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Angles between planes in the hexagonal and tetragonal crystal systems
Micron, 1970, 2:59-61 with I plate
59
Angles between planes in the hexagonal and tetragonal crystal systems D. W. HOGAN* and D. J. DYSON (British Steel Corporation, Swinden Laboratories, Moorgate, Rotherham, Yorkshire, U.K.)
Manuscript received May 12, 1970
A short note introducing a new computer listing of inter-planar angles in the hexagonal and tetragonal crystal systems with examples of the print out obtained. De br~ves remarques qui introduisent une nouvelle liste pour ordinateur des angles formgs par des plans diff~rents dans les systkmes de cristaux hexagonaux et tetragonaux, avec des exemples de r~sultats obtenus de l'ordinateur. Eine kurze Anmerkung zur Einfiihrung eines neuen Rechenautomatverzeichnis zwischenfla'chlicher Winkel in den hexagonalen und tetragonalen Kristallsystemen zusammen mit Beispielen der erhaltbaren Abdr#cke.
The solution of single crystal electron diffraction patterns requires a comparison of the observed interplanar spacings and angles with the corresponding calculated values for phases which are believed to be present. Lists of interplanar spacings are readily available (ASTM publication: Powder Diffraction File, 1: 20); so too are tables of angles between pairs of planes in the cubic system for which h, k and 1~5 (Peavler and Lenusky, 1965; Andrews, Dyson and Keown, 1967). Tables of angles between certain pairs of low index planes in the tetragonal and hexagonal systems are also available (Andrews, Dyson and Keown, 1967; Frounfelker and Hirthe, 1962; Frounfelker, Seitz and Hirthe, 1962; Taylor and Lieber, 1954; Rarey, Stringer and Edington, 1966; Sellmyer, 1965). These enable stereographic projections to be constructed for certain c/a values which in turn can be used to determine the angle between any two poles which are plotted. To eliminate the construction of stereographic projections a computer programme has been written to extend considerably the data available for angles between planes in the hexagonal and tetragonal systems. The print-out, part of which is shown in Figure 1, lists angles for c/a values between 0.3, 3.0, 0.3 and 5.0 respectively. Angles for intermediate c/a values are readily obtained by interpolation. For each c/a value several angles are given. The angles have been calculated using the formulae :
* Present Address: Rolls Royce Ltd., Aero Engine Division, Metallurgical Research Department, Elton Road,
Derby, England.
60
HOGAN and DYSON
Hexagonal system, h,h2÷klk2+ ~(h,k2-l-k~h2) ÷ 3a2 (1,12) cos ~ -
[
/
h,2q- k2q- htk , + ~ca.i- (12)]
Tetragonal system,
1
Tcr/~)]
[h2@k22@hzk2@ 3a2
]2,
1
a 2 (h~h2÷k~kz) +-~5 (lI12)
cos N -
,/
[ al--f (h~ + k12)+~2 (1~)]
[-~2 (h2-? k2) + ~2 (12)]}
A twelve fold multiplicity of the hexagonal system occurs because any permutation of h, k and i alters only the numerator of the above equation. Each of the six permutations can be combined with a positive or negative (a/c)2 term. Similar arguments show that there is an eight fold mutliplicity in the tetragonal system. Twenty-five values of hlkll I between 001 and 221 have been paired with sixty-five values ofh2k212 between 010 and 551 for each system. It should be noted that with the appropriate transformation it is possible to convert the Miller indices of planes in the rhombohedral system into indices of equivalent planes in the hexagonal system. Each possible angle between any two families of planes is coded on the output. Where more than one angle of the same magnitude is found these code letters are set equal and only a single value is printed. The output together with interplanar angles in the cubic system will shortly be available (Hogan and Dyson, 1970). REFERENCES ANDREWS, K. W., DYSON, D. J. and KEowN, S. R., 1967. Interpretation of electron diffraction patterns. Hilger and Watts, London. FROUNVELKER,R. E. and HmTHE, W. M., 1962. Crystallographic data for the tetragonal crystal system. Trans. A.I.M.E., 224: 196-198. FROUNVEI~KER,R. E., SEITZ, M. A. and HIRTHE, W. M., 1962. Crystallographic data for the hexagonal crystal system. Nucl. Sci. and Engin., 14: 192-196. FROUNFELKER,R. E., SEITZ, M. A. and HIRTHE,W. M., 1962. Index to the powder diffraction file, 1-20. A S T M Philadelphia, U.S.A. HOGAN, D. W. and DYSON, D. J., 1970. Interplanar angles in the cubic, tetragonal and hexagonal systems. Structural Publications Ltd., Shakespeare Road, London, N3-In Press.
PEAVI.ER, R. J. and LENUSKY, 1965. I.M.D. Special Report No. 8. Am. Inst. Mining Met., and Pet. Eng. RAREY, C. R., STRINGER, J. and EDINGTON, J. W., 1966. Crystallographic techniques for the interpretation of transmission electron micrographs of hexagonal metals. Trans. A.I.M.E., 236: 811-812.
