坂倉輝俊・井田
セラミックス基盤工学研究センター年報 (2010). Vol. 10, 39-42

解説隆・石澤伸夫

グラムシャリエ級数展開法を用いた非調和熱振動解析とその応用
坂倉輝俊・井田 隆・石澤伸夫
名古屋工業大学セラミックス基盤工学研究センター〒 507-0071 岐阜県多治見市旭ヶ丘 10-6-29

Anharmonic Thermal Vibration Analysis Employing Gram-Charlier Series
Expansion and Its Application
Terutoshi Sakakura, Takashi Ida, Nobuo Ishizawa
Ceramics Research Laboratory, Nagoya Institute of Technology
10-6-29 Asahigaoka, Tajimi, Gifu 507-0071 JAPAN
Methods to project atomic displacement parameters in reciprocal space to probability density function (PDF) in direct space andto convert PDF into the effective one particle potential have been described. Since the PDF expanded with Gram-Charlier series(GCE) is the most widely used model to express anharmonicity of atomic thermal motion, theories on GCE are covered in detailwith their application.
Key words: Anharmonic vibration, Gram-Charlier series expansion (GCE), Effective one-particle potential (Vopp), Probabilitydensity function (PDF)ℂ⺰ߪߓ߼ߦ
⺞๺᷷ᐲ࿃ሶ

X✢‫ޔ‬ਛᕈሶ✢‫ޔ‬㔚ሶ✢╬ߩ࿁᛬ᵄߩᝄ᏷ߪᢔੂ૕ኒᐲಽᏓߩࡈ࡯࡝ࠛᄌ឵ߢ⴫ߐࠇࠆ‫ޕ‬ේሶᩭኒᐲߩേ⊛ಽ

GCEᴺߦߟ޿ߡ⺑᣿ߔࠆ೨ߦ⺞๺ᝄേࡕ࠺࡞ߩ႐วᏓߦଐሽߔࠆ㗄ߪ᷷ᐲ࿃ሶߣ๭߫ࠇ‫ޔ‬ේሶᩭߩሽ࿷⏕ߩ Pdf ෸᷷߮ᐲ࿃ሶߦߟ޿ߡ⺑᣿ߔࠆ‫⺞ޕ‬๺ᝄേࠍ઒₸ኒᐲಽᏓ㑐ᢙ Pdf (probability density function) ߩࡈ࡯ቯߒ‫ޔ‬Pdf ߦ Gauss ಽᏓࠍ↪޿ࠆߣ⺞๺᷷ᐲ࿃ሶࠍᓧࠆ‫৻ޕ‬ᰴ⁛┙ߥ3ᣇะ u'i ߩ౔‫ߥ┙⁛ߦޘ‬ᮡḰ஍Ꮕǻ'i࡝ࠛᄌ឵ߣߒߡ
 T (h ) =+∞ࠍਈ߃ߡ⇣ᣇᕈࠍ⴫⃻ߔࠆߣ㧔㕖╬ᣇᕈ᷷ᐲ࿃ሶ㧕‫⏕ޔ‬−∞₸ኒᐲಽᏓ㑐ᢙ Pdf ෸᷷߮ᐲ࿃ሶ T ߪᰴߢਈ߃ࠄࠇ

 Pdf (u) exp(2Ǹihu)du        (1)ߢਈ߃ࠄࠇࠆ‫ޕ‬ᓥߞߡ‫ޔ‬ㅒⓨ㑆ߢ᷷ᐲ࿃ሶࠍ♖ኒൻߔࠆࠆ‫ޕ‬ߎߣߪ‫ޔ‬ታⓨ㑆ߢේሶᩭߩሽ࿷⏕₸ኒᐲಽᏓࠍ᳞߼ࠆ੐

