© 2020 Tsunetomo Yamada

Part 3,Structure Analysis

© 2020 Tsunetomo Yamada

Structure Factor Formula

© 2020 Tsunetomo Yamada

Structure factor formula

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Bµ<latexit sha1_base64="PLnTSob2yCxAPpW1E5mamc8QWXM=">AAAB7nicbVBNSwMxEJ34WetX1aOXYBE8ld0q6LHUi8cK9gPatWTTbBuaZJckK5SlP8KLB0W8+nu8+W9M2z1o64OBx3szzMwLE8GN9bxvtLa+sbm1Xdgp7u7tHxyWjo5bJk41ZU0ai1h3QmKY4Io1LbeCdRLNiAwFa4fj25nffmLa8Fg92EnCAkmGikecEuukdv0x68l02i+VvYo3B14lfk7KkKPRL331BjFNJVOWCmJM1/cSG2REW04FmxZ7qWEJoWMyZF1HFZHMBNn83Ck+d8oAR7F2pSyeq78nMiKNmcjQdUpiR2bZm4n/ed3URjdBxlWSWqboYlGUCmxjPPsdD7hm1IqJI4Rq7m7FdEQ0odYlVHQh+Msvr5JWteJfVqr3V+VaPY+jAKdwBhfgwzXU4A4a0AQKY3iGV3hDCXpB7+hj0bqG8pkT+AP0+QNjcY+a</latexit>

pµ<latexit sha1_base64="8Cqw9fx4hyQG9XYAYj8IJubG38A=">AAAB7nicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgh6LXjxWsB/QriWbZtvQJBuSrFCW/ggvHhTx6u/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61TJJqQpsk4YnuRNhQziRtWmY57ShNsYg4bUfj25nffqLasEQ+2ImiocBDyWJGsHVSWz1mPZFO++WKX/XnQKskyEkFcjT65a/eICGpoNISjo3pBr6yYYa1ZYTTaamXGqowGeMh7ToqsaAmzObnTtGZUwYoTrQradFc/T2RYWHMRESuU2A7MsveTPzP66Y2vg4zJlVqqSSLRXHKkU3Q7Hc0YJoSyyeOYKKZuxWREdaYWJdQyYUQLL+8Slq1anBRrd1fVuo3eRxFOIFTOIcArqAOd9CAJhAYwzO8wpunvBfv3ftYtBa8fOYY/sD7/AGqPY/I</latexit>

: Position: Atomic displacement parameter (ADP): Occupancy

Fµ0

<latexit sha1_base64="k0VMQZAQu5+pevJhZPNSyTH5zWk=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBU9mtgh6LgnisYD+kXUs2zbahSXZJskJZ+iu8eFDEqz/Hm//GdLsHbX0w8Hhvhpl5QcyZNq777RRWVtfWN4qbpa3tnd298v5BS0eJIrRJIh6pToA15UzSpmGG006sKBYBp+1gfD3z209UaRbJezOJqS/wULKQEWys9HDzmPZEMu27/XLFrboZ0DLxclKBHI1++as3iEgiqDSEY627nhsbP8XKMMLptNRLNI0xGeMh7VoqsaDaT7ODp+jEKgMURsqWNChTf0+kWGg9EYHtFNiM9KI3E//zuokJL/2UyTgxVJL5ojDhyERo9j0aMEWJ4RNLMFHM3orICCtMjM2oZEPwFl9eJq1a1Tur1u7OK/WrPI4iHMExnIIHF1CHW2hAEwgIeIZXeHOU8+K8Ox/z1oKTzxzCHzifP5JZkEE=</latexit>

fµ(he)<latexit sha1_base64="EqSrcvf/K9R9zf3ubfuYWI8YSFE=">AAAB/nicbVDJSgNBEO2JW4zbqHjyMhiEeAkzUdBj0IvHCGaBbPR0apImPQvdNWIYBvwVLx4U8ep3ePNv7CRz0MQHBY/3qqiq50aCK7TtbyO3srq2vpHfLGxt7+zumfsHDRXGkkGdhSKULZcqEDyAOnIU0IokUN8V0HTHN1O/+QBS8TC4x0kEXZ8OA+5xRlFLffPI6yUdP05LHYRHdL1klPbgrG8W7bI9g7VMnIwUSYZa3/zqDEIW+xAgE1SptmNH2E2oRM4EpIVOrCCibEyH0NY0oD6objI7P7VOtTKwvFDqCtCaqb8nEuorNfFd3elTHKlFbyr+57Vj9K66CQ+iGCFg80VeLCwMrWkW1oBLYCgmmlAmub7VYiMqKUOdWEGH4Cy+vEwalbJzXq7cXRSr11kceXJMTkiJOOSSVMktqZE6YSQhz+SVvBlPxovxbnzMW3NGNnNI/sD4/AGHMpXY</latexit>

: Fourier integral: atomic scattering factor

μ-th occupation domain

F (h) =X

µ

X

{R|t}µ

fµ(he)pµ exp[�Bµ(he)2/4]

⇥ exp[2⇡ih(Rrµ + t)]Fµ0 (R

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{R|t}µ<latexit sha1_base64="NJQEyDhauHxAvS0XAIi/1BIReQ0=">AAAB8nicbVBNSwMxEM3Wr1q/qh69BIvgqexWQY9FLx6r2A/oriWbZtvQbLIks0JZ+zO8eFDEq7/Gm//GtN2Dtj4YeLw3w8y8MBHcgOt+O4WV1bX1jeJmaWt7Z3evvH/QMirVlDWpEkp3QmKY4JI1gYNgnUQzEoeCtcPR9dRvPzJtuJL3ME5YEJOB5BGnBKzU9bO7J/AnD36c9soVt+rOgJeJl5MKytHolb/8vqJpzCRQQYzpem4CQUY0cCrYpOSnhiWEjsiAdS2VJGYmyGYnT/CJVfo4UtqWBDxTf09kJDZmHIe2MyYwNIveVPzP66YQXQYZl0kKTNL5oigVGBSe/o/7XDMKYmwJoZrbWzEdEk0o2JRKNgRv8eVl0qpVvbNq7fa8Ur/K4yiiI3SMTpGHLlAd3aAGaiKKFHpGr+jNAefFeXc+5q0FJ585RH/gfP4AoS+Reg==</latexit>

: Symmetry operators of space group which create new occupation domains in a unit cell from the independent one.

Yamamoto, Akiji, "Crystallography of quasiperiodic crystals."Acta Cryst. A 52, 4, 509-560, (1996)

© 2020 Tsunetomo Yamada

540 CRYSTALLOGRAPHY OF QUASIPERIODIC CRYSTALS

and all faces of calaverite are shown to have small integral Miller indices. However, the importance of the dense lattice plane asserted by the Bravais-Friedel law has not been confirmed. In modulated structures and quasicrystals, some lattice planes are not periodic. In particular, in icosahedral quasicrystals, all lattice planes are not periodic. Therefore, the direct application of the Bravais-Friedel law is impossible. However, the law of rational indices can be applicable the same as in the modulated structure. Two examples are shown in Fig. 27 (Tsai, Inoue & Masumoto, 1987, 1989). The first found stable icosahedral quasicrystal AI-Cu-Li shows a triacontahedral morphology, indicating that the faces are indexable as 110000 etc., which is perpendicular to the twofold axes (Dubost, Lang, Tanaka, Stainfort & Audier, 1986). On the other hand, that of face-centered icosahedral AI-Cu-Fe is dodecahedral (Fig. 27a). This has faces with indices 100000 etc. that are perpendicular to the fivefold axes. Decagonal A1-Ni-Co (Fig. 27b) has decagonal-columnar shape with ten faces that are normal to the twofold axes. In the decagonal lattice, there are two twofold axes along 10000 and 10010 but which index is related to the prism face has not been clarified. In all cases, the crystal faces seem to be related to strong reflections. In fact, in i-AI-Cu-Li and i-A1--Cu-Fe, the strongest reflections are 110000 and 100000, while the strongest reflections of d-A1-Ni-Co are 00002 and 13420. The latter is parallel to the twofold axes 10010. Therefore, a face parallel to the tenfold axis may be 100i0.

10. Structure-factor calculations The structure-factor calculations of quasiperiodic tilings have been performed by several people (Duneau & Katz, 1985; Kalugin, Kitayev & Levitov, 1985a,b; Elser, 1986; Jari6, 1986; Pavlovitch & K16man, 1987). Analytic expressions of general quasicrystals having polygonal or polyhedral occupation domains are given by Yamamoto & Ishihara (1988), Yamamoto & Hiraga (1988) and Yamamoto (1992b).

The structure factors of quasicrystals can be calcu- lated based on the section method. In this method, the diffraction patterns are regarded as the projection of the Fourier spectra in nD space onto the external space. In quasicrystals, the correspondence between the diffraction spots and the reciprocal-lattice points in nD space is one to one. Therefore, the calculation of the structure factor is reduced to the calculation of the Fourier spectra in nD space. The structure factor is the Fourier integral of the electron density in a unit cell of nD space. This is reduced to the calculation of the Fourier integral of the occupation domain in the internal space and the phase factor related to the location of the domain. It is usually assumed that the occupation domains of quasicrystals are polygonal for polygonal quasicrystals and polyhedral for

icosahedral quasicrystals, since this is the case for those of quasiperiodic tilings. They are divided into triangles or tetrahedra, the Fourier integrals of which are given analytically. Thus, we obtain the analytical expression for the structure factor. In a rough approximation, a polygon or a polyhedron can be approximated by a circle or a sphere. In fact, for the pentagonal Penrose tiling or Stampfli tiling and 3D Penrose tiling, this is a good approximation (see Figs. 18c, 19 and 23c). In such cases, the structure factor can easily be calculated. In all cases, it has the following form:

F ( h ) = E E f l ' ( h e ) p ~ ' e x p [ - B " ( h ~ ) 2 / 4 ] {Rlt}"

× exp[2r ih . (Rr" + t ) ] F ~ ( R - l h ) , (48)

where the position, temperature factor and occupancy of the #th independent occupation domain are repre- sented by # ' , B v and pV. F0~(h) and fV(h e) are the Fourier integral and atomic scattering factor of the #th occupation domain. The sum of {Rlt}~' runs over the symmetry operators of the space group which create new occupation domains in a unit cell from the independent ones. If the occupation domain is a circle with a radius r, its Fourier integral is

F~'(h) = 2 V J l ( q o ) / q o (49)

with V - - 71-r 2 and q0 - 27rhir, where J1 is the Bessel function of first order. Similarly, the Fourier integral of a sphere with radius r is given by

Fo'(h) = 3V[sin(qo) - qo COS(qO)]/(qo) 3 (50)

with V = 47rr3/3. For polygonal or polyhedral domains, which are de-

composed into several triangles or tetrahedra, F~'(h) is calculated by using the site symmetry from their independent parts. Since the Fourier integral is linear, Fg' is given by the summation of Fourier integrals of triangles or tetrahedra. Provided that the occupation domain consists of v independent triangles or tetrahedra, it is given by

Fo"(h/ E ' - ' = F~i (R h), (51) i = l R'

where R' is the rotational part of the site-symmetry operator, which runs over all site-symmetry operators. The Fourier integral of a triangle defined by the vector e I and e 2 (see Fig. 28a) is given by (Jari6, 1986; Ishihara & Yamamoto, 1988)

Foi (h ) = V{q , [exp( iq2 ) - 1] - q2[exp(iql) - 1 ] } / q l q 2 ( q l - q2) , (52)

Provided that the occupation domain consists of ν independent tetrahedra, the Fourier integral is given by

540 CRYSTALLOGRAPHY OF QUASIPERIODIC CRYSTALS

and all faces of calaverite are shown to have small integral Miller indices. However, the importance of the dense lattice plane asserted by the Bravais-Friedel law has not been confirmed. In modulated structures and quasicrystals, some lattice planes are not periodic. In particular, in icosahedral quasicrystals, all lattice planes are not periodic. Therefore, the direct application of the Bravais-Friedel law is impossible. However, the law of rational indices can be applicable the same as in the modulated structure. Two examples are shown in Fig. 27 (Tsai, Inoue & Masumoto, 1987, 1989). The first found stable icosahedral quasicrystal AI-Cu-Li shows a triacontahedral morphology, indicating that the faces are indexable as 110000 etc., which is perpendicular to the twofold axes (Dubost, Lang, Tanaka, Stainfort & Audier, 1986). On the other hand, that of face-centered icosahedral AI-Cu-Fe is dodecahedral (Fig. 27a). This has faces with indices 100000 etc. that are perpendicular to the fivefold axes. Decagonal A1-Ni-Co (Fig. 27b) has decagonal-columnar shape with ten faces that are normal to the twofold axes. In the decagonal lattice, there are two twofold axes along 10000 and 10010 but which index is related to the prism face has not been clarified. In all cases, the crystal faces seem to be related to strong reflections. In fact, in i-AI-Cu-Li and i-A1--Cu-Fe, the strongest reflections are 110000 and 100000, while the strongest reflections of d-A1-Ni-Co are 00002 and 13420. The latter is parallel to the twofold axes 10010. Therefore, a face parallel to the tenfold axis may be 100i0.

