H-K) (28)
or ~b. ~ ~bK+~b.-K (29)
or
4~-. +4~K +~.-K ~0.
Relation (29) is identical to (16), the triplet relation. Thus by introducing
the obvious argument that the most important terms in Sayre's equation
(27) must reflect the phase &H the triplet relation is found.
In the event that only a number of larger terms in (27) are available the
scaling constant f/gV has no meaning. Nevertheless most likely the phase
information included in these terms is correct and thus an expression such
as
EkIF~:F.-KIexp i(qSK+4.-K) (29)
exp iq~. = lY.KIFKF.-~Iexp i(05K+4.-K)I
in which K ranges over a limited number of terms may be very helpful.
The so called tangent formula (Katie and Hauptman, 1956)
Y.K E3 sin (OSK+ ~b.-K). (31)
tan q~. =Y.K E., cos (4~K+4~,-~:)
in which the signs of numerator and denominator are used to determine
the quadrant of the phase ~b., is closely related to (30). This formula is used
in almost all direct method procedures.
13

The Positive Quartet Relation
The triplet relation, although a two-dimensional phase relation, is very
successful in solving three-dimensional crystal structures. Nevertheless, it
may be more appropriate to try to solve structures with three-dimensional
phase relationships, the quartet relations.
The positive quartet relation is formulated as:
~bH+~b,~ +4,t +~b_H_~_t =0 (32)
for large E 4 : N -l [EnEKELE-H-r-LI.
Analoguous to the treatment of the triplet relation now three strong
reflections H, K and L are combined and the electron density must be
found in the sets of planes of Fig: 12. As a result the electron density will
be found near the points of intersection of the three planes which are
indicated for only a few planes from the sets in Fig. 13. For a strong reflection
-H-K-L it is much more likely that its plane of maximum electron
density will run through the points of intersection (Fig. 14a) than that it
will clear these points (Fig. 14b). From Fig. 14a the quartet relation (32)
follows as straightforward as the triplet relation from Fig. 10a (Schenk,
1981).
This quartet relation, however, is not as strong as the triplet relation
because of the factor N -z in E4. It will be recalled that in E3 a term N-~
J
I
I
Fig. 12. If the reflections H, K and L are strong, the electron density will probably lie in the
neighbourhood of the three sets of equidistant planes defined by H, K and L.
14

Y
Fig. 13. The electrondensity will befound near the points of intersection of the three sets of planes
H, K and L.
appears. The reliability is improved by combining the quartet with an
identical one constructed from two triplets:
q~. +4~K + q~-.-K ~0
, ~b.+ 6-.-K-L+4~.+K =0
-1-~b. +~bK +~bL+~b-n--K--L ~0 (33)
which holds for large E. and large IEn+K]. This can be understood by
drawing the H +K reflection in Fig. 13, as indicated in Fig. 1ha. Reflection
H +K strong indicates that the electron density will be found near the
intersection of H and K and thus a large [EH÷K[ is an additional indication
that quartet (32) is true. In Figs. 15b and 15c similar situations are sketched
for the other two cross terms H + L and K + L, which both leads to sums
of triplets analogous to (33) involving the phases 4~H+t. and q~K÷Lrespec-
tively. So in conclusion a large value of E4 and large le.+KI, IZ.+LI and
[EK+L[ are indications that the positive quartet relation (32) is likely to be
true, and thus positive quartets are controlled by the magnitudes of 7
structure factors.
The Negative Quartet Relation
In the event that the sum of the four phases is equal to ~:
(J)H 2C(~K 4-C~L+(/)-H-K-L: "71" (34)
the resulting relationship is 4 referred to as the negative quartet relation
15

• I "...
~ ~ ~ ~ ~- III
0 "-
/
I
J
I1 ~ ~,~
I1 ~ "~.,.
".. / I
)-...
/ "-
I
J 1
J
I " ....".. ~."..~. / II III1lI lI "... . ~'~'~. .
I I
L, I
--. --..
N
Fig. 14. If reflection -H- K- L has also a large intensity, it is more likely that its planes of
maximum intensity run through the points of intersection of H, K and L (a) than that they run
clear of them (b).
and such relationship exist for reasonably strong intensities for H, K, L
and -H - K - L.
The planes of maximum for the 4 reflections involved in relation (34) are
indicated in Fig. 16. It can be seen that for all indicated positions three out
of four planes intersect. If atoms are located at these points the resulting
unitary structure factors of H, K, L and -H-K- L will be 0.5, because
three atoms lie in the planes and one lies halfway between. Thus for a
negative quartet relation the reflections H, K, L and -H-K-L will in
16

