◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆
❚❘❸◆● ❚❍⑩■ ❍❆■ ▼❖❉❊ ❑➌❚ ❍ÑP P❍❷◆ ✣➮■ ❳Ù◆●
❚❍➊▼ ❇❆ ❱⑨ ❇❰❚ P
ì
t ỵ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✣↕✐ ❤å❝ ❍✉➳
✯❊♠❛✐❧✿ tr✉♦♥❣♠✐♥❤❞✉❝❅❞❤s♣❤✉❡✳❡❞✉✳✈♥
❚â♠ t➢t✿ ❇➔✐ ❜→♦ ♥➔② tr➻♥❤ ❜➔② ♠ët ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❝❤ó♥❣ tæ✐ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t
♣❤✐ ❝ê ✤✐➸♥ ❜➟❝ t❤➜♣ ✈➔ ❜➟❝ ❝❛♦ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣
t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣✳ ❑➳t q✉↔ ❦❤↔♦ s→t ❝❤♦ t❤➜② tr↕♥❣ ♥➔② t❤➸ ❤✐➺♥
t➼♥❤ ❝❤➜t ♥➨♥ tê♥❣ ❤❛✐ ♠♦❞❡ ♥❤÷♥❣ ❤♦➔♥ t♦➔♥ ❦❤ỉ♥❣ ❝â t➼♥❤ ❝❤➜t ♥➨♥ ❤✐➺✉ ❤❛✐
♠♦❞❡✳ ❑❤✐ ❦❤↔♦ s→t sü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③✱ ❝❤ó♥❣ tỉ✐ ♥❤➟♥
t❤➜② tr↕♥❣ t❤→✐ ♥➔② ❤♦➔♥ t♦➔♥ ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③✳ ❍ì♥
♥ú❛✱ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❤❛✐ ♠♦❞❡✱ ❦➳t q✉↔ ❦❤↔♦ s→t
❝❤♦ t❤➜② tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥
tê♥❣ t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ị♠ ð ♠å✐ ❜➟❝ t❤➜♣ ✈➔ ❝❛♦✱ tr♦♥❣ ✤â ❜➟❝ ❝➔♥❣
❝❛♦ t❤➻ ❝➜♣ ✤ë ♣❤↔♥ ❦➳t ❝❤ò♠ ❝➔♥❣ t❤➸ ❤✐➺♥ ♠↕♥❤ ❤ì♥✳
❚ø ❦❤â❛✿ ◆➨♥ tê♥❣ ❤❛✐ ♠♦❞❡✱ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡✱ sü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝
❈❛✉❝❤② ✲ ❙❝❤✇❛r③✱ ♣❤↔♥ ❦➳t ❝❤ò♠✳
✶
●■❰■ ❚❍■➏❯
❱✐➺❝ t↕♦ r❛ ❝→❝ tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ õ ỵ rt q trồ t ố ợ ♥❣➔♥❤
❦❤♦❛ ❤å❝ ❧÷đ♥❣ tû✱ ✈➻ ❦❤↔ ♥➠♥❣ ù♥❣ ❞ư♥❣ ❝õ❛ ú ỷ ỵ tổ t ữủ tỷ ữ
t tố ở tr tổ t t t t ỗ tớ ❣✐↔♠ ♥❤✐➵✉ t➼♥ ❤✐➺✉ ✈➔ t➠♥❣
✤ë tr✉♥❣ t❤ü❝ ❝õ❛ t❤æ♥❣ t✐♥ ❬✶❪✳ ❇➯♥ ❝↕♥❤ ✤â✱ ❝→❝ tr↕♥❣ t❤→✐ ♥➔② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✤➸ →♣
❞ư♥❣ ✈➔♦ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤÷ q ữủ tỷ tt ữủ tỷ t ỵ ❝❤➜t r➢♥
❬✷❪✳ ❈→❝ tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ ♥❣➔② ❝➔♥❣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ t✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔
♣❤→t tr✐➸♥✱ ✈ỵ✐ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ ♥❤÷ tr↕♥❣ t❤→✐ ♥➨♥✱ tr↕♥❣ t❤→✐
❦➳t ❤đ♣ ✤è✐ ①ù♥❣✱ ♣❤↔♥ ✤è✐ ①ù♥❣✳ ❱➔♦ ✶✾✾✶✱ ❆❣❛r✇❛❧ ✈➔ ❚❛r❛ ✤➣ ✤➲ t ỵ tữ tr
t t ủ t t s õ ữủ t ỵ ự ❦❤↔♦ s→t ❬✹✱ ✺❪✳
◆❣♦➔✐ t❤➯♠ ♣❤♦t♦♥ ❝á♥ ❝â ❜ỵt ♣❤♦t♦♥ ✈➔♦ ❝→❝ tr↕♥❣ t❤→✐ ❦➳t ❤đ♣ ❝ơ♥❣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉
❬✻❪ ✈➔ ❝❤♦ r❛ ✤í✐ ❝→❝ tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ ♠ỵ✐ ❦❤→❝✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝
tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ t❤➯♠ ✈➔ ❜ỵt ♣❤♦t♦♥ ❧➔ ✈✐➺❝ ❤➳t sù❝ q✉❛♥ trå♥❣✱ ❦❤æ♥❣ ❝❤➾ t↕♦ r❛ ❝→❝
tr↕♥❣ t❤→✐ ♣❤✐ ờ ợ ỏ r ữợ ❝ù✉ ♠ỵ✐ ❝❤♦ ❝→❝ ♥❤➔ ❦❤♦❛
❤å❝✳ ❚ø ❦➳t q✉↔ t❤✉ ✤÷đ❝✱ ♥❣÷í✐ t❛ ù♥❣ ❞ư♥❣ ❝→❝ tr↕♥❣ t❤→✐ ♥➔② ✈➔♦ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝
❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✣↕✐ ❤å❝ ❍✉➳
■❙❙◆ ✶✽✺✾✲✶✻✶✷✱ ❙è ✸✭✺✺✮✴✷✵✷✵✿ tr✳✹✵✲✺✵
◆❣➔② ♥❤➟♥ ❜➔✐✿ ✺✴✽✴✷✵✶✾❀ ❍♦➔♥ t❤➔♥❤ ♣❤↔♥ ❜✐➺♥✿ ✶✴✸✴✷✵✷✵❀ ◆❣➔② ♥❤➟♥ ✤➠♥❣✿ ✶✺✴✸✴✷✵✷✵
✹✶
◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳
❦ÿ t❤✉➟t✱ ♥❤➜t ❧➔ ✤è✐ ✈ỵ✐ ❦❤♦❛ ❤å❝ t❤ỉ♥❣ t✐♥ ❧÷đ♥❣ tû ✈➔ ♠→② t➼♥❤ ❧÷đ♥❣ tû✳ ◗✉→ tr➻♥❤
