In the Classroom
Teaching Space Group Symmetry through Problems1 George L. Hardgrove Department of Chemistry, St. Olaf College, Northfield, MN 55057-1098 The identification of point groups of molecules and ions and the applications of group theory to bonding and spectroscopy have long been an important part of undergraduate curricula in chemistry. There has been much less emphasis on space groups, which describe the symmetry of crystals. With the rise in interest in chemical reactions and NMR spectroscopy in the solid state there is a real need to introduce students to this area. Although relatively few students will actually determine crystal structures from diffraction measurements, many chemists will be consumers of the results of such studies. A knowledge of crystal symmetry will make the visualization of atomic arrangements from published structures much easier. Students can get a practical introduction to crystal symmetry by fitting the atoms or ions of a compound into the positions in the unit cell permitted by the space group. This paper presents a number of student exercises of this type. I was introduced to this teaching methodology by Wheatley’s book (1) . Some other resources on teaching crystallography include the Symposium on Teaching Crystallography (2) from this Journal and its accompanying articles. Kettle and Norrby discuss the Bravais lattices (3). Hathaway (4) describes a model building exercise involving a space group. Course designs are discussed by Bond and Carrano (5) and Donnay and Donnay (6). Useful computer programs using lattice symmetry have been written by Gunter and Downs (7) for drilling holes in balls for models, Abad-Zapatero and O’Donnell (8) for three-dimensional displays of crystal symmetry, and Phillips (9) for showing the relationship between the diffraction pattern and the lattice symmetry. A crystal is characterized by repeating patterns of atoms extending in three-dimensional space. It can be thought of as a network of lattice points whose surroundings are alike by translation along three axes in space. Unit cells always have lattice points at the eight corners, and some cells such as body-centered, face-centered, and end-centered have additional lattice points. Space groups contain symmetry operations characteristic of point groups (rotation axes, mirror planes, etc.). They also have translation operations characteristic of lattices. The symmetry of any molecule or complex ion can be described by the operations of a point group. Similarly, the symmetry of each crystalline substance can be described by the operations of one of the 230 space groups. The description of space groups contains two unfamiliar features. First, crystal symmetry is described using the Hermann– Mauguin (H-M) notation rather than the Schoenflies (S) notation, which is more familiar to chemists. Rotation axes C2 and C3 are 2 and 3 in H-M notation, and improper axes are described as rotoinversions such as 3¯ and 4¯ . Reflection planes σ are described as m. A more complete explanation of H-M notation is given in the book by Stout and Jensen (10). While point groups are described by listing their symmetry operations in character tables, space groups are described by listing the fractional coordinates of equivalent positions that result by operating on an arbitrary point x,y,z by each of the symmetry operations in the group. The equivalent positions and diffraction characteristics of each of the 230 space groups are available in the International Tables for Crystallography (11). A shorter compilation of a
few of the space groups has been prepared for teaching purposes (12), but it does not include all the groups used in the exercises described in this article. In a structural determination of a crystalline substance, the Bragg reflections in the diffraction pattern are assigned Miller indices (hk,), and the intensity of X-ray scattering for each hk, reflection is measured. The systematic absence of certain reflections along with relationships among the intensities give clues to aid the assignment of a particular space group for the structure giving this pattern. The objective of these exercises is for the student to determine the number of formula units per cell and to place the atoms of the compound into symmetry-related positions in the unit cell. Such placement should be consistent with normal bonding considerations for those atoms. NaCl The systematic absences in the diffraction pattern for NaCl indicate a face-centered lattice. The symmetry of the intensities of the hk, reflections is consistent with the space group Fm3¯ m. This group has 192 symmetry operations, and a partial list of the equivalent positions is given in Table 1 in the row labeled 192,. This space group can be thought of as a kaleidoscope. If you place one atom at an arbitrary location x,y,z that is not located on a symmetry element, then 191 additional atoms will be generated. The number of equivalent positions generated will be smaller if an atom is placed on a symmetry element such as a mirror plane or rotation axis or at the intersection of several such symmetry elements. Such locations are called special positions. The different types of special positions for Fm3¯ m are shown in Table 1 with labels a through , along with the point group symmetry about each such site.
Table 1. Equivalent General and Special Positions for Space Group Fm 3¯ ma for Face-Centered Lattice (0,0,0; 0,1 /2, 1/ 2; 1 /2,0, 1/2; 1/ 2, 1/2,0)+ No. of Label Positions
Site Symmetry
Equivalent Positions
192
,
1
x,y,z; z,x,y; y,z,x; etc.
