!/0stafg#Š‹Œdaçf€Å · !/0stafg#Š‹Œdaçf€Å hhhŽèê‘gŽè’ “ m. baake, "a...
TRANSCRIPT
!/0STAFG#Š‹ŒdAçf€Å
HHHŽèê‘gŽè’“ M. Baake, "A Guide to Mathematical Quasicrystals", arXiv:math-ph/9901014v1 20 Jan 1999.“”W. Steurer and S. Deloudi, "Crystallography of Quasicrystals", Springer series in materials science 126, Springer, 2009, ISBN 978-3-642-01899-2.“ A. Yamamoto, Acta Crystallogr., A52 (1996) 509.!"The Physics of Quasicrystals", edited by P.J. Steinhardt and S. Ostlund, World Scientific, 1987 ISBN 9971-50-226-7.“”•–—˜g2!/0A™š›7ST<”œùžŸ ¡¢£i“”¤¥”¦g§¨©ªg«”¬g®¯”°g2!/0AŠG<”±²’³
´µG¶d·¸¹Hº»H¼
Outline 1. !"#$%&'()*$%+,-. 2. !/0*12/03145678+,9:;<=>?@A 3. !BC;#D12E.!/0AFGHH3.1HInflationIJK!L%AMNOHH3.2HPQRBCST*AUVHH3.3HDodecagonalWL%=MKX+Y9:;AZ[ 4. \]^D12E.!/0AST 5. D20_`!/0 5.1 aAbc 5.2 D20_`!/0AFG
FdAef31ghijklm*n.oFApqr
s@tuvgwxyt8*zu1g2\]=M{|!/0=}?~�<€ÅPQR‚ƒw3=„7…†‡ˆ‰K*zuX
―�1�―
½¾”!"#$%&'()*$%+,-. ¿ÀAÁÂ
ÃÄ=+Y?Kz*IJ{|3QRA+,9:;=ÅÆKz*ˆ3ÇKXJ.M. Cowley, "Diffraction physics" 2nd edition, North-Holland, 1981.
zAÈÉIsÊK
hijËlmAÂ
ÌÍ
!
" u,v( )2
= " #u,#v( )2
~�31+,ÎÏAÐшeÒ56KX
+,#bÂAOÓÔÕFÖ×ØÙkÚ
ÛAÎ8ÜÝ
D20_`!/0 D12E.!/0
2. !/0*12/03145678+,9:;<=>?@A
Þ]ß!àÔ1+áâÚ/23bcãÊÅ6KäßFåæ7çèéêë
ìíAîGF#2+áï
Zn80Mg5Sc15 Mn74Cr10Ni1Si15
5+ð[ñA$%+,-. 12+ð[ñA$%+,-.
òIóKJ…IgGôæ7!/0A+,f1"UF3‰KX
―�2�―
!/0Aüý;
!/0Quasicrystals 2QR!/0Ô1ðOÓIBCæÚ
D8E., D10E., D12E.!/0Octagonal, Decagonal and Dodecagonal quasicrystals
þÿ/0(!/0}!A"#Š‹gŠ;AŽ$Š‹Ú
3QR!/0D20_`!/0Icosahedral quasicrystal
/0dæ7efHWL%#!BCæ7%&9:;HHHHHH +,9:;H'¹æ7STÔWL%=()?K@AÚ#HHHHHH*%×+,£ig-+.
S/R0AefH]1“\]g¡+23×,gPÐ%g-4ö5^....
zz€ÅA›631gw7WL%=8N98gQIa6=*%3()t|g!/0AST=G‚?KX
3. !BC;#D12E.!/0AFGDO.*DõE.A£öÙ=÷{|•_=øùKúû
―�3�―
DO.*D3E.ABCæ£öj¢ùg?7:;þÿ/0
<ú#BC;==x78£öÙ>N@‰KA€?r@A23xÅù<I?‰K81gBCæ37vBCæI?
“DE=FÊ|1GH*8…IJ“KLMA£öÙ=÷…z*€ÅNOKPQÏDïE.*DO.A£öÙ>N1RS3178X
D12E.!/0AWL%#!BCTUÙ*+¢G.£öj¢ùTUÙ
M.Baake et al., Physica A, 191Ô1992Ú 554. H HHHH. Kawamura, Prog. Theor. Phys. 70 (1983) 352.