3

Pdf (u' ) = ∏ߦ╬ଔߢ޽ࠆ‫ޕ‬෩ኒߦߪX✢‫ޔ‬㔚ሶ✢ߩ႐วߦߪߘࠇࠄߩᢔੂ૕ߩኒᐲಽᏓᭂᄢ૏⟎ߩሽ࿷⏕₸ኒᐲಽᏓࠍ᳞߼ࠆ੐ߦ⋧ᒰߔࠆ߇‫ޔ‬ߘࠇߪේሶᩭߩሽ࿷⏕₸ኒᐲಽᏓ=ߦ╬ߒ޿ߣ⠨߃ߡ᭴ࠊߥ޿‫ޕ‬
Pdf ߦߤߩ᭽ߥዷ㐿㑐ᢙ♽ࠍ↪޿ࠆ߆ߦࠃࠅ‫ޔ‬᭽‫ߥޘ‬᷷ᐲ࿃ሶࡕ࠺࡞ࠍ┙ߡࠆ੐߇ߢ߈ࠆ߇‫ޔ‬ᐢߊ↪޿ࠄࠇߡσ 'i( 1  u'iexp−2  σ 'i2π){T (h' ) = ∏ exp − 2π 2 (σ ' i h' i )i =12 (2)det Ȉ' −1 1
exp− u' t Ȉ' −1 u'(2π ) 3 2
3{}2}      (3)= exp − 2Ǹ2 h' t Ȉ' h'޿ࠆ Pdf ߪ⺞๺ᝄേㄭૃߩ▸࿐ౝߢߪ Gauss 㑐ᢙ‫ޔ‬㕖⺞๺ᝄേㄭૃߢߪ

i =1



1

Gram-Charlier series expansionૉߒ‫ޔ‬u' ߪ3ߟߩ⁛┙ߥᝄേᣇะߦᴪߞߚࡌࠢ࠻࡞ࠍၮ

㧔GCE㧕ߢ޽ࠆ‫ߩઁ ޕ‬㕖⺞๺᷷ᐲ࿃ሶࡕ࠺࡞ߣߒߡߪ‫ޔ‬ᐩߣߒߚታⓨ㑆ࡌࠢ࠻࡞‫ޔ‬h' ߪߎߩታⓨ㑆ၮᐩߦኻߔ

৻☸ሶࡐ࠹ࡦࠪࡖ࡞ᴺ╬߇޽ࠆ߇‫ߡࠇࠊ૶ࠅ߹޽ޔ‬ࠆࡌࠢ࠻࡞⋧෻♽ࠍၮᐩߣߒߚㅒⓨ㑆ࡌࠢ࠻࡞ߢ޽ࠆ‫ޕ‬޿ߥ޿‫ᧄޕ‬ⓂߢߪGCEᴺ3)-6)ߩℂ⺰ߣන⚿᥏♖ኒ᭴ㅧ⸃Ǜ' ߪಽᢔ౒ಽᢔⴕ೉ߢኻⷺᚑಽߩߺ߇ǻ i2 ߥࠆ୯ࠍᨆᴺߢGCEᴺࠍ↪޿ߚታ଀ߦߟ޿ߡ⺑᣿ߔࠆ‫ޕ‬߽ߟ‫ޕ‬ၮᐩߩㆬ߮ᣇࠍ3ߟߩ⁛┙ߥᝄേᣇะ߆ࠄᩰሶਗ1), 2)

ㅴࡌࠢ࠻࡞ a, b, c ߦㆬ߮⋥ߔߦߪ‫ޔ‬ᩰሶਗㅴࡌࠢ࠻࡞
― 39 ―グラムシャリエ級数展開法を用いた非調和熱振動解析とその応用ࠍၮᐩߣߒߡ⴫⃻ߒߚࡌࠢ࠻࡞ࠍ u ߣߒߡ‫ޔ‬u'=Ru㧔ㅒⓨ㑆ߢߪ h'=R-1h 㧕ߥࠆၮᐩᄌ឵ࠍⴕߥ߃߫ࠃ޿‫ޕ‬ૉߒ‫ޔ‬R ߪ3ⴕ3೉ߩᄌ឵ⴕ೉ߢ޽ࠆ‫ޔࠅࠃߦࠇߎޕ‬ᑼ(2)෸߮(3)ߪߘࠇߙࠇᑼ(4)෸߮(5)ߦᄌ឵ߐࠇࠆ‫ޕ‬( )det Ȉ −1
 1
exp− u t Ȉ −1 u   
3(2π )
 2
 Pdf (u) ={} (4) T (h) = exp − 2Ǹ h Ȉh          2t2

