USER'S INSTRUCTIONS FOR PROGRAM TSCALE ROBERT H. BLESSING HAUPTMAN-WOODWARD INSTITUTE 73 HIGH STREET BUFFALO, NEW YORK 14203 TELEPHONE: (716) 856-9600 E-MAIL: Blessing@HWI.Buffalo.Edu ------------------------------------------------------------------------ PROGRAM TSCALE ANALYZES STANDARD REFERENCE REFLECTION INTENSITIES AS FUNCTIONS OF ACCUMULATED X-RAY EXPOSURE TIME. IT FITS LINEAR, QUADRATIC, OR CUBIC POLYNOMIALS TO THE INTENSITY VS. TIME DATA FOR EACH STANDARD. THE PROGRAM CHOOSES AMONG THE POLYNOMIAL ORDERS BY MEANS OF THE STATISTICAL F-TEST FOR THE SIGNIFICANCE OF AN ADDED TERM. IF THE BEHAVIOR OF ANY STANDARD IS TOO COMPLICATED TO BE FITTED BY ONE OF THE POLYNOMIALS, THE USER CAN, IN A SECOND RUN OF THE PROGRAM, CUT THE DATA FOR THAT STANDARD INTO TIME SEGMENTS SHORT ENOUGH TO BE ADEQUATELY FITTED BY THE LOW ORDER POLYNOMIALS. THE USER CAN ALSO EXCLUDE ANY OR ALL MEASUREMENTS OF A GIVEN STANDARD FROM THE POLYNOMIAL FITTING, AND ELIMINATE FROM THE DATA SET ANY REFLECTION MEASUREMENTS DEEMED UNRELIABLE. RESULTS OF THE ANALYSIS ARE PRESENTED IN TABLES AND PLOTS OF THE INTENSITY VS. TIME DATA AND THE FITTED POLYNOMIALS. WHEN THE USER IS SATISFIED WITH THE POLYNOMIAL FITTING, THE POLYNOMIALS ARE USED TO CALCULATE SCALE FACTORS TO SCALE THE INTENSITY DATA EITHER TO THE MID-TIME OF THE EXPERIMENT OR TO SOME OTHER REFERENCE TIME SELECTED BY THE USER. THE SCALE FACTOR APPLIED TO A GIVEN REFLECTION IS A WEIGHTED AVERAGE OF THE SCALE FACTORS CALCULATED FROM THE SEVERAL POLYNOMIALS FITTED TO THE SEVERAL STANDARDS. THE AVERAGE CAN BE WEIGHTED TOWARD THE STANDARD WHOSE RECIPROCAL LATTICE VECTOR DIRECTION IS NEAREST THAT OF THE GIVEN REFLECTION, SO THAT THE SCALING IS ANISOTROPIC. THIS CAN BE USEFUL WHEN RADIATION DAMAGE HAS CAUSED INTENSITIES TO DECAY ANISOTROPICALLY. THE AVERAGE CAN ALSO BE WEIGHTED TOWARD THE STANDARD THAT IS NEAREST IN INTENSITY TO THE GIVEN REFLECTION, SO THAT STRONG DATA ARE SCALED RATHER MORE BY STRONG STANDARDS AND WEAK DATA BY WEAK STANDARDS. THIS CAN BE HELPFUL FOR SUBSEQUENT REFINEMENT OF EXTINCTION PARAMETERS WITH DATA SETS WITH SEVERE EXTINCTION. FINALLY, THE AVERAGE CAN ALSO BE WEIGHTED TOWARD THE STANDARD THAT IS NEAREST IN SIN(THETA)/LAMBDA TO THE GIVEN REFLECTION, SO THAT HIGH-ANGLE DATA ARE SCALED RATHER MORE BY HIGH-ANGLE STANDARDS AND LOW-ANGLE DATA BY LOW-ANGLE STANDARDS. THIS CAN BE HELPFUL WHEN CHANGES IN EXPERIMENTAL TEMPERATURE OR CRYSTAL DECAY AFFECT HIGH-ANGLE DATA MORE THAN LOW-ANGLE DATA. ------------------------------------------------------------------------ THE Q-SUM FUNCTION --- - --- -------- THE PROGRAM WILL PREPARES PLOTS OF INTENSITY VS. TIME AND Q-SUM VS. TIME