- Email: [email protected]
The irreducible representation matrices of the icosahedral point groups I and Ih
Superlattices
and Microstructures,
391
Vol. 3. No. 4, 1987
THE IRREDUCIBLE
REPRESENTATION MATRICES OF THE ICOSAHEDRAL GROUPS I AND Ih
POINT
Wei Min Hu", Jin Long Yang, Jing Zhou, and Fang Shu Department of Physics, University of Science and Technology of China Hefei, The People's Republic of China (Received 20 April 1987)
The irreducible representation matrices of the icosahedral point groups I and Ih are presented. These matrices are useful in calculating the electronic structures of the icosahedral quasicrystals.
1. Introduction It is well known that space group symmetry an extremly important role in energy band plays Such calculations for crystalline solids.[l] are fundamental calculations in understanding those solids from the microscopic point of view. In last of the few years, the discovery quasicrystals has stimulated a lot of work concentrating new exciting on these materials.[Z] But very little work has been done on the electronic structure of quasicrystals. Quasicrystals, unlike crystalline solids, do not have three-dimensional space group symmetry. However, in order to do electronic structure calculations for quasicrystals, point group symmetry can be used. Most quasicrystals discovered so far have the icosahedral point group symmetry. Unfortunately, to the knowledge of the authors, the irreducible representation matrices of the icosahedral point groups have never been published before, probably because before the discovery of the quasicrystals, the icosahedral point groups were considered to be of 'no physical interest, since they do not occur in Nature as symmetry groups of molecules."[3] Recently we completed calculations of all the irreducible representation matrices for the icosahedral point groups I and Ih, which, we believe, will be useful in electronic structure calculations for quasicrystals.
2. Representation
about one of the 6 axes of fifth order. (In Fig.1, for example, axis Oz is such an axis.) We order those elements as the second element to the thirteenth element. The third class also contains 12 elements, 6(C5 m 28 c5 m 3' for m-1,2,---6.), those are either countericiockwise or clockwise rotations of (4s/5) about one of the 6 axes of fifth order. We order those elements as the fourteenth element to the twenty-fifth element. The fourth class contains 20 elements; those are either counter-clockwise or clockwise rotations of 2~/3 about one of the 10 axes of third order, lU(C3 m 19 c3,m 2 m-1,2,----10). (In Fig. 1, for exampl&, axis bo is such an axis.) We order those elements as the twenty-sixth element to the forty-fifth element. The fifth class contains 15 elements. Those are rotations of * about one of the 15 axes of second order (In Fig. 1, for example, axis Oy). We order those elements as the forty-sixth to the sixtieth element.
Matrices
The point group I contains 60 elements divided among 5 classes. We order these 60 elements as following: The first class contains only the identity. We assign it as the first element. The second class contains 12 elements, those aZ~C~i%~r c5 al 4, with m-1,2,----6.), counter-clockwise or clockwise rotations of 2x/5 * Present address: Department of Physics, University of Notre Dame, Notre Dame, IN 46556, U.S.A.
0749-6036/87/040391+08S02.00/0
Fig. 1. Icosahedral structure and axes used in this paper. Ox, Oy, and Oz are the Cartesian axes. Oa is a third-order axis.
0 1987 Academic Press Limited
392
Superlattices TABLE I. Character
E
A Tl
1 ;
E2 H
4 5
6G5,m,l 6%,m,4
table for the group I
6%,m,2 6c5,m,3
For
1 0
1 -1
(l+J5)/2 -1 0
0 1 -1
-1 0 1
Vol. 3, No. 4, 1987
the 5th element we have
_____G-____ ______~_____----T1-----T2--40 41-20 -40.14 43 -17 14 49 27 -3 16-39-26 33 -14 5 48 -41 5-31 41 -5 21-21 -33 2 32-37 -7 43-31 28 -20-48-28 -38-19 17-41 -26-29 17 28 37 19 46 14 -5 39 13 28-17 32 .16 47 13 29 2
1oc 1oc;:;;; 15c*,m
(I-j5),2
and Microstructures,
For the 6th element we have (l-J5)/2 -1 0
_____G_-___ ---T2-----T1--24-36 43 12-52-20 -43 0 24 36 19 46 31 -27-21 48 -31 0 19-46 -50 0 28 -50 O-28 -12-52-20 0 -27 21-48 0
--____H_____--3 42 0 O-42 16 15-27-48 11 39 34 12 20 25 26-18-43 24-23 33-22 31-19-30
For the 7th element we have Having defined these elements of the group, we the multiplication table of the group I, obtain given in Appendices I and II. Mathematically the multiplication table defines the group. The character table of the group I is given in Table I, and has been published before [4]. irreducible calculations, the In many are much more useful. matrices representation representation irreducible Determining the matrices is more complicated and difficult than determining the character table. Here we present all the irreducible representation matrices of five the point group I; each element has matrices: One representation irreducible (Tl and one-dimensional, two three-dimensional and one four-dimensional one (G), T2)v five-dimensional (H). The first element of the group is the identity, which is represented by unit matrices. The A representation is always unity for any element, so we only need give the irreducible representation matrices of Tl, T2, G, and H. They are as follows.
For the irreducible representation the 2nd element we have ---T1-----T2--21-52 0 -46 36 0 52 21 0 -36-46 0 0 01 0 01
_____G_____ 21-52 0 0 52 21 0 0 0 O-46-36 0 0 36-46
matrices
of
______H_______ 10 0 0 0 O-46-36 0 0 0 36-46 0 0 0 0 0 21 52 0 0 o-52 21
For the 3rd element we have -____G_____ "-T2--"-=1--21 52 0 -46-36 0 21 52 0 0 -52 21 0 36-46 0 -52 21 0 0 0 01 0 01 0 O-46 36 0 O-36-46
---___H_______ 10 0 0 0 O-46 36 0 0 O-36-46 0 0 0 0 0 21-52 0 0 0 52 21
______H_______ -3 16 42 15 O-27 O-48 .42 11
39 26 33 34-18-22 12-43 31 20 24-19 25-23-30
For the 8th element we have ---T1--49-27-20 27 21 48 -20-48 28
_____G_____ _-____H_______ ---T2--38-19 43 9 19-28-48 -3-33 26-39-16 19-46-31 -19-46-31 0 -33 56-61-13 -6 43 31-28 -28 31-35 27 -26 61 9-28 37 48 O-27 21 39 13-28-35-54 -16 -6-37 54 59
For the 9th element we have _____G_____ ______H_______ ---T2-----T1--49 27-20 38 19 43 9-19-28 48 -3-33-26 39-16 -27 21-48 -19-46 31 19-46 31 0 -33 56 61 13 -6 -20 48 28 43-31-28 -28-31-35-27 26-61 9-28-37 -48 0 27 21 -39-13-28-35 54 -16 -6 37-54 59 For the 10th element we have _____G_____ ---T2-----T1--24-19-50 12 27-50 -43 31-12 27 36 46 0 52-21 0 0 0 52 21 43-31 28 -20-48-28 24-19-20 48 -36-46 0 0
-_____H_______ -3 42 0 0 -42
16-39-26 33 15-34 18-22 27 12-43-31 48 20 24 19 11-25 23-30
For the 11th element we have _____G_____ ---T2-----T1--24 36 43 12 52-20 -43 0 24-36 -3 42 0 -19 46-31 27-21-48 31 O-19-46 16 15 27 -50 0 28 -50 O-28 -12 52-20 0 -39-34 12 27 21 48 0 -26 18-43 33-22-31
O-42 48 11 20-25 24 23 19-30
For the 12th element we have
For the 4th element we have _____G_____
_____G_____ ---T1-----T2--24 19-50 12-27-50 -43-31-12-27 -36 46 0 -52-21 0 0 O-52 21 43 31 28 -20 48-28 24 19-20-48 36-46 0 0
______H_______
---T2-----T1--40-14 43 -40-41-20 -17 41-38 19 -3-33-26 39-16 41 5-31 -14 5-48 14 -5-19 46 16 2-29 13 47 -20 48 28 43-31-28 49 27 17 lh -39 32 17 28 13 27-21-41 -5 -26-37 28-17 29 33 -7 37 32 2
______H_______ _____G_____ ---T2-----T1--40-41-20 -40 14 43 -17-14 49-27 -3 16 39 26 33 14 5-48 41 5 31 -41 -5-27-21 -33 2-32 37 -7 43 31 28 -20 48-28 -38 19 17 41 26 29 17 28-37 -19 46-14 -5 -39-13 28-17-32 -16 47-13-29 2
Superlattices
and Microstructures,
For the 13th element we have
For the 21st element we have
__-__G_____
---T1--‘ ---T2--40 14 43 -40 41-20 -17-41-38-19 -41 5 31 14 5 48 -14 -5 19 46 -20-48 28 43 31-28 49-27 17-14 -27-21 41 -5
______H_______
-3-33 26-39-16 16 2 29-13 47 39-32 17 28-13 26 37 28-17-29 33 -7-37-32 2
For the 14th element we have ---T1‘--46-36 36-46 0
0
0 0 1
---T2--21-52 52 21 0
0
393
Vol. 3, No. 4, 1987
---T1--38 19-43 -19-46-31 -43 31-28
_____G_____
0 -46-36 0 0 0 36-46 0 0 1 0 0 21 52 0 O-52 21
_____G_____ ---T1-----T2---46 36 0 21 52 0 -46 36 0 0 -36-46 0 -52 21 0 -36-46 0 0 0 01 0 01 0 0 21-52 0 05221
______H_______ 10 0 0 0 02152 0 0 O-52 21 0 0 0 0 O-46 36 0 0 O-36-46
______H_______ 10 0 0 0 0 21-52 0 0 0 52 21 0 0 0 0 o-46-36 0 0 0 36-46
For the 16th element we have _____G_____
---Tl-----T2--12 27 50 24 19 50 -20-48-24 19 52-21 0 -36 46 0 0 O-36-46 20 48-28 -43-31 28 12 27-43 31 -52 21 0 0
______H_______ -3-33 42-30 0 19 0 31 -42-22
26-39-16 23 25 11 24-20 48 43 12-27 18 34 15
For the 17th element we have _____G_____
______H_______
---T2--24-36-43 -20 0 12-52 -3 42 0 O-42 19 46-31 -48 0 27 21 -33-30 19 31-22 50 0 28 -24-36-43 0 26 23 24 43 18 19-46 31 0 -39 25-20 12 34 -16 11 48-27 15
_____G_____
______H_______
17-14-49-27 -3 lb-39-26 33 41 -5 27-21 -33 2 32-37 -7 38 19-17 41 26 29-17-28-37 19 46 14 -5 -39-13-28 17-32 -16471329 2 For the 19th element we have _____G_____
---T1----‘T2---40-14-43 40-41 20 17 41 38 -41 5 31 14 5 48 -14 -5 19 20 48-28 -43-31 28 -49 27-17 -27-21 41
-3 16 39 26 33
16-39-26 33 59-54-37 -6 54-35 28-13 37 28 9 61 -6 13-61 56
_____c_____ ---T2---‘-Tl---40 14-43 40 41 20 17-41 38-19 41 5-31 -14 5-48 14 -5-19 46 20-48-28 -43 31 28 -49-27-17-14 27-21-41 -5
______H_______ -3-33-26 39-16 16 2-29 13 47 39-32-17-28-13 26 37-28 17-29 33 -7 37 32 2
For the 23rd element we have ---T1-‘-40 41 20 14 5-48 -43-31-28