❦❤↔♦ s→t ❝→❝ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t
❤đ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ ✤➣ ✤÷đ❝ ♠ët sè t→❝ ❣✐↔ t❤ü❝ ❤✐➺♥ ❬✼✱ ✽❪✳ ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❝❤ó♥❣ tỉ✐
✤÷❛ r❛ tr↕♥❣ t❤→✐ ❣å✐ ❧➔ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t ợt ởt
t tờ ữ s
|
ab
= N (
a3 + ˆb) (|α a |β
b
✭✶✮
− |β a |α b ) ,
tr♦♥❣ ✤â
Nα,β ={|α|6 + |β|6 + 9(|α|4 + |β|4 ) + 19(|α|2 + |β|2 ) + 12 + 2Re[α3 β + αβ 3 ]
1
− exp[−|α − β|2 ](2Re[α∗3 β 3 + 9α∗2 β 2 + 19α∗ β + α3 β + αβ 3 ] + 12)}− 2 ,
✭✷✮
❧➔ ❤➺ sè ❝❤✉➞♥ ❤â❛✱ a
ˆ† ✈➔ ˆb ❧➛♥ ❧÷đt ❧➔ t♦→♥ tû s✐♥❤ ✤è✐ ✈ỵ✐ ♠♦❞❡ a ✈➔ t♦→♥ tû ❤õ② ✤è✐ ✈ỵ✐
♠♦❞❡ b✱ ❘❡❬❩❪ ❧➔ ❦➼ ❤✐➺✉ ♣❤➛♥ t❤ü❝ ❝õ❛ ♠ët số ự ởt tr t ợ ữ
ữủ ♥❣❤✐➯♥ ❝ù✉✱ ❦❤↔♦ s→t✳ ❱➻ ✈➟②✱ tr♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t
♣❤✐ ❝ê ✤✐➸♥ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥
tê♥❣ t❤ỉ♥❣ q✉❛ t➼♥❤ ❝❤➜t ♥➨♥ tê♥❣ ❤❛✐ ♠♦❞❡ ✈➔ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡✱ sü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣
t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③ ✈➔ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ ❝❛♦ ❤❛✐ ♠♦❞❡✳
✷
❚➑◆❍ ❈❍❻❚ ◆➆◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■ ❍❆■ ▼❖❉❊ ❑➌❚ ❍ÑP P❍❷◆ ✣➮■ ❳Ù◆●
❚❍➊▼ ❇❆ ❱⑨ ❇❰❚ ▼❐❚ P❍❖❚❖◆ ❚✃◆●
✷✳✶✳ ◆➨♥ tê♥❣ ❤❛✐ ♠♦❞❡
❍❛✐ ❦✐➸✉ ♥➨♥ ❜➟❝ ❝❛♦ ❧➔ ♥➨♥ tê♥❣ ✈➔ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡ ✤➣ ✤÷đ❝ ❍✐❧❧❡r② ❬✾❪ ✤÷❛ r❛ ✈➔♦ ♥➠♠
✶✾✽✾✳ ❚❤❡♦ ✤â✱ ♠ët tr↕♥❣ t❤→✐ ✤÷đ❝ ❣å✐ ❧➔ ♥➨♥ tê♥❣ ♥➳✉ tr✉♥❣ ❜➻♥❤ tr♦♥❣ tr↕♥❣ t❤→✐ ✤â
t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝
∆Vˆϕ
tr♦♥❣ ✤â Vˆϕ =
1
2
2
= Vˆϕ2 − Vˆϕ
2
<
1
n
ˆa + n
ˆb + 1 ,
4
✭✸✮
eiϕ a
ˆ†ˆb† + e−iϕ a
ˆˆb ✱ n
ˆa = a
ˆ† a
ˆ✱n
ˆ b = ˆb†ˆb ❧➛♥ ❧÷đt ❧➔ t♦→♥ tû sè ❤↕t ❝õ❛ ❤❛✐
♠♦❞❡ a ✈➔ b✳ ✣➸ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ ❦❤↔♦ s→t t❛ ✤➦t S ❧➔ t❤❛♠ sè ♥➨♥ tê♥❣ ❝â ❞↕♥❣
S = Vˆϕ2 − Vˆϕ
2
−
1
(ˆ
na + n
ˆ b + 1) .
4
✭✹✮
▼ët tr↕♥❣ t❤→✐ ❣å✐ ❧➔ ♥➨♥ tê♥❣ ❤❛✐ ♠♦❞❡ ♥➳✉ t❤❛♠ sè S < 0✳ ❱➻ α ✈➔ β ❧➔ ❝→❝ sè ♣❤ù❝ ♥➯♥
t❛ ✤➦t α = ra exp (iϕa )✱ β = rb exp (iϕb ) ✈➔ φh,k,m = hϕ + kϕa + mϕb ✱ ✈ỵ✐ h, k, m ❧➔ ❝→❝ sè
♥❣✉②➯♥✱ ra , rb , ϕa , ϕb ❧➔ ❝→❝ sè t❤ü❝✳ ❙û ❞ö♥❣ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ♣❤↔♥ ✤è✐ ①ù♥❣
t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ ✈➔♦ ❝ỉ♥❣ t❤ù❝ ✭✹✮ t❛ ♥❤➟♥ ✤÷đ❝
1 2
S = Nαβ
{2{(ra8 rb2 + ra2 rb8 + 15(ra6 rb2 + ra2 rb6 ) + 61(ra4 rb2 + ra2 rb4 ) + 240ra2 rb2 ) cos(φ2,−2,−2 )
4
+ (ra5 rb3 + 6ra3 rb3 + 6ra rb3 ) cos(φ2,1,−1 ) + ra5 rb3 cos(φ2,−5,−3 ) + ra3 rb5 cos(φ2,−3,−5 )
✹✷
❍➬ ◆●➴❈ ❚❘❯◆●✱ ❚❘×❒◆● ▼■◆❍ ✣Ù❈
+ (ra3 rb5 + 6ra3 rb3 + 6ra3 rb ) cos(φ2,−1,1 ) − x(2ra5 rb5 (cos(φ2,−5,1 ) + cos(φ2,1,−5 ))
+ 15ra4 rb4 (cos(φ2,−4,0 ) + cos(φ2,0,−4 )) + 61ra3 rb3 (cos(φ2,−3,−1 ) + cos(φ2,−1,−3 ))
+ (ra3 rb5 + 6ra3 rb ) cos(φ2,−1,1 ) + 120ra2 rb2 cos(φ2,−2,−2 ) + ra5 rb3 cos(φ2,−5,−3 )
+ ra3 rb5 cos(φ2,−3,−5 ) + (ra5 rb3 + 6ra rb3 ) cos(φ2,1,−1 ) + 6(ra2 rb4 + ra4 rb2 ) cos(φ2,0,0 )}
+ 2(ra8 rb2 + ra2 rb8 ) + 31(ra6 rb2 + ra2 rb6 ) + 137(ra4 rb2 + ra2 rb4 ) + 350ra2 rb2 + 139(ra2