2+áïIJKäßFåæêë#D12E.AVW=Xn3/Y*gZ2[oA3E4E£j¢ùˆ\6KX3E4EA]F14#áï
^_AË¢Ø4`i3¬Â¶?Ë¢Ø4`i13E4EA]Fˆ4#áï3ab
cd33E4EA߈ªe?KGQ1bÇ7f
―�4�―
M.Baake, R.Klitzling andM. Schlottmann,Physica A 191
(1992) 554.X. Zeng and G.Unger, Phil. Mag.86 (2006) 1093.
3.1.HInflationIJK!L%AMNO
2+áïgIhbtgifID12E.=^_
StampfliTUÙ312LMAD12E.Ajk=+¢G.I„7…X
P. Stampfli, Helv.Phys. Acta, 59
(1986) 1260.
SchlottmannÅATUÙ31gD12E.Ajk=l[oAifALM3m\ÐÊ?K
m\ÐÊ1
m\ÐÊ2
!BC;ÔäßFåæ7çèéêëÚ
Z2[oAD3E.I1n57DO.3]gD3E.7]
Z2[oADO.I1n57DO.7]gD3E.16]
!
Sn+1
Tn+1
"
# $
%
& ' = A
Sn
Tn
"
# $
%
& '
!
A =7 3
16 7
"
# $
%
& '
!
n"#, 7 3
16 7
$
% &
'
( ) S
T
$
% & '
( ) = *
S
T
$
% & '
( )
oß!1„åAAìípAªq:
!
" = 7+ 4 3 = 2+ 3( )2
Zn[oIsÊKDO.AFSn*DõE.AFTnIq8|Ar¶Í
sATUÙ31gtAç5Aß@2+áïgHÔç5AIJ1÷:7€{xAIg2+áïˆ3|ÇxuÚ
―�5�―
3.2.HPQRBCST*AUV
“ƒ C‚IJKL%fAjkgQIjk56xL%fAŠGFEÛAñ„ {r// | r‚†‡‚} HƒC‚=…wvˆÂ?K*g2LMAç5L*S3ŠGFEˆøùÅ6KX“ŠGFE/ÐIßr?Kphason‰D=Z[?K*gBC^å=M6KXHHHHHHH {r// | r‚+#r‚†‡‚},H#r‚= Dr// HHH
: Quasiperiodic tiling, : Approximant, : Random tiling
ŠRAxùg2QRDOL%€Å1QRWL%=MKm\=è@KX‹ÇˆîGFAŒAXzçŽÔr//, r‚)=ˆÂ?KXÔòAxùgr//, r‚=Ö×ØÙ*t|~èÚÚ
ŠGFE
phasonFE
L: LongS: Short
A. Katz and M. Duneau, J. Physique, 47 (1986) 181.V. Elser, Acta Crystallogr., A42 (1986) 36.A.Yamamoto, Acta Crystallogr., A52 (1996) 509.
!
L(r//, r" ) = # r $ r
nm( )n,m=-%
%
& , r = r//
+ r", rnm
= a(ne1+me
2)
22QRDOL%v½QRWL%<Aw*ùH
!
W (r//, r" ) =
1
0
# $ %
!
r" # C" Am\
a6wx
!
"(r//) = L(r
//, r# )W (r//, r# )dr#$%
%
&
xytge1*e21DBXz{|gngm1}F3‰KXUFL1BCL%=~?A3gaAhijËlm1�L%*7KX?7:;ghgkˆ}F3‰K�L%fyÊ3p=@qX7sgu1�FEA€_Ö×ØÙ3‰KX
!
F L(r)[ ] =1
a2
"(u#uhk
h,k=#$
$
% ), uhk
=1
a(he
1+ ke
2)
HL%ÂFaA2QRDOL%1"UFABC^å*t|~56KXaAUFL1L%f3yÊp==;ga6wx3103‰KXwxg€_Ö×ØÙrAŠGFE*phasonFE/Ð=ga6Å6r//*r‚*?KX
HWL%=~?UF!1gƒ=~?UFW=UFLI€Ê|L%fAjk=
„78gQIŠGFEIñ„t|…Å6KX
―�6�―
!
"(u// ) = F [#(r// )]
= L(r//, r$ )W (r//, r$ )dr$%&
&
'( )%&
&
' exp %2(ir// )u//( )dr//
= F L(r//, r$ )W (r//, r$ )[ ]}{u$=0
= F L(r//, r$ )[ ]{ *F W (r//, r$ )[ ]}u$=0
=1
a2
+(u%uhk
h.k=%&
&
, )- . /
*F W (r//, r$ )[ ]}u$=0
š{|gWL%1UÙ£UFA+,f=NOKX*1›œùžÐ!a1DOL%AªtAç5
WL%AhijËlmgHH+,ÎÏ
AhijËlm"(u//)AŸ
!
