22.101 Applied Nuclear Physics (Fall 2006)

Lecture 2 2 (12/4/06) Nuclear Decays

References :

W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967), Chap 4.

A nucleus in an excited state is uns table becau se it can alway s undergo a trans i tion (d ecay) to a lo wer-energy state of the same nucleus . Such a transition will be accom p anied by the em ission of gamm a radiation . A nucleus in either an excited or ground state also can un dergo a tran sition to a lo wer-energy state of another nucleus .

This decay is accom p lished by the em ission of a particle such as an alph a, electron o r positron, with or without subsequent gamma e m ission. A nucleus which undergoes a transition spontaneously , that is, without being supplie d with additional energy as in bom b ardm e n t, is said to be radioactive. It is found experim e ntally that naturally occurring radioactive nuclides em it one or m o re of the three types of radiations,

 particle s,   particles, and  rays. Measurem ents of the energy of the nuclear

radiation provide the m o st direct inform ati on on the energy-level structure of nuclides. One of the most extensive com p ilations of ra dioisotope data and deta iled nuclea r leve l diagram s is the Table of Isotopes , edited by Lederer, Hollander and Perlm a n.

In this chapter we will s upplem ent our previou s discuss i ons of beta decay and radioactiv e decay by briefly exam ining the study of decay constants, selection rules, and som e aspects of  ,   , and  decay energ e tics.

Alpha Decay

Most rad i oactive substances are  e m itters. Most nuclides with A > 150 are unstable against  decay.  decay is v e ry unlikely for light nu clid es. The decay constan t decreases expo nentially with d ecreasing Q-value, here called the decay energy,   ~ exp(  c / v ) , where c is a co nstant and v the speed of the  partic le,

v 

. The m o me ntum and energy conservati on equations are quite straightforward

in this case, as can be seen in Fig. 20.1.

Fig. 20.1. Particle em ission and nuclear recoil in - decay .

p  p

D

 0 (20.1)

M c 2  ( M

c 2  T

)  ( M 

c 2  T 

) (20.2)

Both kinetic energies are sm all enough that non-relativistic energy- m o m e nt um relations m a y be used,

T  p 2 / 2 M

 p 2 / 2 M

  M 

/ M D

 T 

(20.3)

Treating the decay as a reaction the correspond in g Q-value b ecom e s

Q    M  ( M  M )  c 2

= T D  T 

 M D  M  T  A T

(20.4)

 A  4 

This shows that the kinetic energy of the -particle is always les s than Q  . Since Q  > 0 ( T  is nec e ss arily positiv e), it f o llows that -decay is an exo t herm ic process. The

various energies involved in the decay process can be displayed in an energy-level diagram shown in Fig. 20.2. One can s ee at a glance how th e rest m a sses and

Fig. 20.2. Energy-level diagram for -decay.

the kine tic e n ergies com b ine to en su re ene r gy co nservation. W e will see in the nex t lectu r e th at energy-level diagram s are also us ef ul in depic ting collision - in duced nucle ar reactions. T h e separation energy S  is the work ne cessary to se parate an -particle from the nucleus,

S    M ( A  4 , Z  2 )  M   M ( A , Z )  c 2

= B ( A , Z )  B ( A  4 , Z  2 )  B ( 4 , 2 ) =  Q  (20.5)

One can use the sem i -empirical m a ss form ula to determ ine whether a n u cleus is stable agains t -decay. In th is way one finds Q  > 0 for A > 150. Eq.(20.5) also shows that when the daughter nucleus is m a gi c, B(A-4,Z-2) is large, and Q  is large. Conversely, Q  is sm all when the pa ren t nucleu s is m a gic.

Estima ting -decay Constant

An estim ate of the decay constan t can be m a de by treating th e decay as a barrier penetration problem , an approach proposed by Gam o w (1928) and also by Gurney and Condon (1928). The idea is to assum e the -particle already exists as a particle inside the daughter nucleus where it is confined by th e Coulom b potential, as illustrated in F i g.