N.
I ii
I II
i; 11
i/
I,,' eN
I
Fig. 15. lf H, K, L and -H- K - L are strong and the electron density will be found near their
points of intersecting, the H + K reflection, indicated in (a) with a dotted line, is expected to be
strong, conuersely a large /EH.K/ supports the quartet relation (II). Fig.'s (b) and (c) give the
analogous situations with respect to the H + L and K + L reflections.
17

Fig. 16. In the case of the negative quartet relation the planes of maximum electron density of
H, K, L and -H-K-L run as indicated here. At distinct positions three planes intersect and
there most likely the electron density will be found.
general not be found amongst the very strongest. The next question to be
answered is: what is the intensity of reflection H + K, if the electron density
is located near the market points of Fig. 16. From Fig. 17 it can be easily
seen that H + K will have a small [El-magnitude: equal numbers of points
of electron density concentration lie on the H + K-planes and halfway in
between. As the same holds for the other cross terms it can be stated that
the negative quartet relation (34) is likely to be true for reasonably large
values of E4 and small IE,+KI, IEH+~[and IEK+,.I.
S
i •
/ it
• - .... "-~" _'~_._~'Jl
"--.
.......... "-® ' ~ _~
,~, --~ _
Fig. 17. [s fig. 16 with the reflection H + K indicated by means of dotted lines. It is easily checked
that this reflection is weak, because the electron density is distributed in equal amounts on and
between the planes.
18

How Numerous are the Reliable Triplets and Quartets
In the following table numbers of relations are given together with their
percentage of correct indications for triplets, quartets and negative quartets
above variable thresholds of respectively the triplet product E3 and a quartet
product E* (Schenk, 1973). The numbers are given for a a~za-steroid with
N = 40, in space group P]'.
Triplets
% Positive quartets Negative quartets
nr correct
E~ relations relations E4 nr % nr %
6.0 21 100 6.0 185 100
4.0 143 100 4.0 1213 I00
3.0 353 100 3.0 3295 100 I I00
2.5 583 99.8 2.5 5813 99.8 2 100
2.0 980 99.7 2.0 10,006 99.5 17 I00
1.5 1823 99.2 1.5 13,114 98.8 38 I00
1.0 3395 96.9
As can be seen many relations are available to solve this small N = 40
structure. As a rule the number of useful tn.'plets, and quartets diminishes
as N increases; this effect is quite noticeabl'e for quartets. :
One comment regarding the use of negative quartets. If phase relationships
such as the triplet relation
4~H+q~K +~-H-K =0.
are used exclusively and there is no translational symmetry, the trivial
solution with all phases ~H = 0 is the most consistent one. To find phases
equal to ~"(e.g. in space group P1) it is necessary to use relations of the type
: ~bH+~bK +...~ ~'.
Thus relations such as negative quartets (34), although few in number, play
an important role in these Structure determinations.
Direct Methods in Action
The first direct method, by means of which structures were solved, was
the symbolic addition method. This method originates from Gillis (1948),
however, due to the work of Karle and Karle (1966) it developed to a
standard method. The problem can be defined as how do we find rn phases,
provided there are n phase relationships (n >>m). In the first place a few
(~<3) phases can be chosen to fix the origin and then, using phase relation-
ships, new phases can be derived from these three. In general it will not be
19

