USER'S INSTRUCTIONS FOR PROGRAMS REFPK AND BGLP --- ROBERT H. BLESSING HAUPTMAN-WOODWARD INSTITUTE 73 HIGH STREET BUFFALO, NEW YORK 14203 TELEPHONE: (716) 856-9600 ELECTRONIC MAIL: Blessing@HWI.Buffalo.Edu ------------------------------------------------------------------------ PROGRAM REFPK EXAMINES AND ANALYZES OMEGA OR THETA/TWO-THETA SCANNED BRAGG REFLECTION PROFILES. PEAK POSITIONS AND PEAK WIDTHS ARE FOUND FOR THE REFLECTIONS THAT PASS SEVERAL SIGNAL-TO-NOISE TESTS. THE DISPLACEMENTS OF THESE PEAKS FROM THE MIDPOINTS OF THEIR SCANS AND THE BASE WIDTHS OF THESE PEAKS ARE USED AS OBSERVATIONS FOR LEAST-SQUARES CALCULATIONS OF PARAMETERS THAT ALLOW PEAK POSITIONS AND PEAK WIDTHS, WHICH ARE SCATTERING ANGLE DEPENDENT AND ANISOTROPICALLY VARIABLE, TO BE CALCULATED FOR ALL REFLECTIONS, INCLUDING THOSE WITH SMALL SIGNAL-TO-NOISE RATIOS. PROGRAM BGLP APPLIES CORRECTIONS FOR COINCIDENCE COUNTING LOSSES; CALCULATES THE PEAK LIMITS FOR EACH REFLECTION USING THE PARAMETERS FROM PROGRAM REFPK; FITS A LEAST-SQUARES STRAIGHT LINE TO THE BACKGROUND OUTSIDE THE PEAK LIMITS; SUBTRACTS THE BACKGROUND, DIVIDES OUT THE LORENTZ AND POLARIZATION FACTORS, AND INTEGRATES THE NET REFLECTION INTENSITY BETWEEN THE PEAK LIMITS; AND CALCULATES THE ESTIMATED STANDARD DEVIATION OF THE NET INTENSITY. ------------------------------------------------------------------------ LOCATION OF INTENSITY-WEIGHTED PEAK CENTROIDS -------- -- ------------------ ---- --------- IN ORDER TO LOCATE THE INTENSITY-WEIGHTED PEAK CENTROID, INITIAL PEAK LIMITS ARE OBTAINED USING A RELATIVELY NARROW MOVING WINDOW OF WIDTH W TO FIND THE SMALLEST X FOR WHICH THE COUNT RATE INTEGRATED FROM X TO X + W SIGNIFICANTLY EXCEEDS THAT FROM X - W TO X, AND THE LARGEST X FOR WHICH THE COUNT RATE FROM X TO X - W EXCEEDS THAT FROM X TO X + W. X X X X X X X X X X X X X X X X X X X XX X X X X X* *X XXXXX*XXXXX*XXXX * * XXX*XXXXX*XXX * * * * * * * * * * * * -----+-----+-----+--------------------------------+-----+-----+--- X1-W X1 X1+W X2-W X2 X2+W THE BACKGROUND IS ESTIMATED BY FITTING A LEAST-SQUARES STRAIGHT LINE TO THE COUNT RATE PROFILE OUTSIDE THE INITIAL PEAK LIMITS, AND THE PEAK CENTROID WEIGHTED BY THE INTENSITY ABOVE THE ESTIMATED LINEAR BACKGROUND IS TAKEN TO CORRESPOND TO THE BRAGG ANGLE, THETA, FOR THE INTENSITY-WEIGHTED AVERAGE WAVELENGTH, LAMBDA(K-ALPHA-BAR). THE POSITIONS OF THETA(K-ALPHA-1) AND THETA(K-ALPHA-2) ARE CALCULATED FROM THE POSITION OF THE THE PEAK CENTROID. ------------------------------------------------------------------------ LOCATION OF PEAK LIMITS -------- -- ---- ------ THE PEAK LIMITS ARE LOCATED BY THE METHOD OF LEHMANN AND LARSEN, WHICH SEEKS THE TWO POINTS IN THE SCAN PROFILE BETWEEN WHICH INTEGRATION OF THE NET INTENSITY GIVES MINIMUM SIGMA(I)/I. (IN PRACTICE, THE ROUTINE ACTUALLY SEEKS TO MAXIMIZE THE INVERSE RATIO I/SIGMA(I) IN ORDER TO AVOID THE PROBLEM OF DIVISION BY A ZERO VALUE OF I.) SEE M. S. LEHMANN AND F. K. LARSEN, ACTA CRYST., A30, 580 (1974); AND R. H. BLESSING, P. COPPENS, AND P. BECKER, J. APPL. CRYST., 7, 448 (1974). THE MINIMUM SIGMA(I)/I CRITERION IS INHERENTLY BIASED. IT FINDS PEAK LIMITS THAT ARE SLIGHTLY WITHIN THE PEAKS AND THUS SLIGHTLY UNDERESTIMATES THE AREA UNDER THE PEAKS. LEHMANN AND LARSEN HAVE SHOWN THAT THE BIAS OF THE PEAK LIMITS IS PROPORTIONAL TO 1/(2 + R), WHERE R IS THE INTEGRATED PEAK-TO- BACKGROUND RATIO. THE PEAK LIMITS FOUND BY THE ALGORITHM ARE CORRECTED ACCORDINGLY IN PROGRAM REFPK, BEFORE THE LEAST-SQUARES CALCULATIONS OF THE PEAK WIDTH PARAMETERS. ------------------------------------------------------------------------ SIGNAL-TO-NOISE TESTS ------ -- ----- ----- ALL REFLECTIONS ARE SCREENED IN ORDER TO SELECT OBSERVATIONS FOR THE LEAST-SQUARES CALCULATIONS OF THE PEAK POSITION AND PEAK WIDTH PARAMETERS. REFLECTIONS FOR THE LEAST-SQUARES CALCULATIONS ARE SELECTED ON THE BASIS OF THE FOLLOWING TESTS: 1. THE ROUTINE FOR CALCULATING THE INTENSITY-WEIGHTED PEAK CENTROID MUST FIND A SIGNIFICANTLY POSITIVE PEAK ABOVE THE ESTIMATED LINEAR BACKGROUND, I.E., I .GE. CUTOFF*SIGMA(I), WHERE TYPICALLY CUTOFF = 3. 