2007 ᐕ 7 17 ᣣฃℂ
* ╙ 35 ࿁ᣣᧄ⚿᥏ᚑ㐳࿖ౝળ⼏㧔 2005 ᐕ 8 㧘᧲ᐢፉ Ꮢ㧕ߦ߅ߡ৻ㇱ⊒
** ⑺↰ᄢቇᎿቇ⾗Ḯቇㇱ᧚ᢱᎿቇ⑼㧚 Department of Materials Science and Engineering, Faculty of Engineering and Resource Science, Akita University
⎇ⓥႎ๔ ᒁ߈ߍ Cu Ꮧ⭯ว㊄⚿᥏ߦ߅ߌࠆォᒻᚑߩ ଦㅴߣᛥ *
ᷡ᳓㓶ੑ
**Promotion and Suppression of Dislocation Formation in Czochralski-Grown Copper Dilute Alloy Crystals*
Yuji Imashimizu
**Abstract
Some peculiar dislocation distributions formed in the Czochralski-grown crystals of Cu-0.05~0.2at%Al and Cu-0.02~0.1 at%Ni alloys were examined by use of etching techniques which distinguish the kinds of dislocation.
For the copper-aluminium dilute alloy crystals, the dislocation formation was suppressed resulting in grown crystals of low dislocation density compared with the copper crystal, while the substructure of an array of polygon walls parallel to a ޛ 112 ޜ direction or a polygon boundary crossing with an array of slip traces at right angle were occasionally formed. On the other hand, the slip traces in one or two ޛ 110 ޜ directions were prominently formed for the copper–nickel dilute alloy crystals. These effects suggest that some macroscopic compositional stress which probably arised from the difference in solute concentration between the center and periphery overlapped