ΚΩΝΣΤΑΝΤΙΝΟΣ ΚΑΡΑΚΑΤΣΑΝΗΣ

ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ ΚΩΝΣΤΑΝΤΙΝΟΣ ΚΑΡΑΚΑΤΣΑΝΗΣ Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης Τμήμα Φυσικής Τομέας Πυρηνικής Φυσικής και Στοιχειωδών Σωματιδίων Θ 2017

ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ Υποβλήθηκε στο Τμήμα Φυσικής, Τομέας Πυρηνικής Φυσικής και Φυσικής Στοιχειωδών Σωματιδίων, Σ.Θ.Ε., Α.Π.Θ. Στο πλαίσιο του προγράμματος ΥΠΟΤΡΟΦΙΕΣ ΑΡΙΣΤΕΙΑΣ Ι.Κ.Υ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΕΛΛΑΔΑ - ΠΡΟΓΡΑΜΜΑ SIEMENS. ΕΞΕΤΑΣΤΙΚΗ ΕΠΙΤΡΟΠΗ Γεώργιος Λαλαζήσης (Επιβλέπων) Καθηγητής Τμήματος Φυσικής Σ.Θ.Ε./Α.Π.Θ Χαράλαμπος Μουστακίδης (Μέλος τριμελούς συμβουλευτικής επιτροπής) Επίκουρος καθηγητής Τμήματος Φυσικής Σ.Θ.Ε./Α.Π.Θ Θεόδορως Γαϊτάνος (Μέλος τριμελούς συμβουλευτικής επιτροπής) Επίκουρος καθηγητής Τμήματος Φυσικής Σ.Θ.Ε./Α.Π.Θ Νικόλαος Βλάχος Καθηγητής Τμήματος Φυσικής Σ.Θ.Ε./Α.Π.Θ Αναστάσιος Πέτκου Αναπληρωτής Καθηγητής Τμήματος Φυσικής Σ.Θ.Ε./Α.Π.Θ Διονύσης Μπονάτσος Διευθυντής έρευνας, Ινστιτούτο Πυρηνικής και Σωματιδιακής Φυσικής του ΕΚΕΦΕ «Δημόκριτος» Γεώργιος Σουλιώτης Αναπληρωτής Καθηγητής Τμήματος Χημείας Ε.Κ.Π.Α.

= 20

= 20

N = 20

A 160 190 6 + N = 104 8 N = 106 A 250

v ext ( ) v [ρ] ρ(r) v [ρ] = F HK [ρ] + d 3 v ext ( )ρ(r) F HK

v ext ( ) ρ gs (r) v [ρ] F HK [ρ] F HK [ρ] v [ρ] ρ(r) v ext ( ) ρ(r) v KS ρ gs ( ) N ρ gs ( ) = ρ(r) := ϕ i ( ) 2, [ ] 2 2m + v KS ϕ i ( ) = ϵ i ϕ i ( ). F HK [ρ] i F HK [ρ] = T s [ρ] + E H [ρ] + E xc [ρ],

T s [ρ] E H [ρ] E xc [ρ] T s [ρ] + E H [ρ] v KS ( ) v KS ( ) = δ(e H + E xc ), δρ V F (, ) E xc [ρ] ϕ i ( ) ĥ = [ 2 2m + v KS ] = δe[ˆρ] δ ˆρ,

ˆρ(, ) = N ϕ( )ϕ i ( ). i ˆρ 2 = ˆρ. ρ gs ( ) = ˆρ(, )) ρ gs ( ) v KS ( ) ˆρ ρ gs ( ) E[ˆρ] E[ˆρ] = F HK [ρ gs ] F HK [ρ gs ] ˆρ j µ ( ) = (ρ( ), ( )) ρ( )

V 12 (ρ) E[ˆρ] E[ˆρ] = Φ T + V 12 (ρ) Φ, Φ ˆρ = Φ a a Φ V 12 (ρ) ρ

ρ( ) = s ˆρ( s, ) 2 2 2 2 S = 0 S = 1 ρ( ) = ρ 0 ( ) + σ ρ 1 ( ). E[ρ 0, ρ 1 ] ρ( ) ρ 1 l j > = l + 1/2 j > = l 1/2

j > j > π = 0 1 ν = 1 0 τ 3 ν = 1 2 ν τ 3 π = 1 2 π τ 3 = 1 2 1 0 0 1.

τ ν = π, τ + ν = 0, τ π = 0, τ + π = ν, τ + = 0 1 0 0, τ = 0 0 1 0. τ = {τ 1, τ 2, τ 3 }. A A T = τ (i) i=1 T 3 = A τ (3), i=1 T 3 = 1 (N Z). 2 T 3 = 0 ρ = ρ + = ρ n + ρ p ρ = ρ + = ρ n ρ p

j α k = n U nk c k + V nkc n. (c, c ) ˆρ ˆk ρ n,n = Φ c n c n Φ, k n,n = Φ c n c n Φ, Φ R = ˆρ ˆk ˆk 1 ˆρ R E[R] = E MF [ˆρ] + E pair [ˆκ].

ĥ ˆ ˆ U = E k U. ĥ V k V k ĥ D = δe δ ˆρ, k ˆ = δe δˆκ, k U( ) V ( ) Ĥ(ˆρ) ĥ ˆ c2 p 2 + (Mc 2 ) 2 Mc 2 + p2 2M.

S 400 S +350 V + S 50 V S 750 100 1 2 π ± π 0

Z(4430) P + c (4380) P + c (4450)

ω σ E/A

ρ

ρ α α

E[ρ] ρ 10 5 v ext ( )

ρ I ( ) = ρ I ( + ) i = 1 A 1/A ρ(r) E[ρ] ˆF = d 3 rf( )ρ( ),

ĥ E[ρ]

σ L = ψ [ iγ µ µ M g σ σ g ω γ µ ω µ g ρ γ µ τ ρ µ eγ µ A µ ] ψ + 1 2 µ σ µ σ U σ (σ) 1 4 Ωµν Ω µν + U ω (ω µ ) 1 4 R µν R µν + U ρ ( ρ µ ) 1 4 F µν F µν,

M m m (m = σ, ω, ρ) g m e Ω µν = µ ω ν ν ω µ, R µν = µ ρ ν ν ρ µ, F µν = µ A ν ν A µ µ ν (x µ ) = (t, x, y, z). A µ (x) U σ (σ) U ω (ω µ ) U ρ (ˆρ µ )

U σ (σ) = 1 2 m2 σσ 2 + 1 3 g 2σ 3 + 1 4 g 3σ 4, U ω (ω µ ) = 1 2 m2 ωω µ ω µ + 1 4 c 3 ( ω µ ω µ ) 2, U ρ ( ρ µ ) = 1 2 m2 ρ ρ µ ρ µ + 1 4 d 3 ( ρ µ ρ µ ) 2. [ ] H = d 3 r H = d 3 ϕ m r π m L(r). t ϕ m (ϕ m = ψ, σ, ω ν, ρ ν, A ν ) L π m = [ ϕ m / t ]. H = H ψ + H σ + H ω + H ρ + H A + H m H ψ = ψ [α p + βm] ψ, H σ = 1 2 σ σ + U σ(σ), H ω = 1 2 ω µ ω µ U ω (ω), H ρ = 1 2 ρ µ ρ µ U ρ (ρ), H A = 1 2 A µ A µ, H = g σ σ ψψ + g ω ω µ ψγ µ ψ + g ρ ρ µ ψγ µ τψ + ea µ ψγ µ ψ

k ψ(x) = ψ k (x)c k ψ (x) = ψk(x)c k, k k c k k ψ k c k c k {c k, c k } = δ kk {c k, c k } = {c k, c k } = 0. Φ = A i c i 0 Φ Φ = 1. Φ c k c k Φ Φ ϕc k c k Φ Φ ϕ Φ ϕ σ ω µ ρ µ A µ ϕ Φ ϕ Φ = ϕ Φ ϕc k c k Φ = ϕ Φ c k c k Φ, ˆρ kk = Φ c k c k Φ. Φ E RMF (ˆρ, ϕ) = Φ H Φ,

] Φ H Φ = [ĥ0 ˆρ + d 3 r [ + β ( g σ σ + g ω ω µ γ µ + g ρ τ ρ µ γ µ + ea µ γ µ) ] ˆρ { 1 2 σ σ + U σ(σ) + 1 2 ω µ ω µ U ω (ω) + 1 2 ρ µ ρ µ U ρ (ρ) + 1 2 A µ A µ }, ˆρ ĥ 0 = α p + M. ψ a (x) = ψ a (r)e iϵat. ˆρ(r, r ) = A ψ i (r) ψ i (r ). i τ 3 ρ µ 3 ρ µ ψ k ϕ δ ( E (ˆρ, ϕ) (Λˆρ) ) = 0, Λ ϵ k ψ k ϵ k [ α (p V ) + V 0 + β(m + S) ] ψ k (r) = ϵ k ψ k (r),

S V µ S(r) = g σ σ(r), V µ (r) = g ω ω µ (r) + g ρ τ 3 ρ µ 3(r) + ea µ (r). σ ω µ ρ µ 3 A µ σ + U σ(σ) = g σ ρ s, ω µ + U µ ω (ω) = g ω j µ, ρ µ 3 + U µ ρ (ρ 3 ) = g ρ j µ 3, A µ = ej c µ, U σ = U σ σ, U µ ω = U ω, U µ ω ρ = U ρ. µ ρ 3µ ρ s j µ = (ρ v, j) j µ 3 = (ρ 3, j 3 ) j µ c = (ρ c, j c ) ρ s (r) = Φ ψψ Φ = j µ (r) = Φ ψγ µ ψ Φ = j µ 3 (r) = Φ ψγ µ τ 3 ψ Φ = j c µ (r) = Φ ψγ µ 1 τ 3 ψ Φ = 2 A i=1 A i=1 A i=1 A i=1 ψ i (r)ψ i (r), ψ i (r)γ µ ψ i (r), ψ i (r)γ µ τ 3 ψ i (r), ψ i (r)γ µ 1 τ 3 ψ i (r). 2 E = E + E σ + E ω + E ρ + E + E, E kin E σ E ω E ρ

