7. R adi oac ti ve dec a y
7.1 Gamma de c a y
7 .1 .1 C la s s ic a l th eo r y o f r a d ia tio n
7 .1 .2 Qu a n t u m mec h a n ic a l th eo r y
7 .1 .3 E xten s io n to M u ltip o les
7 .2 .1 R ea c tio n s a n d p h en o men o lo g y
7 .2 .2 C o n s er v a tio n la w s
7 .2 .3 F er mi’s T h eo r y o f B eta Dec a y
Rad ioactiv e d eca y is th e p r o ces s in wh ic h an u n s tab le n u cleu s sp on tan eou s ly los es en er gy b y emittin g ion izin g p ar ticles an d r ad iation . Th is d eca y , or los s of en er gy , r es u lts in an atom of on e t y p e, called th e p aren t n u clid e, tr an s for min g to an atom of a d iff er en t t y p e, n amed th e d au gh ter n u clid e.
Th e th r ee p r in cip al mo d es of d eca y ar e called th e alp h a, b eta an d gamma d eca y s . W e alr ead y in tr o d u ced th e gen er al p r in cip les of r ad ioactiv e d eca y in S ection 1.3 an d w e s tu d ied mor e in d ep th alp h a d eca y in S ection 3.3 . In th is c h ap ter w e con s id er th e oth er t w o t y p e of r ad ioactiv e d eca y , b eta an d gamma d eca y , mak in g u s e of ou r k n o wled ge of q u an tu m mec h an ics an d n u clear s tr u ctu r e.
7. 1 Gamma dec a y
G amma d eca y is th e th ir d t y p e of r ad ioactiv e d eca y . Un lik e th e t w o oth er t y p es of d eca y , it d o es n ot in v olv e a c h an ge in th e elemen t. It is ju s t a s imp le d eca y fr om an ex cited to a lo w er (gr ou n d ) s tate. In th e p r o ces s of cou r s e s ome en er gy is r eleas ed th at is car r ied a w a y b y a p h oton . S imilar p r o ces s es o ccu r in atomic p h y s ics , h o w ev er th er e th e en er gy c h an ges ar e u s u ally m u c h s maller , an d p h oton s th at emer ge ar e in th e v is ib le s p ectr u m or x -r a y s .
Th e n u clear r eaction d es cr ib in g gamma d eca y can b e wr itten as
A X ∗ → A X + γ
Z Z
wh er e ∗ in d icates an ex cited s tate.
W e h a v e s aid th at th e p h oton car r ies a w a y s s ome en er gy . It als o car r ies a w a y momen tu m, an gu lar momen tu m an d p ar it y (b u t n o mas s or c h ar ge) an d all th es e q u an tities n eed to b e con s er v ed . W e can th u s wr ite an eq u ation for th e en er gy an d momen tu m car r ied a w a y b y th e gamma-p h oton .
�
F r om s p ecial r elativ it y w e k n o w th at th e en er gy of th e p h oton (a mas s les s p ar ticle) is
E = m 2 c 4 + p 2 c 2 → E = pc
2 m
≪ ≈
(wh ile for mas s iv e p ar ticles in th e n on -r elativ is tic limit v c w e h a v e E mc 2 + p 2 .) In q u an tu m mec h an ics w e h a v e s een th at th e momen tu m of a w a v e (an d a p h oton is w ell d es cr ib ed b y a w a v e) is p = h k with k th e w a v e n u m b er . Th en w e h a v e
E = h k c = h ω k
2 m
Th is is th e en er gy for p h oton s wh ic h als o d efi n es th e fr eq u en cy ω k = k c (comp ar e th is to th e en er gy for mas s iv e p ar ticles , E = n 2 k 2 ).
k
G amma p h oton s ar e p ar ticu lar ly en er getic b ecau s e th ey d er iv e fr om n u clear tr an s ition s (th at h a v e m u c h h igh er en er gies th an e.g. atomic tr an s ition s in v olv in g electr on ic lev els ). Th e en er gies in v olv ed r an ge fr om E ∼ . 1 ÷ 10M eV, giv in g k ∼ 10 − 1 ÷ 10 − 3 fm − 1 . Th an th e w a v elen gth s ar e λ = 2 π ∼ 100 ÷ 10 4 fm, m u c h lon ger th an th e t y p ical n u clear
d imen s ion s .
G amma r a y s p ectr os cop y is a b as ic to ol of n u clear p h y s ics , for its eas e of ob - E i, I i, Π i
s er v ation (s in ce it’s n ot ab s or b ed in air ), accu r ate en er gy d eter min ation an d
E γ =ħω=E i- E f Π γ =Π i Π f L γ
in for mation on th e s p in an d p ar it y of th e ex cited s tates .
Als o, it is th e mos t imp or tan t r ad iation u s ed in n u clear med icin e.
7.1.1 Cl as s i c al theo ry of radi ati on
F r om th e th eor y of electr o d y n amics it is k n o wn th at an acceler atin g c h ar ge
E f, I f, Π f
r ad iates . Th e p o w er r ad iated is giv en b y th e in tegr al of th e en er gy fl u x (as F ig . 4 2 : S c h ema tic s o f g a mma d ec a y
giv en b y th e P o y n tin g v ector ) o v er all s olid an gles . Th is giv es th e r ad iated
p o w er as :
P =
2 e 2 | a | 2
3
c 3
wh er e a i s th e acceler ation . Th is is th e s o-called Lar mor for m u la for a n on -r elativ is tic acceler ated c h ar ge.