+ rb2 ) + ra8 + rb8 + 16(ra6 + rb6 ) + 73(ra4 + rb4 ) + 48 + 2(2ra5 rb3 + 7ra3 rb3 + ra5 rb
+ 4ra3 rb ) cos(φ0,3,1 ) + 2(2ra3 rb5 + 7ra3 rb3 + ra rb5 + 4ra rb3 ) cos(φ0,1,3 ) − x(350ra2 rb2
+ 48 + 2[ra4 rb4 cos(φ0,4,−4 ) + 73ra2 rb2 cos(φ0,2,−2 ) + 31ra4 rb4 cos(φ0,2,−2 )
+ (137ra3 rb3 + 139ra rb ) cos(φ0,1,−1 ) + (2ra3 rb5 + 4ra rb3 ) cos(φ0,1,3 ) + ra4 rb2 cos(φ0,4,0 )
+ (2ra5 rb5 + 16ra3 rb3 ) cos(φ0,3,−3 ) + (2ra5 rb3 + 4ra3 rb ) cos(φ0,3,1 ) + ra2 rb4 cos(φ0,0,4 )
+ 7(ra2 rb4 + ra4 rb2 ) cos(φ0,2,2 )])} − {(16(ra6 + rb6 ) + ra8 + rb8 + 73(ra4 + rb4 )
+ 103(ra2 + rb2 ) + 48 + ra6 rb2 + ra2 rb6 + 9(ra4 rb2 + ra2 rb4 ) + (8ra3 rb + 2ra5 rb ) cos(φ0,3,1 )
+ 38ra2 rb2 + 2ra3 rb3 (cos(φ0,3,1 ) + cos(φ0,1,3 )) + (8ra rb3 + 2ra rb5 ) cos(φ0,1,3 )
− x(2ra4 rb4 cos(φ4,0,0 ) + 32ra3 rb3 cos(φ3,0,0 ) + (2ra4 rb2 + 2ra2 rb4 ) cos(φ0,2,2 )
+ (146ra2 rb2 + 2ra4 rb4 ) cos(φ2,0,0 ) + (206ra rb + 18ra3 rb3 ) cos(φ1,0,0 ) + 48
+ 2ra4 rb2 cos(φ0,4,0 ) + 2ra2 rb4 cos(φ0,0,4 ) + 8ra3 rb cos(φ0,3,1 ) + 8ra rb3 cos(φ0,1,3 )
1 4
{((2ra2 rb4 + 6ra2 rb2 ) cos(φ1,0,2 ) + 2(ra7 rb + rb7 ra + 12(ra5 rb
+ 38ra2 rb2 )} − Nαβ
4
+ rb5 ra ) + 48ra rb + 37(ra3 rb + rb3 ra )) cos(φ1,−1,−1 ) + (2rb3 ra + 2ra3 rb ) cos(φ1,−1,−1 )
+ (2ra4 rb2 + 6ra2 rb2 ) cos(φ1,2,0 ) + 2ra4 rb2 cos(φ1,−4,−2 ) + 2ra2 rb4 cos(φ1,−2,−4 )
− x(2ra2 rb4 cos(φ1,0,2 ) + 2ra2 rb4 cos(φ1,−2,−4 ) + 6(ra3 rb + rb3 ra ) cos(φ1,1,1 )
+ 2ra4 rb2 cos(φ1,2,0 ) + 2ra4 rb4 (cos(φ1,−4,2 ) + cos(φ1,2,−4 )) + 2ra4 rb2 cos(φ1,−4,−2 )
+ 24ra3 rb3 (cos(φ1,−3,1 ) + cos(φ1,1,−3 )) + 96ra rb cos(φ1,−1,−1 ) + 74ra2 rb2 (cos(φ1,−2,0 )
✭✺✮
+ cos(φ1,0,−2 )))}2 ,
tr♦♥❣ ✤â
Nα,β =(ra6 + rb6 + 9(ra4 + rb4 ) + 19(ra2 + rb2 ) + 2ra3 rb cos(φ0,3,1 ) + 2ra rb3 cos(φ0,1,3 )
+ 12 − x(2ra3 rb3 cos(φ0,3,−3 ) + 18ra2 rb2 cos(φ0,2,−2 ) + 38ra rb cos(φ0,1,−1 )
1
+ 2ra3 rb cos(φ0,3,1 ) + 2ra rb3 cos(φ0,1,3 ) + 12)− 2 ,
✭✻✮
✈ỵ✐ x = exp[−ra2 − rb2 + 2ra rb cos(0,1,1 )]
ỗ t st sỹ ử tở ừ t❤❛♠ sè ❙ ✈➔♦ ❜✐➯♥ ✤ë ❦➳t ❤đ♣ rb ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❦❤↔♦
s→t ❧➔ ra = rb , ϕ = π2 , ϕa = π, ϕb = π2 ✈➔ 0 rb 5 ỗ t t t t❤❛♠ sè ♥➨♥
tê♥❣ S ❧✉ỉ♥ ❜➨ ❤ì♥ ❤♦➦❝ ❜➡♥❣ 0✱ ♥❣❤➽❛ ❧➔ ❧✉æ♥ ①✉➜t ❤✐➺♥ q✉→ tr➻♥❤ ♥➨♥ tê♥❣ tr♦♥❣ tr↕♥❣
t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣✳ ❑❤✐ ❝→❝ t❤❛♠ sè
ra ✱ rb ❝➔♥❣ t➠♥❣ t❤➻ t❤❛♠ sè ❙ ❝➔♥❣ ➙♠✱ ♥❣❤➽❛ ❧➔ q✉→ tr➻♥❤ ♥➨♥ tê♥❣ ❝➔♥❣ t❤➸ ❤✐➺♥ rã✳ ❱ỵ✐
◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳
✹✸
❍➻♥❤ ✶✿ ❑❤↔♦ s→t sü ♣❤ö t❤✉ë❝ ❝õ❛ t❤❛♠ sè ❙ ✈➔♦ ❜✐➯♥ ✤ë ❦➳t ❤ñ♣ rb ❝õ❛ tr↕♥❣ t❤→✐
❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣✳
✤✐➲✉ ❦✐➺♥ ❦❤↔♦ s→t tr➯♥✱ ❝❤ù♥❣ tä tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔
❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ ❝â t➼♥❤ ♥➨♥ tê♥❣✳ ❑❤✐ ❝→❝ t❤❛♠ sè ra ✱ rb ❝➔♥❣ t➠♥❣ t❤➻ t➼♥❤ ♥➨♥ tê♥❣
❝õ❛ tr↕♥❣ t❤→✐ ❝➔♥❣ t❤➸ ❤✐➺♥ rã✳
✷✳✷✳ ◆➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡
❚❤❡♦ ❍✐❧❧❡r② ❬✾❪ ♠ët tr↕♥❣ t❤→✐ ✤÷đ❝ ❣å✐ ❧➔ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡ ♥➳✉ ❦❤✐ tr✉♥❣ ❜➻♥❤ tr♦♥❣
tr↕♥❣ t❤→✐ ✤â t❤ä❛ ♠➣♥ ❜➜t ✤➤♥❣ t❤ù❝
ˆ2 − W
ˆϕ
W
ϕ
2
−
1
(|ˆ
na − n
ˆ b |) < 0.