FOR EACH SEGMENT OF EACH STANDARD. THE Q-SUM FUNCTION CORRESPONDING TO A FUNCTION Y = Y(X) GIVEN AS DATA POINTS [X(I), Y(I)] IS DEFINED AS Q[X(N)] = SUM (I=1,N) [Y(I) - Y0], N=1,2,...,NMAX, WHERE Y0 IS THE MEAN VALUE OF Y. IF Y(X) IS LINEAR, THEN Y = A + B*X Y0 = A + B*X0, AND Q[X(N)] = SUM (I=1,N) B*[X(I) - X0] Q[X(N)] = B*SUM (I=1,N) [X(I) - N*B*X0]. IF THE X-VALUES ARE EQUALLY SPACED AT INTERVALS OF C UNITS OF X, SUM (I=1,N) X(I) = SUM (I=1,N) (C*I) = C*SUM (I=1,N) I = C*N*(N + 1)/2, BUT, AS THE X VALUES ARE EQUALLY SPACED, N = X(N)/C, AND SUM (I=1,N) X(I) = X(N)*[X(N)/C + 1])/2 SUM (I=1,N) (XI) = [X(N)**2 + C*X(N)]/(2*C) THUS, IF Y(X) IS LINEAR, THEN Q(X) IS QUADRATIC. AND, IF THE LINE Y(X) HAS NEGATIVE SLOPE, THEN THE CURVE Q(X) OPENS DOWN. Y(X) Q(X) . . . . . . . . . * . . *.* . * . . * . * Y0.............*...... . . . . * . * . * . . * . . . . * . * . * . . . . . . . . . . . * . * . . . . ..........................X ..........................X . . . . X0 X0 TO DECIDE WHETHER AND WHERE TO CUT THE DATA INTO TIME SEGMENTS, THE USER SHOULD EXAMINE THE INTENSITY VS. TIME AND Q-SUM VS. TIME PLOTS TOGETHER. POINTS OF DISCONTINUITY THAT SEEM TO BE INTERSECTIONS BETWEEN BOTH-UP OR BOTH-DOWN PARABOLIC SEGMENTS OF THE Q-SUM FUNCTION, AND POINTS OF INFLECTION THAT SEEM TO BE INTERSECTIONS BETWEEN ONE-UP AND ONE-DOWN PARABOLIC SEGMENTS OF THE Q-SUM FUNCTION SHOULD BE CONSIDERED AS POSSIBLE TIMES AT WHICH TO CUT THE DATA. E.G., Q(X) . * * Q(X) . . . * * . . . . * . . . . * . . . *.* * . * * . . * . . * * . . * . . . . . * . * * . . * . . . . . . * *. . . . . * . . * . .* .* . . * . * * . . . . ..........................X ..........................X . . . . XCUT XCUT IN GENERAL, NOT ALL THE TIME CUTS SUGGESTED BY THE Q-SUM PLOTS WILL BE NEEDED, BECAUSE THE Q(X) DISCONTINUITIES AND INFLECTIONS INDICATE LINEAR SEGMENTS OF Y(X), BUT THE PROGRAM WILL ALSO FIT QUADRATIC OR CUBIC Y(X) IF THEY ARE STATISTICALLY APPROPTIATE. MOREOVER, NOT ALL POSSIBLE CUTS ARE EXPERIMENTALLY SIGNIFICANT, AND THE USER SHOULD EXAMINE THE INTENSITY VS. TIME PLOTS AND THE EXPERIMENTAL RECORDS TO DECIDE WHICH, IF ANY, CUTS TO MAKE. ------------------------------------------------------------------------ ANISOTROPIC AND INTENSITY- AND SIN(THETA)/LAMBDA-DEPENDENT SCALING ----------- --- --------- --- ----------------- --------- ------- THE SCALE FACTOR APPLIED TO A GIVEN REFLECTION MEASUREMENT IS A WEIGHTED AVERAGE OF THE SCALE FACTORS CALCULATED FOR THE TIME OF THE GIVEN MEASUREMENT FROM THE SEVERAL SCALING POLYNOMIALS. THE WEIGHT FOR THE I-TH SCALE FACTOR IN THIS AVERAGING IS WI = W0*W1*W2*W3, WHERE THE FOUR WEIGHTING FACTORS ARE DEFINED AS