10. Structure-factor calculations The structure-factor calculations of quasiperiodic tilings have been performed by several people (Duneau & Katz, 1985; Kalugin, Kitayev & Levitov, 1985a,b; Elser, 1986; Jari6, 1986; Pavlovitch & K16man, 1987). Analytic expressions of general quasicrystals having polygonal or polyhedral occupation domains are given by Yamamoto & Ishihara (1988), Yamamoto & Hiraga (1988) and Yamamoto (1992b).

The structure factors of quasicrystals can be calcu- lated based on the section method. In this method, the diffraction patterns are regarded as the projection of the Fourier spectra in nD space onto the external space. In quasicrystals, the correspondence between the diffraction spots and the reciprocal-lattice points in nD space is one to one. Therefore, the calculation of the structure factor is reduced to the calculation of the Fourier spectra in nD space. The structure factor is the Fourier integral of the electron density in a unit cell of nD space. This is reduced to the calculation of the Fourier integral of the occupation domain in the internal space and the phase factor related to the location of the domain. It is usually assumed that the occupation domains of quasicrystals are polygonal for polygonal quasicrystals and polyhedral for

icosahedral quasicrystals, since this is the case for those of quasiperiodic tilings. They are divided into triangles or tetrahedra, the Fourier integrals of which are given analytically. Thus, we obtain the analytical expression for the structure factor. In a rough approximation, a polygon or a polyhedron can be approximated by a circle or a sphere. In fact, for the pentagonal Penrose tiling or Stampfli tiling and 3D Penrose tiling, this is a good approximation (see Figs. 18c, 19 and 23c). In such cases, the structure factor can easily be calculated. In all cases, it has the following form:

F ( h ) = E E f l ' ( h e ) p ~ ' e x p [ - B " ( h ~ ) 2 / 4 ] {Rlt}"

× exp[2r ih . (Rr" + t ) ] F ~ ( R - l h ) , (48)

where the position, temperature factor and occupancy of the #th independent occupation domain are repre- sented by # ' , B v and pV. F0~(h) and fV(h e) are the Fourier integral and atomic scattering factor of the #th occupation domain. The sum of {Rlt}~' runs over the symmetry operators of the space group which create new occupation domains in a unit cell from the independent ones. If the occupation domain is a circle with a radius r, its Fourier integral is

F~'(h) = 2 V J l ( q o ) / q o (49)

with V - - 71-r 2 and q0 - 27rhir, where J1 is the Bessel function of first order. Similarly, the Fourier integral of a sphere with radius r is given by

Fo'(h) = 3V[sin(qo) - qo COS(qO)]/(qo) 3 (50)

with V = 47rr3/3. For polygonal or polyhedral domains, which are de-

composed into several triangles or tetrahedra, F~'(h) is calculated by using the site symmetry from their independent parts. Since the Fourier integral is linear, Fg' is given by the summation of Fourier integrals of triangles or tetrahedra. Provided that the occupation domain consists of v independent triangles or tetrahedra, it is given by

Fo"(h/ E ' - ' = F~i (R h), (51) i = l R'

where R' is the rotational part of the site-symmetry operator, which runs over all site-symmetry operators. The Fourier integral of a triangle defined by the vector e I and e 2 (see Fig. 28a) is given by (Jari6, 1986; Ishihara & Yamamoto, 1988)

Foi (h ) = V{q , [exp( iq2 ) - 1] - q2[exp(iql) - 1 ] } / q l q 2 ( q l - q2) , (52)

: Rational part of the site-symmetry operator

: Fourier integral of a tetrahedron

540 CRYSTALLOGRAPHY OF QUASIPERIODIC CRYSTALS

and all faces of calaverite are shown to have small integral Miller indices. However, the importance of the dense lattice plane asserted by the Bravais-Friedel law has not been confirmed. In modulated structures and quasicrystals, some lattice planes are not periodic. In particular, in icosahedral quasicrystals, all lattice planes are not periodic. Therefore, the direct application of the Bravais-Friedel law is impossible. However, the law of rational indices can be applicable the same as in the modulated structure. Two examples are shown in Fig. 27 (Tsai, Inoue & Masumoto, 1987, 1989). The first found stable icosahedral quasicrystal AI-Cu-Li shows a triacontahedral morphology, indicating that the faces are indexable as 110000 etc., which is perpendicular to the twofold axes (Dubost, Lang, Tanaka, Stainfort & Audier, 1986). On the other hand, that of face-centered icosahedral AI-Cu-Fe is dodecahedral (Fig. 27a). This has faces with indices 100000 etc. that are perpendicular to the fivefold axes. Decagonal A1-Ni-Co (Fig. 27b) has decagonal-columnar shape with ten faces that are normal to the twofold axes. In the decagonal lattice, there are two twofold axes along 10000 and 10010 but which index is related to the prism face has not been clarified. In all cases, the crystal faces seem to be related to strong reflections. In fact, in i-AI-Cu-Li and i-A1--Cu-Fe, the strongest reflections are 110000 and 100000, while the strongest reflections of d-A1-Ni-Co are 00002 and 13420. The latter is parallel to the twofold axes 10010. Therefore, a face parallel to the tenfold axis may be 100i0.

10. Structure-factor calculations The structure-factor calculations of quasiperiodic tilings have been performed by several people (Duneau & Katz, 1985; Kalugin, Kitayev & Levitov, 1985a,b; Elser, 1986; Jari6, 1986; Pavlovitch & K16man, 1987). Analytic expressions of general quasicrystals having polygonal or polyhedral occupation domains are given by Yamamoto & Ishihara (1988), Yamamoto & Hiraga (1988) and Yamamoto (1992b).

The structure factors of quasicrystals can be calcu- lated based on the section method. In this method, the diffraction patterns are regarded as the projection of the Fourier spectra in nD space onto the external space. In quasicrystals, the correspondence between the diffraction spots and the reciprocal-lattice points in nD space is one to one. Therefore, the calculation of the structure factor is reduced to the calculation of the Fourier spectra in nD space. The structure factor is the Fourier integral of the electron density in a unit cell of nD space. This is reduced to the calculation of the Fourier integral of the occupation domain in the internal space and the phase factor related to the location of the domain. It is usually assumed that the occupation domains of quasicrystals are polygonal for polygonal quasicrystals and polyhedral for

icosahedral quasicrystals, since this is the case for those of quasiperiodic tilings. They are divided into triangles or tetrahedra, the Fourier integrals of which are given analytically. Thus, we obtain the analytical expression for the structure factor. In a rough approximation, a polygon or a polyhedron can be approximated by a circle or a sphere. In fact, for the pentagonal Penrose tiling or Stampfli tiling and 3D Penrose tiling, this is a good approximation (see Figs. 18c, 19 and 23c). In such cases, the structure factor can easily be calculated. In all cases, it has the following form:

F ( h ) = E E f l ' ( h e ) p ~ ' e x p [ - B " ( h ~ ) 2 / 4 ] {Rlt}"

× exp[2r ih . (Rr" + t ) ] F ~ ( R - l h ) , (48)

where the position, temperature factor and occupancy of the #th independent occupation domain are repre- sented by # ' , B v and pV. F0~(h) and fV(h e) are the Fourier integral and atomic scattering factor of the #th occupation domain. The sum of {Rlt}~' runs over the symmetry operators of the space group which create new occupation domains in a unit cell from the independent ones. If the occupation domain is a circle with a radius r, its Fourier integral is

F~'(h) = 2 V J l ( q o ) / q o (49)

with V - - 71-r 2 and q0 - 27rhir, where J1 is the Bessel function of first order. Similarly, the Fourier integral of a sphere with radius r is given by

Fo'(h) = 3V[sin(qo) - qo COS(qO)]/(qo) 3 (50)

with V = 47rr3/3. For polygonal or polyhedral domains, which are de-

composed into several triangles or tetrahedra, F~'(h) is calculated by using the site symmetry from their independent parts. Since the Fourier integral is linear, Fg' is given by the summation of Fourier integrals of triangles or tetrahedra. Provided that the occupation domain consists of v independent triangles or tetrahedra, it is given by

Fo"(h/ E ' - ' = F~i (R h), (51) i = l R'

where R' is the rotational part of the site-symmetry operator, which runs over all site-symmetry operators. The Fourier integral of a triangle defined by the vector e I and e 2 (see Fig. 28a) is given by (Jari6, 1986; Ishihara & Yamamoto, 1988)

Foi (h ) = V{q , [exp( iq2 ) - 1] - q2[exp(iql) - 1 ] } / q l q 2 ( q l - q2) , (52)

Structure factor formulaYamamoto, Akiji, "Crystallography of quasiperiodic crystals."Acta Cryst. A 52, 4, 509-560, (1996)

© 2020 Tsunetomo Yamada

ei3<latexit sha1_base64="2I8HRh0hNI8JyvkaVAva4YO3aeI=">AAAB+3icbVBNS8NAEN3Ur1q/Yj16CRbBU0lawR4LXjxWsB/QxrLZTtqlmw92J9IS8le8eFDEq3/Em//GbZuDtj4YeLw3w8w8LxZcoW1/G4Wt7Z3dveJ+6eDw6PjEPC13VJRIBm0WiUj2PKpA8BDayFFAL5ZAA09A15veLvzuE0jFo/AB5zG4AR2H3OeMopaGZnmAMEPPTyF7THk2TOvZ0KzYVXsJa5M4OamQHK2h+TUYRSwJIEQmqFJ9x47RTalEzgRkpUGiIKZsSsfQ1zSkASg3Xd6eWZdaGVl+JHWFaC3V3xMpDZSaB57uDChO1Lq3EP/z+gn6DTflYZwghGy1yE+EhZG1CMIacQkMxVwTyiTXt1psQiVlqOMq6RCc9Zc3SadWderV2v11pdnI4yiSc3JBrohDbkiT3JEWaRNGZuSZvJI3IzNejHfjY9VaMPKZM/IHxucP0SyU6A==</latexit>

ei2<latexit sha1_base64="K1zceYqd0qAkQlzDdxxy2ZeCZtk=">AAAB+3icbVBNS8NAEN3Ur1q/Yj16CRbBU0mqYI8FLx4r2FZoY9hsJ+3SzQe7E2kJ+StePCji1T/izX/jts1BWx8MPN6bYWaenwiu0La/jdLG5tb2Tnm3srd/cHhkHle7Kk4lgw6LRSwffKpA8Ag6yFHAQyKBhr6Anj+5mfu9J5CKx9E9zhJwQzqKeMAZRS15ZnWAMEU/yCB/zHjuZY3cM2t23V7AWidOQWqkQNszvwbDmKUhRMgEVarv2Am6GZXImYC8MkgVJJRN6Aj6mkY0BOVmi9tz61wrQyuIpa4IrYX6eyKjoVKz0NedIcWxWvXm4n9eP8Wg6WY8SlKEiC0XBamwMLbmQVhDLoGhmGlCmeT6VouNqaQMdVwVHYKz+vI66TbqzmW9cXdVazWLOMrklJyRC+KQa9Iit6RNOoSRKXkmr+TNyI0X4934WLaWjGLmhPyB8fkDz6eU5w==</latexit>

ei1<latexit sha1_base64="g8lidGl3l26A19tN/qRoTfPY3Qc=">AAAB+3icbVBNS8NAEN3Ur1q/Yj16CRbBU0mqYI8FLx4r2FZoY9hsJ+3SzQe7E2kJ+StePCji1T/izX/jts1BWx8MPN6bYWaenwiu0La/jdLG5tb2Tnm3srd/cHhkHle7Kk4lgw6LRSwffKpA8Ag6yFHAQyKBhr6Anj+5mfu9J5CKx9E9zhJwQzqKeMAZRS15ZnWAMEU/yCB/zHjuZU7umTW7bi9grROnIDVSoO2ZX4NhzNIQImSCKtV37ATdjErkTEBeGaQKEsomdAR9TSMagnKzxe25da6VoRXEUleE1kL9PZHRUKlZ6OvOkOJYrXpz8T+vn2LQdDMeJSlCxJaLglRYGFvzIKwhl8BQzDShTHJ9q8XGVFKGOq6KDsFZfXmddBt157LeuLuqtZpFHGVySs7IBXHINWmRW9ImHcLIlDyTV/Jm5MaL8W58LFtLRjFzQv7A+PwBziKU5g==</latexit>

