LETTERS
TO THE
lectures a n d practical work given, for example, in this
U n i v e r s i t y , for the degree of M. Sc. (Crystallography),
b y specialists in each field.
To be able to r u n a n X - r a y equipment, to measure a n d
i n t e r p r e t X - r a y photographs, to complete a structure
analysis a n d write a paper on it is not a sufficient t r a i n i n g
for a m o d e r n crystallographer, a n y more t h a n to be able
to c a r r y out a complicated chemical analysis is sufficient
t r a i n i n g for a chemist. Most of us know this, b u t have n o t
faced the implications, because we prefer research to
pedagogy. B u t this is not the w a y in which the best
research is u l t i m a t e l y done. P e r h a p s it is a p i t y t h a t we
call our science 'Crystallography', w h e n it is really the
s t u d y of the solid state, w i t h all t h a t t h a t implies.
EDITOR
875
I a m clear t h a t while t r a i n e d chemists, biochemists,
physicists, geologists, engineers or m a t h e m a t i c i a n s m a y
e v e n t u a l l y m a k e good crystallographers, t h e t r a i n i n g of
a really first-class crystallographer m u s t include somet h i n g of all these sciences a n d does, in fact, merit m u c h
more careful p l a n n i n g t h a n it has h i t h e r t o had. Crystallog r a p h y could be a first-degree subject in itself, w i t h these
other subjects as necessary or desirable ancillaries, a n d
w i t h branches of c r y s t a l l o g r a p h y as subsequent fields for
specialization.
Meanwhile I m a i n t a i n t h a t the places where even a
p a r t i a l l y a d e q u a t e t r a i n i n g in c r y s t a l l o g r a p h y can be
o b t a i n e d are few indeed, n o t more perhaps t h a n a dozen.
I should be glad to be proved wrong.
Notes and News
Announcements and other items of crystallographic interest will be published under this heading at the discretion of the
Editorial Board. Copy should be sent direct to the British Co-editor (R. C. Evans, Crystallographic Laboratory,
Cavendish Laboratory, Cambridge, England).
International Union of Crystallography
T h e U n i o n has received the following most generous
donations as contributions towards the expenses of its
publications:
F r o m the N e t h e r l a n d s Organization for P u r e R e s e a r c h
(Z.W.O.) the s u m of fl. 15,000 (approximately £1,500)
for Structure Reports.
F r o m Messrs C I B A A. G. the sum of Swiss fr. 2,000
( a p p r o x i m a t e l y £165) for Acta Crystallographica.
C o m m i s s i o n on
Crystallographic N o m e n c l a t u r e
T h e E x e c u t i v e Committee has accepted the recommendation of the Commission t h a t E. W. Nuffield (Canada)
should be co-opted on to the Commission.
Conference on Defects in Crystalline Solids
The H. H. Wills P h y s i c a l L a b o r a t o r y of the U n i v e r s i t y
of Bristol, E n g l a n d , in co-operation w i t h the Intern a t i o n a l U n i o n of P u r e a n d Applied Physics (particularly
its Commission on the P h y s i c s of the Solid State) a n d
w i t h The I n s t i t u t e of Physics, is organizing a conference
on 'Defects in Crystalline Solids' from 13 to 17 J u l y 1954
in Bristol. While n o t excluding other subjects in the field
the organizers propose to give particular a t t e n t i o n to
defects such as dissolved atoms, vacancies a n d 2'-centres,
to microwave resonance m e t h o d s of investigating their
properties, a n d to the w a y in which t h e y re-act w i t h
dislocations. Thus dislocations will be discussed in their
chemical aspects, as influencing diffusion a n d precipita-
t i o n in t h e solid state, r a t h e r t h a n in relation to plastic
flow.
I t is hoped t h a t a n u m b e r of a u t h o r s from overseas
will personally present t h e i r papers, a n d w i t h this in
m i n d the Conference has been a r r a n g e d to follow immedia t e l y after t h e General Assembly of the I n t e r n a t i o n a l
U n i o n of P u r e a n d Applied Physics.