4
✭✼✮
✣➸ ✤ì♥ ❣✐↔♥ ❝❤♦ ✈✐➺❝ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♥➨♥ ❤✐➺✉✱ t❛ ✤➦t t❤❛♠ sè ♥➨♥ ❤✐➺✉ D ❝â ❞↕♥❣
ˆ ϕ2 − W
ˆϕ
D= W
ˆϕ =
tr♦♥❣ ✤â W
1
2
2
−
1
(|ˆ
na − n
ˆ b |) ,
4
✭✽✮
eiϕ a
ˆˆb† + e−iϕ a
ˆ†ˆb ✱ n
ˆa = a
ˆ† a
ˆ, n
ˆ b = ˆb†ˆb ❧➛♥ ❧÷đt ❧➔ t♦→♥ tû sè ❤↕t ❝õ❛ ❤❛✐
♠♦❞❡ a ✈➔ b✳ ▼ët ❝→❝❤ t÷ì♥❣ tü ♣❤➛♥ ♥➨♥ tê♥❣✱ ❦❤✐ ❦❤↔♦ s→t t➼♥❤ ♥➨♥ ❤✐➺✉ ❝õ❛ tr↕♥❣ t❤→✐
❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ t❛ t❤✉ ✤÷đ❝ ❦➳t q✉↔ ❝ư
t❤➸ ♥❤÷ s❛✉
1 2
{(2ra2 rb8 + 30ra2 rb6 + 120ra2 rb4 + 120ra2 rb2 + 2ra4 rb2 ) cos(φ2,−2,2 ) + (2ra5 rb3 + 12ra3 rb3
D = Nαβ
4
+ 12ra rb3 ) cos(φ2,−1,−3 ) + 2ra3 rb5 cos(φ2,−1,5 ) + 2ra5 rb3 cos(φ2,5,−1 ) + (2ra3 rb5 + 12ra3 rb3
+ 12ra3 rb ) cos(φ2,−3,−1 ) + (2ra8 rb2 + 30ra6 rb2 + 120ra4 rb2 + 120ra2 rb2 + 2ra2 rb4 ) cos(φ2,2,−2 )
− x(2ra3 rb7 cos(φ2,−3,3 ) + 2ra rb7 cos(φ2,1,3 ) + 2ra7 rb3 cos(φ2,3,−3 ) + 2ra7 rb cos(φ2,3,1 )
+ (120ra rb5 + 2ra5 rb ) cos(φ2,−1,1 ) + (120ra5 rb + 2ra rb5 ) cos(φ2,1,−1 ) + 30ra2 rb6 cos(φ2,−1,1 )
+ 30ra6 rb2 cos(φ2,1,−1 ) + (2ra3 rb5 + 12ra3 rb ) cos(φ2,−3,−1 ) + 12ra2 rb4 cos(φ2,−2,−2 ) + 12ra4 rb2
cos(φ2,−2,−2 ) + (2ra5 rb3 + 12ra rb3 ) cos(φ2,−1,−3 ) + 120(ra4 + rb4 ) cos(φ2,0,0 )) + 2(ra8 rb2
+ ra2 rb8 ) + 31(ra6 rb2 + ra2 rb6 ) + 137(ra4 rb2 + ra2 rb4 ) + 350ra2 rb2 + 120(ra2 + rb2 ) + ra8 + rb8
✹✹
❍➬ ◆●➴❈ ❚❘❯◆●✱ ❚❘×❒◆● ▼■◆❍ ✣Ù❈
+ 64(ra4 + rb4 ) + 15(ra6 + rb6 ) + (4ra5 rb3 + 14ra3 rb3 + 2ra5 rb + 6ra3 rb ) cos(φ0,3,1 ) + (4ra3 rb5
+ 14ra3 rb3 + 2ra rb5 + 6ra rb3 ) cos(φ0,1,3 ) + 36 − x(2ra4 rb4 cos(φ0,4,−4 ) + (4ra5 rb5 + 30ra3 rb3 )
cos(φ0,3,−3 ) + 128ra2 rb2 cos(φ0,2,−2 ) + 350ra2 rb2 + 36 + (4ra3 rb5 + 6ra rb3 ) cos(φ0,1,3 )
+ 2ra2 rb4 cos(φ0,0,4 ) + (4ra5 rb3 + 6ra3 rb ) cos(φ0,3,1 ) + (62ra4 rb4 + 274ra3 rb3 + 240ra rb )
cos(φ0,1,−1 ) + 2ra4 rb2 cos(φ0,4,0 ) + 14(ra2 rb4 + ra4 rb2 ) cos(φ0,2,2 ) − |(36 + ra8 + rb8 + 15(ra6
+ rb6 ) + 62(ra4 + rb4 ) + (2ra5 rb + 6ra3 rb − 2ra3 rb3 ) cos(φ0,3,1 ) − 34ra2 rb2 − (ra6 rb2 + ra2 rb6 )
+ 72(ra2 + rb2 ) + (2ra rb5 + 6ra rb3 − 2ra3 rb3 ) cos(φ0,1,3 ) − 9(ra4 rb2 + ra2 rb4 ) − x(36 − 34ra2 rb2
+ (124ra2 rb2 − 2ra4 rb4 ) cos(φ2,0,0 ) + 2ra4 rb4 cos(φ4,0,0 ) + 2ra2 rb4 cos(φ0,0,4 ) + 2ra4 rb2
cos(φ0,4,0 ) + 6ra3 rb cos(φ0,3,1 ) + 30ra3 rb3 cos(φ3,0,0 ) + (144ra rb − 18ra3 rb3 ) cos(φ1,0,0 )
1 4
{(2ra7 rb + 24ra5 rb + 72ra3 rb
− (2ra2 rb4 + 2ra4 rb2 ) cos(φ0,2,2 ) + 6ra rb3 cos(φ0,1,3 )))|} − Nαβ
4
+ 48ra rb + 2ra rb3 ) cos(φ1,1,−1 ) + 2ra4 rb2 cos(φ1,4,0 ) + (2ra rb7 + 24ra rb5 + 72ra rb3 + 48ra rb
+ 2ra3 rb ) cos(φ1,−1,1 ) + 2ra2 rb4 cos(φ1,0,4 ) + (2ra4 rb2 + 2ra2 rb4 + 12ra2 rb2 ) cos(φ1,−2,−2 )
− x(2ra rb5 cos(φ1,1,3 ) + (72ra rb3 + 2ra3 rb ) cos(φ1,−1,1 ) + 48(ra2 + rb2 ) cos(φ1,0,0 ) + 2ra3 rb3
cos(φ1,−1,−3 ) + (72ra3 rb + 2ra rb3 ) cos(φ1,1,−1 ) + 2ra3 rb3 cos(φ1,−3,−3 ) + 2ra5 rb cos(φ1,3,1 )
+ 2ra5 rb3 cos(φ1,3,−3 ) + 24ra2 rb4 cos(φ1,−2,2 ) + 24ra4 rb2 cos(φ1,2,−2 ) + 12ra2 rb2 cos(1,2,2 )
+ 2ra3 rb5 cos(1,3,3 ))}2 .
ỗ t ✷ ❦❤↔♦ s→t t❤❛♠ sè D t❤❡♦ ❜✐➯♥ ✤ë ❦➳t ủ rb ữợ ợ
✷✿ ❑❤↔♦ s→t sü ♣❤ö t❤✉ë❝ ❝õ❛ t❤❛♠ sè ❉ ở t ủ rb ữợ
ừ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣✳
s→t ❧➔ ra = 2rb , ϕa = 0, ϕb = π2 , 0 ≤ rb ≤ 3 ✈➔ 0 ≤ ϕ ≤ 2π ✳ ỗ t t t t số
D ❧✉ỉ♥ ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ ✵ ✈➔ ❝❤ó♥❣ ❝â t➼♥❤ t t ữợ
t số ra ✱ rb ❝➔♥❣ t➠♥❣ t❤➻ t❤❛♠ sè D ❝➔♥❣ ❞÷ì♥❣✱ tù❝ ❧➔ ❝➔♥❣ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t
♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡✳ ữ ợ st ữ tr tr t❤→✐ t❤➯♠ ❜❛ ✈➔ ❜ỵt
♠ët ♣❤♦t♦♥ tê♥❣ ❤♦➔♥ t♦➔♥ ❦❤ỉ♥❣ ❝â t➼♥❤ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡✳
✹✺
◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳
✸
❙Ü ❱■ P❍❸▼ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❈❆❯❈❍❨✲❙❈❍❲❆❘❩ ❱⑨ ❚➑◆❍ P❍❷◆ ❑➌❚
❈❍Ị▼ ❈Õ❆ ❚❘❸◆● ❚❍⑩■ ❍❆■ ▼❖❉❊ ❑➌❚ ❍ĐP P❍❷◆ ✣➮■ ❳Ù◆● ❚❍➊▼ ❇❆
❱⑨ ❇❰❚ ▼❐❚ P❍❖❚❖◆ ❚✃◆●
✸✳✶✳ ❙ü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③
❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✕ r trữớ ủ ố ợ trữớ ❝ê ✤✐➸♥ ❝â
❞↕♥❣
a
ˆ†2 a
ˆ2
I=
ˆb†2ˆb2
1
2
− 1 ≥ 0.