O

: Fourier integral of a tetrahedron

V = ei1 · [ei2 ⇥ ei3]<latexit sha1_base64="bNGqER1hNdBBex46lUvfkuPPfBs=">AAACInicbZDLSsNAFIYnXmu9VV26CRbBVUlaQV0IRTcuK9gLNLFMpift0MmFmROxhD6LG1/FjQtFXQk+jNM2C9v6w8DPd87hzPm9WHCFlvVtLC2vrK6t5zbym1vbO7uFvf2GihLJoM4iEcmWRxUIHkIdOQpoxRJo4AloeoPrcb35AFLxKLzDYQxuQHsh9zmjqFGncNG4dBAe0fNTGN3zju2wboTtGVZ2kAegZljF7RSKVsmayFw0dmaKJFOtU/h0uhFLAgiRCapU27ZidFMqkTMBo7yTKIgpG9AetLUNqd7pppMTR+axJl3Tj6R+IZoT+ncipYFSw8DTnQHFvpqvjeF/tXaC/rmb8jBOEEI2XeQnwsTIHOdldrkEhmKoDWWS67+arE8lZahTzesQ7PmTF02jXLIrpfLtabF6lcWRI4fkiJwQm5yRKrkhNVInjDyRF/JG3o1n49X4ML6mrUtGNnNAZmT8/AKf2KWS</latexit>

q4 = q2 � q3, q5 = q3 � q1, q6 = q2 � q1<latexit sha1_base64="nw7j7ECQCvMR1u/lIRvK7JPS7vQ=">AAACDnicbVC7TsMwFHV4lvIKMLJYVJUYoEra8liQKlgYi0QfUhtFjuu0Vh0nsR2kquoXsPArLAwgxMrMxt/gtBmg5UqWzuNeXd/jRYxKZVnfxtLyyuraem4jv7m1vbNr7u03ZZgITBo4ZKFoe0gSRjlpKKoYaUeCoMBjpOUNb1K/9UCEpCG/V6OIOAHqc+pTjJSWXLMYu9Wr2C2fxm7lBMbumSYVTeyUnGeO7ZoFq2RNCy4COwMFkFXdNb+6vRAnAeEKMyRlx7Yi5YyRUBQzMsl3E0kihIeoTzoachQQ6Yyn50xgUSs96IdCP67gVP09MUaBlKPA050BUgM576Xif14nUf6lM6Y8ShTheLbITxhUIUyzgT0qCFZspAHCguq/QjxAAmGlE8zrEOz5kxdBs1yyK6XyXbVQu87iyIFDcASOgQ0uQA3cgjpoAAwewTN4BW/Gk/FivBsfs9YlI5s5AH/K+PwBP1OZEg==</latexit>

qj = 2⇡hi · eij(j = 1, 2, 3)<latexit sha1_base64="7KFXd01x2a5adGM2jDlpSDGP1IM=">AAACGnicbZDLSgNBEEV7fMb4irp00xiECCHMREE3gujGZQRjApk49HRqTMeeh901YhjyHW78FTcuFHEnbvwbOzGCrwsNl1NVVNf1Eyk02va7NTE5NT0zm5vLzy8sLi0XVlbPdJwqDnUey1g1faZBigjqKFBCM1HAQl9Cw788GtYb16C0iKNT7CfQDtlFJALBGRrkFZwrr7dfdRPhItygH2TdwblweSfGLwAGeD1a6u075Wp5e8srFO2KPRL9a5yxKZKxal7h1e3EPA0hQi6Z1i3HTrCdMYWCSxjk3VRDwvglu4CWsRELQbez0WkDumlIhwaxMi9COqLfJzIWat0PfdMZMuzq37Uh/K/WSjHYa2ciSlKEiH8uClJJMabDnGhHKOAo+8YwroT5K+VdphhHk2behOD8PvmvOatWnO1K9WSneHA4jiNH1skGKRGH7JIDckxqpE44uSX35JE8WXfWg/VsvXy2TljjmTXyQ9bbB271oGk=</latexit>

Fµ0i

<latexit sha1_base64="xWpQu9eQ/0ZN0WzkXR+L6Sgmuuc=">AAAB83icbVDLSsNAFL3xWeur6tLNYBFclaQKuiwK4rKCfUATy2Q6aYdOJmEeQgn5DTcuFHHrz7jzb5y2WWjrgQuHc+7l3nvClDOlXffbWVldW9/YLG2Vt3d29/YrB4dtlRhJaIskPJHdECvKmaAtzTSn3VRSHIecdsLxzdTvPFGpWCIe9CSlQYyHgkWMYG0l//Yx82OT9zOX5f1K1a25M6Bl4hWkCgWa/cqXP0iIianQhGOlep6b6iDDUjPCaV72jaIpJmM8pD1LBY6pCrLZzTk6tcoARYm0JTSaqb8nMhwrNYlD2xljPVKL3lT8z+sZHV0FGROp0VSQ+aLIcKQTNA0ADZikRPOJJZhIZm9FZIQlJtrGVLYheIsvL5N2vead1+r3F9XGdRFHCY7hBM7Ag0towB00oQUEUniGV3hzjPPivDsf89YVp5g5gj9wPn8AJFGRwA==</latexit>

Fµ0i(h) = �iV [q2q3q4 exp(iq1)q3q1q5 exp(iq2)

+q1q2q6 exp(iq3)q4q5q6]/(q1q2q3q4q5q6)<latexit sha1_base64="Yot7FA1cdn9cJJZnCbK1n1u4q18=">AAACYnicbZFfT9swFMWdMBjLBhR4ZA/RqkmtEF3SFtgLEtqkaY9MWgtSk1mOe0MtnH/2DaKK8iX3xhMv+yBzQzttsCtZOv7dc2X7OMql0Oh595a99mJ94+XmK+f1m63tndbu3lhnpeIw4pnM1FXENEiRwggFSrjKFbAkknAZ3Xxe9C9vQWmRpd9xnkOYsOtUxIIzNIi25l9+VEFS1rTyRN0JEO4wiqtZ3T07EuNJQfsFHRR0GMBd3hEF9bvN3i/o8Qr1u0HgHDbMmE9WeGCcQ2MzKPzQWbUHf2CXttpez2vKfS78pWiTZV3Q1s9gmvEygRS5ZFpPfC/HsGIKBZdQO0GpIWf8hl3DxMiUJaDDqomodt8bMnXjTJmVotvQvycqlmg9TyLjTBjO9NPeAv6vNykx/hhWIs1LhJQ/HhSX0sXMXeTtToUCjnJuBONKmLu6fMYU42h+xTEh+E+f/FyM+z1/0Ot/G7bPPy3j2CQH5B3pEJ+cknPylVyQEeHkwVq3tq0d65ft2Lv2/qPVtpYz++Sfst/+BqdUs5A=</latexit>

Structure factor formulaYamamoto, Akiji, "Crystallography of quasiperiodic crystals."Acta Cryst. A 52, 4, 509-560, (1996)

© 2020 Tsunetomo Yamada

• Position• ADP• Occupancy• Ratio of elements

for each occupation domains

Once structure model is built, it can be optimized to best fit the experimental data (minimization of Q).

Q =X

h

wh(|Fo|2 � |Fc|2)2<latexit sha1_base64="e3D3z6rmnuwUDD3XjU/kSInX3l0=">AAACDXicbZDLSsNAFIYn9VbrLerSzWAV6sKSREE3QlEQly3YC7RpmEwn7dDJhZmJUtK+gBtfxY0LRdy6d+fbOGmz0OoPBz7+cw4z53cjRoU0jC8tt7C4tLySXy2srW9sbunbOw0RxhyTOg5ZyFsuEoTRgNQllYy0Ik6Q7zLSdIdXab95R7igYXArRxGxfdQPqEcxkspy9IPaRUfEvjOA96pK42snCSfjrnWcEk7pqGs5etEoG1PBv2BmUASZqo7+2emFOPZJIDFDQrRNI5J2grikmJFJoRMLEiE8RH3SVhggnwg7mV4zgYfK6UEv5KoCCafuz40E+UKMfFdN+kgOxHwvNf/rtWPpndsJDaJYkgDPHvJiBmUI02hgj3KCJRspQJhT9VeIB4gjLFWABRWCOX/yX2hYZfOkbNVOi5XLLI482AP7oARMcAYq4AZUQR1g8ACewAt41R61Z+1Ne5+N5rRsZxf8kvbxDdXSmsU=</latexit>

wh<latexit sha1_base64="+HFvdca+Lesy34Bp6BlZ18fG7gE=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbMLuRCmhP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXJtRKwecJxwP6IDJULBKFrp/qk37JXKbsWdgSwTLydlyFHvlb66/ZilEVfIJDWm47kJ+hnVKJjkk2I3NTyhbEQHvGOpohE3fjY7dUJOrdInYaxtKSQz9fdERiNjxlFgOyOKQ7PoTcX/vE6K4ZWfCZWkyBWbLwpTSTAm079JX2jOUI4toUwLeythQ6opQ5tO0YbgLb68TJrVindeqd5dlGvXeRwFOIYTOAMPLqEGt1CHBjAYwDO8wpsjnRfn3fmYt644+cwR/IHz+QNe1I3a</latexit>

wh =1

�2<latexit sha1_base64="AISU4GHGjD66PyPG21zgC6x5n+o=">AAAB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JUQTdC0Y3LCvYBTQyT6aQdOjMJMxOlxCz8FTcuFHHrb7jzb5y2WWjrgQuHc+7l3nvChFGlHefbWlhcWl5ZLa2V1zc2t7btnd2WilOJSRPHLJadECnCqCBNTTUjnUQSxENG2uHwauy374lUNBa3epQQn6O+oBHFSBspsPcfgsGFF0mEMzfPPEX7HN3V8sCuOFVnAjhP3IJUQIFGYH95vRinnAiNGVKq6zqJ9jMkNcWM5GUvVSRBeIj6pGuoQJwoP5vcn8Mjo/RgFEtTQsOJ+nsiQ1ypEQ9NJ0d6oGa9sfif1011dO5nVCSpJgJPF0UpgzqG4zBgj0qCNRsZgrCk5laIB8iEoU1kZROCO/vyPGnVqu5JtXZzWqlfFnGUwAE4BMfABWegDq5BAzQBBo/gGbyCN+vJerHerY9p64JVzOyBP7A+fwAJV5Yc</latexit>

: weight

: standard deviation�<latexit sha1_base64="uveq51XskeZ/BmBbyb/DEzkG8yU=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5idzCZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNaJMornQnwoZyJmnTMstpJ9EUi4jTdjS+nfntJ6oNU/LBThIaCjyULGYEWye1eoYNBe6XK37VnwOtkiAnFcjR6Je/egNFUkGlJRwb0w38xIYZ1pYRTqelXmpogskYD2nXUYkFNWE2v3aKzpwyQLHSrqRFc/X3RIaFMRMRuU6B7cgsezPxP6+b2vg6zJhMUkslWSyKU46sQrPX0YBpSiyfOIKJZu5WREZYY2JdQCUXQrD88ipp1arBRbV2f1mp3+RxFOEETuEcAriCOtxBA5pA4BGe4RXePOW9eO/ex6K14OUzx/AH3ucPna2PJw==</latexit>

Structure factor formulaYamamoto, Akiji, "Crystallography of quasiperiodic crystals."Acta Cryst. A 52, 4, 509-560, (1996)

© 2020 Tsunetomo Yamada

Diffraction Intensities Measurement

© 2020 Tsunetomo Yamada

XtaLAB Synergy CrysAlisPRO

If you wish to use, contact to Prof. Takakura (A02)

Rigaku Oxford Diffraction Rigaku Oxford Diffraction

Diffraction experiment

© 2020 Tsunetomo Yamada

Diffraction experiment

Data collection & analysis

• indexing• integration of intensities• corrections (absorption, etc)

Reflection listIndices, Intensity, sigmaI(h)

<latexit sha1_base64="qezJY6gAoGu7C/OGd3aJ5VMGe1w=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJexGQY9BL3qLYB6QLGF2MpsMmX040xsMS77DiwdFvPox3vwbJ8keNLGgoajqprvLi6XQaNvf1srq2vrGZm4rv72zu7dfODhs6ChRjNdZJCPV8qjmUoS8jgIlb8WK08CTvOkNb6Z+c8SVFlH4gOOYuwHth8IXjKKR3LtSB/kTen46mJx1C0W7bM9AlomTkSJkqHULX51exJKAh8gk1brt2DG6KVUomOSTfCfRPKZsSPu8bWhIA67ddHb0hJwapUf8SJkKkczU3xMpDbQeB57pDCgO9KI3Ff/z2gn6V24qwjhBHrL5Ij+RBCMyTYD0hOIM5dgQypQwtxI2oIoyNDnlTQjO4svLpFEpO+flyv1FsXqdxZGDYziBEjhwCVW4hRrUgcEjPMMrvFkj68V6tz7mrStWNnMEf2B9/gBxppHj</latexit>