A E = d 3 r ψ i (r)(α p + βm)ψ i(r), i=1 E σ = 1 2 g σ d 3 r ρ s (r)σ(r), E ω = 1 2 g ω d 3 r j µ (r)ω µ (r), E ρ = 1 2 g ρ d 3 r j 3µ (r)ρ µ 3(r), E = 1 2 e d 3 r j cµ (r)a µ (r). { E = d 3 r U σ 1 2 σu σ U ω + 1 2 ω µu µ ω U ρ + 1 } 2 ρ 3µU µ ρ. g σ (ρ v ) g ω (ρ v ) g ρ (ρ v ) ρ v [ α (p V ) + V 0 (r) + Σ 0 R(r) + β ( M + S(r) )] ψ k (r) = ε k ψ k (r), Σ 0 R Σ 0 R = g σ ρ v ρ s σ + g ω ρ v ρ v ω 0 + g ρ ρ v ρ 3 ρ 0 3. E.. = 1 2MA ˆP 2.., A Pˆ c.m. 2 = A i ˆp i E = E + E...

( ψoγψ), O {1, τ}, Γ {1, γ µ, γ 5, γ 5 γ µ, σ µ,ν }, ψ τ Γ 4 4 ψoψ L = L free + L 4f + L hot + L der + L em

L free = ψ(iγ µ µ M)ψ, L 4f = 1 2 α Σ( ψψ)( ψψ) 1 2 α V ( ψγ µ ψ)( ψγ µ ψ) 1 2 α T S( ψ τψ)( ψ τψ) 1 2 α T V ( ψ τγ µ ψ)( ψ τγ µ ψ), L hot = 1 3 β S( ψψ) 3 1 4 γ S( ψψ) 4 1 4 γ V [( ψγ µ ψ)( ψγ µ ψ)] 2, L L = 1 2 δ S ν ( ψψ) ν ( ψψ) 1 2 δ V ν ( ψγ µ ψ) ν ( ψγ µ ψ) 1 2 δ T S ν ( ψ τψ) ν ( ψ τψ) 1 2 δ T V ν ( ψ τγ µ ψ) ν ( ψ τγ µ ψ) L = 1 4 F µν F µν ψea µ γ µ ψ. A µ F µν e α S α V α T S α T V β S γ S γ V δ S δ V δ T S δ T V α β γ δ S V T S V T S T V δ α T S δ T S

γ 5 γ 5 γ µ 00 H = L ϕ i ϕi L, ϕ i H = d 3 rh { = d 3 r ψ [α p + M] ψ + 1 2 α S( ψψ)( ψψ) + 1 2 α V ( ψγ µ ψ)( ψγ µ ψ) + 1 2 α T V ( ψ τγ µ ψ)( ψ τγ µ ψ) + 1 3 β S( ψψ) 3 + 1 4 γ S( ψψ) 4 + 1 4 γ V [( ψγ µ ψ)( ψγ µ ψ)] 2 + 1 2 δ S( ψψ) ( ψψ) + 1 2 δ V ( ψγ µ ψ) ( ψγ µ ψ) + 1 2 δ T V ( ψ τγ µ ψ) ( ψ τγ µ ψ) + ψeγ µ A µ ψ + 1 2 A µ A µ }. A E Φ H Φ = d 3 r ψ i (α p + βm)ψ i i=1 + 1 2 α Sρ 2 s + 1 3 β Sρ 3 s + 1 4 γ Sρ 4 s + 1 2 δ Sρ s ρ s + 1 2 α V j µ j µ + 1 4 γ V (j µ j µ ) 2 + 1 2 δ V j µ j µ + 1 2 α T V j 3µ j µ 3 + 1 2 δ T V j 3µ j µ 3 +ea µ j µ p + 1 2 A µ A µ }, ρ s (r) j µ (r) j µ 3 (r) j µ c (r)

ψ i [α (p V ) + V 0 + β(m + S)]ψ k = ε k ψ k. S(r) V µ (r) S(r) = Σ s, V µ (r) = Σ µ + τ 3 Σ µ 3, Σ s Σ µ Σ µ 3 Σ s = α S ρ s + β S ρ 2 s + γ S ρ 3 s + δ S ρ s, Σ µ = α V j µ + γ V (j µ j µ )j µ + δ V j µ + ea µ, Σ µ 3 = α T V j µ 3 + δ T V j µ 3. A µ j V A E = d 3 x ψ i (α p + βm) ψ i + 1 2 α Sρ 2 s + 1 2 α V ρ 2 v + 1 2 α T V ρ 2 3 i=1 + 1 3 β Sρ 3 s + 1 4 γ Sρ 4 s + 1 4 γ V ρ 4 v + 1 2 δ Sρ s ρ s + 1 2 δ V ρ v ρ v + 1 2 δ T V ρ 3 ρ 3 + 1 2 ea0 ρ c }. E = E + E... ˆρ

ˆρ 2 = ˆρ ρ(r) ph pp J = 0

ĥ ˆ (pp) Φ α + k = n U nk c + n + V nk c n, n r, s, t p = f, g U V ˆρ ˆκ ˆρ nn = Φ c n c n Φ, ˆκ nn = Φ c n c n Φ. E [ˆρ, ˆκ] = E [ˆρ] + E [ˆκ], E [ˆρ] E [ˆκ] = 1 ˆκ n 4 1 n n 1 1n 1 V pp n 2 n 2 ˆκ n2 n, 2 n 1 n 1 n 2 n 2 n 1 n 1 V pp n 2 n 2 U V ĥ D λ ˆ ˆ ĥ D + λ U k V k = E k U k V k. ĥd

λ ˆ ˆ n1 n = 1 n 1 2 1 n 1 V pp n 2 n 2 ˆκ n2 n. 2 n 2 n 2 E k (U k, V k ) E k > 0 (Vk, U k ) E k E k E k α k Φ = 0 E k > 0 Φ = α k. E k >0 Φ Φ ˆρ nn = E k >0 VnkV n k,, ˆκ nn = VnkU n k. E k >0 a p

1200 E a > 1200 ĥ M 2M E p > 0 E a < 0 Φ = α p α a. E p>0 E a<0 ˆρ nn = E p>0 ˆκ nn = E p>0 VnpV n p + VnaV n a, E a<0 VnpU n p + VnaU n a. E a<0 V na ˆκ ˆ ˆ n1 p 1,n 1 p 1 = 1 2 n 2 p 2,n 2 p 2 n 1 p 1, n 1p 1 V pp n 2 p 2, n 2p 2 ˆκ n2 p 2,n 2 p 2. p 1, p 2, p 3, p 4 f, g

U k = f (U) k ig (U) k V k = f (V ) k ig (V ) k. U k V k ˆκ ˆ n1 f,n 1 f = 1 n 2 1 f, n 1f V pp n 2 f, n 2f a ˆκ n2 f,n 2 f. n 2 n 2 ˆ fg ˆ gf ˆ gg V pp G J = 0 ˆ ˆ U k (r) V k (r) u k v k u2 k v 2 k = 1 2 [ 1 ± ϵ k λ (ϵk λ) 2 + 2 ].

ˆ = 1 2 [ E(N + 2) E(N + 1) (E(N + 1) E(N)) ] vk 2 vk 2 G G G G

1 S 0 k 2 dk (k) = 0 2π k V 1 S 0 k (k ) 2 2E(k ), k V 1 S 0 k = Gp(k)p(k ). p(k) = e a2 k 2 G a G = 728 3 a = V pp (r 1, r 2, r 1, r 2) = Gδ(R R )P (r)p (r ), R = 1 2 (r 1 + r 2 ) r = 1 2 (r 1 r 2 ) P (r) p(k) 1 /2a2 P (r) = (4πa 2 e r2. ) 3/2 δ(r R ) δ(r R ) pp n 1 n 2 V pp n 1n 2 a = n 1 n 2 V pp n 1n 2 n 1 n 2 V pp n 2n 1

N n 1 n 2 V pp n 1n 2 a = G Wn N 1 n 2 W N n. 1 n 2 N ˆ N ˆ n1 n 2 = G P N Wn N 1 n 2 P N = 1 2 (W N ˆκ) N N E = G PNP N. N

β ĥ D λ ˆ ˆ ĥ D + λ U k V k = E k U k V k ĥ D = α + β(m + S) + V 0 A λ B ff 0 B T C λ 0 0 ff 0 A + λ B 0 0 B T C + λ f U g U f V g V = E f U g U f V g V. ρ nn ρ nñ ρñn ρññ = f V n f V n i f V n g Ṽ n i g V ñ f V n g V ñ g Ṽ n, n ñ