B o a r d a n d lodging will be provided in Wills H a l l
(a s t u d e n t hall of residence) on special terms, or a t hotels.
The Conference is open to a n y scientist interested in
this field, subject to t h e limitations of accommodation.
F u r t h e r particulars m a y be o b t a i n e d from t h e Secret a r y , H . H . Wills P h y s i c a l L a b o r a t o r y , R o y a l F o r t ,
Bristol 8, E n g l a n d , or from the Secretary, The I n s t i t u t e
of Physics, 47, Belgrave Square, L o n d o n S.W. 1, E n g l a n d .
Those wishing to a t t e n d the Conference are asked to
a p p l y to t h e former, m a r k i n g the envelope '1954 Conference' a n d s t a t i n g w h e t h e r t h e y wish to be a c c o m m o d a t e d
a t Wills H a l l or a t a n hotel a n d for w h a t n i g h t s accomm o d a t i o n is required.
Photoelasticity and Photoplasticity
The I n t e r n a t i o n a l U n i o n of Theoretical a n d Applied
Mechanics annpunces t h a t a Colloquium on the above
subject will be held in Brussels, Belgium, from 29 to 31
J u l y 1954. The U n i o n also announces t h a t a Colloquium
on the Solid State will be held in Madrid, Spain, during
1955.
Crystallographers interested are invited to a t t e n d
these meetings; f u r t h e r information m a y be obtained
from the Secretary of the U n i o n (F. H . v a n den Dungen,
48 avenue de l'Arbal@te, Boitsfort, Brussels, Belgium).
876
Book Reviews
Works intended for notice in this column should be sent direct to the Editor (P. P. Ewald, Polytechnic Institute of Brooklyn,
99 Livingston Street, Brooklyn 2, N . Y . , U.S.A.). As far as practicable book,s will be reviewed in a country different
from that of publication.
Computing Methods and the Phase P r o b l e m
in X-ray Crystal Analysis. E d i t e d b y RAY
PEPINSKY. P p . ix + 390. P e n n s y l v a n i a S t a t e College:
X - r a y C r y s t a l A n a l y s i s L a b o r a t o r y . 1952. P r i c e
$ 7.50.
The reviewer ~articularly valued the o p p o r t u n i t y of rereading this volume for the present notice, a n d reflection
has led to an enhanced opinion of its worth. Although
primarily intended as a report of contributions to a
conference held at P e n n s y l v a n i a State College in April
1952, the descriptions of the X - R A C a n d S-FAC a n d of
the theoretical work inspired b y these machines have
m u c h p e r m a n e n t value a n d are certainly w o r t h y of
record.
An opening address, b y Pepinsky, in which various
sponsors receive thanks, leads to generalizations on the
n a t u r e of the information implicit in X - r a y single-crystal
diffraction data. A review paper, b y Bijvoet, on mathematical a n d computational problems of X - r a y analysis
follows; this contains a large n u m b e r of un-precise
formulae, for example eq. (1) (p. 10), eq. (3) (p. 15),
eq. (1) (p. 16), and particularly the description of leastsquare methods. Papers b y Beevers, Patterson, a n d
Buerger consider various aspects of implication or image
theory. The contribution of Beevers is, in the reviewer's
opinion, the most attractive, a n d forms one of the clearest
expositions of the subject in print. A paper, b y lYlacGill a r r y , on relations between H a r k e r - K a s p e r inequalities
a n d Buerger equalities makes the (too often forgotten)
point t h a t b o t h methods are likely to be useless on complicated structures. Goedkoop follows w i t h some relations
between structure factors (a minor error occurs in eq.
