5. N uc l ea r Struc ture
5.1 Charac te ris tic s of the n uc le ar forc e
5.2 The De ute ron
5 .2 .1 R ed u c ed H a milto n ia n in th e c en t er -o f-ma s s fr a me
5 .2 .3 Deu ter o n exc it ed s ta te
5 .2 .4 S p in d ep en d en c e o f n u c lea r fo r c e
5 .3 .1 S h ell s tr u c tu r e
5 .3 .2 N u c leo n s H a milto n ia n
5 .3 .3 S p in o r b it in ter a c tio n
5 .3 .4 S p in p a ir in g a n d v a len c e n u c leo n s
5. 1 Cha rac teris tic s of the nuc lea r fo rc e
In th is p ar t of th e cou r s e w e w an t to s tu d y th e s tr u ctu r e of n u clei. Th is in tu r n s will giv e u s in s igh t on th e en er gies an d for ces th at b ou n d n u clei togeth er an d th u s of th e p h en omen a (th at w e’ll s tu d y later on ) th at can b r eak th em ap ar t or cr eate th em.
In or d er to s tu d y th e n u clear s tr u ctu r e w e n eed to k n o w th e con s titu en ts of n u clei (th e n u cleon s , th at is , p r oton s an d n eu tr on s ) an d tr eat th em as Q M ob jects . F r om th e p oin t of v iew of Q M as w e s tu d ied u n til n o w, w e w an t fi r s t to k n o w wh at is th e s tate of th e s y s tem (at eq u ilib r iu m). Th u s w e w an t to s olv e th e time-in d ep en d en t S c h r ¨ od in ger eq u ation . Th is will giv e u s th e en er gy lev els of th e n u clei.
Th e ex act n atu r e of th e for ces th at k eep togeth er th e n u cleu s con s titu en ts ar e th e s tu d y of q u an tu m c h r omo d y n amics , th at d es cr ib es an d lo ok for th e s ou r ce of th e s tr on g in ter action , on e of th e fou r fu n d amen tal in ter action s , alon g with gr a v itation , th e electr omagn etic for ce an d th e w eak in ter action . Th is th eor y is w ell-b ey on d th is cou r s e. Her e w e w an t on ly to p oin t ou t s ome of th e p r op er ties of th e n u cleon -n u cleon in ter action :
– A t s h or t d is tan ces is s tr on ger th an th e Cou lom b for ce: w e k n o w th at n u clei comp r is e tigh tly p ac k ed p r oton s , th u s to k eep th es e p r oton s togeth er th e n u clear for ce h as to b eat th e Cou lom b r ep u ls ion .
– Th e n u clear for ce is s h or t r an ge. Th is is s u p p or ted b y th e fact th at in ter action s amon g e.g. t w o n u clei in a molecu le ar e on ly d ictated b y th e Cou lom b for ce an d n o lon ger b y th e n u clear for ce.
– Not all th e p ar ticles ar e s u b jected to th e n u clear for ce (a n otab le ex cep tion ar e electr on s )
– Th e n u clear for ce d o es n ot d ep en d at all on th e p ar ticle c h ar ge, e.g. it is th e s ame for p r oton s an d n eu tr on s .
– Th e n u clear for ce d o es d ep en d on s p in , as w e will p r o v e in th e cas e of th e d eu ter on .
– Ex p er imen ts can r ev eal oth er p r op er ties , s u c h as th e fact th at th er e is a r ep u ls iv e ter m at v er y s h or t d is tan ces an d th at th er e is a comp on en t th at is an gu lar -d ep en d en t (th e for ce is th en n ot c entr al an d an gu lar momen tu m is n ot con s er v ed , alth ou gh w e can n eglect th is to a fi r s t ap p r o x imation ).
W e will fi r s t s ee h o w th es e c h ar acter is tics ar e r efl ected in to th e Hamilton ian of th e s imp les t (n on -tr iv ial) n u cleu s , th e d eu ter on . Th is is th e on ly n u cleu s th at w e can attemp t to s olv e an aly tically b y for min g a fu ll mo d el of th e in ter action b et w een t w o n u cleon s . Comp ar in g th e mo d el p r ed iction with ex p er imen tal r es u lts , w e can v er ify if th e c h ar acter is tics of th e n u clear for ce w e d es cr ib ed ar e cor r ect. W e will th en later s tu d y h o w th e n u clear for ce p r op er ties s h ap e th e n atu r e an d comp os ition of s tab le an d u n s tab le n u clei.
69
5. 2 The Deuteron
5.2.1 R educ ed Hami l toni an i n the c enter- of- mas s frame
W e s tar t with th e s imp les t p r ob lem, a n u cleu s for med b y ju s t on e n eu tr on an d on e p r oton : th e d eu teron . W e will at fi r s t n eglect th e s p in s of th es e t w o p ar ticles an d s olv e th e en er gy eigen v alu e p r ob lem (time-in d ep en d en t S c h r ¨ od in ger eq u ation ) for a b ou n d p -n s y s tem. Th e Hamilton ian is th en giv en b y th e k in etic en er gy of th e p r oton an d th e n eu tr on an d b y th eir m u tu al in ter action .
1 2 1 2
n
p
H = 2 m p ˆ n + 2 m p ˆ p + V nuc ( | x p − x n | )
Her e w e s tated th at th e in ter action d ep en d s on ly on th e d is tan ce b et w een th e t w o p ar ticles (an d n ot for ex amp le th e an gle...)
W e cou ld tr y to s olv e th e S c h r ¨ od in ger eq u ation for th e w a v efu n ction Ψ = Ψ ( R x p , R x n , t ). Th is is a w a v efu n ction th at tr eats th e t w o p ar ticles as fu n d amen tally in d ep en d en t (th at is , d es cr ib ed b y in d ep en d en t v ar iab les ). Ho w ev er , s in ce th e t w o p ar ticles ar e in ter actin g, it migh t b e b etter to con s id er th em as on e s in gle s y s tem. Th en w e can u s e a d iff er en t t y p e of v ar iab les (p os ition an d momen tu m).
{ } → { }
R =
W e can mak e th e tr an s for mation fr om R x p , R x n R R , R r wh er e R R d es cr ib es th e aver age p os ition of th e t w o p ar ticles (i.e. th e p os ition of th e total s y s tem, to b e accu r ately d efi n ed ) an d R r d es cr ib es th e r elativ e p os ition of on e p ar ticle wr t th e oth er :
R m p → x p + m n → x n
m p + m n
cen ter of mas s
R r = R x p − R x n r elativ e p os ition
�
W e can als o in v er t th es e eq u ation s an d d efi n e R x p = x p ( R R , R r ) an d R x n = x n ( R R , R r ). Als o, w e can d efi n e th e cen ter of mas s momen tu m an d r elativ e momen tu m (an d v elo cit y ):
p R c m = p R p + p R n
p R r = ( m n p R p − m p p R n ) / M Th en th e (clas s ical) Hamilton ian , u s in g th es e v ar iab les , r ead s
2 M
c m
2 µ
r
nuc
H = 1 p 2 + 1 p 2 + V ( | r | )
m p + m n
wh er e M = m p + m n an d µ = m p m n is th e r ed u ced mas s . No w w e can ju s t wr ite th e q u an tu m v er s ion of th is
clas s ical Hamilton ian , u s in g
∂ ∂
in th e eq u ation
p ˆ c m = − i r
∂ R R
p ˆ r = − i r ∂ R r
1 2 1 2
H = 2 M p ˆ c m + 2 µ p ˆ r + V nuc ( | r ˆ | )
No w, s in ce th e v ar iab les r an d R ar e in d ep en d en t (s ame as r p an d r n ) th ey comm u te. Th is is als o tr u e for p c m an d r
c m
c m
(an d p r an d R ). Th en , p c m comm u tes with th e wh ole Hamilton ian , [ p R ˆ , H ] = 0. Th is imp lies th at p R ˆ is a con s tan t
2 M c m
of th e motion . Th is is als o tr u e for E c m = 1 p R ˆ 2 , th e en er gy of th e cen ter of mas s . If w e s olv e th e p r ob lem in th e
cen ter -of-mas s fr ame, th en w e can s et E c m = 0 an d th is is n ot ev er goin g to c h an ge. In gen er al, it mean s th at w e can ign or e th e fi r s t ter m in th e Hamilton ian an d ju s t s olv e
2
r 2
H D = − 2 µ ∇ r + V nuc ( | R r | )
H
In p r actice, th is cor r es p on d s to h a v in g ap p lied s ep ar ation of v ar iab les to th e or igin al total S c h r ¨ od in ger eq u ation . Th e Hamilton ian D (th e d eu ter on Hamilton ian ) is n o w th e Hamilton ian of a s in gle-p ar ticle s y s tem, d es cr ib in g th e motion of a r ed u ced mas s p ar ticle in a cen tr al p oten tial (a p oten tial th at on ly d ep en d s on th e d is tan ce fr om th e or igin ). Th is motion is th e motion of a n eu tr on an d a p r oton r elativ e to eac h oth er . In or d er to p r o ceed fu r th er w e n eed to k n o w th e s h ap e of th e cen tr al p oten tial.
