M a s s a ch u s e tts
I n s titu te o f
T e ch n o logy
22.02
I ntroduction
t o
A p pli ed N uclear P hysics
S p rin g 2012
P rof. P aola Capp ella ro
N uclea r S cience a nd Engineer ing D epa r t m ent
[This page intentionally blank.]
Contents
1 I n tro duc tion to Nuc le ar Ph ys ic s 5
1 .1 .1 T er min o lo g y 5
1 .1 .2 U n its, d imen sio n s a n d p h ysic a l c o n sta n ts 6
1 .1 .3 N u c lea r R a d iu s 6
1 .2 B in d in g en erg y a n d S emi-emp iric a l ma s s fo rm u la 6
1 .2 .1 B in d in g en erg y 6
1 .2 .2 S emi-emp iric a l ma ss fo rm u la 7
1 .2 .3 Lin e o f S ta b ilit y in th e C h a rt o f n u c lid es 9
1 .3 R a d io a c tiv e d ec a y 1 1
1 .3 .1 A lp h a d ec a y 1 1
1 .3 .2 B eta d ec a y 1 3
1 .3 .3 Ga mma d ec a y 1 5
1 .3 .4 S p o n ta n eo u s fi ss io n 1 5
1 .3 .5 B ra n c h in g R a tio s 1 5
2 I n tro duc tion to Q uan tum Me c hanic s 17
2 .1 La w s o f Qu a n tu m M ec h a n ic s 1 7
2 .2 S ta tes , o b serv a b les a n d eig en v a lu es 1 8
2 .2 .1 P ro p erties o f eig en fu n c tio n s 2 1
2 .2 .2 R eview o f lin ea r A lg eb ra 2 2
2 .3 Mea su remen t a n d p ro b a b ilit y 2 4
2 .3 .1 W a v efu n c tio n c o lla p se 2 5
2 .3 .2 P o s itio n mea su remen t 2 5
2 .3 .3 Mo men tu m mea su remen t 2 6
2 .3 .4 E xp ec ta tio n v a lu es 2 6
2 .4 E n erg y eig en v a lu e p r o b lem 2 8
2 .4 .1 F ree p a rtic le 2 9
2 .5 Op era to rs, C o mm u ta to rs a n d U n c erta in t y P rin c ip le 3 0
2 .5 .1 C o mm u ta to r 3 0
2 .5 .2 C o mm u tin g o b s erv a b les 3 2
2 .5 .3 U n c erta in t y p rin c ip le 3 3
3 Sc atte ring, T unne ling and A lpha De c a y 35
3 .1 R eview : E n erg y eig en v a lu e p ro b lem 3 5
3 .2 U n b o u n d P ro b lems in Qu a n tu m M ec h a n ic s 3 6
3 .2 .1 I n fi n ite b a rrier 3 6
3 .2 .2 Fin ite b a rrier 3 8
3 .3 A lp h a d ec a y 4 1
3 .3 .1 E n er g etic s 4 1
3 .3 .2 Qu a n tu m mec h a n ic s d es c rip tio n o f a lp h a d ec a y 4 2
4 E ne rgy L e v e ls 47
4 .1 B o u n d p ro b lems 4 7
4 .1 .1 E n er g y in S qu a re in fi n ite w ell (p a rtic le in a b o x) 4 7
4 .1 .2 Fin ite squ a re w ell 5 0
4 .2 Qu a n tu m Mec h a n ic s in 3 D: A n g u la r mo men tu m 5 3
4 .2 .1 S c h r ¨ o d in g er equ a tio n in s p h eric a l c o o rd in a tes 5 3
4 .2 .2 A n g u la r mo men tu m o p era to r 5 4
4 .2 .3 S p in a n g u la r mo men tu m 5 6
4 .2 .4 A d d itio n o f a n g u la r mo men tu m 5 7
4 .3 S o lu t io n s to th e S c h r ¨ od in g er equ a tio n in 3 D 6 0
4 .3 .1 T h e H yd ro g en a to m 6 1
4 .3 .2 A to mic p erio d ic stru c tu re 6 2
4 .3 .3 T h e H a rmo n ic Osc illa to r P o ten tia l 6 3
4 .4 I d en tic a l p a rtic les 6 7
4 .4 .1 B o so n s, fermio n s 6 7
4 .4 .2 E xc h a n g e o p era to r 6 7
4 .4 .3 P a u li exc lu sio n p rin c ip le 6 7
5 Nuc le ar Struc ture 69
5 .1 C h a ra c teristic s o f th e n u c lea r fo rc e 6 9
5 .2 T h e Deu tero n 7 0
5 .2 .1 R ed u c ed H a milto n ia n in th e c en ter-o f-ma ss fra me 7 0
5 .2 .2 Gro u n d sta te 7 0
5 .2 .3 Deu tero n exc ited sta te 7 2
5 .2 .4 S p in d ep en d en c e o f n u c lea r fo rc e 7 3
5 .3 N u c lea r mo d els 7 4
5 .3 .1 S h ell stru c tu r e 7 4
5 .3 .2 N u c leo n s H a milto n ia n 7 5
5 .3 .3 S p in o rb it in tera c tio n 7 8
5 .3 .4 S p in p a irin g a n d v a len c e n u c leo n s 8 0
6 Time E v olution in Q uan tum Me c hanic s 83
6 .1 T ime-d ep en d en t S c h r ¨ od in g er equ a tio n 8 3
6 .1 .1 S o lu tio n s to th e S c h r ¨ o d in g er equ a tio n 8 3
6 .1 .2 U n ita ry E v o lu tio n 8 5
6 .2 E v o lu tio n o f w a v e-p a c k ets 8 5
6 .3 E v o lu tio n o f o p era to rs a n d exp ec ta tio n v a lu es 8 7
6 .3 .1 H eisen b erg E qu a tio n 8 8
6 .3 .2 E h ren fest’s th eo rem 8 8
6 .4 F ermi’s Go ld en R u le 9 0
7 Radioac tiv e de c a y 93
7 .1 Ga mma d ec a y 9 3
7 .1 .1 C la ssic a l th eo ry o f ra d ia tio n 9 4
7 .1 .2 Qu a n tu m mec h a n ic a l th eo ry 9 5
7 .1 .3 E xten s io n to Mu ltip o les 9 7
7 .1 .4 S elec tio n R u les 9 8
7 .2 B eta d ec a y 1 0 0
7 .2 .1 R ea c tio n s a n d p h en o men o lo g y 1 0 0
7 .2 .2 C o n ser v a tio n la w s 1 0 1
7 .2 .3 F ermi’s T h eo ry o f B eta Dec a y 1 0 1
8 A pplic ations of Nuc le ar Sc ie nc e 105
8 .1 I n tera c tio n o f ra d ia tio n w ith ma tter 1 0 5
8 .1 .1 C ro ss S ec tio n 1 0 5
8 .1 .2 N eu tro n S c a tterin g a n d A b so rp tio n 1 0 6
8 .1 .3 C h a rg ed p a rtic le in tera c tio n 1 1 0
8 .1 .4 E lec tro ma g n etic ra d ia t io n 1 1 7
1. Intro duc ti on to N uc l ea r P hys i c s
1.1 B as ic Conc e pts
1 .1 .1 T er min o lo g y
1 .1 .2 U n its, d imen sio n s a n d p h ysic a l c o n s ta n ts
1 .1 .3 N u c lea r R a d iu s
1.2 B inding e ne rgy and Se mi-e mpiric al mas s form ula
1 .2 .1 B in d in g en erg y
1 .2 .2 S emi-emp iric a l ma ss fo rm u la
1 .2 .3 Lin e o f S ta b ilit y in th e C h a rt o f n u c lid es
1.3 Radioac tiv e de c a y
1 .3 .1 A lp h a d ec a y
1 .3 .2 B eta d ec a y
1 .3 .3 Ga mma d ec a y
1 .3 .4 S p o n ta n eo u s fi s sio n
1 .3 .5 B ra n c h in g R a tio s
1. 1 Bas ic Conc epts
In th is c h apter w e r ev iew s o me n otatio n s an d b a s ic concepts in Nu clea r Ph y s ics . Th e c hap ter is mean t to s etu p a co mmon la n gu a ge for the r es t of th e mater ia l w e will co v er a s w ell a s r is in g q u es tion s that w e will a n s w er la ter o n .