Ex amp le. As an imp or tan t ex amp le w e con s id er an electr ic d ip ole. An ele ctr ic d ip ole can b e con s id er ed as an os cillatin g c h ar ge, o v er a r an ge r 0 , s u c h th at th e electr ic d ip ole is giv en b y d ( t ) = q r ( t ). Th en th e eq u ation of motion is
an d th e acceler ation
Av er aged o v er a p er io d T = 2 π /ω , th is is
r ( t ) = r 0 cos ( ω t )
a = r ¨ = − r 0 ω 2 cos ( ω t )
a 2 = dta ( t ) = r 2 ω 4
� � ω ∫ T 1
2 π 0 2 0
F in ally w e ob tain th e r ad iativ e p o w er for an electr ic d ip ole:
2
1 e 2 ω 4
P E 1 = 3 c 3
| ' r 0 |
A . Electroma gn etic mu ltip oles
In or d er to d eter min e th e clas s ical e.m. r ad iation w e n eed to ev alu ate th e c h ar ge d is tr ib u tion th at giv es r is e to it. Th e electr os tatic p oten tial of a c h ar ge d is tr ib u tion ρ e ( r ) is giv en b y th e in tegr al:
V ( ' r ) = 1 ∫
ρ e ( r ' ′ )
4 π ǫ 0 V o l ′ | ' r − r ' ′ |
| ! r − r ! ′ |
r r r
J ( /
W h en tr eatin g r ad iation w e ar e on ly in ter es ted in th e p oten tial ou ts id e th e c h ar ge an d w e can as s u me th e c h ar ge (e.g. a p ar ticle!) to b e w ell lo calized ( r ′ ≪ r ). Th en w e can ex p an d 1 in p o w er s er ies . F ir s t, w e ex p r es s ex p licitly th e n or m | ' r − r ' ′ | = √ r 2 + r ′ 2 − 2 r r ′ cos ϑ = r 1 + r ′ 2 − 2 r ′ cos ϑ . W e s et R = r ′ an d ǫ = R 2 − 2 R cos ϑ : th is is a s mall q u an tit y , giv en th e as s u mp tion r ′ ≪ r . Th en w e can ex p an d :
1 1 1
= √
= 1 ( 1 − 1 3
2 − 5 ǫ 3 + . . . )
ǫ + ǫ
| ' r − r ' ′ | r 1 + ǫ r
Rep lacin g ǫ with its ex p r es s ion w e h a v e:
2 8 16
1 √ 1 = 1 ( 1 − 1 ( R 2 −
3
2 R cos ϑ ) + ( R
2 − 2 R cos ϑ ) 2 − 5 ( R 2 − 2 R cos ϑ ) 3 + . . . )
r 1 + ǫ r 2
=
r
1 + [ − 2 R + R cos ϑ ] + [ 8 R
— 2 R
1 ( 1 2 3 4 3 3
8
cos ϑ + R
2
cos
3 2 2
16
ϑ ] + [ − 16
+
R
8
5 R 6 15 5
15 4 2
cos ( ϑ ) −
R
4
cos ( ϑ ) + R
2
cos ( ϑ )] + . . .
5 3 3 )
=
r
1 + R cos ϑ + R
— 2
+ R
—
+ . . .
1 ( 2 ( 3 cos 2 ϑ 1 ) 3 ( 5 cos 3 ( ϑ ) 3 cos ( ϑ ) ) )
2
2
2
W e r ecogn ized in th e co efficien ts to th e p o w er s of R th e Legen d r e P oly n omials P l (cos ϑ ) (with l th e p o w er of R l , an d n ote th at for p o w er s > 3 w e s h ou ld h a v e in clu d ed h igh er ter ms in th e or igin al ǫ ex p an s ion ):
∞
1 1 1 0 l
1 0 ∞ ( r ′ ) l
r √ 1 + ǫ = r
R P l (cos ϑ ) = r r
l = 0 l = 0
P l (cos ϑ )
W ith th is r es u lt w e can as w ell calcu late th e p oten tial:
V ( ' r ) = ρ ( ' r ′ ) P l (cos ϑ ) d r ' ′
4 π ǫ 0 r V o l ′ r l = 0
1 1 ∫
1 0 ∞ ( r ′ ) l
r
Th e v ar iou s ter ms in th e ex p an s ion ar e th e m u ltip oles . Th e few lo w es t on es ar e :
1 1 4 π ǫ 0 r
∫
V o l ′
ρ ( ' r ′ ) d r ' ′ = Q
4 π ǫ 0 r
M on op ole
1 1 ∫
ρ ( ' r ′ ) r ′ P (cos ϑ ) d r ' ′ = 1 1 ∫
ρ ( ' r ′ ) r ′ cos ϑ d r ' ′ =
r ˆ · d '
D ip ole
∫
∫
1
4 π ǫ 0 r 2 V o l ′
1 1
4 π ǫ 0 r 2 V o l
ρ ( ' r ′ ) r ′ 2 P 2 (cos ϑ ) d r ' ′ = ρ ( ' r ′ ) r ′ 2
1 1
′ 4 π ǫ 0 r 2
cos 2 ϑ − d r ' ′ Q u ad r u p ole
3 1
4 π ǫ 0 r 3 V o l ′
4 π ǫ 0 r 3
V o l ′ 2 2
( )
Th is t y p e of ex p an s ion can b e car r ied ou t as w ell for th e magn etos tatic p oten tial an d for th e electr omagn etic, time-d ep en d en t fi eld .
A t lar ge d is tan ces , th e lo w es t or d er s in th is ex p an s ion ar e th e on ly imp or tan t on es . Th u s , in s tead of con s id er in g th e total r ad iation fr om a c h ar ge d is tr ib u tion , w e can ap p r o x imate it b y con s id er in g th e r ad iation ar is in g fr om th e fi r s t few m u ltip oles : i.e. r ad iation fr om th e electr ic d ip ole, th e magn etic d ip ole, th e electr ic q u ad r u p ole etc.
Eac h of th es e r ad iation ter ms h a v e a p ecu liar an gu lar d ep en d en ce. Th is will b e r efl ected in th e q u an tu m mec h an ical tr eatmen t b y a s p ecifi c an gu lar momen tu m v alu e of th e r ad iation fi eld as s o ciated with th e m u ltip ole. In tu r n s , th is will giv e r is e to s election r u les d eter min ed b y th e ad d ition r u les of an gu lar momen tu m of th e p ar ticles an d r ad iation i n v olv ed in th e r ad iativ e p r o ces s .
7.1.2 Quantum mec hani c al theo ry
In q u an tu m mec h an ics , gamma d eca y is ex p r es s ed as a tr an s ition fr om an ex cited to a gr ou n d s tate of a n u cleu s . Th en w e can s tu d y th e tr an s ition r ate of s u c h a d eca y v ia F er mi’s G old en r u le
2 π
W = h |� ψ f
| V ˆ | ψ i � | 2 ρ ( E f )
Th er e ar e t w o imp or tan t in gr ed ien ts in th is for m u la, th e d en s it y of s tates ρ ( E f ) an d th e in ter action p oten tial V ˆ .
n y
n
n x
A . Den sit y of sta tes
d E f
Th e d en s it y of s tates is d efi n ed as th e n u m b er of a v ailab le s tates p er en er gy : ρ ( E f ) = d N s , wh er e N s is th e n u m b er of s tates . W e h a v e s een
dn
at v ar iou s time th e con cep t of d egen er acy : as eigen v alu es of an op er ator can b e d egen er ate, th er e migh t b e mor e th an on e eigen fu n ction s h ar in g th e s ame eigen v alu es . In th e cas e of th e Hamilton ian , wh en th er e ar e d egen er acies it mean s th at mor e th an on e s tate s h ar e th e s ame en er gy .