FOLLOWS: W0 = 1/SIGMA(I)**2 WHERE SIGMA(I)**2 IS THE VARIANCE OF THE I-TH SCALE FACTOR AS CALCULATED FROM THE VARIANCE-COVARIANCE MATRIX FROM THE LEAST- SQUARES FIT OF THE COEFFICIENTS OF THE I-TH SCALING POLYNOMIAL. IF IANISO .EQ. 0, W1 = 1 IF IANISO .NE. 0, W1 = 1/(1 + C1**2*(1 - COS(PHI)**2)) WHERE COS(PHI) = (H*HI)/(H*HI), H AND HI ARE THE RECIPROCAL - -- - -- LATTICE VECTORS FOR THE GIVEN REFLECTION AND THE I-TH STANDARD REFLECTION, RESPECTIVELY, AND PHI IS THE ANGLE BETWEEN THE TWO VECTORS. (H*HI) REPRESENTS THE DOT PRODUCT OF THE VECTORS, AND - -- (H*HI) REPRESENTS THE PRODUCT OF THE VECTOR MAGNITUDES. IF IYDIFF .EQ. 0, W2 = 1 IF IYDIFF .NE. 0, W2 = 1/(1 + C2*(Y - YI)**2) WHERE Y IS THE UNSCALED INTENSITY OF THE GIVEN REFLECTION AND YI IS THE INTENSITY CALCULATED FROM THE I-TH SCALING POLYNOMIAL AT THE TIME THE GIVEN REFLECTION WAS MEASURED. IF ISDIFF .EQ. 0, W3 = 1 IF ISDIFF .NE. 0, W3 = 1/(1 + C3*(H - HI)**2) WHERE, AS ABOVE, H AND HI ARE THE RECIPROCAL LATTICE VECTOR MAGNITUDES (H = DSTAR = 2*SIN(THETA)/LAMBDA) FOR THE GIVEN REFLECTION AND THE I-TH STANDARD REFLECTION, RESPECTIVELY. REASONABLE ORDERS OF MAGNITUDE FOR THE WEIGHTING COEFFICIENTS ARE C1 = 10, C2 = 1, AND C3 = 10. IANISO .NE. 0 ALLOWS THE SCALING TO BE WEIGHTED TOWARD THE STANDARD REFLECTION WITH RECIPROCAL LATTICE VECTOR DIRECTION NEAREST THAT OF THE GIVEN REFLECTION. IYDIFF .NE. 0 ALLOWS THE SCALING TO BE WEIGHTED TOWARD THE STANDARD REFLECTION THAT IS NEAREST IN INTENSITY TO THE GIVEN REFLECTION. ISDIFF .NE. 0 ALLOWS THE SCALING TO BE WEIGHTED TOWARD THE STANDARD REFLECTION THAT IS NEAREST IN SIN(THETA)/LAMBDA TO THE GIVEN REFLECTION. THE USER SHOULD CONSIDER SETTING IANISO OR IYDIFF OR ISDIFF .NE. 0 ONLY IF THE STANDARD REFLECTIONS PROVIDE A GOOD SAMPLING OF RECIPROCAL LATTICE DIRECTIONS AND THE INTENSITY RANGE OF THE DATA. THIS NORMALLY REQUIRES NINE OR MORE DIFFERENT STANDARD REFERENCE REFLECTIONS -- THREE STRONG (ONE VERY STRONG), THREE MEDIUM, AND THREE WEAK -- ALL WELL-DISTRIBUTED OVER RECIPROCAL SPACE. ------------------------------------------------------------------------ MODIFIED VALUES OF THE ESTIMATED STANDARD DEVIATIONS -------- ------ -- --- --------- -------- ---------- EACH REFLECTION IS SCALED ACCORDING TO YCORR = Y/F, WHERE F IS THE WEIGHTED AVERAGE INVERSE SCALING FACTOR F = SUM(WI*FI)/SUM(WI) AND WI = W0*W1*W2*W3 AS DESCRIBED ABOVE. THE VARIANCE OF THE AVERAGED F IS TAKEN TO BE SIGMA(F)**2 = SUM(WI**2*SIGMA(FI)**2)/(SUM(WI))**2 OR SIGMA(F)**2 = (1/(N - 1))*SUM(WI*(FI - F)**2)/SUM(WI), WHICHEVER IS LARGER. THE SCALED VARIANCE FOR THE REFLECTION IS THEN GIVEN BY SIGMA(YCORR)**2 = (SIGMA(Y)**2 + P**2*Y**2)/F**2 + (Y**2/F**2)*SIGMA(F)**2, WHERE P, THE PROPORTIONALITY CONSTANT FOR INSTRUMENTAL INSTABILITIES (MC CANDLISH, STOUT, AND ANDREWS, 1975. ACTA CRYST. A31, 245.), IS A MEASURE OF THE EXCESS SCATTER OF THE STANDARD REFERENCE REFLECTIONS, OVER AND ABOVE THEIR STATISTICAL SCATTER, ABOUT THE SMOOTH TREND OF THEIR TIME DEPENDENCE. I.E., P**2 = (SUM((YMEAS - YCALC(T))**2) - SUM(SIGMA(YMEAS)**2))/ SUM(YMEAS**2). ------------------------------------------------------------------------ PROGRAM LIMITS: ------- ------ 20 STANDARD REFLECTIONS 10 SEGMENTS FOR EACH STANDARD 500 MEASUREMENTS IN EACH SEGMENT 20 (RANGES OF) SERIAL NUMBERS AND 20 (RANGES OF) EXPOSURE TIMES FOR STANDARD MEASUREMENTS TO BE OMITTED FROM THE ANALYSIS 50 (RANGES OF) SERIAL NUMBERS AND 50 (RANGES OF) EXPOSURE TIMES FOR (NON-STANDARD) MEASUREMENTS TO BE REJECTED FROM THE DATA SET ------------------------------------------------------------------------ INPUT FILES REQUIRED: ----- ----- -------- REFLECTION DATA FILE (NAME.BLP FROM PROGRAM BGLP) ---------- ---- ---- AN UNFORMATTED FILE IN WHICH EACH RECORD CONTAINS: 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 (HR) ----- -------- ---- ---- II, IH, IK, AND IL ARE INTEGER*4 WORDS; THE OTHER VALUES ARE REAL WORDS. THE DIFFRACTOMETER ANGLES ARE NOT USED BY THE PROGRAM, BUT THEY ARE TRANSFERRED TO THE OUTPUT FILE FOR POSSIBLE LATER USE IN ABSORPTION AND EXTINCTION CORRECTION CALCULATIONS. CONTROL DATA FILE 'TSCALE.DAT' ------- ---- ---- ========== THE USER MUST SUPPLY AT LEAST RECORDS 1 - 4 FOLLOWED BY SIX BLANK RECORDS. 1. TITLE (A) JOB TITLE 2. FILE1 (A) NAME OF INPUT REFLECTION DATA FILE 3. FILE2 (A) NAME FOR OUTPUT FILE OF SCALED DATA 4. CONTROL VARIABLES (8I10): ITABL = 0 DO NOT 1 DO PRINT TABLES OF INTENSITY VS. TIME DATA FOR (EACH SEGMENT OF) EACH STANDARD. IPLOT = 0 DO NOT 1 DO PRINT ON THE LINE PRINTER PLOTS OF INTENSITY VS. TIME FOR (EACH SEGMENT OF) EACH STANDARD. IQSUM = 0 DO NOT 1 DO PRINT ON THE LINE PRINTER PLOTS OF Q-SUM VS. TIME FOR (EACH SEGMENT OF) EACH STANDARD. IGRAPH = 0 DO NOT 1 DO GRAPH THE INVERSE SCALING FUNCTIONS FOR THE SEVERAL STANDARDS. IAPPLY = 0 DO NOT 1 DO APPLY SCALING FACTORS AND WRITE THE OUTPUT FILE OF SCALED DATA. IANISO .EQ. 0 DO NOT .NE. 0 DO APPLY SCALING FACTORS WITH ANISOTROPIC WEIGHTING. C1 = IANISO IYDIFF .EQ. 0 DO NOT .NE. 0 DO APPLY SCALING FACTORS WITH WEIGHTING BY INTENSITY DIFFERENCE. C2 = IYDIFF ISDIFF .EQ. 0 DO NOT .NE. 0 DO APPLY SCALING FACTORS WITH WEIGHTING BY SIN(THETA)/LAMBDA DIFFERENCE. C3 = ISDIFF RECOMMENDED CONTROL VARIABLES FOR A FIRST RUN ARE: 1, 1, 0, 1, 0, 0, 0, 0 5. LATTICE PARAMETERS (6F10.0) A, B, C, ALPHA, BETA, GAMMA LATTICE PARAMETERS