c l ¤-b c l ¤ { ¿ ?; c" e m ì Å m ¹ ÅV R Ë ú n Þ £ ; V ê s? 08 ý Yrast ô p § Ö ¨ ; c .U ; c 6 X ¢

(1)

b

c l ¤-b c l ¤ { ¿ ?; c" e m ì Å m ¹ ÅV R Ë ú n Þ £ ; V ê s? 08 ý Yrast ô p § Ö ¨ ; c .U ; c 6 X ¢

ä Ã Å 8 ýÇ X Ø ß O Ë] M ö «Å k Ä

»¨ £* å · ò 6 B £ · * × < . > r )

@ / < Æ § Ó ü t o < Æõ , ; 402-751

(2010¸ 1 Z 4 13{ 9 ~ Ã Î6 £ §, þ j7 á x Ã º& ñ : r 2010¸ 2 Z 4 5{ 9 ~ Ã Î6 £ §)

s

7 Hë H \ " f H Û ¼ 2 ; _ > r â + « > / B Nd _ Ä »´ ò$ í ` ¦ 7 £ x l 0 A # Ã º- Ã º Ù þ \ " f ì ø Í ì ø Í $ í

× æF G © I _ Yrast [ þ t> p u \ -t \ s / B Nd ` ¦ & h 6 x % i . ì ø Í $ í × æF G © I \ & h 6 x K : r ( Û ¼

2 ;\ _ > r t · ú § H) õ ü < q § # ^ ¦ M :, ì ø Í ì ø Í $ í × æF G s \ H \ -t Ø æì r y Z O # Q 4

R e t 3 l w Ù ¼ Ð, Û ¼ 2 ; _ > r â + « > / B Nd s Û ¼ 2 ;\ _ > r t · ú § H â + « >/ B Nd \ q K 8 Ä »o ¦ ½ + É Ã

º \ O . t ë ß s Qô Ç ½ ¨\ ¦ : x # ì ø Í ì ø Í $ í × æF G © I _ Yrast [ þ t> p u \ -t # Qb G>

×

æF G \ _ > r H t · ú Ã º e . ¢ ¸ô Ç y y _ × æF G © I \ ´ ú ð r Û ¼ 2 ;\ _ > r t · ú § H õ ü < q § K

^ ¦ M :, Ã º × æF G © I ¢ ¸ H f . Ë Ã º × æF G © I ^ \ ´ ú ð r Û ¼ 2 ; _ > r õ H s ` & h É r > hÃ º_

\ ¦ s 6 x % i H 8 £ ¤ \ " f Õ ª ´ ò6 x$ í ` ¦ ° ú H .

Ùþ

d#Q: Yrast \-t â+«>/BNd, Û¼2;\ _>r Hâ+«>/BNd, Ãº-ÃºÙþ_ ìøÍìøÍ$í ×æFG©I_ Yrast [

þ

t>pu\-t, ÙþÃº

Spin-dependent Empirical Formula for the Yrast Excitation Energies of the Unnatural Parity States in Even-even Nuclei

Dooyoung Kim · Dongwoo Cha · Jin-Hee Yoon

^∗

Department of Physics, Inha University, Inchon 402-751 (Dated: Received 13 January 2010, in final form 5 February 2010)

We test the validity of the spin-dependent empirical formula for the Yrast excitation energies of the unnatural parity states in even-even nuclei. Compared to those for the natural parity states, the energies are not spaced enough to be differentiated by using the spin-dependent empirical formula.

Therefore, we do not find the spin-dependent empirical formula to be more advantageous than the spin-independent one. However, a trial to fit all even or odd multipole state Yrast excitation energies yields information on the spin dependence of the Yrast excitation energies, and the fit for all energies is as good as a separate fit for each multipole state energy.

PACS numbers: 21.10.Re, 23.20.Lv

Keywords: Empirical formula, Spin-dependent empirical formula, Yrast unnatural parity state excitation energy, Valence nucleon numbers

∗E-mail: [email protected]

-113-

(2)