(1.2) (p. 61), which, however, does not invalidate the
s u b s e q u e n t discussion) ; the well-known Banerjee analysis
is repeated for the second time (it occurred previously in
Bijvoet's paper) and, assuming t h a t the phases of F
values are known, is used to derive relations between
structure factors. After deriving various s t a n d a r d inequalities a n d d e t e r m i n a n t a l relations, a section on the
possibility of practical application concludes w i t h the
remarks t h a t 'with growing n u m b e r of atoms a point is
reached b e y o n d which no s t r u c t u r e determination is
possible, independent of the methods u s e d ' - - a n observation w i t h which the reviewer heartily a g r e e s - - a n d
t h a t ' . . . m a t h e m a t i c a l tools are available for a n 'automatic' structure determination . . . ', which seems more
speculative. N e x t follows a paper b y F r i e d m a n on
m a t h e m a t i c a l aspects of the phase problem in which the
s t a t e m e n t appears (p. 75) t h a t n o n - n e g a t i v i t y does n o t
define a unique density distribution; there follows a
derivation of several inequalities depending upon nonnegativity, a t o m i c i t y a n d bounded positivity. E q u a t i o n
(10) (p. 77) is misprinted in a w a y t h a t m a y cause confusion. A further paper b y Bijvoet deals w i t h the isomorphous-replacement m e t h o d of phase determination,
a n d the ambiguities which result from its application to
non-centrosymmetric structures; a short description of
Bijvoet's elegant absorption-edge technique for eliminating this difficulty concludes the paper.
W h a t m a y be called the 'phase section' concludes w i t h
a note by A h m e d on the diffuse-scattering technique of
phase determination first suggested b y the present reviewer. A further subgroup of papers on analogue X - r a y
computers follows: I~obertson, Beevers & Robertson,
a n d R a m s a y & Lipson describe mechanical devices for
Fourier synthesis a n d structure-factor calculation. The
last authors give a pessimistic picture of the use of the
H~gg machine at Manchester a n d Beevers & l~obertson
have produced an excellent description of classical integrator theory a3 applied to one-dimensional Fourier
summation.
Bunn, in an exquisitely composed paper, considers the
relative advantages of the F l y ' s E y e a n d X - r a y microscope optical analogue calculators a n d P e p i n s k y ' s X - R A C
and S-FAC, a n d this should be read b y all aspiring
machine designers. Hanson, Taylor & Lipson discuss
their technique for introducing phase a n d amplitude into
the X - r a y microscope, and produce some excellent
photographic illustrations, although it is a p i t y t h a t Fig.
6(e) (p. l l 7 ) has not been given the same orientation as
(a) and (b). One wonders w h y the mica-plate technique
has worked so well a t Manchester whereas Perutz, at
Cambridge, (mentioned by Bunn) obtained only poor
results.
The final sub group of this section deals w i t h digital
computing methods. A paper b y Cox describes the b i n a r y
and t e r n a r y punched-card methods used a t Leeds, which,
however, require the use of a gang p u n c h - - a n item of
equipment not available w i t h all installations. Hughes,
in a paper too vague to be really useful, describes the
punched-card installation at the California I n s t i t u t e of
Technology, a n d gives the interesting estimate of $40
for the cost of a three-dimensional synthesis on this
equipment. The penultimate paper of t h e first section,
b y Ordway, deals w i t h high-speed digital computers and,
in particular, with the application of SEAC to crystalstructure analysis. An introduction to the preparation of
problems for a machine of this t y p e a n d a functional
description of the SEAC are followed b y a n analysis of
the calculation of dak~, Fhkl a n d fhkz b y t h e machine.
I t is interesting to notice t h a t , even wi~h this v e r y fast
machine, w i t h 1000 terms a two-dimensional synthesis
requires 1 hr. and a three-dimensional synthesis 8 hr.
On the other h a n d the calculation of 1000 F values takes
only 20 rain. The cost of computing is quoted a t $1 per
minute. The section concludes w i t h a note b y P e r u t z on
protein analysis, in which the use of Fourier transforms
of chains is discussed.
