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High-Resolution Electron Microscopy$

John C. H. Spence

Print publication date: 2008

Print ISBN-13: 9780199552757

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199552757.001.0001

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(p.377) APPENDIX 1

(p.377) APPENDIX 1

Source:
High-Resolution Electron Microscopy
Publisher:
Oxford University Press

The following FORTRAN program calculates values of the defocus Δf, spherical aberration constant C s and astigmatism constants from optical diffractogram ring radii Sn. It is based on the method of Krivanek (Optik, 45, 97) and was written by M. A. O'Keefe. As written, it is restricted to the case of large underfocus settings, but is easily modified for overfocus (Δf positive) values, as described below.

The method depends on the fact that minima will occur in the diffractogram intensity (given by eqn (3.36) if

(A1.1) APPENDIX 1

Here N is any integer (including zero). On dividing by u 2, we find that a plot of N/u 2 against u 2 forms a straight line with slope (C s λ 3/2) and intercept (Δ). The program uses linear regression to find the slope and intercept (and hence C s and Δf) from the values of N and Sn supplied. The ring radii Sn must be specified in reciprocal Angstroms, and are related to the measured ring radii by eqns (10.5) or (10.6). Input data includes a title and the microscope accelerating voltage in kilovolts. Then follows one line of input data for each maximum or minimum, containing, in column 2, the number of the ring, in column 4 a zero (for a minimum) or a one (for a maximum), and, in the next sixteen columns, the radius of this maximum or minimum (in reciprocal Angstroms). The method of numbering the rings is indicated by the numbers across the page in Fig. A1.1. These numbers correspond to –N in eqn (A1.1). (The program actually works with a new n = 2N for convenience, evaluated after line 10, so that the slope and intercept are twice the values given above.)

The correct numbering of the rings requires some care. All the worked examples in Krivanek (1976) follow the system indicated in Fig. A1.1, which is correct only for (p.378)

APPENDIX 1

Fig. A1.1 Optical diffraction pattern of an electron image of a thin carbon foil suitable for use with the FORTRAN program given. The pattern shows negligible astigmatism and drift (see Section 10.7). Numbers running across the pattern are the numbers NRING needed as input to the program. The maximum of the first (inner) ring has not been used since its position is difficult to determine. The numbers running down the pattern are the quantities n = 2NRING – MAXMIN evaluated after line 10 of the program. The intensity in this pattern is given by eqn (3.36), together with the damping effects of partial coherence and chromatic aberration discussed in Chapter 4, Instrumental aberration constants are obtained by the program from patterns such as these.

large underfocus (Δf negative). Figure A1.2 shows χ(u)/π plotted for three defocus values with C s = 1.8 mm at 100 kV. The values of N from eqn (A1.1) are indicated on the curves. It is seen that there are essentially three cases: (1) Positive focus, for which N = 1, 2, 3, 4, ; (2) Near Scherzer focus, for which the first value of N must be 0 (N = 0, 1, 2, 3, …); and (3) Larger negative focus, in which case (for the example shown) N = –1, –2, –3, 3, 2, 1, 0, 1, 2, 3, 4,. … The turning point occurs at the stationary phase condition (see eqn (5.66).

(A1.2) APPENDIX 1

Here the declining values of N start to increase, and may or may not repeat at the minimum value (e.g. 3 above). The stationary phase condition may sometimes be identified on a diffractogram by a broad intense ring, however if this condition coincides with χ(u) = –mπ, a broad absence of intensity will result. In practice, (p.379)

APPENDIX 1

Fig. A1.2 The function χ(u) plotted in units of π for three focus settings Δf = –980 (Scherzer focus), –2100 and +800 Å as indicated. Values of N in eqn (A1.1) are indicated. Here C s = 1.8 mm and the accelerating voltage is 100 kV.

as shown in Fig. A3.1, the turning point may be driven beyond the resolution of the microscope by choosing from eqn (A1.2)

(A1.3) APPENDIX 1

where u 0 is the highest spatial frequency in the diffractogram. Then the values of N will increase monotonically by unit increments with negative sign. These are the conditions for which the FORTRAN program is written. An additional internal consistency test also exists, since the program returns values of the standard deviation for slope and intercept. Because of the method of analysis, these values should not be taken as the errors in C s and Δf. They may, however, be used to confirm that a good straight-line fit has been obtained, and that the values of N have therefore been correctly chosen.