Ĥ + C 2µ ( ˆQ 2µ q 2µ ) 2, µ=0,2 Ĥ ˆQ 2µ ˆQ 20 = 2z 2 x 2 y 2 ˆQ22 = x 2 y 2. q 2µ C 2µ β (r, θ, ϕ) r j m z π t = ±1/2 ψ k (r, s, t) = f k(r)φ lk j k m k (θ, ϕ, s) ig k (r)φ lk j k m k (θ, ϕ, s) χ tk (t). Φ ljm ljm Φ ljm (θ, ϕ, s) = [χ 1 (s) Y l (θ, ϕ)] jm. 2

l ( l) j π l = j ± 1 ( 2, 1 l = j 2, π = ( )l, κ = ± j + 1 ). 2 M + S(r) + V (r) r κ+1 r r κ 1 r M S(r) + V (r) f k g k = ϵ k f k g k S(r) V (r) R nl (r, b 0 ) b 0. n f k (r) = n ñ f n (i) R nlk (r, b 0 ), g k (r) = ñ g (i) ñ R ñ l k (r, b 0 ). R nl (r, b 0 ) = b 3/2 0 R nl (ξ) = b 3/2 0 N nl ξ l L l+1/2 n (ξ 2 )e ξ2 /2, ξ = r/b 0 n = 0, 1, 2,... L m n (ξ 2 ) 2n! N nl = (l + n + 1/2)!. n ñ N = 2n +l Ñ = 2ñ + l Ñ = N + 1 α = nljm α = ñ ljm

A αα = C α α = B α α = 0 0 0 r 2 drr nl (r) [ M + S(r) + V (r) ] R n l(r), r 2 drrñ l(r) [ M S(r) + V (r) ] Rñ l(r), ( r 2 drr nl (r) r κ 1 ) r Rñ l(r). V C (r) = e2 d 3 r ρ p (r ) 4π r r. r r r = 2 r r, V C (r) = e2 d 3 r r r r ρ p (r ). 8π ( ) V C (r) = e2 dr r 2 3r + r 2 d 2 ρ p (r ) 4 r dr 2. 0 ϕ = σ, ω, ρ ( ) 2 r 2 2 r r + m2 ϕ ϕ(r) = S ϕ (r). n b n b ϕ(r) = ϕ n R n0 (r, b 0 ), S ϕ (r) = s ϕ nr n0 (r, b 0 ). n=0 n=0

n b N B = 2n b n b n H nn ϕ n = s ϕ n H nn = b 2 0 δ nn (2n + 3/2) + b 2 0 δ nn +1 (n + 1)(n + 3/2) +b 2 0 δ n n+1 (n + 1)(n + 3/2). H j z π k K k f + k (r, z)e iλ ϕ f k ψ k (r, s, t) = (r, z)e iλ +ϕ ig + k (r χ tk (t),, z)e iλ ϕ ig k (r, z)e iλ +ϕ Λ ± = K k ± 1/2 (r, z, ϕ) V (z, r ) = 1 2 mω2 zz 2 + 1 2 mω2 r 2.

ħω ħω z β 0 ħω z = ħω 0 e 5 4π β 0, ħω = ħω 0 e 1 5 2 4π β 0. ħ ħ b z =, b =, mω z mω b 2 b z = b 3 0 ħω 0 β 0 α = n z n r Λm s, n z n r z r Λ m s z j z z K = Λ + m s, π = ( 1) nz+λ. ξ = z/b z η = r 2 /b2 Φ α (r, s) = φ nz (z, b z )φ Λ n r (r, b ) eiλϕ 2π χ ms (s) φ nz (z, b z ) = N n z H nz (ξ)e ξ2 /2, bz φ Λ n r (r, b ) = N Λ n r b 2η Λ /2 L Λ n r (η)e η/2, H nz (ξ) L Λ n r (η) N nz = 1 ( π2 n znz!), N n Λ n r! r = (n r + Λ )!.

f k (r, s, t) = 1 f + k (r, z)e iλ ϕ α max 2π f k (r = f, z)e iλ α (i) Φ α (r, s)χ tk (t), +ϕ g k (r, s, t) = 1 2π g+ k (r, z)e iλ ϕ g k (r, z)e iλ +ϕ α α max = α g (i) α Φ α(r, s)χ tk (t), α max α max N = 2n r + Λ + n z N max N max + 1 A αα = δ ΛΛ δ msm s dzφ nz (z)φ n z (z) 0 dr r φ Λ n r (r )φ Λ n r (r ) [ M + S(r, z) + V (r, z) ], C α α = δ Λ Λ δ ms m dzφñz (z)φñ s z (z) dr r (r φ Λñr )φ Λñ r (r ) 0 [ M S(r, z) + V (r, z) ], B α α = δ Λ Λδ ms m s δ nrñr ( 1) 1 ) ñz (δ 1/2 ms nzñ b z 1 z 2 δ nz n zñ z+1 2 +δ ms m s 1δ nzñz dr r φ Λ n r (r ) r φ Λñr (r ) Λ 0 +δ ms m s+1δ nzñz 0 dr r φ Λ n r (r ) r φ Λñr (r ) + Λ 0 0 φ Λ n r (r )φ Λñr (r ) φ Λ n r (r )φ Λñr (r ). V C (r) = e2 d 3 r r r r ρ p (r ). 8π ϕ ( ) V C (r, z) = e2 dr 2π r dz 4r r d(r, z)e ρ p (r d(r, z), z ) 0

d(r, z) = (z z ) 2 + (r + r )2 ϕ = σ, ω, ρ ( ) 2 1 2 r 2 r r z + 2 m2 ϕ ϕ(r, z) = S ϕ (r, z). ϕ(r, z) = n b n zn r ϕ nzn r φ nz (z, b z )φ 0 n r (r, b ). β 0 ħω 0 n b n z n r H nznrn z n r = δ nrn δ r n zn z 1 [ δ nrn r 2 b 2 z +δ nzn z b 2 H nzn rn z n r ϕ n z n r = sϕ n zn r b 2 z ] (n z + 1/2) + b 2 (2n r + 1/2) + m 2 ϕ ( (nz + 1)n zδ nzn z 2 + n z (n z + 1)δ nzn z +2 ) (n rδ nrn r 1 + n r δ nrn r+1). H

ρ(r) ρ(r, r ) T = 0

j > = l π + 1 2 j < = l ν + 1 2

V T = ( τ 1 τ 2 )S 12 V (r) τ 1,2 ( ) V r S 12 S 12 = 3( s 1 r/r)( s 2 r/r) ( s 1 s 2 ), s 1,2 r r S 12 = 3([ s 1 s 2 ] (2) [ r r] (2) /r 2 ), = 24π/5([ s 1 s 2 ] (2) Y (2) [ ] K K [ r r] (2) /r 2 = 8π/15Y (2). S = 1

V (r) S 12 = 2 S 12 = 1 ρ ω V OP EP ( k) = 4πf π 2 ( σ 1 τ m 2 1 τ k)( σ 2 k) 2. π k 2 + m 2 π V OP EP ( k) = 4πf 2 π 3m 2 π τ 1 τ 2 3( σ 1 k)( σ 2 k) σ 1 σ 2 k 2 k 2 + m 2 π + σ 1 σ 2 ( 1 m2 π k 2 + m 2 π fπ 2 = 0.08 m π = 138 k σ 1 σ 2 δ( r 1 r 2 ) V OP EP V OP EP ( r) = f 2 πm π τ 1 τ 2 [ ( 1 3m π r + 1 (3m π r) 2 + 1 (3m π r) 3 ), ) ] e mπr S 12 + 13 σ e mπr 1 σ 2. m π r

ρ ρ V ρ ( k) = 4πf 2 ρ m 2 ρ ( σ 1 τ 1 τ k)( σ 2 k) 2. k 2 + m 2 ρ ( ) V ρ ( r) = fρ 2 1 m ρ τ 1 τ 2 3m ρ r + 1 (3m ρ r) + 1 e mρr S 2 (3m ρ r) 3 12 ( ) + 2 3 σ e mρr 1 σ 2 m ρ r 4π. m 3 ρδ( r) m ρ = 770 f 2 ρ m ρ = 4.86 ρ ω V ω ( k) = 4πf 2 ω m 2 ω ( σ 1 τ 1 τ k)( σ 2 k) 2. k 2 + m 2 ω ( ) 1 V ω ( r) = fωm 2 ω [ 3m ω r + 1 (3m ω r) + 1 e mωr S 2 (3m ω r) 3 12 + 2 ( e m ωr 3 σ 1 σ 2 m ω r 4π ) ]. m 3 ωδ( r) ω m ω = 783

π ρ ω σ ω δ ρ ρ ρ ρ

ρ ω E RHF [ˆρ, ϕ] σ ϕ m = σ, ω, ρ, A E π [ˆρ] E RHF [ˆρ, ϕ] = E RMF [ˆρ, ϕ] + E π [ˆρ]. L π = 1 2 ( ) µ π µ π m 2 π π 2, m π = 138

fπ 2 = λfπ 2(free) λ m π g ω g ρ m σ m ω m ρ g σ g 2 (fm 1 ) g 3 λ χ 2 L pv = f π m π ψγπ γ µ µ π τψ f π λ f π λ m σ σ

g σ g ω g ρ g 2 g 3 σ λ (λ = 1) λ = 0 χ 2 300 200 χ 2 100 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λ χ 2 χ 2 λ λ = 0 λ π1h 11/2

π1g 7/2

f ext (, t) i t Ψ(t) = (Ĥ + f ext(t)) Ψ(t) Ψ(t) Ψ(0) ρ(r, t) = Ψ(t) N δ(r r i ) Ψ(t) i f ext (, t) ϕ(r, t) (i = 1... N) i t ϕ(r, t) = [ 2 /2m + v KS [ρ](r, t)ϕ i (r, t) u KS [ρ](r, t) ρ(r, t) = N ϕ i (r, t) 2, i

t v KS [ρ](r, t) t ρ(r, t) v xc v KS [ρ](r, t) = v[ρ](r, t) + v xc [ρ](r, t) + f ext (r, t), v v[ρ](r, t) = v[ρ](r, t) + v xc [ρ](r, t), Σ(r, t) ρ(r, t) ρ(r, t) v s [ρ](r) E[ρ ] f ext (r, t) v[ρ](r, t) ρ s ( ) ρ(r, t) = ρ s ( ) + δρ(, t) f ext v[ρ s + δρ](, t) = v[ρ s ](r) + dt d 3 r V [ρ s ](r, t, r, t )δρ(r, t ). V V (r, r, t t δu(, t) ) = δρ(, t ), ρ=ρs