70
V(r)
R 0 = 2.1fm E 0 =-2.2M eV
r
- V 0 =-35M eV
5.2.2 G round s tate
W h at ar e th e mos t imp or tan t c h ar acter is tics of th e n u clear p oten tial? It is k n o wn to b e v er y stron g an d sh ort ran ge . Th es e ar e th e on ly c h ar acter is tics th at ar e of in ter es t n o w; als o, if w e limit ou r s elv es to th es e c h ar acter is tics an d b u ild a s imp le, fi ctitiou s p oten tial b as ed on th os e, w e can h op e to b e ab le to s olv e ex actly th e p r ob lem.
≈ − −
If w e lo ok ed at a mor e comp lex , alb eit mor e r ealis tic, p oten tial, th en mos t p r ob ab ly w e can n ot fi n d an ex act s olu tion an d w ou ld h a v e to s imp lify th e p r ob lem. Th u s , w e ju s t tak e a v er y s imp le p oten tial, a n u clear s q u ar e w ell of r an ge R 0 2 . 1 f m an d of d ep th V 0 = 35 M eV .
W e n eed to wr ite th e Hamilton ian in s p h er ical co or d in ates (for th e r ed u ced F ig . 3 1 : N u c lea r p o ten tia l
v ar iab les ). Th e k in etic en er gy ter m is giv en b y :
r
2
r 2 r 2 1 ∂
( 2 ∂ )
∂ r
r 2
1 ∂ (
∂ ) 1
∂ 2
r 2 1 ∂
( 2 ∂ ) L ˆ 2
∂ r
− 2 µ ∇ r = − 2 µ r 2 ∂ r
— 2 µr 2 s in ϑ ∂ ϑ
s in ϑ
∂ ϑ
+ s in 2 ϑ ∂ ϕ 2
= − 2 µ r 2 ∂ r
r
+ 2 µr 2
wh er e w e u s ed th e an gu lar momen tu m op er ator (for th e r ed u ced p ar ticle) L ˆ 2 . Th e S c h r ¨ od in ger eq u ation th en r ead s
r
� r 2 1 ∂
( 2 ∂ )
∂ r
L ˆ 2 �
H
+ 2 µr 2 + V nuc ( r ) Ψ n,l ,m ( r , ϑ , ϕ ) = E n Ψ n,l ,m ( r , ϑ , ϕ )
W e can n o w als o c h ec k th at [ L ˆ 2 , ] = 0. Th en L ˆ 2 is a con s tan t of th e motion an d it h as common eigen fu n ction s with th e Hamilton ian .
l
W e h a v e alr ead y s olv ed th e eigen v alu e p r ob lem for th e an gu lar momen tu m. W e k n o w th at s olu tion s ar e th e s p h er ical h ar mon ics Y m ( ϑ , ϕ ):
l l
L ˆ 2 Y m ( ϑ , ϕ ) = r 2 l ( l + 1) Y m ( ϑ , ϕ )
Th en w e can s olv e th e Hamilton ian ab o v e with th e s ep ar ation of v ar iab les meth o d s , or mor e s imp ly lo ok for a s olu tion
l
l
Ψ n,l ,m = ψ n,l ( r ) Y m ( ϑ , ϕ ):
m L ˆ 2 [ Y m ( ϑ , ϕ )] m
− 2 µ r 2 ∂ r r ∂ r
Y l ( ϑ , ϕ ) + ψ n,l ( r )
2 µr 2
= [ E n − V nuc ( r )] ψ n,l ( r ) Y l
( ϑ , ϕ )
u s in g th e eigen v alu e eq u ation ab o v e w e h a v e
− 2 µ r 2 ∂ r
r
r 1 ∂
∂ ψ ( r ) c c
c c c c c
2 ( 2
n,l
∂ r
) m c
r 2 l ( l + 1) c Y m ( c ϑ , ϕ ) m c
l
c Y l c ( ϑ , ϕ ) + ψ n,l ( r )
l
2 µr 2
= [ E n − V nuc ( r )] ψ n,l ( r ) c Y l c ( ϑ , ϕ )
an d th en w e can elimin ate Y m to ob tain :
r
+ V nuc ( r ) +
r 2 1 d ( 2 d ψ n,l ( r ) )
d r
r 2 l ( l + 1)
2 µr 2
ψ n,l ( r ) = E n ψ n,l ( r )
No w w e wr ite ψ n,l ( r ) = u n,l ( r ) /r . Th en th e r ad ial p ar t of th e S c h r ¨ od in ger eq u ation b ecomes
r 2 d 2 u
r 2 l ( l + 1)
− 2 µ d r 2 + V nuc ( r ) + 2 µ r 2
u ( r ) = E u ( r )
with b ou n d ar y con d ition s
u nl (0) = 0 → ψ (0) is fi n ite
u nl ( ∞ ) = 0 → b ou n d s tate
Th is eq u ation is ju s t a 1D S c h r ¨ od in ger eq u ation in wh ic h th e p oten tial V ( r ) is r ep laced b y an eff ecti v e p oten ti al
r 2 l ( l + 1)
V ef f ( r ) = V nuc ( r ) + 2 µr 2
th at p r es en ts th e ad d ition of a cen tr ifu gal p oten tial (th at cau s es an ou t w ar d for ce).
71
V(r)
R 0 = 2.1fm
E 0 =-2.2M eV
r
- V 0 =-35M eV
V(r)
R 0 = 2.1fm
E 0 =- 2.2M eV
r
- V 0 =-35M eV
F ig . 3 2 : N u c lea r p o ten tia l fo r l / = / 0 . Left , n u c lea r p o ten t ia l a n d c en tr ifu g a l p o ten t ia l. R ig h t, th e effec tiv e p o ten t ia l.
Notice th at if l is lar ge, th e cen tr ifu gal p oten tial is h igh er . Th e gr ou n d s tate is th en fou n d for l = 0. In th at cas e th er e is n o cen tr ifu gal p oten tial an d w e on ly h a v e a s q u ar e w ell p oten tial (th at w e alr ead y s olv ed ).
Th is giv es th e eigen fu n ction s
k 2 1 ∂ 2
− 2 µ r ∂ r + V nuc ( r )
u 0 ( r ) = E 0 u 0 ( r )
an d
u ( r ) = A s in ( k r ) + B cos ( k r ) , 0 < r < R 0
u ( r ) = C e − κ r + D e κ r , r > R 0
Th e allo w ed eigen fu n ction s (as d eter min ed b y th e b ou n d ar y con d ition s ) h a v e eigen v alu es fou n d fr om th e o d d -p ar it y s olu tion s to th e eq u ation
with
− κ = k cot ( k R 0 )
(with E 0 < 0).
k 2 = 2 µ ( E + V ) κ 2 = − 2 µ E
r 2
r 2
0
0
0
4 2 2 k
≤ ≥ ≥
Recall th at w e fou n d th at th er e w as a min im u m w ell d ep th an d r an ge in or d er to h a v e a b ou n d s tate. T o s atis fy th e con tin u it y con d ition at r = R 0 w e n eed λ / 4 R 0 or k R 0 1 2 π = π . Th en R 0 π .