1.1.1 T ermi nol ogy
A g iv en atom is s p ecifi ed b y th e n u m b er o f
- n eutr on s : N
- p r o to n s : Z
- electr on s : th er e ar e Z electr on in n eu tr a l ato ms
A to ms of th e s a me el em ent h a v e s a me atomic n u m b er Z. They ar e n o t all eq u al, h o w ev er . I s otop es of th e s ame elemen t h a v e d iff er en t # of neu tr ons N .
Z
Is otop es ar e d enoted b y A X N or mo r e often b y
A X
Z
92
wh er e X is the c h emical s y m b ol a n d A = Z + N is the mas s n u m b er . E.g .: 235 U , 238 U [th e Z n u m b er is r edu n d a n t,
th u s it is often omitted].
W h en talk ing of d iff er en t n u clei w e ca n r efer to th em a s
• Nu clid e: atom/n ucleu s with a s p ecifi c N a n d Z.
• Is ob a r : n uclides with s ame mas s # A ( / = Z , N ).
• Is oton e: n uclid es with s ame N , / = Z .
• Is omer : s ame n uclide (b u t d iffer en t ener gy s tate).
1.1.2 U ni ts , di mens i ons and phys i c al c ons tants
×
Nu clear en er gies ar e meas ur ed in p o w er s o f th e un it El e ctr onvol t : 1eV = 1 . 6 10 − 19 J . Th e electr o n v olt cor r es p on d s to th e kin etic ener g y ga in ed b y an electr on acceler a ted th r ou gh a p o ten tia l d iffer en ce o f 1 v o lt. Nu clear en er gies ar e u s ually in th e r a n ge of M eV (meg a-electr on v o lt, or 10 6 eV).
×
Nu clear ma s s es ar e meas ur ed in ter ms o f th e atomic mas s u nit : 1 am u o r 1u = 1 . 66 10 − 27 k g . O n e am u is equ iv alen t to 1 / 12 of the mas s of a neu tr al gr ou n d-s tate a tom o f 12 C. S in ce electr o n s ar e m u c h lig h ter th an p r o to n s and n eu tr ons (a n d p r oto n s an d n eu tr on s h a v e s imilar mas s ), o n e n u cleon h a s mas s of a b ou t 1 am u .
Beca u s e o f th e ma s s -en er gy eq u iv a len ce, w e will o ften exp r es s mas s es in ter ms of ener g y un its . T o co n v er t b et w een ener g y (in M eV) an d ma s s (in am u ) th e co n v er s ion factor is of cour s e the s p eed of lig h t s q uar e (s in ce E = mc 2 ). In thes e u n its w e h a v e: c 2 = 9 31 . 502 M eV/ u .
- Pr oton mas s : 938 . 280 M eV/c 2 .
- Neu tr o n mas s : 93 8 . 57 3M eV/c 2 .
- Electr on mas s : 0 . 51 1M eV/c 2 .
Note: y ou can fin d mo s t of th es e v a lues in K r a n e (an d onlin e!)
S cales of magnitu d e for t y pical len gth s ar e fem to meter (1fm= 10 − 15 m) a ls o called F er mi (F ) an d Angs tr om 1 ˚ A = 10 − 10 m (fo r a to mic p r op er ties ) wh ile t yp ical time s cales s pan a v er y b r oa d r an ge.
× ×
× ×
Ph y s ical co n s ta n ts that w e will en co u n ter in clud e th e s p eed o f ligh t, c = 29 9 , 79 2 , 45 8 m s − 1 , the electr on c har g e, e = 1 . 60 2176 48 7 10 − 19 C, th e Plan c k con s tan t h = 6 . 62 6068 96 10 − 34 J s and k , Av o gad r o’s n u m b er N a = 6 . 02 214 179 10 23 mol − 1 , th e p er mittiv it y of v acu u m ǫ 0 = 8 . 85 418 781 7 10 − 12 F m − 1 (F = F ar a d a y) an d man y oth er s . A go o d r efer en ce (o n lin e) is NIST: h ttp:/ /p h y s ics .n is t.g o v / cu u/in d ex .h tml
Th er e y o u can als o fi n d a to o l to con v er t en er gy in d iff er en t u n its : h ttp:/ /p h y s ics .n is t.go v/cuu /Cons tan ts /en er gy .h tml
1.1.3 Nuc l ea r R adi us
Th e r ad iu s of a n ucleu s is n o t w ell d efi n ed , s in ce w e can not d es cr ib e a n u cleu s a s a r igid s p h er e with a g iv en r ad iu s . Ho w ev er , w e can s till h a v e a p r a ctical defi n itio n for th e r an g e at wh ic h th e den s it y of th e n u cleon s in s id e a n u cleu s app r o xima te our s imp le mo del o f a s p h er e for man y ex p er imen ta l s ituations (e.g. in s catter in g exp er imen ts ).