ρ s t t R(r, t, r, t ) δρ(r, t) = d 3 r dt R(r, r, t t )f ext (r, t ), R(r, r, ω) = R 0 (r, r, ω) + d 3 r 1 d 3 r 2 R 0 (r, r 1, ω)v (r 1, r 2, ω)r(r 2, r, ω). R 0 ρ s (r) f ext v[ρ](r, t) ρ = ρ s E[ρ s ] v[ρ](r, t) ρ(r, t) ρ(r, t) t v s (r) v[ρ s ] KS (r) v[ρ](r, t) v s [ρ s ](r, t) ρs=ρ(t). v[ρ] ρ(r, t) ρ s (r) ρ(r, t) t ρ s (r) = ρ(r, t)

v[ρ](r, t) v s [ρ s ] E[ρ s ] V ad (r, r, t t δe[ρ s ) = δ s (r)δ s (r ) δ(t t ), ρs=ρ(t) Σ(ω) Σ(ω) = Σ + Σ (e) (ω). Σ Σ (e) (ω)

Z = 82 N = 126 Z = 114 Z = 120 Z = 126 N = 162 N = 184 Z = 120 N = 162 Z = 120

± G(ξ, ξ ; ε) = n (Ψ(ξ)) 0n (Ψ (ξ )) n0 ε (E n (N+1) E0 N ) + iδ + m (Ψ (ξ )) 0m (Ψ(ξ)) m0 ε + (E m (N 1) E0 N ) iδ, (Ψ (ξ)) n0 = Φ (N+1) n Ψ (ξ) Φ (N) 0, (Ψ (ξ)) m0 = Φ (N 1) m Ψ(ξ) Φ (N) 0, δ +0 Φ (N) 0 Φ (N) n n N E (N) 0 E n (N) ξ n

G η 1η 2 k 1 k 2 (ε) = δ k1 k 2 δ η1 η 2 Gη 1 k 1 (ε), Gη 1 k 1 (ε) = 1 ε η 1 E k1 + iη 1 δ, δ +0. ψ η k H RHB ψ η k = ηe k ψ η k ξ = {k, η} k η ± 1 E k (ε H RHB ) G(ε) = 1 H RHB H RHB = hd m λ, h D + m + λ λ m h D = αp + β(m + Σ). Σ) Σ(r) = m Γ m ϕ m (r) ϕ m Γ m

Σ (e) (ε) ( ) ε H RHB Σ (e) (ε) G(ε) = 1. ψ η k 2 2 ( η=±1 k (ε η 1 E k1 )δ η1 ηδ k1 k Σ (e)η 1η k 1 k ) (ε) G ηη 2 kk 2 (η) = δ η1 η 2 δ k1 k 2, η, η i Σ (e) ψ η k Σ (e) Σ (e)η 1η k 1 k (ε) = η=±1 η µ=±1 k,µ δ ηµ,ηγ ηµ;η 1η µ;k 1 k γηµ;η 2η µ;k 2 k, δ +0. ε ηe k η µ (Ω µ iδ) k k µ Ω µ η µ = ±1 γ ηµ;η 1η 2 µ;k 1 k 2 µ

{kη} G η k (ε) = ν S η(ν) k ε ηe (ν) k, E (ν) k ε ηe k Σ (e)ηη kk (ε) = 0. ε S +(ν) k S (ν) k S η(ν) k = S (ν) k 1 dσ(e)ηη kk dε (ε) ε=e (ν) k 1, = = = k E k {ν} λ±e (ν) k S (ν) k = vk 2 (ν) S k λ S (ν) k = (1 vk 2 (ν) ) S k v 2 ν S (ν) k = 1, E k = ν E (ν) k S (ν) k, Σ (e)

Z = 2, 8, 20, 40, 82 N = 2, 8, 20, 50, 82, 126

f(r)l s l j > = l + 1 2 j < = l 1 2 j >

ρ δ N

34 N = 20 34 34 2s 1/2 ( 1p) 34 2s 1/2 34 1f 7/2 2p 3/2 2p 1/2 1f 5/2 35 d, p N = 20 2p 3/2 2p 1/2 36 34 2p 1f N = 20 40 36 34 38 1f 7/2 2p 3/2 2p 1/2 1f 5/2

2p 1f V ω S σ δ ρ V S.O. = W (p σ) W = 1 (V S) 2Meff 2 M eff = M 1 (V S) 2 ψ

ψ(r) = f(r) g(r), M + S + V σp σp M S + V f = (M + ϵ) f g g,. ϵ (2M + ϵ + S V ) g = q 2M + ϵ + S V σpf, { } q σp 2M + ϵ + S V σp + V + S f = ϵf. ϵ ϵ (ϵ + S V )/2M S V S V

V S (ϵ + S V )/2M 40 (ϵ S V )/2M M (bmr) V + S 5 ϵ = 0 ϵ 7 (ϵ S V )/2M 4 M eff (r) = M 1 (V (r) S(r)) 2 ϵ/2m eff 0.6 ϵ { } σp q σp + V + S f = ϵf. 2M eff { } 1 p 2M eff (r) p + V pot(r) + 1 (2M) ( V ls(r))(p σ) f 2 i = ϵ i f i. M eff (r) V pot (r) = V (r) + S(r) V ls (r) = M (V (r) S(r)). M eff (r)

( ) 1 1 2M 2 r r V ls(r) ls. C i = gi 2 /m 2 i V S = (C ω + C σ )(ρ p + ρ n ) + τ 3 (C ρ + C δ )(ρ p ρ n ) C i = g 2 i /m 2 i i = σ, ω, δ, ρ C i = α S, α V, α T S, α T V W τ = W 1 ρ τ + W 2 ρ τ τ W 1 W 2 W 1 W 2 1 + 2 C ρ + C δ C ω + C σ M r C ρ +C δ 10 20 C ρ C δ C ρ C δ δ C δ = 0 C ρ +C δ S V V (SO) 12 (r 12 ) = iw 0 (σ 1 + σ 2 ) (ˆk δ(r 12 )ˆk)

r 12 = r 1 r 2 ˆk = (i/2)( 1 2 ) W 0 W τ (r) = W 1 ρ τ + W 2 ρ τ τ. W 1 W 2 W 1 = 2. W 2 ˆP τ = 1(1+ˆτ 2 1 ˆτ 2 ) x V SO = iw 0 1 2 (1 + x ˆP τ )(σ 1 + σ 2 )ˆk δ(r 12 )ˆk. W 1 = W 0 (1 + x w )/2 W 2 = W 0 /2 x w

ħω = 41A 1/3 N F = N B = 20 1f 5/2 38 36 1f 5/2 2p 1/2 34 f 5/2 38 36 p 1/2 34 1f 5/2 2p 1/2 34 1f 5/2 2p 1/2 40 34

80 1f5/2 2f5/2 3f5/2 4f5/2 60 Energy (MeV) 40 20 0 8 10 12 14 16 18 20 Number of shells 80 1f5/2 2f5/2 3f5/2 4f5/2 60 Energy (MeV) 40 20 0 8 10 12 14 16 18 20 Number of shells f 5/2 38 36

80 1f5/2 2f5/2 3f5/2 4f5/2 60 Energy (MeV) 40 20 0 60 40 8 10 12 14 16 18 20 Number of shells 1p1/2 2p1/2 3p1/2 4p1/2 Energy (MeV) 20 0-20 6 8 10 12 14 16 18 20 22 Number of shells f 5/2 p 1/2 34

= 2 0.2 2p 1/2 1f 5/2 Square of w.f. (fm -3 ) 0.1 0 0.2 0.1 40 Ca 34 Si 0 0 2.5 5 7.5 10 r (fm) 1f 5/2 2p 1/2 40 34 N = 20 δ

40 Z = 20 1d 3/2 38 36 2s 1/2 34 40 36 34 40 36 34 34 ( 1p) 33 2s 1/2 34 36 (2s 1/2 ) = 1.53 1f 7/2 2p 3/2 2p 1/2 1f 5/2 34 34 d, p 35 2p = 2p 1/2 2p 3/2 36 34

proton density (fm -3 ) 0.1 0.05 DD-ME2 NL3 DD-PC1 40 Ca 0 2 4 r (fm) proton density (fm -3 ) 0.1 0.05 DD-ME2 NL3 DD-PC1 36 S 0 2 4 r (fm) proton density (fm -3 ) 0.1 0.05 DD-ME2 NL3 DD-PC1 34 Si 0 2 4 r (fm) 40 36 34

proton density (fm -3 ) 0.1 0.05 DD-ME2 NL3 DD-PC1 40 Ca 0 2 4 r (fm) proton density (fm -3 ) 0.1 0.05 DD-ME2 NL3 DD-PC1 36 S 0 2 4 r (fm) proton density (fm -3 ) 0.1 0.05 DD-ME2 NL3 DD-PC1 34 Si 0 2 4 r (fm) 40 36 34

proton density (fm -3 ) 0.2 0.15 0.1 0.05 DD-ME2 NL3 DD-PC1 40 Ca 0 2 4 r (fm) proton density (fm -3 ) 0.2 0.15 0.1 0.05 DD-ME2 NL3 DD-PC1 36 S 0 2 4 r (fm) proton density (fm -3 ) 0.2 0.15 0.1 0.05 DD-ME2 NL3 DD-PC1 34 Si 0 2 4 r (fm) 40 36 34