a
ˆ†ˆb†ˆbˆ
a
✭✶✵✮
❙ü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ ♥❣❤➽❛ ❧➔ I < 0✱ ❝❤ù♥❣ tä tr↕♥❣ t❤→✐ ❦❤↔♦ s→t ❧➔ ♣❤✐ ❝ê
✤✐➸♥✳ ✣è✐ ✈ỵ✐ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ợt ởt t tờ
t t ữủ t q ừ t❤❛♠ sè I ♥❤÷ s❛✉✿
I ={{ra10 + rb10 + 21(ra8 + rb8 ) + 138(ra6 + rb6 ) + 330(ra4 + rb4 ) + 250(ra2 + rb2 ) + ra4 rb2 + ra2 rb4
+ (2ra7 rb + 12ra5 rb + 12ra3 rb ) cos(φ0,3,1 ) + (2ra rb7 + 12ra rb5 + 12ra rb3 ) cos(φ0,1,3 ) + 72
− x(72 + 2ra5 rb5 cos(φ5,0,0 ) + (504ra rb + 2ra3 rb3 ) cos(φ1,0,0 ) + 42ra4 rb4 cos(φ4,0,0 ) + 2ra3 rb5
cos(φ0,1,−5 ) + 2ra5 rb3 cos(φ0,5,−1 ) + 276ra3 rb3 cos(φ3,0,0 ) + 660ra2 rb2 cos(φ2,0,0 ) + 12ra3 rb
cos(φ0,3,1 ) + 12ra rb3 cos(φ0,1,3 ) + 12ra4 rb2 cos(φ0,4,0 ) + 12ra2 rb4 cos(φ0,0,4 ))} × {6(ra4 + rb4 )
+ 2ra3 rb5 cos(φ0,3,1 ) + 2ra5 rb3 cos(φ0,1,3 ) + ra6 rb4 + ra4 rb6 + 18ra4 rb4 + 18(ra4 rb2 + ra2 rb4 ) + (ra6
+ rb6 ) − x(2ra3 rb3 cos(φ3,0,0 ) + 2ra5 rb3 cos(φ0,1,3 ) + 2ra3 rb5 cos(φ0,3,1 ) + (2ra5 rb5 + 36ra3 rb3 )
1
cos(φ1,0,0 ) + 12ra2 rb2 cos(φ2,0,0 )) + 18ra4 rb4 }} 2 /{ra8 rb2 + ra2 rb8 + 15(ra6 rb2 + ra2 rb6 ) + 18(ra2
+ rb2 ) + 64(ra4 rb2 + ra2 rb4 ) + 156ra2 rb2 + (2ra3 rb5 + 6ra3 rb3 ) cos(φ0,1,3 ) + (2ra5 rb3 + 6ra3 rb3 )
cos(φ0,3,1 ) − x((6ra2 rb4 + 6ra4 rb2 ) cos(φ0,2,2 ) + 2ra5 rb5 cos(φ3,0,0 ) + 30ra4 rb4 cos(φ2,0,0 )
+ (128ra3 rb3 + 36ra rb ) cos(φ1,0,0 ) + 156ra2 rb2 + 2ra3 rb5 cos(φ0,1,3 ) + 2ra5 rb3 cos(φ0,3,1 ))} − 1.
✭✶✶✮
❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ð ✭✶✵✮✱ ♥➳✉ t❤❛♠ sè I tr♦♥❣ ❜✐➸✉ t❤ù❝ ✭✶✶✮ ♥❤➟♥ ❣✐→ trà ➙♠ t❤➻ tr↕♥❣ t❤→✐
❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tờ t tự
r ỗ t ✸ ❦❤↔♦ s→t t❤❛♠ sè I t❤❡♦ ❜✐➯♥ ✤ë ❦➳t ❤đ♣ rb ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❦❤↔♦
s→t ❧➔ ra = rb , 0 ≤ rb ≤ 1, ϕ = ϕa = , b = 0 ỗ t t t❤❛♠ sè I ♥❤➟♥
❣✐→ trà tr♦♥❣ ❦❤♦↔♥❣ −1 ≤ I ≤ 0✳ ❑❤✐ t❤❛♠ sè ra , rb ❝➔♥❣ ❣✐↔♠ t❤➻ t❤❛♠ sè I ❝➔♥❣ ➙♠✳
❈→❝ ❦➳t q✉↔ ✤â ❝❤♦ t❤➜② tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët
♣❤♦t♦♥ tê♥❣ ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✕ ❙❝❤✇❛r③ ✈➔ ❦❤✐ ra , rb ❝➔♥❣ ❣✐↔♠ t❤➻ I ❝➔♥❣
➙♠ ✈➔ ❞➛♥ t✐➳♥ ✈➲ ❣✐→ trà ♥❤ä ♥❤➜t ❧➔ −1✱ ♥❣❤➽❛ ❧➔ sü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✕
❙❝❤✇❛r③ t❤➸ ❤✐➺♥ ❝➔♥❣ ♠↕♥❤✳
✹✻
❍➬ ◆●➴❈ ❚❘❯◆●✱ ❚❘×❒◆● ▼■◆❍ ✣Ù❈
❍➻♥❤ ✸✿ ❑❤↔♦ s→t sü ♣❤ö t❤✉ë❝ ❝õ❛ t❤❛♠ sè ■ ✈➔♦ ❜✐➯♥ ✤ë ❦➳t ❤ñ♣ rb ❝õ❛ tr↕♥❣ t❤→✐
❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣✳
✸✳✷✳ ❚➼♥❤ ♣❤↔♥ ❦➳t ❝❤ị♠
◆➠♠ ✶✾✾✵✱ ❈❤✐♥❣ ❚s✉♥❣ ▲❡❡ ❬✶✵❪ ✤➣ ✤÷❛ r❛ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ò♠ ✈➔ t❤❛♠ sè Ra,b (l, p)
✤➦❝ tr÷♥❣ ❝❤♦ t➼♥❤ ❝❤➜t ✤â✳ ❚❤❡♦ ỉ♥❣✱ ♠ët tr↕♥❣ t❤→✐ ❝â t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❦❤✐ t❤❛♠
sè Ra,b (l, p) t❤ä❛ ♠➣♥
(l+1) (p−1)
n
ˆb
(p−1) (l+1)
n
ˆb
+ n
ˆa
n
ˆa
Rab (l, p) =
(l) (p)
(p) (l)
n
ˆa n
ˆb
✭✶✷✮
− 1 < 0,
+ n
ˆa n
ˆb
✈ỵ✐ l, p ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✭l ≥ p > 0✮✱ n
ˆa = a
ˆ† a
ˆ, n
ˆ b = ˆb†ˆb ❧➔ t♦→♥ tû sè ❤↕t ❝õ❛ ❤❛✐
♠♦❞❡ a✱ b✱ ❦❤✐ ✤â t❤❛♠ sè Ra,b (l, p) ð ✭✶✷✮ ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉✿
a
ˆ†(l+1) a
ˆ(l+1)ˆb†(p−1)ˆb(p−1) + a
ˆ†(p−1) a
ˆ(p−1)ˆb†(l+1)ˆb(l+1)
Rab (l, p) =
− 1.