3. THE ROUTINE FOR LOCATING THE PEAK LIMITS MUST FIND TRUE MINIMA, SIGMA(I)/I, AT EACH SIDE OF THE PEAK. ------------------------------------------------------------------------ CALCULATION OF PEAK POSITIONS ----------- -- ---- --------- THE LEAST-SQUARES FIT OF PEAK POSITIONS BY PROGRAM REFPK IS SIMPLY A RECALCULATION OF THE DIFFRACTOMETER ORIENTATION MATRIX USING AS ORIENTATION ANGLE OBSERVATIONS THE OMEGA VALUES OF THE INTENSITY- WEIGHTED PEAK CENTROIDS ALONG WITH THE DIFFRACTOMETER VALUES OF CHI AND PHI. THIS GIVES AN ORIENTATION MATRIX ADJUSTED FOR THE OMEGA PEAK DISPLACEMENTS AND AVERAGED OVER THE CHI AND PHI SETTINGS USED OVER THE WHOLE COURSE OF THE DATA SET MEASUREMENT. EXPERIENCE HAS SHOWN THAT THE RECALCULATED ORIENTATION MATRIX IS USEFUL ONLY WHEN DISPLACEMENTS OF THE PEAK CENTROIDS FROM THE MIDPOINTS OF THE SCANS ARE SMALL AND ONLY, OR MAINLY, ORIENTATION DEPENDENT (I.E., DUE TO FIXED, SMALL TRANSLATIONAL OR ROTATIONAL MISORIENTATION OF THE SPECIMEN CRYSTAL), AND NOT SIGNIFICANTLY TIME DEPENDENT AS WELL (I.E., NOT DUE TO CHANGES IN ORIENTATION CAUSED BY CRYSTAL SLIPPAGE). IF THERE WERE MANUAL OR AUTOMATIC REORIENTATIONS OF THE CRYSTAL DURING THE INTENSITY MEASUREMENTS, THE RECALCULATED ORIENTATION MATRIX WILL BE A KIND OF AVERAGE OVER THE SEVERAL ORIENTATIONS. IF THERE WERE LARGE CHANGES IN ORIENTATION, MORE THAN A FEW DEGREES, THEN THE DATA SET SHOULD BE SUBDIVIDED INTO BATCHES OF APPROXIMATELY CONSTANT ORIENTATION, AND THE BATCHES PROCESSED SEPARATELY, BECAUSE THE PEAK WIDTH ANISOTROPY PARAMETERS ARE ORIENTATION DEPENDENT. THE RECALCULATED ORIENTATION MATRIX WILL NOT, IN GENERAL, GIVE ACCURATE LATTICE PARAMETERS. IT IS USED ONLY TO ESTIMATE AN OMEGA OFFSET PEAK POSITION FOR SCANS OF WEAK REFLECTIONS WITH NO SIGNIFICANT PEAK ABOVE BACKGROUND, AND IT IS USED FOR THIS PURPOSE ONLY WHEN IT IS WELL-DETERMINED, I.E., ONLY WHEN THE CRYSTAL MISORIENTATION IS SMALL. OTHERWISE, THE SCAN MIDPOINT IS ASSUMED TO BE THE BEST ESTIMATE OF THE PEAK POSITION IN THE WEAK REFLECTION SCANS. ------------------------------------------------------------------------ CALCULATION OF PEAK WIDTHS ----------- -- ---- ------ THE LEAST-SQUARES FIT OF PEAK WIDTHS BY PROGRAM REFPK ASSUMES THAT THE REFLECTION PEAKS RESULT FROM A CONVOLUTION OF INSTRUMENTAL AND SPECTRAL DISTRIBUTION FUNCTIONS WITH AN ANISOTROPIC CRYSTAL SIZE AND MOSACITY DISTRIBUTION FUNCTION. X X X X X X X X X X X X X X X X X X X XX X X X X X XXXXXXXXXXX XXXXXXXXXXXX XXX XXXX +-----+------------+--------+----+--------+-----------+---------+ I1 XE L1 X1 X0 X2 L2 I2 <--------> <-----------> W1 W2 W1 AND W2 ARE BASE WIDTHS OF THE HALF-PEAKS BELOW THETA(ALPHA-1) AND ABOVE THETA(ALPHA-2), RESPECTIVELY. THEY ARE EACH ASSUMED TO BE REPRESENTABLE AS EITHER W = [Z(K)*Q(J,K)*Z(J)]**(1/2) + T*TAN(THETA) (1) OR W = [Z(K)*Q(J,K)*Z(J) + (T*TAN(THETA))**2]**(1/2) (2) OR W = [Z(K)*Q(J,K)*Z(J) - 2*F*T*TAN(THETA)/TAN(THETA(M)) + T*(TAN(THETA)/TAN(THETA(M)))**2]**(1/2) (3) THE TWO HALF-PEAKS ARE TREATED SEPARATELY, BECAUSE THE REFLECTION PEAKS ARE NOT, IN GENERAL, SYMMETRIC AND THE SPECTRAL LINES DO NOT HAVE THE SAME WIDTH. EQUATIONS (1) AND (2) APPLY IN THE CASE OF A BETA-FILTERED X-RAY BEAM OR A BEAM FROM