Diffraction Intensity

Sample

<100 μm

© 2020 Tsunetomo Yamada

Structure analysis

© 2020 Tsunetomo Yamada

• Phase retrieval

• Fourier synthesis (electron density map)

• Model construction

• Refinement

• Visualization & interpretation

Structure analysis

© 2020 Tsunetomo Yamada

Phase problem

F (h)<latexit sha1_base64="J52xGEq3njS7JiMOVomCN7b1z2U=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJexGQY9BQTxGMA9IljA7mU2GzD6c6Q2GJd/hxYMiXv0Yb/6Nk2QPmljQUFR1093lxVJotO1va2V1bX1jM7eV397Z3dsvHBw2dJQoxusskpFqeVRzKUJeR4GSt2LFaeBJ3vSGN1O/OeJKiyh8wHHM3YD2Q+ELRtFI7m2pg/wJPT8dTM66haJdtmcgy8TJSBEy1LqFr04vYknAQ2SSat127BjdlCoUTPJJvpNoHlM2pH3eNjSkAdduOjt6Qk6N0iN+pEyFSGbq74mUBlqPA890BhQHetGbiv957QT9KzcVYZwgD9l8kZ9IghGZJkB6QnGGcmwIZUqYWwkbUEUZmpzyJgRn8eVl0qiUnfNy5f6iWL3O4sjBMZxACRy4hCrcQQ3qwOARnuEV3qyR9WK9Wx/z1hUrmzmCP7A+fwBs9pHg</latexit>

I(h)<latexit sha1_base64="qezJY6gAoGu7C/OGd3aJ5VMGe1w=">AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJexGQY9BL3qLYB6QLGF2MpsMmX040xsMS77DiwdFvPox3vwbJ8keNLGgoajqprvLi6XQaNvf1srq2vrGZm4rv72zu7dfODhs6ChRjNdZJCPV8qjmUoS8jgIlb8WK08CTvOkNb6Z+c8SVFlH4gOOYuwHth8IXjKKR3LtSB/kTen46mJx1C0W7bM9AlomTkSJkqHULX51exJKAh8gk1brt2DG6KVUomOSTfCfRPKZsSPu8bWhIA67ddHb0hJwapUf8SJkKkczU3xMpDbQeB57pDCgO9KI3Ff/z2gn6V24qwjhBHrL5Ij+RBCMyTYD0hOIM5dgQypQwtxI2oIoyNDnlTQjO4svLpFEpO+flyv1FsXqdxZGDYziBEjhwCVW4hRrUgcEjPMMrvFkj68V6tz7mrStWNnMEf2B9/gBxppHj</latexit>

Atomis Structure (electron density ρ)⇢(r)<latexit sha1_base64="xZHUrQbCYnPybrbYRvYSKkFmlmU=">AAAB+XicbVBNTwIxEO3iF+LXqkcvjcQEL2QXTfRI9OIRE0ESlpBu6UJDt920s0Sy4Z948aAxXv0n3vw3FtiDgi+Z5OW9mczMCxPBDXjet1NYW9/Y3Cpul3Z29/YP3MOjllGppqxJlVC6HRLDBJesCRwEayeakTgU7DEc3c78xzHThiv5AJOEdWMykDzilICVeq4b6KGqBMCeIIwyPT3vuWWv6s2BV4mfkzLK0ei5X0Ff0TRmEqggxnR8L4FuRjRwKti0FKSGJYSOyIB1LJUkZqabzS+f4jOr9HGktC0JeK7+nshIbMwkDm1nTGBolr2Z+J/XSSG67mZcJikwSReLolRgUHgWA+5zzSiIiSWEam5vxXRINKFgwyrZEPzll1dJq1b1L6q1+8ty/SaPo4hO0CmqIB9doTq6Qw3URBSN0TN6RW9O5rw4787HorXg5DPH6A+czx+J/ZOY</latexit>

Diffraction Intensity

Structure Factor

Loss of phase informationcomplex conjugate

I(h) / F (h)F ⇤(h) = |F (h)|2<latexit sha1_base64="GszLws0mZr40X8/PawJvx1sXQNs=">AAACLHicbVDJSgNBEO2JW4xb1KOXwSBED2EmCnoRgoGgtwhmgcwk9HR6kiY9C901Ypjkg7z4K4J4MIhXv8POcsjig4LX71XRVc8JOZNgGCMtsba+sbmV3E7t7O7tH6QPj6oyiAShFRLwQNQdLClnPq0AA07roaDYczitOb3i2K89UyFZ4D9BP6S2hzs+cxnBoKRWuviQtYC+gOPG3eG5FYoghEAvzYul5sX883aw4A6a+VY6Y+SMCfRVYs5IBs1QbqU/rHZAIo/6QDiWsmEaIdgxFsAIp8OUFUkaYtLDHdpQ1McelXY8OXaonymlrbuBUOWDPlHnJ2LsSdn3HNXpYejKZW8s/uc1InBv7Jj5YQTUJ9OP3IjrKo9xcnqbCUqA9xXBRDC1q066WGACKt+UCsFcPnmVVPM58zKXf7zKFO5mcSTRCTpFWWSia1RA96iMKoigV/SOvtBIe9M+tW/tZ9qa0GYzx2gB2u8fSwOoPQ==</latexit>

⇢(r) =1

V

ZF (h) exp(�2⇡ih · r)dh

<latexit sha1_base64="JokFrMofb9zPi1+hOIsLplRMdf8=">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</latexit>

F (h) =

Z⇢(r) exp(2⇡ih · r)dr

<latexit sha1_base64="N1BYiANdfKWkVEBXFuOm+BFv/+8=">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</latexit>

© 2020 Tsunetomo Yamada

Phase retrieval

Charge Flip method　Oszlanyi G., Suto A. (2004): Acta Crystal. A60, 134Low-Density Elimination method　Shiono M.et al (1992) Acta Crystal. A48, 451

Reflection list

RomdamPhases F

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FT-1g

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FT

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Modification

New electron density＞

＞Phases

Electron density

Phasing algorithm

SUPERFLIPlodemac H. Takakura et al, Phys. Rev. Lett. 86 (2001) 236-239

Palatinus L. (2004): Acta Crystallogr. A60, 604-610,

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Software

© 2020 Tsunetomo Yamada

Structure analysis of i-Sc12Zn88

© 2020 Tsunetomo Yamada

When the golden mean (! ¼ ð1 þffiffiffi5

pÞ=2) appears in the

phason strain matrix it is changed to a rational number, and aperiodic ‘rational’ AP is obtained (Elser & Henley, 1985; Ishii,1989; Janssen, 1991; Gratias et al., 1995). In that case, as amatter of comparison the " value for a 1/1 and a 2/1 cAPwould be equal to % 0.23 and 0.09, respectively.

It is also interesting to compare the results with thoseobserved in the i-R–Zn–Mg (R= Y, Tb, Ho) phases for which adetailed study has been carried out (Letoublon et al., 2000).Some samples did not show any Bragg peak shift, whereas foursamples did show a significant peak shift, larger when seen onthe fivefold axis and with a " value of the order % 0.01, i.e.slightly larger than in the present sample.

The present results demonstrate that the diffraction patternof the i-ScZn7.33 departs slightly but distinctly from the perfecticosahedral symmetry, unlike in the case of i-Sc–Zn–Mg forinstance (de Boissieu et al., 2005). We cannot say whether thestructure is locked on a periodic AP or not, but from the valueof the phason strain, the smallest periodic AP compatible withthe present findings would have a cubic unit cell with a latticeparameter larger than 90 A.

In the next step, we have investigated the Bragg peakbroadening by measuring the longitudinal and transverseBragg peak width for a selection of about 16 reflections lyingin a broad Qpar and Qperp range along the twofold and fivefoldaxes. Each reflection was first centred and then measured withtwo different settings in transverse and longitudinal geome-tries. The FWHM of these reflections measured along thelongitudinal and transverse directions were plotted as afunction of Qpar and Qperp (see Fig. S4 in the supportinginformation). Whereas there is no visible trend for theevolution of the FWHM as a function of Qpar, a clear lineardependence is observed when plotted at the FWHM as afunction of Qperp. We observe, however, an anisotropicbroadening, with a faster increase of the FWHM as a functionof Qperp when measured in transverse geometry, e.g. the slope# of the linear relationship of the FWHM of the fivefoldreflections as a function of Qperp is equal to 3.8 & 10% 3 and 6.7& 10% 3 for longitudinal and transverse directions, respectively.This is larger than what was found in the i-Al–Pd–Mn forwhich the value of # is equal to 2 & 10% 3 and 4 & 10% 4 for theas-grown and annealed samples, respectively (Letoublon et al.,2001; Gastaldi et al., 2003). In addition, when comparing as-cast and annealed i-ScZn7.33 samples, the # value for thelongitudinal width for the fivefold reflections is found to be 4.3& 10% 3 and 3.8 & 10% 3, respectively. This result indicates thatthe annealing indeed reduces the phason strain distribution inthe sample. On the other hand, they are larger than what wasfound in the i-Al–Pd–Mn for which the value of # is equal to 2& 10% 3 and 4 & 10% 4 for the as grown and annealed samples,respectively (Gastaldi et al., 2003).

The origin of this linear phason strain distribution is still anopen question. It is known that dislocations in quasicrystalsare accompanied by both a phonon and a phason straindistribution (Lubensky et al., 1986; Socolar & Wright, 1987). Itcan thus be guessed that the present phason strain distributionis related to the presence of dislocations, but further experi-

ments are required to validate this hypothesis. In the followingwe will treat the results within the icosahedral space group.This approach is jusned because of the use of a high-resolutiondiffraction setup.

4. Atomic structure of i-ScZn7.33

4.1. Synchrotron single-crystal X-ray diffraction data

Fig. 4 shows the reciprocal-space sections of twofold,threefold and fivefold planes, reconstructed from the non-attenuated diffraction data for the annealed i-ScZn7.33 samplecollected on the CRISTAL beamline. A few strong reflectionsare indicated using the N/M indexing scheme after Cahn et al.(1986). More than 290 000 Bragg peaks were observed,leading to 4778 unique reflections in a Q-range up to 16 r.l.u.The maximum Qperp value necessary for the indexing was lessthan 3 r.l.u. On the twofold section, a large amount of diffusescattering is present around strong Bragg reflections such asthe 20/32 and 18/29 on the twofold section. As will be shownlater, such strong diffuse scattering is due to long-wavelengthphason fluctuations in the sample. In addition, we notice smalldisplacements of some weak reflections from their idealposition, which can be explained by the presence of the linearphason strain in the sample as shown in the previous section.

research papers

IUCrJ (2016). 72 Tsunetomo Yamada et al. ' Atomic structure and phason modes of i-Sc–Zn 5 of 12

Figure 4Reciprocal-space sections reconstructed from the diffraction data for theannealed i-ScZn7.33 sample, perpendicular to (a) two-, (b) three- and (c)fivefold axes. The lines on the twofold plane indicate two-, three- andfivefold axes. (d) and (e) are magnified pictures of (a) showing the diffusescattering and peak shift, respectively. The reflections labeled by A, B andC, respectively are indexed as 20/32y, 18/29 and 20/32x reflections afterCahn et al. (1986). The white arrowheads indicate the peaks shifted fromthe ideal positions. The red arrowheads indicate streaks due to thesaturation.

Files: m/gq5006/gq5006.3d m/gq5006/gq5006.sgml GQ5006 FA IU-1611/27(17)5 () GQ5006 PROOFS M:FA:2016:72:4:0:0–0

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When the golden mean (! ¼ ð1 þffiffiffi5

pÞ=2) appears in the

phason strain matrix it is changed to a rational number, and aperiodic ‘rational’ AP is obtained (Elser & Henley, 1985; Ishii,1989; Janssen, 1991; Gratias et al., 1995). In that case, as amatter of comparison the " value for a 1/1 and a 2/1 cAPwould be equal to % 0.23 and 0.09, respectively.

It is also interesting to compare the results with thoseobserved in the i-R–Zn–Mg (R= Y, Tb, Ho) phases for which adetailed study has been carried out (Letoublon et al., 2000).Some samples did not show any Bragg peak shift, whereas foursamples did show a significant peak shift, larger when seen onthe fivefold axis and with a " value of the order % 0.01, i.e.slightly larger than in the present sample.

