|
Warning | |
---|---|
To fix the origin of the crystal, the distances to 3 non-coplanar faces must be kept invariant. The program does not choose these faces automatically. They should be as mutually perpendicular as possible, and sufficiently large to have well-characterized Miller indices. However, when the invariant faces have been chosen, weights on these observations are automatically set to zero. |
To take into account systematic errors due to magnification effect, refined distances can be constrained, or a correlation value between observations can be used. In the latter case, the variance-covariance matrix V is no longer diagonal, but the weighting matrix remains diagonal even in the case of 1/ weights.
Becker & Coppens extinction parameters can be applied or refined, either isotropically or anisotropically. Following Zachariasen (1967), one can distinguish two types of secondary extinction. Type I refers to crystals in which the extinction is dominated by the mosaic distribution (described by g ), whereas effective particle size (related to the parameter ) is the critical parameter describing extinction in type II crystals. For anisotropic extinction, g and become tensorial quantities as defined according Thornley and Nelmes (1974).
Several extinction refinement options are available: g and can be refined separately choosing type I or type II respectively. In this case, primary extinction is neglected. g and can also be refined simultaneously (but only g can be tensorial) allowing primary extinction to be taken into account. Either a Lorentzian or a Gaussian distribution of the mosaic spread g can be selected.
A short summary of used expressions is given below. Full details can be found in Becker & Coppens (1974a, 1974b, 1975) and Zachariasen (1967).
and for general extinction:
then
and finally:
The parameters g and are related respectively to the half-width of the mosaic spread distribution expressed in radians and to the size of the mosaic blocks expressed in cm by
The values of g and given in the program listing are multiplied by 104.
(Pseudo-) merohedral twin laws may be determined
automatically and entered onto the bdf by
CREDUC
for use by LSLS. Twin laws may also be entered
directly in LSLS by use of
twinop lines. By default
one twin law
viz. the identity operation is
assumed. For centrosymmetric space groups all twin laws are
generated by
CREDUC
or must be entered by way of
twinop lines. For
non-centrosymmetric space groups, each twin law generated
by
CREDUC
or entered by way of a
twinop line corresponds
to a pair of twin laws:
viz. the one indicated and the one
related to it through a center of inversion. The two laws
of the inversion-related pair in non-centrosymmetric space
groups have serial numbers i and i+n where n is the number
twin matrices on the bdf or
twinop lines given in
LSLS.
Twinning (
) is treated by refining the volume fractions
x
of each twin component and automatically
constraining their sum to be 1.0.twi
For non-centrosymmetric structures, attention has to
be paid to two points.
Primo, anomalous dispersion should be
applied.
Secundo, the bdf should be prepared
with each reflection and its Friedel opposite in separate
packets:
on the
SORTRF line.sepfrl
The Becker and Coppens extinction theory used in LSLS does not take twinning into account, implying that the coupling constant between the incident and diffracted beams is forced to be symmetric under the lattice-point group symmetry.
The following example illustrates the way to handle
twins in the non-centrosymmetric space group
P 23.
CREDUC
generated 2 (pairs of) twin laws:
( 1 0
0)
|
( 1 0
0)
|
(-1 0
0)
|
|||
( 0 1
0)
|
for (1) Identity |
( 0 1
0)
|
and (3) Centre |
( 0 -1
0)
|
|
( 0 0
1)
|
( 0 0
1)
|
( 0 0
-1)
|
|||
( 0 -1
0)
|
( 0 -1
0)
|
( 0 1
0)
|
|||
(-1 0
0)
|
for (2) 2-fold axis |
(-1 0
0)
|
and (4) Mirror |
( 1 0
0)
|
|
( 0 0
1)
|
( 0 0
-1)
|
( 0 0
1)
|
|||
In order to refine only the enantiomorph-polarity parameter in this case starting with a value of 0.5, the following lines are necessary:
twinop *10 1 0.5 0.5 noref twi(2) twi(4)
Assuming that refinement of dispersion parameters has
been turned on by
on the
LSLS line, dispersion
parameters can be addressed individually by
rd
or
re
for the real or the imaginary parts, or as a
set with
im
.dsp
Neutron scattering factor refinement is achieved by turning on dispersion refinement and keeping imaginary parts invariant.
The overall displacement parameter is automatically
refined if at least one atom's displacement parameter type
is set to overall. It can be held invariant by using
noref
.uov
Atomic positional parameters are always refined unless the atom is in a special position or they are explicitly constrained. noref <atom> fixes all atomic parameters of the named atom.
The program can be used to refine overall,
isotropic or anisotropic displacement parameters. In
mixed mode, the displacement-parameter type of an atom is
set in accordance to the value on the bdf. However it can
be modified by the use of
ref and
noref lines. After
refinement, the mode used is written on the output bdf.
If the displacement factor is non-positive definite, the
program will either stop (
), or reset it to a positive value (
ts
), or continue with non-physical values (
tr
).tp
ADDATM
generates symmetry constraints for atoms on special
positions which are written on logical record
lrcons:
. See the documentation of
ADDATM
for details.