96
k
m
x,x,z; z,x,x; x,z,x; etc. 0, y,z; z ,0,y; y,z,0; etc.
96
j
m
48
I
mm
1/
48
h
mm
0, x,x; x,0,x; x,x ,0; etc.
48
g
mm
x, 1/ 4,1/ 4; 1/ 4,x, 1/ 4; etc. x,x,x; x,x¯ ,x¯ ; x¯ ,x,x¯ ; etc.
32
f
3m
24
e
4mm
24
d
mmm
8
c
4¯ 3 m
1
4
b
4
a
m3¯ m m3¯ m
1/
2,x,x;
x, 1/2, x; x,x,1/ 2; etc.
– x,0,0; x¯ ,0,0; 0, x,0; 0,x,0 0,0,x; 0,0,x¯ 0, 1/4 ,1/ 4; 1/ 4,0,1/ 4; 1/4, 1/ 4,0; 0, 1/4 ,3/ 4; 3/ 4,0,1/ 4; 1/4, 3/ 4,0
/ 4,1 /4, 1/ 4; 3/ 4,3 /4, 3/ 4 1 1 2, /2 , / 2
0,0,0
aHahn,
T. International Tables for Crystallography Vol. A: Space-Group Symmetry ; Kluwer: Dordrecht, Netherlands, 1992.
Vol. 74 No. 7 July 1997 • Journal of Chemical Education
797
In the Classroom
Figure 1. Structure of K 2SiF6 .
The analysis of the NaCl structure begins with the calculation of Z, which is the number of formula units per unit cell: dVN A Z= fw where d is the density in g/cm 3, V the volume of the unit cell in cm3, NA Avogadro’s number, and fw the formula weight in g/mol. For NaCl the crystal is cubic so that a = b = c and the measured cell dimension a = 5.640 Å. The measured density is 2.164 g/cm3. The calculated value of Z is 4.004, so we can say that there are four formula units per cell. Thus there are 4 Na+ ions and 4 Cl{ ions per cell. Looking at the equivalent general and special positions for this space group in Table 1 we cannot place the Na+ ion in the 192, general positions because we do not have 192 of them. They can be placed only in the 4a or 4b special positions. For the 4a set the equivalent positions are then 0,0,0; 0,1/2,1/2; 1/ 2,0, 1/ 2; and 1/ ,1/ ,0, as indicated by the face-centering operations 2 2 listed at the top of Table 1. The symbol m3¯ m tells us that the symmetry at these sites is octahedral (O h) as we expect for NaCl. The symbol is a contraction of m4 3¯ m2 , which tells us we have m4 (C4h) symmetry in the directions along the x,y,z axes, 3¯ (S 6) symmetry in the directions along the body diagonals, and m2 (C2h) symmetry in the directions along the face diagonals. We can then place the 4 Cl{ ions at the 4b positions. An equivalent arrangement results if the positions of the Na+ and Cl{ ions are reversed. CaF2 There are four CaF2 formula units per cell in space group Fm3¯ m. The four Ca2+ ions go in either the 4a or the 4b positions with m3¯ m or Oh symmetry. The only positions ¯ m or Td symmeavailable for the F{ ions are the 8c with 43 try, as we would expect for these tetrahedral holes.
Figure 2. Structure of C6H12 N4 (hexamethylenetetramine).
octahedral SiF 62{ ion. The other possibility is the 24e positions with 4mm (C4v) symmetry. These ions would lie along the x,y,z axes in the + and { directions all with the same fractional coordinate. Note that x¯ = {x . The latter positions fit our expectations of an octahedral arrangement of F{ ions about Si4+. The structure is shown in Figure 1. Hexamethylenetetramine, C6H12 N4 ¯ m, This compound crystallizes in the space group I 43 for which the equivalent positions and site symmetries are listed in Table 2. There are 2 molecules per cell. The 12 C atoms can be assigned to either the 12d positions with 4¯ (S4) symmetry or the 12e positions with mm (C2v) symmetry. A carbon atom bonded to 2 H atoms and 2 N atoms cannot have S4 symmetry but it can have C 2v symmetry, so these atoms are assigned to the 12e positions. The 24 H atoms can be placed in either the 24f positions with 2 (C2 ) symmetry or the 24g with m (σ) symmetry. It is unlikely that hydrogens in a CH2 group lie on a 2-fold axis, but they can lie on a mirror plane. The 8 N atoms fit into the 8c positions with (C 3v) symmetry. Note that the centers of the two mol¯ m (Td) symecules can be placed in the 2a positions with 43 metry. This requires that the symmetry of the molecule must be Td. Thus the crystallographic symmetry imposes restraints on the molecular symmetry for this molecule. The structure is shown in Figure 2. Table 2. Equivalent General and Special Positions for Space Group I 4¯ 3m a for Body-Centered Lattice (0,0,0; 1/2, 1/ 2,1/ 2)+ No. of Label Positions
K2SiF6 There are four K2 SiF6 formula units per cell in space group Fm3¯ m. The Si4+ ions go into either the 4a or the 4b positions but not both. The K+ ions go into the 8c tetrahedral positions. For the 24 F{ ions we have two choices. If we choose the 24d positions, the fluoride ions would be required to have mmm (D2h) symmetry, which is impossible for an
798
Site Symmetry
Equivalent Positions
48
h
1
x,y,z; z,x,y; y,z,x; etc.