I. " e Â ] Ø

Ù þ

_ $ í | 9 ` ¦ ½ ©" î H X < # Q Ù þ \ " f Õ ª $ í | 9 s # Qb G

>

ì r í÷ & H t ¶ ú ( R Ð H כ É r _ p e H { 9 s . % i & h Ü

¼ Ð Ù þ _ ì ø Í ¦ & h | 9 | ¾ Ó/ B Nd É r Õ ª @ /³ ð& h \ V ½ + É Ã º e

[1]. s Qô Ç ~ ½ ÓZ O É r Ù þ _ : £ ¤& ñ ô Ç $ í | 9 s Ù þ _ r & h

t ³ ð(| 9 | ¾ ÓÃ º ~ ª $ í Ã º ¢ ¸ H × æ$ í Ã º 1 p x) \ # Q b

G> ' a s ÷ & H t \ @ /ô Ç & ñ Ð\ ¦ · ú 9º ¡ § Ü ¼ Ð+ Õ ª $ í

| 9

_ H" é ¶` ¦ Æ Ò& ñ ½ + É Ã º e H ~ ½ ÓZ O ` ¦ ] j/ B N ô Ç . þ j H : r

½ ¨Õ ªÒ ¨ \ " f H Ã º- Ã º Ù þ \ " f_ ì ø Í $ í × æF G

© I (natural parity multipole state) Ð_ Yrast [ þ t> p u \

-t \ ¦ | 9 | ¾ ÓÃ º, ª $ í Ã º(valence proton number),

×

æ$ í Ã º(valence neutron number)_ < ÊÃ º Ð ³ ð & ³ #

â

+ « >/ B Nd ` ¦ ¸Ø ¦ % i [2]. Õ ª + ' e ± ² ú É r 7 Hë H` ¦ : x

#

s / B Nd s ì ø Í $ í Ã º × æF G © I ÷ r m

ì ø Í $ í f . Ë Ã º × æF G © I _ Yrast [ þ t> p u \ -t \ ¸ ¸ ú

&

h

6 x| ¨ c Ã º e 6 £ §` ¦ Ð% i Ü ¼ 9 [3,4], ¢ ¸ô Ç # Q × æF G © I

\

/ B N: x& h Ü ¼ Ð & h 6 x ½ + É Ã º e H 1 l x{ 9 ô Ç ° ú כ` ¦ ¹ 1 Ôl 0 A

# Û ¼ 2 ;\ _ > r H Ã º& ñ ) a â + « > / B Nd ` ¦ ] jî ß ¦, s

\

¦ Ã º × æF G õ f . Ë Ã º × æF G y y \ & h 6 x ô Ç õ \ ¦ ] j r

% i [5]. s ô Ç ½ ¨\ ¦ : x # Ã º- Ã º Ù þ \ " f_

ì ø Í $ í × æF G © I \ " f Yrast [ þ t> p u \ -t | 9 | ¾ Ó Ã

º, ª $ í Ã º, × æ$ í Ã ºë ß Ü ¼ Ð & h ] X y [ O " î ½ + É Ã º e

H ª e ` ¦ µ 1 ß) ? /% 3 Ü ¼ 9, s Qô Ç Ó ü t o | ¾ Ó[ þ t _ % i ½ + É\ @ / K

" f ¸ 7 H _ % i [6].

þ

j H \ H s Qô Ç â + « > / B Nd ` ¦ ì ø Í ì ø Í $ í × æF G

© I (unnatural parity multipole state)\ " f Yrast [ þ t> p u

\

-t \ ¸ & h 6 x ½ + É Ã º e H t \ @ /ô Ç ½ ¨ ' K &

[7]. ì ø Í ì ø Í $ í × æF G © I Ð_ s H l $ í

s (electric transition) l $ í s (magnetic transition) \ _ K & ñ ÷ & 9, " f Õ ª ì r í ª

#

# Q " { 9 ' a$ í ` ¦ l @ / l # Q§ > . Õ ª Q s ½ ¨\

Ø Ô ì ø Í ì ø Í $ í × æF G © I \ " f 8 £ ¤& ñ ) a Yrast [ þ t

>

p

u \ -t H © { © y Ô ¦ ½ ©g Ë :ô Ç ¸_ þ v` ¦ Ðe \ ¸ Ô ¦ ½ ¨

¦, â + « > / B Nd s Õ ª ' 1 l x` ¦ & h ] X > ¸ ú [ O " î ¦ e .