By con s id er in g th e n u cleu s + r ad iation to b e en clos ed in a ca v it y of v olu me L 3 , w e h a v e for th e emitted p h oton a w a v efu n ction r ep r es en ted b y th e s olu tion of a p ar ticle in a 3D b o x th at w e s a w in a Pr ob lem S et.
As for th e 1D cas e, w e h a v e a q u an tization of th e momen tu m (an d h en ce of th e w a v e-n u m b er k ) in or d er to fi t th e w a v efu n ction in th e
b o x . Her e w e ju s t h a v e a q u an tization in all 3 d ir ection s : F ig . 4 3 : Den s it y o f s t a tes : c o u n tin g th e s ta t es
( 2 D )
2 π 2 π 2 π
k x =
L n x , k y = L n y , k z = L n z ,
( 2 π ) 3
(with n in teger s ). Th en , goin g to s p h er ical co or d in ates , w e can cou n t th e n u m b er of s tates in a s p h er ical s h ell b et w een
n an d n + dn to b e dN s
= 4 π n 2 dn . Ex p r es s in g th is in ter ms of k , w e h a v e dN s
= 4 π k 2 dk L 3
. If w e con s id er ju s t a
=
s mall s olid an gle dΩ in s tead of 4 π w e h a v e th en th e n u m b er of s tate dN s
fi n ally ob tain th e d en s it y of s tates :
L 3
( 2 π ) 3
k 2 dk dΩ . S in ce E = h k c = h ω , w e
dN s L 3
2 dk L 3 k 2
ω 2 L 3
ρ ( E ) =
dE
= (2 π ) 3 k
dE dΩ = (2 π ) 3 h c dΩ = h c 3 (2 π ) 3 dΩ
B . Th e vecto r p oten tia l
Nex t w e con s id er th e p oten tial cau s in g th e tr an s ition . Th e in ter action of a p ar ticle with th e e.m. fi eld can b e ex p r es s ed
'
A
ˆ in ter ms of th e v ector p oten tial of th e e.m. fi eld as :
V ˆ =
e ' ˆ ˆ
mc A · p '
ˆ '
ˆ
wh er e p ' is th e p ar ticle’s momen tu m. Th e v ector p oten tial A in Q M is an op er ator th at can cr e ate or annihil ate
p h oton s ,
A =
( a ˆ k e
' ˆ 0 2 π h c 2
k
V ω
k
i ! k · ! r
+ a ˆ k e
) ' ǫ k
† − i ! k · ! r
wh er e a ˆ k ( a ˆ k † ) an n ih ilates (cr eates ) on e p h oton of momen tu m ' k . Als o, ' ǫ k is th e p olar ization of th e e.m. fi eld . S in ce gamma d eca y (an d man y oth er atomic an d n u clear p r o ces s es ) is ab le to cr eate p h oton s (or ab s or b th em) it mak es s en s e th at th e op er ator d es cr ib in g th e e.m. fi eld w ou ld b e ab le to d es cr ib e th e cr eation an d an n ih ilation of p h oton s .
∝
Th e s econ d c h ar acter is tic of th is op er ator ar e th e ter ms e − i ! k · ! r wh ic h d es cr ib e a p lan e w a v e, as ex p ected for e.m. w a v es , with momen tu m h k an d fr eq u en cy c k .
C. Dip ole tra n sition fo r ga mma d eca y
T o calcu late th e tr an s ition r ate fr om th e F er mi’s G old en r u le,
2 π
h
f
W = |� ψ
| V ˆ | ψ ⟩ | 2 ρ ( E ) ,
⟨ | | ⟩
i
f
w e ar e r eally on ly in ter es ted in th e matr ix elemen t ψ f V ˆ ψ i , wh er e th e in itial s tate d o es n ot h a v e an y p h oton , an d th e fi n al h as on e p h oton of momen tu m h k an d en er gy h ω = h k c . Th en , th e on ly elemen t in th e s u m ab o v e for th e v ector p oten tial th at giv es a n on -zer o con tr ib u tion will b e th e ter m ∝ a ˆ † k , with th e ap p r op r iate ' k momen tu m:
e V if = mc
2 π h c 2
V ω k
' ǫ k ·
( p ' ˆ e − i ! k · ! r )
2 n
Th is can b e s imp lifi ed as follo w. Remem b er th at [ p ' ˆ 2 , ' r ˆ ] = − 2 i h p ' ˆ . Th u s w e can wr ite, p ' ˆ =
i [ p ' ˆ 2 , ' r ˆ ] = im [ p ! ˆ 2 , ' r ˆ ] =
n
n
2 m
nuc
2 m
nuc
n
nuc
im [ p ! ˆ 2 + V
( ' r ˆ ) , ' r ˆ ]. W e in tr o d u ced th e n u clear Hamilton ian H
= p ! ˆ 2 + V
( ' r ˆ ): th u s w e h a v e p ' ˆ = im [ H
, ' r ˆ ].
2 m
nuc
T ak in g th e ex p ectation v alu e
f
i
h
f
nuc
⟨ ψ | p ' ˆ | ψ ⟩ = im � ⟨ ψ | H
i
' r ˆ | ψ ⟩ − ⟨ ψ
| ' r ˆ H
| ψ ⟩ �
f
nuc
i
an d r emem b er in g th at | ψ i,f ⟩ ar e eigen s tates of th e Hamilton ian , w e h a v e
f
i
h
f
⟨ ψ | p ' ˆ | ψ ⟩ = im ( E
i
— E ) ⟨ ψ
| ' r ˆ | ψ ⟩ = imω
⟨ ψ | ' r ˆ | ψ ⟩ ,
f
i
k
f
i
wh er e w e u s ed th e fact th at ( E f − E i ) = h ω k b y con s er v ation of en er gy . Th u s w e ob tain
V if = mc V ω
imω ' ǫ k · ' r e
= i
' ǫ k ·
' r e
e 2 π h c 2 ( ˆ − i ! k · ! r ) � 2 π h e 2 ω k
k
V
( ˆ − i ! k · ! r )
· ≪
can mak e an ex p an s ion in s er ies : e − ! k · ! r ∼ L 1 ( − i ' k · ' r ) l = L 1 ( − ik r cos ϑ ) l . Th is s er ies is v er y s imilar in mean in g
W e h a v e s een th at th e w a v elen gth s of gamma p h oton s ar e m u c h lar ger th an th e n u clear s ize. Th en ' k ' r 1 an d w e
l l !
to th e m u ltip ole s er ies w e s a w for th e clas s ical cas e.