A noref line can fix any parameter. The general form of the arguments is:
(p1/p2)(a1/a2) fixes from atomic parameter p1 to atomic parameter p2 of all atoms in list between a1 and a2 inclusive. (p1/p2) can be replaced by p1 if only one parameter is to be fixed, and a1 can replace (a1/a2) if there is only one target atom.
Atomic parameters are :
,
x
,
y
,
z
,
u
,
u11
,
u22
,
u33
,
u12
,
u13
,
u23
pop
a1/a2 fixes all atomic parameters from atom a1 to atom a2. a1/a1 is abbreviated a1.
When an atom name is identical to an atom type name, then a1 and a2 will hold for atom types instead of individual atoms.
g1(n1/n2) fixes general parameters g1, which can be
,
uov
,
skf
,
twi
,
shp
,
ext
,
dsp
or
re
, from n1 to n2. If only one parameter is
kept invariant, n1 can be used for n1/n2.im
For dispersion refinement,
fixes real and imaginary parts and it is
equivalent to (
dsp
/
re
).im
The syntax of the
line is identical to that of
noref. When mixed
displacement mode has been selected on the
LSLS line,
ref /
noref lines can be used
to change atomic displacement types.
ref U converts the
displacement type to isotropic, while
ref U11/U23 set the
displacement type to anisotropic.ref
ref U(N) (U11/U23)(N2) ref U(N1) U(N3/N6)
These examples show how to set to isotropic the displacement parameter type for all 6 nitrogen atoms, except for atom N2. There is a difference in the resulting displacement parameters values between these two examples. In the first example, all nitrogens are first changed to isotropic, and then the displacement factor of atom N2 is reset to an anisotropic tensor. In the second example, the displacement parameter of atom N2 is not changed at all.
Warning | |
---|---|
If atom types Y and U appear in the atom type list, atomic parameters y and& u must be lowercase. |
As for invariant parameters, symmetry constraints are
read from the input bdf logical record
lrcons:
. Linear constraints between parameters can also
by introduced by
constr lines :
p = Q + f(1)*r(1) + ... + f(n)*r(n) where : | |
p = object (dependent) parameter (same format as ref/noref lines) | |
Q = constant term | |
f(i) = ith multiplicative term | |
r(i) = ith reference (independent) parameter (same format as ref/noref lines). |
The specification of the parameters is the same as in ref / noref lines, but object and reference parameters must be uniquely determined. It should be noted that the constant terms Q and f(i) are mandatory, and blanks are not allowed.
Warning | |
---|---|
Origin fixing is carried out by shift-limiting restraints, following the algorithm of Flack & Schwarzenbach (1988). Additional constraints should not be used in LSLS in polar space-groups. |
The normal-equations matrix is constructed for independent variables only to minimize storage size and inversion time. The constraints modify the derivative vector, which leads to an unconstrained least-squares problem. The dependent parameters contribute to the derivatives vector of the reference parameters , which become:
The dependent parameter estimated standard deviation is then given by (4.76):
.
Restraints are additional stereochemical pseudo-observations used to restrict interatomic geometry. For a discussion of the relations between rigid bodies and restraints, see Didisheim & Schwarzenbach (1987). Distance, bond-angle, dihedral-angle and rigid-bond restraints are provided. If M and M are respectively the direct and reciprocal metric tensors, x is the position of atom i expressed in the crystal coordinates, the displacement tensor of atom i expressed in the same basis, , and V the volume of the cell, then restraints are given by (4.77):
When the normal-equations matrix is ill-defined, it is possible to force it into a positive-definite form, which is easily invertible. The usual way to do this is to use shift-limiting restraints which increase the diagonal element of the variable leading to the degeneracy. Two possibilities are available in LSLS: (a) when the weight w of the restraint is greater than 1, the restraint equation adds the quantity to the minimized function (4.68), so the diagonal element becomes + w. This type of shift-limiting restraint may then be viewed as a pseudo-observation whose value is taken to be the variable's value in cycle n, and the restraint imposes that in cycle n+1 should be "close to" its former value. (b) If w is smaller than 1, it is interpreted as a Levenberg-Marquardt parameter (see Press, Flannery, Teukolsky & Vettering (1986)). There is no pseudo-observational justification to this method where the diagonal element is multiplied by a w independent of the . It should be pointed out that tests to determine the optimal Levenberg-Marquardt parameter are not carried out by LSLS.
LSLS provides 3 different weighting schemes: the usual minimal variance w = 1/ , unit weights w = 1, or weights read from the bdf (idn 1900 is taken as the square root of the weight on Y ); this latter possibility allows the user to define his/her own weighting scheme. In any case, the goodness of fit and the parameter variance-covariance matrix are computed according to (4.70) and (4.71).
Weights on |F| are computed from weights on the refined quantity Y (either or I ) by (4.78) where LPT is set to 1.0 if Y = | F | . These | F | weights are used in the calculation of the weighted R-factor.
A reflection can be omitted from the refinement under several conditions: if it is flagged as suffering form extinction (rcode 3), if its weight is 0, if its weighted difference | is greater than some user defined value, or if is less than a user defined threshold. These unused reflections do not contribute to the matrices, Durbin-Watson statistic, and goodness-of-fit. Less-than reflections are those for which less than the cutoff value, and those with rcode = 2. In the latter case, the reflections are discarded only if is greater than or equal to Y . The status code is never updated by LSLS.