ì

ø Í ì ø Í $ í × æF G © I \ " f Yrast [ þ t> p u \ -t H

ì ø Í $ í Ã º × æF G © I \ q K × æF G ç ß \ -t

H s \ O s q 5 p w ô Ç ß ¼l Ð ] t 9e . s H ì ø Í

$ í f . Ë Ã º × æF G © I ü < ¸ q 5 p w . " f ì ø Í

$ í

f . Ë Ã º × æF G © I \ " f s p ¸ ú & h 6 x ) a â + « > / B Nd É r

#

QÖ ¼ & ñ ¸ ì ø Í ì ø Í $ í © I \ ¸ ¸ ú & h 6 x ÷ &o l @ /

÷

&% 3 . Û ¼ 2 ; _ > r â + « > / B Nd É r # Qb ? s 7 Hë H \ " f H Ô

¦ ½ ©g Ë :ô Ç ' 1 l x` ¦ Ðs H ì ø Í ì ø Í $ í © I \ ¸ × æF G

\

© ' a\ O s Û ¼ 2 ;\ _ > r H 1 l x{ 9 ô Ç ° ú כÜ ¼ Ð [ O " î s

0 p x ½ + É כ t , 7 £ ¤ Û ¼ 2 ; _ > r â + « > / B Nd ` ¦ & h 6 x # s

[ þ t Ó ü t o | ¾ Ó_ ' 1 l x` ¦ & h ] X y l Õ ü t ½ + É Ã º e H t \ ¦ · ú

Ð 9 ô Ç .

1 10

0 50 100 150 200 250

1 10

(a) Even multipoles

2^- 4^- 6^- 8^- 10^- (b) Odd multipoles

Mass number A Excitation Energy Ex (MeV)

1⁺ 3⁺ 5⁺ 7⁺ 9⁺

Fig. 1. Measured excitation energies of the lowest unnat- ural parity states in even-even nuclei. The upper part is for the even multipole states and the lower part is for the odd multipole states. The experimental values are form Ref. [8]. Open circles(dark yellow in color version) are 2

⁻

or 1

⁺

, solid circles(blue in color) are 4

⁻

or 3

⁺

, open triangles(green in color) are 6

⁻

or 5

⁺

, solid triangles(red in color) are 8

⁻

or 7

⁺

, open squares(black in color) are 10

⁻

or 9

⁺

for even or odd multipole unnatural parity states.

II. ä Ã Å 8 ýÇ X Ø ß O Ë] M ö «Å k ÄÊ Ý § X ì Ä Ó Þ

s

] j t 8 £ ¤& ñ ) a ¸ H Ã º- Ã º Ù þ \ " f, ì ø Í ì ø Í

$ í

× æF G © I _ Yrast [ þ t> p u \ -t Fig. 1\ ü <e

. s X <s ' H Ref. [8] \ " f : r כ s . Ã º(2

⁻

∼ 10

⁻

) ü < f . Ë Ã º(1

⁺

∼ 9

⁺

) _ × æF G © I \ @ /K y y 0 Aü <

A \ Õ ª 94 R e Ü ¼ 9, y y _ × æF G © I H É r l

ñ( : r ! Q \ H Ò oõ l ñ) Ð ³ ð & ³÷ &# Q e . Ã º

× æF G _ â Ä º 2

⁻

H " é ¶ Ü ¼ Ð, 4

⁻

H 5 Å q s ð ø Í " é ¶ Ü ¼ Ð, 6

⁻

H 5 Å q s [ j ¸ Ð, 8

⁻

É r 5 Å q s ð ø Í [ j ¸ Ð, 10

⁻

É r 5 Å q s

W 1 ¸ Ð Õ ª§ 4 Ü ¼ 9, f . Ë Ã º × æF G _ â Ä º 1

⁺

H " é ¶ Ü ¼

Ð, 3

⁺

H 5 Å q s ð ø Í " é ¶ Ü ¼ Ð, 5

⁺

H 5 Å q s [ j ¸ Ð, 7

⁺

É r 5 Å q s

ð ø Í [ j ¸ Ð, 9

⁺

É r 5 Å q s W 1 ¸ Ð Õ ª§ 4 . ^ & h Ü ¼ Ð

Ð ¤` ¦ M : { 9 & ñ ô Ç ½ ©g Ë :$ í ` ¦ Ðs o l @ / l H j Ë µ H ì

r í\ ¦ Ð .

(3)

s

X <s \ & h 6 x K : r â + « > / B Nd É r 6 £ § õ ° ú .