The best way to use the program for the calibration of a new microscope is to record several through-focus series at focus settings satisfying eqn (A1.3) (using manufacturer's data for C s and U 0), from which the smallest focal step increment (and C s) can be deduced. The Scherzer focus required for structure imaging can then be obtained by counting ‘clicks’ from the minimum contrast position, given by eqn (6.26) (see also Section 12.5).

(p.380)

A FORTRAN program for finding defocus and spherical aberration constants from measured optical diffractograms ring radii

C

PROGRAM CSEPS

C

TO DETERMINE CS(MM) AND DEFOCUS (ANGSTROMS) GIVEN RING DIAMETERS

C

FROM DIFFRACTOGRAMS OF MICROGRAPHS OF THIN AMORPHOUS MATERIALS

DIMENSION NAME(13),X(20),Y(20)

C

READ IN VOLTAGE IN KV

C

READ (5,100) NAME,VOLTK

C

CALCULATE WAVELENGTH

WAVL = 0.387818/SQRT(VOLTK * (1.0 + 0.978459E –3* VOLTK))

WAVCUB = (WAVL **3)* 1.0E7

70 N = 0

20 CONTINUE

C

READ IN NUMBER OF RING (1 FOR FIRST, 2 FOR SECOND. …),

C

WHETHER RING IS MAXIMUM (MAXIMIN = 1) OR MINIMUM (MAXIMIN = 0)

C

RADIUS OF RING IN RECIPROCAL ANGSTROMS

READ(5,110)NRING,MAXMIN,URAD

IF(NRING,NE.0) GO TO 10

IF(N.EQ.0) GO TO 80

GO TO 30

10 CONTINUE

IF(N.EQ.0) WRITE(6,200) NAME, VOLTK,WAVL

N = N + 1

USQ = URAD *URAD

X(N) = USQ

Y(N) = –FLOAT(2 * NRING-MAXMIN)/USQ

C

CHECK WHETHER MAXIMUM OR MINIMUM

IF(MAXMIN.EQ.0) GO TO 25

IF(MAXMIN.EQ.1) GO TO 35

WRITE(6,260)

GO TO 80

25 WRITE(6,220) NRING, URAD, X(N), Y(N)

GO TO 20

35 WRITE(6,210) NRING,URAD,X(N), Y(N)

GO TO 20

30 CONTINUE

C

IF N.GT.1 THEN FIT CURVE

IF(N.FT.1) GO TO 40

WRITE(6,230)

GO TO 70

40 CONTINUE

SUMX = 0.0

SUMY = 0.0

SUMX2 = 0.0

SUMXY = 0.0

SUM2 = 0.0

FN = FLOAT(N)

C

SUM OVER N POINTS

DO 50 1 = 1,N

XI = X(I)

YI = Y(I)

SUMX = SUMX + XI

SUMY = SUMY + YI

SUMX2 = SUMX2 + XI * XI

SUMXY = SUMXY + XI * YI

SUMY2 = SUMY2 + YI * YI

C

CALCULATE CURVE PARAMETERS

50 CONTINUE

DELTA = FN * SUMX2 — SUMX * SUMX

A = (SUMX2 * SUMY — SUMX * SUMXY)/DELTA

B = (SUMXY * FN — SUMX * SUMY)/DELTA

EPS = A/(2.0 * WAVL)

CS = B/WAVCUB

C = FN –2.