34 W 1 /W 2 40 36 34 N = 20 2s 1/2 40 38 36 34 = 1 7/2 1 5/2 = 2 3/2 2 1/2 40 38

40 36 34 40 36 36 34 f p f p p f 36 34 40 36 36 34 40 36 34 40 2 3/2 2 1/2 36 34 1f 5/2 36 SF = 0.36 34 SF = 0.32 40 38 36 34 7/2 p f

40 38 36 34 W 1 W 2 δ 40 36 36 34 f p f p δ f p p f

f p δ 36 34 34 40 36 36 34 40 36

(a) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 40 Ca 7.21 1.70 (b) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 38 Ar 1.77 6.90 (c) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 S 1.80 6.43 (d) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 0.71 6.08 34 Si NL3 (a) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 40 Ca 7.40 1.71 (b) 1f 2p 7/2 3/2 2p 1/2 1f 5/2 38 Ar 1.72 7.04 (c) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 S 1.65 6.52 (d) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 0.87 6.12 34 Si DD-ME2 (a) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 40 Ca 7.83 1.77 (b) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 38 Ar 1.74 7.57 (c) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 S 7.12 1.64 (d) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 0.88 34 Si 6.61 DD-PC1-1 0 1 2 3 4 5 6 7 8 9 E(MeV) 2p 3/2 2p 1/2 1f 5/2 1f 5/2 40 38 36 34

3 2.5 2 NL3 NL3* FSUGold DD-ME2 DD-MEδ DD-PC1 PC-PF1 SLy5 D1S Expt. E (MeV) 1.5 1 0.5 (p) 0 9 8.5 8 40 38 36 Mass number A 34 NL3 NL3* FSUGold DD-ME2 DD-MEδ DD-PC1 PC-PF1 SLy5 D1S Expt. 7.5 E (MeV) 7 6.5 6 5.5 (f) 5 40 38 36 Mass number A 34 p f A

34 36 34 34 V S.O. = W (p σ). W W τ = W 1 ρ τ + W 2 ρ τ τ. l ls 40 38 36 34

36 34 W W 1 W 2 34 W 1/W 2 1.07 W 1/W 2 = 2 f 40 36

40 2s 1/2 36 34 34

ˆ (3) (N) = 1 [B(N 1, Z) + B(N + 1, Z) 2B(N, Z)], 2 (3) C (N) = 1 [B(N, Z) + B(N 2, Z) 2B(N 1, Z)] 2

38 36 34 (3) C f p 40 36 36 34 2s 1/2 36 34 36 34 38 36 34 38 36 2s 1/2 34

0.2 ρ tot. No pairing With pairing 0.15 0.1 38 Ar 0.05 ρ p Densities (fm -3 ) 0 0.2 0.15 0.1 0.05 ρ p ρ tot. 36 S 0 0.2 ρ tot. 0.15 0.1 34 Si 0.05 ρ p 0 1 2 3 4 5 r (fm) ρ ρ p 38 36 34

38 f p 36 f p 38 2s 1/2

36 34 (2S 1/2 ) δ 2s 1/2 36 34 2p 3/2 2p 1/2 34 f p 2s 1/2

v fac p f 2s 1/2 36 34 38 36 38 36 36 34 36 34 p

(a) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 40 Ca 7.21 1.69 (b) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 38 Ar 1.64 6.92 (c) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 S 6.46 1.68 (d) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 0.80 34 Si 5.94 NL3 (a) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 40 Ca 7.40 1.71 (b) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 38 Ar 7.08 1.64 (c) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 S 1.57 6.55 (d) 1f 2p 7/2 3/2 2p 1/2 1f 5/2 0.94 34 Si 6.00 DD-ME2 (a) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 40 Ca 7.83 1.77 (b) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 1.67 38 Ar 7.58 (c) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 S 7.14 1.56 1f 2p 7/2 3/2 2p (d) 1/2 1f 5/2 0.96 34 Si 6.52 DD-PC1-1 0 1 2 3 4 5 6 7 8 9 E(MeV)

3 2.5 NL3 NL3* FSUGold DD-ME2 DD-MEδ DD-PC1 PC-PF1 Exp. 2 E (MeV) 1.5 1 0.5 (p) 0 8 40 38 36 Mass number A 34 7.5 7 E (MeV) 6.5 6 5.5 (f) 5 40 38 36 Mass number A 34

40 38 36 34 δ 40 36 36 34 f p f p δ 2s 1/2 36 34

36 34 (2S 1/2 ) δ 2s 1/2 36 34 p 36 34 p 2s 1/2 v fac v fac = 0

v fac = 2 2p π(2s 1/2 ) 36 34 π(2s 1/2 ) 36 34 36 34 v fac = 0.5 2p 3/2 2p 1/2 π(2s 1/2 ) v fac 36 34 36 v fac 0.60 v fac 0.5 v fac 1.10 1.20 34 v fac 36 v fac 1.2 34 v fac 2

36 34 36 34 36 34 36 34 36 34 36 34 2p 3/2 2p 1/2 (2s 1/2 ) v fac 36 34

2 neutron 2p SO splitting (MeV) 1.8 1.6 1.4 1.2 1 0.8 36 S 34 Si proton (2s 1/2 ) occupancy 0.6 2 1.5 1 0.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 v fac 36 S 34 Si 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 v fac 2p 3/2 2p 1/2 (2s 1/2 ) v fac 36 34

2 neutron 2p SO splitting (MeV) 1.8 1.6 1.4 1.2 1 0.8 36 S 34 Si proton (2s 1/2 ) occupancy 0.6 2 1.5 1 0.5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 v fac 36 S 34 Si 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 v fac

2 neutron 2p SO splitting (MeV) 1.8 1.6 1.4 1.2 1 0.8 36 S 34 Si 0.6 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 v fac proton (2s 1/2 ) occupancy 2 1.5 1 0.5 36 S 34 Si 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 v fac

2p π(2s 1/2 ) p (SO(p)) = 0.20 (2s 1/2 ) = 0.28 (SO(p)) = 0.22 (2s 1/2 ) = 0.29 (SO(p)) = 0.22 (2s 1/2 ) = 0.29 π(2s 1/2 ) 36 34 2s 1/2 p 2s 1/2 p

SO splitting reduction (%) 70 60 50 40 30 20 10 NL3 NL3* FSUGold DD-ME2 DD-MEδ DD-PC1 PC-PF1 Exp. 0 0 0.5 1 1.5 2 (2s 1/2 ) 2p 1/2 2p 3/2 2s 1/2 36 34

σ ω ρ 34 36

j > j >;< = l ± 1/2 j > j < l 2p 1/2 2p 3/2 1f 5/2 1f 7/2 T 2013 2c 2013

40 38 36 34 T 2013 2c 2013 40 36 36 34 f p f p T 2013 2c 2013 f p

(a) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 40 Ca 7.87 1.92 (b) 1f 2p 7/2 3/2 2p 1/2 1f 5/2 38 Ar 1.74 6.81 (c) 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 S 5.80 1.64 (d) 34 Si 1f 2p 7/2 3/2 2p 1/2 1f 5/2 0.66 5.12 NL3RH2-1 0 1 2 3 4 5 6 7 8 9 E(MeV) 40 36 36 34 40 36 j < π1d 3/2 j > ν1f 7/2 40 j <; ν1f 5/2 40 π1d 3/2 36 f 40 36 j >; ν2p 3/2 j <; ν2p 1/2

4 2 NL3RH2 NL3 1f 5/2 0 E (MeV) -2-4 1f 7/2-6 -8 (I) (II) 4 20 18 16 Proton number Z 14 2 0 2p 1/2 E (Mev) -2-4 2p 3/2-6 -8 (II) 20 18 16 Proton number Z 14 1f 7/2 2p 3/2 2p 1/2 1f 5/2

3 2.5 NL3 NL3+fr. NL3RHF0.5 Exp. 2 E (MeV) 1.5 1 0.5 (p) 0 8 40 38 36 Mass number A 34 7.5 7 E (MeV) 6.5 6 5.5 (f) 5 40 38 36 Mass number A 34

36 34 f p p 2s 1/2 p γ µ 12 = 34 14 V 23 δρ µ 34, 14 V 23 δρ µ 34 Σ(ω) Σ (e) 12 (ω) = kµ ( γ µ 1k γµ 2k ω ε 3 Ω µ + iη + γ µ k1 γµ k2 ω ε 3 + Ω µ iη ), Ω µ k Σ

Σ(r, r ; ω) = Σ(r)δ(r r ) + Σ (e) (r, r ; ω) Σ ĥ = αp + β(m + S) + V = αp + βm + Σ. (ε ĥ Σe (ε))g(ε) = β, (ε ε k Σ e k(ε))g k (ε) = 1. k ε (λ) k λ k S (λ) k λ k = 2 12.0/ A 36 34 1f 7/2 2p 3/2 2p 1/2 1f 5/2 ε (λ) k

36 34 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 34

8 6 NL3* MF 36 S NL3* PVC Energy (MeV) 4 2 0-2 -4-6 1f 5/2 2p 1/2 2p 3/2 1f 7/2 8 NL3* MF 34 Si NL3* PVC 6 Energy (MeV) 4 2 0-2 -4 1f 5/2 2p 1/2 2p 3/2 1f 7/2 1f 7/2 2p 3/2 2p 1/2 1f 5/2 36 34

2p 3/2 36 1f 5/2 1f 7/2 36 34 43 39 36 34 2p 1/2 2p 3/2 1f 5/2 1f 7/2 37 35 1f 5/2