A s imple for m u la th a t lin ks th e n ucleu s r ad iu s to th e n um b er of n u cleon s is th e em pir ic al r adiu s for m u l a :
R = R 0 A 1 / 3
1. 2 Binding energy and Semi- empiric al mas s fo rmula
1.2.1 Bi ndi ng energy
p
Tw o imp o r tan t n u clear pr op er t th at w e w a n t to s tud y ar e th e n uclear b in d in g en er gy a n d the mas s of n u clid es . Y ou cou ld th in k th at s ince w e k n o w th e mas s es of th e p r oton and th e n eu tr on , w e co u ld s imp ly fi n d th e mas s es o f a ll
n uclides with th e s imp le for m u la: m N
= ? Z m
+ N m n
. Ho w ev er , it is s een exp er imen ta lly th a t th is is not the cas e.
F r o m s p ecia l r elativ it y theor y , w e k no w th at to ea c h ma s s cor r es p on ds s o me en er gy , E = mc 2 . Th en if w e ju s t s um u p th e ma s s es of all th e con s tituen ts of a n ucleu s w e w ou ld ha v e h o w m u c h en er gy they r epr es en t. Th e mas s o f a n u cleu s is als o r ela ted to its in tr in s ic ener gy . It th u s mak es s en s e th at th is is not only the s u m of its co n s titu en t ener g ies , s in ce w e exp ect th a t s ome o th er en er gy is s p en t to k eep th e n u cleu s togeth er . If the en er gy w er e eq u al, then it w ouldn ’t b e fa v o r able to h a v e b o u n d n u clei, and a ll th e n u clei w ould b e un s tab le, co n s tan tly c han g in g fr om th eir b oun d s ta te to a s u m o f p r o to n s an d n eu tr ons .
Z
Th e b in ding en er gy of a n u cleu s is th en g iv en b y th e d iff er en ce in ma s s en er gy b et w een th e n ucleu s an d its co n s titu en ts . F or a n ucleu s A X N th e bind in g en er gy B is g iv en b y
�
B = Z m p + N m n − m N ( A X ) c 2
—
Ho w ev er , w e w an t to ex p r es s th is q u an tit y in ter ms of ex p er imen tally acces s ib le q u a n tities . Th u s w e wr ite th e n u clea r mas s in ter ms of th e atomic mas s , th a t w e ca n meas u r e, m N ( A X ) c 2 = [ m A ( A X ) Z m e ] c 2 + B e , wh er e m A ( A X ) is th e atomic mas s of th e n u cleus . W e fu r th er n eglect the electr on ic b in d in g ener gy B e b y s ettin g m N ( A X ) c 2 = [ m A ( A X ) − Z m e ] c 2 .
W e fi n a lly ob tain th e exp r es s ion for the n u clear b in d in g en er gy :
B = Z m p + N m n − [ m A ( A X ) − Z m e ] } c 2
8
6
4
2
50
100
150
200
250
Fig . 1 : B in d in g en erg y p er n u c leo n ( B / A in MeV vs. A ) o f sta b les n u c lid es (R ed ) a n d u n s ta b le n u c lid es (Gra y).
Q u an tities of in ter es t a r e als o the n eutr on an d p r oto n s ep a r a tion ener g ies :
Z
Z
S n = B ( A X N ) − B ( A − 1 X N − 1 )
S p = B ( A X N ) − B ( A − 1 X N )
Z Z − 1
wh ic h ar e th e analogo u s o f th e ion iza tion ener g ies in a to mic p h ys ics , r efl ectin g the en er gies of the val enc e n ucleons . W e will s ee th a t th es e ener g ies s h o w s ign a tu r es of th e s h ell s tr u ctu r e of n u clei.
1.2.2 Semi - empi ri c al mas s fo rmul a
Th e b in d in g en er gy is u s u ally p lo tted a s B / A or bind ing en er g y p er n ucleon. This illu s tr ates th at the b in d in g en er gy is o v er all s imply p r op or tion a l to A , s in ce B / A is mos tly cons ta n t.
Th er e ar e h o w ev er co r r ectio n s to th is tr en d . Th e d ep en d en ce of B / A on A (a n d Z ) is cap tu r ed b y th e s emi- em pir ic al m as s for m u l a . Th is fo r m u la is b a s ed on fir s t p r inciple cons id er atio n s (a mo d el for th e n uclear for ce) an d on ex p er i men tal ev id en ce to fi nd th e ex act p a r ameter s d efi ning it. In th is mo d el, th e s o-called l i q u i d -d ro p mo d el, all n ucleons ar e un ifo r mly dis tr ib u ted in s id e a n u cleu s an d ar e b ou nd tog eth er b y th e n uclear for ce wh ile th e Cou lom b in ter action ca u s es r ep u ls ion amon g p r o tons . Ch ar a cter is tics o f th e n uclear fo r ce (its s hor t r a n ge) and of the Co u lom b in ter action exp lain p a r t of the s emi-emp ir ica l mas s fo r m u la. Ho w ev er , o th er (s ma ller ) cor r ection s ha v e b een in tr o du ced to tak e in to acco u n t v ar ia tions in th e bin ding en er gy th a t emer ge b eca u s e of its q u a n tu m-mec h a n ical n a tu r e (a n d th a t giv e r is e to th e n u cl ear sh el l mo d el ).
Th e s emi-emp ir ical mas s for m u la (SEM F ) is
M ( Z , A ) = Z m ( 1 H ) + N m n − B ( Z , A ) /c 2
wh er e th e b in d in g en er g y B ( Z , A ) is g iv en b y the fo llo win g fo r m u la:
2
B ( A, Z ) = a A − a A − a Z ( Z − 1) A − a
2 / 3
v
s
c
− 1 / 3
sy m
( A − 2 Z )
A
+ δ a A
p
− 3 / 4
|
v o lu m ր e s u r ↑ fa ce W e will n o w s tu dy eac h ter m in th e S EM F . |
Co u ↑ lo m b |
s y m ↑ metr y |
p a ir i n տ g |
|
7 |
A . V o lu me term
Th e fi r s t ter m is th e v olu me ter m a v A that d es cr ib es h o w th e b in d in g ener g y is mo s tly pr op o r tio n a l to A . W h y is that s o ?
— ∼
Remem b er th at th e bind in g en er gy is a meas u r e of the in ter action amon g n ucleons . Since n u cleo n s ar e clos ely p ac k ed in th e n u cleu s an d the n u clear fo r ce h a s a v er y s h o r t r ange, eac h n u cleo n end s u p in ter acting o n ly with a few n eig h b or s . This mean s th at in dep en d en tly o f the to ta l n u m b er of n ucleons , eac h on e o f th em co n tr ib u te in th e s ame w a y . Th us th e for ce is n ot pr op o r tio n al to A ( A 1) / 2 A 2 (th e total # of n u cleon s o n e n ucleon ca n in ter act with ) b u t it’s s imp ly p r op o r tional to A . Th e cons ta n t o f p r op o r tionalit y is a fi ttin g par a meter th at is foun d ex p er imen ta lly to b e a v = 1 5 . 5M eV.
