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%
\ \vfil
\centerline�\gros Enhancements in Powder Pattern Indexing�
\medskip
\vskip 2cm
\centerline�By Daniel TAUPIN�\centerline�Laboratoire de Physique des Solides, as
\centerline�B\�atiment 510, Centre Universitaire, 91405 Orsay, France�
\vfil
\centerline�\bf Abstract�
\medskip
The indexing strategy of our fifteen year old powder pattern indexing programme
Several enhancements lead to an increased reliability and much shorter computing
patterns, especially monoclinic.
\vfil
\centerline�\sl \today�
\eject
\newcount\eqnum
\def\nweq#1�\global\advance\eqnum by 1\relax\eqno(\number\eqnum#1)�
\def\ideq#1� \eqno(\number\eqnum#1)�
\def\refeq#1�(\number#1)�
%
\def\section#1�\removelastskip\bigskip\penalty -4000\noindent�\bf #1�\medskip\pe
\def\subsect#1�\removelastskip\medskip\penalty -2000�\bf #1�\medskip\penalty +20
\section�I. Introduction.�
Half a dozen programmes at least are now available to assign indices to X-ray po
of unknown lattice cell:
Werner (1964),
 Taupin (1968, 1973), Visser (1969), Lou\"er \& Lou\"er (1972), Kohlbeck \& H\"o
 Werner \ital�et al.� (1985), etc..
These programmes and others have been discussed by Shirley (1978) who gives some
%\check
\par
Since the problem of retrieving line indices in a powder pattern is not at all t
each of these programmes has a distinct strategy. Some of them take account of s
between line position (Visser, 1969) and quickly find the solution, when they su
Others prefer a systematic strategy, using dichotomy (Lou\"er \& Lou\"er, 1972)
a tentatively exhaustive set of index combinations (Taupin, 1973).
\par
Although generally successful, our latter programme (Taupin, 1973) appeared to b
defective, namely it sometimes wasted hours of computing time trying hopeless in
(e.g. index combinations which would necessarily lead to cell volumes beyond the
\section�II. Recalling the bases of out indexing strategy.�
Let $n$ be the number of unknown parameters defining the crystal cell ($n$=1, 2,
tetragonal, orthorhombic, monoclinic and triclinic systems, respectively);
then all possible combinations of $n$ index triplets $\accol�hk\ell�$ are tentat
of $n$ lines of the \expal\ diagram (usually the $n$ lowest-angle lines) and, fo
the following linear system is solved:
%\check
%\bye
$$h_i�2A+k_i�2B+\ell_i�2C+2h_ik_iD+2k_i\ell_iE+2\ell_ih_iF=Q_i\quad;\quad i=1,n\
\xdef\glosys�\refeq\eqnum�
where
$$Q_i=1/d�2_i\nweq��$$
$$A=\vec�a�*���2�\quad;\quad B=\vec�b�*���2�\quad;\quad C=\vec�c�*���2�\nweq��$$
$$D=\vec�a�*�\cdot\vec�b�*�\quad;\quad E=\vec�b�*�\cdot\vec�c�*�\quad;\quad F=\v
with additional constraints for all lattices but triclinic, e.g.: $A=B=C$, $D=E=
tetragonal, etc..\Footnote�Although the common use is now to call them $\beta$ a
we prefer to denote as $\gamma$ and $c$ the particular angle and the particular
the monoclinic lattices.�
\par For each set of $n$ triplets $\accol�hk\ell�$ there is a solution, i.e. a p
obviously indexes the $n$ lines (the \ital�base� lines) taken, but seldom all th
diagram. If it does, one has found one of the possible solutions of the problem.
\par
If the problem were handled as crudely as described above, one would have to per
number of trials, and even now -- in the last eighties -- this would lead to unr
time spent.
It is therefore necessary to introduce constraints and checks which short cut us
redundant combinations as soon as possible.
\section�III. The combinatorial strategy and the constraints.�
Since the optimization of higher symmetry
 (i.e. cubic, hexagonal and tetragonal) lattice seeking is not crucial owing to
trials to be performed,
while these lattices require special handling of the indices $h$, $k$ and $\ell$
the stategy implemented in the case of orthorhombic, monoclinic and triclinic la
\par�\bf III.1 Varying $h$, $k$, $\ell$ for the base lines.�\par
Since it is definitely impossible to make the indices $\accol�hk\ell�$ run from
compelled to assign limits to the $\accol�hk\ell�$ of the base lines.