0.75 0.5 0.25 SF 0.75 0.5 0.25 (a) 1f 7/2 2p 2p 3/2 1/2 36 S 2.0 (b) 36 S 1f 7/2 2p 3/2 2.28 2p 1/2 6.30 <1f 5/2 > 1f 5/2 EXP. PVC+NL3* (c) 36 S 1f 7/2 2p 3/2 2p 1/2 1f 5/2 1.80 6.32 NL3* 0.75 0.5 0.25 SF 0.75 0.5 0.25 (d) 34 Si (e) 34 Si 1/2 1.10 <1f 1f 7/2 2p 2p 3/2 1f 7/2 2p 3/2 1.40 2p 1/2 5.28 1f 5/2 5/2 > EXP. PVC+NL3* (f) 34 Si 1f 7/2 2p 3/2 0.85 2p 1/2 5.77 1f 5/2 NL3* -1 0 1 2 3 4 5 6 7 8 9 E(MeV) 37 35

36 34 36 34 f p 1f 2p 1f 7/2 2p 3/2 2p 1/2 2p 1f 7/2 2p 1/2 2p 3/2 1f 5/2 40 38 36 34

δ 2p 3/2 2p 1/2 36 34 34 34 38 1d 3/2 f 36 2s 1/2

p 34 2s 1/2 36 40 36 36 34 2p 3/2 2p 3/2 36 34

N = 20 2p 3/2 2p 1/2 34 N = 162

A 180 A > 100

180

E τ f T λ i 2 /( E) 2λ+1, λ T λ E

10 16 180m 178m2

Ενδιάμεση κατάσταση Μετάπτωση απομείωσης Ισομερής Βασική κατάσταση

180 J π = 9 J π = 1 + 2 8 T 1/2 = 8.2 β 178m2 T 1/2 = 31 178 178m2 178m2 J π = K π = 16 + J π = 13 J π = K π = 8 L = 3 K = 8 ν = K L = 5 178m2 180m

180m 180m 180m 178m2 180m 178m2 178 177m E = 970 T 1/2 = 160 242m E = 49 T 1/2 = 141 177 242m

1 N = 82 Z = 50 A 130 N = 82 Z = 64 A 150 N = 126 Z = 82 A 210 A 180

K Ω J j Ω Ω Ω

J π = K π = i (πi ) Ωi E k (ϵk ϵ F ) 2 + 2 ϵ k ϵ F 178 Z = 73 N = 105 2( p + n ) 3 178 K π = 21 22 + ν 4 Ω Z = 72

2p Φ Φ c + l c l α + k = l U lk c + l + V lk c l, α = U + V + c = W + c, α + V T U T c + c + α + k, α k W

U lk V lk ĥ D λ ˆ ˆ ĥ D + λ U k V k = E k U k V k. π k K k K +K K Φ 0

α k Φ 0 = 0 E k > 0 Φ 0 = α k, E k >0 Φ 0 Φ 1 Φ 1 = α 1 Φ 0 = α 1 α k α 1 k Φ 2 = α 1α 2 Φ 0 (α 1, α 2,..., α N ) N α 1 = α 1, α 2 = α 2,..., α N = α N. (α 1,..., α N, α 1,..., α N ) α 1 α 1, α 2 α 2 (U l1, V l1 ) (Vl1, U l1 ) (U l2, V l2 ) (Vl2, U l2 ) W α 1 α 2 α 1, α 2 α 1, α 2 2 n 1 K K = ± K 1 ± K 2 ± ± K n, π = n πi

K K = i K i K k +K K K K N = 162 K K max = K 1 + K 2 K min = K 1 K 2 K max K min K K max K min

+K K ρ κ K ρ = ρ M M + 1 2 (U k b U T k b κ = κ M M 1 2 (U k b V T k b V k b V T k b ), + V k b U T k b ), U kb V kb U V A 160 190 Z 72 N 104

A 160 190 6 : 5/2 [512], 7/2 + [633] Z 70 74 N 100 108 6 + : 5/2 [512], 7/2 [514] 6 + : 5/2 + [402], 7/2 + [404] 8 : 9/2 + [624], 7/2 [514] 8 : 9/2 [514], 7/2 + [404] 10 : 9/2 [505], 11/2 + [615] Z 76 N 110 116 10 + : 9/2 + [624], 11/2 + [615] 10 + : 9/2 [514], 11/2 [505] 12 + : 11/2 + [615], 13/2 + [606] Z 102 108 N 150 164 8 : 7/2 + [624], 9/2 [734] 8 : 7/2 [514], 9/2 + [624] 8 : 7/2 + [613], 9/2 [734] 10 : 9/2 + [615], 11/2 [725] 10 : 9/2 [505], 11/2 + [615] n z n ρ m l m l N = n z + 2n ρ + m l = n x + n y + n z. m l s z

j z Ω = m l + m s = m l ± 1 2. Ωπ[Nn z m l ], π π = ( 1) N Ω β 0.25 6 + 8 6 + 8 8 178 8 ν 2 7/2 [514] 9/2 + [624] N = 106 6 + 5/2 [512], 7/2 [514] Z = 72 8 174 184 186 8 π 2 7/2 + 9/2 6 + ν 2 7/2 [514] 5/2 [512] N = 104

A 160 190 2577 709.9 1867 8 N = 106 6 + Z = 72

j j + 1/2 β 2 Q 20 9 Q 20 = 5π AR2 0β 2. Z 70 74 N 100 108 Z 76 N 110 116 Z 102 108 N 150 164 nl j j + 1/2

A 160 190 j 174 176 178 Z = 72 N = 102 106 174 176 178 186 Z = 76 N = 110 254 Z = 102 N = 152 b 2 0.2 0.3 6 + N = 104 6 + N = 104 ν5/2 [512] ν7/2 [514] π9/2 [514] π7/2 + [404] A > 100 170 176 N = 104 172 174 178 180

126-5 3p 1/2 7/2- [503] 9/2- [505] 1/2- [510] 1/2+ [651] 3/2- [512] 3p 3/2 1i 13/2 2f 5/2 9/2+ [624] E s.p. (MeV) Fermi 7/2+ [633] 5/2- [512] 2f 7/2 7/2- [514] 1/2- [521] 5/2+ [642] 1/2+ [640] 3/2+ [651] 1h 9/2-10 3/2- [521] 11/2- [505] 5/2- [523] 0 0.25 g.s. 0.5 β2 174

A 160 190 3/2- [532] 82 5/2+ [402] 9/2- [514] 3s 1/2-5 1/2- [541] E s.p. (MeV) 2d 3/2 1h 11/2 7/2+ [404] 1/2+ [411] Fermi 7/2- [523] 3/2+ [411] 2d 5/2 5/2- [532] 0 0.25 g.s. 0.5 β2 174

7/2- [503] 9/2- [505] -5 3p 3/2 1/2- [510] 3/2- [512] 1i 13/2 2f 5/2 9/2+ [624] E s.p. (MeV) Fermi 7/2+ [633] 5/2- [512] 2f 7/2 7/2- [514] 1/2- [521] 3/2+ [651] 1h 9/2-10 1/2+ [640] 3/2- [521] 5/2+ [642] 11/2- [505] 0 0.25 g.s. 0.5 β2 176

A 160 190 5/2- [523] 82 3/2+ [402] 1/2- [550] 3/2- [532] 5/2+ [402] E s.p. (MeV) -5 3s 1/2 9/2- [514] 1/2- [541] 2d 3/2 1h 11/2 7/2+ [404] 1/2+ [411] 7/2- [523] 2d 5/2 3/2+ [411] 0 0.25 g.s. 0.5 β2 176

126-5 3p 1/2 3p 3/2 11/2+ [615] 7/2- [503] 9/2- [505] 1/2- [510] 3/2- [512] 1i 13/2 2f 5/2 9/2+ [624] E s.p. (Mev) Fermi 7/2+ [633] 5/2- [512] 7/2- [514] 2f 7/2 5/2+ [642] 1/2+ [640] 1/2- [521] -10 1h 9/2 3/2- [521] 3/2+ [651] 0 0.25 0.5 β2 178

A 160 190 82-5 5/2+ [402] 3/2- [532] 1/2- [541] 3s 1/2 E s.p. (MeV) 2d 3/2 9/2- [514] 7/2+ [404] Fermi 1h 11/2 1/2+ [411] 2d 5/2 7/2- [523] 3/2+ [411] -10 5/2- [532] 1/2- [530] 3/2- [541] 0 0.25 g.s. 0.5 β2 178

-5 126 7/2- [503] 9/2- [505] 3p 1/2 3p 3/2 11/2+ [615] 1/2- [510] E s.p. (MeV) 2f 5/2 Fermi 9/2+ [624] 3/2- [512] 5/2- [512] 7/2+ [633] 2f 7/2-10 5/2+ [642] 0 0.1 0.2 g.s. 0.3 0.4 0.5 β2 186

A 160 190 82 3/2- [532] -5 3s 1/2 5/2+ [402] 1/2- [541] Fermi E s.p. (MeV) 2d 3/2 9/2- [514] 7/2+ [404] 1h 11/2 2d 5/2 1/2+ [411] 7/2- [523] -10 3/2+ [411] 5/2- [532] 0 0.1 0.2 g.s. 0.3 0.4 0.5 β2 186

1/2- [761] 2g 7/2-5 11/2- [725] 7/2+ [613] 3/2+ [622] E s.p. (MeV) 2g 7/2 9/2+ [615] Fermi 1j 15/2 9/2- [734] 5/2+ [622] 0 0.1 0.2 0.3 0.4 0.5 β2 254

A 160 190 5/2- [512] 9/2- [505] 1/2+ [615] 2f 7/2 9/2+ [624] E s.p. (MeV) 1i 13/2 Fermi 1/2- [521] 3/2- [521] 7/2- [514] 7/2+ [633] -5 5/2+ [642] 1/2+ [400] 0 0.2 g.s. 0.4 β2 254