Th is v alu e is s ma ller th a n th e bin ding en er g y of th e n u cleo n s to th eir neigh b o r s a s d eter min ed b y the s tr en g th of th e n uclear (s tr on g) in ter a ction. It is fou nd (a n d w e will s tu dy mor e later ) th at th e en er gy b in d in g on e n ucleon to the oth er n u cleons is o n th e or der of 5 0 M eV. The to tal b in d in g en er gy is in s tead th e d iff er en ce b et w een th e in ter a ctio n of a n ucleon to its n eig h b or an d th e k in etic en er g y of th e n u cleo n its elf. As for electr on s in a n ato m, the n u cleon s a r e fer mion s , th us th ey can not all b e in th e s a me s ta te with zer o k in etic en er gy , b u t th ey will fi ll u p a ll th e kin etic en er gy lev els a ccor d in g to P a u li’s ex clus ion p r in cip le. Th is mo d el, wh ic h tak es in to a cco u n t th e n uclear bind in g en er gy a n d the k in etic en er gy d u e to th e fi llin g o f s h ells , in deed g iv es a n accu r a te es timate fo r a v .
B . S u rfa ce term
Th e s u r face ter m, a s A 2 / 3 , als o b a s ed on the s tr o n g for ce, is a cor r ection to the v o lume ter m. W e ex pla in ed the v o lume ter m as ar is in g fr o m th e fact that eac h n u cleon in ter a cts with a con s tan t n u m b er of n u cleon s , in d ep en d en t o f
A . W h ile th is is v a lid for n u cleon s deep with in th e n ucleu s , th os e n u cleon s on th e s u r face of the n ucleu s h a v e few er n ear es t neigh b o r s . Th is ter m is s imilar to s u r face for ces th a t ar is e fo r examp le in d r o p lets of liq uid s , a mec h a n is m that cr eates s ur face ten s io n in liq u id s .
2 / 3
S in ce th e v o lu me for ce is p r op or tion a l to B V ∝ A , w e ex p ect a s u r fa ce fo r ce to b e ∼ ( B V ) 2 / 3 (s ince the s u r face S ∼ V ). Als o the ter m m u s t b e s u b tr acted fr o m th e v olu me ter m and w e ex p ect th e co efficien t a s to h a v e a s imilar or d er o f magnitu d e as a v . In fa ct a s = 1 3 − 18 M eV.
C. Cou lomb term
− −
Th e thir d ter m a c Z ( Z 1) A − 1 / 3 d er iv es fr om th e Coulom b in ter action amon g p r oton s , an d of cou r s e is p r op o r tio n al to Z . Th is ter m is s ub tr acted fr om th e v olu me ter m s in ce th e Co u lo m b r epu ls io n ma k es a n u cleu s con tain in g man y p r oto n s les s fa v o r able (mor e ener g etic).
T o mo tiv ate the for m o f th e ter m and es timate th e co efficien t a c , th e n u cleu s is mo deled as a u n ifo r mly c h a r ged s p her e. The p oten tia l en er gy of s u c h a c h ar ge d is tr ibu tion is
1 3 Q 2
E =
4 π ǫ 0 5 R
3 R
s in ce fr om th e u nifor m d is tr ib u tio n in s id e th e s p h er e w e h a v e th e c h a r g e q ( r ) = 4 π r 3 ρ = Q ( r d 3 an d th e p oten tia l
ener g y is th en :
1 ∫ q ( � r → ) 1 ∫ q ( � → r ) 1 ∫ R q ( ( r )
E = dq ( � r ) = d 3 3 � → r ρ = dr 4 π r 2 ρ
4 π ǫ 0 | � r | 4 π ǫ ǫ 0 | � r | 4 π ǫ 0 0 r
dr =
1 ∫ R 3 Q r � 3 1 ! 1 ∫ R 3 Q 2 r 4 1 3 Q 2
4 π ǫ ǫ 0
0 0
4 π R 3 R r 4 π ǫ 0
0 0
R 6
4 π ǫ 0 5 R
—
Us in g th e emp ir ica l r adiu s fo r m u la R = R 0 A 1 / 3 and th e total c h ar ge Q 2 = e 2 Z ( Z 1) (r efl ectin g th e fact th at th is ter m will app ear only if Z > 1, i.e. if th er e ar e a t leas t t w o pr oton s ) w e h a v e :
Q 2 e 2 Z ( Z − 1)
0
R = R A 1 / 3
wh ic h giv es the s h a p e of th e Cou lom b ter m. Th en th e con s ta n t a c
ca n b e es timated fr o m a c
3 e 2
≈
5 4 π ǫ 0 R 0
, with
R 0 = 1 . 25fm, to b e a c ≈ 0 . 69 1 M eV, n ot far fr om th e ex p er imen tal v alu e.
V olume
15
V olume + Sur fac e
10
5
50
100
150
200
V olume + Sur fac e+C oulomb V olume + Sur fac e+C oulomb+A symmetr y
Fig . 2 : S E MF fo r sta b le n u c lid es. W e p lo t B ( Z , A ) / A vs. A . T h e v a rio u s ter m c o n trib u tio n s a re a d d ed o n e b y o n e to a rriv e a t th e fi n a l fo r m u la .
D. S ymmetry term
Th e Cou lom b ter m s eems to in d ica ted th a t it w o u ld b e fa v or ab le to ha v e les s p r oton s in a n u cleus and mor e n eu tr o n s . Ho w ev er , this is n ot th e ca s e a n d w e h a v e to in v ok e s ometh in g b ey o n d th e liq uid-d r op mo d el in o r d er to ex p la in the fact th at w e h a v e r ou g h ly th e s a me n u m b er o f n eu tr ons an d pr o tons in s tab le n u clei. Ther e is th u s a cor r ection ter m in th e S EM F wh ic h tr ies to ta k e in to accoun t th e s y mmetr y in pr o tons an d n eu tr on s . This cor r ection (an d th e follo win g on e) ca n on ly b e ex p lain ed b y a mo r e co mp lex mo d el o f th e n u cleus , th e sh el l mo d el , tog ether with th e q u a n tu m-mec han ica l ex cl u s ion pr incipl e , th at w e will s tu d y later in the cla s s . If w e w er e to a d d mo r e n eu tr on s , th ey will h a v e to b e mo r e ener g etic, th u s incr eas in g th e to ta l en er gy of the n u cleu s . Th is incr eas e mor e th an off -s et th e Coulo m b r ep u ls ion, s o th at it is mor e fa v or ab le to h a v e an a p pr o xima tely equ al n u m b er of p r o to n s an d n eutr on s .