The meaning of the printed status codes is the
following:
is for reflections without their Friedel
mate,
F
is for systematically absent reflections.
Reflections flagged with these codes are always used in the
refinement.
A
is for a reflection suffering from
extinction,
E
for less-than reflections, i.e. which
smaller than the
threshold,
L
for reflections with a weight of zero, and
W
for a rejected reflection for which
is greater than
the cutoff. Reflections marked with these codes are always
excluded from the refinement.
R
is for unobserved reflections; it corresponds
to rcode = 2. The asterisk
U
marks reflections which do not contribute to
the matrix.*
The unweighted and weighted R-factors are printed by the program. The unweighted R-factor is defined as:
and the weighted R-factor by:
For the 4 R-factors printed by the program:
(1) marked "Of contributing reflections" is calculated with R, Y = |F| and M = NREF.
(2) marked "Of weighted contributing reflections" is calculated with wR, Y = |F| and M = NREF.
(3) marked "Of Observations" is calculated with R; for refinement on |F| , Y = |F| ; for refinement on I, Y = I; and M = NOBS.
(4) marked "Of Weighted Observations" is calculated with wR; for refinement on |F| , Y = |F| ; for refinement on I, Y = I; and M = NOBS.
Reads parameter- and observation- information from the input archive bdf
Writes refined parameters on output archive
bdf
Optionally writes variance-covariance matrix to
file
cmx
Optionally writes parameters to file
pch
title Gd3Ru4Al12 compid gra CIFENT cifin STARTX upd sgname -P 6C 2C REFCAL neti inst 1 excl LSABS ADDATM extinc typ1 is gaus 0.005 atom Al1 0.1622 0.3244 0.5763 0.008 atom Gd 0.1928 0.3856 0.25 0.007 atom Ru1 0.5 0 0 0.007 atom Al3 0.3333 0.6667 0.0119 0.008 atom Al4 0 0 0.25 0.008 atom Ru2 0 0 0 0.007 atom Al2 0.5637 0.1274 0.25 0.008 0.95 atom Ru3 0.5637 0.1274 0.25 0.008 0.05 LSLS cy 5 ad ax pp tr rs sm noref shp(1) shp(3) shp(5) noref pop ref pop(Ru3) pop(Al2) constr x(Ru3)=0.0+1.0*x(Al2) constr u(Ru3)=0.0+1.0*u(Al2) constr pop(Ru3)=1.0-1.0*pop(Al2) REGFE sr 1 nan tab finish
To refine the crystal shape, the absorption program
LSABS is run just after the conversion from raw data to net
intensities. No averaging is done, and the refinement is
made on all intensities. 3 faces are kept invariant. Ru3
and Al2 are constrained to the same position, and the same
displacement parameter. The stereochemical disorder of
these atoms is taken into account constraining the total
population of the site to be equal to 1. The
variance-covariance matrix is saved to be used in the
calculation of distances by
REGFE
.
compid coanp CIFENT cifin STARTX upd latice n p symtry x,y,z symtry -x,y,1/2+z symtry -x,1/2+y,-z symtry x,1/2+y,1/2-z NEWCEL transf 0 0 1 1 0 0 0 1 0 CREDUC REFCAL excl fsqr ffac SORTRF ord lkh aver 1 sepfrl f2rl ADDATM Atomic parameters omitted here for brevity LSLS cy 1 f2 cs 3.0 noref C N O H LSLS cy 5 f2 cs 3.0 rp ep FOURR MAPLST finish
In this example, the space group is changed by
NEWCEL
, then twins are searched (
CREDUC
), the data are converted to |
F |
and averaged in separate packet for Friedel pairs.
The scale factor is refined alone for one cycle, and then
all structural parameters are refined, including
enantiomorph (absolute structure) parameter. When the
initial value of the scale factor is far from the real one,
it can be useful to refine the scale factor alone in a
first refinement step.
Becker, P.J. & Coppens, P. (1974). Acta Cryst. A30, 129-147.
Becker, P.J. & Coppens, P. (1974). Acta Cryst. A30, 148-153.
Becker, P.J. & Coppens, P. (1975). Acta Cryst. A31, 417-425.
Bernardinelli, G. & Flack, H.D. (1985). Acta Cryst. A41, 500-511.
Didisheim, J.-J. & Schwarzenbach, D. (1987). Acta Cryst. A43, 226-232.
Flack, H.D. (1983). Acta Cryst. A39, 876-881.
Flack, H.D. & Schwarzenbach, D. (1988). Acta Cryst. A44, 499-506.
Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T. (1986). Numerical Recipes. Cambridge University Press. Cambridge.
Thornley, F.R. & Nelmes, R.J. (1974). Acta Cryst. A30, 748-757.
Schwarzenbach, D. et al. (1989). Acta Cryst. A45, 63-75.
Zachariasen, W.H. (1967). Acta Cryst. 23, 558-564.