_____G_____ ______H_______ ---T2--40-14-43 17 14-49 27 -3 lb 39 26 33 41 5 31 -41 -5-27-21 -33 2-32 37 -7 20-48 28 38-19-17-41 -26-29-17-28 37 -19 46-14 -5 39 13-28 17 32 -16 47-13-29 2
For the 24th element we have _____G_____ ______H_______ ---T2-----T1--12-52 20 24 36-43 -20 0 12 52 -3 42 0 O-42 -27-21-48 -19 46 31 48 O-27 21 -33-30-19-31-22 50 O-28 50 0 28 -24 36-43 0 -26-23 24 43-18 -19-46-31 0 39-25-20 12-34 -16 11-48 27 15 For the 25th element we have _____G_____ ---T2------G--12-27 50 24-19 50 -20 48-24-19 -52-21 0 36 46 0 0 0 36-46 20-48-28 -43 31 28 12-27-43-31 52 21 0 0
______H_______ -3-33-26 39-16 42-30-23-25 11 o-19 24-20-48 O-31 43 12 27 -42-22-18-34 15
For the 26th element we have
For the 18th element we have ---Tl-----T2---40-41 20 40 14-43 -14 5 48 -41 5-31 -43 31-28 20 48 28
______H______-
For the 22nd element we have
For the 15th element we have
---T1--12 52 20 27-21 48 50 O-28
_____G_____ .--T2--49-27 20 -35 27 28 31 27 21-48 -27 21-48 0 20 48 28 28 48 9-19 -31 0 19-46
______H_______
19 -3-33 26-39-16 46 lb 2 29-13 47 14 -39 32-17-28 13 -5 -26-37-28 17 29 33 -7-37-32 2
For the 20th element we have _____G_____
_____G_____ ______H_______ ---T2--‘-‘T1--a-41 43 51 14-20 35 27 40 14 -3 lb 39 26 33 41 -5-31 -14 -5-48 -27-21 41 -5 16-58 57-44 4 43 31 28 -20 48-28 40-41 -9-19 -39-57 -5 0 60 -14 -5 19 46 -26 44 0 -5-55 33 4-60 55 62 For the 27th element we have ---T1-----T2--8 41 43 51-14-20 -41 -5 31 14 -5 48 43-31 28 -20-48-28
_____G-____
______H_______
35-27 40-14 27-21-41 -5 40 41 -9 19 14 -5-19 46
-3 16-39-26 33 16-58-57 44 4 39 57 -5 O-60 26-44 0 -5 55 33 4 60-55 62
For the 28th element we have ______H_______
--‘Tl-----T2--38-19-43 49 27 20 -35-27 28-31 -3 16 39 19-46 31 -27 21 48 27 21 48 0 16 59 54 -43-31-28 20-48 28 28-48 9 19 -39-54-35 31 O-19-46 -26-37 28 33 -6-13
26 33 37 -6 28 13 9-61 61 56
---T1---28-31 43 48 0 31 -20 48 28
_____G_____ ---T2--28 48-20 -17 41 8-41 31 0 48 -14 5-41 5 43-31-28 -51-14 17 14 14 541 5
_____-H_______ -3-33-26 39-16 16 -4 18-25-45 39-25-28-20 25 26-18 43 28 18 33 63-18 25 -4
394
Superlattices
For the 29th element we have _____G_---_
_____G_____
---T1-----T2---12-52-20 -24 36 43 20 O-12 52 -3 42 0 O-42 27-21 48 19 46-31 -48 0 27 21 -33-30 19 31-22 -50 0 28 -50 O-28 24 36 43 0 -26-23-24-43-18 19-46 31 0 39-25 20-12-34 -16 11 48-27 15
______H_______
-3-33-26 39-16 42-30-23-25 11 0 19-24 20 48 0 31-43-12-27 -42-22-18-34 15
_____G_____
-3-33 26-39-16 42-30 23 25 11 O-19-24 20-48 O-31-43-12 27 -42-22 18 34 15
For the 33rd element we have
_____G-____ --____H_______ ---T2-----Tl---3 42 0 O-42 -24 36-43 -12 52 20 43 O-24-36 19 46 31 -27-21 48 -31 0 19-46 16 15-27-48 11 50 O-28 50 0 28 12 52 20 0 -39-34-12-20-25 -27 21-48 0 -26 18 43-24 23 33-22 31-19-30
---T2-----Tl--28-48 20 -28-31-43 -31 0 48 48 O-31 -43-31-28 20-48 28
20 O-12-52 -3 42 0 O-42 48 O-27 21 -33-30-19-31-22 24-36 43 0 26 23-24-43 18 -19-46-31 0 -39 25 20-12 34 -16 11-48 27 15
For the 34th element we have ___T ___ _____G_____ 1 ---T2---28-48-20 28-31 43 -17 14-51-14 -3 16-39-26 33 31 O-48 -48 0 31 -41 5 14 5 -33 -4 25 18 63 43-31 28 -20-48-28 8 41 17-41 26-18-28 43 18 41 5-14 5 -39 25-20 28-25 -16-45-25-18 -4 For the 35th element we have _____G_____
---%-----T2---28 31 43 28-48-20 -17-41 8 41 -3-33 26-39-16 -48 O-31 -31 O-48 14 5 41 5 16 -4-18 25-45 -20-48 28 43 31-28 -51 14 17-14 -39 25-28-20-25 -14 5-41 5 -26 18 43 28-18 33 63 18-25 -4 For the 36th element we have ---T2-----Tl---3-33 26-39-16 51-14 20 8-41-43 -9-19-40-41 4 14 -5-48 41 -5 31 19 46 14 -5 -33 62-55-60 20 48-28 -43-31 28 -40-14 35-27 -26 55 -5 O-44 41 -5 27-21 39 60 0 -5-57 -16 4 44 57-58
_____G_____
______H_______
17 14 51-14 -3 16 39 26 33 41 5-14 5 -33 -4-25-18 63 26-18 28-43 18 -8 41-17-41 -41 5 14 5 -39 25 20-28-25 -16-45 25 18 -4
For the 41st element we have
_____G_____
_____G_____
-3 16-39-26 33 42 15-34 18-22 O-27-12 43 31 o-48-20-24-19 -42 11-25 23-30
For the 40th element we have
For the 32nd element we have
---Tl-----T2---12 52-20 -24-36 43 -27-21-48 -19 46 31 -50 0 28 -50 O-28
_____c-____ ---Tl-----T2---24 19 50 -12-27 50 43-31 12-27 36 46 0 52-21 0 0 0 52 21 -43 31-28 20 48 28 -24 19 20-48 -36-46 0 0
For the 39th element we have
For the 31st element we have
---Tl-----T2---12-27-50 -24-19-50 20 48 24-19 52-21 0 -36 46 0 0 O-36-46 -20-48 28 43 31-28 -12-27 43-31 -52 21 0 0
____-(___-____H_ ---T2---‘-Tl--51 14 20 8 41-43 -9 19-40 41 -3-33-26 39-16 -14 -5 48 -41 -5-31 -19 46-14 -5 -33625560 4 20-48-28 -43 31 28 -40 14 35 27 26-55 -5 0 44 -41 -5-27-21 -39-60 0 -5 57 -16 4-44-57-58 For the 38th element we have
For the 30th element we have
---%-----T2---12 27-50 -24 19-50 20-48 24 19 -52-21 0 36 46 0 0 0 36-46 -20 48 28 43-31-28 -12 27 43 31 52 21 0 0
Vol. 3, No. 4. 7987
For the 37th element we have ___-__H_______
---T1-----T2---28 48-20 28 31 43 -17-14-51 14 -3 16 39 26 33 -31 0 48 48 O-31 41 5-14 5 -33 -4-25-18 63 43 31 28 -20 48-28 8-41 17 41 -26 18-28 43-18 -41 5 14 5 39-25-20 28 25 -16-45 25 18 -4
_____G_____
and Microstructures,
---T2-----Tl--28-31-43 -28 48 20 -48 O-31 -31 O-48 20 48-28 -43-31 28
_____G_____ 17 41 -8-41 14 5 41 5 51-14-17 14 -14 5-41 5
______H_______
-3-33 26-39-16 16 -4-18 25-45 39-25 28 20 25 26-18-43-28 18 33 63 18-25 -4
For the 42nd element we have _____G_____ ---T2-----Tl--28 31-43 -28-48 20 17-41 -8 41 -3-33-26 39-16 48 0 31 31 0 48 -14 5-41 5 16 -4 18-25-45 20-48-28 -43 31 28 51 14-17-14 .39 25 28 20-25 14 5 41 5 -26 18-43-28-18 33 63-18 25 -4 For the 43rd element we have _____G_____ ---T2-----Tl--28 48 20 -28 31-43 17-14 51 14 -3 16-39-26 33 31 O-48 -48 0 31 -41 5 14 5 -33 -4 25 18 63 -43 31-28 20 48 28 -8-41-17 41 -26 18 28-43-18 41 5-14 5 39-25 20-28 25 -16-45-25-18 -4 For the 44th element we have +--Tl-----T2---24-36-43 -12-52 20 -19 46-31 27-21-48 50 O-28 50 0 28
_____G_____ 43 O-24 36 31 O-19-46 12-52 20 0 27 21 48 0
-3 42 0 O-42 lb 15 27 b8 11 39 34-12-20 25 26-18 43-24-23 33-22-31 19-30
Superlattices
and Microstructures,
For the 53rd element we have
For the 45th element we have _____G_____ ---T1-----T2---24-19 50 -12 27 50 43 31 12 27 -36 46 0 -52-21 0 0 O-52 21 -43-31-28 20-48 28 -24-19 20 48 36-46 0 0
______H_______ -3 42 0 0 -42
16 39 26 33 15 34-18-22 27-12 43-31 48-20-24 19 11 25-23-30
For the 46th element we have _____G_____
--‘T1-----T2---38 19 43 -49-27-20 35 27-28 31 19-46 31 -27 21 48 27 21 48 0 43 31 28 -20 48-28 -28 48 -9-19 31 O-19-46
-3 16 39 26 33
16 39 26 33 59 54 37 -6 54 35-28-13 37-28 -9 61 -6-13 61 56
For the 48th element we have ---%-----T2---28 O-50 28 O-50 50 O-28 0 o-1 0 o-1 0 0 0 0142 -50 0 28 -50 O-28 -28 O-50 0 010 0
---Tl---51 14-20 14 -5-48 -20-48 28
---T2---8 41 43 41 -5 31 43 31-28
9 19 40 41
-3 42 0 O-42 6 0 0 4 0 0 28 50 0 0 0 50-28 0 -42 4 0 0 6
_____G_____ ______H_______ ---T2-----T1---38-19 43 -49 27-20 35-27-28-31 -3 16-39-26 33 -19-46-31 27 21-48 -27 21-48 0 16 59-54-37 -6 43-31 28 -20-48-28 -28-48 -9 19 -39-54 35-28 13 -31 0 19-46 -26-37-28 -9-61 33 -6 13-61 56 For the 51st element we have
_____G_____ -9 19 19-46 28 31 -48 0
______H_____-_
28-48 -3-33-26 39-16 31 0 -33 56 61 13 -6 35 27 -26 61 -9 28 37 27 21 39 13 28 35-54 16 -6 37-54 59
---T2-----Tl--46 36 0 -21 52 0 36-46 0 52 21 0 0 o-1 0 o-1
_____G_____ 46 36 0 0 36-46 0 0 0 O-21-52 0 O-52 21
______H_______ 10 0 0 0 0 21 52 0 0 0 52-21 0 0 0 0 0 46-36 0 0 O-36-46
_____G_____ ---T2-----Tl---21 52 0 46-36 0 -21 52 0 0 52 21 0 -36-46 0 52 21 0 0 0 o-1 0 0 46 36 0 o-1 0 0 36-46
______H_______ 10 0 0 0 O-46-36 0 0 O-36 46 0 0 0 0 O-21-52 0 0 O-52 21
---Tl---10 0 010 0 o-1
---T2---10 0 010 0 o-1
_____G_____
______"____-__
-10 0 0 010 0 0 o-1 0 0 0 01
10 0 0 010 0 0 O-10 0 0 o-1 0 0 0 01
0 0 0 0
For the 59th element we have ______H_______ -3 42 0 O-42 6 0 0 4 0 o-28-50 0 0 O-50 28 0 -42 4 0 0 6
For the 52nd element we have _____G_____
---T2-----Tl---49-27 20 -38-19-43 -27 21-48 -19-46 31 20-48-28 -43 31 28
For the 58th element we have
For the 50th element we have
---T2-----Tl--28 0 50 -28 0 50 -50 0 28 0 o-1 0 0 0 0142 o-1 0 50 O-28 50 0 28 28 0 50 0 010 0
-3 16 39 26 33 16-58 57-44 4 39 57 5 O-60 26-44 0 5 55 33 4-60 55 62
For the 57th element we have ______H_______
19 40 41 -3-33 26-39-16 46 14 -5 -33 62-55-60 4 14-35 27 26-55 5 0 44 -5 27-21 -39-60 0 5 57 -16 4 44 57-58
_____G_____
______H_______
For the 56th element we have ______H_______
For the 49th element we have _____G_____
_____G_____ ---T2-----Q---8 41-43 -51-14 20 -35-27-40-14 41 -5-31 -14 -5-48 -27-21 41 -5 20-48 28 -40 41 9 19 -43-31-28 -14 -5 19 46
For the 55th element we have ______H_______
-‘-Tl-----T2---51-14-20 -8-41 43 -3-33-26 39-16 9-19 40-41 -14 -5 48 -41 -5-31 -19 46-14 -5 -33 62 55 60 4 -20 48 28 43-31-28 40-14-35-27 -26 55 5 O-44 -41 -5-27-21 39 60 0 5-57 -16 4-44-57-58
_____G_____
_____G_____ ______H_______ ---T2-----T1---8-41-43 -51 14 20 -35 27-40 14 -3 16-39-26 33 -41 -5 31 14 -5 48 27-21-41 -5 16-58-57 44 4 -43 31-28 20 48 28 -40-41 9-19 -39-57 5 0 60 14 -5-19 46 -26 44 0 5-55 33 4 60-55 62 For the 54th element we have
______H_______
For the 47th element we have _____G_____
395
Vol. 3, No. 4, 1987
---Tl---21-52 0 -52 21 0 0 o-1
_____G_____ ---T2--46 36 0 -21-52 0 0 36-46 0 -52 21 0 0 0 o-1 0 0 46-36 0 O-36-46
______H_______ 10 0 0 0 O-46 36 0 0 0 36 46 0 0 0 0 O-21 52 0 0 0 52 21'
For the 60th element we have ______H_______
---T2-----T1---49 27 20 -38 19-43 -9-19 28 48 -3-33 26-39-16 27 21 48 19-46-31 -19-46-31 0 -33 56-61-13 -6 20 48-28 -43-31 28 28-31 35-27 26-61 -9 28-37 48 O-27 21 -39-13 28 35 54 -16 -6-37 54 59
_____G_____ ---T2---"T1---21-52 0 46-36 0 0 46-36 0 -36-46 0 -52 21 0 -36-46 0 0 0 o-1 0 o-21 52 0‘0 -1 0 05221
______H_______ 10 0 0 0 0 21-52 0 0 O-52-21 0 0 0 0 0 46 36 0 0 0 36-46
396
Superlattices
numbers in irreducible Here the the are symbols with the representation matrices following meanings: O-O, l-l, 2-l/10, 3-l/5, 4-2/5, 5-l/2, 6-3/5.
