The diffraction from one- and two-dimensional aperiodic structures is studied by using Fibonacci and other aperiodic gratings produced by several methods. By examining the laser diffraction patterns obtained from these gratings, the effects of aperiodic order on the diffraction pattern was observed and compared to the diffraction from real quasicrystalline surfaces. The correspondence between diffraction patterns from two-dimensional gratings and from real surfaces is demonstrated.

1.
R.
McGrath
,
J.
Ledieu
,
E. J.
Cox
, and
R. D.
Diehl
, “
Quasicrystal surfaces: Structure and potential as templates
,”
J. Phys.: Condens. Matter
14
,
R119
R144
(
2002
).
2.
R.
Penrose
, “
Pentaplexity
,”
Math. Intell.
2
,
32
37
(
1979
).
3.
T. H. Garland, Fascinating Fibonaccis: Mystery and Magic in Numbers (Dale Seymour, Palo Alto, CA, 1987).
4.
R. McGrath, J. Ledieu, E. J. Cox, and R. D. Diehl, “Tilings and coverings of quasicrystal surfaces,” in Coverings of Discrete Quasiperiodic Sets: Theory and Applications to Quasicrystals, edited by P. Kramer and Z. Papadopolos (Springer, Berlin, 2003), pp. 257–268.
5.
J.
Ledieu
,
R.
McGrath
,
R. D.
Diehl
,
T. A.
Lograsso
,
D. W.
Delaney
,
Z.
Papadopolos
, and
G.
Kasner
, “
Tiling of the fivefold surface of Al70Pd21Mn9,
Surf. Sci.
492
(
3
),
L729
L734
(
2001
).
6.
Z.
Papadopolos
,
G.
Kasner
,
J.
Ledieu
,
E. J.
Cox
,
N. V.
Richardson
,
Q.
Chen
,
R. D.
Diehl
,
T. A.
Lograsso
,
A. R.
Ross
, and
R.
McGrath
, “
Bulk termination of the quasicrystalline fivefold surface of Al70Pd21Mn9,
Phys. Rev. B
66
(
18
),
184207
184213
(
2002
).
7.
J. M.
Dubois
, “
The applied physics of quasicrystals
,”
Phys. Scr., T
T49
,
17
23
(
1993
).
8.
B. C.
Anderson
,
P. D.
Bloom
,
K. G.
Baikerikar
,
V. V.
Shear
, and
S. M.
Mallapragada
, “
Al–Cu–Fe quasicrystal/ultra-high molecular weight polyethylene composites as biomaterials for acetabular cup prosthetics
,”
Biomaterials
23
,
1761
1768
(
2002
).
9.
J. W. Cahn, “Epilogue,” in Proceedings of the 5th International Conference on Quasicrystals, edited by C. Janot and R. Mosseri (World Scientific, Singapore, 1996), p. 807.
10.
M. Senechal, Quasicrystals and Geometry (Cambridge U.P., Cambridge, 1995).
11.
R.
Lifshitz
, “
The rebirth of crystallography
,”
Z. Kristallogr.
217
,
342
343
(
2002
).
12.
R.
Lifshitz
, “
Quasicrystals: A matter of definition
,”
Found. Phys.
33
(
12
),
1703
1711
(
2003
).
13.
R.
Lifshitz
, “
The symmetry of quasiperiodic crystals
,”
Physica A
232
,
633
647
(
1996
).
14.
“Report of the Executive Committee for 1991, International Union of Crystallography,” Acta Crystallographica 48, 922–946 (1992).
15.
L.
Pauling
, “
Apparent icosahedral symmetry is due to directed multiple twinning of cubic crystals
,”
Nature (London)
317
,
512
514
(
1985
).
16.
R. D.
Diehl
,
J.
Ledieu
,
N.
Ferralis
,
A. W.
Szmodis
, and
R.
McGrath
, “
Low-energy electron diffraction from quasicrystal surfaces
,”
J. Phys.: Condens. Matter
15
(
3
),
R63
R81
(
2003
).
17.
C. Janot, Quasicrystals: A Primer (Clarendon, Oxford, 1992).
18.
M.
Gierer
,
M. A.
Van Hove
,
A. I.
Goldman
,
Z.
Shen
,
S.-L.
Chang
,
P. J.
Pinhero
,
C. J.
Jenks
,
J. W.
Anderegg
,
C.-M.
Zhang
, and
P. A.
Thiel
, “
Fivefold surface of quasicrystalline AlPdMn: Structure determination using low-energy electron diffraction
,”
Phys. Rev. B
57
,
7628
7641
(
1998
).
19.
N.
Ferralis
,
K.
Pussi
,
E. J.
Cox
,
M.
Gierer
,
J.
Ledieu
,
I. R.
Fisher
,
C. J.
Jenks
,
M.