The present results demonstrate that the diffraction patternof the i-ScZn7.33 departs slightly but distinctly from the perfecticosahedral symmetry, unlike in the case of i-Sc–Zn–Mg forinstance (de Boissieu et al., 2005). We cannot say whether thestructure is locked on a periodic AP or not, but from the valueof the phason strain, the smallest periodic AP compatible withthe present findings would have a cubic unit cell with a latticeparameter larger than 90 A.

In the next step, we have investigated the Bragg peakbroadening by measuring the longitudinal and transverseBragg peak width for a selection of about 16 reflections lyingin a broad Qpar and Qperp range along the twofold and fivefoldaxes. Each reflection was first centred and then measured withtwo different settings in transverse and longitudinal geome-tries. The FWHM of these reflections measured along thelongitudinal and transverse directions were plotted as afunction of Qpar and Qperp (see Fig. S4 in the supportinginformation). Whereas there is no visible trend for theevolution of the FWHM as a function of Qpar, a clear lineardependence is observed when plotted at the FWHM as afunction of Qperp. We observe, however, an anisotropicbroadening, with a faster increase of the FWHM as a functionof Qperp when measured in transverse geometry, e.g. the slope# of the linear relationship of the FWHM of the fivefoldreflections as a function of Qperp is equal to 3.8 & 10% 3 and 6.7& 10% 3 for longitudinal and transverse directions, respectively.This is larger than what was found in the i-Al–Pd–Mn forwhich the value of # is equal to 2 & 10% 3 and 4 & 10% 4 for theas-grown and annealed samples, respectively (Letoublon et al.,2001; Gastaldi et al., 2003). In addition, when comparing as-cast and annealed i-ScZn7.33 samples, the # value for thelongitudinal width for the fivefold reflections is found to be 4.3& 10% 3 and 3.8 & 10% 3, respectively. This result indicates thatthe annealing indeed reduces the phason strain distribution inthe sample. On the other hand, they are larger than what wasfound in the i-Al–Pd–Mn for which the value of # is equal to 2& 10% 3 and 4 & 10% 4 for the as grown and annealed samples,respectively (Gastaldi et al., 2003).

The origin of this linear phason strain distribution is still anopen question. It is known that dislocations in quasicrystalsare accompanied by both a phonon and a phason straindistribution (Lubensky et al., 1986; Socolar & Wright, 1987). Itcan thus be guessed that the present phason strain distributionis related to the presence of dislocations, but further experi-

ments are required to validate this hypothesis. In the followingwe will treat the results within the icosahedral space group.This approach is jusned because of the use of a high-resolutiondiffraction setup.

4. Atomic structure of i-ScZn7.33

4.1. Synchrotron single-crystal X-ray diffraction data

Fig. 4 shows the reciprocal-space sections of twofold,threefold and fivefold planes, reconstructed from the non-attenuated diffraction data for the annealed i-ScZn7.33 samplecollected on the CRISTAL beamline. A few strong reflectionsare indicated using the N/M indexing scheme after Cahn et al.(1986). More than 290 000 Bragg peaks were observed,leading to 4778 unique reflections in a Q-range up to 16 r.l.u.The maximum Qperp value necessary for the indexing was lessthan 3 r.l.u. On the twofold section, a large amount of diffusescattering is present around strong Bragg reflections such asthe 20/32 and 18/29 on the twofold section. As will be shownlater, such strong diffuse scattering is due to long-wavelengthphason fluctuations in the sample. In addition, we notice smalldisplacements of some weak reflections from their idealposition, which can be explained by the presence of the linearphason strain in the sample as shown in the previous section.

research papers

IUCrJ (2016). 72 Tsunetomo Yamada et al. ' Atomic structure and phason modes of i-Sc–Zn 5 of 12

Figure 4Reciprocal-space sections reconstructed from the diffraction data for theannealed i-ScZn7.33 sample, perpendicular to (a) two-, (b) three- and (c)fivefold axes. The lines on the twofold plane indicate two-, three- andfivefold axes. (d) and (e) are magnified pictures of (a) showing the diffusescattering and peak shift, respectively. The reflections labeled by A, B andC, respectively are indexed as 20/32y, 18/29 and 20/32x reflections afterCahn et al. (1986). The white arrowheads indicate the peaks shifted fromthe ideal positions. The red arrowheads indicate streaks due to thesaturation.

Files: m/gq5006/gq5006.3d m/gq5006/gq5006.sgml GQ5006 FA IU-1611/27(17)5 () GQ5006 PROOFS M:FA:2016:72:4:0:0–0

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Diffraction patterns of p-type i-ScZn7.33

4057 unique reflection (F/sig>3)

CRISTAL beamline at the synchrotron SOLEIL

twofold threefold fivefold

Pm35<latexit sha1_base64="WVohS1q2xaXIn67WSQJfTQVrOU4=">AAAB+XicbVDLSsNAFL2pr1pfUZdugkVwVZJW0WXRjcsK9gFtKJPppB06Mwkzk0IJ+RM3LhRx65+482+cpllo64HLPZxzL3PnBDGjSrvut1Xa2Nza3invVvb2Dw6P7OOTjooSiUkbRyySvQApwqggbU01I71YEsQDRrrB9H7hd2dEKhqJJz2Pic/RWNCQYqSNNLTtFh8ESKaNLG/X2dCuujU3h7NOvIJUoUBraH8NRhFOOBEaM6RU33Nj7adIaooZySqDRJEY4Skak76hAnGi/DS/PHMujDJywkiaEtrJ1d8bKeJKzXlgJjnSE7XqLcT/vH6iw1s/pSJONBF4+VCYMEdHziIGZ0QlwZrNDUFYUnOrgydIIqxNWBUTgrf65XXSqde8Rq3+eFVt3hVxlOEMzuESPLiBJjxAC9qAYQbP8ApvVmq9WO/Wx3K0ZBU7p/AH1ucPba+Thg==</latexit>

Space group

a=5.012(3) Å

Yamada, Tsunetomo, et al. "Atomic structure and phason modes of the Sc–Zn icosahedral quasicrystal." IUCrJ 3.4 (2016): 247-258.

© 2020 Tsunetomo Yamada

Reconstructed electron density of p-type i-ScZn7.33

Pm35<latexit sha1_base64="WVohS1q2xaXIn67WSQJfTQVrOU4=">AAAB+XicbVDLSsNAFL2pr1pfUZdugkVwVZJW0WXRjcsK9gFtKJPppB06Mwkzk0IJ+RM3LhRx65+482+cpllo64HLPZxzL3PnBDGjSrvut1Xa2Nza3invVvb2Dw6P7OOTjooSiUkbRyySvQApwqggbU01I71YEsQDRrrB9H7hd2dEKhqJJz2Pic/RWNCQYqSNNLTtFh8ESKaNLG/X2dCuujU3h7NOvIJUoUBraH8NRhFOOBEaM6RU33Nj7adIaooZySqDRJEY4Skak76hAnGi/DS/PHMujDJywkiaEtrJ1d8bKeJKzXlgJjnSE7XqLcT/vH6iw1s/pSJONBF4+VCYMEdHziIGZ0QlwZrNDUFYUnOrgydIIqxNWBUTgrf65XXSqde8Rq3+eFVt3hVxlOEMzuESPLiBJjxAC9qAYQbP8ApvVmq9WO/Wx3K0ZBU7p/AH1ucPba+Thg==</latexit>

Space group

a=5.012(3) Å

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( 12 ,12 ,

12 ,

12 ,

12 ,

12 )

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( 12 , 0, 0, 0, 0, 0)<latexit sha1_base64="NbKtrXizsh1x3Ml+hEyN3N2gBDI=">AAACAXicbVDLSgMxFL1TX7W+Rt0IboJFqCBlpgp2WXDjsoJ9QDuUTJppQzOZIckIZagbf8WNC0Xc+hfu/BvTdgRtPZcLh3PuJbnHjzlT2nG+rNzK6tr6Rn6zsLW9s7tn7x80VZRIQhsk4pFs+1hRzgRtaKY5bceS4tDntOWPrqd+655KxSJxp8cx9UI8ECxgBGsj9eyjUlcHEpPUnaSVybnzU2c9u+iUnRnQMnEzUoQM9Z792e1HJAmp0IRjpTquE2svxVIzwumk0E0UjTEZ4QHtGCpwSJWXzi6YoFOj9FEQSdNCo5n6eyPFoVLj0DeTIdZDtehNxf+8TqKDqpcyESeaCjJ/KEg40hGaxoH6TFKi+dgQTCQzf0VkiE0g2oRWMCG4iycvk2al7F6UK7eXxVo1iyMPx3ACJXDhCmpwA3VoAIEHeIIXeLUerWfrzXqfj+asbOcQ/sD6+Aa8M5Rx</latexit>

C :V :

E :

Isostructure to CdYb

TY, et al. "Atomic structure and phason modes of the Sc–Zn icosahedral quasicrystal." IUCrJ 3.4 (2016): 247-258.

CC

C

© 2020 Tsunetomo Yamada

Reconstructed electron density of p-type i-ScZn7.33

Tsai-type clusterTY, et al. "Atomic structure and phason modes of the Sc–Zn icosahedral quasicrystal." IUCrJ 3.4 (2016): 247-258.

C

© 2020 Tsunetomo Yamada

Atomic structure of p-type i-ScZn7.33

Figure S5

0 100 200 300 400 500 6000

100

200

300

400

500

600

Fobs

Fcal

c

Rw = 0.0529R = 0.1090

0.001 0.01 0.1 1 10 1000.001

0.01

0.1

1

10

100

Fobs

Fcal

c

Rw = 0.0529R = 0.1090

Structure refinement CdYb modelsoftware qcdiff

TY, et al. "Atomic structure and phason modes of the Sc–Zn icosahedral quasicrystal." IUCrJ 3.4 (2016): 247-258.

© 2020 Tsunetomo Yamada

Software

© 2020 Tsunetomo Yamada

CrysAlisPRO

If you wish to use, contact to Prof. Takakura (A02)

Rigaku Oxford Diffraction

Diffraction Data

© 2020 Tsunetomo Yamada

QUASI - Software package for structure analysis

Yamamoto, A. (2008). Science and technology of advanced materials, 9(1), 013001.

http://wcp-ap.eng.hokudai.ac.jp/yamamoto/

© 2020 Tsunetomo Yamada

PyQC - Python tools for Quasi-Crystallography

>intersection=pod.Intersection(obj1,obj2)>obj3,obj4,obj5=intersection.using_tetrahedron()

obj1 obj2obj3(obj1 and obj2)

obj4 (obj1 not obj2)

PyQC will be released in the feature.

Python libraries for model constructions (only for icosahedral case):• Boolean operation• Delaunay triangulation (Tetrahedralization)• Interactive• Error free

Intersect & Subtract

6D coordinates are expressed in (a+b*TAU)/c, where a,b and c are integer.

obj5 (obj2 not obj1)

© 2020 Tsunetomo Yamada

Exercise

© 2020 Tsunetomo Yamada

Fibonacci Chain under Phason Strain

✓d0

1

d02

◆=

a2Dp⌧2 + 1

✓⌧ 11 ⌧

◆✓1 u0 1

◆✓a1a2

◆

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Phason matrix :

1/1 Approximant

T i =

✓Id U i

0 Id

◆

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Fibonacci lattice

Unit vectors under phason strain

’

’

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a2<latexit sha1_base64="utrxihzR+/YZjCtES1vVHkyyy9U=">AAAB9XicbVBNS8NAEN3Ur1q/qh69BIvgqSRVsMeCF48V7Ae0sWy2k3bpZhN2J2oJ+R9ePCji1f/izX/jts1BWx8MPN6bYWaeHwuu0XG+rcLa+sbmVnG7tLO7t39QPjxq6yhRDFosEpHq+lSD4BJayFFAN1ZAQ19Ax59cz/zOAyjNI3mH0xi8kI4kDzijaKT7PsIT+kFKs0FaywblilN15rBXiZuTCsnRHJS/+sOIJSFIZIJq3XOdGL2UKuRMQFbqJxpiyiZ0BD1DJQ1Be+n86sw+M8rQDiJlSqI9V39PpDTUehr6pjOkONbL3kz8z+slGNS9lMs4QZBssShIhI2RPYvAHnIFDMXUEMoUN7fabEwVZWiCKpkQ3OWXV0m7VnUvqrXby0qjnsdRJCfklJwTl1yRBrkhTdIijCjyTF7Jm/VovVjv1seitWDlM8fkD6zPH/0okss=</latexit>

Linear phason strain

d0i =

mX

i=1

Qija0j =

mX

i=1

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Determine the phason matrix for q/p approximant.Q1

© 2020 Tsunetomo Yamada

q/p：1/0, 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, ...