with the thermal stress during pulling of the copper dilute alloy crystal, and caused the asymmetric dislocation distributions depending on the deviation of the crystal center from the rotation axis. 㩷
㩷
㩷 㪈㪅㩷 ✜㩷 ⸒㩷
㪈㪅㪈㩷 ᒁ䈐䈕⚿᥏䈱ォᒻᚑ䈮䈿䈜ṁ⾰ᷝടലᨐ㩷 Cz 㧔ᒁ߈ߍ㧕 Si ⚿᥏
(1)㧘 LEC 㧔ᶧኽᱛᒁ߈
ߍ㧕 GaAs ⚿᥏
(2)㨪(4)߅ࠃ߮ InP ⚿᥏
(2)ߢߪ㧘ㆡᒰߥṁ
⾰ࠍ࠼ࡊߔࠆߣ⢒ᚑ⚿᥏ߩォኒᐲ߇⪺ߒߊૐᷫ
ߔࠆߎߣ߇⍮ࠄࠇߡࠆ㧚ਛߢ߽㧘 LEC GaAs ⚿᥏ߩ
࠼ࡄࡦ࠻ߦࠃࠆォኒᐲߩૐᷫߪ㧘ૐォኒᐲൻ ߩ㔍ߒ GaAs ⚿᥏ࠍήォൻߔࠆലᨐ߇ࠆߎߣ ߆ࠄ㧘ߘߩᯏ᭴ߦߟߡᄙߊߩ⎇ⓥ߇ߥߐࠇߚ㧚ߎ ࠇ߹ߢ㧘ṁ⾰ේሶߪ㧘 ( γ ) ォߣߩ⋧↪ߦࠃࠅ ᾲᔕജߦࠃࠆォߩㆇേࠍᛥߔࠆ
(3),(5)~(7)㧚 ( δ ) ォ
Ḯࠍ࿕⌕ߔࠆ
(8)㧚 ( ε ) ὐᰳ㒱㓸วߦࠃࠆォ࡞ࡊ ߥߒォߩᩭᒻᚑࠍᛥߔࠆ
(9)㧚 ( ζ ) ⁁ォߩ
ㆇേߦࠃࠆォᶖṌࠍଦㅴߔࠆ
(10)㨪(12)㧘ߥߤߩ ᯏ᭴߇ឭ᩺ߐࠇߚ㧚ߎߩ߁ߜ㧘 ( γ ) ߩᯏ᭴ߦ㑐ߒߡ ߪ㧘⥃⇇ߖࠎᢿᔕജߩ᷹ቯ⚿ᨐߦࠃࠅ⺑ߐࠇࠆߣ
߁ႎ๔
(13),(14)ߣᔅߕߒ߽⺑ߢ߈ߥߣߔࠆႎ๔
(15),(16)
߇ߥߐࠇߡࠆ㧚ᓟ⠪ߢߪ㧘 LEC GaAs ⚿᥏ߩ
ォᒻᚑߪᾲᔕജߩᄢ߈ߐߩߺߦଐሽߒߥߣ߁
ታ㧘߹ߚ㧘ᓸ㊂ߩṁ⾰ᷝടߦࠃࠆ⚿᥏ߩ⎬ൻߪᾲ ᔕജߦࠃࠆォᒻᚑࠍᛥߔࠆߩߦචಽߢߥߣߔ ࠆታ㛎⚿ᨐߦࠃࠅ㧘 ( ε ) ߩᯏ᭴߿ṁ⾰ේሶߣὐᰳ㒱 ߣߩ⋧↪ࠍㅢߓߡォᒻᚑߪᛥߐࠇࠆߣ߁ ᯏ᭴߇⼏⺰ߐࠇߡࠆ㧚ߎߩࠃ߁ߦᒁ߈ߍ⚿᥏ߩ ṁ⾰ᷝടߦࠃࠆォኒᐲૐᷫߩᯏ᭴ߦߟߡߪ⎇ⓥ
⠪㑆ߦ⇣ߥࠆ⸃߇ࠅᔅߕߒ߽ࠄ߆ߢߪߥ㧚 㪈㪅㪉㩷 㪚㫌Ꮧ⭯ว㊄⚿᥏䈱ォᒻᚑ㩷
⪺⠪ࠄߪߎࠇ߹ߢ㧘ᒁ߈ߍᴺߦࠃࠅ৻ቯߩ᧦ઙ 㧔⚐ൻ Ar ᳇ᵹਛ , Ⲣᶧ㊂ 600 g 㧘ᒁ߈ߍㅦᐲ 1.7 10
㧙5m/s, ⚿᥏࿁ォㅦᐲ 1 rad/s, ᚑ㐳ᣇ [111] 㧕ߩ ਅߢ⋥ᓘ⚂ 10 mm, 㐳ߐ⚂ 70 mm ߩ Cu ⚿᥏߅ࠃ߮⒳ޘ ߩṁ⾰Ớᐲߩ Cu- Ꮧ⭯ Al ว㊄⚿᥏㧘 Cu- Ꮧ⭯ Au ว㊄⚿
᥏߅ࠃ߮ Cu- Ꮧ⭯ Ni ว㊄⚿᥏ߩᒁ߈ߍࠍⴕ㧘⢒ᚑ
⚿᥏ߩᚑ㐳ᣇะߦု⋥ߥ (111) ᢿ㕙ߦ߅ߌࠆォಽ Ꮣ ࠍ ⣣ 㘩 ᴺ ߦ ࠃ ࠅ ⺞ ߴ 㧘 ᰴ ߩ ࠃ ߁ ߥ ⚿ ᨐ ࠍ ᓧ ߚ
(17)~(20)
㧚
Cu ⚿᥏ߦߪ 5~7 10
8m
㧙2ߩኒᐲߩォ߇ᒻᚑߐࠇ㧘 ォߩᄢㇱಽߪಽᢔߒߡࠆ߇㧘ォࡇ࠶࠻߇ޛ 110 ޜ ߔߴࠅᣇะߦ㈩ߒߚߔߴࠅ᭴ㅧ㧔એᓟ̌ߔߴࠅォ
