L Least Squares Refinement information

L cards drive the various Least Squares Refinement programs of CCSL. For a general introduction to Least Squares the reader is referred to Chapter 5.

Each L card has its own format, not particularly connected with any other L card.

An L card starts with L then a CCSL-word, which determines what else is on the card. There is no predetermined sequence for L cards. It would be usual, but not essential, to type all L cards starting with the same CCSL-word consecutively.

DATA FOLLOWING ALLOWED CCSL WORDS:

The general CCSL-words MODE, REFI, SCAL, TFAC and WGHT are described first. Then the group FIX, VARY, RELA and FUDG concerned with Least Squares parameters, followed by the group SLAK, ATOM, BOND, ANGL, EQUA, EQUB, LINE and TORS concerned with geometric slack constraints. Finally there is a new (2008 Mark4.4) group comprising DATA and SORC specific to multi-source least squares refinement.

MODE
Data

An integer, MODE, specifying the mode in which reflection data will be given. This is interpreted by main programs SFLSQ, MAGLSQ etc. The integer MODER (in the range 0 to 8) is set to MODE modulo 100 and MODOBS to MODE/100. MODER specifies the format if the data to be read as follows:

MODER=0

The user will supply a routine QLSQIN(K,NOMORE), which will read into COMMON /OBSCAL/ the necessary quantities. This is only necessary if the data are in some format unknown to the system.

MODER=1

h,k,l, Gobs, (Scale number) (Code)

MODER=2

h,k,l, Gobs, W, (Scale number) (Code)

MODER=3

h,k,l, Gobs, $\sigma$, (Scale number) (Code)

MODER=4

h,k,l, Gcalc, Gobs, $\sigma$, (as output by several main programs)

MODER=5 (obsolete)

h,k,l, Gobs, $\sigma$, C1, C2, C3, C4, (Scale number)
to be used when extinction corrections are applied. See Becker and Coppens, Acta Cryst A30 p129:

$$\begin{array}{llllll} C1 &= &(\lambda^3\overline\tau /V^2 \si... ...heta\\ C3 &= &A(\theta)& &C4 = &B(\theta)\ [-2ex] \end{array}$$

MODER=6

h,k,l, Acalc, Bcalc, Gobs, $\sigma$, (Scale number) (Code)

MODER=7

As mode 3 but h,k,l, are floating point numbers

MODER=8 (obsolete)

As mode 5 but h,k,l, are floating point numbers

MODER=9

h,k,l, (floating),Gobs, $\sigma$, In,Out,Code

MODER=10

h,k,l, (fixed),Gobs, $\sigma$, In,Out,Code
Modes 9 and 10 are for SNP polarisation matrix measurements, Gobs is the scattered polarisation and $\sigma$ its estimated standard deviation. In and Out are integers which define the incident and scattered polarisation directions ( $\pm 1,2,3=\pm x,y,z$). Code=0 if $hkl$ are the same as for the previous observation, Code=1 if they are different.

MODER=11

h,k,l, (any),2theta,INT(up) ESD(up) INT(down) ESD(down) for magnetised powder data. GOBS is alternately INT(up) and INT(down) for the same $hkl$ (free format).

The reflection indices, $h,k,l$ are in format I5 (for MODER=1-6 and 10) or in F8 (for MODER=7-9). The numbers Gobs, $\sigma$ its standard deviation, W the weight to be applied to this observation, and the four C values are all in format F10. The integers In, Out, Scale number and Code are in I5, and are given in parentheses when they are optional.
If MODOBS = 0, Gobs and Dobs give the value of the structure amplitude and its standard deviation even for IREF=2.
If MODOBS = 1, Gobs and Dobs give the value of the measured integrated intensity and its standard deviation even for IREF=1
Note that MODOBS=1 is only valid for IREF less than 4.

The Scale number indicates which of several scale factors applies to this observation. If it is read as zero it is set to 1, so if there is only one scaling region this number may be omitted from the data. In flipping ratio or polarised neutron asymmetry refinements the scale number may be used to indicate measurements made with different beam polarisation.
The Code is at present only used if MODER is 9 or 10, but the user may wish to use it in other cases in his own main program.

Assumptions made

if no L MODE card is given, MODER=3.