20.3. The decay constant is th en th e proba bility per unit tim e that it can tunnel throu gh the poten tia l,

Fig. 20.3. Tunneling of an  particle through a nuclear Coulomb barrier.

 ~  v  P (20.6)

 R

 

where v is th e rela tive spe e d of the and the daughter nucleus, R is the radius of the daughter nucleus, and P the tran sm ission coefficient. Eq.(20.6) is a standard form for describing tunneling probability in th e for m of a rate. The prefactor  / R is the attem p t frequency, the rate at w h ich the particle trie s to tunnel through the barrier, and P is the probability of tunneling for each try. Recall from our study of barrier pen e tration (cf.

Chap 5, eq. (5.20)) that the transm iss i on coefficient can be written in th e form

P ~ e   ( 2 0 . 7 )

2 r 2  

  1 / 2

   dr 2 m V ( r )  E

r 1

2 b 

 2 Z e 2

  1 / 2

= dr  2   D  Q  

ħ R    r   

(20.8)

w ith   M  M D /( M   M D ) . The integral can be evaluated ,

  8 Z

e 2 

   

D cos 1

ħ v

y ( 1

y ) 1 / 2

(20.9)

where y = R/b = Q  /B, B = 2Z D e 2 /R, Q    v 2 / 2  2 Z e 2 / b . Typically B is a few tens or m o re Mev, while Q  ~ a few Mev. One can therefore inv oke the th ick barrier approxim a tion, in which case b >> R (or Q  << B), and y << 1. Then

cos  1 ~    1 y 3 / 2  ... (20.10)

2 6

the square b r acket in (20 . 9) becom e s

and

  ~   2 2

 O ( y 3 / 2 ) (20.11)

4  Z e 2

16 Z

e 2  R  1 / 2

  D  D  

(20.12)

ħ v ħ v  b 

So the expression for the decay cons tant becom e s

  v   4  Z D e  8 

2

2   1 / 2 

 exp 



ħ v ħ

Z D e R 



(20.13)

where is the reduced m a ss. Since G a m o w was the first to study this problem , the exponent is som e ti m e s known as the Ga m o w factor G.

To illus t ra te the applica t ion of (20.13) we conside r estim a ting the decay co nstant

of the 4.2 Mev -par tic le em itted by U 238 . Ignoring the sm all recoil effects, we can write

T ~ 1  v 2  v ~ 1.4 x 10 9 cm /s, ~ M

 2 

R ~ 1.4 (234) 1/3 x 10 -1 3 ~ 8.6 x 10 -1 3 cm

4  Z e 2

 D  

173 ,

8  Z e

2  R

 1 / 2  83

ħ v ħ

Thus

P  e  90 ~ 10  39 (20.14)

As a result o u r estim a te is

  ~ 1 . 7 x 10  18 s -1 , or t 1/ 2 ~ 1.3 x 10 10 yrs

The experim e ntal half-life is ~ 0.45 x 10 10 yrs. C onsidering our estim a te is very rough, the agreem e n t is rather rem a rkable. In general one should not expect to predict   to be better than the correct o r der of m a gnitude (say a factor of 5 to 10). Notice that in our exam ple, B ~ 30 Mev and Q  = 4.2 Mev. Also b = RB/ Q  = 61 x 10 -1 3 cm . So the th ick

barrier approxim a tion, B >> Q  or b > > R, is indee d well jus tif ied.

The theoretical exp r ess i on for the decay c onstant provides a basis for an em pirical relation between the half-life a nd th e decay energy. Since t 1/ 2 = 0.693/ , we have from (20.13)

ℓ n ( t

1 / 2

)  ℓ n  0 . 693 R / v   4  Z

e 2 / ħ v  8  Z

D ħ

e 2  R  1 / 2 (20.15)

We note R ~A 1/3 ~ Z 1 / 3 , so th e las t te rm varies with Z D like Z 2 / 3 . Also, in the second

D D

trerm v 

. Therefore (20.15) suggests the following relation,

log( t

1 / 2

)  a  b

(20.16)

with a and b being param e ters depending only on Z D . A relation of this f o rm is known as the Geige r - N uttall rule.