H ij (u) = wi w j − sij 3

H ijk (u) = wi w j wk − 3 wi s jk

4

H ijkl (u) = wi w j wk wl − 6 wi w j s kl + 3 s ij s kl

5H ijklm (u) = wi w j wk wl wm

(5)− 10 wi w j wk s lm + 15 wi s jk s lmߎߎߢ‫ޔ‬ǛǛ ߪ᷷ᐲ࿃ሶߩB⴫⸥ࠍ↪޿ࠇ߫‫ޔ‬Ǜ
Ǜ=2Ǹ2Bߢ޽ࠆ‫ࠆ޿↪ޔߦᦝޕ‬ၮᐩߩ⚵ࠍ a, b, c ߆ࠄ‫ࠄࠇߘޔ‬ࠍ1Åࠍ㧝ߣߒߡⷙᩰൻߒߚන૏ࡌࠢ࠻࡞ â , b̂ , ĉ ߦᄌ6H ijklmn (u) = wi w j wk wl wm wn − 15 wi w j wk wl s mn
+ 45 wi w j s kl s mn − 15 s ij s kl s mn឵ߔࠆߣ‫ޔ‬Å2 ࠍන૏ߣߔࠆ‫ޔ‬ಽᢔ౒ಽᢔⴕ೉ U 㧔᷷ᐲ࿃ሶߩU⴫⃻㧕ࠍ↪޿ߡ⴫ߖߡ‫ޔ‬       (10)ߢ޽ࠆ‫ ߢߎߎޕ‬wi=(Ǜu)i ‫ޔ‬sij ߪᑼ(6)ߢᣢ಴ߩࠃ߁ߦ

Ȉ = s ij = U ij a *i a * j           Ǜߩᚑಽߢ޽ࠆ‫ޛߚ߹ޕ‬wisjk‫ߪ╬ޜ‬ᷝ߃ሼࠍᎼ࿁ߐ
ⴕ೉Ǜ

(6)

ߖߚ㗄ߩᐔဋࠍขࠆߎߣࠍᗧ๧ߒ‫߫߃଀ޔ‬3ᰴߩࠛ࡞ࡒ࡯࠻ᄙ㗄ᑼߦ಴ߡߊࠆ
‫ޛ‬wisjk‫ޜ‬ߪ (wisjk + wjski + wksij)/3 ߢߢ޽ࠆ‫ޕ‬޽ࠆ‫ޕ‬1ᰴߣ2ᰴߩ㗄ࠍ⋭⇛ߒߚ GCE ߦࠃࠆ Pdf ߪ৻⥸ൻ Chevyshev-Hermite ᄙ㗄ᑼࠍ↪޿ߡ‫ޔ‬)%'ߦࠃࠆ㕖⺞๺᷷ᐲ࿃ሶ
ࡐ࠹ࡦࠪࡖ࡞ߩ㕖⺞๺ᕈ෸߮ᦝߥࠆ⇣ᣇᕈࠍዉ౉ߔ
ࠆὑߦGCEᴺ߇ࠃߊ↪޿ࠄࠇ‫ޔ‬

Pdf GCE (u) = (1 − c Di +i

12!

+ 51! c ijklm H ijklm + 61! c ijklmn H ijklmn  (11)

ijc Di D j− c Di D j D k
13!

Pdf GCE (u) = (1 + 31! c ijk H ijk + 41! c ijkl H ijkl+ ") Pdf har (u)ijk+ 41! c ijkl Di D j D k Dl    ߣ⴫ߐࠇࠆ‫ࠛ࡝࡯ࡈߩߘߚ߹ޕ‬ᄌ឵ߢ޽ࠆ᷷ᐲ࿃ሶߪ

(7)(2Ǹi ) 3 ijk
c hi h j hk
3!
(2Ǹi ) 4 ijkl
c hi h j hk hl+    (12)4!
5(2Ǹi ) ijklm+c hi h j hk hl hm
5!
+ ") exp − 2Ǹ2 h t ȈhT (h) = (1 ++ ") Pdf har (u)ߢਈ߃ࠄࠇࠆ‫ ߢߎߎޕ‬c ijkl... ߪዷ㐿ଥᢙ‫ޔ‬Pdfhar ߪᑼ(4)ߢ⴫ߐࠇࠆ GaussಽᏓ㑐ᢙ‫ޔ‬Di ߪ ui ᣇะߩ஍ᓸಽṶ▚ሶ Di=/ui ߢ޽ࠆ‫ޕ‬හߜ‫ޔ‬Gram-Charlier 㜞ᰴ⚖ᢙዷ㐿ߪᑼ(4)ߩ⺞๺ᝄേࡕ࠺࡞ߩ Gauss ಽᏓ㑐ᢙߣߘߩ{㜞ᰴ஍ዉ㑐ᢙߣߩ๺ߢ⴫⃻ߒߚ߽ߩߢ޽ࠆ‫ߥߒ߆ߒޕ‬߇ࠄ‫⚿ޔ‬᥏⸃ᨆߢߪߎߩ߁ߜ1ᰴߣ2ᰴߩ㗄ߪ↪޿ߥ޿‫ޕ‬}ߣ⴫ߔ੐߇಴᧪ࠆ‫ޕ‬