The second section of the book consists of a description,
b y Pepinsky, of the history, design, construction a n d
future of the X-RAC. The discussion is, in general,
pleasantly w r i t t e n and it is well illustrated. An amusing
piece of information (p. 171) is t h a t the X - R A C was
conceived during a carousel (roundabout) ride in a Boston
a m u s e m e n t park. The S-FAC (structure-factor a u t o m a t i c
calculator) a n d d a t a storage facilities which are to be
added to X-RAC will m a k e the installation one of v e r y
great power, although P e p i n s k y ' s description of 'The
Compleat Crystallographer' is perhaps over optimistic
BOOK
REVIEWS
(p. 272). A c o m m e n t which m a y be made is upon the
n o t a t i o n used for Fourier syntheses a n d structure factors:
there appears no good reason for the introduction of such
unfamiliar a n d inelegant expressions as t h a t given on
p. 261 (eq. (15)) w h e n t h e s t a n d a r d a n d accepted symbols
are available.
The final section of the book is composed of nine appendices. The first, b y Pepinsky, describes the preparation
of problems for the X - R A C a n d reproduces s t a n d a r d
forms for use b y those fortunate enough to have access
to the machine. The second appendix, also b y Pepinsky,
deals w i t h the extension of the range of indices available
on t h e X-RAC. The use of a non-standard n o t a t i o n leads
(p. 295, eq. (2)) to the remarkable result:
877
Data for X-ray Analysis. Volume I. Charts
for Solution of Bragg's Equation. By W. P x m
RISH a n d B . W . IRWIN. P p . 108. E i n d h o v e n : P h i l i p s .
1953. P r i c e $2.
Data for X-ray Analysis. Volume II. Tables
for Computing the Lattice Constant of
Cubic Crystals. B y W . PARRISH, M. G. EKSTEIN
a n d B. W . I R w i n . P p . 90. E i n d h o v e n : P h i l i p s . 1953.
P r i c e $2.
These volumes are the first of a series which is intended
to supplement the International Tables b y describing
m e t h o d s for facilitating certain routine calculations of
r a t h e r specialized interest. If the s t a n d a r d of these first
two volumes is maintained, it seems t h a t the complete
series will be v e r y useful indeed to those who deal chiefly
_F(R) -~ exp [-- 2rtiR. t]. F(R) .
w i t h t h e applied side of X - r a y diffraction.
The first volume is of interest to those who are conThe t h i r d appendix, b y MacGillavry & Pepinsky,
cerned w i t h identification of materials b y means of their
considers the application of conditions on Q(x, y, z) (e.g.
powder patterns. I t consists entirely of a large n u m b e r
non-negativity) to the X - R A C a n d the possibility of sign
of graphs of spacing against angle (0 a n d 20) for the K a
determination. A p p e n d i x 4, b y MacGillavry & Pepinsky,
lines of Me, Cu, Co, Fe and Cr radiations; where necessary,
analyses the calculation of 2' values b y sampling the
separate curves are drawn for K a 1, Ka2 a n d their weighted
continuous d e n s i t y distribution a t points on a Iattice.
means. The graphs are drawn on e x t r e m e l y open scales,
This technique, originally due to Beevers, is shown to be
a n d so are n o t subject to the usual criticism t h a t graphical
capable of accurate results so long as the lattice chosen
methods are n o t v e r y accurate. On the other hand, the
h a s a t least four times as m a n y points as there are
large scales necessitate several pages for each radiation,
(H, K) values. A valuable point is made on p. 312,
a n d one would h a v e t h o u g h t t h a t the information could
which can be condensed to ' I n ---- Out', a n d should be
have been presented more compactly in t a b u l a r form.
n o t e d b y all protein crystallographers.