2
A
Th e s hap e o f the s y mmetr y ter m is ( A − 2 Z ) . It can b e mo r e eas ily u n d er s to o d b y con s id er in g th e fact th a t th is ter m go es to zer o for A = 2 Z and its eff ect is s maller fo r lar ger A (wh ile fo r s ma ller n u clei th e s y mmetr y eff ect is mor e imp or tan t). Th e co efficien t is a sy m = 2 3M eV
E. P a irin g term
Th e fi nal ter m is link ed to th e ph y s ica l ev id en ce th at lik e-n u cleo n s ten d to p a ir off. Th en it means th at th e b in d in g ener g y is g r ea ter ( δ > 0) if w e h a v e a n ev en -ev en n u cleus , wh er e a ll the n eu tr on s an d a ll the p r oto n s a r e p air ed -off . If w e h a v e a n ucleu s with b oth a n o d d n u m b er of neu tr ons an d of p r oto n s , it is th us fa v or ab le to con v er t on e o f the pr oton s in to a n eu tr on s or v ice-v er s a (of cou r s e, ta k in g in to a cco u n t th e oth er cons tr ain ts ab o v e). Th u s , with all oth er fa cto r cons ta n t, w e h a v e to s u btr a ct ( δ < 0) a ter m fr o m th e b ind in g en er g y for o d d -o d d con fi g u r a tion s . F in a lly , for ev en -o dd co n fi g u r ation s w e do not ex p ect a n y in fluen ce fr o m this p air in g en er gy ( δ = 0). Th e p a ir in g ter m is then
+ a p A − 3 / 4 ev en -ev en
+ δ a p A − 3 / 4 = 0 ev en -o d d
− a p A − 3 / 4 o dd -o d d with a p ≈ 34 M eV. [S ometimes th e for m ∝ A − 1 / 2 is a ls o fou nd ].
1.2.3 L i ne of Stabi l i t y i n the Cha rt of nuc l i des
A 1 + 1 A − 1 / 3 a c
By taking th e fi r s t d er iv ativ e wr t Z w e ca n calcula te th e optima l Z s u c h th a t th e ma s s is minim u m. W e ob tain :
Z m in =
4 a sy m
4 a
2 1 + 1 A 2 / 3 a c
sy m
≈
A 1
2 / 3
a c ) − 1 A
1 2 / 3 a c )
1 + + A
2 4
a sy m
≈ 2 1 − 4 A
a sy m
2
≪
≈ ≈
wh ic h giv es Z A at s mall A , b u t h a s a co r r ectio n for lar g er A s u c h th at Z 0 . 41 A for h ea v y n u clei. [ Note th e app r o ximation an d s er ies ex pan s ion is tak en b ecau s e a c a sy m ]
If w e p lot Z / A v s . A the n u clid es lie b et w een 1/ 2 a n d 0 . 41. Th er e is a lin e of s tab ilit y , follo wing th e s ta b le is otop es (r ed in fi g u r e 4 and bla c k in figu r e 3 ). Th e is otop es ar e th en v a r iou s ly lab eled , for examp le h er e b y th eir lifetime.
In ter a ctiv e info r matio n is a v a ilable at h ttp:/ /www.n n d c.bn l.go v / c h a r t/ .
© Brookhaven National Laboratory. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse .
Fig . 3 : C h a rt o f n u c lid es fro m h ttp :/ / w w w .n n d c .b n l.g o v/ c h a rt/ . E a c h n u c lid e is c o lo r-la b eled b y its h a lf-life (b la c k fo r s ta b le n u c lid es)
150
20 0.35
100
50
120
100
80
60
40
250
200
50
100
150
200
250
0.55
0.50
0.45
0.40
Fig . 4 : N u c lid e c h a rt (o b ta in ed w ith th e s o ft w a re M a th ema tic a ). Left: Z vs. A , R ig h t: Z / A vs. A . I n red , sta b le n u c lid es . T h e b la c k lin e r ep resen ts Z = A / 2 .
1. 3 R adioac tive dec a y
Rad ioactiv e d eca y is the pr o ces s in wh ic h an u n s ta b le n u cleus s p on taneous ly los es en er gy b y emittin g io n izin g par ticles and r ad iatio n . Th is d eca y , or los s o f ener g y , r es u lts in an a to m of one t y p e, ca lled th e p aren t n uclide, tr ans fo r min g to a n atom o f a d iff er en t t y p e, n amed th e d au g h ter n uclide.
—
Th e thr ee p r incipal mo d es of d eca y ar e ca lled th e alp h a , b eta and g amma d eca ys . W e will s tud y th eir d iffer en ces an d exact mec h a n is ms la ter in th e clas s . Ho w ev er th es e d eca y mo d es s h a r e s o me common featu r e th at w e d es cr ib e n o w. W h at thes e r a d ioactiv e d eca y s d es cr ib e ar e fu nd amen ta lly qu a n tu m p r o ces s es , i.e. tr a n s ition s amo n g t w o q u a n tum s tates . Th u s , th e r a d io activ e d eca y is s ta tis tica l in n a tu r e, an d w e ca n o n ly des cr ib e th e ev olu tio n of th e ex p ecta tion v a lu es o f qu an tities of in ter es t, for ex amp le th e n um b er of atoms th at d eca y p er un it time. If w e o b s er v e a s ingle u n s ta b le n u cleu s , w e can not kn o w a p r ior i wh en it will deca y to its dau g h ter n u clid e. Th e time a t wh ic h th e d eca y h a p p en s is r and om, th u s at eac h in s tan t w e can h a v e th e p ar en t n u clid e with s ome p r ob a b ilit y p an d th e d a u gh ter with p r o b abilit y 1 p . This s to c h a s tic p r o ces s can o n ly b e d es cr ib ed in ter ms of the q u a n tum mec h anical ev olu tio n of th e n u cleu s . Ho w ev er , if w e lo o k at an en s em b le o f n u clei, w e ca n p r ed ict at eac h in s tan t th e a v er a ge n u m b er of p a r en t an d a u gh ter n uclides .