Rather than assigning separate limits to $h$, $k$ and $\ell$
it is obviously wiser to limit the value of
\hpkpl\ and to make this arbitrary limit proportional to
the $Q$ of the �\sl base\/� lines.
In practice, $h$, $k$ and $\ell$ do not vary separately for each base line, but
table of triplets $hk\ell$ �\sl ordered against� \hpkpl, that is, ordered agains
likelihood. Thus, simple lattices with low indices are sought first, and unlikel
volumes are examined later, if needed.
However, this way of limiting the indices which we have used for years, exhibits
several crystal cell symmetries are tested; in fact, the number of lines satisfy
$$ h�2+k�2+\ell�2\le K\nweq��$$
in a triclinic lattice is twice the number in a monoclinic, which in turn is twi
and so on.
Conversely, there is no obvious reason why the probability that the low index li
below the \expal\ threshold should be different in any of these systems.
Therefore, we think it is wiser not to state a limit for the actual value of \hp
but to state -- for example -- that the first line should be labelled with a tri
choosen within the $N$ first ones (ordered by increasing \hpkpl).
\par
This latter number has a rather obvious probabilistic meaning: for a cell whose
same order of magnitude, assigning the label of the first line to be within the
in the table means that we expect that the structural extinction rules or dampin
will not remove all the $N$ first possible labels from the \expal\ data.
Thus a value of $N$ ranging from 15 to 25 is quite reasonable, especially when I
-- i.e. with separate $hk\ell$ tables -- as we do now.
Concerning the limit of the next base lines, it must be pointed out that, while
have to be extrapolated proportionally to $Q$, the
value of $N$ (which is akin to a number of reciprocal unit cells in a given elli
should be extrapolated proportionally to $Q��3/2�$, in other words, to $1/d�3$.
\par�\bf III.2 The constraints on the reciprocal cell parameters.�\par
%\check
In addition to the normal constraints resulting from the symmetry of the cell, e
the generality of the programme is not restricted by imposing the following cons
$$A\ge B \ge C > 0\quad;\quad (orthorhombic)\nweq�a�$$
$$A\ge B \ge 2D \ge 0\quad;\quad (monoclinic)\ideq�b�$$
$$A\ge B\ge C > 2E\ge 0\quad; B\ge 2D\ge 0\quad; C\ge F\ge 0\quad;\quad(triclini
\xdef\redcon�\refeq\eqnum�
%\check
The positivity of all the unknowns $A$, $B$, $C$, $D$, $E$ and $F$
leads to a first simple elimination rule:
if a line $Q_1$ is tentatively indexed by $h_1k_1\ell_1$ and
 a line $Q_2$ by $h_2k_2\ell_2$ and if $Q_2>Q_1$, then
it is impossible to have:
$$\bunch h_1�2\ge h_2�2\text� and �
 k_1�2\ge k_2�2\text� and �
 \ell_1�2\ge \ell_2�2\text� and �
\quad\quad\cr\quad\quad
 h_1k_1\ge h_2k_2\text� and �
 k_1\ell_1\ge k_2\ell_2\text� and �
 \ell_1h_1\ge \ell_2h_2\quad.\quad\endbunch\nweq��$$
\xdef\elimhkl�\refeq\eqnum�%
%\check
Therefore, any triplet which violates this condition with respect to the indices
lines of smaller $Q$ must be immediately rejected.
\par
Besides -- and this is an important novelty -- one does not need to have assigne
index triplets to all the first $n$ \expal\ lines before beginning to solve the
Rather, it is advisable to begin eliminating unknowns between tentative equation
two lines have been assigned index triplets.
Then, due to the high frequency of zeros in the indices of the low-angle lines,
one or several unknowns if often available once only 2 or 3 lines have been tent
If this results -- for example -- in $B$ being greater than $A$ in a monoclinic
then all trials using these tentative labels have to be rejected (owing to redun
constraints \redcon) and thousands of useless trials can thus be short cut.
\par
Another constraint is of major importance, namely the maximum volume of the unit
the minimum volume of the reciprocal unit cell.
A first value of this limit is normally given by the programme user, but it can
dynamically restricted by information considerations (see Taupin, 1988), especia
one good lattice has been found.
%\check
Since the volume of the reciprocal unit cell is readily deduced from the paramet
$$V��*2�=ABC+2DEF-CD�2-AE�2-BF�2\nweq��$$
\xdef\volumeq�\refeq\eqnum�
%\check
it is sufficient to locate $A$ to $F$ within intervals to get an estimation of t
within which $V�*$ lies, and reciprocally when taking account of the conditions
\par
Additional efficiency can be also attained by introducing reasonable constraints
of the cell parameters $a$, $b$ and $c$.