F or ex amp le, for l = 0 w e ob tain :
l l !
V if =
V
' r · ' ǫ k
r 2 π h e 2 ω k ( ˆ )
( ) ( )
wh ic h is th e d ip olar ap p r o x imation , s in ce it can b e wr itten als o u s in g th e electr ic d ip ole op er ator e ' r ˆ . Th e an gle b et w een th e p olar ization of th e e.m. fi eld an d th e p os ition ' r ˆ is ' r ˆ · ' ǫ = ' r ˆ s in ϑ
Th e tr an s ition r ate for th e d ip ole r ad iation , W ≡ λ ( E 1) is th en :
λ ( E 1) =
2 π ˆ
2 ω 3
( ˆ ) 2 2
0 0 3
h |� ψ f | V | ψ i ⟩ | ρ ( E f ) = 2 π c 3 h | ' r | s in ϑ dΩ
an d in tegr atin g o v er all p os s ib le d ir ection of emis s ion ( J 2 π dϕ J π (s in 2 ϑ ) s in ϑ dϑ = 2 π 4 ):
λ ( E 1) = 3 h c 3 |
' r
|
4 e 2 ω 3
( ˆ ) 2
| ' r |
M u ltip ly in g th e tr an s ition r ate (or p h oton s emitted p er u n it time) b y th e en er gy of th e p h oton s emitted w e ob tain th e r ad iated p o w er , P = W h ω :
P =
3
4 e 2 ω 4
c 3
( ˆ ) 2
Notice th e s imilar it y of th is for m u la with th e clas s ical cas e:
1 e 2 ω 4
P E 1 = 3 c 3
| ' r 0 |
( )
2
W e can es timate th e tr an s ition r ate b y u s in g a t y p ical en er gy E = h ω for th e p h oton emitted (eq u al to a t y p ical en er gy d iff er en ce b et w een ex cited an d gr ou n d s tate n u clear lev els ) an d th e ex p ectation v alu e for th e d ip ole ( | ' r ˆ | ∼
R nuc ≈ r 0 A 1 / 3 ). Th en , th e tr an s ition r ate is ev alu ated to b e
e 2 E 3
2 2 / 3
14 2 / 3 3
λ ( E 1) = h c ( h c ) 3 r 0 A = 1 . 0 × 10 A E
≈ ×
(with E in M eV). F or ex amp le, for A = 64 an d E = 1M eV th e r ate is λ 1 . 6 10 15 s − 1 or τ = 10 − 15 (fem tos econ d s !) for E = 0 . 1M eV τ is on th e or d er of p icos econ d s .
O b s . Becau s e of th e lar ge en er gies in v olv ed , v er y fas t p r o ces s es a r e ex p ected in th e n uclear d eca y fr om ex cited s tates , in accor d an ce with F er mi’s G old en r u le an d th e en er gy /time u n cer tain t y r elation .
7.1.3 Extens i on to Mul ti p ol es
W e ob tain ed ab o v e th e tr an s ition r ate for th e electr ic d ip ole, i.e. wh en th e in ter action b et w een th e n u cleu s an d th e
±
e.m. fi eld is d es cr ib ed b y an electr ic d ip ole an d th e emitted r ad iation h as th e c h ar acter of electr ic d ip ole r ad iation . Th is t y p e of r ad iation can on ly car r y ou t of th e n u cleu s on e q u an tu m of an gu lar momen tu m (i.e. Δl = 1, b et w een ex cited an d gr ou n d s tate). In gen er al, ex cited lev els d iff er b y mor e th an 1 l , th u s th e r ad iation emitted n eed to b e a h igh er m u ltip ole r ad iation in or d er to con s er v e an gu lar momen tu m.
A . Electric Mu ltip oles
W e can go b ac k to th e ex p an s ion of th e r ad iation in ter action in m u ltip oles :
~ 0
V ˆ 1
l !
ˆ
' ˆ l
( i k · ' r )
Th en th e tr an s ition r ate b ecomes :
l
λ ( E l ) = l [(2 l + 1)!!] 2 h c
8 π ( l + 1) e 2 ( E ) 2 l + 1 ( 3 ) 2
h c
l + 3
( ˆ ) 2 l
( )
c
| ' r |
Notice th e s tr on g d ep en d en ce on th e l q u an tu m n u m b er . S ettin g again | ' r ˆ | ∼ r 0 A 1 / 3 w e als o h a v e a s tr on g d ep en d en ce on th e mas s n u m b er .
Th u s , w e h a v e th e follo win g es timates for th e r ates of d iff er en t electr ic m u ltip oles :
- λ ( E 1 ) = 1 . 0 × 10 14 A 2 / 3 E 3
- λ ( E 2) = 7 . 3 × 10 7 A 4 / 3 E 5
- λ ( E 3) = 34 A 2 E 7
- λ ( E 4) = 1 . 1 × 10 − 5 A 8 / 3 E 9
B . Ma gn etic Mu ltip oles
Th e e.m. p oten tial can als o con tain magn etic in ter action s , lead in g to magn etic tr an s ition s . Th e tr an s ition r ates can b e calcu lated fr om a s imilar for m u la:
λ ( M l ) = l [(2 l + 1)!!] 2 h c h c l + 3
µ p − l + 1
8 π ( l + 1) e 2 E 2 l + 1 ( 3 ) 2
( ˆ ) 2 l − 2 � h
( 1 )�
m p c
l + 1
c
| ' r |
wh er e µ p is th e magn etic momen t of th e p r oton (an d m p its mas s ). Es timates for th e tr an s ition r ates can b e fou n d b y s ettin g µ p − 1 ≈ 10:
- λ ( M 1) = 5 . 6 × 10 13 E 3
- λ ( M 2) = 3 . 5 × 10 7 A 2 / 3 E 5
- λ ( M 3) = 16 A 4 / 3 E 7
- λ ( M 4) = 4 . 5 × 10 − 6 A 2 E 9
7.1.4 Sel ec ti on R ul es
Th e an gu lar momen tu m m u s t b e con s er v ed d u r in g th e d eca y . Th u s th e d iff er en ce in an gu lar momen tu m b et w een th e in itial (ex cited ) s tate an d th e fi n al s tate is car r ied a w a y b y th e p h oton emitted . An oth er con s er v ed q u an tit y is th e total p ar it y of th e s y s tem.