SELLMYER, D. J., 1965. Crystallographic angles tbr hexagonal crystals 1 . 1 0 ~ 1 . 9 0 . Trans. A.I.M.E., 233: 436-437. TAYLOR, A. and LIEBER, S., 1954. Crystallographic angles for hexagonal metals. J. Metals, 6: 190-192.
47,68 72.82 67.80 54.12 49.90 71.06 61.98 60.08
A B C D E F G H
50.25 83.35 76.31 88.76 53.19 81.02 69.05 66.56
50.70 $8.55 82.72 61.23 54.36 88.55 73.82 70.77
50.)4 SI.92 87.70 62.81 54.41 85.#0 77.25 73.68
4A.7U 76.29 R~.27 63.12 83.84 80.32 79.83 75.79
A B ] 8 _~ ~" G H
47.PI 71.38 84.90 63.33 52.95 75.97 81.86 77.58
4 5 . 4 <) 67.03 82.05 63.30 51,89 72.!7 83.48 78.61
43.7[ 63.14 79.59 63.14 50.77 68.8[ 84.82 79.59
41.95 59.64 77.46 62.91 49.64 65.83 85.92 80.58
#O.?Z 56.46 75.59 62.63 48.53 63.17 86.86 81.02
a B 2 D E F G H
?,rl.)8 53.56 73.95 62.33 47.47 60.78 87.65 81.56
37.00 50.92 72.51 62.04 46.47 58.6/+ 83.33 82.00
35.80 48.50 71.23 61.75 48.53 56.70 88.92 82.3~
34.39 46.2R 70.09 61.48 4/+.66 5#.95 89.42 32.71
~2.76 44.23 69.08 61.22 /+3.85 53.36 39.87 ~P.99 A
B C 0 E F g H
31.52 42.34 48.17 60.97 43.11 51.92 89.74 83.23
29.24 38.9~ A6.63 60,54 41.78 49.42 ~9.09 83.62
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63.54 72.72 81.46
58.20 70.04 80.17
51.67 65.58 78.07
Figure I
55.25 67.67 79.05
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34.72
E
Part of printouts for c/a~O 3 and c / a = 1.633
48.43 63.76 77.22
% =
56.31 52.57 49.25 77.OP 69.6? 67.6 ° ~9.~5 88.61 57.0o 81.5 e gO.O0 7~.60 69.6R 69.68 69.63 ..... ============
40.7
A
C/z~=1.633 0.3 0.5 0.6 0.7 0.8 0.9 1,0 I.i i.? 1.3 1.4 1.8 1.6 1.7 1.8 ...................................................................................................................................
351
123
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59
Angles between planes in the hexagonal and tetragonal crystal systems D. W. HOGAN* and D. J. DYSON (British Steel Corporation, Swinden Laboratories, Moorgate, Rotherham, Yorkshire, U.K.)
Manuscript received May 12, 1970
A short note introducing a new computer listing of inter-planar angles in the hexagonal and tetragonal crystal systems with examples of the print out obtained. De br~ves remarques qui introduisent une nouvelle liste pour ordinateur des angles formgs par des plans diff~rents dans les systkmes de cristaux hexagonaux et tetragonaux, avec des exemples de r~sultats obtenus de l'ordinateur. Eine kurze Anmerkung zur Einfiihrung eines neuen Rechenautomatverzeichnis zwischenfla'chlicher Winkel in den hexagonalen und tetragonalen Kristallsystemen zusammen mit Beispielen der erhaltbaren Abdr#cke.
The solution of single crystal electron diffraction patterns requires a comparison of the observed interplanar spacings and angles with the corresponding calculated values for phases which are believed to be present. Lists of interplanar spacings are readily available (ASTM publication: Powder Diffraction File, 1: 20); so too are tables of angles between pairs of planes in the cubic system for which h, k and 1~5 (Peavler and Lenusky, 1965; Andrews, Dyson and Keown, 1967). Tables of angles between certain pairs of low index planes in the tetragonal and hexagonal systems are also available (Andrews, Dyson and Keown, 1967; Frounfelker and Hirthe, 1962; Frounfelker, Seitz and Hirthe, 1962; Taylor and Lieber, 1954; Rarey, Stringer and Edington, 1966; Sellmyer, 1965). These enable stereographic projections to be constructed for certain c/a values which in turn can be used to determine the angle between any two poles which are plotted. To eliminate the construction of stereographic projections a computer programme has been written to extend considerably the data available for angles between planes in the hexagonal and tetragonal systems. The print-out, part of which is shown in Figure 1, lists angles for c/a values between 0.3, 3.0, 0.3 and 5.0 respectively. Angles for intermediate c/a values are readily obtained by interpolation. For each c/a value several angles are given. The angles have been calculated using the formulae :
* Present Address: Rolls Royce Ltd., Aero Engine Division, Metallurgical Research Department, Elton Road,
Derby, England.