̍ߣ⒓㧕ࠍߺ㧘⚿᥏ㄝㇱߩォኒᐲߪਛᔃ
8
今清水雄二ㇱߦߊࠄߴߡ㜞
(17)㧚ߎࠇߦኻߒߡ 0.1 㨪 0.2 at%Al ࠍᷝടߒߚ Cu- Ꮧ⭯ Al ว㊄⚿᥏ߢߪߔߴࠅォߩᒻ ᚑ߇ᛥߐࠇ㧘߹ߚ㧘⚿᥏ߩਛᔃㇱ߆ࠄㄝㇱ߳ඨ ᓘᣇะߦォኒᐲ߇Ⴧᄢߔࠆะ߇ᛥߐࠇ㧘ォ
ኒᐲߪ㧝ᩴએૐᷫߔࠆ
(18)㧚 Cu- Ꮧ⭯ Au ว㊄⚿᥏ߦ ߅ߡ߽ォኒᐲૐᷫߩṁ⾰Ớᐲߩ▸࿐߅ࠃ߮ૐᷫ
₸ߪ⇣ߥࠆ߇㧘ห᭽ߩലᨐ߇ࠆ
(19)㧚৻ᣇ㧘 0.02 㨪 0.1 at% ߩ Ni ࠍᷝടߒߚ Cu- Ꮧ⭯ Ni ว㊄⚿᥏ߩォኒ ᐲߪᒁߍ⚿᥏ߏߣߦ߫ࠄߟ߈߇ᄢ߈ߊ㧘 Ni ᷝടߩ ォᒻᚑߦ߅ࠃ߷ߔലᨐߪࠄ߆ߢߪߥ߇㧘ߔߴ ࠅォߩᒻᚑ߇ଦㅴߐࠇࠆะ߇ࠅ㧘ォኒᐲߪ ᐔဋ⊛ߦߪ Cu ⚿᥏ࠃࠅߒࠈ㜞
(20)㧚
ߎߩࠃ߁ߦ Cu ⚿᥏ߦ߅ߡ߽ GaAs ߿ InP ⚿᥏ߣห
᭽ߦ㧘ᓸ㊂ߩṁ⾰ᷝടߦࠃࠅォᒻᚑߪᛥߐࠇ㧘 ߘߩലᨐߪṁ⾰ర⚛ߩ⒳㘃߅ࠃ߮Ớᐲߦࠃߞߡ⇣ߥ ࠆ
(18)㨪(21)㧚ߒ߆ߒߥ߇ࠄ㧘ߕࠇߩ Cu Ꮧ⭯ว㊄⚿᥏ ߣ߽ṁ⾰Ớᐲࠍ㜞ߊߔࠆߣォߩ㓸Ⓧߒߚࠢࠬ࠲
߅ࠃ߮ォႺ⇇߇ᒻᚑߐࠇ㧘ォኒᐲߪჇᄢߔ ࠆ㧚ߎߩലᨐߪ㧘ォኒᐲߩჇᄢߔࠆਅ㒢ߩṁ⾰Ớ ᐲ߇ฦว㊄⚿᥏ߩⲢᶧᚑ㐳ߦ߅ߌࠆ⚵ᚑ⊛ㆊ಄ළࠍ
ߔ⥃⇇Ớᐲߩផቯ୯ߦ߶߷৻⥌ߔࠆߎߣ߆ࠄ㧘
ߕࠇ߽⚵ᚑ⊛ㆊ಄ළߦࠃࠆ࡞⁁ᨆ᭴ㅧߩዉߦ ߣ߽ߥ߁ォᒻᚑߩߚߣ⠨߃ࠄࠇࠆ
(18)㨪(20)㧚
ㅀߩ⚿ᨐߦ߅ߡ㧘⚐ Cu ⚿᥏ߩォߪߘߩඨᓘ ᣇะߩಽᏓ߆ࠄਥߦᾲᔕജߩ↪ߦࠃࠅᒻᚑߐࠇߚ ߣផኤߐࠇࠆ
(17)㧚߹ߚ㧘ૐỚᐲߩ Cu- Ꮧ⭯ Al ว㊄
ࠆߪ Cu- Ꮧ⭯ Au ว㊄⚿᥏ߦ߅ߡߔߴࠅォߩ ᒻᚑ߇ᛥߐࠇ㧘߹ߚ㧘ォኒᐲ߇ඨᓘᣇะߦჇᄢ ߔࠆะ߇ᛥߐࠇ㧘ᐔဋ⊛ߦૐᷫߔࠆߩߪᾲᔕജ ߦࠃࠆォᒻᚑ߇ᛥߐࠇࠆߚߣ⠨߃ࠄࠇࠆ㧚ߒ ߆ߒߥ߇ࠄ㧘৻ᣇߦ߅ߡ㧘 0.1 㨪 0.2 at%Al ࠍૐ Ớᐲߩว㊄⚿᥏ߩਛߦߪ⚐ Cu ⚿᥏ߦߪᒻᚑߐࠇߥ
ࠃ߁ߥ㧘߹ߚ㧘 1at% એߩỚᐲߩ⚿᥏ߦᒻᚑߐࠇ ࠆ⚵ᚑ⊛ㆊ಄ළߦࠃࠆṁ⾰ᨆࠍߣ߽ߥ߁Ⴚ⇇ߣ ߪ࠲ࠗࡊߩ⇣ߥࠆ㧘Ⴚ⇇᭴ㅧࠆߪႺ⇇ߣߔ ߴࠅォࠍ․⇣ߥォ᭴ㅧ߇ᒻᚑߐࠇࠆߎߣ߇
ࠆ㧚ߐࠄߦ㧘 Cu ́ Ni ว㊄ߢߪ⚵ᚑ⊛ㆊ಄ළࠍߎ ߐߥૐỚᐲߩ 0.02 㨪 0.1 at% ⚿᥏ߦߔߴࠅォ߇⪺
ߒߊᒻᚑߐࠇ㧘ォኒᐲߪ Cu ⚿᥏ߣห⒟ᐲߥߒ
ߒࠈ߿߿㜞ߊߥࠆะ߇ࠆ㧚ߎࠇࠄߩߎߣߪૐỚ ᐲߩ Cu Ꮧ⭯ว㊄⚿᥏ߢߪᓸ㊂ߩṁ⾰ᷝടߪ⚐ Cu ⚿ ᥏ߩォᒻᚑࠍᛥߔࠆ߇㧘⚵ᚑ⊛ㆊ಄ළߦࠃࠆṁ
⾰ߩᨆ᭴ㅧߩዉߣߪ⇣ߥࠆᯏ᭴ߦࠃߞߡォᒻ ᚑࠍଦㅴߔࠆ႐วߩࠆߎߣࠍ␜ߔ㧚
ᧄ⎇ⓥߢߪ㧘ᒁ߈ߍ Cu Ꮧ⭯ว㊄⚿᥏ߦ߅ߌࠆォ
ᒻᚑߩᛥߣ⚵ᚑ⊛ㆊ಄ළߦࠃࠆṁ⾰ᨆߣߪ⇣
ߥࠆォᒻᚑߩଦㅴലᨐߦߟߡ⍮ࠍᓧࠆߚߦ㧘 Cu- Ꮧ⭯ Al ว㊄⚿᥏ߦ߅ߌࠆ․⇣ߥォ᭴ㅧ߅ࠃ߮
Cu- Ꮧ⭯ Ni ว㊄⚿᥏ߦ߅ߌࠆ⪺ߒߔߴࠅォߩᒻ ᚑࠍߒߊ⺞ߴ㧘ᓸ㊂ṁ⾰ේሶߩォᒻᚑߦ߅ࠃ߷