A . P a rit y ch a n ge
Th e p ar it y of th e gamma p h oton is d eter min ed b y its c h ar acter , eith er magn etic or electr ic m u ltip ole. W e h a v e
Π γ ( E l ) = ( − 1) l
Π γ ( M l ) = ( − 1) l − 1
Electr ic m u ltip ole M agn etic m u ltip ole
→
Th en if w e h a v e a p ar it y c h an ge fr om th e in itial to th e fi n al s tate Π i Π f th is is accou n ted for b y th e emitted p h oton as :
Π γ = Π i Π f
Th is of cou r s e limits th e t y p e of m u ltip ole tr an s ition s th at ar e allo w ed giv en an in itial an d fi n al s tate.
ΔΠ = n o → Ev en Electr ic, O d d M agn etic
ΔΠ = y es → O d d Electr ic, Ev en M agn etic
B . A n gu la r momen tu m
F r om th e con s er v ation of th e an gu lar momen tu m:
' ˆ ' ˆ ' ˆ
I i = I f + L γ
th e allo w ed v alu es for th e an gu lar momen tu m q u an tu m n u m b er of th e p h oton , l , ar e r es tr icted to
l γ = | I i − I f | , . . . , I i + I f
O n ce th e allo w ed l h a v e b een fou n d fr om th e ab o v e r elation s h ip , th e c h ar acter (magn etic or electr ic) of th e m u ltip ole is fou n d b y lo ok in g at th e p ar it y .
In gen er al th en , th e mos t imp or tan t tr an s ition will b e th e on e with th e lo w es t allo w ed l , Π . High er m u ltip oles ar e als o p os s ib le, b u t th ey ar e goin g to lead to m u c h s lo w er p r o ces s es .
|
M u ltip olar it y |
An gu lar M omen tu m l |
P ar it y Π |
M u ltip olar it y |
An gu lar M omen tu m l |
P ar it y Π |
|
M 1 M 2 M 3 M 4 M 5 |
1 2 3 4 5 |
+ - + - + |
E1 E2 E3 E4 E5 |
1 2 3 4 5 |
- + - + - |
T a b le 3 : A n g u la r mo men tu m a n d p a r it y o f th e g a mma m u ltip o les
C. Domin a n t Deca y Mo d es
In gen er al w e h a v e th e follo win g p r ed iction s of wh ic h tr an s ition s will h ap p en :
1. Th e lo w es t p er mitted m u ltip ole d omin ates
~
2. Electr ic m u ltip oles ar e mor e p r ob ab le th an th e s ame magn etic m u ltip ole b y a factor 10 2 (h o w ev er , wh ic h on e is goin g to h ap p en d ep en d s on th e p ar it y )
λ ( E l ) 2
λ ( M l ) ≈ 10
3. Emis s ion fr om th e m u ltip ole l + 1 is 10 − 5 times les s p r ob ab le th an th e l -m u ltip ole emis s ion .
λ ( E l ) λ ( M l )
λ ( E , l + 1) ≈ 10 − 5 , λ ( M , l + 1) ≈ 10 − 5
4. Com b in in g 2 an d 3, w e h a v e:
λ ( E , l + 1) λ ( M , l + 1)
≈ 10 − 3 , ≈ 10 − 7
λ ( M l ) λ ( E l )
Th u s E 2 comp etes with M 1 wh ile th at’s n ot th e cas e for M 2 v s . E 1
D. In tern a l con version
W h at h ap p en if n o allo w ed tr an s ition s can b e fou n d ? Th is is th e cas e for ev en -ev en n u clid es , wh er e th e d eca y fr om th e 0 + ex cited s tate m u s t h ap p en with ou t a c h an ge in an gu lar momen tu m. Ho w ev er , th e p h oton alw a y s car r ies s ome an gu lar momen tu m, th u s gamma emis s ion is imp os s ib le.
Th en an oth er p r o ces s h ap p en s , called i n tern al con v ersi on :
Z
Z
A X ∗ → A X + e −
Z
wh er e A X is a ion ized s tate an d e − is on e of th e atomic electr on s .
Bes id es th e cas e of ev en -ev en n u clei, in ter n al con v er s ion is in gen er al a comp etin g p r o ces s of gamma d eca y (s ee K r an e for mor e d etails ).
7. 2 Beta dec a y
Th e b eta d eca y is a r ad ioactiv e d eca y in wh ic h a p r oton in a n u cleu s is con v er ted in to a n eu tr on (or v ice-v er s a). In th e p r o ces s th e n u cleu s emits a b eta p ar ticle (eith er an electr on or a p os itr on ) an d q u as i-mas s les s p ar ticle, th e n eu tri n o .
Courtesy of Thomas Jefferson National Accelerator Facility - Office of Science Education. Used with permission.
F ig . 4 4 : B eta d ec a y s c h ema tic s
Recall th e mas s c h ain an d Beta d eca y p lots of F ig. 7 . W h en s tu d y in g th e b in d in g en er gy fr om th e S EM F w e s a w th at at fi xed A th er e w as a min im u m in th e n u clear mas s for a p ar ticu lar v alu e of Z . In or d er to r eac h th at min im u m, u n s tab le n u clid es u n d er go b eta d eca y to tr an s for m ex ces s p r oton s in n eu tr on s (an d v ice-v er s a).