possible to phase all reflections in this way and hence a suitable reflection
(large ]E[, many relationships with large E3) is given a symbolic phase and
again the relationships are used to find new phases in terms of the already
known ones. Usually it will be necessary to choose several symbols in order
to phase most of the strong reflections. Finally the numerical values of the
symbols are determined (e.g. by using negative quartet relations) and from
the known phases a Fourier map can be calculated. This process is known
as the symbolic addition method. Most structures are now solved by multi:
solution tangent refinement procedures, which use many starting sets of
numerical phases and the tangent formula (31) to extend and refine the
phases. The correct solution may then be selected by using figures of merit,
based e.g. on the internal consistency of the triplet-relations, or on the
negative quartets.
Additional Literature
In the preceding chapters the main object was to clarify the basis of the
present direct methods. In this chapter a brief guide to additional literature
is given.
This triplet relation originates from the early fifties and was implicitly
present in the important papers by Harker and Kasper (1948), Karle and
Hauptman (1950) and Sayre (1952). For the centrosymmetric case it was
explicitly formulated by Sayre (1952), Cochran (1952), Zachariasen (1952)
and Hauptman and Karle (1953). The latter authors gave it its probability
basis, which was independently derived by Kitaigorodsky (1954) as well.
The noncentrosymmetric case was formulated first by Cochran (1955).
Another useful expression related to the Y-2relation is the tangent formula
(31) derived by Karle and Hauptman (1956).
A very important development was the use of symbols for tackling the
set of triplet relations (1) in order to lind the phases. Symbols are assigned
to unknown phases such that a successful phase extension can be carried
out. Later in the process in most cases the numerical values of the symbols
can be determined. The use of symbols was first introduced by Gillis (1948)
and later successfully applied by Zachariasen (1952) and Rumanova (1954),
but due to the work of Karle and Karle (1963, 1966) the method could
develop to a standard technique in crystallography. In particular the first
structure determination of a non-centrosymmetric structure (Karle and
Karle, 1964) proved the value of direct methods. The method has recently
been described in detail by J. Karle (1974) and Schenk (1980a). The latter
gives also some exercises. For centrosymmetric structures the symbolic
addition procedure has been automized amongst others by Beurskens (I 965),
Germain and Woolfson (1968), Schenk (I 969), Ahmed (I 970), Dewar (1970),
and Stewart (1970).
20

In noncentrosymmetric structures the programming problems are much
greater and therefore the number of successful automatic program systems
is smaller, examples are the systems of Dewar (1970) and: the interactive
system SIMPEL (Overbeek and Schenk, 1978).
Nowadays most of the structures are solved by multisolution tangent
refinement procedures, which use many sets of numerical phases to start
with and the tangent refinement (31) to extend and refine the phases. Th'e
most widely used procedure of this sort is the computer package MULTAN
(Germain and Woolfson, 1968; Main, 19:78; Main, 1980).
The positive seven-magnitude quartet relationship (32) was first formu-
lated by Schenk (1973) and at the same time a two-dimensional analog}~
of the negative quartet relationship proved to be useful. (Schenk and d~e,
Jong, 1973; Schenk, 1973b). The negative quartet in theory and practice"
was then published by Hauptman (1974) and Schenk (1974). In the latter
paper the first Figure of Merit based on negative quartets was successfully
formulated and tested. Theories concerning 7 magnitude-quartets were
developed later, among which the one of Hauptman (1975:) is best estab-
lished. Applications of quartet.s include their use in starting set procedures
and figures of merit, further brief details of which can be found in a recent
review article (Schenk, 1980b).
a~'"
!.
References
Ahmed, F. R., in F. R. Ahmed (ed.), Crystallographic Computing, pp. 55-57, Copenhagen,
Munksgaard (1970).
Beurskens, P. T., Thesis, Utrecht (1965).
Cochran, W., Acta Cryst. 5 (1952), 65--68.
Cochran, W., Acta Crvst. 8 (1955), 473--478.
Dewar, R. B. K., in F. R. Ahmed (ed.), Crystallographic Computing, pp. 63-65, Copenhagen,
Munksgaard (1970).
Germ~in. G. and Woolfson, M. M., Acta Crvst. B24 (1968), 91-96.
Gillis, J., Acta Cryst. 1 (1948), 174--179.
Harker, D. and Kasper, J. S., Acta Cryst. I (1948), 70-75.
Hauptman, H. and Karle, J., ACA Monograph No. 3, Pittsburgh, Polycrystal Book Service
(1953).
Hauptman, H., Acta Crvst. A31 (1975), 680--687.
Hauptman, H., Acta Cryst. A30 (1974), 472--477.
Karle, I. L. and Karle, J., Acta Cryst. 16 (1963), 969-975.
Karle, I. L. and Karle, J., Acta Cryst. 17 (1964), 835-841.
Karle, J., in International Tables for X-ray Crystallography, Vol. IV, Section 6, pp. 337-358,
Birmingham, The Kynoch Press (1974).
Karle, J. and Hauptman, H., Acta Cryst. 3 (1950), 181-187.
Karle, J. and Hauptman, H., Acta Cryst. 9 (1956), 635-651.
Karle, J. and Karle, I. L., Acta Cryst. 21 (1966), 849-859.
Kitaigorodskii, A. I., DokL Acad. Nauk SSSR 94 (1954); Trudy Inst. Crystallogr. 10 (1954), 27.
21