They can obviously be stated by the programme user
 (practically, we ask for constraints on $1/a�*$, $1/b�*$ and $1/c�*$) but,
even if he does not want to give them, these upper and lower bounds may be deriv
of \expal\ lines with a good reliability.
Practically, one can take account of the fact that indices greater than 2 are ve
first line in orthorhombic, monoclinic and triclinic lattices, and set the lower
to approximately $Q_1$ divided by the maximum value allowed for $h_1�2+k_1�2+\el
which can be deduced from $N$,
 unless explicitely specified.
\par
%\check
Conversely, extra-flat tile-shaped cells are also very unfrequent.
This can be interpreted in setting the upper bounds of $A$, $B$, $C$ to some val
$Q_�20�$.
\par
%\check
Thus, most of the strategic combinatorial work has to deal with inequalities.
\par�\bf III.3 Tactical considerations.�\par
 Each time a tentative triplet is assigned to a new \expal\ line, the following
\item�a)� the tentative triplet is compared to those previously assigned to the
it should be rejected according to \elimhkl.
\item�b)�the previously stored equations are �\sl formally\/�
 substituted in the new equation resulting from the assignment of this triplet t
By ``formally'' we mean that the actual values of the $\accol�Q_k�$ of the previ
numerically used here, but that we build for each newly assigned triplet a linea
%\check
$$\alpha A+\beta B+\gamma C ... = p_1Q_1 + p_2Q_2 + p_3Q_3 ... \nweq��$$
\xdef\formeq�\refeq\eqnum�
in which only $\alpha$, $\beta$, $\gamma$, ..., $p_1$, $p_2$, $p_3$, ... are num
\item�c)� If $\alpha$, $\beta$, $\gamma$, ... are all equal to zero -- this mean
triplet was not \indpt\ of the previous ones -- the minimum and maximum value of
of this linear condition are computed, according to the allowed discrepancies st
these lines, namely $\epsilon_1$, $\epsilon_2$, $\epsilon_3$, etc..
If these values do not enclose zero, the triplet is rejected and another one is
If they do enclose zero, the triplet is considered as eligible for this \expal\
dropped since it was not \indpt\ of the previous ones; thus other triplets have
level to the next higher angle line (note that all this process is recursive).
This test is in fact of major importance: if it was not correctly done, the prog
indefinitely (or crash) in falling into the trap consisting in assigning -- for
all the $\accol�hk0�$ triplets to all the given lines of an orthorhombic materia
have been rejected by this test.
\item�d)� If at least one of the $\alpha$, $\beta$, $\gamma$, ... is not zero,
then the maximum and the minimum possible values of the left hand member of \for
using the present state of knowledge of the possible ranges of $A$, $B$, ..., $F
to the maximum and minimum possible values of the second hand member computed as
In case of incompatibility, another triplet is tentatively assigned to the same
the same checking process is performed.
In case of compatibility, the resulting equation is kept but further computation
to try to restrict the intervals where the unknowns are supposed to be located.
\item�e)�
If, at these stage, the number of equations kept is less than the required numbe
using maximum or minimum of the other unknowns depending upon the sign of $\alph
the formal relationskip \formeq\ is used to cyclically update (i.e. restrict) th
If some update happens, the new intervals are matched to equation \volumeq\ and
Of course, updating intervals may result in contradictions which in turn lead to
of the tentative triplet as above.
 %\check
\item�f)� If the tentative triplet appears to be eligible, but the number of equ
is less than the required number $n$, then the whole process is recursively star
in order to assign tentative triplets to the next higher \expal\ line.
The number of levels of this recursive process is, however, limited in order to
pathological cases where -- for example -- all the \expal\ lines could be indexe
yielded by some high volume cell. In practice, we limit the number of \ital�drop
also called the \ital�degenerescence index� to a certain number $n_�deg�$, so th
number of base lines considered simultaneously never exceeds $n+n_�deg�$.
Thus, if the tentative triplet would lead the number of base lines to exceed thi
this triplet is merely rejected and a new one is tried at the same level (step a
\item�g)� Conversely, if the number of \indpt\ equations is equal to $n$,
then a complete indexing work is done, which may or may not succeed in a satisfa
If it succeeds the minimum \ital�information merit� (see Taupin, 1988) is update
overall value \ital�minus 10� (this means that any further lattice found whose \
times the \prty\ of the best one
                 the best one will be rejected). In turn, this often leads to a
maximum direct cell volume, i.e. an increase of the minimum reciprocal cell volu
in shrinking the allowed intervals for the unknowns.