and Microstructures,
Acknowledgment -- The authors are grateful Professor John D. Dow for his assistance preparing the manuscript.
to in
REFERENCES "Quantum Theory of Molecules and Solids" McGraw-Hill Book Company,Inc.(1963). [7-lD. Shechtman, I. Blech, D. Gratias, and J. W. (1984); D. Cahn, Phys. Rev. Lett. 12, 1951 Levine and P. J Steinhardt, Phys.s= ---A 53, 2477 (1984). Quantum [31 L. D. Landau and E. M. Lifshitz, Mechanics, (Pergamon Press Ltd., London, 1962), p. 328. Point Group [41 J. A. Salthouse and M. J. Ware, Character Tables and Related Data (Cambridge University Press, Cambridge,72)
111 J:C.Slater
Appendix
C5,l '5,2 C5,3 C5,4 C5,5 C5.6 $1 n-
specified
by
their
(0, 0, 1);
(sin A cos 36'. sin A sin 36', cos A); (-sin A cos 72', sin A sin 72', cos A); (-sin A, 0, cos A); (-sin A cos 72',-sin A sin 72', cos A); (sin A cos 36',-sin A sin 36', cos A); (sin B, 0, cos B); (cos 72' sin B, sin 72' sin B, cos B); (-cos 36O sin B, sin 36' sin B, cos B) C3'4 - (-cos 36' sin B,-sin 36' sin B, cos B) C3'5 - (cos 72' sin B,-sin 72' sin B, cos B); '3'6 - (sin C, 0, cos C); C3'7 - (cos 72' sin C, sin 72' sin C, cos C); '3'8 - (-cos 36' sin C, sin 36O sin C, cos C) sin C,-sin 36' sin C, cos C) C3'9 - (-cos 36' n C3'10 - (cos 72" sin C,-sin 72" sin C, cos C); C2'1 - (cos 36' sin D, sin 36' sin D, cos D); C2'2 - (-cos 72' sin D, sin 72' sin D, cos D); '2'3 - (-sin D, 0, cos D); '2'4 - (-cos 72' sin D,-sin 72' sin D, cos D); '2'5 - (cos 36' sin D,-sin 36' sin D, cos D); '2'6 - (sin E, 0, cos E); '2'7 - (cos 72' sin E, sin 72' sin E, cos E); - (-cos 36' sin E, sin 36' sin E, cos E); '2:8 - (-cos 36' sin E,-sin 36O sin E, cos E); C2,9 C2,lO - (cos 72' sin E,-sin 72O sin E, cos E); C2,ll - (sin 72', cos J2', 0) C2,12 - (sin 36', cos 36", 0); '2,13 - (0, 1, 0): cos 36', 0); and '2,14 - (-sin 36', 0 '2,15 - (-sin 72 ( cos 72 O, 0).
c;l; -
Here we have: ~-63.43'; B-37.38'; C-79.18'; D-31.72O: and E-58.29O.
I: Notation
of the element means the n m,p rotation by Psp/n about the counter-clockwise mth order axis C, m. For n-5, we have m-l, 2, . . . , 6; for n-3, we have m-l, 2, . ... 10; for 2, . . . . n-2, we have -1, 2, . . . . 15; and p-l, n-l. C
4. 1987
The elements of the group I are assigned numerical symbols as follows (E-l means that E is assigned the numerical symbol 1.):
The rotation axes can be direction cosines:
57-(5+2J5)li2 10, 58-(2J5-1)/l?, 59-(7+J5)/20, 60-(10-2J5)1/4,5 61-(50+22/5) j2/20, 62-(2/5+1)/10, and 63-(3/5-l)/lO. As a result -19 stands for -[(5-J5)/10]1/2/2, for example. The point group Ih can be considered as the direct product of the point group Ci and the point group I: Ih-IxC . The 120 elements of the point group Ih can c e divided into 10 classes. Therefore the group Ih will have 10 different irreducible representations. After we have given the irreducible representation matrices of the point group I, the irreducible representations are composed of of the point group Ih, which 1200 matrices can be easily obtained.
Vol. 3, No
Appendix
II: Multiplication
Table
The multiplication table for the elements (symbolized by numbers) can be written as a matrix Mi.i, where i and j run from 1 to 60. That matrig is:
Superlattices
and Microstructures,
397
Vol. 3, No. 4. 1987
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 14 1 26 8 28 10 30 12 32 4 34 6 15 3
3 1 15 11 29 13 31 5 33 7 35 9 27 2 14
4 28 9 16 1 38 12 6 36 2 43 26 11 47 31
5 10 27 1 17 8 29 42 3 39 13 7 37 34 50
6 30 11 28 13 18 1 40 4 8 38 2 45 48 33
7 12 29 9 39 1 19 10 31 44 3 41 5 26 46
8 32 13 2 37 30 5 20 1 42 6 10 40 49 35
9 4 31 43 7 11 41 1 21 12 33 36 3 28 47
10 34 5 12 42 2 39 32 7 22 1 44 8 50 27
11 6 33 38 3 45 9 13 43 1 23 4 35 30 48
12 26 7 36 10 4 44 2 41 34 9 24 1 46 29
13 8 35 6 27 40 3 37 11 5 45 1 25 32 49
14 15 2 50 30 46 32 47 34 48 26 49 28 3 1
15 3 14 35 47 27 48 29 49 31 50 33 46 1 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 52 36 17 4 57 26 38 51 28 56 46 43 19 41
17 39 51 5 16 42 46 57 27 52 37 29 56 44 24
18 53 38 47 45 19 6 59 28 40 52 30 58 21 43
19 41 52 31 58 7 18 44 47 59 29 53 39 36 lb
20 54 40 32 60 48 37 21 8 56 30 42 53 23 45
21 43 53 54 41 33 60 9 20 36 48 56 31 38 18
22 55 42 44 54 34 57 49 39 23 10 58 32 25 37
23 45 54 58 33 55 43 35 57 11 22 38 49 40 20
24 51 44 60 34 36 55 26 59 50 41 25 12 17 39
25 37 55 40 50 60 35 51 45 27 59 13 24 42 22
26 46 12 51 2 16 34 28 24 14 36 50 4 29 7
27 5 50 13 46 37 15 17 35 29 25 3 51 10 34
28 47 4 46 6 52 2 18 26 30 16 14 38 31 9
29 7 46 3 52 5 47 39 15 19 27 31 17 12 26
30 48 6 14 40 47 8 53 2 20 28 32 18 33 11
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
51 37 52 39 53 41 54 43 55 45 50 13 46 5 47
38 52 40 53 42 54 44 55 36 51 4 46 6 47 8
17 5 58 44 40 60 32 54 50 35 46 3 52 7 18
4 16 30 47 54 33 58 45 41 60 12 51 2 46 32
46 27 19 7 60 36 42 57 34 55 14 35 47 3 53
43 57 6 18 32 48 55 35 60 37 36 17 4 52 2
26 51 47 29 21 9 57 38 44 59 34 25 14 27 48
57 39 45 59 8 20 34 49 51 27 16 29 38 19 6
36 56 28 52 48 31 23 11 59 40 24 37 26 17 14
52 29 59 41 37 56 10 22 26 50 28 15 18 31 40
56 42 38 58 30 53 49 33 25 13 51 5 16 39 28
28 46 53 31 56 43 39 58 12 24 2 50 30 15 20
25 40 17 42 19 44 21 36 23 38 35 6 27 8 29
45 18 37 20 39 22 41 24 43 16 11 28 13 30 5
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
5 1 22 34 45 25 30 20 15 33 29 9 39 12 58
1 4 32 14 23 35 18 40 31 21 7 36 10 26 22
15 35 7 1 24 26 37 17 32 22 48 23 31 11 41
33 23 1 6 34 14 25 27 20 42 21 57 9 38 12
34 24 15 27 9 1 lb 28 39 19 22 59 49 25 33
22 41 44 21 35 26 25 lb 115 8 29 26 11 14 1 17 18 29 30 57 59 19 20 23 24 59 56 11 50
19 31 24 36 27 17 1 10 28 14 18 48 59 21 25
20 32 43 23 28 18 15 31 13 1 37 10 56 22 16
30 14 21 33 16 38 29 19 1 12 8 34 20 49 56
37 8 57 22 18 59 48 21 35 11 27 1 17 10 52
6 28 20 48 57 23 19 59 9 36 1 26 8 14 42
27 13 39 10 59 24 20 56 49 23 15 11 29 1 19
11 38 8 30 22 49 59 25 21 56 9 16 1 28 10
50 25 29 5 41 12 56 16 22 58 49 45 15 13 31
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
7 48 9 49 11 24 40 16 42 18 44 20 36 22 38
48 10 49 12 50 43 17 45 19 37 21 39 23 41 25
41 30 21 14 33 51 13 57 10 45 24 8 56 34 23
15 22 35 44 25 9 56 6 52 20 31 57 11 19 40
9 20 43 32 23 26 25 52 5 59 12 37 16 10 58
47 34 15 24 27 21 42 11 58 8 53 22 33 59 13
3 54 11 22 45 12 60 28 17 53 7 56 4 39 18
53 2 48 26 15 56 5 23 44 13 60 10 54 24 35
29 49 3 55 13 41 20 4 57 30 19 54 9 58 6
21 8 54 2 49 16 27 58 7 25 36 5 57 12 55
19 14 31 50 3 60 8 43 22 6 59 32 21 55 11
33 42 23 10 55 4 51 18 29 60 9 17 38 7 59
10 31 12 33 4 55 18 51 20 52 22 53 24 54 16
32 7 34 9 26 23 52 25 53 17 54 19 55 21 51
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
24 18 60 47 21 27 11 42 2 23 50 6 37 14 54
50 58 25 19 60 3 43 8 28 54 15 38 13 47 20
4 60 16 20 57 14 55 29 13 44 2 25 46 8 39
28 24 46 60 17 48 22 3 45 10 30 55 15 40 5
13 40 17 43 4 35 6 30 5 57 lb 45 18 47 8 10 42 19 41 7 48 14 49 12 51 24 43 31 3 14 5 37 52 36 12 33 2 32 7 17 51 38 47 15 2
56 13 42 6 32 52 15 44 9 50 lb 3 39 4 34
58 46 19 27 7 40 2 21 49 4 45 14 53 35 9
23 17 58 5 44 6 26 53 15 36 11 46 18 3 41
44 47 41 15 9 25 6 56 32 38 55 30 60 49 43
49 39 55 7 24 11 16 40 47 56 33 52 45 31 60
12 53 36 48 43 50 45 17 8 58 34 40 51 32 57
14 44 50 41 51 33 57 13 18 42 48 58 35 53 37
1 21 4 54 38 34 59 46 37 19 10 60 26 42 52
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
27 29 31 33 35 25 17 19 21 23 60 56 57 58 59
28 30 32 34 26 16 18 20 22 24
29 19 53 48 15 27 39 59 20 49 37 42 22 55 25