E

x

= αA

^−γ

+ β

p

e

^−λ^p^N^p

+ β

n

e

^−λⁿ^Nⁿ

. (1)

#

l " f A H | 9 | ¾ ÓÃ ºs 9, N

p

( ¢ ¸ H N

n

) É r ª $ í Ã

º(¢ ¸ H × æ$ í Ã º) Ð" f þ jü @ F Y ` \ > r F H ª $ í Ã

º(¢ ¸ H × æ$ í Ã º) ª $ í ½ ¨" í (¢ ¸ H × æ$ í ½ ¨" í )_ Ì Ã

º × æ É r ° ú כ` ¦ × þ ô Ç . α, γ, β

p

, β

_n

, λ

_p

, λ

_n

É r x h AÜ ¼ Ð

& ñ ÷ & H [ þ t s . s â + « > / B Nd \ " f H s X O > ¸¿ º 6 > h_ \ ¦ 6 x H X <, s ° ú כ[ þ t É r y y _ × æF G © I

\ @ /K x h A÷ &# Q [ jà Ô Ð & ñ ) a . 7 £ ¤, y y _ × æ F

G © I H ¸¿ º 6> h_ ° ú כ` ¦ ° ú H . s õ Ref.

[7] \ ü <e Ü ¼ 9, s \ ¦ Table 1 \ 6 x % i . s ³ ð\ ¦

Ð y y _ × æF G © I \ @ /K ¸¿ º 6> h_ ° ú כõ Õ

ª\ É r χ

²

° ú כõ y × æF G © I \ > r F H X <s Ì Ã

º(N

⁰

) ü <e . X <s H þ j è_ χ

²

° ú כ` ¦ ë ß 7 á ¤ ¸ 2

¤ x h A % i H X <, # l " f 6 x ô Ç χ

²

° ú כ É r A ü < ° ú .

χ

²

= 1 N

0

N0

X

i=1

log E

^cal_x

(i) − log [E

_x^exp

(i)]

2

. (2)

Ð: x _ χ

²

° ú כ @ / X <s \ log ° ú כ` ¦ 2 [ô Ç s Ä » H ° ú É r

× æF G ? /\ " f ¸ \ -t _ s Ã ºz \ " f Ã ºÑ þ C \ s Ø

Ôl M :ë H s . s â Ä º log ° ú כ` ¦ 2 [ t · ú §Ü ¼ H \ -t

% ò

% i ` ¦ 0 AÅ Ò Ð x h Aô Ç ! l rs ÷ &# Q É r \ -t % ò % i \ " f

H ] j@ / Ð ) a õ \ ¦ Å Òt 3 l w ô Ç . s X O > x h Aô Ç õ Ð

¹

1 Ô · p [ þ t` ¦ s 6 x # > í ß ) a ° ú כs Fig. 2(b)ü < Fig.

3(b) \ Õ ª 94 R e Ü ¼ 9, @ /^ & h Ü ¼ Ð ^ & h ì r í\ ¦ ¸ ú [ O

" î ¦ e . ³ ð_ t } × ¦ \ ü <e H “All” _ _ p

H f . Ë Ã º ¢ ¸ H Ã º × æF G _ ¸ H X <s \ @ /ô Ç χ

²

° ú כ

`

¦ _ p ô Ç . 7 £ ¤, y X <s ' _ ÐÕ ª _ ] jY L` ¦ 8 ú x ½ + Ëô Ç Ê

ê, ^ X <s Ì Ã º Ð è H ° ú כs .

s

] j s X O > y y _ × æF G \ É r ° ú כ` ¦ ° ú

H @ / Ã º f . Ë Ã º × æF G ^ \ / B N: x Ü ¼ Ð & h 6 x ½ + É Ã º e

H ° ú כ` ¦ ¹ 1 Ô Ð . s [ þ t É r × æF G \ _ > r K

9, s \ ¦ Û ¼ 2 ; _ > r â + « >/ B Nd s ô Ç . Û ¼ 2 ; _ > r

â

+ « >/ B Nd \ " f H d (1)\ " f α @ / \ α

0

J

^a

\ ¦, γ @ / \ γ

0

J

^c

\ ¦ 6 x ô Ç . ¢ ¸ô Ç λ

_p(n)

@ / \ λ

⁰_p(n)

/ √

J \ ¦ 6 x ô Ç

. 7 £ ¤,

E

x

= α

0

J

^a

A

^−γ⁰^J^c

+ β

p

e

^−λ⁰^p^N^p^/

√

J

+ β

n

e

^−λ⁰ⁿ^Nⁿ^/

√ J

(3) s

. ¢ ¸ô Ç s x h A\ " f 6 x ÷ & H χ

²

° ú כ É r 0 A_ & ñ _ d õ 1

l x{ 9 ì r ¸_ N

0

H ^ X <s ' _ Ì Ã ºs . s õ

Table 2\ ü <e .