"(r//)
zAè@O=2PQRBCL% v”2QRg3QRWL%<Ih¡t|¢q?KXÔPQRL%*ƒWˆ+Y9:;=£xt|gaz€Å+,9:;ˆNOKXÚ
A. Katz and M. Duneau, J. Physique, 47 (1986) 181.HV. Elser, Acta Crystallogr., A42 (1986) 36.
5ÅIgPQRBC/0AR€¤¥Iƒ=ˆÂt|g*%^_=8N9…X!/0AST‚ƒ*1g+,ÎÏ｜"(u//)｜2€ÅPQR/0AST=¥ùK’X
A. Yamamoto, Acta Crystallogr., A52 (1996) 509.
M. de Boissieu, Lectures on Quasicrystals (1994)I¦§
2ñ„AhijËlm<à22QRhijËlmA¨_<
!
A =
cos"
6#sin
"
60 0
sin"
6cos
"
60 0
0 0 cos#5"
6#sin
#5"
6
0 0 sin#5"
6cos
#5"
6
$
%
& & & & & & & &
'
(
) ) ) ) ) ) ) )
=1
2
3 #1 0 0
1 3 0 0
0 0 # 3 1
0 0 #1 # 3
$
%
& & & &
'
(
) ) ) )
3.3HDodecagonalWL%=MKX+Y9:;AZ[4QRFE31g12+9:AL%gdodecagonalL%ˆxy1qê‘?K’ˆ“Å6|8KXaAdodecagonalL%1gw”A2qAgenerator €ÅS/56KXH. Brown et al., "Crystallographic Groups of Four-Dimensional Space", John Wiley & Sons, Inc, 1978, ISBN0-471-03095-3
!
R12
=
0 0 0 "1
1 0 0 0
0 1 0 1
0 0 1 0
#
$
% % % %
&
'
( ( ( (
!
R2
=
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
"
#
$ $ $ $
%
&
' ' ' '
#R12112+A+Y•M31‰KˆgDBXz{|A+Y„å3178X
!
R12
12= E
w”31gR12I–ÿ3DBXz{|I{—vw”A„å=è@KX
P.W.Leung, C.L.Henley, G.V.Chester, Phys. Rev. B, 39 (1989) 446.W. Steurer and S. Deloudi, "Crystallography of Quasicrystals", Springer, 2009.
ŠGFE3A+Y
phasonFE3A+Y
˜‡#2qA2QRFEIs8|12+9:;ˆ™ê56|8KX
―�7�―
R12*A=U©—ÊK„å B
!
A = BR12B"1
!
B =1
2
2 3 1 0
0 1 3 2
"2 3 "1 0
0 1 " 3 2
#
$
% % % %
&
'
( ( ( (
„åR12*A1–ÿ
zz3g„åB1
B1dodecagonal^3AçŽ=DBXz^Ilm?K„å7A3g
!
1
0
0
0
"
#
$ $ $ $
%
&
' ' ' '
(
1
0
)1
0
"
#
$ $ $ $
%
&
' ' ' '
!
0
1
0
0
"
#
$ $ $ $
%
&
' ' ' '
(
3 /2
1/2
3 /2
1/2
"
#
$ $ $ $
%
&
' ' ' '
!
0
0
1
0
"
#
$ $ $ $
%
&
' ' ' '
(
1/2
3 /2
)1/2
) 3 /2
"
#
$ $ $ $
%
&
' ' ' '
!
0
0
0
1
"
#
$ $ $ $
%
&
' ' ' '
(
0
1
0
1
"
#
$ $ $ $
%
&
' ' ' '
DodecagonalA{|Ö×ØÙ1gzAJ…Ilm56KX
ª ª ª
ª
!
B"1
=1
2 3
3 "1 " 3 1
2 0 2 0
0 2 0 "2
"1 3 "1 3
#
$
% % % %
&
'
( ( ( (
R2A–ÿlm
!
R2
=
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
"
#
$ $ $ $
%
&
' ' ' '
!