The second volume is more diversified, a n d p r e s u m a b l y
Appendix 5, b y Pepinsky, discusses the use of positive
owes its existence to the difficulties sometimes m e t w i t h
kernels in t e r m i n a t i n g a Fourier series at a small n u m b e r
in identifying cubic substances from the A.S.T.M. index;
of t e r m s w i t h o u t producing diffraction effects, a n d gives
if a powder p h o t o g r a p h can be recognized as t h a t of a
particular a t t e n t i o n to the form { ( l + c o s x)/2} m.
cubic compound, identification is often more certain if
Appendix ~, b y Calderon & Pepinsky, extends the
its lattice constant, a, can be accurately meast~ed.
n o n - n e g a t i v i t y criterion of Herglotz, which is t h e basis
The tables m a k e use of the equation a = ]/N. ½2/sin 0,
of polynomial a n d i n e q u a l i t y theory, a n d leads n a t u r a l l y
where N = h ~+ k 2+ l 2. Values of ~/N. ½4 to six decimal
to A p p e n d i x 7, b y the same authors. I n the latter, the
places are given for the wavelengths of the al, a~, a n d fll
bounded-polynomial m e t h o d is discussed, a n d t h e poslines of the K radiations of Cu, Ni, Co, Fe a n d Cr, for
sibility of using the X - R A C in a steepest-descent process
values of N up to 378. Division of this q u a n t i t y b y the
is suggested.
observed value of sin 0 for a given line gives the a p p a r e n t
Appendix 8, again b y Calderon a n d Pepinsky, convalue of a derived from t h a t line. The results m a y t h e n
siders the uniqueness of solutions a n d concludes t h a t
be plotted against sin 2 0, or ½ (cos ~ 0/sin 0 + c o s ~ 0/0),
n o n - n e g a t i v i t y a n d finite total mass are sufficient to
extrapolation to 0 = 90 ° giving the true value of a;
define a unique solution in the centrosymmetric case
b o t h these functions are presented.
of a non-19eriodic structure, b u t not otherwise, a result
A n extensive list of lattice constants of cubic substances
which has been t a c i t l y assumed b y Bragg a n d P e r u t z
is included, together w i t h representations of typical
in their work on haemoglobin.
powder patterns, in the form of lines of v a r y i n g heights,
Appendix 9, b y Sayre, concludes the book; it forms an
d r a w n a t appropriate positions on log d scales.
excellent s u m m a r y of Fourier transform theory, as
I t is perhaps u n f o r t u n a t e t h a t the values of waveapplied to structure analysis, a n d contains some account
lengths used in t h e computations (J. Sci. Instrum. (1947),
of sampling m e t h o d s for structure-factor calculation.
24, 27) are n o t those more recently recommended (Acta
A m i n o r blemish is the use of a n o n - s t a n d a r d notation.
Cryst. (1950), 3, 400), b u t since the difference are only
A l t h o u g h m a n y of the papers contained in the book
of the order of 0-001%, a n d t h e u n c e r t a i n t y of t h e wavehave now appeared elsewhere, a n d despite the numerous
lengths is a b o u t 0.003 %, the difference is negligible for
misprints, t h e work can be thoroughly recommended to all
most work.
research workers in X - r a y structur~ analysis. No other
For books intended for considerable use, the bindings,
complete account of P e p i n s k y ' s X - R A C is readily availof t h i n flexible card, seem r a t h e r i n a d e q u a t e a n d will
able, a n d this alone makes the book w o r t h reading.
p r o b a b l y n o t s t a n d up well to the use t h e y will receive.
Moreover, most of the other papers in the collection are
On the other hand, the price is low, a n d laboratories
ones which crystallographers will like to have gathered
which m a k e considerable use of the books should have
together for easy reference. Considering t h e style of
no compunction in replacing t h e m when t h e y become
binding (limp card) the cost of the book is high.
outworn.
H . LIPSON
A. D. BOOTH College of Technology
Manchester 1
~Birkbeck College Resea~'ch Laboratory
England
London W. C. 1, England