60
HOGAN and DYSON
Hexagonal system, h,h2÷klk2+ ~(h,k2-l-k~h2) ÷ 3a2 (1,12) cos ~ -
[
/
h,2q- k2q- htk , + ~ca.i- (12)]
Tetragonal system,
1
Tcr/~)]
[h2@k22@hzk2@ 3a2
]2,
1
a 2 (h~h2÷k~kz) +-~5 (lI12)
cos N -
,/
[ al--f (h~ + k12)+~2 (1~)]
[-~2 (h2-? k2) + ~2 (12)]}
A twelve fold multiplicity of the hexagonal system occurs because any permutation of h, k and i alters only the numerator of the above equation. Each of the six permutations can be combined with a positive or negative (a/c)2 term. Similar arguments show that there is an eight fold mutliplicity in the tetragonal system. Twenty-five values of hlkll I between 001 and 221 have been paired with sixty-five values ofh2k212 between 010 and 551 for each system. It should be noted that with the appropriate transformation it is possible to convert the Miller indices of planes in the rhombohedral system into indices of equivalent planes in the hexagonal system. Each possible angle between any two families of planes is coded on the output. Where more than one angle of the same magnitude is found these code letters are set equal and only a single value is printed. The output together with interplanar angles in the cubic system will shortly be available (Hogan and Dyson, 1970). REFERENCES ANDREWS, K. W., DYSON, D. J. and KEowN, S. R., 1967. Interpretation of electron diffraction patterns. Hilger and Watts, London. FROUNVELKER,R. E. and HmTHE, W. M., 1962. Crystallographic data for the tetragonal crystal system. Trans. A.I.M.E., 224: 196-198. FROUNVEI~KER,R. E., SEITZ, M. A. and HIRTHE, W. M., 1962. Crystallographic data for the hexagonal crystal system. Nucl. Sci. and Engin., 14: 192-196. FROUNFELKER,R. E., SEITZ, M. A. and HIRTHE,W. M., 1962. Index to the powder diffraction file, 1-20. A S T M Philadelphia, U.S.A. HOGAN, D. W. and DYSON, D. J., 1970. Interplanar angles in the cubic, tetragonal and hexagonal systems. Structural Publications Ltd., Shakespeare Road, London, N3-In Press.
PEAVI.ER, R. J. and LENUSKY, 1965. I.M.D. Special Report No. 8. Am. Inst. Mining Met., and Pet. Eng. RAREY, C. R., STRINGER, J. and EDINGTON, J. W., 1966. Crystallographic techniques for the interpretation of transmission electron micrographs of hexagonal metals. Trans. A.I.M.E., 236: 811-812.
SELLMYER, D. J., 1965. Crystallographic angles tbr hexagonal crystals 1 . 1 0 ~ 1 . 9 0 . Trans. A.I.M.E., 233: 436-437. TAYLOR, A. and LIEBER, S., 1954. Crystallographic angles for hexagonal metals. J. Metals, 6: 190-192.
47,68 72.82 67.80 54.12 49.90 71.06 61.98 60.08
A B C D E F G H
50.25 83.35 76.31 88.76 53.19 81.02 69.05 66.56
50.70 $8.55 82.72 61.23 54.36 88.55 73.82 70.77
50.)4 SI.92 87.70 62.81 54.41 85.#0 77.25 73.68
4A.7U 76.29 R~.27 63.12 83.84 80.32 79.83 75.79
A B ] 8 _~ ~" G H
47.PI 71.38 84.90 63.33 52.95 75.97 81.86 77.58
4 5 . 4 <) 67.03 82.05 63.30 51,89 72.!7 83.48 78.61
43.7[ 63.14 79.59 63.14 50.77 68.8[ 84.82 79.59
41.95 59.64 77.46 62.91 49.64 65.83 85.92 80.58
#O.?Z 56.46 75.59 62.63 48.53 63.17 86.86 81.02
a B 2 D E F G H
?,rl.)8 53.56 73.95 62.33 47.47 60.78 87.65 81.56
37.00 50.92 72.51 62.04 46.47 58.6/+ 83.33 82.00
35.80 48.50 71.23 61.75 48.53 56.70 88.92 82.3~
34.39 46.2R 70.09 61.48 4/+.66 5#.95 89.42 32.71
~2.76 44.23 69.08 61.22 /+3.85 53.36 39.87 ~P.99 A
B C 0 E F g H
31.52 42.34 48.17 60.97 43.11 51.92 89.74 83.23
29.24 38.9~ A6.63 60,54 41.78 49.42 ~9.09 83.62
28.23
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47.70
45.45
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63.54 72.72 81.46
58.20 70.04 80.17
51.67 65.58 78.07
Figure I
55.25 67.67 79.05
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:
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34.72
E
Part of printouts for c/a~O 3 and c / a = 1.633
48.43 63.76 77.22
% =
56.31 52.57 49.25 77.OP 69.6? 67.6 ° ~9.~5 88.61 57.0o 81.5 e gO.O0 7~.60 69.6R 69.68 69.63 ..... ============
40.7
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351
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