6 + ν5/2 [512] ν7/2 [514] 178,180 π5/2 + [402] π7/2 + [404] ν5/2 [512] ν7/2 [514] π5/2 + [402] π7/2 + [404] E qp,k ϵ k E qp = (ϵ k λ) 2 + 2 k, λ k

A 160 190 Neutrons -5 1/2- [510] 9/2+ [624] 1/2- [510] 3/2- [512] 9/2+ [624] 1/2- [510] 3/2- [512] 9/2+ [624] 1/2- [510] 3/2- [512] Energy (MeV) -6-7 -8-9 -10-11 -4 Fermi 170Hf 5/2- [512] 7/2- [514] 7/2+ [633] 1/2- [521] 5/2+ [642] 3/2- [521] 3/2+ [651] 11/2- [505] 5/2- [523] 1/2+ [640] 11/2+ [615] 7/2- [503] 9/2- [505] Fermi 172Hf 5/2- [512] 7/2- [514] 7/2+ [633] 1/2- [521] 5/2+ [642] 3/2- [521] 3/2+ [651] 11/2- [505] 1/2+ [640] Neutrons 11/2+ [615] 7/2- [503] 9/2- [505] 174Hf 5/2- [512] 7/2- [514] Fermi 7/2+ [633] 1/2- [521] 5/2+ [642] Fermi 3/2+ [651] 3/2+ [651] 3/2- [521] 1/2+ [640] 1/2+ [640] 3/2- [521] 11/2- [505] 11/2- [505] 9/2- [505] 7/2- [503] 11/2+ [615] 176Hf 9/2+ [624] 5/2- [512] 7/2+ [633] 7/2- [514] 5/2+ [642] 1/2- [521] 9/2- [505] 7/2- [503] 11/2+ [615] -5 Energy (MeV) -6-7 1/2- [510] 3/2- [512] 9/2+ [624] Fermi 1/2- [510] 3/2- [512] Fermi 9/2+ [624] Fermi 1/2- [510] 3/2- [512] 9/2+ [624] Fermi 1/2- [510] 3/2- [512] 9/2+ [624] -8-9 5/2- [512] 5/2- [512] 7/2+ [633] 7/2+ [633] 5/2- [512] 7/2+ [633] 7/2- [514] 7/2- [514] 5/2+ [642] 5/2+ [642] 7/2- [514] 178Hf 180Hf 182Hf 184Hf 5/2- [512] 7/2+ [633] 7/2- [514] N = 98 112

Protons Energy (MeV) -4-5 -6-7 -8-9 -10-11 -12-1 -2-3 -4-5 -6-7 -8-9 3/2+ [402] 1/2- [530] 3/2- [532] 5/2+ [402] 9/2- [514] 1/2- [541] Fermi 1/2+ [411] 7/2+ [404] 7/2- [523] 3/2+ [411] 5/2- [532] 5/2+ [413] 3/2+ [402] 1/2- [530] 5/2+ [402] 3/2- [532] 9/2- [514] 1/2- [541] Fermi 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 5/2- [523] 3/2+ [402] 1/2- [530] 5/2+ [402] 3/2- [532] 9/2- [514] 1/2- [541] Fermi 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 170Hf 172Hf 174Hf 176Hf 5/2+ [402] 3/2- [532] 1/2- [541] 9/2- [514] Fermi 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 1/2- [530] 3/2+ [402] 5/2- [523] 3/2- [532] 5/2+ [402] 1/2- [541] 9/2- [514] Fermi Protons 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 1/2- [530] 1/2+ [400] 3/2+ [402] 5/2- [523] 3/2- [532] 1/2- [541] 5/2+ [402] 9/2- [514] Fermi 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 1/2- [530] 178Hf 180Hf 182Hf 184Hf 5/2- [523] 1/2- [550] 3/2+ [402] 3/2- [532] 5/2+ [402] 1/2- [541] 9/2- [514] Fermi 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 1/2+ [400] 3/2+ [402] 5/2- [523] 3/2- [532] 1/2- [541] 5/2+ [402] 9/2- [514] Fermi 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541]

A 160 190 ν5/2 [512] ν7/2 [514] 6 + 170 176 (3) (N) = 1 [B(N 1, Z) + B(N + 1, Z) 2B(N, Z)], 2 (5) (N) = 1 [B(N + 2, Z) 4B(N + 1, Z) 8 + 6B(N + 1, Z) 4B(N 1, Z) + B(N 2, Z)],

D 5 D 3 D 5 D 3 ν5/2 [512] ν7/2 [514] 170 176 π5/2 + [402] π7/2 + [404] 178,180 6 + 6 + 176

A 160 190 4 DD-ME2 D^3 DD-ME2 D^5 DD-PC1 D^3 DD-PC1 D^5 Expt. 3 Energy (MeV) 2 1 0 170 172 174 176 178 180 Hf isotope 6 +

6 + Z = 72 D 3 D 5 D 3 D 5 D 3 D 5 170 2 174 176 176 0.5 ν5/2 [512] ν7/2 [514] 176 π5/2 + [402] π7/2 + [404]

A 160 190 6 + 10 6 + D 3 D 5 D 3 D 5 176 N = 104 6 + N = 104 172 174 176 178 180 ν5/2 [512]

ν7/2 [514] ν5/2 [512] ν5/2 [512] ν7/2 [514] ν7/2 + [633] ν7/2 [514] 6 + 6 + D 3 D 5 6 + 6 + 172 0.5 174

A 160 190 Energy (MeV) Energy (MeV) -4-5 -6-7 -8-9 -2-3 -4-5 -6-7 -8-9 -10-11 3/2- [512] 1/2- [510] 9/2+ [624] 5/2- [512] Fermi 7/2- [514] 7/2+ [633] 1/2- [521] 5/2+ [642] 11/2- [505] 3/2+ [651] 3/2- [521] 1/2+ [640] 3/2- [512] 1/2- [510] 9/2+ [624] 5/2- [512] Fermi 7/2- [514] 7/2+ [633] 1/2- [521] 5/2+ [642] 3/2+ [651] 1/2+ [640] 11/2- [505] 3/2- [521] Neutrons 1/2- [510] 3/2- [512] 9/2+ [624] 5/2- [512] Fermi 7/2+ [633] 7/2- [514] 1/2- [521] 5/2+ [642] 3/2+ [651] 1/2+ [640] 1/2+ [651] 9/2- [505] 1/2- [510] 3/2- [512] 9/2+ [624] 5/2- [512] Fermi 7/2+ [633] 7/2- [514] 1/2- [521] 5/2+ [642] 172Er 174Yb 176Hf 178W 180Os 1/2- [550] 5/2+ [402] 3/2- [532] 9/2- [514] 1/2- [541] 7/2+ [404] 1/2+ [411] Fermi 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 5/2- [523] 3/2+ [402] 1/2- [550] 5/2+ [402] 3/2- [532] 9/2- [514] 1/2- [541] 7/2+ [404] Fermi 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 1/2- [530] 5/2+ [413] Protons 5/2- [523] 1/2- [550] 3/2+ [402] 5/2+ [402] 3/2- [532] 1/2- [541] 9/2- [514] Fermi 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 1/2- [530] 5/2- [523] 1/2- [550] 3/2+ [402] 5/2+ [402] 3/2- [532] 9/2- [514] Fermi 1/2- [541] 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 172Er 174Yb 176Hf 178W 180Os 7/2- [503] 3/2+ [642] 11/2+ [615] 9/2- [505] 1/2+ [651] 1/2- [510] 3/2- [512] 9/2+ [624] 5/2- [512] Fermi 7/2+ [633] 7/2- [514] 1/2- [521] 3/2+ [402] 1/2- [550] 5/2+ [402] 3/2- [532] Fermi 9/2- [514] 1/2- [541] 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] N = 104

6 + N = 104 D 3 D 5 D 3 D 5 2.5 2 DD-ME2 D^3 DD-ME2 D^5 DD-PC1 D^3 DD-PC1 D^5 Expt. Energy (MeV) 1.5 1 0.5 0 172 Er 174 Yb 176 Hf 178 W 180 Os N=104 isotone 6 + N = 104

A 160 190 8 N = 106 A 160 190 8 N = 106 N = 106 8 N = 106 6 + N = 102 N = 106 8 174 182 184 186 188 8 ν 2 7/2 [514] 9/2 + [624] N = 106 8 ν9/2 + [624] ν7/2 [514] 174 ν5/2 [512] 176 182 ν7/2 + [633] ν1/2 [521]