2 R 0
In or d er to fi n d a b ou n d s tate, w e n eed th e p oten tial en er gy to b e h igh er th an th e k in etic en er gy V 0 > E k in . If w e k n o w R 0 w e can u s e k ≥ π to fi n d
r 2 π 2 π 2 r 2 c 2 π 2
(191 M eV f m ) 2
V 0 > 2 µ 4 R 2 = 8 µ c 2 R 2 = 8 469 M eV (2 . 1 f m ) 2 = 23 . 1 M eV
0 0
—
W e th u s fi n d th at in d eed a b ou n d s tate is p os s ib le, b u t th e b in d in g en er gy E 0 = E k in V 0 is q u ite s mall. S olv in g n u mer ically th e tr as cen d en tal eq u ation for E 0 w e fi n d th at
E 0 = − 2 . 2 M eV
Notice th at in ou r p r o ced u r e w e s tar ted fr om a mo d el of th e p ote n tial th at in clu d es th e r an ge R 0 an d th e s tr en gth V 0 in or d er to fi n d th e gr ou n d s tate en er gy (or b in d in g en er gy ). Ex p er imen tally in s tead w e h a v e to p er for m th e in v er s e p r o ces s . F r om s catter in g ex p er imen ts it is p os s ib le to d eter min e th e b in d in g en er gy (s u c h th at th e n eu tr on an d p r oton get s ep ar ated ) an d fr om th at, b as ed on ou r th eor etical mo d el, a v alu e of V 0 can b e in fer r ed .
5.2.3 Deuteron exc i ted s tate
Ar e b ou n d ex cited s tates for th e d eu ter on p os s ib le?
Con s id er fi r s t l = 0. W e s a w th at th e b in d in g en er gy for th e gr ou n d s tate w as alr ead y s mall. Th e n ex t o d d s olu tion
w ou ld h a v e k = 3 π = 3 k 0 . Th en th e k in etic en er gy is 9 times th e gr ou n d s tate k in etic en er gy or E 1 = 9 E 0 =
2 R 0 k in k in
×
9 32 . 8 M eV = 295 . 2 M eV . Th e total en er gy th u s b ecomes p os itiv e, th e in d ication th at th e s tate is n o lon ger b ou n d
(in fact, w e th en h a v e n o lon ger a d is cr ete s et of s olu tion s , b u t a con tin u u m of s olu tion s ).
n 2 l ( l + 1)
0
Con s id er th en l > 0. In th is cas e th e p oten tial is in cr eas ed b y an amou n t
2 µR 2
≥ 18 . 75 M eV (for l = 1). Th e
p oten tial th u s b ecomes s h allo w er (an d n ar r o w er ). Th u s als o in th is cas e th e s tate is n o lon ger b ou n d . Th e d eu teron h as on l y on e b ou n d state.
72
5.2.4 Spi n dep endenc e of nuc l ea r fo rc e
Un til n o w w e n eglected th e fact th at b oth n eu tr on an d p r oton p os s es s a s p in . Th e q u es tion r emain s h o w th e s p in in fl u en ces th e in ter action b et w een th e t w o p ar ticles .
Th e total an gu lar momen tu m for th e d eu ter on (or in gen er al for a n u cleu s ) is u s u ally d en oted b y I . Her e it is giv en b y
R ˆ R ˆ R ˆ R ˆ
I = L + S p + S n
R ˆ R ˆ R ˆ R ˆ R ˆ
F or th e b ou n d d eu ter on s tate l = 0 an d I = S p + S n = S . A p r ior i w e can h a v e S = 0 or 1 (r ecall th e r u les for
R 1
ˆ ad d ition of an gu lar momen tu m, h er e S p,n = 2 ).
Th er e ar e ex p er imen tal s ign atu r es th at th e n u clear for ce d ep en d s on th e s p in . In fact th e d eu ter on is on ly fou n d with
R
ˆ
S = 1 (mean in g th at th is con fi gu r ation h as a lo w er en er gy ).
S p
· R
Th e s imp les t for m th at a s p in -d ep en d en t p oten tial cou ld as s u me is V spin ∝ R ˆ ˆ (s in ce w e w an t th e p oten tial to
S n
b e a s calar ). Th e co efficien t of p r op or tion alit y V 1 ( r ) / r 2 can h a v e a s p atial d ep en d en ce. Th en , w e gu es s th e for m for
p
th e s p in -d ep en d en t p oten tial to b e V spin = V 1 ( r ) / r 2 S R ˆ
ˆ
· R
S n . W h at is th e p oten tial for th e t w o p os s ib le con fi gu r ation s
of th e n eu tr on an d p r oton s p in s ?
R R R 2
ˆ ˆ ˆ
Th e con fi gu r ation ar e eith er S = 1 or S = 0. Let u s wr ite S = r S ( S + 1) in ter ms of th e t w o s p in s :
R ˆ 2 R ˆ 2 R ˆ 2 R ˆ R ˆ
Th e las t ter m is th e on e w e ar e lo ok in g for :
S = S p + S n + 2 S p · S n
R ˆ R ˆ 1 � R ˆ 2
R ˆ 2
R ˆ 2 �
ˆ 2 R ˆ 2
R ˆ 2
S p · S n = 2
S − S p − S n
Becau s e S
op er ator s 10 :
an d S p , S n comm u te, w e can wr ite an eq u ation for th e ex p ectation v alu es wr t eigen fu n ction s of th es e
R ˆ
R ˆ
R ˆ R ˆ r 2
S p · S n = � S , S p , S n , S z | S p · S n | S , S p , S n , S z � =
2
R
R
4
2 2
z
s in ce S p,n = 1 , w e ob tain
2 ( S ( S + 1) − S p ( S p + 1) − S n ( S n + 1))
ˆ ˆ r 2 (
3 ) + n 2
T r ip let S tate,
S = 1 , 1 1 , m )
S p · S n = 2
S ( S + 1) − 2 = − 3 n 2
S in glet State,
S = 0 , 1 , 1 , 0 )
4
4
| −
4
2
2
If V 1 ( r ) is an attr activ e p oten tial ( < 0), th e total p oten tial is V nuc | S = 1 = V T = V 0 + 1 V 1 for a tr ip let s tate, wh ile its s tr en gth is r ed u ced to V nuc S = 0 = V S = V 0 3 V 1 for a s in glet s tate. Ho w lar ge is V 1 ?
—
≈ − −
V 0 + V 1 = V T
W e can comp u te V 0 an d V 1 fr om k n o win g th e b in d in g en er gy of th e tr ip let s tate an d th e en er gy of th e u n b ou n d vir tu al s tate of th e s in glet (s in ce th is is v er y clos e to zer o, it can s till b e ob tain ed ex p er imen tally ). W e h a v e E T = 2 . 2M eV (as b efor e, s in ce th is is th e ex p er imen tal d ata) an d E S = 77k eV. S olv in g th e eigen v alu e p r ob lem for a s q u ar e w ell, k n o win g th e b in d in g en er gy E T an d s ettin g E S 0, w e ob tain V T = 35M eV an d V S = 25M eV (Notice th at of cou r s e V T is eq u al to th e v alu e w e h ad p r ev iou s ly s et for th e d eu ter on p oten tial in or d er to fi n d th e cor r ect b in d in g en er gy of 2 . 2M eV, w e ju s t –wr on gly – n eglected th e s p in ear lier on ). F r om th es e v alu es b y s olv in g a s y s tem of t w o eq u ation s in t w o v ar iab les :
1
4
4
V 0 − 3 V 1 = V S
w e ob tain V 0 = − 32 . 5M eV V 1 = − 10M eV. Th u s th e s p in -d ep en d en t p ar t of th e p oten tial is w eak er , b u t n ot n egligib le.
10 N o t e th a t o f c o u r s e w e u s e th e c o u p le d r ep r es en ta tio n s in c e th e p r o p er t ies o f th e d eu ter o n , a n d o f its s p in -d ep en d en t en er g y , a r e s et b y th e c o mmo n s ta te o f p r o to n a n d n eu tr o n
5. 3 Nuc lea r mo dels
In th e cas e of th e s imp les t n u cleu s (th e d eu ter iu m, with 1p -1n ) w e h a v e b een ab le to s olv e th e time in d ep en d en t S c h r ¨ od in ger eq u ation fr om fi r s t p r in cip les an d fi n d th e w a v efu n ction an d en er gy lev els of th e s y s tem —of cou r s e with s ome ap p r o x imation s , s imp lify in g for ex amp le th e p oten tial. If w e tr y to d o th e s ame for lar ger n u clei, w e s o on w ou ld fi n d s ome p r ob lems , as th e n u m b er of v ar iab les d es cr ib in g p os ition an d momen tu m in cr eas es q u ic k ly an d th e math p r ob lems b ecome v er y comp lex .