ߔലᨐߦߟߡᬌ⸛ߒߚ㧚 㩷 㪉㪅㩷 ታ㛎ᣇᴺ㩷
వߩ⎇ⓥ
(17)㨪(21)ߦ߅ߡ৻ቯߩ᧦ઙਅߢᒁ߈ߍ
ߚ Cu Ꮧ⭯ว㊄⚿᥏ߩ߁ߜ , ૐỚᐲߩ Cu-0.05 㨪 0.2at%
Al ว㊄⚿᥏߅ࠃ߮ Cu-0.02 㨪 0.1at%Ni ว㊄⚿᥏ߩᚑ㐳 ᣇะߦု⋥ߥ (111) ᢿ㕙ߩ৻ㇱ㧘ߐࠄߦᧃ┵ㇱߩ (111) ᢿ㕙ߩ․⇣ߥォಽᏓࠍォ⣣㘩ᴺߦࠃߞߡ⺞ߴߚ㧚 ᧄታ㛎ߢߪߣߊߦォ⣣㘩ᶧߦ⁁ォߣࠄߖࠎォ
ߣࠍ್ߔࠆ⣣㘩ᶧ㧔એᓟ㧘ਣᎹᶧߣ⒓㧕
(22)߅ ࠃ߮ᱜ⽶ߩ⁁ォࠍ್ߔࠆ⣣㘩ᶧ㧔એ㒠㧘 Young ᶧߣ⒓㧕
(23),(24)ࠍ↪ߡ㧘ዉߐࠇߚォߩ⒳㘃ࠍ
⺞ߴߚ㧚࿃ߺߦ Cu ⚿᥏ࠍਣᎹᶧߦࠃߞߡ⣣㘩ߒߚ
(111) ᢿ㕙ࠍૐ₸㧔ኻ‛ࡦ࠭ 10 㨪 12 㧕ߩశቇ㗼
ᓸ㏜ߦࠃࠅᬌ㏜ߔࠆߣ㧘ࠄߖࠎォ⟎ߦߪᲧセ⊛
ᷓ㤥ࠦࡦ࠻ࠬ࠻ߩࡇ࠶࠻㧔㤥ࡇ࠶࠻ߣ⒓ߔࠆ㧕 ߇㧘߹ߚ㧘⁁ォ⟎ߦߪᲧセ⊛ᵻ⊕ࠦࡦ࠻
ࠬ࠻ߩࡇ࠶࠻㧔⊕ࡇ࠶࠻ߣ⒓ߔࠆ㧕߇ⷰኤߐࠇࠆ㧚
৻ᣇ㧘 Young ᶧߦࠃߞߡ⣣㘩ߒߚ႐วߦߪᱜߩ⁁
ォ⟎ߦߪ㤥ࡇ࠶࠻߇㧘⽶ߩ⁁ォ⟎ߦߪ߿
߿ዊߐ⊕ࡇ࠶࠻߇ⷰኤߐࠇࠆ
(24)㧚
3. ታ㛎⚿ᨐ
㪊㪅㪈㩷㪚㫌㪄Ꮧ⭯㪘㫃ว㊄⚿᥏䈮䈍䈔䉎․⇣䈭Ⴚ⇇᭴ㅧ㩷 㪊㪅㪈㪅㪈㩷㪚㫌㪄Ꮧ⭯㪘㫃ว㊄⚿᥏䈱ォᒻᚑ㩷
ᒁ߈ߍߚ⚐ Cu ߅ࠃ߮ Cu- Ꮧ⭯ Al ว㊄ߩ⚿᥏
(18)ߩ ォኒᐲࠍ Al Ớᐲߦኻߒߡࡊࡠ࠶࠻ߒߚ࿑ࠍ࿑㧝ߦ㧘
߹ߚ㧘ߔߴࠅォߩᒻᚑ㗫ᐲߦ߷ߔ Al ᷝടߩലᨐ ࠍ࿑㧞ߦ␜ߔ㧚ߎࠇࠄߩ⚿ᨐߪ㧘ᓸ㊂ߩ Al ᷝടߪߔ ߴࠅォߩᒻᚑࠍᛥߒォኒᐲࠍૐਅߐߖࠆ߇㧘 ṁ⾰Ớᐲࠍ㜞ߊߔࠆߣႺ⇇᭴ㅧߩዉߣߣ߽ߦォ
ኒᐲߪჇᄢߔࠆߎߣࠍ␜ߒߡࠆ㧚㤥ࡑࠢߪߘ ߁ߒߚႺ⇇ߩᒻᚑߐࠇߚ⚿᥏ߩォኒᐲߢࠆߎ ߣࠍ␜ߔ߇㧘Ყセ⊛㜞Ớᐲ (1 㨪 2 at%Al) ߩ⚿᥏ߩႺ
⇇ߪ⍴ߊ߆ߟ㓸࿅⊛ߦᒻᚑߐࠇ㧘ᢿ㕙ߩਛᄩㇱߩߺ ߩ႐วߩઁ㧘ਛ㑆ㇱࠆߪㄝ߆ࠄਛᄩߦ߆ߌߡ ਇဋ৻ߦᒻᚑߐࠇ㧘න㐳ߐߚࠅߩォࡇ࠶࠻ᢙ ߪᲧセ⊛ᄙ㧚ߎࠇࠄߩ⚿᥏ߦߪ⚵ᚑ⊛ㆊ಄ළࠍ↢
ߕࠆ᧦ઙਅߢ⢒ᚑߒߚ⚿᥏ߦዉߐࠇࠆ․ᓽ⊛ߥォ
ࠢࠬ࠲᭴ㅧ߇ⷰኤߐࠇ㧘೨▵㧔 1.2 㧕ߢㅀߴߚ
Akita University
࿑ 1 ᒁ߈ߍᴺߦࠃࠅ⢒ᚑߒߚ Cu- Ꮧ⭯ Al ว㊄⚿᥏ߩォኒᐲߩ Al Ớᐲ ଐሽᕈ㧘
٤: ήႺ⇇ 㧘
٨:Ⴚ⇇ᒻᚑ㧚
ࠃ߁ߦ㧘Ⴚ⇇ߪ⚵ᚑ⊛ㆊ಄ළߦߣ߽ߥ߁ṁ⾰ᨆ ߦࠃࠅዉߐࠇߚォ᭴ㅧߢࠆߣ⠨߃ࠄࠇࠆ
(18)㧚 ߎࠇߦኻߒߡ࿑㧝ߩૐ Al Ớᐲ㧔 0.05 㨪 0.2 at% 㧕⚿᥏ ߩォኒᐲࠍ㤥ࡑࠢߢ␜ߒߚ⚿᥏ߦߪ࠲ࠗࡊߩ߿
߿⇣ߥࠆႺ⇇߇ᒻᚑߐࠇߚ㧚߹ߚ࿑㧞ߪ Cu- Ꮧ⭯
Al ว㊄⚿᥏ߢߪߔߴࠅォߩᒻᚑ㗫ᐲߪో⊛ߦૐ
߇㧘৻ㇱ⚿᥏ߦߔߴࠅォ߇ᒻᚑߐࠇߚߎߣࠍ␜
ߒߡࠆ㧚ߎࠇࠄߩ⚿᥏ߪ࿑㧝ߩႺ⇇ࠍォ
ኒᐲߩ߿߿㜞⚿᥏ߦኻᔕߔࠆ㧚એਅߦߘߩࠍ␜
ߔ㧚
㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷 㩷
㪊㪅㪈㪅㪉㩷 Ⴚ⇇㩷
․⇣ߥォಽᏓߩ߭ߣߟߩߣߒߡ 0.05at%Al ว
㊄⚿᥏ߩ৻ᢿ㕙ࠍਣᎹᶧߢ⣣㘩ߒⷰኤߒߚォࡇ࠶
࠻ಽᏓߩ౮⌀ࠍ࿑㧟ߦ␜ߔ㧚Ⴚ⇇ߪ (a) ߩࠃ߁ߦᲧ セ⊛㐳ߊ⚿᥏ᢿ㕙ߩᐢ▸࿐ߦಽᏓߒߡ߅ࠅ㧘 1 at%
એߩ Al Ớᐲߩ⚿᥏ߦ⚵ᚑ⊛ㆊ಄ළߦࠃࠅዉߐࠇ ࠆᨆ᭴ㅧࠍߣ߽ߥ߁Ⴚ⇇
(18)ߣߪ࠲ࠗࡊ߇⇣ߥ ࠆ㧚ૐ₸ߩ㗼ᓸ㏜౮⌀ (b) ࠃࠅ㧘Ⴚ⇇ࠍ᭴ᚑߔࠆ ォࡇ࠶࠻ߪ߅߽ߦ⊕ࡇ࠶࠻ߢࠅ㧘ォߪਃⷺᒻ ࡇ࠶࠻ߩㄝߦု⋥ߥޛ 112 ޜᣇะߦ㈩ߒߡࠆߎߣ ߇ಽ߆ࠆ . ߎࠇࠄߩߎߣ߆ࠄ , Ⴚ⇇ߪ࿕ᶧ⇇㕙ߦ ᐔⴕߥࡃࠟࠬࡌࠢ࠻࡞ࠍߔࠆߔߴࠅ♽ߩ
⁁ォߩౣ㈩ߦࠃߞߡᒻᚑߐࠇߚࡐࠧࡦ㧔ዊ
ⷺ㧕Ⴚ⇇ߢࠆߣ⠨߃ࠄࠇࠆ㧚㩷 㪊㪅㪈㪅㪊㩷 䊘䊥䉯䊮Ⴚ⇇㩷
․⇣ߥႺ⇇ߩᒻᚑߣߒߡ㧘 0.2 at%Al ᷝട⚿᥏ߩ ᢿ㕙ߦⷰኤߐࠇߚォಽᏓߩੑߟߩࠍ࿑㧠 (a) ߅ ࠃ߮ (b) ߦ㧘߹ߚߘߩ৻ㇱߩૐ₸ߩ㗼ᓸ㏜౮⌀ࠍ࿑
㧡 (a) 㧘 (b) ߅ࠃ߮ (c) ߦ␜ߔ㧚࿑㧠 (a) ߢߪ৻ߟߩޛ 112 ޜ ᣇะߦᐔⴕ㈩ߒߚႺ⇇߇ᒻᚑߐࠇߚ㧚ߎࠇࠄ
Ⴚ⇇ߪ࿑㧡 (a) ߦ␜ߔࠃ߁ߦ⊕ࡇ࠶࠻㧔⁁ォ㧕 ߆ࠄ᭴ᚑߐࠇߡ߅ࠅ㧘৻⟲ߩࡐࠧࡦႺ⇇ߩߢ
ࠆߎߣ߇ࠊ߆ࠆ㧚࿑㧠 (b) ߩ႐ว㧘৻ߟߩࡐࠧࡦႺ
⇇ߣߘࠇߦု⋥ߥߔߴࠅォ߆ࠄߥࠆ᭴ㅧ߇ⷰኤ ߐࠇߚ㧚หߓ▎ᚲࠍਣᎹᶧߣ Young ᶧߦࠃߞߡ⣣㘩
࿑㧞 ᒁ߈ߍ Cu ⚿᥏߅ࠃ߮ Cu- Ꮧ⭯ Al ว㊄⚿᥏ߩ
ⷰኤᢿ㕙㧔 18 એ㧕ߦ߅ߌࠆߔߴࠅォߩᒻᚑ㗫 ᐲ , ήࡑࠢ㧦ᱴߤߥ㧘ࡂ࠶࠴ࡦࠣ㧦ዋ㧘࠳
ࡉ࡞ࡂ࠶࠴ࡦࠣ㧦⪺ߒ㧚
10
今清水雄二࿑㧟 Cu-0.05at%Al ⚿᥏ߦᒻᚑߐࠇߚႺ⇇㧘 (a) Ⴚ⇇ᒻᚑߩో⊛ߥ᭽⋧㧘 (b) Ⴚ⇇
ߪ⊕ࡇ࠶࠻㧔⁁ォ㧕ߦࠃࠅ᭴ᚑߐࠇߡ
ࠆ㧚
࿑㧠 Cu-0.2at%Al ⚿᥏ߦᒻᚑߐࠇߚ․⇣ߥ ォಽᏓߩో⊛ߥ᭽⋧㧚 (a) ᐔⴕ㈩ߒ ߚႺ⇇㧘 (b) Ⴚ⇇ߣߔߴࠅォ㧚
࿑㧡 Cu-0.2at%Al ว㊄⚿᥏ߦᒻᚑߐࠇߚ․⇣ߥォಽᏓߩ৻ㇱ㧘 (a) ࿑ 4 (a) ߩႺ⇇㧔ਣᎹ
ᶧ⣣㘩㧕 㧘 (b) ࿑ 4 (b) ߩႺ⇇ߣߔߴࠅォߩ৻ㇱ㧔ਣᎹᶧ⣣㘩㧕 㧘 (c) (b) ߣห৻▎ᚲ 㧔 Young ᶧ⣣㘩㧕 㧚
Akita University
ߒૐ₸ਅߢⷰኤߒߚ౮⌀ߩ࿑㧡 (b) ߅ࠃ߮ (c) ࠃࠅ㧘
Ⴚ⇇ࠍ᭴ᚑߔࠆォߪਣᎹᶧ⣣㘩ߢߪ⊕ࡇ࠶࠻ࠍᒻᚑ
ߒ㧘߹ߚ Young ᶧ⣣㘩ߢߪߘߩᄢㇱಽߩࡇ࠶࠻ᒻᚑߪ
⍎ߢߪߥ㧚ߎࠇߪ⽶ߩ⁁ォߦኻᔕߔࠆᓟ⠪ߦ ࠃࠆ⊕ࡇ࠶࠻ߪᵻߊዊߐߚߢࠆߣ⠨߃ࠄࠇࠆ㧚
৻ᣇ㧘Ⴚ⇇ߦု⋥ߥޛ 110 ޜᣇะߦ㈩ߒߚߔߴࠅォ
ࠍ᭴ᚑߔࠆォߪਣᎹᶧ⣣㘩ߢߪ⊕ࡇ࠶࠻㧔࿑㧡 (b) 㧕㧘
߹ߚ Young ᶧ⣣㘩ߢߪ㤥ࡇ࠶࠻㧔࿑㧡 (c) 㧕ࠍᒻᚑߒ㧘
ᱜߩ⁁ォߢࠆߎߣ߇ࠊ߆ࠆ㧚