ˆ†p a
ˆpˆb†lˆbl
a
ˆ†l a
ˆlˆb†pˆbp + a
✭✶✸✮
◆➳✉ t❤❛♠ sè R(l, p) ❝➔♥❣ ➙♠ t❤➻ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ị♠ ❤❛✐ ♠♦❞❡ t❤➸ ❤✐➺♥ ❝➔♥❣ ♠↕♥❤✳ ❇➙②
❣✐í t❛ ❦❤↔♦ s→t ❝ư t❤➸ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ị♠ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐
①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ t❤ỉ♥❣ q✉❛ t➼♥❤ sè tờ qt a
l a
lbpbp ữ
ỵ r số ❤↕♥❣ ❝á♥ ❧↕✐ ❝â tr♦♥❣ ❜✐➸✉ t❤ù❝ ✭✶✸✮ ✤➲✉ ✤÷đ❝ s✉② r❛ tø sè ❤↕♥❣ tê♥❣ q✉→t
♥➔②✳ ❚❤ü❝ ❤✐➺♥ ♠ët sè ♣❤➨♣ ❜✐➳♥ ✤ê✐ t❛ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ♥❤÷ s❛✉✿
2(l+3)
2
a
ˆ†l a
ˆlˆb†pˆbp = Nαβ
(ra2(l+3) rb2p + ra2p rb
2(l+1)
+ 18)(ra2(l+1) rb2p + ra2p rb
2(l−1)
(ra2(l−1) rb2p + ra2p rb
2(l−2)
rb
2(l+2)
) + (6l + 9)(ra2(l+2) rb2p + ra2p rb
) + (15l2 + 30l
) + (20l3 + 30l2 + 22l + 6))(ra2l rb2p + ra2p rb2l ) + (15l4 + 3l2 )
) + ra2l rb2p+2 + rb2l ra2p+2 + (6l5 − 15l4 + 12l3 − 3l2 )(ra2(l−2) rb2p + ra2p
2(l−3)
) + l2 (l − 1)2 (l − 2)2 (ra2(l−3) rb2p + ra2p rb
) + (2ra2l+3 rb2p+1 + 6lra2l+1 rb2p+1 + 6l(l − 1)
ra2l−1 rb2p+1 + (2l3 − 6l2 + 4l)ra2l−3 rb2p+1 ) cos(φ0,3,1 ) + (2rb2l+3 ra2p+1 + 6lrb2l+1 ra2p+1 + 6l(l − 1)
◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳
✹✼
rb2l−1 ra2p+1 + (2l3 − 6l2 + 4l)rb2l−3 ra2p+1 ) cos(φ0,1,3 ) − x((12l + 18)ral+p+2 rbl+p+2 + 2ral+p+3 rbl+p+3
cos((n + 3)φ0,1,−1 ) + (30l2 + 60l + 36)ral+p+1 rbl+p+1 cos((n + 1)φ0,1,−1 ) + (40l3 + 60l2 + 44l
+ 12)ral+p rbl+p cos(nφ0,1,−1 ) + (12l + 18)ral+p+2 rbl+p+2 ) cos((n + 2)φ0,1,−1 ) + (40l3 + 60l2 + 44l
+ 12)ral+p rbl+p + (30l2 + 60l + 36)ral+p+1 rbl+p+1 + 6lral+p+2 rbl+p cos(φ0,n+2,−(n−2) ) + (2l3 − 6l2
+ 4l)ral+p rbl+p−2 cos(φ0,−(n−4),n ) + 6lral+p rbl+p+2 cos(φ0,n−2,−(n+2) ) + 2l2 (l − 1)2 (l − 2)2 ral+p−3
rbl+p−3 cos((n − 3)φ0,1,−1 ) + 2ral+p+1 rbl+p+1 cos((n − 1)φ0,1,−1 ) + 2ral+p+3 rbl+p+1 cos(φ0,n+3,−(n−1) )
+ 2ral+p+1 rbl+p+3 cos(φ0,n−1,−(n+3) ) + (12l5 − 30l4 + 24l3 − 6l2 )ral+p−2 rbl+p−2 cos((n − 2)φ0,1,−1 )
+ (6l2 − 6l)ral+p−1 rbl+p+1 cos(φ0,n−3,−(n−1) ) + (30l4 + 6l2 )ral+p−1 rbl+p−1 cos((n − 1)φ0,1,−1 )
+ 6l(l − 1)ral+p+1 rbl+p−1 cos(φ0,n−1,−(n+3) ) + (2l3 − 6l2 + 4l)ral+p rbl+p−2 cos(φ0,n,−(n−4) ),
✭✶✹✮
✈ỵ✐ n = l − p✳ ❇➙② ❣✐í t❛ s➩ ❦❤↔♦ s→t ❝→❝ tr÷í♥❣ ❤đ♣ ❝ư t❤➸ ♥❤÷ s❛✉✿
❛✮ ❚r÷í♥❣ ❤ñ♣ (l = 4, p = 3)❀ (l = 5, p = 3)❀ (l = 6, p = 3)✱ ✈ỵ✐ ❝ò♥❣ ✤✐➲✉ ❦✐➺♥ ❦❤↔♦ s→t ❧➔
π
rb = ra3 ✱ ϕ = ϕa = 0 ✈➔ ϕb = ✳
4
❍➻♥❤ ✹✿ ❑❤↔♦ s→t sü ♣❤ư t❤✉ë❝ ❝õ❛ R(4, 3) ✧✤÷í♥❣ ❧✐➲♥ ♥➨t✧ ✱ R(5, 3) ✧✤÷í♥❣ ❝❤➜♠
❝❤➜♠✧✱ R(6, 3) ✧✤÷í♥❣ ❝❤➜♠ ❣↕❝❤✧ ✈➔♦ ❜✐➯♥ ✤ë rb ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣
♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣✳
❜✮ ❚÷ì♥❣ tü✱ t❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ (l = 4, p = 2)❀ (l = 5, p = 2)❀ (l = 6, p = 2)✱ ✈ỵ✐ ❝ị♥❣
π
✤✐➲✉ ❦✐➺♥ ❦❤↔♦ s→t ❧➔ rb = ra3 ✱ ϕ = ϕa = 0 ✈➔ ϕb = ✳
4
✹✽
❍➬ ◆●➴❈ ❚❘❯◆●✱ ❚❘×❒◆● ▼■◆❍ ✣Ù❈
❍➻♥❤ ✺✿ ❑❤↔♦ s→t sü ♣❤ư t❤✉ë❝ ❝õ❛ R(4, 2) ✧✤÷í♥❣ ❧✐➲♥ ♥➨t✧✱ R(5, 2) ✧✤÷í♥❣ ❝❤➜♠
❝❤➜♠✧✱ R(6, 2) ✤÷í♥❣ ✧✤÷í♥❣ ❝❤➜♠ ❣↕❝❤✧ ✈➔♦ ❜✐➯♥ ✤ë rb ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t
❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣✳
❝✮ ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ ❦❤↔♦ s→t ❧➔ ra = rb ✱ ϕ = ϕa = 3ϕb ✈➔ ϕb =
✭l = 3, p = 2✮✱ ✭l = 5, p = 4✮ ✈➔ ✭l = 6, p = 5✮✳
π
✱ t❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣
3
❍➻♥❤ ✻✿ ❑❤↔♦ s→t sü ♣❤ư t❤✉ë❝ ❝õ❛ R(3, 2) ✧✤÷í♥❣ ❧✐➲♥ ♥➨t✧✱ R(5, 4) ✧✤÷í♥❣ ❝❤➜♠
❝❤➜♠✧✱ R(6, 5) ✤÷í♥❣ ✧✤÷í♥❣ ❝❤➜♠ ❣↕❝❤✧ ✈➔♦ ❜✐➯♥ ✤ë rb ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t
❤đ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tờ
ỗ t t ♥❤➟♥ t❤➜② t❤❛♠ sè R{l, p} ✤➲✉ ①✉➜t ♣❤→t tø ❣✐→ trà −1✱ ✤➙②
❧➔ ❣✐→ trà ❝ü❝ t✐➸✉ ❝õ❛ ♥â✱ t❤❛♠ sè R{l, p} < 0 ❦❤✐ ❝→❝ t❤❛♠ sè ra , rb r➜t ❜➨✱ ✈➔ ❦❤✐ ❝→❝ t❤❛♠
sè ra , rb t➠♥❣ t❤➻ t❤❛♠ sè R{l, p} ❞➛♥ t✐➳♥ ✤➳♥ 0 ✈➔ ♥➳✉ t❤❛♠ sè ra , rb ❧ỵ♥ ❤ì♥ ♠ët ❣✐→ trà
♥➔♦ ✤â t❤➻ R{l, p} ≥ 0✳ ❈→❝ ❦➳t q✉↔ ♥➔② ❝❤ù♥❣ tä tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ♣❤↔♥ ✤è✐
①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ ❝â t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ò♠ ❦❤✐ ❝→❝ t❤❛♠ sè ra , rb
r➜t ❜➨✱ ♥❣❤➽❛ ❧➔ tr↕♥❣ t❤→✐ ✤â ♠❛♥❣ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ra , rb t➠♥❣
❞➛♥ t❤➻ t❤❛♠ sè R{l, p} ❝ô♥❣ t➠♥❣ ❞➛♥ ✤➳♥ 0✱ ❦❤✐ ✤â t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❝õ❛ tr↕♥❣ t❤→✐
◆●❍■➊◆ ❈Ù❯ ❈⑩❈ ❚➑◆❍ ❈❍❻❚ P❍■ ❈✃ ✣■➎◆ ❈Õ❆ ❚❘❸◆● ❚❍⑩■✳✳✳
✹✾
❞➛♥ ❜✐➳♥ ♠➜t✳ ◆➳✉ t✐➳♣ tö❝ t➠♥❣ ra , rb ữủt q ởt tr ợ õ t t❤❛♠ sè
R{l, p} ≥ 0✱ ♥❣❤➽❛ ❧➔ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣
t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ ❦❤ỉ♥❣ ❝á♥✱ t❤❛② ✈➔♦ ✤â tr↕♥❣ t❤→✐ ♠❛♥❣ t➼♥❤ ❝❤➜t ❝ê ✤✐➸♥✳
✹
❑➌❚ ▲❯❾◆
❇➔✐ ❜→♦ ♥➔② tr➻♥❤ ❜➔② ❦➳t q✉↔ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♥➨♥ tê♥❣ ❤❛✐ ♠♦❞❡✱ ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡✱
sü ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③ ✈➔ t➼♥❤ ♣❤↔♥ ❦➳t ❝❤ị♠ ❜➟❝ ❝❛♦ ❝õ❛ tr↕♥❣ t❤→✐
❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣✳ q tr
st t t ỗ t t❤ỉ♥❣ q✉❛ ❝→❝ t❤❛♠ sè✱ ❝❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ ❦➳t q✉↔ ♥❤÷ s❛✉✿
❚❤ù ♥❤➜t✱ ❦➳t q✉↔ ❦❤↔♦ s→t ❝❤♦ t❤➜② tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛
✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t ♥➨♥ tê♥❣ ❤❛✐ ♠♦❞❡ ♠↕♥❤ ✈➔ ❤♦➔♥ ❤♦➔♥ ❦❤æ♥❣
❝â t➼♥❤ ❝❤➜t ♥➨♥ ❤✐➺✉ ❤❛✐ ♠♦❞❡✳
❚❤ù ❤❛✐✱ t❤æ♥❣ q✉❛ t❤❛♠ sè I ❞ü❛ ✈➔♦ ✤✐➲✉ ❦✐➺♥ ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③
✤➣ ✤÷đ❝ ✤÷❛ r❛ ✈➔ ❦❤↔♦ s→t ❝❤✐ t✐➳t ✈➔ ❝ö t❤➸✳ ❑➳t q✉↔ ❝❤♦ t❤➜② tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t
❤đ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ ❤♦➔♥ t♦➔♥ ✈✐ ♣❤↕♠ ❜➜t ✤➥♥❣ t❤ù❝
❈❛✉❝❤② ✲ ❙❝❤✇❛r③ ♥❣❤➽❛ ❧➔ t➼♥❤ ❝❤➜t ♣❤✐ ❝ê ✤✐➸♥ t❤➸ ❤✐➺♥ t÷ì♥❣ ✤è✐ ♠↕♥❤✳ ◆❣♦➔✐ r❛✱ ❦❤✐
❦❤↔♦ s→t t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ị♠✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤➣ ✤÷❛ r❛ ❜✐➸✉ t❤ù❝ tê♥❣ q✉→t ❝❤♦ t❤❛♠
sè ♣❤↔♥ ❦➳t ❝❤ò♠ tr♦♥❣ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët
♣❤♦t♦♥ tê♥❣✳ ◗✉❛ ✤â✱ ❝❤ó♥❣ tỉ✐ ✤➣ ❦❤↔♦ s→t ❝❤♦ tø♥❣ tr÷í♥❣ ❤đ♣ ❝ư t❤➸ ✈➔ ❦➳t q✉↔ ❧➔ tr↕♥❣
t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ♣❤↔♥ ✤è✐ ①ù♥❣ t❤➯♠ ❜❛ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ tê♥❣ t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t