C

IF NUMBER OF POINTS IS LESS THAN 3 THEN SKIP STD. DEV. CALCULATION

IF(C.LT.1.E – 3) GO TO 60

VNCE = (SUMY2 + A*A*FN + B*B* SUMX2 — 2.0 *(A * SUMY + B * SUMXY

– A * B * SUMX))/C

SIGA = SQRT(VNCE * SUMX2/DELTA)

SIGB = SQRT(VNCE * FN/DELTA)

R = (FN * SUMXY – SUMX * SUMY)/(SQRT(DELTA * (FN * SUMY2 – SUMY * SUMY)))

SIGEPS = SIGA/(2.0 * WAVL)

SIGCS = SIGB/WAVCUB

WRITE(6,240) A,SIGA,B,SIGB,R,EPS,SIGEPS,CS,SIGCS

GO TO 70

60

CONTINUE

WRITE(6,250)A,B,EPS,CS

GO TO 70

80

CONTINUE

CALL EXIT

100

FORMAT(13AG/F10.5)

110

FORMAT(2I2,F12.6)

200

FORMAT(1H1,13AG,/,15HOVOLTAGE (KV) =,F8.2,10X,16HWAVELENGTH (A) =,

$

F9.6,//)

210

FORMAT(1H0,10X,17HRADIUS OF MAXIMUM,12,2H =,F7.4,17H (RECIP.ANGSTROM),

$

20X,3HX =,F7.5,5X,3HY =,F8.2)

220

FORMAT(1HO,10X,17HRADIUS OF MINIMUM,12,2H =,F7.4,17H (RECIP.ANGSTROM),

$

20X,3HX =,F7.5,5X,3HY =,F8.2)

230

FORMAT(1HO,10X,41HONLY ONE POINT IN DATA SET–DATA IGNORED,/)

240

FORMAT(1HO,10X,11HINTERCEPT =,F8.2,2,H(,F5.2,11X,7HSLOPE =

$

,F8.2,2H (,F6.2,1H),11X,12HCORR.COEFF. =,F6.4,///,1HO,50X,20HDEFOCUS

$

(STD.DEV.) =,F9.2,2H (,F6.2,6H) ANG.,5X,15HCS (STD,DEV.) =,F6.3,2H

$

(,F4,3.5H) MM.,//)

250

FORMAT(/,1H0,10X,

$

11HINTERCEPT =,F8.2.5X,7HSLOPE =,F8.2,39X,9HDEFOCUS =

$

,F9.2,5H ANG.,5X,4HCS =,F6.3,4H MM.,//)

260

FORMAT(1HO,20X,’RING NOT LABELED AS MAXIMUM OR MINIMUM’)

Sample data and output from program (100KV)

TEST OF PROGRAM — O.L.'s DATA

VOLTAGE (KV) = 100.00 WAVELENGTH (A) = 0.037013

RADIUS OF MAXIMUM 1 = 0.0770 (RECIP.ANGSTROM) X = 0.00593 Y = −168.66

RADIUS OF MINIMUM 1 = 0.1120 (RECIP.ANGSTROM) X = 0.01254 Y = −159.44

RADIUS OF MAXIMUM 2 = 0.1400 (RECIP.ANGSTROM) X = 0.01960 Y = −153.06

RADIUS OF MINIMUM 2 = 0.1660 (RECIP.ANGSTROM) X = 0.02756 Y = −145.16

RADIUS OF MAXIMUM 3 = 0.1890 (RECIP.ANGSTROM) X = 0.03572 Y = −139.97

RADIUS OF MINIMUM 3 = 0.2150 (RECIP.ANGSTROM) X = 0.04622 Y = −129.80

RADIUS OF MAXIMUM 4 = 0.2420 (RECIP.ANGSTROM) X = 0.05856 Y = −119.53

RADIUS OF MINIMUM 4 = 0.2890 (RECIP.ANGSTROM) X = 0.08352 Y = −95.78

INTERCEPT = −171.83 (0.85) SLOPE = 908.90 (19.57) CORR.COEFF. = 0.9986

DEFOCUS (STD.DEV.) = −2321.26 (11.50) ANG. CS (STD.DEV.) = 1.792 (0.039)MM

(p.381) (p.382)