’

’for q/p approximant

e.g. 1/1 approximant

✓d0

1

d02

◆=

a2Dp⌧2 + 1

✓⌧ ⌧ ⇤ u� 11 u+ ⌧

◆✓a1a2

◆

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✓d0

1

d02

◆=

a2Dp⌧2 + 1

✓⌧ 11 ⌧

◆✓1 u0 1

◆✓a1a2

◆

<latexit sha1_base64="Zk3f9b0gMG0GQV0S3Ky4UqEQmL0=">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</latexit>

(d01 + d0

2)i =

a2Dp⌧2 + 1

�(⌧ ⇤ u� 1) + (u+ ⌧)

<latexit sha1_base64="ksiRIW3hkXNEehbDNgAKU4RtKzY=">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</latexit>

u =1� ⌧

1 + ⌧<latexit sha1_base64="hriP/HNvFiwS2G2ztXD+arOw/8Q=">AAACAHicbVDLSsNAFJ3UV62vqAsXbgaLIIglqYLdCAU3LivYBzShTKaTduhkEuYhlJCNv+LGhSJu/Qx3/o3TNAttPXC5h3PuZeaeIGFUKsf5tkorq2vrG+XNytb2zu6evX/QkbEWmLRxzGLRC5AkjHLSVlQx0ksEQVHASDeY3M787iMRksb8QU0T4kdoxGlIMVJGGthH+sYLBcKpe+EppLPUPc/7wK46NScHXCZuQaqgQGtgf3nDGOuIcIUZkrLvOonyUyQUxYxkFU9LkiA8QSPSN5SjiEg/zQ/I4KlRhjCMhSmuYK7+3khRJOU0CsxkhNRYLnoz8T+vr1XY8FPKE60Ix/OHQs2giuEsDTikgmDFpoYgLKj5K8RjZPJQJrOKCcFdPHmZdOo197JWv7+qNhtFHGVwDE7AGXDBNWiCO9ACbYBBBp7BK3iznqwX6936mI+WrGLnEPyB9fkDRVGWKw==</latexit>

u =q � p⌧

p+ q⌧<latexit sha1_base64="PlconGEkaoXsllSuj5f6QoY250Q=">AAACAnicbZDLSsNAFIYn9VbrLepK3ASLIIglqYLdCAU3LivYCzShTKaTduhkMp2LUEJx46u4caGIW5/CnW/jNM1Cqz8MfPznHM6cP+SUSOW6X1ZhaXllda24XtrY3NresXf3WjLRAuEmSmgiOiGUmBKGm4ooijtcYBiHFLfD0fWs3r7HQpKE3akJx0EMB4xEBEFlrJ59oK/8SECUjs+4r6Cepvx0nEHPLrsVN5PzF7wcyiBXo2d/+v0E6RgzhSiUsuu5XAUpFIogiqclX0vMIRrBAe4aZDDGMkizE6bOsXH6TpQI85hyMvfnRApjKSdxaDpjqIZysTYz/6t1tYpqQUoY1wozNF8UaeqoxJnl4fSJwEjRiQGIBDF/ddAQmkSUSa1kQvAWT/4LrWrFO69Uby/K9VoeRxEcgiNwAjxwCergBjRAEyDwAJ7AC3i1Hq1n6816n7cWrHxmH/yS9fENvV2Xnw==</latexit>

1/1 Approximant

Fibonacci Chain under Phason Strain

Determine the phason matrix for q/p approximant.Q1

internal component of d1'+d2' becomes zero

© 2020 Tsunetomo Yamada

Diffraction Pattern under Phason Strain

d⇤0

1

d⇤0

2

!=

a⇤p⌧2 + 1

✓⌧ 11 ⌧

◆✓1 0�u 1

◆✓a1a2

◆

<latexit sha1_base64="qS83jw/04Zj5eJyuQYWaoMYqfv0=">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</latexit>

1/1 ApproximantFibonacci chain

U i<latexit sha1_base64="W0mAZrrOC1uC5J2+Rhfa7/nmaLc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqYI8FLx4rmLbQxrLZTtulm03Y3Qgl9Dd48aCIV3+QN/+N2zYHbX0w8Hhvhpl5YSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYuizWMSqE1KNgkv0DTcCO4lCGoUC2+Hkdu63n1BpHssHM00wiOhI8iFn1FjJ9x8zPuuXK27VXYCsEy8nFcjR7Je/eoOYpRFKwwTVuuu5iQkyqgxnAmelXqoxoWxCR9i1VNIIdZAtjp2RC6sMyDBWtqQhC/X3REYjradRaDsjasZ61ZuL/3nd1AzrQcZlkhqUbLlomApiYjL/nAy4QmbE1BLKFLe3EjamijJj8ynZELzVl9dJq1b1rqq1++tKo57HUYQzOIdL8OAGGnAHTfCBAYdneIU3RzovzrvzsWwtOPnMKfyB8/kD7IuOug==</latexit>

T e =

✓Id 0�U i Id

◆

<latexit sha1_base64="4nbCqmDen8VvbglhmhnY8/aRfq4=">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</latexit>

d⇤0

i =mX

i=1

(MT e)ijaj<latexit sha1_base64="LVfIahPopI+PZuFBWWpUqbgk6vc=">AAACJnicbVDLSgMxFM34rPVVdelmsIiPRZlRQTeFghs3goJVoS8y6R2NJjNDckcsIV/jxl9x40IRceenmNYKvg4Ezj3nXm7uiTLBNQbBmzcyOjY+MVmYKk7PzM7NlxYWT3WaKwZ1lopUnUdUg+AJ1JGjgPNMAZWRgLPoer/vn92A0jxNTrCXQUvSi4THnFF0UqdUbSLcYhSbrm2bzTXbMdxWmzqXbSP7RTW064cnbQN2w1VX9qudOvPKdkrloBIM4P8l4ZCUyRBHndJTs5uyXEKCTFCtG2GQYctQhZwJsMVmriGj7JpeQMPRhErQLTM40/qrTun6carcS9AfqN8nDJVa92TkOiXFS/3b64v/eY0c472W4UmWIyTsc1GcCx9Tv5+Z3+UKGIqeI5Qp7v7qs0uqKEOXbNGFEP4++S853aqE25Wt451ybW8YR4EskxWyTkKyS2rkgByROmHkjjyQJ/Ls3XuP3ov3+tk64g1nlsgPeO8fL0qnbA==</latexit>

By the phason strain, how the external and internal components of a reflection HK changes?

d⇤1

<latexit sha1_base64="mlxoyQ8JsL8jPzEWZWcBR4gFUZ8=">AAAB+3icbVDLSsNAFJ3UV62vWJduBosgLkpSBbssuHFZwT6gjWEymbRDJw9mbqQl5FfcuFDErT/izr9x2mahrQcuHM65l3vv8RLBFVjWt1Ha2Nza3invVvb2Dw6PzONqV8WppKxDYxHLvkcUEzxiHeAgWD+RjISeYD1vcjv3e09MKh5HDzBLmBOSUcQDTgloyTWrQ2BT8ILMz93Mzh+zy9w1a1bdWgCvE7sgNVSg7ZpfQz+macgioIIoNbCtBJyMSOBUsLwyTBVLCJ2QERtoGpGQKSdb3J7jc634OIilrgjwQv09kZFQqVno6c6QwFitenPxP2+QQtB0Mh4lKbCILhcFqcAQ43kQ2OeSURAzTQiVXN+K6ZhIQkHHVdEh2Ksvr5Nuo25f1Rv317VWs4ijjE7RGbpANrpBLXSH2qiDKJqiZ/SK3ozceDHejY9la8koZk7QHxifP2v+lKY=</latexit>

d⇤2

<latexit sha1_base64="irpbcF7Iu+DvlmPW3nunGWEZ2cY=">AAAB+3icbVBNS8NAEN3Ur1q/Yj16CRZBPJSkCvZY8OKxgv2ANobNZtMu3WzC7kRaQv6KFw+KePWPePPfuG1z0NYHA4/3ZpiZ5yecKbDtb6O0sbm1vVPereztHxwemcfVropTSWiHxDyWfR8rypmgHWDAaT+RFEc+pz1/cjv3e09UKhaLB5gl1I3wSLCQEQxa8szqEOgU/DALci9r5I/ZZe6ZNbtuL2CtE6cgNVSg7ZlfwyAmaUQFEI6VGjh2Am6GJTDCaV4ZpoommEzwiA40FTiiys0Wt+fWuVYCK4ylLgHWQv09keFIqVnk684Iw1itenPxP2+QQth0MyaSFKggy0Vhyi2IrXkQVsAkJcBnmmAimb7VImMsMQEdV0WH4Ky+vE66jbpzVW/cX9dazSKOMjpFZ+gCOegGtdAdaqMOImiKntErejNy48V4Nz6WrSWjmDlBf2B8/gBth5Sn</latexit>

d⇤0

2<latexit sha1_base64="rn9e58hmaAzmXI4B43Qbsu+CH9U=">AAAB/HicbVDLSsNAFJ34rPUV7dLNYBHFRUmqYJcFNy4r2Ae0MUwmk3bo5MHMjRhC/BU3LhRx64e482+cPhbaeuDC4Zx7ufceLxFcgWV9Gyura+sbm6Wt8vbO7t6+eXDYUXEqKWvTWMSy5xHFBI9YGzgI1kskI6EnWNcbX0/87gOTisfRHWQJc0IyjHjAKQEtuWZlAOwRvCD3CzevF/f5+WnhmlWrZk2Bl4k9J1U0R8s1vwZ+TNOQRUAFUapvWwk4OZHAqWBFeZAqlhA6JkPW1zQiIVNOPj2+wCda8XEQS10R4Kn6eyInoVJZ6OnOkMBILXoT8T+vn0LQcHIeJSmwiM4WBanAEONJEtjnklEQmSaESq5vxXREJKGg8yrrEOzFl5dJp16zL2r128tqszGPo4SO0DE6Qza6Qk10g1qojSjK0DN6RW/Gk/FivBsfs9YVYz5TQX9gfP4A1MCU2A==</latexit>

d⇤0

1<latexit sha1_base64="RKgIxB20Hq+QYA2AAWBN6iOW7xQ=">AAAB/HicbVDLSsNAFJ34rPUV7dJNsIjioiRVsMuCG5cV7APaGCaTSTt0MgkzN2II8VfcuFDErR/izr9x+lho64ELh3Pu5d57/IQzBbb9baysrq1vbJa2yts7u3v75sFhR8WpJLRNYh7Lno8V5UzQNjDgtJdIiiOf064/vp743QcqFYvFHWQJdSM8FCxkBIOWPLMyAPoIfpgHhZc7xX1+flp4ZtWu2VNYy8SZkyqao+WZX4MgJmlEBRCOleo7dgJujiUwwmlRHqSKJpiM8ZD2NRU4osrNp8cX1olWAiuMpS4B1lT9PZHjSKks8nVnhGGkFr2J+J/XTyFsuDkTSQpUkNmiMOUWxNYkCStgkhLgmSaYSKZvtcgIS0xA51XWITiLLy+TTr3mXNTqt5fVZmMeRwkdoWN0hhx0hZroBrVQGxGUoWf0it6MJ+PFeDc+Zq0rxnymgv7A+PwB0zaU1w==</latexit>

h<latexit sha1_base64="iK2sHPtXjaZ6VFS2g//jwUZnswo=">AAAB8XicbVBNS8NAEN34WetX1aOXxSJ4KkkV7LHgxWMF+4FtKJvtpF262YTdiVhC/4UXD4p49d9489+4bXPQ1gcDj/dmmJkXJFIYdN1vZ219Y3Nru7BT3N3bPzgsHR23TJxqDk0ey1h3AmZACgVNFCihk2hgUSChHYxvZn77EbQRsbrHSQJ+xIZKhIIztNJDD+EJgzAbTfulsltx56CrxMtJmeRo9EtfvUHM0wgUcsmM6Xpugn7GNAouYVrspQYSxsdsCF1LFYvA+Nn84ik9t8qAhrG2pZDO1d8TGYuMmUSB7YwYjsyyNxP/87ophjU/EypJERRfLApTSTGms/fpQGjgKCeWMK6FvZXyEdOMow2paEPwll9eJa1qxbusVO+uyvVaHkeBnJIzckE8ck3q5JY0SJNwosgzeSVvjnFenHfnY9G65uQzJ+QPnM8fDNiRIQ==</latexit> h0