REFI
Data

An integer, IREF, indicating the type of refinement, that is, the calculated function which is to be compared with the observed data. IREF is consulted by SFLSQ, MAGLSQ etc. Currently recognised values are 1, 2, 4 and 5:

IREF=1 refine on the modulus of the structure factor Fc
IREF=2 refine on the square of Fc
IREF=4 refine on signed Fc (centrosymmetric)
IREF=5 refine on polarised neutron flipping ratio R (magnetic only)
IREF=6 refine on ratio of magnetic to nuclear structure factor: gamma from cryopad (magnetic only)
IREF=7 refine on polarised neutron asymmetries (magnetic only)
IREF=8 refine using elements of the polarisation matrix (snp only)
IREF=9 refine using polarized neutron intensities from powder data (paramagnetic only)

Assumptions made

If no L REFI card is given, IREF=1 is assumed.

SCAL
Data

A scale factor for as many scale regions as are required. These will be indexed 1,2,3 etc as they are read. The scale factor is used to multiply the calculated function. If the numbers will not fit on to the 80-character line, further L SCAL cards may be given.
For IREF = 5 the scale factors are taken to be depolarisation factors which multiply both the Up and Down polarisations

Assumptions made

If no L SCAL cards are given, the program assumes that there is one factor, with value 1.0, to be refined.

TFAC
Data

Overall temperature factor $B$ for structure factors, used as $\exp(-B(\sthl)^2)$ assumed units are Å$^{2}$

Assumptions made

If no L TFAC card is given, no overall temperature factor is used.

WGHT
Data

An integer IWGHT, the number of the weighting scheme required. At present IWGHT must be 1 or 2:

IWGHT=1

use unit weights (no weighting)

IWGHT=2

use the weight as read with the reflections; for MODER=2 the weight is W, and for other modes where $\sigma$ is read, the weight is $1/(\sigma^2)$

Assumptions made

vary with the main program, as appropriate.

Words Concerned with LSQ Parameters

Please refer also to Chapter 5 for the terminology for naming parameters.

FIX
Data

sets of parameter specifications .

In main programs for single crystal structure refinement like SFLSQ, the following names are defined:

Species in family 1, genus 1 : TFAC DOMR MOSC A* B* C* D* E* F* (A*-F* are for geometric slack constraints)

Genus name for family 1, genus 2: SCAL

Species names for family 2, (the structure parameters whose genus name is their atom name):

X Y Z B11 B12 B13 B22 B23 B33 ITF SCAT SITE
and in addition, for magnetic structures,
MU MU1 THET THE1 PHI PHI1 PSI1 PSI2 PSI3 PSI4

A parameter specification is one of the following:

genus name, space(s), species name e.g.

Ca6 X

SCAL 4

species name alone for family 1, genus 1 e.g.

TFAC

the CCSL-word ONLY

this means that all defaults are overridden, and only the parameters which are explicitly listed are to be fixed.

the CCSL-word ALL

followed by some family, genus or species name, as built in to the particular main program reading the L cards.
L FIX ALL SCAL ALL ITF ALL Nb2

the CCSL-word ALL

followed by another CCSL-word;
for SFLSQ etc the CCSL-words are:
XYZ (= X and Y and Z )
BIJ (= B11, B12, B13, B22, B23 and B33 )
XYZT (= X and Y and Z and ITF )
XYZS (= X and Y and Z and SITE )
XYZB (= XYZ and BIJ )
CELL (= A*, B*, C*, D*, E* and F* )
e.g. L FIX ALL BIJ

The parameters specified are fixed in the subsequent refinement.

Assumptions made

If no L FIX cards are given, the main program's own defaults are taken.

VARY
Data

An L VARY card has exactly the same specification as an L FIX card, except that it causes the specified parameters to be varied rather than to be fixed. If the CCSL-word ONLY occurs on any L VARY card, only the parameters which are explicitly listed are varied.

RELA
Data

Each card contains one relationship between parameters (a constraint ) which the user wishes to impose in addition to those which the system will impose automatically.

The integer after RELA is LRELA, the type of relation; at present types 1 and 2 are available, as described in Chapter 5. The data following LRELA are:

LRELA=1

$a_1, p_1, a_2, p_2,$ where the $a$'s are real numbers and the $p$'s are parameter specifications .
Constraint $: a_1\Delta p_1 = a_2\Delta p_2$

LRELA=2

as many of $a_1 p_1, a_2 p_2, a_3 p_3 . .$ as necessary, all on one card.
Constraint $: a_1\Delta p_1 + a_2\Delta p2 +\cdots + a_n\Delta p_n=0$
(Note that the $a_2$ of LRELA=2 is minus that of LRELA=1)

Note

The CCSL-words like ONLY, ALL etc. cannot be used here.
Note also that the constraints apply to the shifts in the parameters, rather than to the parameters themselves

Example

L RELA 1 1 Co SITE 2 Mn SITE
L RELA 2 1.4 SCAL 1 2.8 SCAL 3 -1.9 SCAL 5

Assumptions made

If there are no L RELA cards, only the constraints which are inherent in the symmetry of the problem are applied.