RECORD MAY BE BLANK IF BOTH IANISO = 0 AND ISDIFF = 0. FOR EACH OF THE FIVE RECORD TYPES 6 TO 10, SUPPLY AT LEAST THE BLANK TERMINATOR RECORD. RECORD TYPES 6 TO 8 CONTROL THE CALCULATION OF THE SCALING FUNCTIONS, AND RECORD TYPES 9 AND 10 ALLOW THE USER TO EDIT THE DATA SET. 6. EXPOSURE TIME CUTS (3I4,F10.0) ONE RECORD PER CUT IH, IK, IL, XTCUT (MAXIMUM XTIME FOR THE SEGMENT) BLANK TERMINATOR RECORD ----- ---------- ------ 7. SERIAL NUMBERS OF STANDARD MEASUREMENTS TO BE OMITTED FROM THE CALCULATION OF SCALING FUNCTIONS (3I4,2I10) ONE RECORD PER MEASUREMENT (OR RANGE OF MEASUREMENTS) IH, IK, IL, I1 (OR I1, I2 (INCLUSIVE)) BLANK TERMINATOR RECORD ----- ---------- ------ 8. EXPOSURE TIMES OF STANDARD MEASUREMENTS TO BE OMITTED FROM THE CALCULATION OF SCALING FUNCTIONS (3I4,2F10.0) ONE RECORD PER MEASUREMENT (OR RANGE OF MEASUREMENTS) IH, IK, IL, X1 (OR X1, X2 (INCLUSIVE)) BLANK TERMINATOR RECORD ----- ---------- ------ 9. SERIAL NUMBERS OF DATA TO BE REJECTED FROM THE DATA SET (2I10) ONE RECORD PER MEASUREMENT (OR RANGE OF MEASUREMENTS) I1 (OR I1, I2 (INCLUSIVE)) BLANK TERMINATOR RECORD ----- ---------- ------ 10. EXPOSURE TIMES OF DATA TO BE REJECTED FORM THE DATA SET (2F10.0) ONE RECORD PER MEASUREMENT (OR RANGE OF MEASUREMENTS) X1 (OR X1, X2 (INCLUSIVE)) BLANK TERMINATOR RECORD ----- ---------- ------ 11. REFERENCE TIME FOR THE SCALING (F10.0) IF LEFT BLANK OR ENTERED AS ZERO, THE PROGRAM WILL DEFAULT TO THE MID-TIME OF THE EXPERIMENT. ------------------------------------------------------------------------ DESCRIPTION OF THE OUTPUT ----------- -- --- ------ THE OUTPUT REFLECTION DATA FILE PRODUCED IF IAPPLY = 1 IS AN UNFORMATTED FILE WITH THE SAME RECORD STRUCTURE AS THE INPUT REFLECTION DATA FILE. THE OUTPUT Y = I/LP VALUE IS THE SCALED VALUE, AND THE OUTPUT SIGY = SIGMA(I/LP) IS SCALED AND INCLUDES CONTRIBUTIONS FROM THE INSTRUMENTAL INSTABILITY AND FROM THE ESTIMATED ERROR IN THE SCALING FACTOR. REFERENCE REFLECTIONS DESIGNATED ON RECORDS OF TYPE 7 OR 8 FOR OMISSIOM FROM THE CALCULATION OF THE SCALING FUNCTIONS WILL NOT BE --- OMITTED FROM THE OUTPUT REFLECTION DATA FILE UNLESS THEY HAVE ALSO BEEN DESIGNATED FOR OMISSION FROM THE DATA SET BY MEANS OF RECORDS OF TYPE 9 OR 10. ALSO IF IAPPLY = 1, THE PRINTED OUTPUT INCLUDES A TABLE OF SPECIAL REFLECTIONS OF THE TYPES H00, 0K0, 00L, HH0, 0KK, H0H, AND HHH. FOLLOWING THIS TABLE THE PROGRAM PRINTS THE MINIMUM, MAXIMUM, AND MEAN VALUES OF THE WEIGHTED-AVERAGE SCALING FACTORS APPLIED TO THE REFLECTIONS AND THE MINIMUM, MAXIMUM, AND MEAN VALUES OF THE RELATIVE ERROR IN THE SCALING FACTORS. ------------------------------------------------------------------------