♣❤↔♥ ❦➳t ❝❤ò♠ ❜➟❝ t❤➜♣ ✈➔ ❜➟❝ ❝❛♦✱ tr♦♥❣ ✤â sè ❜➟❝ ❝➔♥❣ t➠♥❣ ♥❤÷♥❣ ❤✐➺✉ ❝→❝ ❜➟❝ ❝➔♥❣
❜➨ t❤➻ t➼♥❤ ❝❤➜t ♣❤↔♥ ❦➳t ❝❤ị♠ ❝➔♥❣ t❤➸ ❤✐➺♥ rã r➺t ❤ì♥✳
▲❮■ ❈❷▼ ❒◆
◆❣❤✐➯♥ ❝ù✉ ♥➔② ✤÷đ❝ t➔✐ trđ ❜ð✐ ◗✉ÿ P❤→t tr✐➸♥ ❦❤♦❛ ❤å❝ ✈➔ ❝æ♥❣ ♥❣❤➺ ◗✉è❝ ❣✐❛ ✭◆❆❋❖❙✲
❚❊❉✮ tr♦♥❣ ✤➲ t➔✐ ♠➣ sè ✶✵✸✳✵✶✲✷✵✶✽✳✸✻✶✳
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖
❬✶❪ ▲✳ ▲✳ ❍♦✉✱ ❳✳ ❋✳ ❈❤❡♥✱ ❳✳ ❋✳ ❳✉ ✭✷✵✶✺✮✱ ✧❈♦♥t✐♥✉♦✉s✲✈❛r✐❛❜❧❡ q✉❛♥t✉♠ t❡❧❡♣♦rt❛t✐♦♥
✇✐t❤ ♥♦♥✲●❛✉ss✐❛♥ ❡♥t❛♥❣❧❡❞ st❛t❡s ❣❡♥❡r❛t❡❞ ✈✐❛ ♠✉❧t✐♣❧❡✲♣❤♦t♦♥ s✉❜tr❛❝t✐♦♥ ❛♥❞
❛❞❞✐t✐♦♥✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✾✶✱ ✵✻✸✽✸✷✳
❬✷❪ ❳✳ ❲✳ ❳✉✱ ❍✳ ❲❛♥❣✱ ❏✳ ❩❤❛♥❣✱ ❨✳ ▲✐✉ ✭✷✵✶✸✮✱ ✧❊♥❣✐♥❡❡r✐♥❣ ♦❢ ♥♦♥❝❧❛ss✐❝❛❧ ♠♦t✐♦♥❛❧
st❛t❡s ✐♥ ♦♣t♦♠❡❝❤❛♥✐❝❛❧ s②st❡♠s✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✽✽✱ ✵✻✸✽✶✾✳
❬✸❪ ●✳ ❙✳ ❆❣❛r✇❛❧ ❛♥❞ ❑✳ ❚❛r❛ ✭✶✾✾✶✮✱ ✧◆♦♥❝❧❛ss✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ st❛t❡s ❣❡♥❡r❛t❡❞ ❜② t❤❡
❡①❝✐t❛t✐♦♥s ♦♥ ❛ ❝♦❤❡r❡♥t st❛t❡✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✹✸✱ ✹✾✷✳
✺✵
❍➬ ◆●➴❈ ❚❘❯◆●✱ ❚❘×❒◆● ▼■◆❍ ✣Ù❈
❬✹❪ ❙✳ ❙✐✈❛❦✉♠❛r ✭✶✾✾✾✮✱ ✧P❤♦t♦♥✲❛❞❞❡❞ ❝♦❤❡r❡♥t st❛t❡s ❛s ♥♦♥❧✐♥❡❛r ❝♦❤❡r❡♥t st❛t❡s✧✱
P❤②s✳ ❆✿ ▼❛t❤✳ ●❡♥✱ ✸✷✱ ✸✹✹✶✳
❏✳
❬✺❪ ❚✳ ▼✳ ❉✉❝ ❛♥❞ ❏✳ ◆♦❤ ✭✷✵✵✽✮✱ ✧❍✐❣❤❡r✲♦r❞❡r ♣r♦♣❡rt✐❡s ♦❢ ♣❤♦t♦♥✲❛❞❞❡❞ ❝♦❤❡r❡♥t
st❛t❡s✧✱ ❖♣t✳ ❈♦♠♠✉♥✳ ✷✽✶✱ ✷✽✹✷✳
❬✻❪ ❈✳ ◆✳ ❇❡♥❧❧♦❝❤✱ ❘✳ ●✳ P❛tr♦♥✱ ❏✳ ❍✳ ❙❤❛♣✐r♦✱ ◆✳ ❏✳ ❈❡r❢ ✭✷✵✶✷✮✱ ✧❊♥❤❛♥❝✐♥❣ q✉❛♥t✉♠
❡♥t❛♥❣❧❡♠❡♥t ❜② ♣❤♦t♦♥ ❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥✧✱ P❤②s✳ ❘❡✈✳ ❆ ✽✻✱ ✵✶✷✸✷✽✳
❬✼❪ ◆❣✉②➵♥ ❱ơ ❚❤ư② ✭✷✵✶✼✮✱ ▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ t ỵ ỵ tt t ỵ
rữớ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✣↕✐ ❤å❝ ❍✉➳✳
❬✽❪ ◆❣✉②➵♥ ❚❤à ❚❤❛♥❤ ữỡ s t ỵ ỵ tt
t ỵ rữớ ồ ữ ✣↕✐ ❤å❝ ❍✉➳✳
❬✾❪ ▼✳ ❍✐❧❧❡r② ✭✶✾✽✾✮✱ ✧❙✉♠ ❛♥❞ ❞✐❢❢❡r❡♥❝❡ sq✉❡❡③✐♥❣ ♦❢ t❤❡ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ❢✐❡❧❞✧✱
✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✹✵✱ ✸✶✹✼✳
P❤②s✲
❬✶✵❪ ❈✳ ❚✳ ▲❡❡ ✭✶✾✾✵✮✱ ✧▼❛♥②✲♣❤♦t♦♥ ❛♥t✐❜✉♥❝❤✐♥❣ ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♣❛✐r ❝♦❤❡r❡♥t st❛t❡s✧✱
P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✹✶✱ ✶✺✻✾✳
❚✐t❧❡✿
◆❖◆❈▲❆❙❙■❈❆▲ P❘❖P❊❘❚■❊❙ ❖❋ ❚❍❊ ❚❍❘❊❊✲P❍❖❚❖◆✲❆❉❉❊❉ ❆◆❉
❖◆❊✲P❍❖❚❖◆✲❙❯❇❚❘❆❈❚❊❉ ❚❲❖✲▼❖❉❊ ❖❉❉ ❈❖❍❊❘❊◆❚ ❙❚❆❚❊
❆❜str❛❝t✿ ■♥ t❤❡ ♣❛♣❡r✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❧♦✇❡r✲♦r❞❡r ❛♥❞ ❤✐❣❤❡r✲♦r❞❡r ♥♦♥❝❧❛ss✐❝❛❧ ♣r♦♣❡r✲
t✐❡s ♦❢ t❤❡ t❤r❡❡✲♣❤♦t♦♥✲❛❞❞❡❞ ❛♥❞ ♦♥❡✲♣❤♦t♦♥✲s✉❜tr❛❝t❡❞ t✇♦✲♠♦❞❡ ♦❞❞ ❝♦❤❡r❡♥t st❛t❡ ❛s
t✇♦✲♠♦❞❡ s✉♠ sq✉❡❡③✐♥❣✱ t✇♦✲♠♦❞❡ ❞✐❢❢❡r❡♥❝❡ sq✉❡❡③✐♥❣✱ ✈✐♦❧❛t✐♦♥ ♦❢ t❤❡ ❈❛✉❝❤②✲❙❝❤✇❛r③
✐♥❡q✉❛❧✐t②✱ ❛♥❞ t✇♦✲♠♦❞❡ ❤✐❣❤❡r✲♦r❞❡r ❛♥t✐❜✉♥❝❤✐♥❣✳ ❚❤❡ r❡s✉❧ts s❤♦✇ t❤❛t t❤✐s st❛t❡ ❡①✲
❤✐❜✐ts t✇♦✲♠♦❞❡ s✉♠ sq✉❡❡③✐♥❣ ❜✉t ❞♦❡s ♥♦t ❡①❤✐❜✐t t✇♦✲♠♦❞❡ ❞✐❢❢❡r❡♥❝❡ sq✉❡❡③✐♥❣✳ ■♥
❛❞❞✐t✐♦♥✱ t❤✐s st❛t❡ ✈✐♦❧❛t❡s t❤❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛❧✐t② ❛♥❞ ❜❡❝♦♠❡s ♥♦♥❝❧❛ss✐❝❛❧ st❛t❡✳
❲❡ ❛❧s♦ s❤♦✇ t❤❛t t❤❡ t❤r❡❡✲♣❤♦t♦♥✲❛❞❞❡❞ ❛♥❞ ♦♥❡✲♣❤♦t♦♥✲s✉❜tr❛❝t❡❞ t✇♦✲♠♦❞❡ ♦❞❞ ❝♦✲
❤❡r❡♥t st❛t❡ ❛♣♣❡❛rs t✇♦✲♠♦❞❡ ❤✐❣❤❡r✲♦r❞❡r ❛♥t✐❜✉♥❝❤✐♥❣ ✐♥ ❛♥② ♦r❞❡r✱ ❛♥❞ t❤❡ ❞❡❣r❡❡ ♦❢
❛♥t✐❜✉♥❝❤✐♥❣ ❜❡❝♦♠❡s ♠♦r❡ ❛♥❞ ♠♦r❡ ♣r♦♥♦✉♥❝❡❞ ✇❤❡♥ ✐♥❝r❡❛s✐♥❣ t❤❡ ❤✐❣❤❡r✲♦r❞❡r✳
❑❡②✇♦r❞s✿
❙✉♠ sq✉❡❡③✐♥❣✱ ❞✐❢❢❡r❡♥❝❡ sq✉❡❡③✐♥❣✱ ✈✐♦❧❛t✐♦♥ ❈❛✉❝❤②✲❙❝❤✇❛r③ ✐♥❡q✉❛❧✐t②✱
t✇♦✲♠♦❞❡ ❛♥t✐❜✉♥❝❤✐♥❣✳