We conclude our brief consideration of -dec ay at this poin t . For further discussions the student should consult Me yerhof (Chap 4) and Evans (Chap 16).

Beta Decay

Beta decay is considered to be a weak interaction since the in teraction potential is

~ 10 -6 that of nuclear interactions, w h ich are generally regarded as strong. Electrom a gnetic and gra v ita tiona l in terac tions a r e inte rm ediate in th is sen s e.  -decay is

the m o st common type of radioactiv e decay, all nuclides not lying in the “valley of stability” ar e unstab l e a g ainst this tr ansiti on. Th e positrons o r ele c tron s e m itted in  - decay have a continuou s energy di stribution, as illustrated in Fig. 20.4 for the decay o f Cu 64 ,

8 8

4 4

0

0 1 2 3

0

0 1 2 3

Momentum p c, 10 3 gauss cm Momentum p c, 10 3 gauss cm

10 10

8 8

4 4

0

0 0.2

0.4

0.6

0

0 0.2

0.4 0.6

Kinetic ener gy T c, Mev Kinetic ener gy T c, Mev

F i g u r e b y M I T O C W .

Fig. 20.4 . Mom entum (a) and energy (b) di stributions of beta decay in Cu 64 . (from Meyerhof)

Cu 64  Zn 64     , T - ( m ax) = 0.57 Mev

 Ni 64      , T + (m a x ) = 0.66 Mev

7

The values o f T  (max) are characteristic of the partic ular radionuclide; they can be considered as signatures.

If we assum e that in  -decay we have only a parent nucleu s , a daughter nu cleus,

and a  -particle, then we would find that the c onservations of energy, linear and angular moe m nta cannot be all satisfied. It was then proposed by Pauli ( 1933) that particles, called neutrino and antineutrino  , also can be e m itted in  -decay. The neutrino

particle has the properties of zero charge, zero (or nearly zero ) m a ss, and intrins i c ang u lar mom e ntum (spin) of ħ / 2 . The detection of the neutrino is unusually difficult because it has a very long m ean-free path. Its exis tence was confirm e d by Reines and Cowan (1953) using the inverse  -decay reaction induced by a neutrin o , p    n    . The em ission of a neutrino (o r antineutrin o ) in th e  -decay process m a kes it possible to satisfy the energy conservation condition with a continuous distribut ion of the kinetic energy of th e em itted  -particle. Also, linear and angular m o m e nta are now conserved.

The energetics of  -decay can be summ arized as

p  p  p

D

 0 ( 2 0 . 1 7 )

M c 2  M

c 2  T 

 T 

electron decay (20.18)

M c 2  M

c 2  T

 T 

 2 m c 2 positron decay (20.19)

where the ex tra rest m a ss term in positron de cay has been discussed previously in Chap 11 (cf. Eq. (11.9)). Recall also that electr on cap ture (EC) is a com p eting process with positron decay, requiring only the condition M P (Z) > M D (Z-1). Fig. 20.4 shows how the energetics can be expressed in th e form of energy-level diag ram s .

8

M P(Z) c 2

Electron capture

M P(Z) c 2

2m 0 c 2

Electron capture

M

M P(Z) c 2

Q e.c.

c 2

Q e.c.

 + decay

Q  +

D(Z-1)

M D(Z+1) c 2 M D(Z-1) c 2

F i g u r e b y M I T O C W .

Fig. 20.5 . Energetics of   decay processes. (from Meyerhof)

Typical decay schem e s for -em itters are shown in Fig. 20.6. For each nu clea r leve l there is an assignm ent of spin and parity. This inf o rm ation is essentia l f o r dete rm ining whether a transition is allowed according to ce rta i n selection rules, a s we will dis cuss

below.