ߘߩℂ↱ߪ1ᰴߩ㗄ߪኒᐲᭂᄢ૏⟎ߩࠪࡈ࠻ࠍᗧ๧ߔࠆߎߣ‫ޔ‬2ᰴߩ㗄ߪ Pdfhar ⥄૕߇ᣢߦ2ᰴᒻᑼࠍ฽ࠎߢ

᦭ല৻☸ሶࡐ࠹ࡦࠪࡖ࡞8GHHGEVKXG޿ࠆὑߦᦨዊੑਸ਼ࡈࠖ࠶࠹ࠖࡦࠣߩᣂߚߥࡄ࡜ࡔ࡯࠲

છᗧߩኒᐲಽᏓߪ Boltzmann ⛔⸘ࠍ઒ቯߔࠆ੐ߢ᦭

ߣߒߡਇㆡߢ޽ࠆὑߢ޽ࠆ‫ޕ‬ല৻☸ሶࡐ࠹ࡦࠪࡖ࡞ߦᄌ឵ߔࠆ੐߇ߢ߈ࠆ‫ߪࠇߎޕ‬㧟ᰴరߩ႐วߦ᜛ᒛߐࠇߚ৻⥸ൻ Veffective = − k B T ln

Chevyshev-Hermite ᄙ㗄ᑼߪ
nH ijk ... (u) = (−) n P −1 (u) Di D j Dk " P (u)    (8) Pdf (u)       (13) Pdf (u 0 )ߣ⴫ߐࠇࠆ3)‫ ߢߎߎޕ‬kB ߪ Boltzmann ቯᢙ‫ޔ‬u0 ߪ㔖ߢਈ߃ࠄࠇࠆ‫ ߢߎߎޕ‬n ߪᰴᢙ‫ޔ‬P(u) ߪᑼ(4)ߩGaussࡌࠢ࠻࡞ߢ޽ࠆ‫ޕ‬ߎߩᑼ߆ࠄ᣿ࠄ߆ߥߩߪኒᐲಽᏓ߇⽶ಽᏓ㑐ᢙ‫ޔ‬P-1(u) ߪ P-1(u)=1/ P(u) ߢ޽ࠆ‫ࠅ߹ߟޕ‬ᑼ(8)୯ࠍขࠆ႐วߦߪ Veffective ߪቯ⟵ߐࠇߥ޿੐ߢ޽ࠆ‫ߒޕ‬ߪ߆ߒߥ߇ࠄ‫ޔ‬
GCEߦࠃࠆ᷷ᐲ࿃ሶࠍᦨዊੑਸ਼ᴺߢࡈࠖ࠶n࠹ࠖࡦࠣߔࠆ႐ว‫ޔ‬Pdf ߇⽶ၞࠍ߽ߟࠃ߁ߦࡈࠖ࠶࠹ࠖ

H ijk ... (u) P (u) = (−) n Di D j Dk " P (u)     (9)ࡦࠣߐࠇࠆ੐߇ߒ߫ߒ߫ߢ޽ࠆ‫ޕ‬ߎࠇߪ⚿᥏᭴ㅧ⸃ᨆߦ

ߦ╬ଔߢ޽ࠆ‫⥸৻ߩߎޕ‬ൻ Chevyshev-Hermite ᄙ㗄ᑼ

GCEᴺࠍ↪޿ࠆ႐วߩᰳὐߩ৻ߟߢ޽ࠆ7)‫ޕ‬ߩౕ૕⊛ߥ⴫⃻ߪ

0H (u) = 1 

1H i (u) = wi ― 40 ―

坂倉輝俊・井田隆・石澤伸夫

ᢙቇ⊛⵬⿷⺑᣿ࡈ࡯࡝ࠛᄌ឵ߐࠇࠆ‫ޕ‬3ᰴరߩ႐ว߽ห᭽ߦߒߡࡈ࡯࡝ߎߎߢߪ Gauss 㑐ᢙ෸߮‫ޔ‬Hermite ᄙ㗄ᑼߣ Gaussࠛᄌ឵น⢻ߢ‫ޔ‬ᑼ(11)ߩࡈ࡯࡝ࠛᄌ឵ߪᑼ(12)ࠍਈ߃ࠆ‫ޕ‬