A PERPENDICULAR GEOMETRY MONOCHROMATOR; EQUATION (3) APPLIES IN THE CASE OF A PARALLEL GEOMETRY MONOCHROMATOR. IN ALL THREE EQUATIONS, INDEX REPETITION IMPLIES SUMMATION. W1 AND W2 HAVE UNITS OF DEGREES OF CRYSTAL ROTATION, THETA. Z(3) ARE COMPONENTS OF A UNIT VECTOR PERPENDICULAR TO THE INCIDENT AND DIFFRACTED BEAM DIRECTIONS. Q(3,3) ARE ELEMENTS OF SYMMETRIC TENSORS THAT ARE RELATED TO THE ANISOTROPY OF THE SIZE AND OF THE MOSAICITY (FAULT STRUCTURE) OF THE SPECIMEN CRYSTAL. (SEE R. J. NELMES, ACTA CRYST., A36, 641- 653 (1980).) T ARE (SCALAR) COEFFICIENTS RELATED TO THE SPECTRAL WIDTH OF THE CHARACTERISTIC ALPHA-1 AND ALPHA-2 LINES. (VALUES OF THE FULL- WIDTH AT HALF-HEIGHT OF THE SPECTRAL LINES ARE GIVEN BY H. C. COMPTON AND S. K. ALLISON, X-RAYS IN THEORY AND EXPERIMENT, VAN NOSTRAND, NEW YORK, 1935, PP. 744-745.) IN EQUATION (3), THETA(M) IS THE MONOCHROMATOR BRAGG ANGLE AND F IS A NUMERICAL CONSTANT WITH A VALUE IN THE RANGE 1 .LE. F .LE. 2 (H. DACHS, 1978, NEUTRON DIFFRACTION, PP. 25 FF. G. BACON, 1975, NEUTRON DIFFRACTION, PP. 101 FF.). ALL THE VARIOUS FIXED, INSTRUMENTAL CONTRIBUTIONS TO THE REFLECTION WIDTH, SUCH AS THE FINITE SIZE AND THE SPATIAL INTENSITY DISTRIBUTION OF THE X-RAY SOURCE, ARE ASSUMED TO BE IMPLICIT IN THE Q TENSORS. ALL THE VARIOUS SCATTERING ANGLE DEPENDENT CONTRIBUTIONS, SUCH AS BEAM DIVERGENCE, SPECTRAL BROADENING, AND MOSAIC BROADENING, ARE ASSUMED TO BE IMPLICIT IN THE T COEFFICIENTS. THE VECTOR COMPONENTS Z(3) AND THE TENSOR COMPONENTS Q(3,3) ARE REFERRED TO A CARTESIAN AXIAL SYSTEM FIXED IN THE CRYSTAL. THIS AXIAL SYSTEM IS THE SAME AS THE SYSTEM, LABELED AD, DEFINED BY WALTER HAMILTON (INTERNATIONAL TABLES FOR X-RAY CRYSTALLOGRAPHY, 1974, VOL. IV, PP. 273-284). EQUATION (1) IMPLIES THAT THE CRYSTAL ANISOTROPY DISTRIBUTION, OF WIDTH [Z(K)*Q(J,K)*Z(J)]**(1/2), AND THE SPECTRAL DISTRIBUTION, OF WIDTH T*TAN(THETA), CONVOLUTE AS WOULD TWO LORENTZIAN (OR CAUCHY) FUNCTIONS, AND EQUATION (2) IMPLIES THAT THE TWO DISTRIBUTIONS CONVOLUTE AS WOULD TWO GAUSSIAN FUNCTIONS. EQUATION (3) IMPLIES CONVOLUTION OF GAUSSIAN DISTRIBUTION FUNCTIONS FOR THE SPECIMEN AND MONOCHROMATOR CRYSTAL MOSAICITIES AND FOR THE BEAM DIVERGENCE. IN THE FOLLOWING SKETCHES OF LORENTZIAN AND GAUSSIAN PEAKS, THE ABCISSAE UNITS ARE (X - X0)/(W/2), WHERE W IS THE FULL WIDTH AT HALF HEIGHT, AND THE ORDINATE UNITS ARE RELATIVE AND NORMALIZED TO A MAXIMUM HEIGHT EQUAL TO TEN HALF-WIDTHS AT HALF-HEIGHT. GAUSSIAN PEAK G G G G G G G G G G G G G G G G G ----+---+---+---+---+---+---+---+---+---+---+---- -5 -2 -1 0 +1 +2 +5 LORENTZIAN PEAK L L L L L L L L L L L L L L L L L L L L L L L L L ----+---+---+---+---+---+---+---+---+---+---+---- -5 -2 -1 0 +1 +2 +5 X L L G G L L G G L L X X L L LG GL L L L L L G G L L G G L L G G L L G G L ----+---+---+---+---+---+---+---+---+---+---+---- -5 -2 -1 0 +1 +2 +5 NOTE THAT A LORENTZIAN PEAK IS SOMEWHAT NARROWER AT THE TOP AND SUBSTANTIALLY WIDER AT THE BOTTOM THAN A GAUSSIAN PEAK WITH THE SAME HEIGHT AND THE SAME WIDTH AT HALF HEIGHT. FOR BETA FILTERED OR PERPENDICULAR GEOMETRY MONOCHROMATOR DATA, PROGRAM REFPK TRIES BOTH EQ. (1) AND EQ. (2) AS MODELS FOR THE BASE WIDTHS OF THE HALF-PEAKS. FOR EACH SET OF HALF-PEAK BASE WIDTHS, AND FOR EACH OF THE TWO MODELS, EQ. (1) AND EQ. (2), THE PROGRAM FITS BY LEAST-SQUARES SCALAR SPECTRAL COEFFICIENTS, T, AND ANISOTROPY TENSOR COEFFICIENTS, Q(3,3). THE MODEL THAT GIVES THE SMALLER ROOT-MEAN-SQUARE ERROR-OF-FIT, RMSD = SQRT(SUM[(WOBS - WCALC)**2]/[NOBS - NPAR]), IS CHOSEN TO CALCULATE THE BASE WIDTHS FOR EACH HALF-PEAK. THE LEAST-SQUARES FITTING IS DONE TN TWO STAGES. FIRST, THE T- COEFFICIENTS AND ISOTROPIC