26 14 49 55 24 36 28 48 23 59 43 38 18 53 21
15 31 21 54 49 50 29 41 56 22 51 17 39 44 24
16 28 14 50 51 56 38 30 49 25 54 23 45 40 20
50 15 33 23 55 24 46 31 43 58 36 16 52 19 41
52 18 30 14 46 17 58 40 32 50 42 22 55 25 37
51 46 15 35 25 60 16 47 33 45 21 43 38 18 53
47 53 20 32 14 46 19 60 42 34 17 39 44 24 51
17 52 47 15 27 37 57 18 48 35 20 54 23 45 40
14 48 54 22 34 26 47 21 57 44 16 52 19 41 36
13 5 7 9 11 45 37 39 41 43 59 60 56 57 58
6 8 10 12 4 38 40 42 44 36 58 59 60 56 57
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
7 44 59 53 31 3
12 34 55 59 41 9 2 49 45 53 11 6 30 48 33
33 43 55 9 4 35 36 26 11 56 51 38 54 56 58 49 54 44 3 11 50 12 2 3 51 50 4 42 37 52 50 49 12 27 35 26 5 13 46 10 8 29 34 32 7
53 60 37 8 30 47 41 51 5 2 29 7 12 26 46
42 57 52 29 5 8 54 38 47 3 30 48 33 11 6
32 54 57 39 10 2 48 43 52 7 28 47 31 9 4
5 39 19 31 3 13 42 58 53 33 40 20 54 23 45
2 32 22 44 12 4 30 54 58 41 38 18 53 21 43
3 7 41 21 33 35 5 44 60 54 25 37 42 22 55
4 2 34 24 36 43 6 32 55 60 23 45 40 20 54
35 3 9 43 23 55 27 7 36 57 24 51 17 39 44
57
58 59 60 56
9 10 11 12 13 14 15
10
55 40 48 13 8 32 49 35
58 45 6 28 52 39 55 13 2 46 10 34 50 27 5
60 51 27 13 40 53 36 46 3 6 31 9 4 28 47
398
Superlattices 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
1 2 3 4 5 6
and Microstructures,
Vol. 3, No. 4, 1987
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
8 9 10 11 12 13 14 15
31 9 47 33 19 3 53 7 48 41 15 21 29 4 28
32 49 8 34 20 14 42 48 10 54 2 22 30 35 13
33 11 48 23 31 35 21 3 54 9 49 43 15 6 30
34 50 10 24 32 26 22 14 44 49 12 55 2 27 5
35 13 49 45 15 25 33 27 23 3 55 11 50 8 32
36 16 41 56 12 43 24 4 60 26 21 51 9 52 19
37 42 25 8 51 20 27 56 13 17 40 5 60 22 55
38 18 43 52 11 58 4 45 16 6 57 28 23 53 21
39 44 17 7 57 10 52 22 29 58 5 19 42 24 51
40 20 45 30 25 53 13 60 6 37 18 8 59 54 23
41 36 19 21 44 9 59 12 53 24 31 60 7 16 52
42 22 37 10 56 32 17 54 5 57 8 39 20 55 25
43 38 21 57 9 23 36 11 56 4 54 16 33 18 53
44 24 39 41 22 12 58 34 19 55 7 59 10 51 17
45 40 23 18 35 59 11 25 38 13 58 6 55 20 54
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
46 29 26 27 28 17 14 52 50 47 51 15 16 7 12
47 31 28 15 18 29 30 19 14 53 46 48 52 9 4
48 33 30 49 53 15 20 31 32 21 14 54 47 11 6
49 35 32 55 48 50 54 15 22 33 34 23 14 13 8
50 27 34 25 14 51 49 46 55 15 24 35 26 5 10
51 17 24 37 26 56 50 16 25 46 60 27 36 39 44
52 19 16 29 38 39 28 58 46 18 17 47 57 41 36
53 21 18 48 59 31 40 41 30 60 47 20 19 43 38
54 23 20 22 21 49 56 33 42 43 32 57 48 45 40
55 25 22 59 49 24 23 50 58 35 44 45 34 37 42
56 57 60 42 36 54 51 43 37 16 20 17 21 58 59
57 58 56 39 43 22 16 23 17 38 42 52 54 59 60
58 59 57 19 23 44 38 55 52 45 39 18 22 60 56
59 60 58 53 55 41 45 24 18 25 19 40 44 56 57
60 56 59 20 24 21 25 36 40 51 53 37 41 57 58
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
23 58 13 40 10 32 24 50 56 17 43 52 11 18 1
24 60 46 17 31 7 43 4 58 18 55 40 50 37 15
58 19 25 60 5 42 12 34 16 46 38 47 45 53 13
60 20 16 57 47 19 33 9 45 6 25 8 51 42 46
18 47 60 21 17 57 7 44 4 26 6 14 40 48 37
42 10 23 55 6 40 14 48 27 3 17 7 57 44 38
2 26 48 15 43 11 52 18 7 41 10 24 32 50 54
29 3 44 12 25 51 8 42 14 49 47 33 19 9 59
9 43 2 28 49 15 45 13 53 20 41 56 12 16 34
14 50 31 3 36 4 17 52 10 44 32 55 48 35 21
54 22 11 45 2 30 50 15 37 5 56 39 43 58 4
12 36 14 46 33 3 38 6 19 53 44 60 34 51 49
39 7 55 24 13 37 2 32 46 15 52 31 58 41 45
21 54 4 38 14 47 35 3 40 8 60 42 36 57 26
47 15 41 9 51 lb 5 39 2 34 30 49 53 33 60
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
13 6 42 32 58 55 53 60 33 43 3 4 5 2 39
35 45 5 8 44 34 60 51 54 57 33 38 3 6 7
55 59 27 37 7 10 36 26 57 52 23 18 35 40 3
59 53 51 56 29 39 9 12 38 28 45 30 25 20 27
40 30 56 54 52 58 31 41 11 4 13 2 37 32 17
8 2 54 49 38 45 47 53 3 9 5 12 42 34 57
3 11 10 2 55 50 40 37 48 54 31 43 7 4 44
49 55 3 13 12 2 51 46 42 39 54 58 33 45 9
44 41 50 51 3 5 4 2 52 47 58 53 55 60 35
53 48 36 43 46 52 3 7 6 2 40 32 60 54 51
10 12 49 50 11 13 28 30 29 31 39 41 22 24 23
7 9 34 26 35 27 6 8 47 48 19 21 44 36 55
31 33 12 4 50 46 13 5 30 32 53 54 41 43 24
48 49 9 11 26 28 27 29 8 10 20 22 21 23 36
32 34 33 35 4 6 46 47 5 7 42 44 54 55 43
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
30 12 14 36 46 54 39 35 59 5 20 44 49 60 27
5 33 1 23 6 44 53 26 56 47 39 21 12 57 28
20 1 32 4 14 57 29 55 41 27 56 7 22 36 50
39 15 7 35 1 59 30 36 54 28 58 48 41 23 4
54 5 22 1 34 38 46 59 31 51 43 29 58 9 24
59 28 53 46 31 37 1 54 34 11 25 2 20 50 33
35 36 57 40 45 56 39 30 59 54 146 36 35 30 39 46 1 21 55 3 26 16 13 6 17 29 2 53 22
27 60 52 23 6 50 13 20 29 58 28 25 40 48 5 7 17 53 21 3 32 2 22 9 16 12 23 48 35 2 29 51 18 43 149 142 31 52 26 45 30 49 1
43 37 57 8 22 28 50 19 3 24 4 27 52 1 44
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
34 19 24 31 36 35 38 37 30 57 49 18 25 48 56
2 41 26 21 16 49 58 27 40 39 32 59 50 20 17
8 9 2 43 28 22 19 50 60 29 42 41 34 56 46
42 3 10 11 2 58 47 24 21 46 57 31 44 43 26
22 29 44 3 12 45 28 60 48 16 23 47 59 33 36
55 52 59 29 41 13 4 20 14 43 35 28 40 15 21
26 59 51 53 56 15 23 5 6 22 14 45 27 30 42
6 36 28 56 52 32 44 15 25 7 8 24 14 37 29
37 11 8 38 30 39 31 34 36 15 17 9 10 16 14
57 27 39 13 10 18 14 41 33 26 38 15 19 11 12
25 38 40 52 53 5 9 32 26 33 27 4 8 46 48
51 45 37 18 20 29 33 10 4 49 46 11 5 28 32
16 25 17 40 42 47 49 7 11 34 28 35 29 6 10
38 51 52 37 39 30 34 31 35 12 6 50 47 13 7
45 16 18 17 19 8 12 48 50 9 13 26 30 27 31
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
38251837,3039343136495724195648 6 27 40 17 120 58 49 41 26 33 2 3 8 29 42 18 23 60 50 43 26 11 2 3 10 47 58 20 25 57 16 45 28 13 2 29 44 48 60 22 57 59 52 40 28 5 12 15 21 34 45 51 59 56 53 22 14 7 4 15 8 29 37 52 56 45 33 24 14 9 34 9 10 31 39 30 38 37 35 16 51 38 26 11 12 15 19 32 40 39 22 41 39 53 52 8 4 27 33 26 55 36 44 21 19 32 28 5 11 46 25 16 24 43 41 49 47 10 6 29 37 52 51 38 36 35 31 34 30 7 42 19 17 18 16 13 9 50 48 12
50 35 45 59 41 26 15 11 18 9 4 28 47 31
21 56 42 32 14 31 36 17 10 46 29 7 12 26
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1 10 44 41 9 11 8 22 59 21 45 40 20 54 23
11 1 12 36 43 23 13 10 24 56 55 25 37 42 22
45 13 1 4 38 58 25 5 12 16 44 24 51 17 39
40 37 5 1 6 18 60 17 7 4 19 41 36 lb 52
8 42 39 7 1 6 20 57 19 9 18 53 21 43 38
10 22 58 19 7 1 32 23 18 31 6 30 48 33 11
9 12 24 60 21 33 1 34 25 20 35 13 8 32 49
23 11 4 16 57 22 35 1 26 17 34 50 27 5 10
59 25 13 6 18 19 24 27 1 28 7 12 26 46 29
20 56 17 5 8 30 21 16 29 1 47 31 9 4 28
44 55 45 18 19 7 34 35 6 47 1 2 14 15 3
41 24 25 40 53 31 12 50 13 30 3 1 2 14 15
21 36 51 37 20 48 9 26 27 8 15 3 1 2 14
54 43 16 17 42 32 33 4 46 5 14 15 3 1 2
22 23 38 52 39 10 49 11 28 29 2 14 15 3 1
7
46 11 18 55 56 26 27 38 47 59 42 51 37 58 29 31 2 20 54 24 10 147 33 38 27 55 32 41 1 47 140 23 51 34 3 28 48 18 5 25 8 19 3
38 28 46 17 42 23 6 14 27 32 49 35 13 8
59 40 30 47 29 44 25 8 14 5 10 34 50 27
16 46 27 37 20 43 28 15 13 48 33 11 6 30
and Microstructures,
391
Vol. 3. No. 4, 1987
THE IRREDUCIBLE
REPRESENTATION MATRICES OF THE ICOSAHEDRAL GROUPS I AND Ih
POINT
Wei Min Hu", Jin Long Yang, Jing Zhou, and Fang Shu Department of Physics, University of Science and Technology of China Hefei, The People's Republic of China (Received 20 April 1987)
The irreducible representation matrices of the icosahedral point groups I and Ih are presented. These matrices are useful in calculating the electronic structures of the icosahedral quasicrystals.