FUDG
Data

Sets of ( parameter specification , Factor), where the parameter specification is as on FIX and VARY cards except that the CCSL-word ONLY is not used here. The Factor is either a real number or a range indicator. If it is a real number it is a simple multiplicative factor for the shift on the specified parameter.

A range indicator has the form GE limit or LE limit where GE means ``greater than or equal to", LE means ``less than or equal to" and limit is a real number giving the limiting value. If the information will not fit on to an 80-character line several L FUDG cards may be given.

Note

The calculated shifts for the specified parameters are adjusted by the fudge factors before they are applied. In the case of range indicators if, after applying the shift, any of the specified parameters are outside the range, it is set equal to limit . The specified parameters may be individuals or groups.

Example

L FUDG ALL SITE 0.6 Na4 ITF 0.8 L FUDG Mn SITE LE 1.0

Assumptions made

If no L FUDG cards are read, no fudge factors are applied.

Words Concerned with Slack Constraints:

The remaining CCSL-words control the imposition of geometric slack constraints.

SLAK
Data

The presence of an L SLAK card asks for geometric slack constraints to be used. The card gives an integer, STYP and a real number, SWGHT.

The units digit of STYP=1 means ``use only slack constraints, and none of the conventional LSQ observations."
The units digit of STYP=2 means ``use both LSQ observations and slack constraints."
The tens digit of STYP gives the weighting scheme required for slack constraints; 1=unit weights, 2=use weight as read, and 3=read $\sigma$, use $1/\sigma^2$.

Each slack constraint is weighted by its own individual weight, usually read from the card which specifies the constraint. If the units digit of STYP=2 it will be weighted, in addition, by multiplication by SWGHT.

ATOM
Data

A new atom name and an Atom Specification .

An L ATOM card is used to assign a name to an atomic position which is equivalent to, but not identical with, one on an A card. Cards like L BOND, L ANGL etc can then refer to this new atom by name. The Atom Specification can be given in one of two formats. In either case it starts with the atom name.

In the first format the new atomic position is specified by 5 integers, $s, l, c_x, c_y, c_z$. $s$ is the serial number of the symmetry operator producing the required position from that on the A card; it is given negatively if the centre of symmetry is also involved, and can be found by running a CCSL job which calls SYMOP and OPSYM(1). $l$ is similarly the number of the lattice translation used. $c_x$, $c_y$, $c_z$ are cell translations in the $x$, $y$ and $z$ directions enabling the position generated by $s$ and $l$ to be put into any of the neighbouring unit cells; their values would usually be 0, +1 or $-$1.

In the other format the atom name is simply followed by the actual $x$ $y$ and $z$ fractional coordinates to be used, and the CCSL works out the values of $s$, $l$ etc.

The two types of format are distinguished by the presence or absence of a decimal point in the x coordinate.

Example

L ATOM Na1A Na1 2 0 0 0 1
L ATOM Mn99 Mn3 .1234 .2345 .5

BOND
Data

A bond name followed by two atom names, and, optionally, a bond length $b_{opt}$ and its allowed deviation $\sigma$. The atom names must occur on either an A card or a L ATOM card. If no bond length is given, this card simply defines the bond name for subsequent use by L ANGL, L EQUB etc cards.

If $b_{opt}$ and $\sigma$ are given, this is a request to apply a slack constraint to the calculated length of the bond $b_{cal}$ of the form:

$$b_{cal} = b_{opt} \pm \sigma$$

Example

L BOND B04 C1 C2 L BOND BMn9 Mn99 Mn9 2.3 0.05

ANGL
Data

An angle name followed by the bond names of two bonds to a common atom and, optionally, an angle $\phi_{opt}$ in degrees and its $\sigma$. The bond names must occur on L BOND cards. If $\phi_{opt}$ is not given, this card simply defines the angle name for subsequent use by L EQUA, L TORS cards.