Q-VALUES ARE FITTED, AND THE TYPE OF CONVOLUTION MODEL IS CHOSEN. THEN, USING THE FITTED T-VALUES, THE ANISOTROPIC Q-COEFFICIENTS ARE FITTED. (THIS TWO-STAGE FITTING IS A VARIANT ON THE PROCEDURE DESCRIBED BY BLESSING [CRYST. REV. 1, 3-58 (1987)], WHICH WAS A ONE-STAGE PROCEDURE THAT ASSUMED FIXED, PRECALCULATED T-VALUES FOR EQS. (1) AND (2), AND FITTED THE T- COEFFICIENTS ALONG WITH THE Q-COEFFICIENTS FOR EQ. (3).) NOTE THAT THE PROGRAM IS DESIGNED TO FIT ONLY THE BASE WIDTHS OF --- ---- --- ---- ------ THE REFLECTION PEAKS. IT DOES NOT FIT LORENTZIAN OR GAUSSIAN --- --- ---------- -- -------- FUNCTIONS TO THE PROFILES OF THE PEAKS, ALTHOUGH IT DOES ASSUME --------- -- --- -------- THAT EITHER LORENTZIAN OR GAUSSIAN CONVOLUTION PROPERTIES ARE ADEQUATE APPROXIMATIONS TO CALCULATE THE WIDTH OF THE CONVOLUTION PRODUCT OF THE CRYSTAL ANISOTROPY DISTRIBUTION AND THE SPECTRAL DISPERSION DISTRIBUTION. ------------------------------------------------------------------------ BACKGROUND SUBTRACTION ---------- ----------- PROGRAM BGLP REPEATS THE SEARCH FOR THE INTENSITY-WEIGHTED PEAK CENTROID FOR EACH REFLECTION. FOR THE VERY WEAKEST REFLECTIONS THAT DO NOT HAVE A SIGNIFICANT PEAK ABOVE BACKGROUND FOR WHICH AN INTENSITY-WEIGHTED CENTROID CAN BE RELIABLY LOCATED, PROGRAM BGLP RESORTS TO THE PEAK POSITION ESTIMATED FROM THE RECALCULATED ORIENTATION MATRIX OR TO THE SCAN MIDPOINT. GIVEN THE PEAK POSITION FOR A REFLECTION, PROGRAM BGLP USES THE PEAK WIDTH PARAMETERS FROM PROGRAM REFPK TO CALCULATE THE PEAK LIMITS. IT FITS A LEAST-SQUARES STRAIGHT LINE TO THE BACKGROUND OUTSIDE THE PEAK LIMITS, AND INTEGRATES THE PEAK ABOVE BACKGORUND BETWEEN THE PEAK LIMITS. FOR SCANS FOR WHICH THE ABSORPTION EDGE OF A BETA FILTER FALLS WITHIN THE SCAN RANGE, A STRUCTURED BACKGROUND MADE-UP OF THREE STRAIGHT-LINE PIECES IS FITTED (SEE R. J. NELMES, ACTA CRYST. A31, 273-279 (1975)). THE STRUCTURED BACKGROUND CORRECTION IS FOUND TO BE SUITABLE FOR THE LOW ANGLE REFLECTIONS FROM CRYSTALS WITH UNIT CELLS OF SMALL TO MODERATE SIZE. FOR THE NUMEROUS VERY LOW ANGLE REFLECTIONS FROM MACROMOLECULAR CRYSTALS, A SIMPLE STRAIGHT-LINE BACKGROUND IS, GENERALLY, A BETTER APPROXIMATION THAN THE MODELING OF THE STRUCTURED BACKGROUND. THIS IS BECAUSE THE SHAPE OF THE BACKGROUND IS OFTEN DETERMINED MORE BY THE SCATTERING FROM THE SPECIMEN CAPILLARY, MOTHER LIQUOR, ETC. THAN BY THE SPECTRAL STRUCTURE DUE TO THE WHITE RADIATION AND THE BETA FILTER ABSORPTION EDGE, WHICH IS OFTEN VERY CLOSE TO THE ALPHA REFLECTION PEAK. ------------------------------------------------------------------------ INPUT FILES REQUIRED: ----- ----- -------- 1. RAW REFLECTION PROFILE DATA FILE 'DATA.RAW' FROM A LOCAL DIFFRACTOMETER TAPE DECODING PROGRAM IS USED BY BOTH PROGRAMS REFPK AND BGLP. 2. CONTROL DATA FILE 'REFPK.DAT' IS DESCRIBED BELOW. 3. AN OPTIONAL CONTROL DATA FILE 'HKLCOND.DAT', WHICH GIVES THE CONDITIONS LIMITING POSSIBLE REFLECTIONS, IS ALSO DESCRIBED BELOW. 4. CONTROL DATA FILE 'BGLP.DAT' IS CREATED BY PROGRAM REFPK. PROGRAM BGLP DOES NOT REQUIRE ANY ADDITIONAL INPUT AND MAY BE RUN IMMEDIATELY FOLLOWING A SATISFACTORY RUN OF PROGRAM REFPK. ------------------------------------------------------------------------ RAW REFLECTION PROFILE DATA FILE 'DATA.RAW' --- ---------- ------- ---- ---- -------- AN UNFORMATTED, BINARY FILE READ BY A SUBROUTINE READ1 IN THE PROGRAMS VIEW, REFPK, AND BGLP. THE RECORDS CONTAIN: JJ MEASUREMENT SERIAL NUMBER JH JK MILLER INDICES JL A1 A2 DIFFRACTOMETER A3 SETTING ANGLES A4 WIDTH SCAN WIDTH (DEGREES OF CRYSTAL ROTATION, THETA) SPEED SCAN SPEED (DEGREES THETA PER MINUTE) X1 (BGL LEFT BACKGROUND COUNT) X2 (PEAK