1. Introduction It is well known that space group symmetry an extremly important role in energy band plays Such calculations for crystalline solids.[l] are fundamental calculations in understanding those solids from the microscopic point of view. In last of the few years, the discovery quasicrystals has stimulated a lot of work concentrating new exciting on these materials.[Z] But very little work has been done on the electronic structure of quasicrystals. Quasicrystals, unlike crystalline solids, do not have three-dimensional space group symmetry. However, in order to do electronic structure calculations for quasicrystals, point group symmetry can be used. Most quasicrystals discovered so far have the icosahedral point group symmetry. Unfortunately, to the knowledge of the authors, the irreducible representation matrices of the icosahedral point groups have never been published before, probably because before the discovery of the quasicrystals, the icosahedral point groups were considered to be of 'no physical interest, since they do not occur in Nature as symmetry groups of molecules."[3] Recently we completed calculations of all the irreducible representation matrices for the icosahedral point groups I and Ih, which, we believe, will be useful in electronic structure calculations for quasicrystals.
2. Representation
about one of the 6 axes of fifth order. (In Fig.1, for example, axis Oz is such an axis.) We order those elements as the second element to the thirteenth element. The third class also contains 12 elements, 6(C5 m 28 c5 m 3' for m-1,2,---6.), those are either countericiockwise or clockwise rotations of (4s/5) about one of the 6 axes of fifth order. We order those elements as the fourteenth element to the twenty-fifth element. The fourth class contains 20 elements; those are either counter-clockwise or clockwise rotations of 2~/3 about one of the 10 axes of third order, lU(C3 m 19 c3,m 2 m-1,2,----10). (In Fig. 1, for exampl&, axis bo is such an axis.) We order those elements as the twenty-sixth element to the forty-fifth element. The fifth class contains 15 elements. Those are rotations of * about one of the 15 axes of second order (In Fig. 1, for example, axis Oy). We order those elements as the forty-sixth to the sixtieth element.
Matrices
The point group I contains 60 elements divided among 5 classes. We order these 60 elements as following: The first class contains only the identity. We assign it as the first element. The second class contains 12 elements, those aZ~C~i%~r c5 al 4, with m-1,2,----6.), counter-clockwise or clockwise rotations of 2x/5 * Present address: Department of Physics, University of Notre Dame, Notre Dame, IN 46556, U.S.A.
0749-6036/87/040391+08S02.00/0
Fig. 1. Icosahedral structure and axes used in this paper. Ox, Oy, and Oz are the Cartesian axes. Oa is a third-order axis.
0 1987 Academic Press Limited
392
Superlattices TABLE I. Character
E
A Tl
1 ;
E2 H
4 5
6G5,m,l 6%,m,4
table for the group I
6%,m,2 6c5,m,3
For
1 0
1 -1
(l+J5)/2 -1 0
0 1 -1
-1 0 1
Vol. 3, No. 4, 1987
the 5th element we have
_____G-____ ______~_____----T1-----T2--40 41-20 -40.14 43 -17 14 49 27 -3 16-39-26 33 -14 5 48 -41 5-31 41 -5 21-21 -33 2 32-37 -7 43-31 28 -20-48-28 -38-19 17-41 -26-29 17 28 37 19 46 14 -5 39 13 28-17 32 .16 47 13 29 2
1oc 1oc;:;;; 15c*,m
(I-j5),2
and Microstructures,
For the 6th element we have (l-J5)/2 -1 0
_____G_-___ ---T2-----T1--24-36 43 12-52-20 -43 0 24 36 19 46 31 -27-21 48 -31 0 19-46 -50 0 28 -50 O-28 -12-52-20 0 -27 21-48 0
--____H_____--3 42 0 O-42 16 15-27-48 11 39 34 12 20 25 26-18-43 24-23 33-22 31-19-30
For the 7th element we have Having defined these elements of the group, we the multiplication table of the group I, obtain given in Appendices I and II. Mathematically the multiplication table defines the group. The character table of the group I is given in Table I, and has been published before [4]. irreducible calculations, the In many are much more useful. matrices representation representation irreducible Determining the matrices is more complicated and difficult than determining the character table. Here we present all the irreducible representation matrices of five the point group I; each element has matrices: One representation irreducible (Tl and one-dimensional, two three-dimensional and one four-dimensional one (G), T2)v five-dimensional (H). The first element of the group is the identity, which is represented by unit matrices. The A representation is always unity for any element, so we only need give the irreducible representation matrices of Tl, T2, G, and H. They are as follows.
For the irreducible representation the 2nd element we have ---T1-----T2--21-52 0 -46 36 0 52 21 0 -36-46 0 0 01 0 01
_____G_____ 21-52 0 0 52 21 0 0 0 O-46-36 0 0 36-46
matrices
of
______H_______ 10 0 0 0 O-46-36 0 0 0 36-46 0 0 0 0 0 21 52 0 0 o-52 21
For the 3rd element we have -____G_____ "-T2--"-=1--21 52 0 -46-36 0 21 52 0 0 -52 21 0 36-46 0 -52 21 0 0 0 01 0 01 0 O-46 36 0 O-36-46
---___H_______ 10 0 0 0 O-46 36 0 0 O-36-46 0 0 0 0 0 21-52 0 0 0 52 21
______H_______ -3 16 42 15 O-27 O-48 .42 11
39 26 33 34-18-22 12-43 31 20 24-19 25-23-30
For the 8th element we have ---T1--49-27-20 27 21 48 -20-48 28
_____G_____ _-____H_______ ---T2--38-19 43 9 19-28-48 -3-33 26-39-16 19-46-31 -19-46-31 0 -33 56-61-13 -6 43 31-28 -28 31-35 27 -26 61 9-28 37 48 O-27 21 39 13-28-35-54 -16 -6-37 54 59
For the 9th element we have _____G_____ ______H_______ ---T2-----T1--49 27-20 38 19 43 9-19-28 48 -3-33-26 39-16 -27 21-48 -19-46 31 19-46 31 0 -33 56 61 13 -6 -20 48 28 43-31-28 -28-31-35-27 26-61 9-28-37 -48 0 27 21 -39-13-28-35 54 -16 -6 37-54 59 For the 10th element we have _____G_____ ---T2-----T1--24-19-50 12 27-50 -43 31-12 27 36 46 0 52-21 0 0 0 52 21 43-31 28 -20-48-28 24-19-20 48 -36-46 0 0
-_____H_______ -3 42 0 0 -42
16-39-26 33 15-34 18-22 27 12-43-31 48 20 24 19 11-25 23-30
For the 11th element we have _____G_____ ---T2-----T1--24 36 43 12 52-20 -43 0 24-36 -3 42 0 -19 46-31 27-21-48 31 O-19-46 16 15 27 -50 0 28 -50 O-28 -12 52-20 0 -39-34 12 27 21 48 0 -26 18-43 33-22-31
O-42 48 11 20-25 24 23 19-30
For the 12th element we have
For the 4th element we have _____G_____
_____G_____ ---T1-----T2--24 19-50 12-27-50 -43-31-12-27 -36 46 0 -52-21 0 0 O-52 21 43 31 28 -20 48-28 24 19-20-48 36-46 0 0
______H_______
---T2-----T1--40-14 43 -40-41-20 -17 41-38 19 -3-33-26 39-16 41 5-31 -14 5-48 14 -5-19 46 16 2-29 13 47 -20 48 28 43-31-28 49 27 17 lh -39 32 17 28 13 27-21-41 -5 -26-37 28-17 29 33 -7 37 32 2
______H_______ _____G_____ ---T2-----T1--40-41-20 -40 14 43 -17-14 49-27 -3 16 39 26 33 14 5-48 41 5 31 -41 -5-27-21 -33 2-32 37 -7 43 31 28 -20 48-28 -38 19 17 41 26 29 17 28-37 -19 46-14 -5 -39-13 28-17-32 -16 47-13-29 2
Superlattices
and Microstructures,
For the 13th element we have
For the 21st element we have
__-__G_____
---T1--‘ ---T2--40 14 43 -40 41-20 -17-41-38-19 -41 5 31 14 5 48 -14 -5 19 46 -20-48 28 43 31-28 49-27 17-14 -27-21 41 -5
______H_______
-3-33 26-39-16 16 2 29-13 47 39-32 17 28-13 26 37 28-17-29 33 -7-37-32 2
For the 14th element we have ---T1‘--46-36 36-46 0
0
0 0 1
---T2--21-52 52 21 0
0
393
Vol. 3, No. 4, 1987
---T1--38 19-43 -19-46-31 -43 31-28
_____G_____
0 -46-36 0 0 0 36-46 0 0 1 0 0 21 52 0 O-52 21
_____G_____ ---T1-----T2---46 36 0 21 52 0 -46 36 0 0 -36-46 0 -52 21 0 -36-46 0 0 0 01 0 01 0 0 21-52 0 05221
______H_______ 10 0 0 0 02152 0 0 O-52 21 0 0 0 0 O-46 36 0 0 O-36-46
______H_______ 10 0 0 0 0 21-52 0 0 0 52 21 0 0 0 0 o-46-36 0 0 0 36-46
For the 16th element we have _____G_____
---Tl-----T2--12 27 50 24 19 50 -20-48-24 19 52-21 0 -36 46 0 0 O-36-46 20 48-28 -43-31 28 12 27-43 31 -52 21 0 0
______H_______ -3-33 42-30 0 19 0 31 -42-22
26-39-16 23 25 11 24-20 48 43 12-27 18 34 15
For the 17th element we have _____G_____
______H_______
---T2--24-36-43 -20 0 12-52 -3 42 0 O-42 19 46-31 -48 0 27 21 -33-30 19 31-22 50 0 28 -24-36-43 0 26 23 24 43 18 19-46 31 0 -39 25-20 12 34 -16 11 48-27 15
_____G_____
______H_______
17-14-49-27 -3 lb-39-26 33 41 -5 27-21 -33 2 32-37 -7 38 19-17 41 26 29-17-28-37 19 46 14 -5 -39-13-28 17-32 -16471329 2 For the 19th element we have _____G_____
---T1----‘T2---40-14-43 40-41 20 17 41 38 -41 5 31 14 5 48 -14 -5 19 20 48-28 -43-31 28 -49 27-17 -27-21 41
-3 16 39 26 33
16-39-26 33 59-54-37 -6 54-35 28-13 37 28 9 61 -6 13-61 56
_____c_____ ---T2---‘-Tl---40 14-43 40 41 20 17-41 38-19 41 5-31 -14 5-48 14 -5-19 46 20-48-28 -43 31 28 -49-27-17-14 27-21-41 -5
______H_______ -3-33-26 39-16 16 2-29 13 47 39-32-17-28-13 26 37-28 17-29 33 -7 37 32 2
For the 23rd element we have ---T1-‘-40 41 20 14 5-48 -43-31-28
_____G_____ ______H_______ ---T2--40-14-43 17 14-49 27 -3 lb 39 26 33 41 5 31 -41 -5-27-21 -33 2-32 37 -7 20-48 28 38-19-17-41 -26-29-17-28 37 -19 46-14 -5 39 13-28 17 32 -16 47-13-29 2
For the 24th element we have _____G_____ ______H_______ ---T2-----T1--12-52 20 24 36-43 -20 0 12 52 -3 42 0 O-42 -27-21-48 -19 46 31 48 O-27 21 -33-30-19-31-22 50 O-28 50 0 28 -24 36-43 0 -26-23 24 43-18 -19-46-31 0 39-25-20 12-34 -16 11-48 27 15 For the 25th element we have _____G_____ ---T2------G--12-27 50 24-19 50 -20 48-24-19 -52-21 0 36 46 0 0 0 36-46 20-48-28 -43 31 28 12-27-43-31 52 21 0 0