7.2.1 R eac ti ons and phenomenol ogy
Th e b eta-d eca y r eaction is wr itten as :
Z
A X N →
A Z + 1
X N ′ − 1
+ e − + ν ¯
Th is is th e β − d eca y . (or n egativ e b eta d eca y ) Th e u n d er ly in g r eaction is :
n → p + e − + ν ¯
wh ic h con v er ts a p r oton in to a n eu tr on with th e emis s ion of an electr on an d an an ti-n eu tr in o. Th er e ar e t w o oth er t y p es of r eaction s , th e β + r eaction ,
Z
A X N
A
→ Z − 1
X N ′ + 1
+ e + + ν ⇐ ⇒ p → n + e + + ν
A
wh ic h s ees th e emis s ion of a p os itr on (th e electr on an ti-p ar ticle) an d a n eu tr in o; an d th e electr on cap tu r e:
Z
A X N + e −
→ Z − 1
X N ′ + 1 + ν
⇐ ⇒ p + e −
→ n + ν
a p r o ces s th at comp etes with , or s u b s titu tes , th e p os itr on emis s ion .
Ex amp l es
ց
30
64 Cu ր
64 Zn + e − + ν ¯ ,
Q β = 0 . 57 M eV
28
29 64 Ni + e + + ν , Q β = 0 . 66 M eV
Th e n eu tr in o an d b eta p ar ticle ( β ± ) s h ar e th e en er gy .
S in ce th e n eu tr in os ar e v er y d ifficu lt to d e tect (as w e will s ee th ey ar e almos t mas s les s an d in ter act v er y w e akl y with matter ), th e elec tr on s /p os itr on s ar e th e p ar ticles d etected in b eta-d eca y an d th ey p r es en t a c h ar acter is tic en er gy s p ectr u m (s ee F ig. 45 ). Th e d iff er en ce b e t w een th e s p ectr u m of th e β ± p ar ticles is d u e to th e Cou lom b r ep u ls ion or attr action fr om th e n u cleu s .
Notice th at th e n eu tr in os als o car r y a w a y an gu lar momen tu m. Th ey ar e s p in -1/2 p ar ticles , with n o c h ar ge (h en ce th e n ame) an d v er y s mall mas s . F or man y y ear s it w as actu ally b eliev ed to h a v e zer o mas s . Ho w ev er it h as b een con fi r med th at it d o es h a v e a mas s in 1998.
© Neil Spooner. All rights reserved. This content is excluded from o ur Creative Commons license. For more information, see http://ocw.mit.edu/fairuse .
7.2.2 Cons ervati on l a ws
F ig . 4 5 : B eta d ec a y s p ec tr a : Dis t r ib u tio n o f mo men tu m (to p p lo ts ) a n d kin etic en er g y (b o tto m) fo r β − (left) a n d β + (r ig h t) d ec a y .
As th e n eu tr in o is h ar d to d etect, in itially th e b eta d eca y s eemed to v iolate en er gy con s er v ation . In tr o d u cin g an ex tr a p ar ticle in th e p r o ces s allo ws on e to r es p ect con s er v ation of en er gy . Bes id es en er gy , th er e ar e oth er con s er v ed q u an tities :
- En ergy : Th e Q v alu e of a b eta d eca y is giv en b y th e u s u al for m u la:
A A
Q β − = [ m N ( X ) − m N ( Z + 1
X ′ ) − m e ] c 2 .
Us in g th e atomic mas s es an d n eglectin g th e electr on ’s b in d in g en er gies as u s u al w e h a v e
A A
Q β − = { [ m A ( X ) − Z m e ] − [ m A ( Z + 1
X ′ ) − ( Z + 1) m e ] − m e } c 2 = [ m A ( A X ) − m A ( A
X ′ )] c 2 .
Z + 1
Th e k in etic en er gy (eq u al to th e Q ) is s h ar ed b y th e n eu tr in o an d th e electr on (w e n eglect an y r ecoil of th e mas s iv e n u cleu s ). Th en , th e emer gin g electr on (r emem b er , th e on ly p ar ticle th at w e can r eally ob s er v e) d o es n ot h a v e a fi x ed en er gy , as it w as for ex amp le for th e gamma p h oton . Bu t it will ex h ib it a s p ectr u m of en er gy (wh ic h is th e n u m b er of electr on at a giv en en er gy ) as w ell as a d is tr ib u tion of momen ta. W e will s ee h o w w e can r ep r o d u ce th es e p lots b y an aly zin g th e Q M th eor y of b eta d eca y .
- Momen tu m : Th e momen tu m is als o s h ar ed b et w een th e electr on an d th e n eu tr in o. Th u s th e ob s er v ed electr on momen tu m r an ges fr om zer o to a max im u m p os s ib le momen tu m tr an s fer .
- An gu l ar momen tu m (b oth th e electr on an d th e n eu tr in o h a v e s p in 1/2)
- P ari t y ? It tu r n s ou t th at p ar it y is n ot con s er v ed in th is d eca y . Th is h in ts to th e fact th at th e in ter action r es p on s ib le v iolates p ar it y con s er v ation (s o it can n ot b e th e s ame in ter action s w e alr ead y s tu d ies , e.m. an d s tr on g in ter action s )
- Ch arge (th u s th e cr eation of a p r oton is for ex amp le alw a y s accomp an ied b y th e cr eation of an electr on )
- L ep ton n u m b er : w e d o n ot con s er v e th e total n u m b er of p ar ticles (w e cr eate b eta an d n eu tr in os ). Ho w ev er th e n u m b er of mas s iv e, h ea v y p ar ticles (or b ar y on s , comp os ed of 3 q u ar k s ) is con s er v ed . Als o th e lep ton n u m b er is con s er v ed . Lep ton s ar e fu n d amen tal p ar ticles (in clu d in g th e electr on , m u on an d tau , as w ell as th e th r ee t y p es of n eu tr in os as s o ciated with th es e 3). Th e lep ton n u m b er is + 1 for th es e p ar ticles an d -1 for th eir an tip ar ticles . Th en an electr on is alw a y s accomp an ied b y th e cr eation of an an tin eu tr in o, e.g., to con s er v e th e lep ton n u m b er (in itially zer o).
7.2.3 F ermi ’s T heo ry of B eta Dec a y
Th e p r op er ties of b eta d eca y can b e u n d er s to o d b y s tu d y in g its q u an tu m-mec h an ical d es cr ip tion v ia F er mi’s G old en r u le, as d on e for gamma d eca y .