TOTAL PEAK COUNT) X3 (BGR RIGHT BACKGROUND COUNT) X4 (RNET NET REFLECTION INTENSITY) X5 (SIGR ESD ON COUNTING STATISTICS ALONE) XTIME X-RAY EXPOSURE TIME (HOURS) JD(96) ARRAY OF RAW STEP SCAN COUNTS JJ IS AN INTEGER*4 WORD; JH, JK, JL, AND JD(96) ARE INTEGER*2 WORDS; AND ALL THE OTHER WORDS ARE REAL*4. JJ IS NEGATIVE FOR STANDARD REFERENCE INTENSITY REFLECTIONS. WIDTH IS NEGATIVE IF THE BEAM ATTENUATOR WAS USED. A1, A2, A3, AND A4 ARE THE SETTING ANGLES AS WRITTEN BY THE DIFFRACTOMETER. NOTE THAT THE ORDER IS IMPORTANT. A1 A2 A3 A4 INT.TAB. TWO-THETA OMEGA CHI PHI BUS.LEV. TWO-THETA OMEGA CHI PHI P3 TWO-THETA OMEGA PHI CHI CAD4 THETA PHI OMEGA KAPPA X1, X2, X3, X4, AND X5 ARE NOT USED BY THE PROGRAMS. IN PROGRAMS VIEW, REFPK, AND BGLP, THE SUBROUTINE READ1 IS CALLED BY A SUBROUTINE READR. SUBROUTINE READR MUST RETURN: ---------- ----- ---- ------ INTEGER*4 VARIABLES: II - MEASUREMENT SERIAL NUMBER IH,IK,IL - REFLECTION INDICES NSTEP - NUMBER OF SCAN STEPS REAL*4 ARRAYS: ANGLES(4) - DIFFRACTOMETER SETTING ANGLES XX(NSTEP) - ROCKING-ANGLE STEPS YY(NSTEP) - COUNT-RATE STEPS VX(NSTEP) - ROCKING-ANGLE VARIANCES VY(NSTEP) - COUNT-RATE VARIANCES REAL*4 VARIABLE: XTIME - X-RAY EXPOSURE TIME THE DIFFRACTOMETER ANGLES MUST BE IN DEGREES, AND THEIR ORDER IS IMPORTANT: DIFF. ANGLES(1) (2) (3) (4) INT.TAB. TWO-THETA OMEGA CHI PHI BUS.LEV. TWO-THETA OMEGA CHI PHI P3 TWO-THETA OMEGA PHI CHI CAD4 THETA PHI OMEGA KAPPA THE ROCKING-ANGLE STEPS MUST INCREASE (FROM NEGATIVE TO POSITIVE) IN ORDER OF INCREASING ABSOLUTE VALUE OF THE BRAGG ANGLE, THETA. RECOMMENDED UNITS: ROCKING ANGLE - DEGREES (THETA) COUNT RATES - COUNTS PER SECOND EXPOSURE TIME - HOURS ------------------------------------------------------------------------ CONTROL DATA FILE 'REFPK.DAT' ------- ---- ---- ========= USER MUST SUPPLY THE DATA OF RECORDS 1 - 8. RECORDS 9 - 20 MAY BE BLANK RECORDS, AND THE PROGRAM WILL SUPPLY DEFAULT DATA. 1. TITLE (A) = JOB TITLE 2. FILE1 (A) = NAME OF RAW REFLECTION PROFILE DATA FILE 3. FILE2 (A) = NAME TO BE GIVEN TO REDUCED DATA OUTPUT FILE FROM PROGRAM BGLP 4. LATTICE CONSTANTS (6F10.0) A(1) = A (ANGSTROMS) A(2) = B A(3) = C A(4) = ALPHA (DEGREES) A(5) = BETA A(6) = GAMMA 5. DIFF (A4) = DIFFRACTOMETER TYPE (H, BL, P3, OR CAD4) H HAMILTON'S DIFFRACTOMETER AXES W. HAMILTON, INTERNATIONAL TABLES FOR X-RAY CRYSTALLOGRAPHY, VOL. IV, 1974, PP. 273-284. BL BUSING'S AND LEVY'S DIFFRACTOMETER AXES W. R. BUSING AND H. A. LEVY, ACTA CRYST., 22, 457-464 (1967). P3 SIEMENS (NEE NICOLET, NEE SYNTEX) P3 DIFFRACTOMETER CAD4 ENRAF-NONIUS CAD4 DIFFRACTOMETER OTHERWISE, DIFFRACTION GEOMETRY IS ASSUMED TO BE FOR A DIFFRACTOMETER AS DEFINED BY HAMILTON. 6. TARGET (A2) = CHEMICAL SYMBOL FOR X-RAY TUBE TARGET (CU, MO, AG, OR OTHER) FOR NEUTRON DATA, ENTER NEUTRON. 7. FILTER (A2) = CHEMICAL SYMBOL FOR BETA FILTER (IF ANY) (NI, ZR, NB, PD, RH, OR OTHER) LEAVE BLANK, OR ENTER NONE, FOR DATA FROM LARGE UNIT CELL, MACROMOLECULAR CRYSTALS, FOR WHICH A SIMPLE STRAIGHT-LINE BACKGROUND IS, GENERALLY, A BETTER APPROXIMATION THAN MODELING THE BACKGROUND STRUCTURE DUE TO THE WHITE RADIATION AND THE BETA FILTER. 