______H_______ -3-33-26 39-16 42-30-23-25 11 o-19 24-20-48 O-31 43 12 27 -42-22-18-34 15
For the 26th element we have
For the 18th element we have ---Tl-----T2---40-41 20 40 14-43 -14 5 48 -41 5-31 -43 31-28 20 48 28
______H______-
For the 22nd element we have
For the 15th element we have
---T1--12 52 20 27-21 48 50 O-28
_____G_____ .--T2--49-27 20 -35 27 28 31 27 21-48 -27 21-48 0 20 48 28 28 48 9-19 -31 0 19-46
______H_______
19 -3-33 26-39-16 46 lb 2 29-13 47 14 -39 32-17-28 13 -5 -26-37-28 17 29 33 -7-37-32 2
For the 20th element we have _____G_____
_____G_____ ______H_______ ---T2--‘-‘T1--a-41 43 51 14-20 35 27 40 14 -3 lb 39 26 33 41 -5-31 -14 -5-48 -27-21 41 -5 16-58 57-44 4 43 31 28 -20 48-28 40-41 -9-19 -39-57 -5 0 60 -14 -5 19 46 -26 44 0 -5-55 33 4-60 55 62 For the 27th element we have ---T1-----T2--8 41 43 51-14-20 -41 -5 31 14 -5 48 43-31 28 -20-48-28
_____G-____
______H_______
35-27 40-14 27-21-41 -5 40 41 -9 19 14 -5-19 46
-3 16-39-26 33 16-58-57 44 4 39 57 -5 O-60 26-44 0 -5 55 33 4 60-55 62
For the 28th element we have ______H_______
--‘Tl-----T2--38-19-43 49 27 20 -35-27 28-31 -3 16 39 19-46 31 -27 21 48 27 21 48 0 16 59 54 -43-31-28 20-48 28 28-48 9 19 -39-54-35 31 O-19-46 -26-37 28 33 -6-13
26 33 37 -6 28 13 9-61 61 56
---T1---28-31 43 48 0 31 -20 48 28
_____G_____ ---T2--28 48-20 -17 41 8-41 31 0 48 -14 5-41 5 43-31-28 -51-14 17 14 14 541 5
_____-H_______ -3-33-26 39-16 16 -4 18-25-45 39-25-28-20 25 26-18 43 28 18 33 63-18 25 -4
394
Superlattices
For the 29th element we have _____G_---_
_____G_____
---T1-----T2---12-52-20 -24 36 43 20 O-12 52 -3 42 0 O-42 27-21 48 19 46-31 -48 0 27 21 -33-30 19 31-22 -50 0 28 -50 O-28 24 36 43 0 -26-23-24-43-18 19-46 31 0 39-25 20-12-34 -16 11 48-27 15
______H_______
-3-33-26 39-16 42-30-23-25 11 0 19-24 20 48 0 31-43-12-27 -42-22-18-34 15
_____G_____
-3-33 26-39-16 42-30 23 25 11 O-19-24 20-48 O-31-43-12 27 -42-22 18 34 15
For the 33rd element we have
_____G-____ --____H_______ ---T2-----Tl---3 42 0 O-42 -24 36-43 -12 52 20 43 O-24-36 19 46 31 -27-21 48 -31 0 19-46 16 15-27-48 11 50 O-28 50 0 28 12 52 20 0 -39-34-12-20-25 -27 21-48 0 -26 18 43-24 23 33-22 31-19-30
---T2-----Tl--28-48 20 -28-31-43 -31 0 48 48 O-31 -43-31-28 20-48 28
20 O-12-52 -3 42 0 O-42 48 O-27 21 -33-30-19-31-22 24-36 43 0 26 23-24-43 18 -19-46-31 0 -39 25 20-12 34 -16 11-48 27 15
For the 34th element we have ___T ___ _____G_____ 1 ---T2---28-48-20 28-31 43 -17 14-51-14 -3 16-39-26 33 31 O-48 -48 0 31 -41 5 14 5 -33 -4 25 18 63 43-31 28 -20-48-28 8 41 17-41 26-18-28 43 18 41 5-14 5 -39 25-20 28-25 -16-45-25-18 -4 For the 35th element we have _____G_____
---%-----T2---28 31 43 28-48-20 -17-41 8 41 -3-33 26-39-16 -48 O-31 -31 O-48 14 5 41 5 16 -4-18 25-45 -20-48 28 43 31-28 -51 14 17-14 -39 25-28-20-25 -14 5-41 5 -26 18 43 28-18 33 63 18-25 -4 For the 36th element we have ---T2-----Tl---3-33 26-39-16 51-14 20 8-41-43 -9-19-40-41 4 14 -5-48 41 -5 31 19 46 14 -5 -33 62-55-60 20 48-28 -43-31 28 -40-14 35-27 -26 55 -5 O-44 41 -5 27-21 39 60 0 -5-57 -16 4 44 57-58
_____G_____
______H_______
17 14 51-14 -3 16 39 26 33 41 5-14 5 -33 -4-25-18 63 26-18 28-43 18 -8 41-17-41 -41 5 14 5 -39 25 20-28-25 -16-45 25 18 -4
For the 41st element we have
_____G_____
_____G_____
-3 16-39-26 33 42 15-34 18-22 O-27-12 43 31 o-48-20-24-19 -42 11-25 23-30
For the 40th element we have
For the 32nd element we have
---Tl-----T2---12 52-20 -24-36 43 -27-21-48 -19 46 31 -50 0 28 -50 O-28
_____c-____ ---Tl-----T2---24 19 50 -12-27 50 43-31 12-27 36 46 0 52-21 0 0 0 52 21 -43 31-28 20 48 28 -24 19 20-48 -36-46 0 0
For the 39th element we have
For the 31st element we have
---Tl-----T2---12-27-50 -24-19-50 20 48 24-19 52-21 0 -36 46 0 0 O-36-46 -20-48 28 43 31-28 -12-27 43-31 -52 21 0 0
____-(___-____H_ ---T2---‘-Tl--51 14 20 8 41-43 -9 19-40 41 -3-33-26 39-16 -14 -5 48 -41 -5-31 -19 46-14 -5 -33625560 4 20-48-28 -43 31 28 -40 14 35 27 26-55 -5 0 44 -41 -5-27-21 -39-60 0 -5 57 -16 4-44-57-58 For the 38th element we have
For the 30th element we have
---%-----T2---12 27-50 -24 19-50 20-48 24 19 -52-21 0 36 46 0 0 0 36-46 -20 48 28 43-31-28 -12 27 43 31 52 21 0 0
Vol. 3, No. 4. 7987
For the 37th element we have ___-__H_______
---T1-----T2---28 48-20 28 31 43 -17-14-51 14 -3 16 39 26 33 -31 0 48 48 O-31 41 5-14 5 -33 -4-25-18 63 43 31 28 -20 48-28 8-41 17 41 -26 18-28 43-18 -41 5 14 5 39-25-20 28 25 -16-45 25 18 -4
_____G_____
and Microstructures,
---T2-----Tl--28-31-43 -28 48 20 -48 O-31 -31 O-48 20 48-28 -43-31 28
_____G_____ 17 41 -8-41 14 5 41 5 51-14-17 14 -14 5-41 5
______H_______
-3-33 26-39-16 16 -4-18 25-45 39-25 28 20 25 26-18-43-28 18 33 63 18-25 -4
For the 42nd element we have _____G_____ ---T2-----Tl--28 31-43 -28-48 20 17-41 -8 41 -3-33-26 39-16 48 0 31 31 0 48 -14 5-41 5 16 -4 18-25-45 20-48-28 -43 31 28 51 14-17-14 .39 25 28 20-25 14 5 41 5 -26 18-43-28-18 33 63-18 25 -4 For the 43rd element we have _____G_____ ---T2-----Tl--28 48 20 -28 31-43 17-14 51 14 -3 16-39-26 33 31 O-48 -48 0 31 -41 5 14 5 -33 -4 25 18 63 -43 31-28 20 48 28 -8-41-17 41 -26 18 28-43-18 41 5-14 5 39-25 20-28 25 -16-45-25-18 -4 For the 44th element we have +--Tl-----T2---24-36-43 -12-52 20 -19 46-31 27-21-48 50 O-28 50 0 28
_____G_____ 43 O-24 36 31 O-19-46 12-52 20 0 27 21 48 0
-3 42 0 O-42 lb 15 27 b8 11 39 34-12-20 25 26-18 43-24-23 33-22-31 19-30
Superlattices
and Microstructures,
For the 53rd element we have
For the 45th element we have _____G_____ ---T1-----T2---24-19 50 -12 27 50 43 31 12 27 -36 46 0 -52-21 0 0 O-52 21 -43-31-28 20-48 28 -24-19 20 48 36-46 0 0
______H_______ -3 42 0 0 -42
16 39 26 33 15 34-18-22 27-12 43-31 48-20-24 19 11 25-23-30
For the 46th element we have _____G_____
--‘T1-----T2---38 19 43 -49-27-20 35 27-28 31 19-46 31 -27 21 48 27 21 48 0 43 31 28 -20 48-28 -28 48 -9-19 31 O-19-46
-3 16 39 26 33
16 39 26 33 59 54 37 -6 54 35-28-13 37-28 -9 61 -6-13 61 56
For the 48th element we have ---%-----T2---28 O-50 28 O-50 50 O-28 0 o-1 0 o-1 0 0 0 0142 -50 0 28 -50 O-28 -28 O-50 0 010 0
---Tl---51 14-20 14 -5-48 -20-48 28
---T2---8 41 43 41 -5 31 43 31-28
9 19 40 41
-3 42 0 O-42 6 0 0 4 0 0 28 50 0 0 0 50-28 0 -42 4 0 0 6
_____G_____ ______H_______ ---T2-----T1---38-19 43 -49 27-20 35-27-28-31 -3 16-39-26 33 -19-46-31 27 21-48 -27 21-48 0 16 59-54-37 -6 43-31 28 -20-48-28 -28-48 -9 19 -39-54 35-28 13 -31 0 19-46 -26-37-28 -9-61 33 -6 13-61 56 For the 51st element we have
_____G_____ -9 19 19-46 28 31 -48 0
______H_____-_
28-48 -3-33-26 39-16 31 0 -33 56 61 13 -6 35 27 -26 61 -9 28 37 27 21 39 13 28 35-54 16 -6 37-54 59
---T2-----Tl--46 36 0 -21 52 0 36-46 0 52 21 0 0 o-1 0 o-1
_____G_____ 46 36 0 0 36-46 0 0 0 O-21-52 0 O-52 21
______H_______ 10 0 0 0 0 21 52 0 0 0 52-21 0 0 0 0 0 46-36 0 0 O-36-46
_____G_____ ---T2-----Tl---21 52 0 46-36 0 -21 52 0 0 52 21 0 -36-46 0 52 21 0 0 0 o-1 0 0 46 36 0 o-1 0 0 36-46
______H_______ 10 0 0 0 O-46-36 0 0 O-36 46 0 0 0 0 O-21-52 0 0 O-52 21
---Tl---10 0 010 0 o-1
---T2---10 0 010 0 o-1
_____G_____
______"____-__
-10 0 0 010 0 0 o-1 0 0 0 01
10 0 0 010 0 0 O-10 0 0 o-1 0 0 0 01
0 0 0 0
For the 59th element we have ______H_______ -3 42 0 O-42 6 0 0 4 0 o-28-50 0 0 O-50 28 0 -42 4 0 0 6
For the 52nd element we have _____G_____
---T2-----Tl---49-27 20 -38-19-43 -27 21-48 -19-46 31 20-48-28 -43 31 28
For the 58th element we have
For the 50th element we have
---T2-----Tl--28 0 50 -28 0 50 -50 0 28 0 o-1 0 0 0 0142 o-1 0 50 O-28 50 0 28 28 0 50 0 010 0
-3 16 39 26 33 16-58 57-44 4 39 57 5 O-60 26-44 0 5 55 33 4-60 55 62
For the 57th element we have ______H_______
19 40 41 -3-33 26-39-16 46 14 -5 -33 62-55-60 4 14-35 27 26-55 5 0 44 -5 27-21 -39-60 0 5 57 -16 4 44 57-58
_____G_____
______H_______
For the 56th element we have ______H_______
For the 49th element we have _____G_____
_____G_____ ---T2-----Q---8 41-43 -51-14 20 -35-27-40-14 41 -5-31 -14 -5-48 -27-21 41 -5 20-48 28 -40 41 9 19 -43-31-28 -14 -5 19 46
For the 55th element we have ______H_______
-‘-Tl-----T2---51-14-20 -8-41 43 -3-33-26 39-16 9-19 40-41 -14 -5 48 -41 -5-31 -19 46-14 -5 -33 62 55 60 4 -20 48 28 43-31-28 40-14-35-27 -26 55 5 O-44 -41 -5-27-21 39 60 0 5-57 -16 4-44-57-58
_____G_____
_____G_____ ______H_______ ---T2-----T1---8-41-43 -51 14 20 -35 27-40 14 -3 16-39-26 33 -41 -5 31 14 -5 48 27-21-41 -5 16-58-57 44 4 -43 31-28 20 48 28 -40-41 9-19 -39-57 5 0 60 14 -5-19 46 -26 44 0 5-55 33 4 60-55 62 For the 54th element we have
______H_______
For the 47th element we have _____G_____
395
Vol. 3, No. 4, 1987
---Tl---21-52 0 -52 21 0 0 o-1
_____G_____ ---T2--46 36 0 -21-52 0 0 36-46 0 -52 21 0 0 0 o-1 0 0 46-36 0 O-36-46
______H_______ 10 0 0 0 O-46 36 0 0 0 36 46 0 0 0 0 O-21 52 0 0 0 52 21'
For the 60th element we have ______H_______
---T2-----T1---49 27 20 -38 19-43 -9-19 28 48 -3-33 26-39-16 27 21 48 19-46-31 -19-46-31 0 -33 56-61-13 -6 20 48-28 -43-31 28 28-31 35-27 26-61 -9 28-37 48 O-27 21 -39-13 28 35 54 -16 -6-37 54 59
_____G_____ ---T2---"T1---21-52 0 46-36 0 0 46-36 0 -36-46 0 -52 21 0 -36-46 0 0 0 o-1 0 o-21 52 0‘0 -1 0 05221
______H_______ 10 0 0 0 0 21-52 0 0 O-52-21 0 0 0 0 0 46 36 0 0 0 36-46
396
Superlattices
numbers in irreducible Here the the are symbols with the representation matrices following meanings: O-O, l-l, 2-l/10, 3-l/5, 4-2/5, 5-l/2, 6-3/5.