If w e call the n u m b er o f r a d io activ e n u clei N , th e n u m b er of d eca ying a to ms p er u n it time is dN /dt . It is fo u n d th a t this r ate is cons ta n t in time and it is pr op or tio n al to th e n u m b er of n u clei thems elv es :
d N
d t
= − λ N ( t )
Th e con s tan t o f p r op or tion a lit y λ is called th e d eca y co n stan t . W e ca n als o r ewr ite the ab o v e eq uatio n as
dN /dt
λ = − N
wh er e th e RHS is th e p r ob a b ilit y p er u n it time fo r one atom to d eca y . Th e fa ct th a t th is p r o b abilit y is a cons ta n t is a c har a cter is tic o f all r ad ioactiv e d eca y . It a ls o lea d s to th e ex p onential l aw of r adio active de c ay :
N ( t ) = N (0 ) e − λt
τ = 1 /λ
W e ca n a ls o defi n e th e mean l i feti me
t 1 / 2 = ln (2) /λ
and th e h al f-l i fe
wh ic h is th e time it ta k es for h a lf of th e atoms to d eca y , an d th e acti v i t y
A ( t ) = λ N ( t )
time δ t s u c h th a t δ t ≪ t
d t
S in ce A ca n als o b e ob tain ed a s d N , th e activ it y ca n b e es tima ted fr o m the n u m b er of d eca ys ΔN d u r ing a s mall
1 / 2
.
A common s itu atio n o ccu r s wh en th e d augh ter n u clid e is als o r a d io activ e. Th en w e h a v e a c h ain of r a d io activ e d eca y s ,
ea c h g o v er n ed b y th eir d eca y la ws . F o r examp le, in a c h ain N 1 → N 2 → N 3 , th e d eca y of N 1 an d N 2 is g iv en b y :
dN 1 = − λ 1 N 1 dt, dN 2 = + λ 1 N 1 dt − λ 2 N 2 dt
An o th er commo n c h a r acter is tic of r ad ioactiv e d eca y s is th a t th ey ar e a w a y for u ns ta b le n u clei to r eac h a mor e ener g etically fa v o r ab le (h en ce s tab le) con figu r ation . In α and β d eca ys , a n u cleu s emits a α or β p a r ticle, tr yin g to ap p r oac h th e mos t s tab le n uclide, while in th e γ d eca y a n excited s ta te deca y s to w ar d th e gr ou nd s ta te with ou t c h anging n u clear s p ecies .
1.3.1 Al pha dec a y
~ −
If w e go b a c k to th e b in din g en er gy p er mas s n u m b er plo t ( B / A v s . A ) w e s ee that ther e is a bu mp (a p eak ) for A 60 10 0. This mea n s th at th er e is a co r r es p o n d in g minim u m (o r en er gy optim u m) ar ou n d th es e n u m b er s . Th en the hea v ier n u clei will w a n t to d eca y to w ar d th is lig h ter n u clid es , b y s h ed d in g s o me p r oto n s an d n eutr on s . M or e s p ecifi cally , the d ecr ea s e in b in d in g ener g y a t h ig h A is d ue to Cou lom b r ep uls io n . Coulo m b r epu ls io n g r o ws in fa ct as Z 2 , m u c h fas ter th a n th e n u clea r fo r ce whic h is ∝ A .
Th is cou ld b e th ou g h t as a s imilar p r o ces s to what hap p en s in th e fi s s io n p r o ces s : fr o m a p ar en t n uclid e, t w o dau g h ter n uclides ar e cr eated . In the α d eca y w e h a v e s p ecifica lly :
A X N − →
A − 4 X ′ + α
Z
2
wh er e α is th e n ucleu s o f He-4: 4 He 2 .
Z − 2 N − 2
Th e α d eca y s h ou ld b e comp eting with o th er p r o ces s es , s u c h a s th e fi s s io n in to equ al d augh ter n uclides , or in to p a ir s in clu d in g 12 C or 16 O th a t ha v e lar g er B / A then α . Ho w ev er α d eca y is us u ally fa v o r ed. In or d er to un d er s tan d th is , w e s tar t b y lo oking at th e ener g etic of the d eca y , b u t w e will need to s tu d y th e q u an tum o r igin of th e deca y to ar r iv e at a fu ll ex pla n atio n .
Image by
2
P article
4 He
Fig . 5 : A lp h a d ec a y sc h ema tic s
MIT OpenCourseWare.
A . En ergetics
− −
In a n aly zing a r a d ioactiv e deca y (or a n y n uclea r r ea ction) an imp or tan t q u a n tit y is Q , th e net en er g y r eleas ed in th e d eca y: Q = ( m X m X ′ m α ) c 2 . Th is is als o eq ual to th e total k in etic en er gy o f th e fr ag men ts , h er e Q = T X ′ + T α (her e a s s umin g th a t th e p a r en t n u clid e is a t r es t).
W h en Q > 0 en er gy is r elea s ed in th e n uclear r eaction , wh ile fo r Q < 0 w e need to p r o v id e en er g y to mak e th e r eaction h ap p en . As in c h emis tr y , w e ex p ect the fir s t r eaction to b e a s p on taneous r eaction , while th e s econd o n e d o es n ot h a p p en in n atu r e with ou t in ter v en tion . (Th e fi r s t r eaction is exo-en er getic th e s eco n d en do-en er getic).
Notice th a t it’s no coin cid en ce th a t it’s called Q . In p r actice giv en s o me r ea gen ts a n d p r o du cts , Q giv e th e qu al ity of
Q α
the r ea ctio n , i.e. h o w en er g etically fa v o r able, h ence p r o b a b le, it is . F o r ex a mp le in th e a lp ha-d eca y lo g ( t 1 / 2 ) ∝ √ 1 ,
wh ic h is th e G eiger -Nu ttall r u le (192 8).
Th e alp h a p ar ticle car r ies a w a y mos t of th e kin etic en er gy (s in ce it is m u c h ligh ter ) a n d b y meas u r in g th is k in etic ener g y ex p er imen tally it is p o s s ib le to k n o w th e mas s es of un s table n u clid es .
W e ca n calcu late Q u s in g th e S EM F . Then :
Q α = B ( A − 4 X N ′ − 2 ) + B ( 4 H e ) − B ( A X N ) = B ( A − 4 , Z − 2) − B ( A, Z ) + B ( 4 H e )
Z − 2 Z
W e ca n a p p r o x imate th e fi nite d iff er en ce with th e r elev a n t gr a d ien t:
Q = [ B ( A − 4 , Z − 2) − B ( A, Z − 2)] + [ B ( A, Z − 2) − B ( A, Z )] + B ( 4 H e ) ≈ = − 4 ∂ B − 2 ∂ B + B ( 4 H e )
α
= 28 . 3 − 4 a v +
a s A − 1 / 3 + 4 a c
1 −
8 Z Z
3
3 A
A 1 / 3
∂ A ∂ Z
— 4 a sy m
1 −
+ 3 a p A − 7 / 4
2 Z 2
A
S in ce w e a r e lo ok in g a t h ea v y n u clei, w e kn o w th a t Z ≈ 0 . 41 A (in s tea d of Z ≈ A/ 2) a n d w e ob tain
Q α ≈ − 36 . 68 + 44 . 9 A − 1 / 3 + 1 . 02 A 2 / 3 ,
wh er e th e s eco n d ter m co mes fr o m th e s u r face co n tr ib u tio n and the la s t ter m is th e Co u lo m b ter m (w e n eg lect th e p a ir ing ter m, s in ce a p r ior i w e do n o t k n o w if a p is zer o or not).