Table 1 õ 2\ ¦ q § # Ð , { 9 é ß Õ ª Ä »´ ò$ í ` ¦ 7

£

x ½ + É Ã º e H χ

²

° ú כs Ã º × æF G _ â Ä º 0.027\ " f

1 10

0 50 100 150 200 250

1 10 Excitation Energy Ex (MeV)

Mass Number A

(c) Spin - Depend.

(b) Spin - Indep.

(a) Data

2^- 4^- 6^- 8^- 10^-

Fig. 2. Yrast excitation energies of the lowest even un- natural parity states in even-even nuclei. The upper part (a) is the experimental data, same figure as the part (a) of Fig. 1. The middle part (b) is the fit with the spin- independent empirical formula, and the lowest part (c) is the fit with the spin-dependent empirical formula. Sym- bol representations are same as those of Fig. 1.

0.035 Ð, f . Ë Ã º × æF G _ â Ä º 0.042\ " f 0.074 Ð ¸F Km 7

£

x % i . Õ ª Q # y 0.1 Ð ¸ É r Ã º Ð" f X <s

\ ¦ & h ] X y ´ ú Æ Ò ¦ e ¦ ½ + É Ã º e . s \ ¦ Õ ªa Ë >Ü ¼ Ð

· p כ s Fig. 2(c)ü < Fig. 3(c)s . Fig. 2ü < 3 É r Table 1 õ Table 2\ ü <e H ° ú כ[ þ t` ¦ s 6 x # y y _ Ù þ

\ @ /K > í ß ô Ç õ s . ] j{ 9 0 A\ H z ´+ « > X <s s 9, î r X < Û ¼ 2 ;\ _ > r t · ú § H â + « >/ B Nd Ü ¼ Ð x h A ô

Ç õ s 9, ] j{ 9 A Û ¼ 2 ; _ > r â + « >/ B Nd ` ¦ 6 x ô Ç

õ s . ¢ ¸ô Ç Fig. 2 H Ã º ì ø Í ì ø Í $ í © I \ @ / ô

Ç õ s 9, Fig. 3 É r f . Ë Ã º ì ø Í ì ø Í $ í © I \ @ /ô Ç

õ s . s ¿ º Õ ªa Ë >_ î r X <ü < A ` ¦ q §K Ð

^ & h Ü ¼ Ð y y _ × æF G © I _ 0 A\ " fÂ Ò' A t _

í H " f t · ú § ¦ ¸ ú l Õ ü t ÷ & ¦ e 6 £ §` ¦ · ú Ã º e .

é

ß t 2

⁻

ü < 4

⁻

, 1

⁺

ü < 3

⁺

+ ' ÷ ¶ 1 p w Ðs s ¿ º X <s

H _ 5 ge # Q ½ ¨ì r s " î S X t · ú § . ^ & h

¸ ª ` ¦ ^ ¦ M :, Û ¼ 2 ; _ > r â + « > / B Nd s © @ /& h Ü ¼ Ð y

×

æF G © I [ þ t s \ \ -t ° ú כ_ Z O # Qf ` ¦ Ø æì r y l Õ ü t

(4)

Table 1. Values of six parameters in Eq.(1) for the Yrast excitation energies E

_x

of the even and odd multipole unnatural parity states. We quoted these parameter values from Ref. [7]. The last rows in two pannels show the overall χ

²

value and total number of data.

J

₁^π

α γ β

p

(β

n

) λ

p

(λ

n

) χ

²

N

0

(MeV) (MeV)

2

⁻₁

48.27 0.73 1.09(1.59) 0.19(0.31) 0.058 246

4

⁻₁

75.04 0.81 1.00(1.27) 0.17(0.24) 0.027 253

6

⁻₁

107.98 0.83 0.77(1.40) 0.19(0.28) 0.019 248

8

⁻₁

277.43 1.00 0.90(1.50) 0.15(0.20) 0.017 230

10

⁻₁

238.48 0.90 1.24(1.76) 0.44(0.25) 0.012 199

All 0.027 1176

1

⁺₁

47.13 0.67 0.54(0.99) 0.76(0.50) 0.079 251

3

⁺₁

49.45 0.76 1.17(1.49) 0.58(0.32) 0.051 236

5

⁺₁

87.00 0.83 1.05(1.25) 0.40(0.24) 0.028 250

7

⁺₁

139.02 0.88 1.19(1.48) 0.28(0.24) 0.021 184

9

⁺₁

172.81 0.86 1.09(1.61) 0.34(0.46) 0.019 159

All 0.042 1080

Table 2. Values of 8 parameters in Eq.(3) for even and odd multipole states.