24
g
m
x,x,z; z,x,x; x,z,x; etc. x, 1/2,0; 0,x ,1/ 2; 1/2,0,x; etc.
24
f
2
12
e
12
d
mm 4¯
8
c
3m
6
b
4¯ 2m
0, 1/2, 1/ 2; 1/2 ,0,1/ 2; 1 /2, 1/ 2,0
2
a
4¯ 3m
0,0,0
aHahn,
x,0,0; 0,x ,0; 0,0,x; etc. 1
/4,1/2,0; 0,1/4,1/2; 1/2,0,1/4; etc. x,x,x; x,x¯ ,x¯ ; ¯x,x,x¯ ; etc.
T. International Tables for Crystallography Vol. A: Space-Group Symmetry; Kluwer: Dordrecht, Netherlands, 1992.
Journal of Chemical Education • Vol. 74 No. 7 July 1997
In the Classroom Table 4. Selected Compounds for Symmetry Group Analysis Space Group Pm3¯ m
Compound
Unit Cell (Å)
Ref
cubic
3.988
a = 4.121
a, p 86
cubic
4.102
a = 5.409
a, p 110
cubic
2.164
a = 5.640
a, p 88
cubic
3.181
a = 5.463
a, p 241
K2SiF6
Fm3¯ m Fm3¯ m
cubic
2.719
a = 8.134
b
CaTiO3
Pm3¯ m
cubic
4.10
a = 3.84
c, pp 390-391
C6H12N4
I 4¯ 3m
cubic
1.345
a = 7.021
d
Pb(NO3)2
Pa 3 Fm3¯ m
cubic
4.545
a = 7.856
e
cubic
1.498
a = 13.05
f
Fm 3 Ia 3¯ d P 4¯ 21m
cubic
2.585
a = 10.512
g
cubic
3.56
a = 11.459
h
tetragonal†
1.817
a = 6.35; c = 3.78
i
¯ 1m P 42 ¯c R3
tetragonal
1.335
a = 5.66; c = 4.71
j
hexagonal‡
2.711
a = 4.99; c = 17.06 c, p 362
R 3¯ R ¯3c
hexagonal
1.115
a = 6.30; c = 11.73 k
hexagonal
3.77
a = 9.36; c = 18.87 l
NaCl CaF2
hexamethylenetetramine
[(CH3)4N]2CeCl6 K3Co(NO2)6 Mg3Al2(SiO4)3 NH4ClO2 CO(NH2)2 urea CaCO3
Cubane, C8H8
Density (g/cm3)
¯ m F43 Fm3¯ m
CsCl ZnS
Figure 3. Structure of C 8H8 (cubane).