㪊㪅㪉㩷 㪚㫌㪄Ꮧ⭯㪥㫀ว㊄⚿᥏䈮䈍䈔䉎䈜䈼䉍ォ᭴ㅧ㩷 ᒁ߈ߍߚ Cu- Ꮧ⭯ Ni ว㊄⚿᥏ߩォኒᐲࠍ Ni Ớ ᐲߦኻߒߡࡊࡠ࠶࠻ߒߚ࿑
(20)ࠍ࿑㧢ߦ㧘߹ߚ㧘ߔߴࠅ ォߩᒻᚑ㗫ᐲߣ Ni Ớᐲߣߩ㑐ଥࠍ࿑㧣ߦ␜ߔ㧚ߎ ࠇࠄߩ࿑߇␜ߔࠃ߁ߦ㧘 0.1at% એਅߩᓸ㊂ߩ Ni ᷝട
⚿᥏ߢߪႺ⇇᭴ㅧ߇ᒻᚑߐࠇࠆߎߣߪᱴߤߥ߇㧘 ߔߴࠅォ߇ᒻᚑߐࠇࠆะ߇ࠅ㧘ᐔဋォኒᐲ ߪ⚐ Cu ⚿᥏ࠃࠅ߿߿㜞
(20)㧚⚐ Cu ⚿᥏߅ࠃ߮ Cu- Ꮧ 㩷
㩷 㩷 㩷 㩷 㩷
࿑㧢 ᒁ߈ߍᴺߦࠃࠅ⢒ᚑߒߚ Cu- Ꮧ⭯ Ni ว㊄⚿᥏ߩォኒᐲߩ Ni Ớᐲଐሽᕈ㧘
٤㧦ήႺ⇇㧘
٨㧦Ⴚ⇇ᒻᚑ㧚 * ߅ࠃ߮ ** ߪૐㅦᐲߢᒁ߈ߍߚ⚿᥏ࠍᗧ
ߔࠆ
20)㧚
࿑㧤 ߔߴࠅォ᭴ㅧߩ (a) ⚐ Cu ⚿ ᥏ (b) Cu-0.1 at%Ni ว㊄⚿᥏㧚
࿑㧣 ᒁ߈ߍ Cu ⚿᥏߅ࠃ߮ Cu- Ꮧ⭯ Ni ว㊄⚿᥏ߩ
ⷰኤᢿ㕙㧔 18 એ㧕ߦ߅ߌࠆߔߴࠅォߩᒻᚑ㗫 ᐲ㧘ήࡑࠢ㧦ᱴߤߥ㧘ࡂ࠶࠴ࡦࠣ㧦ዋ㧘࠳ࡉ
࡞ࡂ࠶࠴ࡦࠣ㧦⪺ߒ㧚 * ૐㅦᐲߢ⢒ᚑߒߚ⚿᥏ߩ
ⷰኤ⚿ᨐࠍ㧚
12
今清水雄二⭯ Ni ว㊄⚿᥏ߩߔߴࠅォߩᒻᚑߩࠍ࿑㧤ߦ␜
ߔ㧚⚐ Cu ⚿᥏ߢߪ (a) ߩࠃ߁ߦ㧘ߔߴࠅォߪߣߊߦ ᒁ߈ߍೋᦼߩ⢋ㇱಽߩᢿ㕙ߦ⪺ߒߊᒻᚑߐࠇࠆ㧚 ߎࠇߦኻߒߡ Cu- Ꮧ⭯ Ni ว㊄⚿᥏ߢߪ (b) ߩࠃ߁ߦ㧘
⢋ㇱಽએᄖߩ৻ቯ⋥ᓘㇱಽߦ߅ߡ߽ߔߴࠅォ߇
⪺ߒߊ㧘ߣߊߦ㧝ࠆߪ㧞ᣇะߦ⇣Ᏹߦᒝߊᒻᚑ ߐࠇࠆะ߇ࠆ㧚
࿑㧥ߪ Cu- Ꮧ⭯ Ni ว㊄⚿᥏ߩᧃ┵ㇱᢿ㕙ࠍ Young ᶧߦࠃߞߡ⣣㘩ߒߚォಽᏓߩశቇ㗼ᓸ㏜౮⌀ߢ
ࠆ㧚 (a) ߦ⚿᥏ᢿ㕙ߩో⊛ߥォಽᏓߩ᭽⋧ࠍ␜ߒ ߚ߇㧘ߘߩ৻ᣇߩㄝㇱߦߪ (b) ߩࠃ߁ߦ⽶ߩ⁁ォ
㧔⊕ዊࡇ࠶࠻㧕߇ᄙߊᒻᚑߐࠇߡ߅ࠅ㧘 (b) ߣߪ
ኻߩㄝㇱ߆ࠄਛᄩㇱಽߦߪ (c) ߅ࠃ߮ (d) ߩࠃ߁ ߦߔߴࠅᣇะߦ㈩ߒߚ߅߽ߦᱜߩ⁁ォ㧔㤥ࡇ
࠶࠻㧕߇ᒻᚑߐࠇߡߚ㧚ߎߩࠃ߁ߦ Cu- Ꮧ⭯ Ni ว㊄
⚿᥏ߢߪᱜ⽶ߩ⁁ォ߇ߦኻߩㄝㇱߦ
ߞߡᒻᚑߐࠇࠆߣߣ߽ߦඨᓘᣇะߦ㕖ኻ⒓⊛ߥߔ ߴࠅ᭴ㅧ߇ᒻᚑߐࠇߚ㧚
㩷
㩷 㪋㪅㩷 ⠨㩷 ኤ㩷 㪋㪅㪈 ․⇣䈭Ⴚ⇇᭴ㅧ䈱ᒻᚑⷐ࿃㩷
Cu- Ꮧ⭯ Al ว㊄ߩ৻ㇱ⚿᥏ߦᒻᚑߐࠇߚࡐࠧࡦ Ⴚ⇇㧔࿑㧠 (a) 㧕ߪ㧘 Cu
(22), Fe-Si
(25), NaCl
(26)ߥߤߩ
⚿᥏ߦᦛߍᄌᒻࠍട߃ߚᓟ㧘߈ߥ߹ߒࠍᣉߒߚ႐ วߦᒻᚑߐࠇࠆࡐࠧࡦႺ⇇ߣห᭽ߢࠆ㧚ߎߩ 㘃ૃᕈࠃࠅ㧘࿑ 4 ߩࡐࠧࡦႺ⇇ߪ㧘࿕ᶧ⇇㕙ઃㄭ ߩ⚿᥏ߦᚑ㐳ᣇะߣᐔⴕߥᔃゲࠍᦛߍゲߣߔࠆ࿑
10(a) ߩࠃ߁ߥඨᓘᣇะߦ㕖ኻ⒓⊛ߥᦛߍᔕജ߇
↪ߒ㧘ߘࠇࠍ✭ߔࠆࠃ߁ߦዉߐࠇߚ⁁ォ߇
ౣ㈩ߒߡᒻᚑߐࠇߚ᭴ㅧߣផ᷹ߐࠇࠆ㧚ߎߩࠃ߁ ߦᐔⴕ㈩ߒߚႺ⇇ߪ 0.05 at%Al ว㊄⚿᥏ߩᧃ┵
ㇱએᄖߩ৻ᢿ㕙ߦ߽ⷰኤߐࠇߚ߇ , ห᭽ߩේ࿃ߦࠃ ߞߡᒻᚑߐࠇߚߣ⠨߃ࠄࠇࠆ .
