4.13. - Least-Squares Refinement of Lattice Parameters
In the previous sections, the procedure described
for determining the orientation matrix required
a knowledge of the lattice parameters of the crystal and
the position of two reflections.
When such information is unknown,
the orientation matrix can be
fit to an unlimited number of observed peak positions
using a least-squares procedure.
(
J. Matthews and R.L. Walker,
Mathematical Methods of Physics,
(Benjamin, Menlo Park, 1970), p. 391.
)
Lattice parameters derived from the fitted orientation matrix
can then be calculated, although such lattice parameters
are not constrained to exhibit any symmetry whatsoever.
In spec's implementation of least square refinement, three
macros are used to create a file that contains the observed peak
positions.
That file is eventually run as a command file, and the least squares
analysis is performed.
The
reflex_beg
macro initializes the reflections
file:
global REFLEX # Variable for file name
# Open the file, save old one as .bak and write header
def reflex_beg '{
if ("$1" == "0") {
if (REFLEX == ")
REFLEX = "reflex"
REFLEX = getval("Reflections file", REFLEX)
} else
REFLEX = "$1"
if (open(REFLEX))
exit
close(REFLEX)
if (file_info(REFLEX, "-s"))
unix(sprintf("mv %s %s.bak", REFLEX, REFLEX))
fprintf(REFLEX,"# %s\n\n_begUB\n\n",date())
}'
The
reflex
macro adds lines to the file that
contain
the (H,K,L)
and (2 ,, , )
of each reflection:
# Add reflection to the file
def reflex '
if ($# != 3) {
print "Usage: reflex H K L"
exit
}
if (REFLEX == ") {
REFLEX = getval("Reflections file", "reflex")
if (REFLEX == ")
exit
}
waitmove; get_angles; calcHKL
fprintf(REFLEX,"H = %g; K = %g; L = %g\n",$1,$2,$3)
{
local i
for (i=0; i<_numgeo; i++)
fprintf(REFLEX,"A[%s]=%9.4f; ",motor_mne(mA[i]),A[mA[i]])
fprintf(REFLEX,"\n")
}
fprintf(REFLEX,"# counts = %g\n", S[DET])
fprintf(REFLEX,"_addUB\n\n")
'
Finally, the
reflex_end
macro puts the proper trailer on the file:
# Add trailer to file
def reflex_end '
fprintf(REFLEX,"_fitUB\n")
printf("Type \"qdo %s\" to calculate new orientation matrix\n",\
REFLEX)
'
When you are ready to calculate the orientation matrix,
simply run the command file.
There is no limit to the number of reflections contained in the
file.
You can also edit the file by hand to add or subtract reflections.
The calculated orientation matrix will remain valid until
you type
calcG,
or invoke a macro that calls
calcG. Those macros are
or0, or1, or_swap and
setlat.
The six original lattice parameters will remain unchanged
when using the
above macros to fit the orientation matrix to the reflections.
The
calcL
macro can be invoked to calculate the lattice parameters
derived from the fitted orientation matrix and place their values
in the appropriate elements of the parameter
array.
The old lattice parameters will be lost.
Here is a sketch of the commands you use to perform
the least squares refinement of the lattice parameters.
1.FOURC> reflex_beg
Reflections file (reflex)? <return>
2.FOURC> (find and move to a reflection ...)
3.FOURC> reflex 2 2 0
4.FOURC> (find and move to another reflection ...)
5.FOURC> reflex 2 0 2
6.FOURC> ...
7.FOURC> reflex_end
Type "qdo reflex" to recalculate orientation matrix.
8.FOURC> qdo reflex
Opened command file `reflex' at level 1.
9.FOURC>
At least three reflections must be used
for the least-squares fitting to work.
To calculate the new lattice
parameters, use the
calcL
macro:
1.FOURC> calcL
2.FOURC> pa
4-Circle Geometry, Omega-Fixed (mode 1), frozen coordinate, sector 0
Primary Reflection (at lambda 1.54):
Omega Chi Phi = 5.00025 90.102 -82.5455
H K L = 2 2 0
Secondary (at lambda 1.54):
Omega Chi Phi = 0 0 90
H K L = 0 1 0
Azimuthal Reference:
H K L = 2 2 0
Lattice Constants:
a b c = 4.12644 4.12295 4.11078
Alpha Beta Gamma = 90.0435 89.9763 90.1221
3.FOURC>
Here is a typical reflections file created by the above macros:
# Wed Jan 31 21:55:01 1990
_begUB
H = 2; K = 2; L = 0
A[tth]= 63.8185; A[th]= 36.9320; A[chi]= 89.8765; A[phi]= -80.0815
# counts = 3456
_addUB
H = 2; K = 0; L = 2
A[tth]= 63.8335; A[th]= 36.8920; A[chi]= 145.4185; A[phi]= 42.8145
# counts = 6345
_addUB
H = 0; K = 2; L = -1
A[tth]= 49.4100; A[th]= 29.6725; A[chi]= 35.8180; A[phi]= 45.9070
# counts = 5634
_addUB
H = 0; K = 2; L = -1
A[tth]= 49.3550; A[th]= 29.7225; A[chi]= 35.9380; A[phi]= 45.9070
# counts = 4563
_addUB
_fitUB
Within this file, the
calc() function codes defined by the macros
_begUB, _addUB and
_fitUB are used to access the C code that performs the least squares operations.
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