Lattice Type
C8H8 cubane RbUO2(NO3)3
Cubane crystallizes in the rhombohea Wyckoff, R. W. G. Crystal Structures; Wiley: New York, 1963; Vol. 1. bLoehlin, J. H. Acta Cryst. dral space group R3¯ with one molecule per 1984, C40, 570. cWyckoff, R. W. G. Crystal Structures; Wiley: New York, 1964; Vol. 2. d Andresen, A. unit cell (see Table 3). For this analysis Acta Cryst. 1957, 10, 107–110. eHamilton, W. Acta Cryst. 1957, 10, 103–107. fCromer, D. T.; Kline, R. J. J. Am. Chem. Soc. 1954, 76, 5282–5283. gWyckoff, R. W. G. Crystal Structures; Wiley: New this structure is described by the correYork, 1965; Vol. 3, p 376. h Zemann, A.; Zemann, J. Acta Cryst. 1961, 14, 835–837. iGillespie, R. B.; sponding hexagonal cell with 3 molecules Sparks, R. A.; Trueblood, K. N. Acta Cryst. 1959, 12, 867–872. jVaughan, P.; Donahue, J. Acta Cryst. per cell. There are 24 C and 24 H atoms 1952, 5, 530–535. kFleischer, E. B. J. Am. Chem. Soc. 1964, 86, 3889-3890. lBarclay, G. A.; Sabine, per cell. We have a problem because no T. M.: Taylor, J. C. Acta Cryst. 1965, 19, 205–209. †For the tetragonal system a = b ≠ c and unit cell volume V = a2 c. ‡For the hexagonal a = b ≠ c; /ab = 120°; unit cell volume V = a2c sin(120°). sets of positions generate 24 atoms. For a cube-shaped molecule it is possible that opposite corners along one of the body diagonals lie on the 3-fold axis. These corner carbons are Table 3. Equivalent General and Special Positions for a placed in the 6c positions with 3 (C 3) symmetry with oppoSpace Group R3¯ for Rhombohedral Lattice Points in a site corner atoms in the +z and {z positions. The attached Hexagonal Setting (0,0,0; 1/ 3, 2/3, 2/ 3; 2/3 ,1/ 3, 1/3)+ hydrogens can be placed in the 6c positions with a different No. of Site Label Equivalent Positions value of z. The 18 remaining C atoms go into the 18f genPositions Symmetry eral positions. The 18 H atoms also go into the 18f positions – 18 f 1 x,y,z;y,x { y,z; y {x ,x,z; with different coordinates x,y,z. The structure is shown in Figure 3. The centers of the molecules are in the 3a posix¯ ,y¯ ,z¯ ; y,y{x,¯z; x{ y,x,z¯ tions with 3¯ (S6 ) symmetry. This implies that the molecule 1/ ,0,0; 0,1 / ,0; 1/ , 1/ ,0 1 9 e 2 2 2 2 must possess S6 symmetry. 1 1 1 1 1 1 1 1 9 d / ,0, / ; 0, / , / ; /2, / 2, /2 2 2 2 2 The data for these structures and a number of other 6 c 3 0,0,z; 0,0,z¯ suitable exercises are given in Table 4. A number of computer graphics packages are available for displaying crys3 3 b 0,0,1 /2 tal structures. The figures in this article were prepared us 3 a 3 0,0,0 ing the CAChe system (13). a Hahn, T. International Tables for Crystallography Volume A: Space-Group Symmetry, Kluwer: Dordrecht, Netherlands, 1992.
Acknowledgment I acknowledge helpful discussion with Lawrence Dahl. Note 1. Presented at the ACS Great Lakes Regional Meeting, 6–8 June 1995.
Literature Cited 1. Wheatley, P. J. The Determination of Molecular Structure, 2nd ed.; Clarendon: Oxford, 1968. 2. Rossi, M.; Berman, H. M. J. Chem. Educ. 1988, 65, 472–473. 3. Kettle, S. F. A.; Norrby, L. J. J. Chem. Educ. 1993, 70, 959–963. 4. Hathaway, B. J. Chem. Educ. 1979, 56, 166–167. 5. Bond, M. R.; Carrano, C. J. J. Chem. Educ. 1995, 72, 451–454. 6. Donnay, G.; Donnay, J. D. H. Am. Mineral. 1978, 63, 840–846.
7. 8. 9. 10.
Gunter, M. E.; Downs, R. T. Am. Mineral. 1991, 76, 293–294. Abad-Zapatero, C.; O’Donnell, T. J. J. Appl. Cryst. 1987, 20, 532–535. Phillips, G. N.; Biophys. J. 1995, 69, 1281–1283. Stout, G. H.; Jensen, L. H. X-ray Structure Determination A Practical Guide: Macmillan: London, 1968; Chapter 3. 11. Hahn, T. International Tables for Crystallography Volume A: SpaceGroup Symmetry;Kluwer: Dordrecht, Netherlands, 1992. 12. Hahn, T. International Tables for Crystallography: Brief Teaching Edition of Volume A: Space-Group Symmetry; Reidel: Dordrecht, Netherlands, 1988. 13. CAChe WorkSystem™ from CAChe Scientific, the computer-aided chemistry division of the Oxford Molecular Group, Inc., Campbell, CA.
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