J α

0

a γ

0

c β

p

(β

n

) λ

p

(λ

n

) χ

²

N

0

(MeV) (MeV)

Even 38.48 0.558 0.719 0.07 0.909(1.529) 0.464(0.535) 0.035 1176 Odd 53.65 0.156 0.738 0.018 1.055(1.47) 0.868(0.556) 0.074 1197

Ç õ \ " f 7 á § 8 ¿ º× ¼ Q4 R Ð .

s

õ \ ¦ 7 á § 8 [ jy ¶ ú ( R Ðl 0 A # y y _

° ú

כ` ¦ Fig. 4 \ Ð Õ ª 9 Ð ¤ . # l " f 5 Å q s l ñ H Û

¼ 2 ;\ _ > r t · ú § H / B Nd \ & h 6 x ô Ç ° ú כs ¦, 5 Å q s ð

ø Í l ñ H Õ ª\ @ /6 £ x ÷ & H Û ¼ 2 ; _ > r / B Nd ` ¦ & h 6 x ô Ç

° ú

כs . ¢ ¸ô Ç, " é ¶ l ñ H Ã º × æF G \ @ /ô Ç ° ú כs 9, W 1 ¸ l ñ H f . Ë Ã º × æF G \ @ /ô Ç ° ú כs . s Õ ª a Ë

>` ¦ Ð @ /^ & h Ü ¼ Ð α\ ¦ ] jü @ô Ç [ þ t É r Û ¼ 2 ; _ > r /

B Nd \ " f _ > r t · ú § H / B Nd \ " f Z > s \ O .

Õ

ª Q α H & h É r J ° ú כ\ " f H ¸ ú { 9 u J° ú כs & f \

s ´ ú §s Z O # Q . : £ ¤ y Ã º × æF G _ â Ä º 8

⁻

10

⁻

_ â Ä º\ Õ ª Q 9, f . Ë Ã º × æF G _ â Ä º 7

⁺

õ 9

⁺

_ â Ä º Õ ª Q . W 1 â Ä º ¸¿ º Û ¼ 2 ; _ > r / B Nd \

"

f_ α° ú כs Û ¼ 2 ;\ _ > r t · ú § H â Ä º_ α° ú כ Ð @ /^

&

h Ü ¼ Ð . α H \ -t ° ú כ` ¦ ^ & h Ü ¼ Ð 0 A A Ð s 1

l

x r v H % i ½ + É` ¦ ô Ç . 7 £ ¤, Õ ª ß ¼l _ ] X @ /° ú כ` ¦ ý at Ä º t

9, Qt [ þ t É r þ jü @y , | 9 ? /\ " f_ o\ ¦ l

Õ ü t ô Ç . " f α° ú כs J_ o\ 3 l w p u × æF G

s Ø æì r y Z O # Qt t 3 l w > ÷ & ¦, s H Ð Õ ªa Ë >\

"

f Ðs 1 p w ^ X <s _ 0 A A ; ¤ s × ¦ # Q× ¼ H ¸_ þ v Ü ¼

Ð è ß . Ref. [9]_ õ ü < q §K Ð , ì ø Í $ í _

2

ü < E

¹⁰

_ \ - t

10C \ " f ß ¼> H 100 C t s H X < q K ì

ø Í ì ø Í $ í _ â Ä º\ H Õ ªë ß p u ß ¼> s t · ú §

H . α_ ° ú כ` ¦ ¶ ú ( R Ð Õ ª ´ òõ \ ¦ f ] X & h Ü ¼ Ð Ö ¼ ´ Ã º e

Õ

ª Q Ã º ì ø Í ì ø Í $ í _ â Ä º Õ ª ° ú כ É r 48 \ " f r K

" f 240 t Ð Ô ¦ õ 5C & ñ ¸\ Ô ¦ õ . " f Û ¼ 2 ; _

> r ¸ H _ 0\ ¾ ú ¦ s ° ú כs 0.5 & ñ ¸_ a° ú כ\ ì ø Í

% ò

÷ &# Q e . 7 £ ¤, J

^a

∼ √

J s Ù ¼ Ð & K ® o √

10 ∼ 3 C & ñ

¸_ ´ òõ µ 1 Ú\ Å Òl 3 l w l M :ë H s . s H f . Ë Ã º × æF G

\

" f ¸ Õ ª@ / Ð H X <, 0.156_ a° ú כs _ p H J _ _ > r ´ òõ _ \ O .