Lindroos
,
R.
McGrath
, and
R. D.
Diehl
, “
Structure of the 10-fold d-Al-Ni-Co quasicrystal surface
,”
Phys. Rev. B
69
,
075410
(
2004
).
20.
D. P. Woodruff and T. A. Delchar, Modern Techniques of Surface Science (Cambridge U.P., Cambridge, 1986).
21.
M. A. Van Hove, W. H. Weinberg, and C.-M. Chan, Low-Energy Electron Diffraction: Experiment, Theory and Surface Structure Determination (Springer-Verlag, Berlin, 1986).
22.
R. Mosseri and J. F. Sadoc, “Two and three dimensional non-periodic networks obtained from self-similar tiling,” in The Structure of Non-Crystalline Materials 1982, edited by P. H. Gaskell, J. M. Parker, and E. A. Davis (Tayor and Francis, London, 1982), pp. 137–150.
23.
A. L.
Mackay
, “
Crystallography and the Penrose pattern
,”
Physica A
114
,
517
522
(
1982
).
24.
S. Y.
Litvin
,
A. B.
Romberger
, and
D. B.
Litvin
, “
Generation and experimental measurement of a one-dimensional quasi-crystal diffraction pattern
,”
Am. J. Phys.
56
(
1
),
72
75
(
1988
).
25.
M.
Boudard
,
M.
de Boussieu
,
C.
Janot
,
G.
Heger
,
C.
Beeli
,
H.-U.
Nissen
,
H.
Vincent
,
R.
Ibberson
,
M.
Audier
, and
J. M.
Dubois
, “
Bulk structure of AlPdMn
,”
J. Phys.: Condens. Matter
4
,
10149
10168
(
1992
).
26.
〈http://alpdmn.phys.psu.edu/gratings.html〉. Links to downloadable images of periodic and quasiperiodic diffractrion gratings.
27.
G. Harburn, C. A. Taylor, and T. R. Welberry, Atlas of Optical Transforms (G. Bell & Sons, London, 1975).
28.
〈http://rsc.anu.edu.au/∼welberry/Optical transform/〉. This site includes information on producing optical transforms as well as slides of various diffractrion gratings.
29.
A. R.
Smith
,
K.-J.
Chao
,
Q.
Niu
, and
C.-K.
Shih
, “
Formation of atomically flat silver films on GaAs with a ‘silver mean’ quasi periodicity
,”
Science
273
,
226
228
(
1996
).
30.
P.
Ebert
,
K.-J.
Chao
,
Q.
Niu
, and
C.-K.
Shih
, “
Dislocations, phase defects, and domain walls in a one-dimensional quasiperiodic superstructure of a metallic thin film
,”
Phys. Rev. Lett.
83
(
16
),
3222
3225
(
1999
).
31.
J.
Ledieu
,
J.-T.
Hoeft
,
D. E.
Reid
,
J.
Smerdon
,
R. D.
Diehl
,
T. A.
Lograsso
,
A. R.
Ross
, and
R.
McGrath
, “
Pseudomorphic growth of a single element quasiperiodic ultrathin film on a quasicrystal substrate
,”
Phys. Rev. Lett.
92
,
135507
(
2004
).
32.
R.
Lifshitz
, “
The square Fibonacci tiling
,”
J. Alloys Compd.
342
,
186
190
(
2002
).
33.
M.
Gierer
,
A.
Mikkelsen
,
M.
Gräber
,
P.
Gille
, and
W.
Moritz
, “
Quasicrystalline surface order on decagonal Al72.1Ni11.5Co16.4: An investigation with spot profile analysis LEED
,”
Surf. Sci. Lett.
463
,
L654
L660
(
2000
).
34.
W.
Steurer
,
T.
Haibach
,
B.
Zhang
,
S.
Kek
, and
R.
Lück
, “
The Structure of decagonal Al70Ni15Co15,
Acta Crystallogr., Sect. B: Struct. Sci.
B49
,
661
675
(
1993
).
35.
M.
Gierer
,
M. A.
Van Hove
,
A. I.
Goldman
,
Z.
Shen
,
S.-L.
Chang
,
C. J.
Jenks
,
C.-M.
Zhang
, and
P. A.
Thiel
, “
Structural analysis of the fivefold symmetric surface of the Al70Pd21Mn9 quasicrystal by low energy electron diffraction
,”
Phys. Rev. Lett.
78
,
467
470
(
1997
).
36.
T.
Cai
,
J.
Ledieu
,
R.
McGrath
,
V.
Fournee
,
T.
Lograsso
,
A.
Ross
, and
P.
Thiel
, “
Pseudomorphic starfish: Nucleation of extrinsic metal atoms on a quasicrystalline substrate
,”
Surf. Sci.
526
(
1–2
),
115
120
(
2003
).
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.