C = BR2B"1
=
0 1 0 0
1 0 0 0
0 0 0 "1
0 0 "1 0
#
$
% % % %
&
'
( ( ( ( ª
ŠGFE*phasonFEga6Å6IsÊK(«=¬?KX4QRDodecagonalL%Af®1D123‰Nga6ˆc2QRFEIs8|™x6|8Kz*ˆÐ€KX
!
a1
=aDD
2
1
0
"1
0
#
$
% % % %
&
'
( ( ( (
!
an
= An"1a
1, n =1 ~ 12ª
HzA…;gnˆ5wsAÖ×ØÙ1nà1~4An./\3~56KXÖ×ØÙA/ÐA¥gsA2qˆŠGFE/Ð=g”A2qˆphasonFE/Ð=¯@KX
DodecagonalL%AL%Ö×ØÙ an
ŠGFE
PhasonFEIƒC‚=ˆÂtL%fAjk=„78gQIaAŠGFE/Ð=°ùK*WL%=M-3ÇKXÔñ„±Ú xytgƒC‚1f®D12=£x?†‡ˆ‰KXÔƒC‚AhijËlmˆ+,ÎÏ=¥ùKz*I²¬Ú
phasonFE
!
an
= an //
+ an"
-150Ï—q+Y30Ï—q+Y
l³i´Alm=@*IL%ÂFaDDAdodecagonalL%=ÂÌ?KX
―�8�―
F. Gähler, PhD thesisDiss. ETH No. 8414,ETH Zürich, 1988.
D12E.Aƒ C‚=÷{xm\
PhasonFE
ŠGFE
{r// | r‚†‡‚}
(1) tiling*t|AóO(2) covering*t|AóO
zA£öÙ¡N=Shield tiling*·Yz*ˆ‰KX
D3E.“DO.£öj¢ù=…KI1h+×£ÙAƒ=÷…’ˆ†‡
M. Baake, "A Guide to Mathematical Quasicrystals", arXiv:math-ph/9901014v1 20 Jan 1999.
!BCST*+¢G.£öj¢ùAµ81gÂw{xƒC‚AíîXzA{!3gcd=¶Œ3ÇKX
*“Å6|8KX
―�9�―
4. \]^D12E.!/0AST
Mn-Cr-Ni-Si\]VI./txD12E.!/0A$%+,-.
12211
A: 1 2 1 0 0, B: 2 3 2 0 0, C: 1 2 2 1 0, D: 1 3 3 1 0, E: 0 0 0 0 2
12+ðOÓIBC;==qXw”31g12+ðOÓÛA¸„ST=WL%Aef3‚ƒ?KX
S. Iwami and T. Ishimasa, Phil. Mag. Lett., 95 (2015) 229.
5¹ºA»F1gBCðOÓ=~?X
12+9:ð€ÅeÒtxMn-Cr-Ni-Si\]A$%&'()
¼fAE½#4.56 Å
$%+,-.
T. Ishimasa et al., Phil.Mag., 95 (2015) 3745.
\~A!/01g!BC?+¢G.£öj¢ù?
―�10�―
hbtx$&)tAç5 a=4.56 Å
$%&'()IóÅ6K£öÙ¡NA»FãÊ
aj// ÔŠGFEÚ
aj‚ ÔphasonFEÚ
mj : index
)IeÒ56K¾|A¿fA€_1g4qAaj//Ö×ØÙA}F©Fn.
/\*t|èÀ56KX
―�11�―
!"
#$%&'()*+,-.$/0123)45,*67890:;<%=>?@%ABCDE)FG<HIJKL01
MNOPQRS?TUJVW,X-Y
43Z[\6$/VABCDE
Phason]^_`]^
a12bc)d
_`]^ephason]^<fg0W,hij)HIklmnoVBp%q-rh@3s)ABCDES?tuv>KL0we.78JV1=>?)3s)DE@%xy.z{0dm|}KL01
T. Ishimasa et al., Phil. Mag., 95
(2015) 3745.
―�12�―
n=202, !=0.82
n: number of vertices
n=307, !=0.84
n=437, !=0.87n=586, !=0.87 n=760, !=0.89
n=953, !=0.94 n=1170, !=0.99 n=1405, !=1.03
ABCcu)~�€Å‚ƒ„…2†‡,ˆ-]^6@%‰ŠV#s)DE<‹x901ƒŒ…2çŽèê{‘’.†‡,ˆ-]^<“ŠKL01
!