If $\phi_{opt}$ and $\sigma$ are given, this is a request to apply a slack constraint to the calculated bond angle of the form:

$$\phi_{cal} = \phi_{opt}\pm \sigma$$

Example

L ANGL PHI6 Bnd1 Bnd2 109.5 0.3

EQUA
Data

Two angle names and a $\sigma$. The angle names must occur on L ANGL cards. This card defines a slack constraint on the values $\phi_1$, $\phi_2$ of the two angles of the form:

$$\phi_1 = \phi_2 \pm \sigma$$

Example

L EQUA Phi4 Phi5 0.4

EQUB
Data

Two bond names and a $\sigma$. The bond names must occur on L BOND cards. This card defines a slack constraint on the values $b_1$, $b_2$ of the two bonds of the form:

$$b_1=b_2\pm \sigma$$

Example

L EQUB Bon1 Bon2 0.001

LINE
Data

Two bond names and a $\sigma$. The bond names must occur on L BOND cards and must have one atom in common. If $b_1 b_2$ are the lengths of the two bonds and $b_3$ the length of the third bond in the triangle, this card defines a slack constraint of the form:

$$b_1+b_2 = b_3\pm\sigma$$

Example

L LINE Bon1 Bon2 0.001

TORS
Data

A torsion angle name , 3 bond names, a torsion angle in degrees $\theta_{opt}$ and its allowed deviation $\sigma$. The bond names must occur on L BOND cards. The torsion angle is defined to be the angle between the plane of bonds 1 and 2 and the plane of bonds 2 and 3. This card defines a slack constraint on the calculated torsion angle $\theta_{cal}$ of the form:

$$\theta_{cal}= \theta_{opt}\pm \sigma$$

Example

L TORS B1 B47 B23 99.9 .1

Words Used in Multi-Source Least Squares (single-crystal)

DATA
Data

A CCSL-word $<name>$ which identifies a set of data. The name of the file containing the data may follow or, if absent, will be asked for interactively.
File names given on the L DATA card card are either absolute path names or are relative to the current working directory. Environment variables are recognised at the start of a path name.

SORC
Data

A CCSL-word $<name>$ which must be the same as one appearing on a L DATA card. The identifier $<name>$ should be followed by one or more CCSL-word/number pairs.

REFI

the number gives the refinement type as defined above.

MODE

the number gives the data format as defined above.

WGHT

the number gives the weight to be given to this set of data.

REFI. MODE and WGHT may all be on the same L SORC card.

or

SCAL

followed by a list of scale-factors for the data set.

DPOP

followed by a list of magnetic domain populations for the data set.

The numbering of magnetic domains is explained in chapter .
When used as least squares parameters the scales and domain populations have genus names which are the data set identifiers and species names starting with either SC or DP and ending with two digits corresponding to the position of the scale factor in the list (SC) or to the domain number (DP). Example

L DATA PA27 $TT/khe27k.pal
L SORC PA27 MODE 9 REFI 8 WGHT 1.0
L DATA SF30 $TT/khe30k.sf
L SORC SF30 MODE 7 REFI 1 WGHT 0.3 SCAL 7.16
L SORC SF30 DPOP 0.25 0.25 0.25 0.25
L SORC PA27 DPOP 0.18 0.07 0.07 0.18 0.18 0.07 0.07 0.18
Z Domain constraints for PA data
L RELA 1 1 PA27 DP08 1 PA27 DP01
L RELA 2 1 PA27 DP01 1 PA27 DP02 1 PA27 DP03 1 PA27 DP04
Z Domain constraints for SF data
L RELA 1 1 SF30 DP07 1 SF30 DP01
L RELA 1 1 SF30 DP05 1 SF30 DP03
L RELA 2 1 SF30 DP01 1 SF30 DP03 1 SF30 DP05 1 SF30 DP07
L FIX SF30 SC01

TELESCOPING OF SIMPLE CARDS:

The REFI , MODE and WGHT information may all occur on the same L card.

Example

L WGHT 2 MODE 1 REFI 4

ROUTINES WHICH READ THE CARDS:

Routines whose names start SFLS in general set up LSQ programs, which includes the reading of L cards. Routines with names starting INPL read a subset of L cards, omitting the FIX, VARY, RELA and FUDG cards and all the cards for geometric constraints.

RDFV

reads L FIX and L VARY cards.

RDRELA

reads L RELA cards.

FUDGIN

reads L FUDG cards.

GEOMIN

reads L SLAK, L ATOM, L BOND, L ANGL, L EQUA, L EQUB, L LINE and L TORS cards.

Called from within routine INPLSF there are:

LLTFAC

to read L TFAC cards.

LLSCAL

to read L SCAL cards.

NOTE:

See also I cards in their application to driving LSQ programs.

Example I card for LSQ:

I NCYC 3 CYC1 10 MCOR 50 CONV 0.05