17 Cl 38 14

t = 38 min 8

1/2

5

t 1/2 = 72 sec

0 +

4

3

2

1

0

18 A 38

7 N 14

e.c. ?

99.4%

[3.5]

 +

0.6%

[7.3]

2m 0 c 2

2

29 Cu 64

12%

4 Be 7

t 1/2 = 53 days

-

1

0

28 Ni 64

F i g u r e b y M I T O C W .

3 Li 7

1/2 -

[3.4]

e.c.

88%

[3.2]

3/2

Fig. 20.6. Energy-level diagram s depicting nuclear trans itio ns involving beta decay.

(from Meyerhof) 9

Experim e ntal half-lives of  -decay hav e values sp read over a v e ry wide ran g e, fro m 10 -3 sec to 10 16 yrs. Generally,   ~ Q 5 . The decay process can not be explained class i cally. The theory o f  -decay was developed by Ferm i (1934) in analogy with the quantum theory of electrom a gnetic d ecay. For a discuss i on o f the elem ents of this theory one can beg i n with Mey e rhof and f o llow the ref e rences g i ven there i n. W e will be content to m e ntion just o n e aspec t of the th eo ry, that concerning the statistical factor

describing the m o mentum and energy distributions of the emitted  particle. Fig. 20.7

shows the nuclear coulomb effects on the m o m e ntum distribution in  -decay in Ca (Z = 20). One can see an enh a ncem ent in the cas e of   -decay and a suppression in the case of   -decay at low m o m e nta. Coulom b effect s on the energy d i stribu tion are even m o re pronounced.

6

4

2

0

0 0.8

1.6 2.4

 -Ray Momentum, 

Fig. 20.7 . Mom e ntum distributions of  -decay in Ca.

F i g u r e b y M I T O C W .

Selection Rules for Beta Decay

Besides energy and linear m o m e ntum conser vation, a nuclear transition must also satisfy angular m o m e ntum and parity conserva tio n. This give s rise to se le ction rules which specif y whether a particular transition between initial and final states, both with specified sp in and parity, is allowed, and if allow e d what m o de of decay is m o st likely.

W e will work out the s e lection rules governing   and -decay. For the former

conservation of angular mom e ntum a nd parity are generally expressed as

10

I P  I D  L   S  (20.17)

   (  1 ) L  (20.18)

where L  is the orbital angular m o m e ntum and S  the intrins i c sp in of the elec tr on- antineutrino system . The m a gn itude of angular mom e ntum ve ctor can take integral values, 0, 1, 2, …, whereas the latter can take on values of 0 and 1 which would

correspond the antiparallel and parallel coupli ng of the electron and neutrino spins.

These two orien t ations will be c a lle d Fermi and Gamow-Teller respectiv ely in what f o llows.

In applying the conservation conditions, th e goal is to find the lowest value of L  that will satisfy (20.17) for whic h there is a corresponding value of S  that is com p atible with (20.18). This then identifies the m o st likely transition among all the allowed transitions. In other words, all the other allowed transitions with higher values of L  ,

which m a ke s them less likely to occur. This is b ecause the d ecay constant is govern ed by the squar e o f a transition m a trix e l em ent, wh ich in term can be written as a series of contributions, one for each L  (recall the discussion of par tial wave expansion in cross

section calculation, Appendix B, where we also argue that the highe r o r d e r par tia l wa ves are less likely the low order ones, ending up with only the s-wave),

  M 2  M ( L  0 ) 2  M ( L  1 ) 2  M ( L  2 ) 2  ... (20.19)

Transitions with L  = 0, 1, 2, … are called allowed, first-borbidden, second- forbidden, …etc. The m a gnitude of the m a trix elem ent squared decreases from one order

to the next higher one by at least a factor of 10 2 . For this reas on we are in teres t ed on ly in the lowes t o r der trans i tion that is allowed.