<latexit sha1_base64="evWO2F59H3Qb4Kg4xKfrl7ljECk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE9laQK9ljw4rGC/YA2lM120i7d7IbdiVhCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXJoIb9LxvZ219Y3Nru7BT3N3bPzgsHR23jEo1gyZTQulOSA0ILqGJHAV0Eg00DgW0w/HtzG8/gjZcyQecJBDEdCh5xBlFK3V7CE8YRtloetEvlb2KN4e7SvyclEmORr/01RsolsYgkQlqTNf3EgwyqpEzAdNiLzWQUDamQ+haKmkMJsjmJ0/dc6sM3EhpWxLdufp7IqOxMZM4tJ0xxZFZ9mbif143xagWZFwmKYJki0VRKlxU7ux/d8A1MBQTSyjT3N7qshHVlKFNqWhD8JdfXiWtasW/qlTvr8v1Wh5HgZySM3JJfHJD6uSONEiTMKLIM3klbw46L86787FoXXPymRPyB87nD3EzkVI=</latexit>

h = hd⇤1 + kd⇤

2<latexit sha1_base64="/EARftW/wCeFvja/sBLqJuahFYw=">AAACG3icbZDLSsNAFIYn9VbrLerSTbAIolCSKNiNUHDjsoK9QBvLZDpph04uzJyIJeQ93Pgqblwo4kpw4ds4TSPU1h8Gfr5zDmfO70acSTDNb62wtLyyulZcL21sbm3v6Lt7TRnGgtAGCXko2i6WlLOANoABp+1IUOy7nLbc0dWk3rqnQrIwuIVxRB0fDwLmMYJBoZ5ud4E+gOslw/Ry+Ov7aS+x0rvkJD0dzTI7Yz29bFbMTMaisXJTRrnqPf2z2w9J7NMACMdSdiwzAifBAhjhNC11Y0kjTEZ4QDvKBtin0kmy21LjSJG+4YVCvQCMjM5OJNiXcuy7qtPHMJTztQn8r9aJwas6CQuiGGhApou8mBsQGpOgjD4TlAAfK4OJYOqvBhligQmoOEsqBGv+5EXTtCvWWcW+OS/XqnkcRXSADtExstAFqqFrVEcNRNAjekav6E170l60d+1j2lrQ8pl99Efa1w83h6K4</latexit>

h = he + hi<latexit sha1_base64="Pif4HnnWfdNLfEIVDp8EGo5aPpI=">AAACEXicbZDLSgMxFIYz9VbrbdSlm8EiFIQyUwW7EQpuXFawF2hryaRn2tDMheSMWIZ5BTe+ihsXirh15863Mb2AtfWHwJf/nENyfjcSXKFtfxuZldW19Y3sZm5re2d3z9w/qKswlgxqLBShbLpUgeAB1JCjgGYkgfqugIY7vBrXG/cgFQ+DWxxF0PFpP+AeZxS11TULbYQHdL1kkF7+4l0C6en8laddM28X7YmsZXBmkCczVbvmV7sXstiHAJmgSrUcO8JOQiVyJiDNtWMFEWVD2oeWxoD6oDrJZKPUOtFOz/JCqU+A1sSdn0ior9TId3WnT3GgFmtj879aK0av3El4EMUIAZs+5MXCwtAax2P1uASGYqSBMsn1Xy02oJIy1CHmdAjO4srLUC8VnbNi6eY8XynP4siSI3JMCsQhF6RCrkmV1Agjj+SZvJI348l4Md6Nj2lrxpjNHJI/Mj5/APHfnvI=</latexit>

h0 = hd⇤0

1 + kd⇤0

2<latexit sha1_base64="pikPgt+pgaDKK/xLsVIQOLPj5PQ=">AAACHnicbZDLSsNAFIYn9VbrLerSTbBIRaEkVbEboeDGZQV7gTaWyXTSDJ1cmDkRS8iTuPFV3LhQRHClb+O0jVCrPwz8851zmDm/E3EmwTS/tNzC4tLySn61sLa+sbmlb+80ZRgLQhsk5KFoO1hSzgLaAAactiNBse9w2nKGl+N6644KycLgBkYRtX08CJjLCAaFevpZF+g9OG7ipaUL7+fST3uJld4mR6X0eDgLK1PY04tm2ZzI+GuszBRRpnpP/+j2QxL7NADCsZQdy4zATrAARjhNC91Y0giTIR7QjrIB9qm0k8l6qXGgSN9wQ6FOAMaEzk4k2Jdy5Duq08fgyfnaGP5X68TgVu2EBVEMNCDTh9yYGxAa46yMPhOUAB8pg4lg6q8G8bDABFSiBRWCNb/yX9OslK2TcuX6tFirZnHk0R7aR4fIQueohq5QHTUQQQ/oCb2gV+1Re9betPdpa07LZnbRL2mf34avo0s=</latexit>

h0 = he0 + hi0<latexit sha1_base64="85KflWkNGuIjZYzTKyR47FTnHeE=">AAACFHicbZDLSgMxFIYz9VbrrerSzWCRCoUyUwW7EQpuXFawF2hryaRn2tDMheSMWIZ5CDe+ihsXirh14c63Mb2AtfWHwJf/nENyficUXKFlfRupldW19Y30ZmZre2d3L7t/UFdBJBnUWCAC2XSoAsF9qCFHAc1QAvUcAQ1neDWuN+5BKh74tzgKoePRvs9dzihqq5sttBEe0HHjQZK//OW7GPJJYf7O80k3m7OK1kTmMtgzyJGZqt3sV7sXsMgDH5mgSrVsK8ROTCVyJiDJtCMFIWVD2oeWRp96oDrxZKnEPNFOz3QDqY+P5sSdn4ipp9TIc3SnR3GgFmtj879aK0K33Im5H0YIPps+5EbCxMAcJ2T2uASGYqSBMsn1X002oJIy1DlmdAj24srLUC8V7bNi6eY8VynP4kiTI3JMTolNLkiFXJMqqRFGHskzeSVvxpPxYrwbH9PWlDGbOSR/ZHz+ADl+n4U=</latexit>

Phason matrix :

Unit vectors under phason strain

Linear phason strain

Q2

© 2020 Tsunetomo Yamada

HK reflection Finbonacci lattice Finonacci lattice under phason strain

External component

Internalcomponent

a⇤

p⌧2 + 1

(⌧H +K)<latexit sha1_base64="52111G1b+/qrAX8rqUyfxryaa2g=">AAACEHicbZC7SgNBFIZn4y3GW9TSZjCI0UDYjYIpAzYBmwjmAtkkzE5mkyGzF2fOCmHZR7DxVWwsFLG1tPNtnFwKTfxh4OM/53Dm/E4ouALT/DZSK6tr6xvpzczW9s7uXnb/oKGCSFJWp4EIZMshignuszpwEKwVSkY8R7CmM7qe1JsPTCoe+HcwDlnHIwOfu5wS0FYve2q7ktCYdM+T2Fb3EuLYBhIl3VLBSpL8hHG1cHPWy+bMojkVXgZrDjk0V62X/bL7AY085gMVRKm2ZYbQiYkETgVLMnakWEjoiAxYW6NPPKY68fSgBJ9op4/dQOrnA566vydi4ik19hzd6REYqsXaxPyv1o7ALXdi7ocRMJ/OFrmRwBDgSTq4zyWjIMYaCJVc/xXTIdEJgc4wo0OwFk9ehkapaF0US7eXuUp5HkcaHaFjlEcWukIVVEU1VEcUPaJn9IrejCfjxXg3PmatKWM+c4j+yPj8AfyKnHk=</latexit>

a⇤

p⌧2 + 1

(�H + ⌧K)<latexit sha1_base64="6wF7/k5sSkP/V3s1QKXiVc71f3o=">AAACEXicbVC7SgNBFJ2NrxhfUUubwSBEg2E3CqYM2ARsIpgHZJMwO5lNhsw+nLkrhGV/wcZfsbFQxNbOzr9x8ig08cCFM+fcy9x7nFBwBab5baRWVtfWN9Kbma3tnd297P5BQwWRpKxOAxHIlkMUE9xndeAgWCuUjHiOYE1ndD3xmw9MKh74dzAOWccjA5+7nBLQUi+bt11JaEy6Z0lsq3sJcWwDiZJuqWAlSf68Wpg88c1pL5szi+YUeJlYc5JDc9R62S+7H9DIYz5QQZRqW2YInZhI4FSwJGNHioWEjsiAtTX1icdUJ55elOATrfSxG0hdPuCp+nsiJp5SY8/RnR6BoVr0JuJ/XjsCt9yJuR9GwHw6+8iNBIYAT+LBfS4ZBTHWhFDJ9a6YDomOCHSIGR2CtXjyMmmUitZFsXR7mauU53Gk0RE6RnlkoStUQVVUQ3VE0SN6Rq/ozXgyXox342PWmjLmM4foD4zPH3IFnLA=</latexit>

a⇤

p⌧2 + 1

(�H + ⌧K)<latexit sha1_base64="6wF7/k5sSkP/V3s1QKXiVc71f3o=">AAACEXicbVC7SgNBFJ2NrxhfUUubwSBEg2E3CqYM2ARsIpgHZJMwO5lNhsw+nLkrhGV/wcZfsbFQxNbOzr9x8ig08cCFM+fcy9x7nFBwBab5baRWVtfWN9Kbma3tnd297P5BQwWRpKxOAxHIlkMUE9xndeAgWCuUjHiOYE1ndD3xmw9MKh74dzAOWccjA5+7nBLQUi+bt11JaEy6Z0lsq3sJcWwDiZJuqWAlSf68Wpg88c1pL5szi+YUeJlYc5JDc9R62S+7H9DIYz5QQZRqW2YInZhI4FSwJGNHioWEjsiAtTX1icdUJ55elOATrfSxG0hdPuCp+nsiJp5SY8/RnR6BoVr0JuJ/XjsCt9yJuR9GwHw6+8iNBIYAT+LBfS4ZBTHWhFDJ9a6YDomOCHSIGR2CtXjyMmmUitZFsXR7mauU53Gk0RE6RnlkoStUQVVUQ3VE0SN6Rq/ozXgyXox342PWmjLmM4foD4zPH3IFnLA=</latexit>

a⇤

p⌧2 + 1

(⌧H +K)� ua⇤

p⌧2 + 1

(�H + ⌧K)<latexit sha1_base64="FvKnMH7GX8PXvc20Zbf9WMgplsc=">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</latexit>

By the phason strain, how the external and internal components of a reflection HK changes?

Q2

Diffraction Pattern under Phason Strain

© 2020 Tsunetomo Yamada

Useful way to visualize the 6D structure

Fivefold Threefold Twofold

d2 + d3 + d4 + d5 + d6<latexit sha1_base64="B12rsy6eyUjp5vcdeM7JoUiJ/ZU=">AAACM3icbVBLS8NAEN7UV62vqEcvwSIIQknaqj0WvIinCvYBbSmb7aZdutmE3YlYQv6TF/+IB0E8KOLV/+C2zaEPBxa+xwyz87khZwps+93IrK1vbG5lt3M7u3v7B+bhUUMFkSS0TgIeyJaLFeVM0Dow4LQVSop9l9OmO7qZ+M1HKhULxAOMQ9r18UAwjxEMWuqZdx2gT+B6cT/pxcXkYp6WFml5kV4u0qukZ+btgj0taxU4KcijtGo987XTD0jkUwGEY6Xajh1CN8YSGOE0yXUiRUNMRnhA2xoK7FPVjac3J9aZVvqWF0j9BFhTdX4ixr5SY9/VnT6GoVr2JuJ/XjsCr9KNmQgjoILMFnkRtyCwJgFafSYpAT7WABPJ9F8tMsQSE9Ax53QIzvLJq6BRLDilQvG+nK9W0jiy6ASdonPkoGtURbeohuqIoGf0hj7Rl/FifBjfxs+sNWOkM8dooYzfP5n5rUU=</latexit>

d1<latexit sha1_base64="NnGMc3/F1IZjNbGZWUvi5vthBGs=">AAAB9XicbVDLSgNBEOyNrxhfUY9eBoPgKexGwRwDXjxGMA9IYpidnU2GzD6Y6VXDsv/hxYMiXv0Xb/6Nk2QPmljQUFR1093lxlJotO1vq7C2vrG5Vdwu7ezu7R+UD4/aOkoU4y0WyUh1Xaq5FCFvoUDJu7HiNHAl77iT65nfeeBKiyi8w2nMBwEdhcIXjKKR7vvIn9D1Uy8bpk42LFfsqj0HWSVOTiqQozksf/W9iCUBD5FJqnXPsWMcpFShYJJnpX6ieUzZhI54z9CQBlwP0vnVGTkzikf8SJkKkczV3xMpDbSeBq7pDCiO9bI3E//zegn69UEqwjhBHrLFIj+RBCMyi4B4QnGGcmoIZUqYWwkbU0UZmqBKJgRn+eVV0q5VnYtq7fay0qjncRThBE7hHBy4ggbcQBNawEDBM7zCm/VovVjv1seitWDlM8fwB9bnDwBNks0=</latexit>

d2 � d4 + d6<latexit sha1_base64="mKiJA/Txhdozeil3I4HW8/aVJIQ=">AAACFXicbVDLSsNAFJ3UV62vqEs3wSIIaklq0S4LblxWsA9oQ5hMJu3QyYOZG7GE/IQbf8WNC0XcCu78G6dtFrX1wMC559zLnXvcmDMJpvmjFVZW19Y3ipulre2d3T19/6Ato0QQ2iIRj0TXxZJyFtIWMOC0GwuKA5fTjju6mfidByoki8J7GMfUDvAgZD4jGJTk6Od9oI/g+qmXOWk1u5gva9nZfHmVOXrZrJhTGMvEykkZ5Wg6+nffi0gS0BAIx1L2LDMGO8UCGOE0K/UTSWNMRnhAe4qGOKDSTqdXZcaJUjzDj4R6IRhTdX4ixYGU48BVnQGGoVz0JuJ/Xi8Bv26nLIwToCGZLfITbkBkTCIyPCYoAT5WBBPB1F8NMsQCE1BBllQI1uLJy6RdrViXlepdrdyo53EU0RE6RqfIQteogW5RE7UQQU/oBb2hd+1Ze9U+tM9Za0HLZw7RH2hfv1eRoCU=</latexit> d2 + d3