" =1
nrj# $ c#
2
j=1
n
%
_`]^<fg0DE)”•v<HJK%Phason]^)–—.˜):;<™š90SmnoV1=)Bp%d<H
I90›#�)DEœ)ùž.8?Se{}V1w>@%5-Ÿ W,X-Y6@{¡%A¢£¤¥hm¦§901
a¨c%a©bc<ªNm‘’JK%33Zt«¤¥hmtu6•01¬®¯¬¨°<@%a¨c)#±ae²Š¢£m³s1
a¨c´µ¨¶%a©bc´©b·¸®):;<ªNm‘’90eFrank-Kasper)‘"¹º¶»g6tu6•01
T. Ishimasa et al., Phil.
Mag., 95 (2015) 3745.
―�13�―
±)¼v24.56 Å
HAADF-STEM½
<:0MORa3bcea¨c)x.$
/01
5. a20º¶ABC
j¾2235 j¾2m352¿À¿
!ÁÂÃ@%xyHÄ{)6%!ÁÅÆ6@m35<$/01
orderƒÇÈ)É…¯60 order¯120
a20º¶HÄÊ
¿À¿
J. Rhyner, PhD thesis
Diss. ETH No. 8451,
ETH Zürich, 1987.
p.39:ËÌÍ
―�14�―
ÎÏ1¯a20º¶!ÁHÄÊ
Zn80Mg5Sc15
Y.Kaneko, Y.Arichika,
T. Ishimasa, Phil. Mag.
Lett., 81 (2001) 777.
D. Shechtman, I Blech,
D. Gratias and J.W. Cahn,
Phys. Rev. Lett., 53
(1984) 1951.
5.1 a20º¶ABC)ÎÏ
33Z€{A¢£
t«{)6%t«[\@ãJL1
ÎÏ2¯BC<ÐÑ90Ò˜¼ÓÔÕÖ.×Ø{³)³Ù0
ÚÛÜmÝÞJVßBCXà!Á6%!Áj)cá.!kÉ<âLSnoV1
ABC
âêBC
M.de Boissieu, S. Francoual, Y. Kaneko, T. Ishimasa, Phys. Rev. Lett. 95 (2005) 105503.
―�15�―
äåææçèæéêëìíëABC)ÜîOPQïðañòº¶
ÎÏó¯ôc.aòõº¶HÄ
Zn-Mg-ScABC)ôc%öc30º¶ƒtriacontahedron)
T. Ishimasa et. al, J. Non-Cryst.
Solid, 334&335 (2004) 1.
T. Ishimasa, Handbook of Metal Physics, Chapter 3, 2008, Elsevier.
äå¿çè¿éêaòõº¶ABC)÷øº<ùúv>Vûv{ü1ôc.aòõº¶HÄekýþg?>0we<ÿ¦
―�16�―
ÎÏ!¯MNOPQ6$0ea"bc'añõbc.#$e‘%
ñ!&"À'
Zn60Mg30Ho10
ñ!&"À' a10bc)()É%€‘’1˜)a10bc)xy<sLK³%w)*+Ê.$?>01
―�17�―
,-5.
3ABCDEFGH
TsaiI
BergmanI
MackayI
T. Ishimasa, "The Science of Complex
Alloy Phases" , TMS, 2005. p 231-249.
J20KLMNO
PQR3S8TUVWXYS8T3ABCZO
DEFGHC[\]^_9`YaYbcde1$fghiHcj'Ckl*mYO*%no7pq0rT9
Al!"#$Al65Cu20Fe15, Al70Pd21Mn9 %&'Zn!"#$Zn60Mg30Ho10(Zn80Mg5Sc15%&'Cu!"#$Cu46Al38Sc16%&)
Cd!"#$Cd85Ca15, Cd84Yb16%&'Ag!"#$Ag42In42Ca16%&')*+Ti!"(Au!"%&
,-6#./%!"0123(456+789
: 4;<=(2006>GrenobleProf. D. Shechtman Nobel Laureate Chemistry 2011
########################2006>?@
―�18�―
5.2 J20KLstuCvw
!1"# 5$%$&G
!
G =
1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 1 0 0 0 0
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
, G
x1
x2
x3
x4
x5
x6
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
=
x1
x3
x4
x5
x6
x2
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
(2) 3$%$&#H
!
H =
0 0 0 0 0 1
1 0 0 0 0 0
0 0 0 0 1 0
0 0 (1 0 0 0
0 0 0 (1 0 0
0 1 0 0 0 0
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
, H
x1
x2
x3
x4
x5
x6
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
=
x6
x1
x5
(x3
(x4
x2
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
G'H(Inversion!)&*# -E"+,-./01(22034567+89
:01; A. Katz and M. Duneau, J. Physique 47 (1986) 181-196.