To illus t ra te how the sele ction rules a r e determ ined, we consid er the tran sition

He 6 ( 0  )  Li 6 ( 1  )

To determ ine the com b ination of L  and S  f o r the f i r s t tr ansition that is allowed, we begin by noting that parity conservation requires L  to be even. Then we see that L  = 0 plus S  = 1 would satisfy both (20.17) and (20.18). Thus the mo st lik ely tra n sition is the transition designated as allowed, G- T . Following the sam e line of argum e nt, one can

arriv e at the following as signm ents.

O 14 ( 0  )  N 14 ( 0  ) allowed, F

n 1 ( 1 / 2  )  H 1 ( 1 / 2  ) allowed, G-T and F

C ℓ 38 ( 2  )  A 38 ( 2  ) first-borbidden, GT and F

Be 10 ( 3  )  B 10 ( 0  ) second-forbidden, GT

Parity Non-conservatio n

The presence of neutrino in  -decay leads to a certain type of non-conservation of parity. It is known that neutrinos have in strinsic spin antiparal lel to their velocity, whereas th e spin orientation of the an tineu tri no is parallel to th eir ve locity ( k eeping in m i nd that and  are different particles). Consider the m i rror experim e nt where a neutrino is moving toward the m i rror from th e left, Fig. 20.8. Applying the inversion symmetry operation

Fig. 20.8. Mirror reflection dem onstrating parity non-sonserving property of neutrino. (from Meyerhof)

( x   x ), the velocity reverses direction, whil e the angular m o m e ntum (spin) does not. Thus, on the other side of the m i rror we have an im age of a particle m oving from the right, but its spin is now parallel to th e veloc ity so it has to be an antin eutrino instead o f a neutrino. This m eans that the property of and  , nam e ly def i nite spin d i re c tion relative to the veloc ity, is not com p atible with pa rity conservation (symm e try under inversion).

For further d i scuss i ons o f beta decay we again ref e r the s t uden t to Mey e rh of and the references given therein.

Gamma Decay

An excited n u cleus can always decay to a lower energy state b y -em i ssion or a com p eting process called internal conversion. In the la tte r th e ex cess nu clear energy is given directly to an atom ic electron which is ej ect e d wi t h a cer t ai n kinetic energy. In

general, co mplicated rearrang em ents of nucleons occur during -decay.

The energetics of -decay is rather s t raightforward. As shown in Fig. 20. 9 a is em itted while the nucleus reco ils.

Fig. 20.9 . Schem a tics of -decay

ħ k  p  0 (20.20)

a

M * c 2  Mc 2  E   T a (20.20)

The recoil e n ergy is usu a lly qu ite s m all,

T  p 2 / 2 M  ħ 2 k 2 / 2 M  E 2 / 2 Mc 2 (20.21)

a a 

Typically, E  ~ 2 Mev, so if A ~ 50, then T a ~ 40 ev. This is g e nera lly neg ligib le.

Decay Constants and Selection Rules

Nuclear ex cited states h a ve half-lives for -em i ssion ranging from 10 -1 6 sec to > 100 years. A rough estim a te of   can be m a de using sem i -classical ideas. From

Maxewell’s equation s on e finds that an accelerated point ch arge e radiates electrom agnetic radiatio n at a rate g i ven by the L a m o r for m ula (cf. Jackson, Classica l Electrod y na mics , Chap 17),

dE 2 e 2 a 2

(20.22)

dt 3 c 3

where a is th e acceleratio n of the charge. Suppose the rad i atin g charge has a m o tion like the sim p le oscillato r ,

x ( t )  x o cos  t (20.23)

where we take x 2  y 2  z 2  R 2 , R being the radius of the nucleus. From (20.23) we

have

o o o

a ( t )  R  2 cos  t (20.24)