and Microstructures,
Acknowledgment -- The authors are grateful Professor John D. Dow for his assistance preparing the manuscript.
to in
REFERENCES "Quantum Theory of Molecules and Solids" McGraw-Hill Book Company,Inc.(1963). [7-lD. Shechtman, I. Blech, D. Gratias, and J. W. (1984); D. Cahn, Phys. Rev. Lett. 12, 1951 Levine and P. J Steinhardt, Phys.s= ---A 53, 2477 (1984). Quantum [31 L. D. Landau and E. M. Lifshitz, Mechanics, (Pergamon Press Ltd., London, 1962), p. 328. Point Group [41 J. A. Salthouse and M. J. Ware, Character Tables and Related Data (Cambridge University Press, Cambridge,72)
111 J:C.Slater
Appendix
C5,l '5,2 C5,3 C5,4 C5,5 C5.6 $1 n-
specified
by
their
(0, 0, 1);
(sin A cos 36'. sin A sin 36', cos A); (-sin A cos 72', sin A sin 72', cos A); (-sin A, 0, cos A); (-sin A cos 72',-sin A sin 72', cos A); (sin A cos 36',-sin A sin 36', cos A); (sin B, 0, cos B); (cos 72' sin B, sin 72' sin B, cos B); (-cos 36O sin B, sin 36' sin B, cos B) C3'4 - (-cos 36' sin B,-sin 36' sin B, cos B) C3'5 - (cos 72' sin B,-sin 72' sin B, cos B); '3'6 - (sin C, 0, cos C); C3'7 - (cos 72' sin C, sin 72' sin C, cos C); '3'8 - (-cos 36' sin C, sin 36O sin C, cos C) sin C,-sin 36' sin C, cos C) C3'9 - (-cos 36' n C3'10 - (cos 72" sin C,-sin 72" sin C, cos C); C2'1 - (cos 36' sin D, sin 36' sin D, cos D); C2'2 - (-cos 72' sin D, sin 72' sin D, cos D); '2'3 - (-sin D, 0, cos D); '2'4 - (-cos 72' sin D,-sin 72' sin D, cos D); '2'5 - (cos 36' sin D,-sin 36' sin D, cos D); '2'6 - (sin E, 0, cos E); '2'7 - (cos 72' sin E, sin 72' sin E, cos E); - (-cos 36' sin E, sin 36' sin E, cos E); '2:8 - (-cos 36' sin E,-sin 36O sin E, cos E); C2,9 C2,lO - (cos 72' sin E,-sin 72O sin E, cos E); C2,ll - (sin 72', cos J2', 0) C2,12 - (sin 36', cos 36", 0); '2,13 - (0, 1, 0): cos 36', 0); and '2,14 - (-sin 36', 0 '2,15 - (-sin 72 ( cos 72 O, 0).
c;l; -
Here we have: ~-63.43'; B-37.38'; C-79.18'; D-31.72O: and E-58.29O.
I: Notation
of the element means the n m,p rotation by Psp/n about the counter-clockwise mth order axis C, m. For n-5, we have m-l, 2, . . . , 6; for n-3, we have m-l, 2, . ... 10; for 2, . . . . n-2, we have -1, 2, . . . . 15; and p-l, n-l. C
4. 1987
The elements of the group I are assigned numerical symbols as follows (E-l means that E is assigned the numerical symbol 1.):
The rotation axes can be direction cosines:
57-(5+2J5)li2 10, 58-(2J5-1)/l?, 59-(7+J5)/20, 60-(10-2J5)1/4,5 61-(50+22/5) j2/20, 62-(2/5+1)/10, and 63-(3/5-l)/lO. As a result -19 stands for -[(5-J5)/10]1/2/2, for example. The point group Ih can be considered as the direct product of the point group Ci and the point group I: Ih-IxC . The 120 elements of the point group Ih can c e divided into 10 classes. Therefore the group Ih will have 10 different irreducible representations. After we have given the irreducible representation matrices of the point group I, the irreducible representations are composed of of the point group Ih, which 1200 matrices can be easily obtained.
Vol. 3, No
Appendix
II: Multiplication
Table
The multiplication table for the elements (symbolized by numbers) can be written as a matrix Mi.i, where i and j run from 1 to 60. That matrig is:
Superlattices
and Microstructures,
397
Vol. 3, No. 4. 1987
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 14 1 26 8 28 10 30 12 32 4 34 6 15 3
3 1 15 11 29 13 31 5 33 7 35 9 27 2 14
4 28 9 16 1 38 12 6 36 2 43 26 11 47 31
5 10 27 1 17 8 29 42 3 39 13 7 37 34 50
6 30 11 28 13 18 1 40 4 8 38 2 45 48 33
7 12 29 9 39 1 19 10 31 44 3 41 5 26 46
8 32 13 2 37 30 5 20 1 42 6 10 40 49 35
9 4 31 43 7 11 41 1 21 12 33 36 3 28 47
10 34 5 12 42 2 39 32 7 22 1 44 8 50 27
11 6 33 38 3 45 9 13 43 1 23 4 35 30 48
12 26 7 36 10 4 44 2 41 34 9 24 1 46 29
13 8 35 6 27 40 3 37 11 5 45 1 25 32 49
14 15 2 50 30 46 32 47 34 48 26 49 28 3 1
15 3 14 35 47 27 48 29 49 31 50 33 46 1 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 52 36 17 4 57 26 38 51 28 56 46 43 19 41
17 39 51 5 16 42 46 57 27 52 37 29 56 44 24
18 53 38 47 45 19 6 59 28 40 52 30 58 21 43
19 41 52 31 58 7 18 44 47 59 29 53 39 36 lb
20 54 40 32 60 48 37 21 8 56 30 42 53 23 45
21 43 53 54 41 33 60 9 20 36 48 56 31 38 18
22 55 42 44 54 34 57 49 39 23 10 58 32 25 37
23 45 54 58 33 55 43 35 57 11 22 38 49 40 20
24 51 44 60 34 36 55 26 59 50 41 25 12 17 39
25 37 55 40 50 60 35 51 45 27 59 13 24 42 22
26 46 12 51 2 16 34 28 24 14 36 50 4 29 7
27 5 50 13 46 37 15 17 35 29 25 3 51 10 34
28 47 4 46 6 52 2 18 26 30 16 14 38 31 9
29 7 46 3 52 5 47 39 15 19 27 31 17 12 26
30 48 6 14 40 47 8 53 2 20 28 32 18 33 11
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
51 37 52 39 53 41 54 43 55 45 50 13 46 5 47
38 52 40 53 42 54 44 55 36 51 4 46 6 47 8
17 5 58 44 40 60 32 54 50 35 46 3 52 7 18
4 16 30 47 54 33 58 45 41 60 12 51 2 46 32
46 27 19 7 60 36 42 57 34 55 14 35 47 3 53
43 57 6 18 32 48 55 35 60 37 36 17 4 52 2
26 51 47 29 21 9 57 38 44 59 34 25 14 27 48
57 39 45 59 8 20 34 49 51 27 16 29 38 19 6
36 56 28 52 48 31 23 11 59 40 24 37 26 17 14
52 29 59 41 37 56 10 22 26 50 28 15 18 31 40
56 42 38 58 30 53 49 33 25 13 51 5 16 39 28
28 46 53 31 56 43 39 58 12 24 2 50 30 15 20
25 40 17 42 19 44 21 36 23 38 35 6 27 8 29
45 18 37 20 39 22 41 24 43 16 11 28 13 30 5
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
5 1 22 34 45 25 30 20 15 33 29 9 39 12 58
1 4 32 14 23 35 18 40 31 21 7 36 10 26 22
15 35 7 1 24 26 37 17 32 22 48 23 31 11 41
33 23 1 6 34 14 25 27 20 42 21 57 9 38 12
34 24 15 27 9 1 lb 28 39 19 22 59 49 25 33
22 41 44 21 35 26 25 lb 115 8 29 26 11 14 1 17 18 29 30 57 59 19 20 23 24 59 56 11 50
19 31 24 36 27 17 1 10 28 14 18 48 59 21 25
20 32 43 23 28 18 15 31 13 1 37 10 56 22 16
30 14 21 33 16 38 29 19 1 12 8 34 20 49 56
37 8 57 22 18 59 48 21 35 11 27 1 17 10 52
6 28 20 48 57 23 19 59 9 36 1 26 8 14 42
27 13 39 10 59 24 20 56 49 23 15 11 29 1 19
11 38 8 30 22 49 59 25 21 56 9 16 1 28 10
50 25 29 5 41 12 56 16 22 58 49 45 15 13 31
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
7 48 9 49 11 24 40 16 42 18 44 20 36 22 38
48 10 49 12 50 43 17 45 19 37 21 39 23 41 25
41 30 21 14 33 51 13 57 10 45 24 8 56 34 23
15 22 35 44 25 9 56 6 52 20 31 57 11 19 40
9 20 43 32 23 26 25 52 5 59 12 37 16 10 58
47 34 15 24 27 21 42 11 58 8 53 22 33 59 13
3 54 11 22 45 12 60 28 17 53 7 56 4 39 18
53 2 48 26 15 56 5 23 44 13 60 10 54 24 35
29 49 3 55 13 41 20 4 57 30 19 54 9 58 6
21 8 54 2 49 16 27 58 7 25 36 5 57 12 55
19 14 31 50 3 60 8 43 22 6 59 32 21 55 11
33 42 23 10 55 4 51 18 29 60 9 17 38 7 59
10 31 12 33 4 55 18 51 20 52 22 53 24 54 16
32 7 34 9 26 23 52 25 53 17 54 19 55 21 51
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
24 18 60 47 21 27 11 42 2 23 50 6 37 14 54
50 58 25 19 60 3 43 8 28 54 15 38 13 47 20
4 60 16 20 57 14 55 29 13 44 2 25 46 8 39
28 24 46 60 17 48 22 3 45 10 30 55 15 40 5
13 40 17 43 4 35 6 30 5 57 lb 45 18 47 8 10 42 19 41 7 48 14 49 12 51 24 43 31 3 14 5 37 52 36 12 33 2 32 7 17 51 38 47 15 2
56 13 42 6 32 52 15 44 9 50 lb 3 39 4 34
58 46 19 27 7 40 2 21 49 4 45 14 53 35 9
23 17 58 5 44 6 26 53 15 36 11 46 18 3 41
44 47 41 15 9 25 6 56 32 38 55 30 60 49 43
49 39 55 7 24 11 16 40 47 56 33 52 45 31 60
12 53 36 48 43 50 45 17 8 58 34 40 51 32 57