\section�IV. Checking and application.�
The new version of the programme has been tested on a set of 112 randomly genera
with artificial erratic deviations supposed to simulate \expal\ imperfections.
These diagrams consisted in 16 specimens of cubic, hexagonal and tetragonal latt
 orthorhombic and monoclinic lattices.
\par
All the 48 higher symmetry lattices but 3 were indexed at the first trial with a
which corresponds for instance to a first line having indices not higher than 3^
Two of the generated patterns exhibited abnormally high indices for the first li
of $N$ (up to 40) to be correctly indexed\Footnote�Whether an
hexagonal pattern starting with the 2^0^2 and 1^1^3 lines is a realistic situati
For another hexagonal lattice, indexing happened to be possible only by increasi
allowed discrepancies by 40^\%\Footnote�One of the new features of our programme
allows the user to increase or reduce the maximm deviations by a common factor,
change them individually in the record corresponding to each given line.�.
\par
All the 32 orthorhombic lattices but 5 were indexed at the first trial with $N=1
assumption the the indices of the first line are not higher than 2^0^1.
Two of them were rather ill-conditioned (bad precision) and spent too much compu
we had to restrict the range of the volume and parameters to achieve the indexin
Two others exhibited first lines with very high indices, so that the parameter $
Finally, one orthorhombic lattice needed an increase of the allowed discrepancie
\medskip
Of the 32 monoclinic patterns, XX were indexed at the first trial
\bigskip ...... \bigskip
\par Besides, the VAX version of the programme was sent to G. Johnson who used i
of the materials recorded in the Powder Diffraction File recorded as ``unindexed
In fact, according to G. Johnson (1988), 25^\% of the unindexed materials was in
\section�V. Availability.�
The indexing programme is written in standard Fortran 77, with a few assembler o
specific supervisor calls for timing controls. It runs presently on VAX/VMS, HP1
IBM/MVS.
If needed, specific versions can be derived for UNISYS/1100, IBM/VM/CMS and PC.
\section�VI. References.�
\def\jac�\ital�\unskip\ J. Appl. Cryst., ��
\def\acta�\ital�\unskip\ Acta Cryst., ��
\def\reference#1��\par\hangindent=\parindent\hangafter 1\noindent�\rm #1�\medski
%
%\check
\reference�Kohlbeck, F. \& H\"orl, E.M. (1976). \jac �\bf 9�, 28-33.�
\reference�Lou\"er, D. \& Lou\"er, M. (1972). \jac �\bf 5�, 271-275.�
\reference�Shirley, R. (1978). \ital�Indexing Powder Diagrams� in \ital�Proc. of
Delft Univ. Press and Oosthoeks, 1978.�
\reference�Taupin, D. (1968). \jac �\bf 1�, 178-181.�
\reference�Taupin, D. (1973). \jac �\bf 6�, 380-385.�
\reference�Taupin, D. (1988). \jac �\bf 21�, \ital�in the press�.�
\reference�Werner, P.E. (1964). \ital�Z. Kryst.,� �\bf 120�, 375-387.�
\reference�Werner, P.E., Eriksson, L. \& Westdahl, M. (1985). \jac �\bf 18�, 367
\bigskip\penalty -5000
\centerline�\bf Appendix. �
\centerline�Practical use of the Powder Diagram Indexing Programme�
\section�A.1 Invoking the programme.�
Typical invocation of the programme is of the form:
\medskip
\par\leftline �\tt PWDCDS \lft input file\rgt,\lft output file\rgt,\lft breakpoi
\rightline� (VMS, RTE-A).�
\par
\par\leftline �\tt // EXEC PGM=PWDCDS,PARM='\lft input file\rgt,\lft output
 file\rgt,\lft breakpoint file\rgt,\lft log file\rgt'�
\rightline�  (IBM-MVS).�
\par
\lft input file\rgt\ is the name of the file containing the input data (this is
 If omitted, the user either prompted to give it, or standard input is used, dep
\lft output file\rgt\ contains the printout after execution (standard output is
On VAX-VMS, RTE-A, and other executive systems which allow dynamical allocation
on IBM-MVS) \lft output file\rgt\ is always a \ital�new� file: if the name alrea
a new derivated file name is created.