㑐ᢙߩⓍߩࡈ࡯࡝ࠛᄌ឵ߦߟ޿ߡߘߩౕ૕⊛ߥᚻ㗅ࠍ)%'ᴺߦࠃࠅᓧࠄࠇߚ2FH)%'෸߮8GHHGEVKXGߩ⸘▚଀⺑᣿ߔࠆ‫ޕ‬⚂925͠ ߦ߅ߌࠆ Pr4Sr1(SiO4)3 㧔hexagonal, P63/m,)CWUU㑐ᢙߩࡈ࡯࡝ࠛᄌ឵

a=9.6914Å, c=7.1872Å㧕ߩ Pr2 (6hᏨ)ߩ㕖⺞๺᷷ᐲ࿃ሶ৻ᰴరߩ႐วߦߟ޿ߡGauss㑐ᢙߩࡈ࡯࡝ࠛᄌ឵ࠍ␜

ࠍ4ᰴ㗄ㄭૃߩGCEᴺߢ⸃ᨆߒߚ‫⚿ߩߘޕ‬ᨐ‫ޔ‬㕖⺞๺ᾲᝄേࡄ࡜ࡔ࡯࠲ߪᰴߩࠃ߁ߦ᳞߹ߞߚ‫ޕ‬ૉߒ‫⸥ޔ‬ㅀߒߡߔ‫ޕ‬exp(-ax ) ߩࡈ࡯࡝ࠛᄌ឵ߪᰴߢਈ߃ࠄࠇࠆ‫ޕ‬
2޿ߥ޿㗄ߪ㔖ߢ޽ࠅ‫⺋ޔ‬Ꮕߪ⋭⇛ߒߚ‫ޕ‬



+∞−∞



{()}

exp − ax 2 − iω x dx  ω 2  +∞iω
= exp − −∞ exp − a x −2 a 4a  2Table 1. Anharmonic atomic displacement parameters up to
 dx

the 4th order of Pr2 at 6h site in apatite-type Pr4Sr1(SiO4)3(hexagonal, P63/m, a=9.6914Å, c=7.1872Å) single crystaliω(ߎߎߢ‫ ޔ‬y = a x −ߣ߅޿ߡ )2 aheated to ~925 ÛC.
2nd terms (Å-2)

 ω 2  +∞2exp−=−∞ exp − y dy
4aa

2
 ω   +∞ +∞122exp−= −∞ −∞ exp − y + z dydz
4aa


{1}

{()}

(ߎߎߢ‫ޔ‬2Ǹrdr = dydz ߣᄌᢙᄌ឵ߒߡ )+∞ ω2 2exp− 2π 0 r exp − r dra 4a 
 ω2 
π=exp−a 4a 

{1}

U12

0.024687

0.025473

0.025065

0.011130

c111

c112

c122

c133

0.000281

0.000086

0.000229-0.000089

c1111

c1112

c1122

c1133

c1222

0.000314

0.0002200.000202-0.000016

0.000214c(2π)ࠍਸ਼ߓࠇ߫‫ࠛ࡝࡯ࡈޔ‬ᄌ឵Ṷ▚ሶࠍ F ߣߒߡ‫ޔ‬

U33

4th terms (104)

ᑼ(14)ߢ a=1/(2ǻ2) ෸߮ ȁ=2Ǹh ߣ⟎߈‫ ޔ‬1 / σ(1233c-0.000008

2222

0.000437c2233

0.000011

c3333

0.000228ߎࠇࠄࠍ↪޿ߡ u ߦ߅ߌࠆ PdfGCE ࠍ⸘▚ߔࠆᚻ㗅ߪᰴߩㅢࠅߢ޽ࠆ‫ޕ‬1) ᑼ(6)ࠍ↪޿ߡ᷷ᐲ࿃ሶߩ2ᰴ㗄߆ࠄ x2 
1exp −= exp −2π 2σ 2 h 2  (15)2 σ2σπ2 FU22

3rd terms (103)
(14)

=U11)