#6xyz{CJ|}~�€CÅ‚ƒ„GUHPVWT96…C�€†Dg_
!
r e
n, n =1 ~ 6
5.3#6xyz{C‡ˆ
$1'#2‰C3xyz{S//$Šwz{'US‹$phasonz{'$2'#S//US‹*Œ8SJ20KLçŽèê‘’“Tkl*‡”T9$3'#S//US‹•–8*orthogonal complement
—S//US‹*Œ8SJ20KLçŽèê‘’“Tkl*‡”T9˜ê&Ckl*
™š›YS8TœùžSŸkl9
!
P//
=1
20
5 1 1 1 1 1
1 5 1 "1 "1 1
1 1 5 1 "1 "1
1 "1 1 5 1 "1
1 "1 "1 1 5 1
1 1 "1 "1 1 5
#
$
% % % % % % %
&
'
( ( ( ( ( ( (
!
P" =1
20
5 #1 #1 #1 #1 #1
#1 5 #1 1 1 #1
#1 #1 5 #1 1 1
#1 1 #1 5 #1 1
#1 1 1 #1 5 #1
#1 #1 1 1 #1 5
$
%
& & & & & & &
'
(
) ) ) ) ) ) )
YX•¡v3Cƒ„0r¢(xC£¤P¥o“9
!
P//P" = 0, P" = E - P
//
!
r r =
r r //
+r r " = P
//(r r )+ P"(
r r )
¦xyz{C‡ˆ( Y•§¨Cƒ„P//
UP‹P&l©ªTœU8l«¬9
A. Katz and M. Duneau, J. Physique, 47 (1986) 181.
―�19�―
C•2xy®KP¯3S8T9 YP5ÅçŽKU°±9 C®K²C}³•&Y0+5Å玴U%TC0(}~“T2‰C5Å´PŠwz{Uphasonz{*µ¢¶T9$·8¸WTU Uê C®K²*CTkl*P//UP‹P¹53o9'6ºC5ÅçŽK»S*‰8S5Å´Pµ¢¶not¼ê½¾H¿CP//UP‹0rT9
6xyNOCçŽKUŠwz{(phasonz{C£¤:5ÅçŽKCÀ!
!
G =
1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 1 0 0 0 0
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
!
GX = X P¥o“ PÁªkl9
!
X =
x1
x2
x3
x4
x5
x6
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
!
X = x1
1
0
0
0
0
0
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
+ x2
0
1
1
1
1
1
"
#
$ $ $ $ $ $ $
%
&
' ' ' ' ' ' '
!
P//(r e 1) =
r e 1//
!
P"(r e 1) =
r e 1"
aÂ6xyÃÄÅNOCNO5v
3xyPenroseGÆhÇÈ*�ÉÊCd6YbJ20KLstuC[\]^_
15.66 Å
BergmanDEFGH
TsaiIDEFGHêsËÌÍ*ÎÏ(Ð{PBergmanDEFGHCÑÒêÓªT
H. Takakura et al., Nature Mater. 6, 2007, 58.k¢‚Ô
TsaiIDEFGH
―�20�―
1.#stu45ÕCÖ×$1'GÆ_ÎÏCØÇgbÙH*knSÚÛ0CŸ45Ü `YU+�€ÝÞÜ$2'stu!"C[2yß
###[\Cà»èê!"á0â%TC•ãäÜ###åætu•12“Têstu•0ç%8!"á9ãêèlÜ###k¢wé*å8stuPêTë*¥o“žçìíU•Ü$3'îO[\Cïµ
2.#stuC[\$1'Úxy]^_0(…ð*ñ8CœÜ $MNOò†_'$2'stu*Œ”TZOÎÏ]^_#$ZOò†_'$3'stu*óôCNOõö
3.#stu2÷Cøjùúû## ühøHg_Cýstuêþÿ{02÷0çTC•ãäÜ
4.#stuCîOÝÞ## w!Â1xyá0•"#£v•($ênS+8%3%&+3S+8%89## š™Cstu0•&lœÜ
5.#stu*óôCŠè•rTCœÜ## î'()ê*ç8(+,-.ê?›8(/0¤vê?›8(rTACstu
•1O23èP4“(....
5Uª*œWSÂ!"ástuC6ˆ©«¬
K. Deguchi et al. , Nature Materials 11 (2012) 1013.
―�21�―