\par
If specified, \lft breakpoint file\rgt\ is used by the programme to store restar
enable it to resume its operations at the point where it has been stopped becaus
time conditions (see parameters below).
If it does not exist, it is created (with restrictions for IBM-MVS); if it exist
and for output. Therefore, it must be deleted or erased when performing a new tr
is not the continuation of a previous one.
\par
 If specified,
\lft log file\rgt\ contains a very short summary of the execution, namely the ti
lattice, its cell parameters and its information merit, and the c.p.u. time alre
If it does not exist, it is created (with restrictions for IBM-MVS); if it exist
the new information is \ital�appended� to the existing file which, therefore, co
of all trials referring the the same \lft log file\rgt.
%\check
\section�A.2 Input data (the \lft input file\rgt\ structure).�
In order to avoid problems with editors (e.g. TSO) which use columns 73:80 for r
significant data are supposed to be stored only in columns 1:72 of the records o
\subsect�A.2.1 First record�
Columns 1:12
contain a set of \ital�option characters� which specify which operations are to
and which kind of data are input below.
\def\otem#1�\item��\tt'#1'���
\otem�A� means that input data will be $2\theta$ angles, rather than $d$ values.
\otem�A1� means that input data will be $\theta$ angles.
\otem�A4� means that input data will be $4\theta$ angles.
\otem�C� means that cubic lattices have to be sought.
\otem�H� means that hexagonal lattices have to be sought.
\otem�L� disables seeking of S, I, F lattices. If specified, only P-lattices are
\otem�M� means that monoclinic lattices have to be sought.
\otem�O� (letter O, not zero) means that orthorhombic lattices have to be sought
\otem�P� causes special ``punch'' output to happen on Fortran unit 7,
which has to be specially allocated before execution (this is a rather obsolete
\otem�Q� means that input data will be $1/d�2=Q$ rather than $d$.
\otem�S� suppresses the normal redundancy test which eliminates lattices which i
lines whose angles are lower than some base lines.
\otem�T� means that tetragonal lattices have to be sought.
\otem�U� means that input data will be $1/d=q$ rather than $d$.
%\check
\otem�V� enables a check that the cell volume is approximately a multiple of the
volume of the elementary molecule. This option implies that the user gives the
density of the sample and its molecular weight.
\otem�X� suppresses the updating of the minimum information merit to the best on
\otem�Z� produce an extensive printout. To be used for maintenance or checking o
\otem�3� means that triclinic lattices have to be sought.
\par
\noindent\ital�Remark: � if none of the options �\tt C�, �\tt H�, �\tt M�, �\tt
are specified, then all lattice symmetries are tried.
\par
Columns 13:72 contain a title which will be reproduced as a heading at the begin
apparently successful indexing, and in the \lft log file\rgt. It is usual to ind
name of the sample in this field.
%\check
\subsect�A.2.2 Second record�
While the first record mainly contained \ital�boolean� parameters, the second on
numerical parameters.
These parameters are in free format, separated by commas, and blanks are meaning
The user can choose to give these parameters as \ital�positional parameters� in
welcome to make no errors in counting the commas when specifying the 16-th param
He can also use \ital�keywords� like �\tt WLTH=1.54018,� to specify the waveleng
\par
Keyword specified parameters have precedence against positional ones.
\par
These parameters are specified as follows:
\par\penalty -2000\relax\removelastskip
%\check
\def\pardesc#1#2�\item��\hbox to 7mm�\hss #1�\hbox to 20mm�\tt
\ \ #2\hss����
%\check
�\parindent 27mm
\pardesc�No.��\ital�keyword��\ital��Description.��
\smallskip
\pardesc�1��NLIN=� Number of \expal\ lines given. May be omitted, then the given
\pardesc�2��NEXT=� Number of lines the programme is allowed to skip if it cannot
stated maximum deviations.
\pardesc�3��NDEG=� Degenerescence parameter: this is the maximum number $n_�deg�
which are allowed to be dependent of the previous ones. For instance, if this pa
this means that the programme will not try to introduce the 9-th line as a base
ones (8=3+5) are not sufficient to determine a trial cell in an orthorhombic lat
The default value is 2; the maximum value is 20-$n$, but it is not wise to use v
\pardesc�4��INDX=�
If not negative, this parameter is the maximum value of
\hpkpl\ for the first line.
If negative, its absolute value gives the number $N$ of missing lines before the
when line triplets are ordered against \hpkpl.