߹ߚ㧘࿑㧠 (b) ߩࠃ߁ߥォ᭴ㅧߪ㧘࿑ 10(b) ߩࠃ߁ߦ㧘 ᚑ㐳⋥ᓟߩ㜞᷷⁁ᘒߦ߅ߡߘࠇ߶ߤᄢ߈ߊߥᏂⷞ
⊛ߥᦛߍᔕജߦࠃߞߡዉߐࠇߚ⽶ߩ⁁ォ߇ౣ㈩
ߒߡᒻᚑߐࠇߚ৻ߟߩႺ⇇ߣޔߘߩᓟߩ಄ළㆊ⒟
ߦ߅ߌࠆ㊰ലᨐߦࠃࠅ㜞᷷ᤨߦᏂⷞ⊛ߥᦛߍᔕജࠍ
↢ߓߚේ࿃߇⸃ᶖߐࠇࠆߣߣ߽ߦኻ╓ภߩᱷ⇐ᔕജ㩷
࿑ 9 Cu-0.1at%Ni ว㊄⚿᥏ߦᒻᚑߐࠇߚߔߴࠅォ᭴ㅧ㧔 Young ᶧ⣣㘩㧕 㧘 (a) ᢿ㕙ో
ߩォಽᏓ㧘 (b) ㄝߩ৻ㇱߦߪ⊕ࡇ࠶࠻㧔⽶ߩ⁁ォ㧕߇ᄙ㧚 (c) (b) ߣ⋧
ኻߔࠆㄝㇱߦߪ㤥ࡇ࠶࠻㧔ᱜߩ⁁ォ㧕߇ᄙߊ㧘 (d) ਛᄩㇱߦߪޛޜߔ ߴࠅᣇะߦ㈩ߒߚ㤥ࡇ࠶࠻㧔ᱜߩ⁁ォ㧕ߣಽᢔߒߚ⊕ࡇ࠶࠻㧔⽶ߩ⁁ォ
㧕߇ⷰኤߐࠇࠆ㧚
Akita University
߇↪ߒ㧘ᱜߩ⁁ォߦࠃࠆߔߴࠅ᭴ㅧࠍᒻᚑߒ
ߚ㧔࿑㧡 (b),(c) 㧕ߎߣࠍ␜ߔߣ⠨߃ࠄࠇࠆ㧚㩷
㪋㪅㪉 㕖ኻ⒓⊛䈭䈜䈼䉍ォ᭴ㅧ䈱ᒻᚑⷐ࿃㩷
࿑㧤 (a) ߩࠃ߁ߥ Cu ⚿᥏ߩߔߴࠅォߩ᭴ㅧߪ㧘 ᚑ㐳⋥ᓟߩ⚿᥏ߦඨᓘᣇะߩ᷷ᐲ൨㈩߇ߣ߽ߥ߁ᾲ㩷 ᔕജࠍ✭ߔࠆࠃ߁ߦォ߇ዉߐࠇߚᓟ㧘಄ළㆊ㩷
⒟ߦ߅ߡ᷷ᐲಽᏓߩဋ৻ൻߦߣ߽ߥᱷ⇐ᔕജ߇㩷
↪ߒ
(27)㧘ᒻᚑߐࠇߚߣ⠨߃ࠄࠇࠆ㧚 Cu- Ꮧ⭯ Ni ว
㊄⚿᥏ߢߪ࿑㧤 (b) ߩࠃ߁ߦ৻ߟߩޛ 110 ޜᣇะߩߔ ߴࠅ᭴ㅧ߇⪺ߒߊᒻᚑߐࠇࠆะ߇ࠆ㧚࿑㧥ߩᢿ 㕙ߩ႐ว㧘৻ߟߩޛ 110 ޜᣇะߦ㈩ߒߚߔߴࠅォ
ߩᒻᚑ㧔࿑ (a) 㧕ߦ߅ߡ㧘৻ㇱㄝㇱઃㄭߦߪ⽶ߩ
⁁ォ㧔⊕ዊࡇ࠶࠻㧕߇ᄙߊዉߐࠇ㧔࿑ (b) 㧕㧘 ߘࠇߣኻߩㄝㇱ߆ࠄਛᄩㇱߦ߆ߌߡᱜߩ⁁
ォ㧔㤥ࡇ࠶࠻㧕߇Ყセ⊛ᄙߊᒻᚑߐࠇߚ㧔࿑ (c) ߅ ࠃ߮ (d) 㧕㧚ߎࠇߪ࿑ 11(a) ߩࠃ߁ߦ㧘ᚑ㐳⋥ᓟߩ㜞᷷
⁁ᘒߦ߅ߡ↢ߓߚᚑ㐳ᣇะߦᐔⴕ߆ߟᔃߒߚゲ ࠍᦛߍゲߣߔࠆᏂⷞ⊛ߥᦛߍ߭ߕߺࠍ✭ߔࠆࠃ߁ ߦㄝઃㄭߩ৻ㇱߦ⽶ߩ⁁ォ߇ዉߐࠇߚᓟ㧘
಄ළㆊ⒟ߦ߅ߡ࿑ 11(b) ߩࠃ߁ߦኻ╓ภߩᱷ⇐
ᔕജ߇↪ߒ㧘ᱜߩ⁁ォߦࠃࠆߔߴࠅ᭴ㅧࠍᒻ ᚑߒߚߎߣࠍߒߡࠆ㧚࿑ 7 ߩ⚿ᨐߪ Cu- Ꮧ⭯ Ni ว
㊄⚿᥏ߢߪߘߩࠃ߁ߥඨᓘᣇะߦ㕖ኻ⒓⊛ߥᏂⷞ⊛
ᔕജߩ↪ߦࠃࠆߔߴࠅォ᭴ㅧࠍ↢ߓ߿ߔߎߣ ࠍ␜ߒߡࠆߣ⠨߃ࠄࠇࠆ㧚
㪋㪅㪊㩷 ᒁ䈐䈕㪚㫌Ꮧ⭯ว㊄⚿᥏䈮↪䈜䉎ᔕജ㩷
㪋㪅㪊㪅㪈㩷 ᷷ᐲ䈍䉋䈶ṁ⾰䈱ਇဋ৻ಽᏓ䈮䉋䉎ᩰሶ䈵䈝䉂㩷 (1) ඨᓘᣇะߩ᷷ᐲᏅߦࠃࠆᾲ߭ߕߺ
Ⲣ ᶧ ߆ ࠄ ᚑ 㐳 ߔ ࠆ ⚿ ᥏ ߦ ߪ 㜞 ᷷ ߩ Ⲣ ᶧ ߆ ࠄ ⒳⚿
᥏ᒁ߈ߍゲ߳ะ߆߁ゲᣇะߩᾲᵹߣ㕙߆ࠄ
࿐߳ᢔߔࠆඨᓘᣇะߩᾲᵹ߇↢ߕࠆ㧚⚿᥏ߪߘ ࠇߦኻᔕߔࠆゲᣇะ߅ࠃ߮ඨᓘᣇะߩ᷷ᐲ൨㈩ࠍ
ߔࠆߚ㧚ᾲ߭ߕߺࠍ↢ߓᾲᔕജߦࠃࠅォ߇ᒻᚑ ߐࠇࠆ㧚ߎߩ߁ߜਛᔃㇱߣㄝㇱߩ᷷ᐲᏅ' T ߦߣ
߽ߥ߁ᄖㇱߩᣇะߩᾲ߭ߕߺߪ㧘ᾲ⤘ᒛଥᢙ 㧔✢⤘ᒛଥᢙ㧕ࠍDߣߔࠆߣ㧘ㄭૃ⊛ߦ
T D '
H
T(1)
ߣߐࠇࠆ㧚ߎࠇߪᚑ㐳ᣇะߦု⋥ᢿ㕙ࠍ⽾ߊ⁁