Ǜⴕ೉ࠍ⸘▚ߒ‫ޔ‬sij ࠍ᳞߼ࠆ‫ޕ‬2) ޽ࠆ u ߦኻߒ w=Ǜuࠍᓧࠆ‫ޕ‬ࠍ⸘▚ߔࠆ‫ޕ‬3) ᑼ(11)ߦ߅޿ߡcijkl... ߇㕖㔖ߩ㗄ߦߟ޿ߡᑼ(10)ࠍ↪޿ߡ Hijkl... ࠍ⸘▚ߔࠆ‫ߦᦝޕ‬ᑼ(4)ߢ
Pdfhar ࠍ⸘▚ߒ‫ޔ‬ᑼ(11)ߦೣࠅ PdfGCE(u) ୯ࠍ᳞߼ࠆ‫ޕ‬*GTOKVGᄙ㗄ᑼߣ)CWUU㑐ᢙߩⓍߩࡈ࡯࡝ࠛᄌ឵ߢ࡞࡯ࡊࠍដߌࠆ੐ߦߥࠆ‫ޕ‬Veffective ߩ⸘▚ߪ◲නߢ‫ޔ‬৻ᰴరߩ႐วߦߟ޿ߡ␜ߔ‫ޕ‬Gauss 㑐ᢙࠍ Pdf(x) ߣ⟎߈‫ޔ‬n ᰴߩ Hermite ᄙ㗄ᑼ nH(x) ߣ Pdf(x) ߩⓍࠍ

f n ( x)= n H ( x) Pdf ( x) ߣ⟎ߌ߫‫ࠛ࡝࡯ࡈߩߘޔ‬ᄌ឵ߪȁ =2Ǹh ࠍ↪޿ߡ‫ޔ‬ߎࠇࠍⶄᢙߩὐߦߟ޿ߡ⸘▚ߔࠆߦߪᚻ㗅2)ߣ3)ߣ

+∞

−∞

f n ( x) exp(iω x)dx ߣᦠߌࠆ‫ޕ‬

ᑼ(13)ߦೣࠅ -kBT ln(Pdf(u)/Pdf(u0)) ࠍ u ߩฦὐߢ⸘▚ߒߡ޿ߊߛߌߢ޽ࠆ‫ޕ‬
Fig.1(a) ߦߪ Pdf ߩ90㧑╬ᢙ୯㕙ࠍ‫ޔ‬Fig.1(b)ߦߪᣇ
ะᲤߩ Veffective ࠍ‫ޔ‬Fig.1(c)ߦߪᣇะᲤߩ Pdf ࠍ␜ߔ‫ޕ‬

fn(x)exp(i ȁ x) ߪ n ߇㔖એᄖߩ႐วߦߪrпߢ㔖ࠍขࠆߩߢ‫ ޔ‬n ࠍ㔖ߦㄭઃߌࠆࠃ߁ߦㇱಽⓍಽࠍ➅ࠅ㄰ߖ߫‫(ޔ‬iω )n

+∞−∞f 0 ( x) exp(iω x) dx ߩߺ߇㕖㔖ߩ㗄ߣߒߡᱷࠆ‫ޕ‬f0(x) ߪ Gauss 㑐ᢙߢ޽ࠆߩߢ‫ߩߎޔ‬㗄ߪᑼ(15)ߦ (i2 Ǹ h)n ࠍਸ਼ߓߚᒻ (i 2πh) exp(−2π σ h ) ߦn222― 41 ―

グラムシャリエ級数展開法を用いた非調和熱振動解析とその応用

4GHGTGPEGU
1)

B. T. M. Willis, Acta Cryst. A25, (1969) 277-300.

2)

K. Tanaka, F. Marumo, Acta Cryst. A39, (1983) 631-

3)

U. H. Zucker, H. Schulz, Acta Cryst. A38, (1982)

641.

563-568.
4)

U. H. Zucker, H. Schulz, Acta Cryst. A38, (1982)

5)

W. F. Kuhs, Acta Cryst. A40, (1984) 133-137.

6)

W. F. Kuhs, Acta Cryst. A48, (1992) 80-98.

7)

C. Scheringer, Acta Cryst. A44, (1988) 343-349.

568-576.

(a)

(b)

(c)
Fig. 1. Pdf and Veffective converted from parameters tabulated
in Table 1; (a) 90% isosurface of Pdf , (b) Veffective along[001] and [100], and (c) Pdf along [001] and [100].

― 42 ―