≥
Th en , the Co u lom b ter m, althou g h s ma ll, mak es Q in cr eas e a t lar ge A . W e fin d th a t Q 0 fo r A � 15 0, an d it is
≈ ≥
Q 6M eV for A = 200 . Alth ough Q > 0, w e fi n d ex p er imen ta lly th a t α d eca y only ar is e for A 20 0.
≈
87
81
F u r th er , tak e fo r ex a mp le F r an ciu m-20 0 ( 200 F r 113 ). If w e calcu late Q α fr om the exp er imen ta lly fo u n d ma s s d iff er en ces w e o b tain Q α 7 . 6M eV (th e pr o du ct is 196 A t). W e ca n d o th e s ame calcula tion for th e h yp oth etical d eca y in to a 12 C a n d r ema in ing fr a gmen t ( 188 Tl 107 ):
Q 1 2 C = c 2 [ m ( A X N ) − m ( A − 12 X ′ ) − m ( 12 C )] ≈ 28 M eV
Z Z − 6 N − 6
Th us th is s econ d r eaction s eems to b e mo r e ener getic, h ence mor e fa v o r able th an the a lph a-d eca y , y et it d o es n o t o ccu r (s o me d eca y s in v olv in g C-12 h a v e b een o b s er v ed , b ut th eir b r an c h in g r a tios ar e m uc h s maller ).
Th us , lo ok in g only at th e en er getic of th e d eca y d o es n o t exp lain s ome q u es tio n s that s u r r ou nd th e alp h a d eca y:
- W h y th er e’s n o 12 C-d eca y ? (or to s o me of th is tigh tly b oun d n u clid es , e.g O -1 6 etc.)
- W h y th er e’s n o s p o n tan eou s fi s s io n in to eq u a l d a u gh ter s ?
- W h y th er e’s alp h a d eca y o n ly for A ≥ 20 0?
Q α
- W hat is th e ex p lan a tion of G eiger -Nu ttall r ule? log t 1 / 2 ∝ √ 1
1.3.2 Beta dec a y
Th e b eta d eca y is a r a d io activ e d eca y in wh ic h a p r o ton in a n u cleu s is co n v er ted in to a n eu tr on (or v ice-v er s a). Th us A is con s tan t, b u t Z and N c h an g e b y 1. In the p r o ces s th e n u cleu s emits a b eta par ticle (eith er an electr on or a p o s itr o n ) and qu as i-ma s s les s p ar ticle, th e n eu tri n o .
Th er e ar e 3 t y p es of b eta d eca y:
Courtesy of Thomas Jefferson National Accelerator Facility - Office of Science Education. Used with permission.
Fig . 6 : B eta d ec a y sc h ema tic s
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 deca y ). Th e u n d er ly in g r eaction is :
n → p + e − + ν ¯
that co r r es p on ds to th e co n v er s ion of a p r oton in to a n eutr on with the emis s ion of an electr on a n d a n an ti-n eu tr in o. Th er e ar e t w o oth er t y p es o f r eaction s , th e β + r eaction ,
Z
A X N
A
→ Z − 1
X N ′ + 1
+ e + + ν ⇐ ⇒ p → n + e + + ν
wh ic h s ees th e emis s io n of a p os itr on (th e electr on an ti-par ticle) and a n eu tr in o; a n d th e electr on ca p tur e:
Z
A X N + e −
A
→ Z − 1
X N ′ + 1 + ν
⇐ ⇒ p + e −
→ n + ν
a p r o ces s th a t comp etes with, or s ub s titu tes , th e p o s itr o n emis s ion .
∝
Reca ll th e mas s o f n u clid e as giv en b y th e s emi-emp ir ical mas s for m u la. If w e k eep A fi x ed , th e S EM F g iv es th e b in d in g ener g y as a fu n ction of Z . Th e only ter m that d ep en ds ex p licitly o n Z is th e Coulo m b ter m. By in s p ection w e s ee th a t B Z 2 . Th en fr om th e S EM F w e h a v e th a t th e mas s es of p os s ib le n u clides with th e s ame mas s n u m b er lie on a par a b ola. Nu clid es lo w er in th e p ar ab o la h a v e s maller M and ar e th u s mor e s table. In o r d er to r ea c h th at min im u m, un s tab le n uclid es u n d er g o a d eca y pr o ces s to tr a n s for m exces s p r o tons in n eutr on s (a n d v ice-v er s a).
A=125
49
I n
56 Ba
50 Sn
55 Cs
51 Sb
53 I
54 X e
52 T e
52 T e
54 X e
A=128
49 I n
57 La
50 Sn
51 Sb
55 Cs
53 I 56 Ba
Fig . 7 : N u c lea r Ma ss C h a in fo r A =1 2 5 , (left) a n d A =1 2 8 (rig h t)
Th e b eta d eca y is the r a d ioactiv e d eca y p r o ces s that can con v er t p r oton s in to n eu tr ons (and vice-v er s a). W e will s tu d y mor e in d ep th th is mec h anis m, bu t h er e w e w an t s imp ly to p o in t ou t ho w this p r o ces s can b e ener g etica lly fa v o r able, and th u s w e ca n p r ed ict wh ic h tr a n s ition s ar e lik ely to o ccur , bas ed o n ly on the SEM F .
F or ex a mple, for A = 12 5 if Z < 52 w e h a v e a fa v or a b le n → p co n v er s ion (b eta deca y ) wh ile fo r Z > 52 w e h a v e
p → n (o r p os itr on b eta d eca y ), s o th at th e s tab le n u clide is Z = 5 2 (tellur iu m).
A . Co n serva tio n la ws
As th e n eu tr ino is h ar d to detect, initia lly th e b eta d eca y s eemed to vio late en er gy co n s er v ation . In tr o d ucin g a n extr a p a r ticle in th e p r o ces s allo ws on e to r es p ect co n s er v a tion of en er gy .
Th e Q v alu e o f a b eta d eca y is giv en b y th e us u a l fo r 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 ma s s es a n d n eg lecting th e electr on ’s b in d in g en er gies as us 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 har ed b y th e n eu tr ino and the electr on (w e n eg lect an y r ecoil of the mas s iv e n u cleu s ). Th en , th e emer g ing electr o n (r emem b er , th e o n ly p ar ticle tha t w e can r eally o b s er v e) d o es n o t h a v e a fix ed ener gy , as it w as for ex ample for th e g amma p h o to n . Bu t it will exh ibit a s p ectr u m o f en er gy (or the n u m b er of electr on at a g iv en ener g y ) a s w ell as a dis tr ib ution 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 a n aly zing th e Q M th eo r y of b eta d eca y .