<latexit sha1_base64="WWRkbhNJ3nsdGwGvE3TuFGQRqYo=">AAACBnicbVDLSsNAFJ3UV62vqEsRgkUQhJK0gl0W3LisYB/QhjCZTtqhk0mYuRFLyMqNv+LGhSJu/QZ3/o3TNovaeuDCmXPuZe49fsyZAtv+MQpr6xubW8Xt0s7u3v6BeXjUVlEiCW2RiEey62NFORO0BQw47caS4tDntOOPb6Z+54FKxSJxD5OYuiEeChYwgkFLnnnaB/oIfpAOMi+tZpeLz1rmmWW7Ys9grRInJ2WUo+mZ3/1BRJKQCiAcK9Vz7BjcFEtghNOs1E8UjTEZ4yHtaSpwSJWbzs7IrHOtDKwgkroEWDN1cSLFoVKT0NedIYaRWvam4n9eL4Gg7qZMxAlQQeYfBQm3ILKmmVgDJikBPtEEE8n0rhYZYYkJ6ORKOgRn+eRV0q5WnFqlendVbtTzOIroBJ2hC+Sga9RAt6iJWoigJ/SC3tC78Wy8Gh/G57y1YOQzx+gPjK9fNhaZjw==</latexit>

d1 � d5<latexit sha1_base64="dXl3MxLtRZG7gyL7I0U0pCcVrV8=">AAACBnicbVDLSsNAFJ3UV62vqEsRgkVwY0mqYpcFNy4r2Ae0IUymk3boZBJmbsQSsnLjr7hxoYhbv8Gdf+O0zaK2Hrhw5px7mXuPH3OmwLZ/jMLK6tr6RnGztLW9s7tn7h+0VJRIQpsk4pHs+FhRzgRtAgNOO7GkOPQ5bfujm4nffqBSsUjcwzimbogHggWMYNCSZx73gD6CH6T9zEud7Hz+eZV5Ztmu2FNYy8TJSRnlaHjmd68fkSSkAgjHSnUdOwY3xRIY4TQr9RJFY0xGeEC7mgocUuWm0zMy61QrfSuIpC4B1lSdn0hxqNQ49HVniGGoFr2J+J/XTSCouSkTcQJUkNlHQcItiKxJJlafSUqAjzXBRDK9q0WGWGICOrmSDsFZPHmZtKoV56JSvbss12t5HEV0hE7QGXLQNaqjW9RATUTQE3pBb+jdeDZejQ/jc9ZaMPKZQ/QHxtcvOrCZkg==</latexit>

5fe 3fe 2fe

5f i 3f i 2f i

d1 � d3 � d5<latexit sha1_base64="fEW6byqHOrQ9pQITHjKtCzyNE6A=">AAACFXicbVDLSsNAFJ3UV62vqEs3wSK40JK0il0W3LisYB/QhjCZTNqhkwczN2IJ+Qk3/oobF4q4Fdz5N07bLPrwwMC559zLnXvcmDMJpvmrFdbWNza3itulnd29/QP98Kgto0QQ2iIRj0TXxZJyFtIWMOC0GwuKA5fTjju6nfidRyoki8IHGMfUDvAgZD4jGJTk6Bd9oE/g+qmXOamVXc6XtcXyOnP0slkxpzBWiZWTMsrRdPSfvheRJKAhEI6l7FlmDHaKBTDCaVbqJ5LGmIzwgPYUDXFApZ1Or8qMM6V4hh8J9UIwpur8RIoDKceBqzoDDEO57E3E/7xeAn7dTlkYJ0BDMlvkJ9yAyJhEZHhMUAJ8rAgmgqm/GmSIBSaggiypEKzlk1dJu1qxapXq/VW5Uc/jKKITdIrOkYVuUAPdoSZqIYKe0St6Rx/ai/amfWpfs9aCls8cowVo339V+aAk</latexit>

Determine the phason matrix for 1/1 approximant.Q3

2D sections of 6D direct lattice

© 2020 Tsunetomo Yamada

Phason Matrix

T i =

✓Id U i

0 Id

◆

<latexit sha1_base64="fgLCpg8M1FBfwCgjdR+MdLyH424=">AAACJ3icbVDLSgMxFM34rOOr6tJNsCiuykwV7EYpuNFdhb6gU0smc9uGZjJDkhHL0L9x46+4EVREl/6J6QPR1gOBwzn33tx7/JgzpR3n01pYXFpeWc2s2esbm1vb2Z3dmooSSaFKIx7Jhk8UcCagqpnm0IglkNDnUPf7lyO/fgdSsUhU9CCGVki6gnUYJdpI7exF5TZlw3Pb86HLRBqHREt2P7Sv22kwxEe4OrI9z3YMH2u2ByL4qWtnc07eGQPPE3dKcmiKcjv74gURTUIQmnKiVNN1Yt1KidSMcjDTEwUxoX3ShaahgoSgWun4ziE+NEqAO5E0T2g8Vn93pCRUahD6ptLs11Oz3kj8z2smulNspUzEiQZBJx91Eo51hEeh4YBJoJoPDCFUMrMrpj0iCdUmWtuE4M6ePE9qhbx7ki/cnOZKxWkcGbSPDtAxctEZKqErVEZVRNEDekKv6M16tJ6td+tjUrpgTXv20B9YX9/TtaVH</latexit>

U i<latexit sha1_base64="W0mAZrrOC1uC5J2+Rhfa7/nmaLc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqYI8FLx4rmLbQxrLZTtulm03Y3Qgl9Dd48aCIV3+QN/+N2zYHbX0w8Hhvhpl5YSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYuizWMSqE1KNgkv0DTcCO4lCGoUC2+Hkdu63n1BpHssHM00wiOhI8iFn1FjJ9x8zPuuXK27VXYCsEy8nFcjR7Je/eoOYpRFKwwTVuuu5iQkyqgxnAmelXqoxoWxCR9i1VNIIdZAtjp2RC6sMyDBWtqQhC/X3REYjradRaDsjasZ61ZuL/3nd1AzrQcZlkhqUbLlomApiYjL/nAy4QmbE1BLKFLe3EjamijJj8ynZELzVl9dJq1b1rqq1++tKo57HUYQzOIdL8OAGGnAHTfCBAYdneIU3RzovzrvzsWwtOPnMKfyB8/kD7IuOug==</latexit>

Linier Phason Matrix

d0i =

mX

i=1

Qija0j =

mX

i=1

(QT i)ijaj<latexit sha1_base64="nhlqjQ9os8nUymtyfHdLR4gN5qQ=">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</latexit>

In the case of 1/1 approximant、internal component of d1'+d2'+d3'-d5' becomes zero.

Twofold

d2 + d3<latexit sha1_base64="WWRkbhNJ3nsdGwGvE3TuFGQRqYo=">AAACBnicbVDLSsNAFJ3UV62vqEsRgkUQhJK0gl0W3LisYB/QhjCZTtqhk0mYuRFLyMqNv+LGhSJu/QZ3/o3TNovaeuDCmXPuZe49fsyZAtv+MQpr6xubW8Xt0s7u3v6BeXjUVlEiCW2RiEey62NFORO0BQw47caS4tDntOOPb6Z+54FKxSJxD5OYuiEeChYwgkFLnnnaB/oIfpAOMi+tZpeLz1rmmWW7Ys9grRInJ2WUo+mZ3/1BRJKQCiAcK9Vz7BjcFEtghNOs1E8UjTEZ4yHtaSpwSJWbzs7IrHOtDKwgkroEWDN1cSLFoVKT0NedIYaRWvam4n9eL4Gg7qZMxAlQQeYfBQm3ILKmmVgDJikBPtEEE8n0rhYZYYkJ6ORKOgRn+eRV0q5WnFqlendVbtTzOIroBJ2hC+Sga9RAt6iJWoigJ/SC3tC78Wy8Gh/G57y1YOQzx+gPjK9fNhaZjw==</latexit>

d1 � d5<latexit sha1_base64="dXl3MxLtRZG7gyL7I0U0pCcVrV8=">AAACBnicbVDLSsNAFJ3UV62vqEsRgkVwY0mqYpcFNy4r2Ae0IUymk3boZBJmbsQSsnLjr7hxoYhbv8Gdf+O0zaK2Hrhw5px7mXuPH3OmwLZ/jMLK6tr6RnGztLW9s7tn7h+0VJRIQpsk4pHs+FhRzgRtAgNOO7GkOPQ5bfujm4nffqBSsUjcwzimbogHggWMYNCSZx73gD6CH6T9zEud7Hz+eZV5Ztmu2FNYy8TJSRnlaHjmd68fkSSkAgjHSnUdOwY3xRIY4TQr9RJFY0xGeEC7mgocUuWm0zMy61QrfSuIpC4B1lSdn0hxqNQ49HVniGGoFr2J+J/XTSCouSkTcQJUkNlHQcItiKxJJlafSUqAjzXBRDK9q0WGWGICOrmSDsFZPHmZtKoV56JSvbss12t5HEV0hE7QGXLQNaqjW9RATUTQE3pBb+jdeDZejQ/jc9ZaMPKZQ/QHxtcvOrCZkg==</latexit>

2fe

2f i

（3×3）

Determine the phason matrix for 1/1 cubic approximant.Q3

Useful way to visualize the 6D structure

0

BBBBBB@

d1

d2

d3

d4

d5

d6

1

CCCCCCA=

ap⌧2 + 1

0

BBBBBB@

1 ⌧ 0 ⌧ �1 0⌧ 0 1 �1 0 ⌧⌧ 0 �1 �1 0 �⌧0 1 �⌧ 0 ⌧ 1�1 ⌧ 0 �⌧ �1 00 1 ⌧ 0 ⌧ �1

1

CCCCCCA

0

BBBBBB@

a1a2a3a4a5a6

1

CCCCCCA

<latexit sha1_base64="vQ+T1uxrvYx1MGyqOK2R9+TaBbw=">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</latexit>

Q = M�1<latexit sha1_base64="ixi/HIAsBLOyhSsSLqxfEyE92Lw=">AAAB+XicbVBNS8NAEN34WetX1KOXxSJ4sSRVsBeh4MWL0IL9gDaWzWbTLt1swu6kUEL/iRcPinj1n3jz37htc9DWBwOP92aYmecngmtwnG9rbX1jc2u7sFPc3ds/OLSPjls6ThVlTRqLWHV8opngkjWBg2CdRDES+YK1/dHdzG+PmdI8lo8wSZgXkYHkIacEjNS37cZtD7gIWPYwfcou3WnfLjllZw68StyclFCOet/+6gUxTSMmgQqiddd1EvAyooBTwabFXqpZQuiIDFjXUEkipr1sfvkUnxslwGGsTEnAc/X3REYirSeRbzojAkO97M3E/7xuCmHVy7hMUmCSLhaFqcAQ41kMOOCKURATQwhV3NyK6ZAoQsGEVTQhuMsvr5JWpexelSuN61KtmsdRQKfoDF0gF92gGrpHddREFI3RM3pFb1ZmvVjv1seidc3KZ07QH1ifPwVXkzo=</latexit>