Default value is -20, i.e. $N=20$.
\pardesc�5��CPUX=� Maximum c.p.u. time, in minutes.
If positive, the programme will stop when this limit is reached. Restart informa
\lft breakpoint file\rgt\ if this file name is present in the command.
If negative, the programme will continue after this limit is reached, but the \l
will be closed, and the output will continue in another file whose name is deriv
one. Thus, the user will be able to look at the recently closed files in order t
see whether something is pgoing wrong and decide whether the programme should co
\pardesc�6��VMAX=� Maximum volume of the unit cell. Default value is 1000 \AA$�3
\pardesc�7��INFM=� Minimum information merit. Only the cattices whose informatio
this parameter are printed and recorded in the \lft log file\rgt.
This minimum is eventually increased to the best merit found, minus 10. This upd
 Default value is 80.
 It is not wise to set it below 40, unless for checking purposes.
Usual good lattices have a merit ranging from 60 to 300.
\pardesc�8��MOLM=� Molecular mass of the sample. Useful only with �\tt V� option
\pardesc�9��DENS=� Density of the sample in $g/cm�3$. Useful only with �\tt V� o
\pardesc�10��DERR=� Relative error on that density. Useful only with �\tt V� opt
\pardesc�11��WLTH=� Wavelength used when recording the pattern. Necessary if inp
Useful to extrapolate deviations when not given in the specific record of any li
\pardesc�12��ERSC=� Error scale factor. If not equal to 1, this parameters is us
given maximum deviations of all lines, without having to modify each record.
Thus, if an indexation seems to fail and no other obvious reasons appear, it may
re-run the programme with �\tt ERSC=1.5� or more.
Conversely, if a great number of bad merit cells appear in the output, it might
with �\tt ERSC=0.7� or less.
Note that this factor is internally restricted when trying large cell volumes wi
\pardesc�13��CHIM=� The minimum value of the expected reduced $\chi�2$. Default
It is not wise to increase it; decreasing it could help finding weird cells, but
c.p.u. time.
\pardesc�14��AMIN=� The minimum value of $1/a�*$, $1/b�*$ and of $1/c�*$.
\pardesc�15��AMAX=� The maximum value of $1/a�*$, $1/b�*$ and of $1/c�*$.
\pardesc�16��BMMX=� The number of ``bad merit'' cells to be accepted before the
stops as if the maximum c.p.u. time condition was reached. The default value is
This parameter is a security which is supposed to warn the user that his data, o
quality he states through his allowed deviations, are so bad that a great number
cells fit them, without reliability.
The corresponding counter is reset at each partial execution, just like c.p.u. c
registers.
\par
�
\noindent\ital�Remark: � using keywords instead of positional values usually lea
record exceeding the 72 column limit. This is circumvented by terminating this r
this last character is then dropped and the next record is considered as a conti
The total limit is 144 characters.
\subsect�A.2.3 Experimental line records.�
There is one record per \expal\ line.
Each record is in free format, with positional values separated by commas.
The values are as follows:
\item�1)� The line position, namely $d$. If options �\tt A�, �\tt A1�, �\tt A2�
is the corresponding angle, if option �\tt Q� is present, the value given is $1/
present the value given is $1/d$.
\item�2)� The maximum deviation of this line, expressed in the same range as the
\item�4,5,6)� The values of $h$, $k$, $\ell$ of that line, if they are known. On
the complete triplet, or nothing.
\item�7)� The particular wavelength with which this line was recorded.
The maximum deviation is necessary only for the first line. If further maximum d
they are extrapolated from the most recently stated, assuming that deviations ar
$\theta$ scale.
\par The presence of the letter �\tt 'P'� or of the question mark �\tt '?'� anyw
means that the programme is allowed to drop this line, provided that the number
does not exceed the value of the second parameter of the second record (�\tt NEX
%\check
\section�A.3 Some advices.�
The default values are fit for the most common trials. It is wise to change them
of some certitude (e.g.: cell volume, maximum and minimum parameters) or when th
actually failed.
\par
The user should remember that multiplying $N$ (�\tt INDX=�) by 2 results in mult
by approximately 16 in the orthorhombic case, and by approximately 50 in the mon
The precision (actual and stated) of the input data is also a very critical para
a factor of 2 in the precision usually results in factors of 30 to 100 in c.p.u.
\par
Therefore, the user is strongly advised to make the first trials with rather res
and to progressively extend the error scale factor, the degenerescence parameter
in case of failure.
\bye