2 π
h
f i
W = |� ψ
| V ˆ | ψ ⟩ | 2 ρ ( E )
f
In gamma d eca y p r o ces s w e h a v e s een h o w th e e.m. fi eld is d es cr ib ed as an op er ator th at can cr eate (or d es tr o y ) p h oton s . Nob o d y ob jected to th e fact th at w e can cr eate th is mas s les s p ar ticles . After all, w e ar e familiar with c h ar ged p ar ticles th at p r o d u ce (cr eate) an e.m. fi eld . Ho w ev er in Q M p h oton s ar e als o p ar ticles , an d b y an alogy w e can h a v e als o cr eation of oth er t y p es of p ar ticles , s u c h as th e electr on an d th e n eu tr in o.
F or th e b eta d eca y w e n eed an oth er t y p e of in ter action th at is ab le to cr eate mas s iv e p ar ticles (th e electr on an d n eu tr in o). Th e in ter action can n ot b e giv en b y th e e.m. fi eld ; mor eo v er , in th e ligh t of th e p os s ib ilities of cr eatin g an d an n ih ilatin g p ar ticles , w e als o n eed to fi n d a n ew d es cr ip tion for th e p ar ticles th ems elv es th at allo ws th es e p r o ces s es . All of th is is ob tain ed b y q u an tu m fi el d th eory an d th e s econ d q u an tization . Q u an tu m fi eld th eor y giv es a u n ifi cation of e.m. an d w eak for ce (electr o-w eak in ter action ) with on e cou p lin g con s tan t e.
Th e in ter action r es p on s ib le for th e cr eation of th e electr on an d n eu tr in o in th e b eta d eca y is called th e w eak i n teracti on an d its on e of th e fou r fu n d amen tal in ter action s (togeth er with gr a v itation , electr omagn etis m an d th e s tr on g in ter action th at k eep s n u cleon s an d q u ar k s togeth er ). O n e c h ar acter is tic of th is in ter action is p ar it y v iolation .
A . Ma trix elemen t
Th e w eak in ter action can b e wr itten in ter ms of th e p ar ticle fi eld w a v efu n ction s :
V int = g Ψ e † Ψ ν ¯ †
∝
wh er e Ψ a ( Ψ a † ) an n ih ilates (cr eates ) th e p ar ticle a , an d g is th e cou p lin g con s tan t th at d eter min es h o w s tr on g th e in ter action is . Remem b er th at th e an alogou s op er ator for th e e.m. fi eld w as a † k (cr eatin g on e p h oton of momen tu m k ).
Th en th e matr ix elemen t
can b e wr itten as :
V if = ⟨ ψ f | H int | ψ i ⟩
∫
V if = g d 3 ' x Ψ p ∗ ( ' x ) [ Ψ e ∗ ( ' x ) Ψ ν ¯ ∗ ( ' x )] Ψ n ( ' x )
† → ∗
(Her e s in ce w e h a v e s calar op er ator s ).
T o fi r s t ap p r o x imation th e electr on an d n eu tr in o can b e tak en as p lan e w a v es :
V if = g d 3 ' x Ψ p ∗ ( ' x ) √ √ Ψ n ( ' x )
∫ e i ! k e · ! x e i ! k ν · ! x
V V
∫
an d s in ce k R ≪ 1 w e can ap p r o x imate th is with
g V if = V
d 3 ' x Ψ p ∗ ( ' x ) Ψ n
( ' x )
W e th en wr ite th is matr ix elemen t as g
V if = V M np
wh er e M np is a v er y comp licated fu n ction of th e n u clear s p in an d an gu lar momen tu m s tates . In ad d ition , w e will u s e in th e F er mi’s G old en Ru le th e ex p r es s ion
2 2
| M np | → | M np | F ( Z 0 , Q β )
wh er e th e F er mi fu n ction F ( Z 0 , Q β ) accou n ts for th e Cou lom b in ter action b et w een th e n u cleu s an d th e electr on th at w e h ad n eglected in th e p r ev iou s ex p r es s ion (wh er e w e on ly con s id er ed th e w eak in ter action ).
B . Den sit y of sta tes
In s tu d y in g th e gamma d eca y w e calcu lated th e d en s it y of s tates , as r eq u ir ed b y th e F er mi’s G old en Ru le. Her e w e n eed to d o th e s ame, b u t th e p r ob lem is comp licated b y th e fact th at th er e ar e t w o t y p es of p ar ticles (electr on an d n eu tr in o) as p r o d u cts of th e r eaction an d b oth can b e in a con tin u u m of p os s ib le s tates .
Th en th e n u m b er of s tates in a s mall ener gy vol u m e is th e p r o d u ct of th e electr on an d n eu tr in o’s s tates :
d 2 N s = dN e dN ν .
Th e t w o p ar ticles s h ar e th e Q en er gy :
Q β = T e + T ν .
F or s imp licit y w e as s u me th at th e mas s of th e n eu tr in o is zer o (it’s m u c h s maller th an th e electr on mas s an d of th e k in etic mas s of th e n eu tr in o its elf ). Th en w e can tak e th e r elativ is tic ex p r es s ion
T ν = c p ν ,
wh ile for th e electr on
e
e
E 2 = p 2 c 2 + m 2 c 4 → E = T e + m e c 2 with T e = √ p 2 c 2 + m 2 c 4 − m e c 2
an d w e th en wr ite th e kin etic en er gy of th e n eu tr in o as a fu n ction of th e electr on ’s ,
T ν = Q β − T e .
Th e n u m b er of s tates for th e electr on can b e calcu lated fr om th e q u an tized momen tu m, u n d er th e as s u mp tion th at th e electr on s tate is a fr ee p ar ticle ( ψ ∼ e i ! k · ! r ) in a r egion of v olu me V = L 3 :
dN e = 4 π k 2 dk e = p 2 dp e
( L ) 3
2 π
4 π V
e
(2 π h ) 3
e
an d th e s ame for th e n eu tr in o,
dN =
4 π V
p 2 dp ,
ν (2 π h ) 3 ν ν
wh er e w e u s ed th e r elation s h ip b et w een momen tu m an d w a v en u m b er : p ' = h ' k .