An oth er d ifficu lt y s tems fr om th e fact th at th e ex act n atu r e of th e n u clear for ce is n ot k n o wn , as th er e’s for ex amp le s ome ev id en ce th at th er e ex is t als o 3-b o d y in ter action s , wh ic h h a v e n o clas s ical an alog an d ar e d ifficu lt to s tu d y v ia s catter in g ex p er imen ts .
Th en , in s tead of tr y in g to s olv e th e p r ob lem ex actly , s tar tin g fr om a micr os cop ic d es cr ip tion of th e n u cleu s con s titu en ts , n u clear s cien tis ts d ev elop ed s ome mo d els d es cr ib in g th e n u cleu s . Th es e mo d els n eed to y ield r es u lts th at agr ee with th e alr ead y k n o wn n u clear p r op er ties an d b e ab le to p r ed ict n ew p r op er ties th at can b e meas u r ed in ex p er imen ts . W e ar e n o w goin g to r ev iew s ome of th es e mo d els .
5.3.1 Shel l s truc ture
A . Th e a tomic sh ell mo d el
Y ou migh t alr ead y b e familiar with th e atomic s h ell mo d el. In th e atomic s h ell mo d el, s h ells ar e d efi n ed b as ed on th e atomic q u an tu m n u m b er s th at can b e calcu lated fr om th e atomic Cou lom b p oten tial (an d en s u in g th e eigen v alu e eq u ation ) as giv en b y th e n u clear ’s p r oton s .
S h ells ar e fi lled b y electr on s in or d er of in cr eas in g en er gies , s u c h th at eac h or b ital (lev el) can con tain at mos t 2 electr on s (b y th e P au li ex clu s ion p r in cip le). Th e p r op er ties of atoms ar e th en mos tly d eter min ed b y electr on s in a n on -comp letely fi lled s h ell. Th is lead s to a p er io d icit y of atomic p r op er ties , s u c h as th e atomic r ad iu s an d th e ion ization en er gy , th at is r efl ected in th e p er io d ic tab le of th e elemen ts . W e h a v e s een wh en s olv in g for th e h y d r ogen
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0.30
0.25
Radius S nm '
0.20
0.15
0.10
0.05
0 20 40 60 80
Z
F ig . 3 3 : A to mic R a d iu s vs Z.
| |
− −
atom th at a q uan tu m s tate is d es cr ib ed b y th e q u an tu m n u m b er s : ψ = n, l , m wh er e n is th e p r in cip le q u an tu m n u m b er (th at in th e h y d r ogen atom w as giv in g th e en er gy ). l is th e an gu lar momen tu m q u an tu m n u m b er (or azim u th al q u an tu m n u m b er ) an d m th e magn etic q u an tu m n u m b er . Th is las t on e is m = l , . . . , l 1 , l th u s togeth er with th e s p in q u an tu m n u m b er , s ets th e d egen er acy of eac h or b ital (d eter min ed b y n an d l < n ) to b e D ( l ) = 2(2 l + 1). His tor ically , th e or b itals h a v e b een called with th e s p ectr os cop ic n otation as follo ws :
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l |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
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S p ectr os cop i n otation |
c s |
p |
d |
f |
g |
h |
i |
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D ( l ) |
2 |
6 |
10 |
14 |
18 |
22 |
26 |
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h is tor ic s tr u ctu r e |
h ea v y n u clei |
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Th e h is tor ical n otation s come fr om th e d es cr ip tion of th e ob s er v ed s p ectr al lin es :
s = s h ar p p = p r in cip al d = d iff u s e f = fi n e
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25
First Ionization Energy [ eV ]
20
15
10
5
0 20 40 60 80
Z
F ig . 3 4 : I o n iz a tio n en er g y vs Z.
O r b itals (or en er gy eigen fu n ction s ) ar e th en collected in to gr ou p s of s imilar en er gies (an d s imilar p r op er ties ). Th e d egen er acy of eac h or b ital giv es th e follo win g (cu m u lativ e) o ccu p an cy n u m b er s for eac h on e of th e en er gy gr ou p :
2 , 1 0 , 18 , 36 , 54 , 70 , 86
Notice th at th es e cor r es p on d to th e w ell k n o wn gr ou p s in th e p e r io d ic tab le.
Th er e ar e s ome d ifficu lties th at ar is e wh en tr y in g to ad ap t th is mo d el to th e n u cleu s , in p ar ticu lar th e fact th at th e p oten tial is n ot ex ter n al to th e p ar ticles , b u t cr eated b y th ems elv es , an d th e fact th at th e s ize of th e n u cleon s is m u c h lar ger th an th e electr on s , s o th at it mak es m u c h les s s en s e to s p eak of or b itals . Als o, in s tead of h a v in g ju s t on e t y p e of p ar ticle (th e electr on ) ob ey in g P au li’s ex clu s ion p r in cip le, h er e matter s ar e comp licated b ecau s e w e n eed to fi ll s h ells with t w o t y p es of p ar ticles , n eu tr on s an d p r oton s .
In an y cas e, th er e ar e s ome comp ellin g ex p er imen tal ev id en ces th at p oin t in th e d ir ection of a s h ell mo d el.
B . Evid en ce of n u clea r sh ell stru ctu re: Tw o-n u cleon sep a ra tion en ergy
Th e t w o-n u cleon s ep ar ation en er gy (2p - or 2n -s ep ar ation en er gy ) is th e eq u iv alen t of th e ion ization en er gy for atoms , wh er e n u cleon s ar e tak en ou t in p air to accou n t for a ter m in th e n u clear p oten tial th at fa v or th e p air ing of n u cleon s . F r om th is fi r s t s et of d ata w e can in fer th at th er e ex is t s h ells with o ccu p ation n u m b er s
8 , 2 0 , 28 , 50 , 82 , 126
Th es e ar e called Magi c n u m b ers in n u clear p h y s ics . Comp ar in g to th e s ize of th e atomic s h ells, w e can s ee th at th e atomic magic n u m b er s ar e q u ite d iff er en t fr om th e n u clear on es (as ex p ected s in ce th er e ar e t w o-t y p es of p ar ticles an d oth er d iff er en ces .) O n ly th e gu id in g p r in cip le is th e s ame. Th e atomic s h ells ar e d eter min ed b y s olv in g th e en er gy eigen v alu e eq u ation . W e can attemp t to d o th e s ame for th e n u cleon s .
5.3.2 Nuc l eons Hami l toni an
` ˛ ¸ x
Th e Hamilton ian for th e n u cleu s is a comp lex man y -b o d y Hamilton ian . Th e p oten tial is th e com b in ation of th e n u clear an d cou lom b in ter action :
i
=
Σ p ˆ 2
Σ Σ e 2
i
H 2 m
i
+ V nuc ( | R x i − R x j | ) +
j ,i ≤ j j ,i ≤ j
| R x i
— R x j |
s u m on p r o t on s o n l y
Th er e is n ot an ex ter n al p oten tial as for th e electr on s (wh er e th e p r oton s cr eate a s tr on g ex ter n al cen tr al p oten tial for eac h electr on ). W e can s till s imp lify th is Hamilton ian b y u s in g mean fi el d th eory 11 .
11 T h is is a c o n c ep t th a t is r elev a n t in ma n y o th er p h ys ic a l s itu a tio n s
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208 Pb |
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64 Ni |
184 114 Ca |
W |
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38 Ar |
102 Mo |
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86 Kr |
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14 C |
132 T e |
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20 8 |
28 50 |
82 |
12 |
6 |
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Ce |
Dy |
Pb Pt Hf |
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Cd |
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O |
4
3
2
S 2p (MeV)
1
0
-1
-2
-3
-4
-5
5
4
3
2
S 2n (MeV)
1
0
-1
-2
-3
-4
-5
0 25 50
75 100
125
150
Nucleon number
Image by MIT OpenCourseWare. After Krane.