8. MONOCR (A2) = MONOCHROMATOR CRYSTAL (IF ANY) (GRAPHITE, QUARTZ, OR OTHER) -- -- (9. MONOCHROMATOR VARIABLES (4F10.0) THM = MONOCHROMATOR BRAGG ANGLE THETA RHOM = MONOCHROMATOR-DIFFRACTOMETER GEOMETRY ANGLE RHO RHO = 0 PARALLEL GEOMETRY RHO = 90 DEGREES PERPENDICULAR GEOMETRY FRACTD = MONOCHROMATOR DYNAMIC DIFFRACTION (PERFECT CRYSTAL) FRACTION SIGFD = SIGMA(FRACTD), ESTIMATED STANDARD DEVIATION OF FRACTD DEFAULT VALUES OF THM ARE STORED IN THE PROGRAM FOR GRAPHITE (0 0 0 2 REFLECTION) AND QUARTZ (1 0-1 1 REFLECTION) MONOCHROMATOR CRYSTALS FOR CU, MO, AND AG X-RAYS. DEFAULT VALUES OF RHO ARE STORED FOR THE NICOLET (SYNTEX) P3 (RHO = 0) AND THE ENRAF-NONIUS CAD4 (RHO = 90 DEGREES) DIFFRACTOMETERS. THE DEFAULT VALUE OF FRACTD IS ZERO, AND A DEFAULT VALUE SIGFD = 0.05*FRACTD WILL BE USED UNLESS THE USER SUPPLIES A NON-ZERO VALUE FOR SIGFD.) (10. WAVELENGTH VALUES (5F10.0) WLA1 = ALPHA-1 WAVELENGTH (ANGSTROMS) WLA2 = ALPHA-2 WAVELENGTH DWLA1 = ALPHA-1 SPECTRAL LINE WIDTH (FWHH ANGSTROMS) DWLA2 = ALPHA-2 SPECTRAL LINE WIDTH EDGE = WAVELENGTH OF K-ABSORPTION EDGE OF BETA FILTER DEFAULT DATA FOR WAVELENGTHS AND SPECTRAL LINE WIDTHS ARE STORED IN THE PROGRAM FOR CU, MO, AND AG K-ALPHA X-RADIATION. DEFAULT DATA FOR K-ABSORPTION EDGE WAVELENGTHS ARE STORED FOR NI, ZR, NB, PD, AND RH BETA FILTERS.) FOR NEUTRON DATA, ENTER WLA1 = WLA2 AND DWLA1 = DWLA2. (11. ESTIMATED ERRORS FOR THE DIFFRACTOMETER SCAN ANGLE AND CLOCK TIMER (2F10.0) SIGTHE (DEGREES THETA) SIGTIM (MICROSECONDS)) (12. COUNTING CONSTANTS FOR COINCIDENCE CORRECTIONS (4F10.0) TAU SIGMA(TAU) ATT SIGMA(ATT) TAU = DEAD TIME OF QUANTUM COUNTING CHAIN (MICROSECONDS) ATT = CORRECTION FACTOR FOR BEAM ATTENUATOR) (13. BEGINNING AND ENDING REFLECTION SERIAL NUMBERS OR X-RAY EXPOSURE TIMES (2I10, 2F10.0) IIMIN, IIMAX = SERIAL NUMBER LIMITS XTMIN, XTMAX = EXPOSURE TIME LIMITS (HOURS) THESE LIMITS CAN BE USED TO PROCESS DATA IN BATCHES. ONLY REFLECTIONS WITHIN GIVEN LIMITS WILL BE PROCESSED. THE DEFAULT PROCEDURE IS TO PROCESS ALL REFLECTIONS. DEFAULT VALUES: IIMIN = -1E5, IIMAX = +1E5; XTMIN = -1.0E5, XTMAX = +1.0E5.) (14. SC1, SC2 (4F10.0). SCAN LIMITS IN DECIMAL FRACTIONS OF SCAN WIDTH. DEFAULT VALUES: SC1 = 0.0, SC2 = 1.0, BUT VALUES OF SC1 .GT. 0 OR SC2 .LT. 1 CAN BE USED TO ALLOW FOR SCANS THAT WERE SO WIDE AS TO OVERLAP NEIGHBORING REFLECTIONS.) (15. WINDOW (F10.0). WIDTH OF THE MOVING WINDOW (DECIMAL FRACTION OF SCAN WIDTH) USED TO FIND INITIAL PEAK LIMITS. DEFAULT VALUE: WINDOW = 0.0833 = 1/12 = 8/96.) (16. CUTOFF (F10.0). E.S.D. MULTIPLIER FOR THE VARIOUS STATISTICAL SIGNIFICANCE TESTS. DEFAULT VALUE: CUTOFF = 3.) (17. FLL (F10.0). MULTIPLIER FOR THE CORRECTION FOR THE LEHMANN- LARSEN BIAS. THE PEAK LIMITS ARE SHIFTED TO L1' = L1 - F*(X1 - L1) AND L2' = L2 + F*(L2 - X2), WHERE F = FLL/(2 + (PK - BG)/BG). DEFAULT VALUE: FLL = 2.) (18. WT2 (F10.0). INTENSITY OF THE ALPHA-2 WAVELENGTH RELATIVE TO ALPHA-1. DEFAULT VALUE: WT2 = 0.5, BUT A CRYSTAL MONOCHROMATOR, DEPENDING ON HOW IT IS ADJUSTED, CAN CHANGE THE ALPHA-2/ALPHA-1 INTENSITY RATIO AND THE INTENSITY-WEIGHTED AVERAGE WAVELENGTH, WL0 = (WL1 + WL2*WT2)/(1 + WT2).) (19. NSMOOTH (I10). NUMBER OF PASSES OF DATA-SMOOTHING FOR PEAK LOCATION. DEFAULT VALUE: NSMOOTH = 2. SET ISMOOTH = -1 TO BY-PASS THE SMOOTHING OF THE SCAN PROFILES, WHICH IS NORMALLY DONE FOR THE PEAK LOCATION PARTS OF THE ANALYSIS. INTENSITY INTEGRATION IS ALWAYS DONE WITH THE UNSMOOTHED DATA.) (20. JMODEL (I10). ALLOWS THE USER TO IMPOSE A CHOICE BETWEEN THE LORENTZIAN (JMODEL = 1) AND GAUSSIAN (JMODEL = 2) CONVOLUTION MODELS. DEFAULT: JMODEL = 0.) ------------------------------------------------------------------------ CONTROL DATA FILE 'HKLCOND.DAT' ------- ---- ---- =========== THIS FILE, WHICH IS OPTIONAL, GIVES THE CONDITIONS LIMITING POSSIBLE REFLECTIONS SO THAT ANY SYSTEMATICALLY ABSENT REFLECTIONS THAT WERE INCLUDED IN THE INTENSITY MEASUREMENTS WILL NOT BE USED TO FIT THE PEAK POSITION AND PEAK WIDTH PARAMETERS IN PROGRAM REFPK. IF THE 'HKLCOND.DAT' FILE IS PRESENT WHEN PROGRAM BGLP IS RUN, THE ABSENT REFLECTIONS WILL NOT BE INCLUDED IN THE OUTPUT REFLECTION DATA FILE FROM PROGRAM BGLP. THE CONDITIONS MUST BE ENTERED IN