Main, P., in H. Schenk, R. Olthof, H. van Koningsveld and G. C. Bassi (eds.), Computing in .
Crystallography, pp. 93-107, Delft, University Press (1978).
Main, P., in R. Diamond, S. Ramasheshan and K. Venkatesan (eds.), Computing in Crystallog-
raphy, pp. 800-813, I. A. S., Bangalore (1980).
Overbeek, A. R. and Schenk, H., in H. Schenk, R. Olthof, H. Van Koningsveld and G. C.
Bassi (eds.), Computing in Crystallography, pp. I08-112, Delft, Delft University Press (1978).
Rumanova, I. M., Dokl. Acad. Nauk_ SSSR 98 (1954), 399.
Sayre, D., Acta Cryst. 5 (1952), 60--65.
Sehenk, H., Transactions of the Kon. Ned. Akad. Wet, Series I, Vol. XXV, 5, Amsterdam,
London, North-Holland Publishing Company (1969).
Schenk, H., Acta Cryst. A28 (1972), 412-422.
Schenk, H., Acta Cryst. A29 (1973a), 77-82.
Schenk, H., Acta Cryst. A30 (1974), 477---482.
Schenk, H., in R. Diamond, S. Ramashan and K. Venkatesan (eds.), Computing in Crystallog-
raphy, pp. 700-722, I.A.S., Bangalore (1980a).
Schenk, H., Ibid. pp. 1000-1018 (1980b).
Schenk, H., Acta Cryst. A37 (1981), 573-578.
Schenk, H. and de Jong, J. G. H. Acta Cryst. A29 (1973), 31-34.
Stewart, J. M., in F. R. Ahmed (ed.), Crystallographic Computing, pp. 71-74, Copenhagen,
Munksgaard (1970).
Zachariasen, W. H., Acta Cryst. 5 (1952), 68-73.
22

International Union of Crystallography
Commission on Crystallographic Teaching
FIRST SERIES PAMPHLETS (1981)
1. A non-mathematical introduction to X-ray diffraction.
2. An introduction to the scope, potential and applications
of X-ray analysis.
3. Introduction to the Calculation of Structure Factors.
4. The Reciprocal Lattice.
5. Close-packed structures.
6. Pourquoi les groupes de Symetrie en Cristallographie.
7. Solving the phase problem when heavy atoms are in
special positions.
8. Anomolous Dispersion of X-rays in Crystallography.
9. Rotation Matrices and Translation Vectors in
Crystallography.
i0. Metric Tensor and Symmetry operations in
Crystallography.
C. A. Taylor
M. Laing
S. C. Wallwork
A. Authier
P. Krishna and D. Pandey
D. Weigel
L. Hohne and
L. Kutchabsky
S. Caticha-Ellis
S. HovmSller
G. Rigault
SECOND SERIES PAMPHLETS (1984)
11. The Stereographic Projection.
12. Projections of Cubic Crystals.
13. Symmetry.
14. Space Group Patterns.
15. Elementary X-Ray Diffraction for Biologists.
16. The Study of Metals and Alloys by X-ray Powder
Diffraction Methods.
17. An Introduction to Direct Methods. The Most
Important Phase Relationships and their
Application in Solving the Phase Problem.
18. An Introduction to Crystal Physics.
19. Introduction to Neutron Powder Diffractometry.
E. J. W. Whittaker
Ian O. A.lagelland
Moreton Moore
L. S. Dent Glasser
W. M. Meier
Jenny P. Glusker
H. Lipson
H. Schenk
Ervin Hartmann
E. Arzi
This selection of booklets represents a sample of teaching approaches at various .
levels (undergraduate and postgraduate) and in various styles. The Commission on
Crystallographic Teaching of the International Union of Crystallography hopes to
develop the series into a large collection from which teachers can make selections
appropriate to their needs and has particularly borne in mind the needs of developing
countries and of the problem of teaching crystallography to students of other
disciplines such as chemistry, biology, etc. and priced as follows: 95p each.
Available from:
University College Cardiff Press P O Box78 Cardiff CF1 1XL Wales, U.K.
Cheques should be made payable to University College Cardiff.

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