To get an av erage rate of energy radiation, we av erage (20.22 ) over a large num ber of oscillation cycles,

 dE 

 2 R 2  4 e 2 

2  

 R 2  4 e 2

 dt 

3 c 3

cos

t avg

3 c 3 (20.25)

  avg

Now we assum e that eac h photon is em itted during a tim e interval (having the physical significan ce of a m ean lifetim e). Then,

 dE 

 dt 

 ħ 



(20.26)

  avg

Equating this with (20.25) gives

  

e 2 R 2 E 3

3 ħ 4 c 3

(20.27)

If we apply this result to a pr ocess in atom ic physics, nam e ly the de-excitation of an atom by electromagnetic em ission, we would take R ~ 10 -8 cm and E  ~ 1 ev, in which case (20.27) gives

  ~ 10 6 sec  1 , or t ~ 7 x 10 -7 sec

On the other hand, if we apply (20.27 ) to nuclear decay, where typically R ~ 5 x 10 -1 3 cm , and E  ~ 1 Mev, we would obtain

  ~ 10 15 sec-1, o r t ~ 3 x 10 -1 6 sec

These result only indicate typical orders of m a gni tude. W h at Eq.(20.27) does not explain is the wide range of values of the half-liv es that h a ve been ob served. For further discussions we again ref e r the student to references such as Meyerhof.

Turning to the question of selection rules for -decay, we can write down the conservation of angular mom e nt a and parity in a for m si m ilar to (20.17) and (20.18),

I i  I f  L  (20.28)

 i   f   (20.29)

Notice that in contrast to ( 20.27) the orbital and spin angul ar mom e nta are incorporated in L  , playing the role of the tota l angular m o m e ntum. Since the photon has spin ħ [for a discussion of photon angular m o m e ntum , see A. S. Davydov, Quantum Mechanics (1965(, pp. 306 and 578], the possible values of L  are 1 (corresponding to the case of zero orbital angular m o m e ntum ), 2, 3, …For the conservation of parity we know the parity of the photon depends on the value of L  . We now encounter two possibilities because in p hoton em ission, which is the pr ocess of electrom a gnetic m u ltipole rad i atio n, one can hav e eith er electric or m a gnetic m u ltipole radiation,

 = (  1 ) L  electric m u ltipole

 (  1 ) L  m a gnetic m u ltipo l e

Thus we can set up the following table,

Radiation	Designation	Val u e o f L 	 
electric dipo le	E1	1	-1
m a gnetic dipole	M1	1	+1
electric quadrupole	E2	2	+1
m a gnetic quadrupole	M2	2	-1
electric octu pole	E3	3	-1

etc.

Sim ilar to th e case of  -decay, the decay constan t can be expressed as a su m of contribution s from each multipole [c f. Blatt and W e isskopf, Theoretical Nuclear Physics ,

p. 627],

     ( E 1 )    ( M 1 )    ( E 2 )  ... (20.30)

provided each contribution is allowed by the se le ction rules. W e are again interested only in the lowest order allowe d transition, and if both E and M trans i tion s are allowed, E w ill dom inate. Take, f o r exam ple, a transition be tw een an initial s t ate w ith spin and parity of 2 + and a final s t ate of 0 + . This transition require s the photon parity to be positiv e, w h ich m eans that f o r an ele c tr ic m u ltipole rad i ation L  would have to be even,

and for a m a gnetic radiation it has to be odd. In view of the initial a nd final spins, we see that angular mom e ntum cons ervation (20.28) requires L  to be 2. Thus, the m o st likely mode of -de cay f o r this transition is E2 . A few other ex am pl es are:

1   0  M1

1  1 

 E1

2 2

9  1 

 M4

2 2

0   0  no -decay allowed

We conclude this discussion of nuclear dec a ys b y the rem a rk that inte rnal conversion (IC) is a com p eting process with -decay. The ato m ic electron ejected has a kinetic energy given by (i gnoring nuclear recoil)

T e  E i  E f  E B (20.31)

where E i  E f is the energy of de-excitation, and E B is the binding energy of the atom ic electron. If we denote by  e the decay constan t for intern al con v ersion, then the to tal decay cons tant for de-ex c itation is

      e (20.32)