F ig . 3 5 : T o p : T w o -p r o to n s ep a r a tio n en er g ies o f is o to n es ( c o n s ta n t N ). B o tto m: t w o -n eu tr o n s ep a r a tio n en er g ies o f is o to p es (c o n s ta n t Z). On th e x-a xis : n u c leo n n u m b er . T h e s u d d en c h a n g es a t t h e ma g ic n u m b er a r e a p p a ren t. F r o m Kr a n e, fi g 5 .2
n
W e can r ewr ite th e Hamilton ian ab o v e b y p ic k in g 1 n u cleon , e.g. th e j th n eu tr on :
or th e k th p r oton :
H j =
p ˆ 2
2 m n
+ V
Σ
i ≤ j
j
nuc
( | R x i
— R x j | )
k
p p ˆ 2
Σ Σ e 2
n
H k = 2 m
+ V nuc ( | R x i − R x k | ) +
i ≤ k i ≤ k
| R x i
— R x k |
s u m on p r o t on s o n l y
th en th e total Hamilton ian is ju s t th e s u m o v er th es e on e-p ar ticle Hamilton ian s :
` ˛ ¸ x
n
p
H = Σ H + Σ H
Th e Hamilton ian s H n an d H
j k
j ( n e u t r on s ) k ( p r ot on s )
p
d es cr ib e a s in gle n u cleon s u b jected to a p oten tial V j ( | R x j | ) — or V j ( | R x j | ) =
j j nuc
nuc c o ul
V j ( | R x j | ) + V j ( | R x j | ) for a p r oton . Th es e p oten tials ar e th e eff ect of all th e oth er n u cleon s on th e n u cleon w e
p ic k ed , an d on ly th eir s u m comes in to p la y . Th e n u cleon w e fo cu s ed on is th en ev olv in g in th e mean fi eld cr eated b y all th e oth er n u cleon s . O f cou r s e th is is a s imp lifi cation , b ecau s e th e fi eld cr eated b y th e oth er n u cleon s d ep en d s als o on th e j th n u cleon , s in ce th is n u cleon in fl u en ces (for ex amp le) th e p os ition of th e oth er n u cleon s . Th is k in d of b ac k -action is ign or ed in th e mean -fi eld ap p r o x imation , an d w e con s id er ed th e mean -fi eld p oten tial as fi x ed (th at is , giv en b y n u cleon s with a fi x ed p os ition ).
~
—
nuc
nuc c o ul
W e th en w an t to ad op t a mo d el for th e mean -fi eld V j an d V j . Let’s s tar t with th e n u clear p oten tial. W e mo d eled th e in ter action b et w een t w o n u cleon s b y a s q u ar e w ell, with d ep th V 0 an d r an ge R 0 . Th e r an ge of th e n u clear w ell is r elated to th e n u clear r ad iu s , wh ic h is k n o wn to d ep en d on th e n u clear mas s n u m b er A , as R 1 . 25 A 1 / 3 fm. Th en V j is th e s u m of man y of th es e s q u ar e w ells , eac h with a d iff er en t r an ge (d ep en d in g on th e s ep ar ation of th e n u cleon s ). Th e d ep th is in s tead almos t con s tan t at V 0 = 50M eV, wh en w e con s id er lar ge-A n u clei (th is cor r es p on d to
1
2
3
4
5
– 10
– 20
– 30
– 40
– 50
F ig . 3 6 : P o t en tia l o b ta in ed fr o m th e s u m o f ma n y r ec t a n g u la r p o ten t ial w ells . B la c k, th e p o ten tia l ra n g e in c r ea s es p r o p o r tio n a lly to th e n u m b er o f n u c leo n s c o n s id er ed . R ed , R ∼ A 1 / 3 . B lu e, h a r mo n ic p o ten tia l, th a t a p p r o xima t es th e d es ir ed p o ten tia l.
th e a v er age s tr en gth of th e total n u cleon p oten tial). W h at is th e s u m of man y s q u ar e w ells ? Th e p oten tial s mo oth s ou t. W e can ap p r o x imate th is with a p ar ab olic p oten tial. [Notice th at for an y con tin u ou s fu n ction , a min im u m can alw a y s b e ap p r o x imated b y a p ar ab olic fu n ction , s in ce a min im u m is s u c h th at th e fi r s t d er iv ativ e is zer o]. Th is t y p e of p oten tial is u s efu l b ecau s e w e can fi n d an an aly tical s olu tion th at will giv e u s a clas s ifi cation of n u clear s tates . O f cou r s e, th is is a cr u d e ap p r o x imation . Th is is th e os cil l ator p oten tial mo d el:
( r 2 )
0
V nuc ≈ − V 0 1 − R 2
No w w e n eed to con s id er th e Cou lom b p oten tial for p r oton s . Th e p oten tial is giv en b y : V
= ( Z − 1) e 2 [ 3 −
r 2 ]
0
c o ul
R 0 2 2 R 2
for r ≤ R 0 , wh ic h is ju s t th e p oten tial for a s p h er e of r ad iu s R 0 con tain in g a u n ifor m c h ar ge ( Z − 1) e .
Th en w e can wr ite an eff ectiv e (mean -fi eld , in th e p ar ab olic ap p r o x imation ) p oten tial as
V e ff = r
R 2 −
2 R 3
− V 0 + 2 R
2 ( V 0 ( Z − 1) e 2 ) 3 ( Z − 1) e 2
0
0
2
0
0
` ≡ 1 m ˛ ¸ ω 2 r 2
x ` ≡ − ˛ ¸ V ′ x
W e d efi n ed h er e a mo d ifi ed n u clear s q u ar e w ell p oten tial V 0 ′ = V 0
3 ( Z − 1) e 2 for p r oton s , wh ic h is s h allo w er th an for
2 R
0
0
2 R 3
n eu tr on s . Als o, w e d efi n ed th e h ar mon ic os cillator fr eq u en cies ω 2 = 2 V 0 − ( Z − 1) e 2 .
0
Th e p r oton w ell is th u s s ligh tly s h allo w er an d wid er th an th e n eu tr on w ell b ecau s e of th e Cou lom b r ep u ls ion . Th is p oten tial mo d el h as limitation s b u t it d o es p r ed ict th e lo w er magic n u m b er s .
Th e eigen v alu es of th e p oten tial ar e giv en b y th e s u m of th e h ar mon ic p oten tial in 3D (as s een in r ecitation ) an d th e s q u ar e w ell:
E = r ω ( N + 3 ) − V ′ .
N 2 0
(wh er e w e tak e V 0 ′ = V 0 for th e n eu tr on ).
2 m r 2
Note th at s olv in g th e eq u ation for th e h ar mon ic os cillator p oten tial is n ot eq u iv alen t to s olv e th e fu ll r ad ial eq u ation , wh er e th e cen tr ifu gal ter m r 2 l ( l + 1) m u s t b e tak en in to accou n t. W e cou ld h a v e s olv ed th at total eq u ation an d fou n d th e en er gy eigen v alu es lab eled b y th e r ad ial an d or b ital q u an tu m n u m b er s . Comp ar in g th e t w o s olu tion s , w e fi n d th at th e h .o. q u an tu m n u m b er N can b e ex p r es s ed in ter ms of th e r ad ial an d or b ital q u an tu m n u m b er s as
N = 2( n − 1) + l
S in ce l = 0 , 1 , . . . n − 1 w e h a v e th e s election r u le fo r Σ l as a fu n ction of N : l = N , N − 2 , . . . (with l ≥ 0). Th e
d egen er acy of th e E N eigen v alu es is th en D ′ ( N ) =
D ( N ) = ( N + 1)( N + 2) wh en in clu d in g th e s p in .
l = N ,N − 2 ,... (2 l + 1) =
1 ( N + 1)( N + 2) (ign or in g s p in ) or
2
W e can n o w u s e th es e q u an tu m n u m b er s to fi ll th e n u clear lev els . Notice th at w e h a v e s ep ar ate lev els for n eu tr on s an d p r oton s . Th en w e can b u ild a tab le of th e lev els o ccu p atio n s n u m b er s , wh ic h p r ed icts th e fi r s t 3 magic n u m b er s .
|
N |
l |
S p ectr os cop ic Notation |
1 2 D ( N ) |
D ( N ) |
Cu m u lativ e of n u cle on s # |
|
0 |
0 |
1 s |
1 |
2 |
2 |
|
1 |
1 |
1 p |
3 |
6 |
8 |
|
2 |
0,2 |
2s ,1d |
6 |
12 |
20 |
|
3 |
1,3 |
2p ,1f |
10 |
20 |
40 |
|
4 |
0,2,4 |
3s ,2d ,1g |
15 |
30 |
70 |
m R 2 2 R 3
�
F or h igh er lev els th er e ar e d is cr ep an cies th u s w e n eed a mor e pr ecis e mo d el to ob tain a mor e accu r ate p r ed iction . Th e oth er p r ob lem with th e os cillator mo d el is th at it p r ed icts on ly 4 lev els to h a v e lo w er en er gy th an th e 50M eV w ell p oten tial (th u s on ly 4 b ou n d en er gy lev els ). Th e s ep ar ation b et w een os cillator lev els is in fact r ω = 2 n 2 V 0 − ( Z − 1) e 2 ≈
m c 2 R 2
� � 0 0
2 n 2 c 2 V 0 . In s er tin g th e n u mer ical v alu es w e fi n d r ω =
0
b et w een os cillator lev els is on th e or d er of 10-20M eV.