8 8 0.3 8 8 N = 106 D 3 D 5 D 3 D 5

A 160 190 Energy (MeV) Energy (MeV) -3-4 -5-6 -7-8 -9-5 -6-7 -8-9 -10-11 7/2- [503] 1/2+ [660] 1/2+ [660] 1/2+ [660] 1/2+ [651] 3/2+ [642] 3/2- [512] 1/2- [510] 9/2+ [624] Fermi 5/2- [512] 7/2- [514] 7/2+ [633] 5/2+ [642] 1/2- [521] 3/2+ [651] 11/2- [505] 1/2+ [640] 1/2- [510] 3/2- [512] 9/2+ [624] Fermi Neutrons 5/2- [512] 7/2+ [633] 7/2- [514] 5/2+ [642] 1/2- [521] 3/2+ [651] 11/2+ [615] 7/2- [503] 9/2- [505] 1/2- [510] 3/2- [512] 9/2+ [624] Fermi 5/2- [512] 7/2+ [633] 7/2- [514] 5/2+ [642] 1/2- [521] 174Er 176Yb 178Hf 180W 7/2- [503] 1/2+ [651] 11/2+ [615] 9/2- [505] 1/2- [510] 3/2- [512] 9/2+ [624] Fermi 5/2- [512] 7/2+ [633] 7/2- [514] 1/2- [521] 5/2+ [642] 3/2+ [651] 3/2- [521] 1/2+ [640] 3/2+ [642] 7/2- [503] 11/2+ [615] 1/2+ [651] 9/2- [505] 1/2- [510] 3/2- [512] 9/2+ [624] Fermi Neutrons 5/2- [512] 7/2+ [633] 1/2- [521] 7/2- [514] 3/2+ [651] 5/2+ [642] 3/2- [521] 3/2+ [642] 7/2- [503] 11/2+ [615] 1/2+ [651] 9/2- [505] 1/2- [510] 3/2- [512] 9/2+ [624] Fermi 5/2- [512] 7/2+ [633] 1/2- [521] 7/2- [514] 5/2+ [642] 182Os 184Pt 186Hg 188Pb 1/2+ [651] 7/2- [503] 11/2+ [615] 9/2- [505] 1/2- [510] 3/2- [512] 9/2+ [624] Fermi 5/2- [512] 7/2+ [633] 3/2+ [642] 1/2+ [651] 7/2- [503] 11/2+ [615] 9/2- [505] 1/2- [510] 3/2- [512] 9/2+ [624] Fermi 5/2- [512] 7/2+ [633] 1/2- [521] 7/2- [514] N = 106

Energy (MeV) Energy (MeV) -3-4 -5-6 -7-8 -9-10 -11-12 -13 0-1 -2-3 -4-5 -6-7 -8 5/2+ [402] 3/2- [532] 1/2- [541] 9/2- [514] 7/2+ [404] 1/2+ [411] Fermi 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 1/2- [530] 5/2+ [413] 5/2- [523] 3/2+ [402] 1/2- [550] 5/2+ [402] 3/2- [532] 1/2- [541] 9/2- [514] 7/2+ [404] Fermi Protons 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 1/2- [530] 5/2+ [413] 1/2- [550] 5/2- [523] 3/2+ [402] 5/2+ [402] 3/2- [532] 1/2- [541] 9/2- [514] Fermi 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 3/2- [541] 1/2- [530] 174Er 176Yb 178Hf 180W 1/2+ [400] 11/2- [505] 1/2- [550] 5/2- [523] 3/2+ [402] 5/2+ [402] 3/2- [532] Fermi 1/2- [541] 9/2- [514] 7/2+ [404] 1/2+ [411] 7/2- [523] Protons 1/2+ [400] 11/2- [505] 1/2- [550] 5/2- [523] 3/2+ [402] 5/2+ [402] Fermi 3/2- [532] 1/2- [541] 9/2- [514] 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 1/2+ [400] 11/2- [505] 1/2- [550] 5/2- [523] 3/2+ [402] Fermi 5/2+ [402] 1/2- [541] 9/2- [514] 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 182Os 184Pt 186Hg 188Pb 5/2- [523] 3/2+ [402] 5/2+ [402] 3/2- [532] 1/2- [541] Fermi 9/2- [514] 7/2+ [404] 1/2+ [411] 7/2- [523] 3/2+ [411] 5/2- [532] 1/2- [550] 1/2+ [400] 11/2- [505] 5/2- [523] 3/2+ [402] Fermi 5/2+ [402] 3/2- [532] 1/2- [541] 9/2- [514] 7/2+ [404] 1/2+ [411]

A 250 ν7/2 [514] ν9/2 + [624] ν9/2 + [624] 8 N = 106 A 250 A 250 250 254

4 3.5 DD-ME2 D^3 DD-ME2 D^5 DD-PC1 D^3 DD-PC1 D^5 Expt. 3 Energy (MeV) 2.5 2 1.5 1 0.5 0 174Er 176Yb 178Hf 180W 182Os 184Pt 186Hg 188Pb N=106 isotone 2 1/2+[660] 7/2-[514] 1/2+[660] 7/2-[514] 7/2-[514] 1/2+[660] 7/2+[633] 7/2-[514] 1/2+[660] 7/2+[633] 9/2-[505] 7/2-[514] 1/2+[660] 7/2+[633] Energy (MeV) 1 7/2+[633] 7/2-[514] 1/2-[510] 3/2-[512] 5/2-[512] 7/2+[633] 7/2-[514] 1/2-[510] 3/2-[512] 5/2-[512] 7/2+[633] 7/2-[514] 7/2+[633] 7/2+[633] 1/2-[510] 1/2-[510] 1/2-[510] 1/2-[510] 3/2-[512] 3/2-[512] 3/2-[512] 3/2-[512] 5/2-[512] 5/2-[512] 5/2-[512] 5/2-[512] 1/2-[510] 1/2-[510] 3/2-[512] 3/2-[512] 5/2-[512] 5/2-[512] 9/2+[624] 9/2+[624] 9/2+[624] 9/2+[624] 9/2+[624] 9/2+[624] 9/2+[624] 9/2+[624] 0 174Er 176Yb 178Hf 180W 182Os 184Pt 186Hg 188Pb 8 N = 106 ν7/2 [514] ν9/2 + [624]

A 250 N = 162 K π K π 5/2 + [622]7/2 [743] 6 3/2 [521]7/2 [514] 5 + 5/2 + [622]1/2 + [631] 3 + 3/2 [521]7/2 + [633] 5 5/2 + [622]9/2 [734] 7 7/2 + [633]7/2 [514] 7 5/2 + [622]9/2 + [615] 7 + 3/2 [521]7/2 + [633] 5 5/2 + [622] + 9/2 + [615] 7 + 7/2 + [633]7/2 [514] 7 9/2 [734]9/2 + [615] 9 3/2 [521]7/2 + [633] 5

K π K π 3/2 + [622]9/2 + [615] 6 + 7/2 + [633]7/2 [514] 7 1/2 + [620]9/2 + [615] 5 + 3/2 [521]7/2 + [633] 5 1/2 + [620]3/2 + [622] 4 + 7/2 + [633]7/2 [514] 7 1/2 + [620]9/2 + [615] 5 + 3/2 [521]7/2 + [633] 5 1/2 + [631]7/2 [743] 4 1/2 [521]3/2 [521] 2 + 7/2 + [624]7/2 [743] 7 1/2 [521]7/2 + [633] 4 5/2 + [622]7/2 [743] 6 1/2 [521]3/2 [521] 2 + 5/2 + [622]9/2 [734] 7 1/2 [521]7/2 + [633] 4 5/2 + [622]9/2 [734] 7 1/2 [521]3/2 [521] 2 + 9/2 + [615]9/2 [734] 9 1/2 [521]7/2 + [633] 4 9/2 + [615]9/2 [734] 9 1/2 [521]3/2 [521] 2 + 5/2 + [622]9/2 [734] 7 1/2 [521]7/2 + [633] 4 3/2 + [622]9/2 + [615] 6 + 1/2 [521]7/2 + [633] 4 9/2 + [615]9/2 [734] 9 1/2 [521]3/2 [521] 2 + 1/2 + [631]7/2 [743] 4 1/2 [521]9/2 [505] 5 + 7/2 + [624]7/2 [743] 7 1/2 [521]9/2 + [624] 5 5/2 + [622]7/2 [743] 6 1/2 [521]9/2 + [624] 5 7/2 [743]9/2 [734] 8 + 1/2 [521]9/2 [505] 5 + 5/2 + [622]9/2 [734] 7 1/2 [521]9/2 + [624] 5 5/2 + [622]9/2 + [615] 7 + 1/2 [521]9/2 [505] 5 + 9/2 + [615]9/2 [734] 9 1/2 [521]9/2 + [624] 5 3/2 + [622]9/2 [734] 6

K π K π 3/2 + [622]9/2 [734] 6 1/2 [521]9/2 + [624] 5 1/2 + [620]9/2 [734] 5 1/2 [521]9/2 [505] 5 + 9/2 + [615]9/2 [734] 9 9/2 [505]9/2 + [624] 9 3/2 + [622]9/2 + [615] 6 + 5/2 [512]9/2 + [624] 7 3/2 + [622]9/2 + [615] 6 + n9/2 + 11/2 [651] 10 1/2 + [620]9/2 + [615] 5 + 9/2 [505]9/2 + [624] 9 5/2 [512]9/2 + [624] 7 1/2 + [620]3/2 + [622] 2 + 5/2 [512]9/2 + [624] 7 1/2 + [620]9/2 + [615] 5 + 9/2 [505]9/2 + [624] 9 3/2 + [622]9/2 + [615] 6 + 1/2 + [620]11/2 [725] 6 5/2 [512]9/2 + [624] 7 3/2 + [622]11/2 [725] 7 9/2 [505]9/2 + [624] 9 9/2 + [615]11/2 [725] 10 7/2 + [613]11/2 [725] 9 5/2 [512]9/2 + [624] 7 1/2 + [620]11/2 [725] 6 9/2 [505]9/2 + [624] 9 Ω

A 160 190

10.1007/3-540-37072-2_1 http://dx.doi.org/10.1007/3-540-37072-2_1

http : / / www. springer. com / gp / book / 9783540212065 D1 π π I = 1 J = 1

T = 0 S π + ψ B Kπ + ψ J/ψp 0 b J/ψK p

ρ

10.1007/978-1-4613-9907-0_2 http://dx.doi.org/ 10.1007/978-1-4613-9907-0_2

nucl-th/0301072 11

76 A 190

ρ

Z = 120

http://dx.doi.org/10.1038/ nphys3916http://10.1038/nphys3916 34 2p 1f 40, 36 34 N = 20 40

j K N = 162 http: //dx.doi.org/10.1038/19911

http://www.nndc.bnl.gov/ensdf/