III. + s Ç Â ] Ø

s

þ

t> p u \ -t \ ¦ Û ¼ 2 ; _ > r â + « >/ B Nd \ H # Ã º

×

(5)

1 10

0 50 100 150 200 250

1 10

(c) Spin - Depend.

(b) Spin - Indep.

(a) Data

Excitation Energy Ex (MeV)

Mass Number A

2⁺ 4⁺ 6⁺ 8⁺ 10⁺

Fig. 3. Yrast excitation energies of the lowest even un- natural parity states in even-even nuclei. (a)Measured Yrast excitation energies, (b) the fit with the spin- independent empirical formula, and (c) the fit with spin- dependent formula. Symbol representations are same as those of Fig. 1.

q

K , \ -t ç ß s ] t 9e H â ¾ Ó` ¦ Ð . Û ¼ 2 ; _

> r â + « > / B Nd É r J

^a

_ _ > r ¸ Ð s Z O # Qf ` ¦ l Õ ü t H X <,

%

3 # Qt H t Ã º° ú כ(a)s Ã ºü < f . Ë Ã º × æF G _ â Ä º y y 0.5558 õ 0.156Ü ¼ Ð" f Õ ª ° ú כs B Ä º . s H ì ø Í

$ í

< f . Ë Ã º × æF G _ â Ä º y y 2.26õ 1.05\ q K É r Ã º u

Ð" f, ì ø Í $ í \ q K Û ¼ 2 ;_ > r ¸ _ \ O 6 £ §` ¦ _

þ

t> p u \ -t \ ¦ l Õ ü t H X < H y y _ × æF G \ & h 6 x

H â + « > / B Nd s Û ¼ 2 ; _ > r â + « >/ B Nd Ð \ -t _ Z O # Q f

_ ' a > \ ¦ ¶ ú ( R Ð H ô Ç t ~ ½ ÓZ O Ü ¼ Ð" f s r ¸ H _ p

e ¦ ½ + É Ã º e Ü ¼ 9, s ÐÂ Ò' ì ø Í ì ø Í $ í × æ F

É

r Õ ª _ p ß ¼t 3 l w ¦ : r t ` ¦ Ã º e .

50 100 150 200 250 300

1 2

0 2 4 6 8 10

0.2 0.4 0.6 0.8

0 2 4 6 8 10

0.2 0.4 0.6 0.8

even even J odd odd J

Multipolarity J

λ

_n

β

_n

α γ

β

_p

λ

_p

Fig. 4. Six parameters(open symbols) in Eq. (1) and the corresponding parameters(filled symbols) in Eq. (3).

Circles(blue in color version) are for even multipoles and squares(red in color version) are for odd multipoles.

P

c p 8 ý ò k >

s

½ ¨ H @ / < Æ §_ t " é ¶ Ü ¼ Ð Ã º' ÷ &% 3 _ þ v m .

Y

c p w Ã U Ø ô

[1] C. F. Weizsacker, Z. Physik, 96, 431 (1935); H. A.

Bethe and R. F. Bacher, Rev. Mod. Phys., 8 193, (1936); E. Feenberg, Rev. Mod. Phys., 19 239, (1947).

[2] E. Ha and D. Cha, Phys. Rev. C75, 057304 (2007).

[3] J.-H. Yoon, E. Ha and D. Cha, J. Phys. G:Nucl. Part.

Phys. 34, 2545 (2007).

[4] G. Jin, J.-H. Yoon and D. Cha, Nucl. Phys. A 812, 58 (2008).

[5] G. Jin, D. Cha and J.-H. Yoon, J. Kor. Phys. Soc.

53, 3483 (2008).

[6] G. Jin and J.-H. Yoon, Sae Mulli 56, 496 (2008).

(6)

c l ¤-b c l ¤ { ¿ ?; c" e   m  ì Å  m ¹ ÅV R Ë ú n Þ £ ; V ê s? 08 ý Yrast ô p § Ö ¨ ; c .U ; c 6  X ¢

b