THE FOLLOWING FORM, ONE RECORD PER CONDITION, NO LEADING OR EMBEDDED BLANKS IN THE RECORDS. NOTE, IN PARTICULAR, THAT THE CYCLIC PERMUTATION ORDER IS USED FOR THE H AND L INDICES; E.G., FOR THE CONDITION FOR AN N-GLIDE PLANE PERPENDICULAR TO THE B-AXIS, ENTER "H0L,L+H=2N". HKL,H+K=2N HKL,K+L=2N HKL,L+H=2N HKL,H+K,K+L,L+H=2N HKL,H+K+L=2N HKL,-H+K+L=3N HKL,H-K+L=3N HKL,H-K=3N HK0,H=2N HK0,K=2N HK0,H+K=2N HK0,H+K=4N 0KL,K=2N 0KL,L=2N 0KL,K+L=2N 0KL,K+L=4N H0L,L=2N H0L,H=2N H0L,L+H=2N H0L,L+H=4N HH(-2H)L,L=2N H(-H)0L,L=2N HHL,L=2N(R-AXES) HHL,L=2N HKH,K=2N HKK,H=2N HHL,2H+L=4N HKH,2H+K=4N HKK,2K+H=4N H00,H=2N H00,H=4N 0K0,K=2N 0K0,K=4N 00L,L=2N 00L,L=4N 000L,L=2N 000L,L=3N 000L,L=6N ------------------------------------------------------------------------ DESCRIPTION OF PROGRAM OUTPUT ----------- -- ------- ------ THE PRINTED OUTPUT FROM PROGRAMS REFPK AND BGLP INCLUDES TABLES ------- ------ THAT LIST REFLECTIONS WITH EXTREME VALUES OF SEVERAL CHARACTER- ISTIC PROPERTIES. THE TABLES ARE PRESENTED IN PAIRS IN WHICH THE FIRST TABLE LISTS THE REFLECTIONS WITH THE SMALLEST VALUES OF A GIVEN PROPERTY AND THE SECOND THOSE WITH THE LARGEST VALUES. THE PROPERTY ON WHICH THE TABLE IS BASED IS STATED IN THE HEADING ABOVE THE TABLE COLUMN HEADINGS, AND THE ENTRIES IN THE TABLE ARE SORTED IN ORDER OF INCREASING VALUE OF THE PROPERTY IN THE TABLES OF SMALLEST VALUES, AND DECREASING VALUE IN THE TABLES OF LARGEST VALUES. THESE TABLES ARE PREPARED IN ORDER TO PROVIDE THE USER WITH A GUIDE FOR VISUAL INSPECTION OF UNUSUAL REFLECTION PROFILES USING ----- --- ------ ---------- -- ------- ---------- -------- ----- PROGRAM VIEW. PROGRAM VIEW IS RUN INTERACTIVELY. IT PLOTS THE ------- ---- REFLECTION INTENSITY PROFILES STORED ON THE RAW REFLECTION DATA FILE, 'DATA.RAW'. THE PROFILE PLOTS ARE DISPLAYED ON THE VIDEO TERMINAL SCREEN, AND MAY ALSO BE PRINTED ON THE LINE PRINTER. THE USER SHOULD EXAMINE A FAIR SAMPLE OF PROFILE PLOTS TO CHECK ------ THAT THE PROGRAMS ARE SETTING THE PEAK LIMITS SATISFACTORILY. THE PROPERTIES TABULATED BY PROGRAM REFPK ARE: DX0O, DISPLACEMENT OF OBSERVED PEAK CENTROID FROM MIDPOINT OF SCAN (DEGREES THETA) D0=X0O-X0C, DIFFERENCE BETWEEN OBSERVED AND CALCULATED CENTROIDS (DEGREES THETA) WC=L2C-L1C, CALCULATED BASE WIDTH (DEGREES THETA) WO-WC=(L2O-L1O)-(L2C-L1C), DIFFERENCE BETWEEN OBSERVED AND CALCULATED FULL-PEAK BASE WIDTH (DEGREES THETA) THE PROPERTIES TABULATED BY PROGRAM BGLP ARE: S=SIN(THETA)/LAMBDA Y=I/LP, LORENTZ AND POLARIZATION CORRECTED NET INTENSITY X0, INTENSITY-WEIGHTED PEAK CENTROID (STEP NUMBER) W=L2-L1, BASE WIDTH OF REFLECTION (DEGREES THETA) PRECEEDING THE TABLES OF REFLECTIONS WITH EXTREME CHARACTERISTICS, THE PRINTED OUTPUT FROM BOTH PROGRAMS REFPK AND BGLP ALSO INCLUDES A TABLE THAT LISTS THE SPECIAL REFLECTIONS OF THE TYPES H00, 0K0, 00L, HH0, 0KK, H0H, AND HHH. ------------------------------------------------------------------------ THE OUTPUT PARAMETERS FILE 'BGLP.DAT' FROM PROGRAM REFPK IS A ------ ---------- ---- -------- FORMATTED, ASCII FILE WRITTEN BY THE FOLLOWING CODE: C C WRITE RESULTS TO OUTPUT FILE, 'BGLP.DAT'. C OPEN (UNIT=IO2,FILE='BGLP.DAT',STATUS='NEW') WRITE (IO2,600) ATIME WRITE (IO2,600) ADATE WRITE (IO2,600) TITLE WRITE (IO2,600) FILE1 WRITE (IO2,600) FILE2 WRITE (IO2,601) (A(I),I=1,6) WRITE (IO2,602) IDIFF,INEUTRON WRITE (IO2,601) THM,RHOM,FRACTD,SIGFD WRITE (IO2,601) EDGE,WLA1,WLA0,WLA2 WRITE (IO2,601) SIGTHE,SIGTIM WRITE (IO2,601) TAU,SIGTAU,ATT,SIGATT WRITE (IO2,601) WINDOW WRITE (IO2,601) CUTOFF WRITE (IO2,602) ISMOOTH WRITE (IO2,605) SC1,SC2 WRITE (IO2,603) IIMIN,IIMAX,XTMIN-0.005,XTMAX+0.005 WRITE (IO2,604) ((U(I,J),J=1,3),I=1,3) WRITE (IO2,604) ((SIGU(I,J),J=1,3),I=1,3) WRITE (IO2,600) MODEL1 WRITE (IO2,604) ((Q1(I,J),J=1,3),I=1,3) WRITE (IO2,604) ((SIGQ1(I,J),J=1,3),I=1,3) WRITE (IO2,604) T1 WRITE (IO2,604) SIGT1 WRITE (IO2,600) MODEL2 WRITE (IO2,604) ((Q2(I,J),J=1,3),I=1,3) WRITE (IO2,604) ((SIGQ2(I,J),J=1,3),I=1,3) WRITE (IO2,604) T2 ( WRITE (IO2,604) F ) ( WRITE (IO2,605) XX0,XW1,XW2 ) 600 FORMAT (1X,A) 601 FORMAT (1X,6F10.5) 602 FORMAT (1X,I10) 603 FORMAT (1X,2I10,2F10.3) 604 FORMAT (3(1X,3E15.8/)) 605 FORMAT (1X,4F10.2) THE DIFFRACTOMETER ORIENTATION MATRIX WRITTEN BY THE PROGRAM IS -------------- ----------- ------ AS DEFINED BY WALTER HAMILTON (1974). INTERNATIONAL TABLES FOR X-RAY CRYSTALLOGRAPHY, VOL. IV, PP. 273-284. IT IS EQUAL TO: THE TRANSPOSE OF A BUSING-LEVY MATRIX PRE-MULTIPLIED BY THE MATRIX --------- -- - ----------- ------ -------------- -- --- ------ ( 0 -1 0 ) ( 1 0 0 ) ; ( 0 0 1 ) THE TRANSPOSE OF A ENRAF-NONIUS CAD4 MATRIX; --------- -- - ------------ ---- ------ THE TRANSPOSE OF A SIEMENS (NICOLET, SYNTEX) P3 MATRIX PRE- --------- -- - ------- ------- ------ -- ------ ---- (-1 0 0 ) MULTIPLIED BY THE MATRIX ( 0 1 0 ) . ---------- -- --- ------ ( 0 0 -1 ) OPTION TO OVERRIDE USE OF CALCULATED PEAK POSITIONS OR WIDTHS ------ -- -------- --- -- ---------- ---- --------- -- ------ AN OPTIONAL LAST RECORD IS WRITTEN TO THE 'BGLP.DAT' FILE BY PROGRAM REFPK IF THE ROOT-MEAN-SQUARE ERROR-OF-FIT FOR THE PEAK POSITIONS EXCEEDS HALF THE ROOT-MEAN-SQUARE PEAK DIPLACEMENT. THIS RECORD SUPPLIES A VALUE XX0 = 0.5 (DECIMAL FRACTION OF THE SCAN WIDTH) FOR THE DEFAULT PEAK POSITION TO BE USED IF THE PEAK CENTROID CANNOT BE WELL DETERMINED. THE USER CAN SUPPLY OR ALTER THIS VALUE, OR SUPPLY VALUES XX0 XW1 XW2 FOR THE DEFAULT PEAK POSITION AND/OR FOR THE PEAK LIMITS, AND THUS OVERRIDE THE VALUES CALCULATED FROM THE RECALCULATED ORIENTATION MATRIX AND THE ANISOTROPIC PEAK WIDTH PARAMETERS FROM PROGRAM REFPK. THE VALUES MUST BE SUPPLIED IN UNITS OF DECIMAL FRACTION OF THE SCAN WIDTH, E.G., PEAK POSITION, XX0 = 0.6, AND PEAK LIMITS, XW1 = 0.25 AND XW2 = 0.95. THE FINAL PEAK LIMITS WILL BE REPOSITIONED ABOUT THE INTENSITY-WEIGHTED PEAK CENTROID, BUT THE PEAK WIDTH WILL BE KEPT FIXED AT (XW2 - XW1). ------------------------------------------------------------------------ THE OUTPUT REFLECTIONS FILE 'DATA.BLP' FROM PROGRAM BGLP IS AN ------ ----------- ---- -------- UNFORMATTED, BINARY FILE IN WHICH THE RECORD STRUCTURE IS: II MEASUREMENT SERIAL NUMBER. NEGATIVE FOR STANDARD REFERENCE REFLECTIONS IH IK MILLER INDICES IL A1 A2 DIFFRACTOMETER A3 SETTING ANGLES A4 Y I/LP SIGY SIGMA(I/LP) XTIME X-RAY EXPOSURE TIME (HOURS) II, IH, IK, AND IL ARE INTEGER*4 WORDS. ALL THE OTHER VALUES ARE REAL*4 WORDS. THE DIFFRACTOMETER SETTING ANGLES ARE AS ORIGINALLY WRITTEN BY THE DIFFRACTOMETER. NOTE THAT THE ORDER IS IMPORTANT. A1 A2 A3 A4 INT.TAB. TWO-THETA OMEGA CHI PHI BUS.LEV. TWO-THETA OMEGA CHI PHI P3 TWO-THETA OMEGA PHI CHI CAD4 THETA PHI OMEGA KAPPA ANGLES TRANSFORMED TO CONFORM TO THE CONVENTIONS DEFINED BY HAMILTON IN THE INTERNATIONAL TABLES FOR X-RAY CRYSTALLOGRAPHY, VOL. IV, 1974, PP. 273-284, ARE USED WITHIN PROGRAMS REFPK AND BGLP, BUT THE UN-TRANSFORMED ANGLES ARE WRITTEN TO THE OUTPUT FILE. THE VALUE OF Y = I/LP = FSQ IS GIVEN IN UNITS OF COUNTS PER SECOND PER DEGREE OF CRYSTAL ROTATION. THE ESTIMATED STANDARD DEVIATION SIGMA(Y) RESULTS FROM A PROPAGATION-OF-ERROR ANALYSIS THAT ALLOWS FOR UNCERTAINTIES DUE TO POISSON COUNTING STATISTICS, SCAN ANGLE SETTING ERRORS, THE COUNTER DEAD TIME CORRECTION, AND THE BEAM ATTENUATOR FACTOR. ------------------------------------------------------------------------