≈ 51 . 5 A − 1 / 3 Th en th e s ep ar ation
5.3.3 Spi n o rbi t i nterac ti on
In or d er to p r ed ict th e h igh er magic n u m b er s , w e n eed to tak e in to accou n t oth er in ter action s b et w een th e n u cleon s . Th e fi r s t in ter action w e an aly ze is th e s p in -or b it cou p lin g.
Th e as s o ciated p oten tial can b e wr itten as
1 R ˆ ˆ
ˆ R ˆ
r 2 V so ( r ) l · R s
wh er e R s an d l ar e s p in an d an gu lar momen tu m op er ator s for a s in gle n u cleon . Th is p oten tial is to b e ad d ed to th e
s in gle-n u cleon mean -fi eld p oten tial s een b efor e. W e h a v e s een p r ev iou s ly th at in th e in ter action b et w een t w o n u cleon s th er e w as a s p in comp on en t. Th is t y p e of in ter action motiv ates th e for m of th e p oten tial ab o v e (wh ic h again is to b e tak en in a mean -fi eld p ictu r e).
W e can calcu late th e d ot p r o d u ct with th e s ame tr ic k alr ead y u s ed :
D → ˆ E
ˆ
l · → s
→ ˆ
= 1 → ˆ 2 2
→ ˆ 2 ˆ 2 k 2 3
( j
— l − → s ) = 2 [ j ( j + 1) − l ( l + 1) − 4 ]
1
wh er e j is th e total an gu lar momen tu m for th e n u cleon . S in ce th e s p in of th e n u cleon is s = 2 , th e p os s ib le v alu es
of j ar e j = l ± 1 . Th en j ( j + 1) − l ( l + 1) = ( l ± 1 )( l ± 1 + 1) − l ( l + 1), an d w e ob tain
2
→ l · → s ˆ
D ˆ E
2 2
=
2
2
1
( l k 2
k 2
an d th e total p oten tial is
− ( l + 1) 2 for j= l- 2
V nuc ( r ) =
2
l + 1
2
2
V
0
— V
so
2
No w r ecall th at b oth V 0 is n egativ e an d c h o os e als o V so n egativ e. Th en :
2
- wh en th e s p in is al i gn ed with th e an gu lar momen tu m ( j = l + 1 ) th e p oten tial b ecomes mor e n egativ e, i.e. th e
w ell is d eep er an d th e s tate mor e tigh tly b ou n d .
- wh en s p in an d an gu lar momen tu m ar e an ti -al i gn ed th e s y s tem’s en er gy is h igh er .
2
Th e en er gy lev els ar e th u s s p lit b y th e s p in -or b it cou p lin g (s ee fi gu r e 37 ). Th is s p littin g is d ir ectly p r op or tion al to th e an gu lar momen tu m l (is lar ger for h igh er l ): ΔE = n 2 (2 l + 1). Th e t w o s tates in th e s ame en er gy con fi gu r ation b u t with th e s p in align ed or an ti-align ed ar e called a d ou b let.
2
Ex amp le: Con s id er th e N = 3 h .o. lev el. Th e lev el 1 f 7 / 2 is p u s h ed far d o wn (b ecau s e of th e h igh l ). Th en its en er gy is s o d iff er en t th at it mak es a s h ell on its o wn . W e h ad fou n d th at th e o ccu p ation n u m b er u p to N = 2 w as 20 (th e 3r d magic n u m b er ). Th en if w e tak e th e d egen er acy of 1 f 7 / 2 , D ( j ) = 2 j + 1 = 2 7 + 1 = 8, w e ob tain th e 4th magic
n u m b er 28.
[Notice th at s in ce h er e j alr ead y in clu d es th e s p in , D ( j ) = 2 j + 1 .]
S in ce th e 1 f 7 / 2 lev el n o w for ms a s h ell on its o wn an d it d o es n ot b elon g to th e N = 3 s h ell an y mor e, th e r es id u al d egen er acy of N = 3 is ju s t 12 in s tead of 20 as b efor e. T o th is d egen er acy , w e migh t ex p ect to h a v e to ad d th e lo w es t lev el of th e N = 4 man ifold . Th e h igh es t l p os s ib le for N = 4 is ob tain ed with n = 1 fr om th e for m u la N = 2( n − 1) + l → l = 4 (th is w ou ld b e 1 g ). Th en th e lo w es t lev el is for j = l + 1 / 2 = 4 + 1 / 2 = 9 / 2 with d egen er acy
3N
2p
1f
1f 5/2
2p 3/2
2p 1/2
2N
1f 7/2
F ig . 3 7 : T h e en er g y lev els fr o m th e h a r mo n ic o s c illa to r lev el ( la b eled b y N ) a r e fi r s t s h ift ed b y th e a n g u la r mo men tu m p o ten t ia l ( 2 p , 1 f ) . E a c h l lev el is th en s p lit b y th e s p in -o r b it in ter a c tio n , w h ic h p u s h es th e en er g y u p o r d o w n , d ep en d in g o n th e s p in a n d a n g u la r mo men tu m a lig n men t
D = 2(9 / 2 + 1) = 10. Th is n ew com b in ed s h ell comp r is es th en 12 + 10 lev els . In tu r n s th is giv es u s th e magic n u m b er 50.
Us in g th es e s ame con s id er ation s , th e s p littin gs giv en b y th e s p in -or b it cou p lin g can accou n t for all th e magic n u m b er s an d ev en p r ed ict a n ew on e at 184:
→ −
- N = 4, 1g 1 g 7 / 2 an d 1 g 9 / 2 . Th en w e h a v e 20 8 = 12 + D (9 / 2) = 10. F r om 28 w e ad d an oth er 22 to ar r iv e at th e magic n u m b er 50.
→
—
- N = 5, 1h 1 h 9 / 2 an d 1 h 11 / 2 . Th e s h ell th u s com b in es th e N = 4 lev els n ot alr ead y in clu d ed ab o v e, an d th e D (1 h 11 / 2 ) = 12 lev els ob tain ed fr om th e N = 5 1 h 11 / 2 . Th e d egen er acy of N = 4 w as 30, fr om wh ic h w e s u b tr act th e 10 lev els in clu d ed in N = 3. Th en w e h a v e (30 10) + D (1 h 11 / 2 ) = 20 + 12 = 32. F r om 50 w e ad d ar r iv e at th e magic n u m b er 82.
→ − −
- N = 6, 1i 1 i 11 / 2 an d 1 i 13 / 2 . Th e s h ell th u s h a v e D ( N = 5) D (1 h 11 / 2 ) + D (1 i 13 / 2 ) = 42 12 + 14 = 44 lev els ( D ( N ) = ( N + 1)( N + 2)). Th e p r ed icted magic n u m b er is th en 126.
→ − −
- N = 7 1 j 15 / 2 is ad d ed to th e N = 6 s h ell, to giv e D ( N = 6) D (1 i 13 / 2 ) + D (1 j 15 / 2 ) = 56 14 + 16 = 58, p r ed ictin g a y et n ot-ob s er v ed 184 magic n u m b er .
Har moni c Oscilla t o r Spin- Or bi t P ot en tia l
l
Specr osc opic
N
Spin- or bit D
M ag ic
. . .
Nota tion Number
6
0
2
4
6
4s 3d 2g 1i
58 184
1i 11/2
1i 13/2
1 3p 3p 1/2
5/2
5 3 2f 2f 3p 3/ 2
2f 7/2
44 126
5 1h
4
3s 1/2
2 2d
2d 3/2
2d 5/2
32 82
4 1g
1g 7/2
1g 9/2
0 3s
1h 9/2
1h 11/2
3
1 2p
1f 5/2
3 1f
2p 1/2 22 50
2p 3/2
1f 7/2
8 28
0 0 1s
1p 1/2
1
1 1p
1p 3/2
1s 1/2
12 20
2
0 2s
2 1d
1d 3/2
2s 1/ 2
1d 5/ 2
6 8
2 2
F ig . 3 8 : S h ell M o d el p r ed ic tio n o f th e ma g ic n u m b er s . Lev el s p littin gs d u e to h .o . lev els , l -qu a n tu m n u m b er a n d s p in -o r b it c o u p lin g . N o t ic e th a t fu r th er v a r ia tio n s in th e p o s it io n o f th e lev els a re a c tu a lly p r es en t (s ee Kr a n e F ig . 5 .6 ). H er e o n ly th e s h iftin g s lea d in g to n ew s h ell g r o u p in g s a r e s h o w n .