14 44 50 41 51 33 57 13 18 42 48 58 35 53 37
1 21 4 54 38 34 59 46 37 19 10 60 26 42 52
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
27 29 31 33 35 25 17 19 21 23 60 56 57 58 59
28 30 32 34 26 16 18 20 22 24
29 19 53 48 15 27 39 59 20 49 37 42 22 55 25
26 14 49 55 24 36 28 48 23 59 43 38 18 53 21
15 31 21 54 49 50 29 41 56 22 51 17 39 44 24
16 28 14 50 51 56 38 30 49 25 54 23 45 40 20
50 15 33 23 55 24 46 31 43 58 36 16 52 19 41
52 18 30 14 46 17 58 40 32 50 42 22 55 25 37
51 46 15 35 25 60 16 47 33 45 21 43 38 18 53
47 53 20 32 14 46 19 60 42 34 17 39 44 24 51
17 52 47 15 27 37 57 18 48 35 20 54 23 45 40
14 48 54 22 34 26 47 21 57 44 16 52 19 41 36
13 5 7 9 11 45 37 39 41 43 59 60 56 57 58
6 8 10 12 4 38 40 42 44 36 58 59 60 56 57
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
7 44 59 53 31 3
12 34 55 59 41 9 2 49 45 53 11 6 30 48 33
33 43 55 9 4 35 36 26 11 56 51 38 54 56 58 49 54 44 3 11 50 12 2 3 51 50 4 42 37 52 50 49 12 27 35 26 5 13 46 10 8 29 34 32 7
53 60 37 8 30 47 41 51 5 2 29 7 12 26 46
42 57 52 29 5 8 54 38 47 3 30 48 33 11 6
32 54 57 39 10 2 48 43 52 7 28 47 31 9 4
5 39 19 31 3 13 42 58 53 33 40 20 54 23 45
2 32 22 44 12 4 30 54 58 41 38 18 53 21 43
3 7 41 21 33 35 5 44 60 54 25 37 42 22 55
4 2 34 24 36 43 6 32 55 60 23 45 40 20 54
35 3 9 43 23 55 27 7 36 57 24 51 17 39 44
57
58 59 60 56
9 10 11 12 13 14 15
10
55 40 48 13 8 32 49 35
58 45 6 28 52 39 55 13 2 46 10 34 50 27 5
60 51 27 13 40 53 36 46 3 6 31 9 4 28 47
398
Superlattices 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
1 2 3 4 5 6
and Microstructures,
Vol. 3, No. 4, 1987
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
8 9 10 11 12 13 14 15
31 9 47 33 19 3 53 7 48 41 15 21 29 4 28
32 49 8 34 20 14 42 48 10 54 2 22 30 35 13
33 11 48 23 31 35 21 3 54 9 49 43 15 6 30
34 50 10 24 32 26 22 14 44 49 12 55 2 27 5
35 13 49 45 15 25 33 27 23 3 55 11 50 8 32
36 16 41 56 12 43 24 4 60 26 21 51 9 52 19
37 42 25 8 51 20 27 56 13 17 40 5 60 22 55
38 18 43 52 11 58 4 45 16 6 57 28 23 53 21
39 44 17 7 57 10 52 22 29 58 5 19 42 24 51
40 20 45 30 25 53 13 60 6 37 18 8 59 54 23
41 36 19 21 44 9 59 12 53 24 31 60 7 16 52
42 22 37 10 56 32 17 54 5 57 8 39 20 55 25
43 38 21 57 9 23 36 11 56 4 54 16 33 18 53
44 24 39 41 22 12 58 34 19 55 7 59 10 51 17
45 40 23 18 35 59 11 25 38 13 58 6 55 20 54
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
46 29 26 27 28 17 14 52 50 47 51 15 16 7 12
47 31 28 15 18 29 30 19 14 53 46 48 52 9 4
48 33 30 49 53 15 20 31 32 21 14 54 47 11 6
49 35 32 55 48 50 54 15 22 33 34 23 14 13 8
50 27 34 25 14 51 49 46 55 15 24 35 26 5 10
51 17 24 37 26 56 50 16 25 46 60 27 36 39 44
52 19 16 29 38 39 28 58 46 18 17 47 57 41 36
53 21 18 48 59 31 40 41 30 60 47 20 19 43 38
54 23 20 22 21 49 56 33 42 43 32 57 48 45 40
55 25 22 59 49 24 23 50 58 35 44 45 34 37 42
56 57 60 42 36 54 51 43 37 16 20 17 21 58 59
57 58 56 39 43 22 16 23 17 38 42 52 54 59 60
58 59 57 19 23 44 38 55 52 45 39 18 22 60 56
59 60 58 53 55 41 45 24 18 25 19 40 44 56 57
60 56 59 20 24 21 25 36 40 51 53 37 41 57 58
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
23 58 13 40 10 32 24 50 56 17 43 52 11 18 1
24 60 46 17 31 7 43 4 58 18 55 40 50 37 15
58 19 25 60 5 42 12 34 16 46 38 47 45 53 13
60 20 16 57 47 19 33 9 45 6 25 8 51 42 46
18 47 60 21 17 57 7 44 4 26 6 14 40 48 37
42 10 23 55 6 40 14 48 27 3 17 7 57 44 38
2 26 48 15 43 11 52 18 7 41 10 24 32 50 54
29 3 44 12 25 51 8 42 14 49 47 33 19 9 59
9 43 2 28 49 15 45 13 53 20 41 56 12 16 34
14 50 31 3 36 4 17 52 10 44 32 55 48 35 21
54 22 11 45 2 30 50 15 37 5 56 39 43 58 4
12 36 14 46 33 3 38 6 19 53 44 60 34 51 49
39 7 55 24 13 37 2 32 46 15 52 31 58 41 45
21 54 4 38 14 47 35 3 40 8 60 42 36 57 26
47 15 41 9 51 lb 5 39 2 34 30 49 53 33 60
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
13 6 42 32 58 55 53 60 33 43 3 4 5 2 39
35 45 5 8 44 34 60 51 54 57 33 38 3 6 7
55 59 27 37 7 10 36 26 57 52 23 18 35 40 3
59 53 51 56 29 39 9 12 38 28 45 30 25 20 27
40 30 56 54 52 58 31 41 11 4 13 2 37 32 17
8 2 54 49 38 45 47 53 3 9 5 12 42 34 57
3 11 10 2 55 50 40 37 48 54 31 43 7 4 44
49 55 3 13 12 2 51 46 42 39 54 58 33 45 9
44 41 50 51 3 5 4 2 52 47 58 53 55 60 35
53 48 36 43 46 52 3 7 6 2 40 32 60 54 51
10 12 49 50 11 13 28 30 29 31 39 41 22 24 23
7 9 34 26 35 27 6 8 47 48 19 21 44 36 55
31 33 12 4 50 46 13 5 30 32 53 54 41 43 24
48 49 9 11 26 28 27 29 8 10 20 22 21 23 36
32 34 33 35 4 6 46 47 5 7 42 44 54 55 43
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
30 12 14 36 46 54 39 35 59 5 20 44 49 60 27
5 33 1 23 6 44 53 26 56 47 39 21 12 57 28
20 1 32 4 14 57 29 55 41 27 56 7 22 36 50
39 15 7 35 1 59 30 36 54 28 58 48 41 23 4
54 5 22 1 34 38 46 59 31 51 43 29 58 9 24
59 28 53 46 31 37 1 54 34 11 25 2 20 50 33
35 36 57 40 45 56 39 30 59 54 146 36 35 30 39 46 1 21 55 3 26 16 13 6 17 29 2 53 22
27 60 52 23 6 50 13 20 29 58 28 25 40 48 5 7 17 53 21 3 32 2 22 9 16 12 23 48 35 2 29 51 18 43 149 142 31 52 26 45 30 49 1
43 37 57 8 22 28 50 19 3 24 4 27 52 1 44
31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
34 19 24 31 36 35 38 37 30 57 49 18 25 48 56
2 41 26 21 16 49 58 27 40 39 32 59 50 20 17
8 9 2 43 28 22 19 50 60 29 42 41 34 56 46
42 3 10 11 2 58 47 24 21 46 57 31 44 43 26
22 29 44 3 12 45 28 60 48 16 23 47 59 33 36
55 52 59 29 41 13 4 20 14 43 35 28 40 15 21
26 59 51 53 56 15 23 5 6 22 14 45 27 30 42
6 36 28 56 52 32 44 15 25 7 8 24 14 37 29
37 11 8 38 30 39 31 34 36 15 17 9 10 16 14
57 27 39 13 10 18 14 41 33 26 38 15 19 11 12
25 38 40 52 53 5 9 32 26 33 27 4 8 46 48
51 45 37 18 20 29 33 10 4 49 46 11 5 28 32
16 25 17 40 42 47 49 7 11 34 28 35 29 6 10
38 51 52 37 39 30 34 31 35 12 6 50 47 13 7
45 16 18 17 19 8 12 48 50 9 13 26 30 27 31
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
38251837,3039343136495724195648 6 27 40 17 120 58 49 41 26 33 2 3 8 29 42 18 23 60 50 43 26 11 2 3 10 47 58 20 25 57 16 45 28 13 2 29 44 48 60 22 57 59 52 40 28 5 12 15 21 34 45 51 59 56 53 22 14 7 4 15 8 29 37 52 56 45 33 24 14 9 34 9 10 31 39 30 38 37 35 16 51 38 26 11 12 15 19 32 40 39 22 41 39 53 52 8 4 27 33 26 55 36 44 21 19 32 28 5 11 46 25 16 24 43 41 49 47 10 6 29 37 52 51 38 36 35 31 34 30 7 42 19 17 18 16 13 9 50 48 12
50 35 45 59 41 26 15 11 18 9 4 28 47 31
21 56 42 32 14 31 36 17 10 46 29 7 12 26
46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
1 10 44 41 9 11 8 22 59 21 45 40 20 54 23
11 1 12 36 43 23 13 10 24 56 55 25 37 42 22
45 13 1 4 38 58 25 5 12 16 44 24 51 17 39
40 37 5 1 6 18 60 17 7 4 19 41 36 lb 52
8 42 39 7 1 6 20 57 19 9 18 53 21 43 38
10 22 58 19 7 1 32 23 18 31 6 30 48 33 11
9 12 24 60 21 33 1 34 25 20 35 13 8 32 49
23 11 4 16 57 22 35 1 26 17 34 50 27 5 10
59 25 13 6 18 19 24 27 1 28 7 12 26 46 29
20 56 17 5 8 30 21 16 29 1 47 31 9 4 28
44 55 45 18 19 7 34 35 6 47 1 2 14 15 3
41 24 25 40 53 31 12 50 13 30 3 1 2 14 15
21 36 51 37 20 48 9 26 27 8 15 3 1 2 14
54 43 16 17 42 32 33 4 46 5 14 15 3 1 2
22 23 38 52 39 10 49 11 28 29 2 14 15 3 1
7
46 11 18 55 56 26 27 38 47 59 42 51 37 58 29 31 2 20 54 24 10 147 33 38 27 55 32 41 1 47 140 23 51 34 3 28 48 18 5 25 8 19 3
38 28 46 17 42 23 6 14 27 32 49 35 13 8
59 40 30 47 29 44 25 8 14 5 10 34 50 27
16 46 27 37 20 43 28 15 13 48 33 11 6 30