ォࠍᒻᚑߔࠆⷐ࿃ߣߥࠆ
(28)㧚ᧄታ㛎ߩᒁ߈ߍⵝ
⟎ߦࠃࠅ Ar ᳇ᵹਛߢ⋥ᓘ 10 mm ߩ⚐ Cu ⚿᥏ࠍ⢒ᚑߔ ࠆߣ߈ߩ⚿᥏ਛᔃㇱߣㄝㇱߩ᷷ᐲᏅ'7ࠍ㧘࠳ࡒ
⹜ᢱߦᾲ㔚ኻࠍขࠅઃߌߡ⋥ធ᷹ቯߒߚ୯ߣਛᔃㇱ ߩゲᣇะߩ᷷ᐲ൨㈩ߩ᷹ቯ୯߆ࠄ㑆ធ⊛ߦផቯߒߚ ୯
(29)ߦࠃࠇ߫㧘ᑼ (1) ߦࠃࠆᾲ߭ߕߺߩ▚୯ߪ 1 ߩࠃ߁ߦߥࠆ㧚 '7ߩ㑆ធ⊛ߥផቯ୯ߪᾲߦࠃࠆ
⸘▚୯
ߦ߶߷৻⥌ߔࠆߩߦኻߒ㧘⋥ធ᷹ቯߒߚ୯ ߇ᄢ߈ߩߪ㕙ߩ᷷ᐲߩ᷹ቯ⺋Ꮕߩߚߣᕁࠊ ࠇࠆ
(29)㧚ߒ߆ߒ㧘ᧄታ㛎ߢᒁ߈ߍߚ⚐ Cu ⚿᥏ߩᚑ 㐳ᣇะߦု⋥ᢿ㕙ߩ⁁ォߩኒᐲ
(18)㧦U § 3 10
8m
㧙2ߪ Billig ߩ㑐ଥᑼ
Ҟ(28)җߦࠃࠇ߫ᓟ⠪ߩ'7߇ਈ߃ ࠆᾲ߭ߕߺߦ⋧ᒰߒ㧘'7ߩ୯߅ࠃ߮'7ߣォኒᐲ ߩቯ㊂⊛㑐ଥߦߟߡߪ⚦ߥᬌ⸛ࠍⷐߔࠆ㧚
Thermal strain
Material D[K
1](1273K) 'T [K] H
T[10
5]
Cu 2.0 10
5 (31)0.2*~5**
(28)0.4~10
D: Linear thermal expansion coefficient,
*Indirectly and ** directly measured values
࿑ 10 Ⴚ⇇ߩᒻᚑࡕ࠺࡞㧘࿕ᶧ⇇㕙⋥ਅߩ⚿᥏ߦ ᦛߍᔕജ㧔⎕✢⍫ශ㧕߇↪ߒߡ⽶ߩ⁁ォ߇ ዉߐࠇ㧘 (a) ዊⷺႺ⇇ߩ ࠆߪ (b) ৻ ߟߩዊⷺႺ⇇ࠍᒻᚑ㧚 (b) ߢߪߐࠄߦ಄ළਛߦ ᱷ⇐ᔕജߦࠃࠆㅒߩᦛߍᔕജ㧔ታ✢⍫ශ㧕߇↪
ߒ㧘ᱜߩ⁁ォߦࠃࠆߔߴࠅ᭴ㅧࠍᒻᚑߔࠆ㧚
࿑ 11 ߔߴࠅォߩᒻᚑࡕ࠺࡞㧚 (a) ࿕ᶧ⇇㕙⋥ਅߩ
⚿᥏ߦᦛߍᔕജ㧔⎕✢⍫ශ㧕߇↪ߒ㧘⽶ߩ
⁁ォ㧕ࠍᒻᚑ㧘 (b) ಄ළਛߦᱷ⇐ᔕജߦࠃ ࠆㅒߩᦛߍᔕജ㧔ታ✢⍫ශ㧕߇↪ߒ㧘ᱜߩ
⁁ォߦࠃࠆߔߴࠅ᭴ㅧࠍᒻᚑߔࠆ㧚
1 Cu ⚿᥏ߦ߅ߌࠆᾲ߭ߕߺ H
Tߩ▚୯㧚
14
今清水雄二(2) ඨᓘᣇะߩṁ⾰ỚᐲᏅߦࠃࠆ⚵ᚑ߭ߕߺ
৻ᣇ㧘 Cz Si ⚿᥏
(32)߅ࠃ߮ LEC InP ⚿᥏
(33)ߥߤߢߪ ᒁ߈ߍ⚿᥏ߩਛᔃߣㄝㇱߣߩ㑆ߦṁ⾰ේሶߩỚ ᐲᏅࠍ↢ߕࠆߎߣ߇⍮ࠄࠇߡࠆޕߎࠇߪᚑ㐳⇇㕙 ߩṁ⾰ේሶߩᢔႺ⇇ጀߩෘߐǬߪ⚿᥏ਛᔃ߆ࠄߩ ඨᓘ〒㔌ߦኻߒߡ৻᭽ߢߥߊ
(34)㧘ലಽ㈩ଥᢙ k
eߩ Ǭଐሽᕈ
(35)ߦࠃࠅṁ⾰߇ඨᓘᣇะߦਇဋ৻ಽᏓߔ ࠆߚߣ⠨߃ࠄࠇࠆ
(36)㧚ߎߩࠃ߁ߥṁ⾰ේሶߩਇဋ
৻ಽᏓߪᩰሶቯᢙߩṁ⾰Ớᐲଐሽᕈߦࠃࠆᩰሶ߭ߕ ߺࠍ↢ߓ
(37)㧘ᾲᔕജߦ㘃ૃߒߚᏂⷞ⊛ߥᔕജ㧔એᓟ㧘
⚵ᚑᔕജߣ⒓㧕ࠍ⺃ዉߔࠆߣ⠨߃ࠄࠇࠆ㧚߹㧘 ⲢᶧỚᐲࠍ C ߣߒߡ㧘ᢔႺ⇇ጀෘߐǬ߇৻᭽ߢߥ
ߎߣߦࠃࠆലಽ㈩ଥᢙߩᏅ ' k
eߦߣ߽ߥ߁⚿
᥏ਛᔃߣㄝㇱߣߩṁ⾰ỚᐲᏅࠍ㧘'C = 'k
eC ߣ߅ ߊ㧚߹ߚ㧘නṁ⾰Ớᐲߚࠅߩᩰሶ߭ߕߺࠍ' a/a ߣߔࠆߣ㧘ṁ⾰ỚᐲᏅߦߣ߽ߥ߁⚵ᚑ߭ߕߺߪ
C a k C a a
a
eC