Th es e p r ed iction s d o n ot d ep en d on th e ex act s h ap e of th e s q u ar e w ell p oten tial, b u t on ly on th e s p in -or b it cou p lin g an d its r elativ e s tr en gth to th e n u clear in ter action V 0 as s et in th e h ar mon ic os cillator p oten tial (w e h ad s een th at th e s ep ar ation b et w een os cillator lev els w as on th e or d er of 10M eV.) In p r actice, if on e s tu d ies in mor e d etail th e
p oten tial w ell, on e fi n d s th at th e os cillator lev els with h ig h er l ar e lo w er ed with r es p ect to th e oth er s , th u s en h an cin g th e gap cr eated b y th e s p in -or b it cou p lin g.
Th e s h ell mo d el th at w e h a v e ju s t p r es en ted is q u ite a s imp lifi ed mo d el. Ho w ev er it can mak e man y p r ed iction s ab ou t th e n u clid e p r op er ties . F or ex amp le it p r ed icts th e n u clear s p in an d p ar it y , th e magn etic d ip ole momen t an d electr ic q u ad r u p olar momen t, an d it can ev en b e u s ed to calcu late th e p r ob ab ilit y of tr an s ition s fr om on e s tate to an oth er as a r es u lt of r ad ioactiv e d eca y or n u clear r eaction s .
Intermediate Intermediate form form with Spin Orbit
4s 2 168
1j 15/2 16 184
184
2d 3/2 4 168
3d 10 166
4s 1/2
2 164
2g 7/2 8 162
2g 18 156
112
1i 26 138
1i 11/2
3d
5/2
126
2g 9/2
12 154
6 142
10 136
1i 13/2 14 126
3p 3/2
2f 7/2
92
3p 6 112
3p 1/2
2 112
4 110
2f
14 106
2 f 5/ 2 6 108
8 100
1h 22 92
3s 2 70
1h 9/2
1h 11/2
82
3s 1/2
10 92
12 82
2 70
2d
1g
10 68
58
40
18 58
2d 3/2
1g 7/2
1f 5/2
1f 7/2
1g 9/2
2d 5/2
4 68
6 64
50
8 58
10 50
2p 6 40
20
8
2
1f 14 34
2p 1/2
2p 3/2
28
20
1d
2 40
6 38
4 32
8 28
2s 2 20
3/ 2
8
2s 1/2
4 20
2 16
1d
10 18
1d 5/2
1p 1/2
6 14
2
2 8
1p 6 8
1p 3/2 4 6
1s 2 2
1s 1/2 2 2
Image by MIT OpenCourseWare. After Krane.
F ig . 3 9 : S h ell M o d el en er g y lev els (fr o m Kr a n e F ig . 5 .6 ) . Left: C a lc u la ted en er g y lev els b a s ed o n p o ten tia l. T o th e r ig h t o f ea c h lev el a r e its c a p a c it y a n d c u m u la tiv e n u m b er o f n u c leo n s u p to th a t lev el. T h e s p in -o r b it in ter a c tio n s p lits th e lev els w ith l > 0 in t o t w o n ew lev els . N o te th a t th e s h ell effec t is qu ite a p p a r en t, a n d ma g ic n u m b er s a r e r ep r o d u c ed exa c tly .
5.3.4 Spi n pai ri ng and val enc e nuc l eons
In th e ex tr eme s h ell mo d el (or ex tr eme in d ep en d en t p ar ticle mo d el), th e as s u mp tion is th at on ly th e las t u n p air ed n u cleon d ictates th e p r op er ties of th e n u cleu s . A b etter ap p r o x imation w ou ld b e to con s id er all th e n u cleon s ab o v e a fi lled s h ell as con tr ib u tin g to th e p r op er ties of a n u cleu s . Th es e n u cleon s ar e called th e v alen ce n u cleon s .
Pr op er ties th at can b e p r ed icted b y th e c h ar acter is tics of th e v alen ce n u cleon s in clu d e th e magn etic d ip ole momen t, th e electr ic q u ad r u p ole momen t, th e ex cited s tates an d th e s p in -p ar it y (as w e will s ee). Th e s h ell mo d el can b e th en u s ed n ot on ly to p r ed ict ex cited s tates , b u t als o to calcu late th e r ate of tr an s ition s fr om on e s tate to an oth er d u e to r ad ioactiv e d eca y or n u clear r eaction s .
As th e p r oton an d n eu tr on lev els ar e fi lled th e n u cleon s of eac h t y p e p air off , y ield in g a zer o an gu lar momen tu m for th e p air . Th is p air in g of n u cleon s imp lies th e ex is ten ce of a p air ing for c e th at lo w er s th e en er gy of th e s y s tem wh en th e n u cleon s ar e p air ed -off .
S in ce th e n u cleon s get p air ed -off , th e total s p in an d p ar it y of a n u cleu s is on ly giv en b y th e las t u n p air ed n u cleon (s ) (wh ic h r es id e(s ) in th e h igh es t en er gy lev el). S p ecifi cally w e can h a v e eith er on e n eu tr on or on e p r oton or a p air n eu tr on -p r oton .
—
Th e p ar it y for a s in gle n u cleon is ( 1) l , an d th e o v er all p ar it y of a n u cleu s is th e p r o d u ct of th e s in gle n u cleon p ar it y . (Th e p ar it y in d icates if th e w a v efu n ction c h an ges s ign wh en c h an gin g th e s ign of th e co or d in ates . Th is is of cou r s e
≥
d ictated b y th e an gu lar p ar t of th e w a v efu n ction – as in s p h er i cal co or d in ates r 0. Th en if y ou lo ok b ac k at th e an gu lar w a v efu n ction for a cen tr al p oten tial it is eas y to s ee th at th e s p h er ical h ar mon ics c h an ge s ign iff l is o d d ). O b s . Th e s h ell mo d el with p air in g for ce p r ed icts a n u clear s p in I = 0 an d p ar it y Π = ev en (or I Π = 0 + ) for all ev en -ev en n u clid es .
A . Od d -Even n u clei
D es p ite its cr u d en es s , th e s h ell mo d el with th e s p in -or b it cor r ection d es cr ib es w ell th e s p in an d p ar it y of all o d d -A n u clei. In p ar ticu lar , all o d d -A n u clei will h a v e h alf-in teger s p in (s in ce th e n u cleon s , b ein g fer mion s , h a v e h alf-in teger s p in ).
Ex amp le: 15 O 7 an d 17 O 9 . (of cou r s e 16 O h as s p in zer o an d ev en p ar it y b ecau s e all th e n u cleon s ar e p air ed ). Th e fi r s t
8 8
8
( 15 O 7 ) h as an u n p air ed n eu tr on in th e p 1 / 2 s h ell, th an l = 1, s = 1 / 2 an d w e w ou ld p r ed ict th e is otop e to h a v e s p in
8
1/2 an d o d d p ar it y . Th e gr ou n d s tate of 17 O 9 in s tead h as th e las t u n p air ed n eu tr on in th e d 5 / 2 s h ell, with l = 2 an d
s = 5 / 2, th u s imp ly in g a s p in 5 / 2 with ev en p ar it y . Both th es e p r ed iction s ar e con fi r med b y ex p er imen ts . Ex amp les : Th es e ar e ev en -o d d n u clid es (i.e. with A o d d ).
123 7 +
→ 51 S b 72 h as 1p r oton in 1g 7 / 2 : → 2 .
133 7 +
→ 51 C s h as 1p r oton in 1g 7 / 2 : → 2 .
35 3
+
→ 17 C l h as 1p r oton in 1d 3 / 2 : → 2 .
29 1
+
→ 14 S i h as 1 n eu tr on in 2s 1 / 2 : → 2 .
28
+
→ 14 S i h as p air ed n u cleon s : → 0 .
Ex amp le: Th er e ar e s ome n u clid es th at s eem to b e ex cep tion s :
→ →
121 5 +
51 S b 70 h as las t p r oton in 2d 5 / 2 in s tead of 1g 7 / 2 :
of th e t w o lev el or d er )
2 (d etails in th e p oten tial cou ld accou n t for th e in v er s ion
147
→ S n 85 h as las t p r oton in 2f 7 / 2 in s tead of 1h 9 / 2 : →
7 − .
79
.
62 2
3
→ 35 B r 44 h as las t n eu tr on in 2p 3 / 2 in s tead of 1f 5 / 2 : → 2 −
→
207
82 P b 125 . Her e w e in v er t 1i 13 / 2 with 3p 1 / 2 . Th is s eems to b e wr on g b ecau s e th e 1i lev el m u s t b e q u ite mor e
en er getic th an th e 3p on e. Ho w ev er , wh en w e mo v e a n eu tr on fr om th e 3p to th e 1i all th e n eu tr on s in th e 1i lev el ar e n o w p air ed , th u s lo w er in g th e en er gy of th is n ew con fi gu r ation .
→ 61 N i 33 1f 5 / 2 → → 2p 3 / 2 → ( 3 − )
28 2
197 3 +
→ 79 Au 118 1f 5 / 2 → → 3p 3 / 2 → ( 2 )
B . Od d -Od d n u clei
O n ly fi v e s tab le n u clid es con tain b oth an o d d n u m b er of p r oton s an d an o d d n u m b er of n eu tr on s : th e fi r s t fou r o d d -o d d n u clid es 2 H, 6 Li, 10 B, an d 14 N. Th es e n u clid es h a v e t w o u n p air ed n u cleon s (or o d d -o d d n u clid es ), th u s th eir
| − |
1 3 5 7
s p in is mor e comp licated to calcu late. Th e total an gu lar momen tu m can th en tak e v alu es b et w een j 1 j 2 an d j 1 + j 2 .
Tw o p r o ces s es ar e at p la y :
1) th e n u clei ten d s to h a v e th e s malles t an gu lar momen tu m, an d
2) th e n u cleon s p in s ten d to align (th is w as th e s ame eff ect th at w e s a w for ex amp le in th e d eu ter on In an y cas e, th e r es u ltan t n u clear s p in is goin g to b e an in teger n u m b er .
C. Nu clea r Ma gn etic R eson a n ce
Th e n u clear s p in is imp or tan t in c h emical s p ectr os cop y an d med ical imagin g. Th e man ip u lation of n u clear s p in b y r ad iofr eq u en cy w a v es is at th e b as is of n u clear magn etic r es on an ce an d of magn etic r es on an ce imagin g. Th en , th e s p in p r op er t y of a p ar ticu lar is otop e can b e p r ed icted wh en y ou k n o w th e n u m b er of n eu tr on s an d p r oton s an d th e s h ell mo d el. F or ex amp le, it is eas y to p r ed ict th at h y d r ogen , wh ic h is p r es en t in mos t of th e liv in g cells , will h a v e s p in 1/2. W e alr ead y s a w th at d eu ter on in s tead h as s p in 1. W h at ab ou t Car b on , wh ic h is als o common ly fou n d in
b iomolecu les ? 12 C is of cou r s e an d ev en -ev en n u cleu s , s o w e ex p ect it to h a v e s p in -0. 13 C 7 in s tead h as on e u n p air ed
6 6
2
n eu tr on . Th en 13 C h as s p in - 1 .
W h y can n u clear s p in b e man ip u lated b y electr omagn etic fi eld s? T o eac h s p in th er e is an as s o ciated magn etic d ip ole, giv en b y :
r
N
µ = g µ N I = γ I
=
wh er e γ N is called th e gy r omagn etic r atio, g is th e g-factor (th at w e ar e goin g to ex p lain ) an d µ N is th e n u clear
magn eton µ N
e n
2 m
≈ 3 × 10 − 8 eV/T (with m th e p r oton mas s ). Th e g factor is d er iv ed fr om a com b in ation of th e
an gu lar momen tu m g-factor an d th e s p in g-factor . F or p r oton s g l = 1, wh ile it is g l = 0 for n eu tr on s as th ey d on ’t
—
h a v e an y c h ar ge. Th e s p in g-factor can b e calcu lated b y s olv in g th e r elativ is tic q u an tu m mec h an ics eq u ation , s o it is a p r op er t y of th e p ar ticles th ems elv es (an d a d imen s ion les s n u m b er ). F or p r oton s an d n eu tr on s w e h a v e: g s,p = 5 . 59 an d g s,n = 3 . 83.
In or d er to h a v e an op er ation al d efi n ition of th e magn etic d ip ole as s o ciated to a giv en an gu lar momen tu m, w e d efi n e it to b e th e ex p ectation v alu e of µ ˆ wh en th e s y s tem is in th e s tate with th e max im u m z an gu lar momen tu m:
⟨ µ ⟩ =
µ N
k ⟨ g l l z + g s s z ⟩ =
µ N
k ⟨ g l j z + ( g s − g l ) s z ⟩
Th en u n d er ou r as s u mp tion s j z = j k (an d of cou r s e l z = k m z an d s z = k m s ) w e h a v e
µ N
⟨ µ ⟩ = k ( g l j k + ( g s − g l ) ⟨ s z ⟩ )
2
2
Ho w can w e calcu late s z ? Th er e ar e t w o cas es , eith er j = l + 1 or j = l − 1 . An d n otice th at w e w an t to fi n d th e
R ˆ R ˆ | ˆ ˆ ˆ
|
p r o jection of S in th e s tate wh ic h is align ed with J , s o w e w an t th e ex p ectation v alu e of
S · J | J . By r ep lacin g th e
| ˆ 2
op er ator s with th eir ex p ectation v alu es (in th e cas e wh er e j z
J
= j r ), w e ob tain
⟨ ⟩
k 1
s z = + 2 for j = l + 2 .
k j 1
⟨ s z ⟩ = − 2 j + 1 for j = l − 2 .
(th u s w e h a v e a s mall cor r ection d u e to th e fact th at w e ar e tak in g an ex p ectation v alu e with r es p ect to a tilted s tate an d n ot th e u s u al s tate align ed with S ˆ z . Remem b er th at th e s tate is w ell d efi n ed in th e cou p led r ep r es en tation , s o th e u n cou p led r ep r es en tation s tates ar e n o lon ger go o d eigen s tates ).
F in ally th e d ip ole is
⟨ µ ⟩ = µ g ( j − 1 ) + g s
2
for j = l + 1 an d
N l 2 2
⟨ µ ⟩ = µ N g l
j ( j + 3 )
2
g l j + 1 −
g s 1 2 j + 1
oth er wis e. Notice th at th e ex act g-factor or gy r omagn etic r atio of an is otop e is d ifficu lt to calcu late: th is is ju s t an ap p r o x imation b as ed on th e las t u n p air ed n u cleon mo d el, in ter action s amon g all n u cleon s s h ou ld in gen er al b e tak en in to accou n t.
D. Mo re comp lex stru ctu res
O th er c h ar acter is tics of th e n u clear s tr u ctu r e can b e ex p lain ed b y mor e comp lex in ter action s an d mo d els . F or ex amp le all ev en -ev en n u clid es p r es en t an an omalou s 2 + ex cited s tate (S in ce all ev en -ev en n u clid es ar e 0 + w e h a v e to lo ok at th e ex cited lev els to lear n mor e ab ou t th e s p in con fi gu r ation .) Th is is a h in t th at th e p r op er ties of all n u cleon s p la y a r ole in to d efi n in g th e n u clear s tr u ctu r e. Th is is ex actly th e ter ms in th e n u cleon s Hamilton ian th at w e h ad d ecid ed to n eglect in fi r s t ap p r o x imation . A d iff er en t mo d el w ou ld th en to con s id er all th e n u cleon s (in s tead of a s in gle n u cleon s in an ex ter n al p oten tial) an d d es cr ib e th eir p r op er t y in a collectiv e w a y . Th is is s imilar to a liq u id d r op mo d el. Th en imp or tan t p r op er ties will b e th e v ib r ation s an d r otation s of th is mo d el.
A d iff er en t ap p r oac h is for ex amp le to con s id er n ot on ly th e eff ects of th e las t u n p air ed n u cleon b u t als o all th e n u cleon s ou ts id e th e las t clos ed s h ell. F or mor e d etails on th es e mo d els , s ee K r an e.
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22.02 Introduction to Applied Nuclear Physics
Spring 2012
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