A t a giv en momen tu m/en er gy v alu e for th e electr on , w e can wr ite th e d en s it y of s tates as
ν
ν
ρ ( p e ) dp e = dN e d T
= 16 π
(2 π h ) 6 p e dp e p ν d T
= 4 π 4 h 6 c 3 [ Q − T e ] p e dp e
d p ν
wh er e w e u s ed : d T ν = c an d p ν = ( Q β − T e ) /c . Th e d en s it y of s tates is th en
ρ ( p e ) dp e = 4 π 4 h 6 c 3 [ Q − T e ] p e dp e = 4 π 4 h 6 c 3 [ Q − ( p 2 c 2 + m 2 c 4 − m e c )] p e dp e
V 2 2 2
V 2 √
2 2 2
e
e
or r ewr itin g th is ex p r es s ion in ter ms of th e electr on k in etic en er gy :
( 2 /
ρ ( T e ) = 4 π 4 h 6 c 3 [ Q − T e ] p e
(as p e dp e = ( T e + m e c 2 ) /c 2 dT e )
dT e
= 4 c 6 π 4 h 6 [ Q − T e ]
T e + 2 T e m e c 2 T e + m e c
2 [ Q − T ] = C F ( Z , Q ) | V f i |
2
Q − ( p 2 c 2 + m 2 c 4 − m e c )
K n o win g th e d en s it y of s tates , w e can calcu late h o w man y electr on s ar e emitted in th e b eta d eca y w ith a given ener gy . Th is will b e p r op or tion al to th e r ate of emis s ion calcu lated fr om th e F er mi G old en Ru le, times th e d en s it y of s tates :
N ( p ) = C F ( Z , Q ) | V f i |
2 p 2 2
2 p 2 � √
2 � 2
an d
c c e e
e
c 5
f i
e
e e
e
e
e
N ( T ) = C F ( Z , Q ) | V | 2 [ Q − T ] 2 √ T 2 + 2 T m c 2 ( T + m c 2 )
Th es e d is tr ib u tion s ar e n oth in g els e th an th e s p e ctr u m of th e emitted b eta p ar ticles (electr on or p os itr on ).
In th es e ex p r es s ion w e collected in th e con s tan t C v ar iou s p ar ameter s d er iv in g fr om th e F er mi G old en Ru le an d d en s it y of s tates calcu lation s , s in ce w e w an t to h igh ligh t on ly th e d ep en d en ce on th e en er gy an d momen tu m. Als o, w e in tr o d u ced a n ew fu n ction , F ( Z , Q ), called th e F er mi fu n ction , th at tak es in to accou n t th e s hap e of th e n u clear w a v efu n ction an d in p ar ticu lar it d es cr ib es th e Cou lom b attr action or r ep u ls ion of th e electr on or p os itr on fr om th e n u cleu s . Th u s , F ( Z , Q ) is d iff er en t, d ep en d in g on th e t y p e of d eca y .
Th es e d is tr ib u tion s w er e p lotted in F ig. 45 . Notice th at th es e d is tr ib u tion s (as w ell as th e d eca y r ate b e lo w) ar e th e p r o d u ct of th r ee ter ms :
c 2
- th e S tatis tical factor (ar is in g fr om th e d en s it y of s tates calcu lation ), p 2 [ Q − T ] 2
- th e F er mi fu n ction (accou n tin g for th e Cou lom b in ter action ), F ( Z , Q )
- an d th e T r an s ition amp litu d e fr om th e F er mi G old en Ru le, | V f i | 2
Th es e th r ee ter ms r efl ect th e th r ee in gr ed ien ts th at d eter min e th e s p ectr u m an d d eca y r ate of in b eta d eca y p r o ces s es .
C. Deca y ra te
Th e d eca y r ate is ob tain ed fr om F er mi’s G old en r u le:
2 π
W = h | V if
| ρ ( E )
2
wh er e ρ ( E ) is th e total d en s it y of s tates . ρ ( E ) (an d th u s th e d eca y r ate) is ob tain ed b y s u mmin g o v er all p os s ib le s tates of th e b eta p ar ticle, as cou n ted b y th e d en s it y of s tates . Th u s , in p r actice, w e n eed to in tegr ate th e d en s it y of s tates o v er all p os s ib le momen tu m of th e ou tgoin g electr on /p os itr on . Up on in tegr ation o v er p e w e ob tain :
∫
V 2
ρ ( E ) = 4 π h c
m a x e
V 2 ( Q mc 2 ) 5
p
2 2 −
dp e [ Q − T e ] p ≈ 4 π h c
4 6 3
0
e 4 6 3 30 c 3
≈
(wh er e w e to ok T e pc in th e r elativ is tic limit for h igh electr on s p eed ). W e can fi n ally wr ite th e d eca y r ate as :
2 π 2
2 π g 2 2 V 2
( Q − mc 2 ) 5
W = h | V if | ρ ( E ) = h V | M np | F ( Z , Q β ) 4 π 4 h 6 c 3 30 c 3
wh er e w e in tr o d uced th e con s tan t
( Q mc 2 ) 5
2 2 −
= G F | M np | F ( Z , Q β ) 60 π 3 h ( h c ) 6
e
1 g m 2 c G F = √ 2 π 3 h 3
n c 137
wh ic h giv es th e s tr en gth of th e w eak in ter action . Comp ar in g to th e s tr en gth of th e electr omagn etic in ter action , as giv en b y th e fi n e con s tan t α = e 2 ∼ 1 , th e w eak is in ter action is m u c h s maller , with a con s tan t ∼ 10 − 6 .
d p
W e can als o wr ite th e d iff er en tial d eca y r ate d W :
e
d W 2 π
= | V
| ρ ( p ) ∝ F ( Z , Q )[ Q − T ] p
2 2 2
d p e
h if e e e
Th e s q u ar e r o ot of th is q u an tit y is th en a lin ear fu n ction in th e n eu tr in o k in etic en er gy , Q − T e :
d p e p 2 F ( Z , Q )
e
s d W 1 ∝ Q − T
e
Th is is th e F er mi-K u r ie r elation . Us u ally , th e F er mi-K u r ie p lot is u s ed to in fer b y lin ear r egr es s ion th e max im u m electr on en er gy ( or Q ) b y fi n d in g th e s tr aigh t lin e in ter cep t.
© Neil Spooner. All rights reserved. This content is excluded from o ur Creative Commons license. For more information, see http://ocw.mit.edu/fairuse .
.
F ig . 4 6 : E xa mp le o f F er mi-Ku r ie p lo t ( s ee a ls o Kr a n e, F ig . 9 .4 , 9 .5 )
MIT OpenCourseWare http://ocw.mit.edu
22.02 Introduction to Applied Nuclear Physics
Spring 2012
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