Ex amp les
64 ր 64 Zn + e − + ν ¯ , Q β = 0 . 57 M eV
29 Cu ց
30
28
64 Ni + e + + ν , Q β
= 0 . 66 M eV
Th e neu tr in o a n d b eta p a r ticle ( β ± ) s h ar e the ener g y . S ince © Neil Spooner. All rights reserved. This content is excluded the n eutr in o s a r e v er y d ifficult to d etect (as w e will s ee th ey from our Creative Commons license. For more information, ar e almos t mas s les s an d in ter act v er y w e akl y with ma tter ), th e see http://ocw.mit.edu/fairuse .
. −
electr on s / p os itr ons ar e th e p a r ticles d etected in b eta -d eca y Fig . 8 : B eta d ec a y sp ec tra : Distrib u tio n o f mo men tu m
(to p p lo ts) a n d kin etic en erg y (b o tt o m) fo r β (left) a n d
and they pr es en t a c h ar acter is tic en er gy s p ectr um (s ee F ig .
8 ).
β + (rig h t) d ec a y .
Th e d iff er en ce b et w een th e s p ectr u m o f th e β ± p a r ticles is du e to th e Cou lom b r epu ls io n or a ttr action fr o m the n ucleu s .
Notice that th e n eu tr in os als o car r y a w a y a n gu lar mo men tu m. Th ey a r e s p in-1 /2 p ar ticles , with n o c h ar ge (hen ce the n a me) a n d v er y s mall ma s s . F o r man y y ear s it w a s actu a lly b eliev ed to h a v e zer o ma s s . Ho w ev er it has b een co n fi r med that it d o es h a v e a mas s in 199 8.
O ther co n s er v ed q u an tities ar e:
- Mo men tu m : Th e mo men tu m is als o s h a r ed b et w een th e electr on an d the n eu tr ino. Th u s th e obs er v ed electr on momen tu m r an g es fr om zer o to a max im u m p os s ib le momen tu m tr a n s fer .
- An g u l ar mo men tu m (b oth th e electr on and th e n eu tr ino h a v e s p in 1/2 )
- P ari t y ? It tu r n s out th at p ar it y is n ot con s er v ed in th is d eca y . Th is hin ts to th e fact th a t th e in ter actio n r es p o n s ib le v iolates p a r it y con s er v a tion (s o it ca n n o t b e the 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 eatio n of a p r oto n is fo r ex amp le a lw a y s a ccomp a n ied b y th e cr eation of a n electr on)
- L ep to n n u m b er : w e d o not co n s er v e th e total n u m b er o f par ticles (w e cr eate b eta and neu tr in os ). Ho w ev er th e n u m b er of mas s iv e, h ea v y par ticles (or b a r y on s , comp os ed o f 3 qu ar k s ) is co n s er v ed . Als o the lepton n u m b er is co n s er v ed . Lep ton s ar e fu n d a men tal p a r ticles (in clud in g th e electr on , m u on an d ta u , as w ell a s th e th r ee t y p es o f n eu tr inos as s o cia ted with th es e 3 ). Th e lep ton n u m b er is + 1 fo r th es e par ticles an d -1 for th eir a n tipar ticles . Th en an electr o n is alw a y s accomp a n ied b y th e cr eation o f a n an tin eutr in o , e.g ., to con s er v e th e lep ton n u m b er (initially zer o).
Alth o u gh th e en er gy in v olv ed in th e d eca y can p r ed ict wh eth er a b eta deca y will o ccu r ( Q > 0), a n d wh ic h t yp e o f b eta d eca y do es o ccu r , th e d eca y r ate can b e qu ite d iff er en t ev en fo r s imilar Q -v alu es . Con s id er fo r ex a mp le 22 Na and 36 Cl. Th ey b oth d eca y b y β d eca y:
22 22 +
11 Na 11 → 10 Ne 12 + β
+ ν , Q = 0 . 22M eV , T 1 = 2 . 6y ear s
2
17
18
2
36 Cl 19 → 36 Ar 18 + β − + ν ¯ Q = 0 . 25M eV , T 1 = 3 × 10 5 y ear s
Ev en if th ey h a v e v er y clo s e Q -v a lu es , ther e is a fi v e or d er magnitu d e in th e lifetime. Th u s w e need to lo ok clo s er to the n u clear s tr u ctu r e in or d er to un d er s tan d thes e diff er en ces .
1.3.3 Gamma dec a y
In the gamma d eca y th e n u clid e is un c h a n ged , b u t it go es fr om a n ex cited to a lo w er en er gy s ta te. Th es e s ta tes ar e ca lled is o mer ic s ta tes . Us u a lly th e r ea ctio n is wr itten as :
Z
N
Z
A X ∗ − → A X N + γ
wh er e th e s tar in d ica te an ex cited s ta te. W e will s tu d y th at the g amma ener g y dep en d s on the en er gy d iff er ence b et w een th es e t w o s ta tes , b ut wh ic h d eca y s ca n h a p p en dep en d , on ce aga in, on th e d etails o f th e n u clea r s tr u ctu r e and o n qu an tu m-mec h anical s election r u les a s s o ciated with th e n u clea r an g u lar momen tu m.
1.3.4 Sp ontaneous fis s i on
S o me n u clei can s p on tan eou s ly un d er go a fi s s ion , ev en ou ts ide the p ar ticu lar cond ition s fou n d in a n uclear r ea ctor . In th e p r o ces s a h ea v y n u clid e s plits in to t w o ligh ter n u clei, o f r ough ly the s ame mas s .
1.3.5 Branc hi ng R ati os
S o me n u clei on ly d eca y via a s in gle p r o ces s , b u t s ometimes th ey can u nd er go man y diff er en t r a d ioactiv e p r o ces s es , that co mp ete on e with th e other . Th e r ela tiv e in ten s ities of th e comp etin g d eca y s ar e called b r an c h ing r atios .
Br an c h in g r atio s ar e ex p r es s ed as p er cen tag e o r s ometimes as p a r tial h a lf-liv es . F o r ex a mp le, if a n ucleu s ca n deca y b y b eta d eca y (an d o th er mo d es ) with a b r an c h in g r a tion b β , th e p ar tial half-life for th e b eta d eca y is λ β = b β λ .
[This page intentionally blank.]
MIT OpenCourseWare http://ocw.mit.edu
22.02 Introduction to Applied Nuclear Physics
Spring 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .