3. Sc atteri ng, T unnel i ng and Al pha D ec a y
3.1 Re vie w: E ne rgy e ige n v alue proble m
3.2 Un b ound Proble ms in Q uan tum Me c hanic s
3 .2 .1 I n fi n ite b a r r ier
3 .3 .2 Qu a n t u m mec h a n ic s d es c r ip tio n o f a lp h a d ec a y
3. 1 R eview : Energy eigenvalue p roblem
Th e time-in d ep en d en t w a v efu n ction ob ey s th e time-in d ep en d en t S c h r ¨ od in ger eq u ation :
H ϕ ( x x ) = E ϕ ( x x )
wh er e E is id en tifi ed as th e en er gy of th e s y s tem. If th e w a v efu nction is giv en b y ju s t its time-in d ep en d en t p ar t, ψ ( x x , t ) = ϕ ( x x ), th e s tate is s tationar y . Th u s , th e time-in d ep en d en t S c h r ¨ od in ger eq u ation allo ws u s to fi n d s tation ar y s tates of th e s y s tem, giv en a cer tain Hamilton ian .
Notice th at th e time-in d ep en d en t S c h r ¨ od in ger eq u ation is n oth in g els e th an th e eigen v alu e eq u ation for th e Hamil ton ian op er ator .
Th e en er gy of a p ar ticle h as con tr ib u tion s fr om th e k in etic en er gy as w ell as th e p oten tial en er gy :
1 2 2 2
H = 2 m ( p ˆ x + p ˆ y + p ˆ z ) + V ( x ˆ , y ˆ , z ˆ )
or mor e ex p licitly :
1 2 ∂ 2 ∂ 2 ∂ 2
H = − 2 m ∂ x 2 + ∂ y 2 + ∂ z 2
+ V ( x , y , z )
wh ic h can b e wr itten in a comp act for m as
1 2
H = − 2 m ∇ + V ( x , y , z )
2
(Notice th at V ( x , y , z ) is ju s t a m u ltip licativ e op er ator , in th e s ame w a y as th e p os ition is ).
In 1D , for a fr ee p ar ticle th er e is n o p oten tial en er gy , b u t on ly k in etic en er gy th at w e can r ewr ite as :
1 2 1 2 ∂ 2
H = 2 m p = − 2 m ∂ x 2
Th e eigen v alu e p r ob lem H w n ( x ) = E n w n ( x ) is th en th e d iff er en tial eq u ation
1 2 ∂ 2 w n ( x )
H w n ( x ) = E n w n ( x ) → − 2 m ∂ x 2 = E n w n ( x )
F or a fr ee p ar ticle th er e is n o r es tr iction on th e p os s ib le en er gies , E n can b e an y p os itiv e n u m b er . Th e s olu tion to th e eigen v alu e p r ob lem is th en th e eigen fu n ction :
w n ( x ) = A s in ( k n x ) + B cos ( k n x ) = A ′ e ik n x + B ′ e − ik n x
wh ic h r ep r es en ts t w o w a v es tr a v elin g in op p os ite d ir ection s .
±
W e s ee th at th er e ar e t w o in d ep en d en t fu n ction s for eac h eigen v alu e E n . Als o th er e ar e t w o d is tin ct momen tu m eigen v alu es k n for eac h en er gy eigen v alu e, wh ic h cor r es p on d to t w o d iff er en t d ir ection s of p r op agation of th e w a v e fu n ction e ± ik n x .
3. 2 U nb ound P roblems in Quantum M ec hanic s
W e will th en s olv e th e time-in d ep en d en t S c h r ¨ od in ger eq u ation in s ome in ter es tin g 1D cas es th at r elate to s catter in g p r ob lems .
3.2.1 I nfini te ba rri er
W e fi r s t con s id er a p oten tial as in F ig. 14 . W e con s id er t w o cas es :
• Cas e A. Th e s y s tem (a p ar ticle) h as a total en er gy lar ger th an th e p oten tial b ar r ier E > V H .
• Cas e B. Th e en er gy is s maller th an th e p oten tial b ar r ier , E < V H .
V(x)
E 1
V H
E 2
R eg ion I
R eg ion II
x
F ig . 1 4 : P o ten tia l fu n c tio n a n d t o ta l en er g y o f th e p a r tic le
Let’s fi r s t con s id er th e clas s ical p r ob lem. Th e s y s tem is a r igid b all with total en er gy E giv en b y th e s u m of th e k in etic an d p oten tial en er gy . If w e k eep th e total en er gy fi x ed , th e k in etic en er gies ar e d iff er en t in th e t w o r egion s :
T I = E T I I = E − V H
If E > V H
, th e k in etic en er gy in r egion t w o is T I I =
2
p
2 m = E − V H
, y ield in g simp ly a r ed u ced v elo cit y for th e p ar ticle.
If E < V H in s tead , w e w ou ld ob tain a n egativ e T I I k in etic en er gy . Th is is n ot an allo w ed s olu tion , b u t it mean s th at
th e p ar ticle can n ot tr a v el in to Region I I an d it’s in s tead con fi n ed in Region I: Th e p ar ticle b ou n ces off th e p oten tial b ar r ier .
In q u an tu m mec h an ics w e n eed to s olv e th e S c h r ¨ od in ger eq u ation in or d er to fi n d th e w a v efu n ction d es cr ib in g th e p ar ticle at an y p os ition . Th e time-in d ep en d en t S c h r ¨ od in ger eq u ation is
1 2 d 2
H ψ ( x ) = − ψ ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) →
− n 2 d 2 ψ ( x ) = E ψ ( x ) in Region I
2 mdx 2
2 2
2 m d x 2
n d ψ ( x )
− 2 m d x 2 = ( E − V H
) ψ ( x ) in Region I I
Th e t w o cas es d iff er b ecau s e in Region I I th e en er gy d iff er en ce ΔE = E − V H is eith er p os itiv e or n egativ e.
A . P ositive en ergy
Let’s fi r s t con s id er th e cas e in wh ic h ΔE = E − V H > 0. In b oth r egion s th e p ar ticle b eh a v es as a fr ee p ar ticle with en er gy E I = E an d E I I = E − V H . W e h a v e alr ead y s een th e s olu tion s to s u c h d iff er en tial eq u ation . Th es e ar e:
I I
ψ I ( x ) = Ae ik x + B e − ik x ψ ( x ) = C e ik ′ x + D e − ik ′ x
n 2 k 2 n 2 k ′ 2
wh er e 2 m = E an d 2 m = E − V H .
W e alr ead y in ter p r eted th e fu n ction e ik x as a w a v e tr a v elin g fr om left to r igh t an d e − ik x as a w a v e tr a v elin g fr om r igh t to left. W e th en con s id er a cas e s imilar to th e clas s ical cas e, in wh ic h a b all w as s en t to w ar d a b ar r ier . Th en th e p ar ticle is in itially d es cr ib ed as a w a v e tr a v elin g fr om left to r igh t in Region I. A t th e p oten tial b ar r ier th e p ar ticle can eith er b e r efl ected , giv in g r is e to a w a v e tr a v elin g fr om r igh t to left in Region I, or b e tr an s mitted , y ield in g a
w a v e tr a v elin g fr om left to r igh t in Region I I. Th is s olu tion i s d es cr ib ed b y th e eq u ation s ab o v e if w e s et D = 0, imp ly in g th at th er e is n o w a v e or igin atin g fr om th e far r igh t.
S in ce th e w a v efu n ction s h ou ld d es cr ib e a p h y s ical s itu ation , w e w an t it to b e a con tin u ou s fu n ction an d with con tin u ou s d er iv ativ e. Th u s w e h a v e to matc h th e s olu tion v alu es an d th eir d er iv ativ es at th e b ou n d ar y x = 0. Th is will giv e eq u ation s for th e co efficien ts , allo win g u s to fi n d th e ex act s olu tion of th e S c h r ¨ od in ger eq u ation . Th is is a b ou ndar y c onditions p r ob lem.
F r om
an d D = 0 w e ob tain th e con d ition s :
with s olu tion s
ψ I (0) = ψ I I (0) an d ψ I ′ (0) = ψ I ′ I (0)
A + B = C , ik ( A − B ) = ik ′ C
B = k − k ′ A, C = 2 k A
k + k ′ k + k ′
W e can fu r th er fi n d A b y in ter p r etin g th e w a v efu n ction in ter ms of a fl u x of p ar ticles . W e th u s fi x th e in comin g w a v e
n k
fl u x to b e Γ wh ic h s ets | A | = V m Γ
(w e can con s id er A to b e a r eal, p os itiv e n u m b er for s imp licit y ). Th en w e h a v e:
k − k ′ mΓ 2 k mΓ
B = J , C = J
k + k ′ 1 k k + k ′ 1 k
W e can als o v er ify th e follo win g id en tit y
wh ic h follo ws fr om:
k | A | 2 = k | B | 2 + k ′ | C | 2
2 ′ 2
| A |
2
′ 2 ′ 2
2 ( k − k ′ ) 2 + 4 k ′ k
k | B |
Let u s m u ltip ly it b y 1 /m :
+ k | C |
= ( k + k ′ ) 2 [ k ( k − k ) + k (2 k ) ] = k | A |
( k + k ′ ) 2
k
1 2 1 k 2 1 k ′ 2
m | A | = m | B | + m | C |
m
| | | |
Recall (p age 26 ) th e in ter p r etation of ψ ( x ) = Ae ik x as a w a v e giv in g a fl u x of p ar ticles ψ ( x ) 2 v = A 2 n k . Th is r elation s h ip s imilar ly h old s for th e fl u x in r egion I I as w ell as for th e r efl ected fl u x . Th en w e can in ter p r et th e eq u alit y ab o v e as an eq u alit y of p ar ticle fl u x :
m
R
m
T
m
Th e in comin g fl u x Γ = n k | A | 2 is eq u al to th e s u m of th e r efl ected Γ = n k | B | 2 an d tr an s mitted Γ = n k | C | 2 fl u xes .
Th e p ar ticle fl u x i s con s er v ed . W e can th en d efi n e th e r efl ection an d tr an s mis s ion co efficien ts as :
Γ = Γ R + Γ T = R Γ + T Γ
wh er e
k B 2
| |
R = k | A | 2 =
k − k ′ 2
k + k ′
, T =
k ′ | C | 2 k | A | 2
2 k 2 k ′ k + k ′ k
=
It’s th en eas y to s ee th at T + R = 1 an d w e can in ter p r et th e r efl ection an d tr an s mis s ion co efficien ts as th e r efl ection
an d tr an s mis s ion p r ob ab ilit y , r es p ectiv ely .
In lin e with th e p r ob ab ilis tic n atu r e of q u an tu m mec h an ics , w e s ee th at th e s olu tion of th e S c h r ¨ od in ger eq u ation d o es n ot giv e u s a p r ecis e lo cation for th e p ar ticle. In s tead it d es cr ib es th e p r ob ab ilit y of fi n d in g th e p ar ticle at an y p oin t in s p ace. G iv en th e w a v efu n ction fou n d ab o v e w e can th en calcu late v ar iou s q u an tit y of in ter es t, s u c h as th e p r ob ab ilit y of th e p ar ticle h a v in g a giv en momen tu m, p os ition an d en er gy .
B . Nega tive en ergy
No w w e tu r n to th e cas e wh er e E < V H , s o th at ΔE < 0. In th e clas s ical cas e w e s a w th at th is imp lied th e imp os s ib ilit y for th e b all to b e in r egion I I. In q u an tu m mec h an ics w e can n ot s imp ly gu es s a s olu tion b as ed on ou r in tu ition , b u t w e n eed again to s olv e th e S c h r ¨ od in ger eq u ation . Th e on ly d iff er en ce is th at n o w in r egion I I w e h a v e
n 2 k ′ ′ 2
2 m = E − V H < 0.
As q u an tu m mec h an ics is d efi n ed in a comp lex s p ace, th is d o es n ot p os e an y p r ob lem (w e can h a v e n egativ e k in etic
en er gie s ev en if th e total en er gy is p os itiv e) an d w e can s olv e for k ′′ s imp ly fi n d in g an imagin ar y n u m b er k ′′ = iκ ,
n 2
κ = V 2 m ( V H − E ) (with κ r eal).
Th e s olu tion s to th e eigen v alu e p r ob lem ar e s imilar to wh at al r ead y s een :
I I
ψ I ( x ) = Ae ik x + B e − ik x ψ ( x ) = C e ik ′ ′ x = C e − κ x ,
wh er e w e to ok D = 0 as b efor e.
| |
| |
Q u an tu m mec h an ics allo ws th e p ar ticle to en ter th e clas s ical for b id d en r egion , b u t th e w a v efu n ction b ecomes a v an is h in g ex p on en tial fu n ction . Th is mean s th at ev en if th e p ar ticle can in d eed en ter th e for b id d en r egion , it can n ot go v er y far , th e p r ob ab ilit y of fi n d in g th e p ar ticle far a w a y fr om th e p oten tial b ar r ier (giv en b y P ( x > 0) = ψ I I ( x ) 2 = C 2 e − 2 κ x ) b ecomes s maller an d s maller .
Again w e matc h th e fu n ction an d its d er iv ativ es at th e b ou n d ar y to fi n d th e co efficien ts :
ψ I (0) = ψ I I (0) → A + B = C
ψ I ′ (0) = ψ I ′ I (0) → ik ( A − B ) = − κC
with s olu tion s
B = k − iκ A, C = 2 k A
k + iκ k + iκ
Th e s itu ation in ter ms of fl u x is in s tead q u ite d iff er en t. W e n o w h a v e th e eq u alit y : k | B | 2 = k | A | 2 :
2 2 2
k | B | 2 = k �
�
= k k 2 + κ 2 = k .
� k − iκ � k + κ
k + iκ
In ter ms of fl u x , w e can wr ite th is r elation s h ip as Γ = Γ R , wh ic h imp lies R = 1 an d T = 0. Th u s w e h a v e n o tr an s mis s ion , ju s t p er fect r efl ection , alth ou gh th er e is a p en etr ation of th e p r ob ab ilit y in th e for b id d en r egion . Th is can b e called an evanes c ent tr an s mitted w a v e.
3.2.2 Fi ni te ba rri er
W e n o w con s id er a d iff er en t p oten tial wh ic h cr eates a fi n ite b ar r ier of h eigh t V H b et w een x = 0 an d L . As d ep icted in F ig. 15 th is p oten tial d iv id es th e s p ace in 3 r egion s . Again w e con s id er t w o cas es , wh er e th e total en er gy of th e p ar ticle is gr eater or s maller th an V H . Clas s ically , w e con s id er a b all in itially in Region I. Th en in th e cas e wh er e
V(x)
E 1
V H
E 2
0
L
x
R eg ion I R eg ion II R eg ion III
F ig . 1 5 : F in it e b a r r ier p o ten tia l
E > V H th e b all can tr a v el ev er y wh er e, in all th e th r ee r egion s , wh ile for E < V H it is goin g to b e con fi n ed in Region I, an d w e h a v e p er fect r efl ection . W e will con s id er n o w th e q u a n tu m mec h an ical cas e.
A . P ositive k in etic en ergy
F ir s t w e con s id er th e cas e wh er e ΔE = E − V H > 0. Th e k in etic en er gies in th e th r ee r egion s ar e
|
Region I |
Region I I |
Region I I I |
|
n 2 k 2 |
n 2 k ′ 2 |
n 2 k 2 |
|
= 2 m = E |
T = 2 m = E − V |
H T = 2 m = |
|
38 |
T E
An d th e w a v efu n ction is
Region I Region I I Region I I I
Ae ik x + B e − ik x C e ik ′ x + D e − ik ′ x E e ik x
(again w e p u t th e ter m F e − ik x = 0 for p h y s ical r eas on s , in an alogy with th e clas s ical cas e s tu d ied ). Th e co efficien ts can b e calcu lated b y con s id er in g th e b ou n d ar y con d ition s .
In p ar ticu lar , w e ar e in ter es ted in th e p r ob ab ilit y of tr an s mis s ion of th e b eam th r ou gh th e b ar r ier an d in to r egion
I I I. Th e tr an s mis s ion co efficien t is th en th e r atio of th e ou tgoin g fl u x in Region I I I to th e in comin g fl u x in Region I (b oth of th es e fl u x es tr a v el to th e Righ t, s o w e lab el th em b y R ):
I
T = I I I = =
k | ψ R | 2 k | E | 2 | E | 2
k | ψ R | 2 k | A | 2 | A | 2
wh ile th e r efl ection co efficien t is th e r atio of th e r efl ected (fr om r igh t to left, lab eled L ) an d in comin g (fr om left to r igh t, lab eled R ) fl u x in Region I:
R = I = =
k | ψ L | 2 k | B | 2 | B | 2
I
k | ψ R | 2 k | A | 2 | A | 2
W e can s olv e ex p licitly th e b ou n d ar y con d ition s :
ψ I (0) = ψ I I (0) ψ I I ( L ) = ψ I I I ( L )
ψ I ′ (0) = ψ I ′ I (0) ψ I ′ I ( L ) = ψ I ′ I I ( L )
an d fi n d th e co efficien ts B , C , D , E ( A is d eter min ed fr om th e fl u x in ten s it y Γ . F r om th e fu ll s olu tion w e can v er ify th at T + R = 1, as it s h ou ld b e p h y s ically .
/
O b s . Notice th at w e cou ld als o h a v e fou n d a d iff er en t s olu tion , e.g. in wh ic h w e s et F = 0 an d A = 0, cor r es p on d in g
to a p ar ticle or igin atin g fr om th e r igh t.
B . Nega tive En ergy
n 2
n 2
In th e cas e wh er e ΔE = E − V H < 0, in r egion I I w e ex p ect as b efor e an imagin ar y momen tu m. In fact w e fi n d Region I Region I I Region I I I
n 2
k = V 2 m E
k ′ = iκ, κ = V 2 m ( V H − E )
k = V 2 m E
An d th e w a v efu n ction is
Region I Region I I Region I I I
Ae ik x + B e − ik x C e − κ x + D e κ x E e ik x
Th e d iff er en ce h er e is th at a fi n ite tr an s mis s ion th r ou gh th e b ar r ier is p os s ib le an d th e tr an s mis s ion co efficien t is n ot zer o. In d eed , fr om th e fu ll s olu tion of th e b ou n d ar y con d ition p r ob lem, w e can fi n d as in th e p r ev iou s cas e th e co efficien ts T an d R an d w e h a v e T + R = 1.
→ ∞
Th er e is th u s a p r ob ab ilit y th at th e p ar ticle tu nnel s th r ou gh th e fi n ite b ar r ier an d ap p ear s in Region I I I, th en con tin u in g to x .
O b s . Alth ou gh w e h a v e b een d es cr ib in g th e s itu ation in ter ms of w a v e tr a v elin g in on e d ir ection or th e oth er , wh at w e ar e d es cr ib in g h er e is n ot a time-d ep en d en t p r ob lem. Th er e is n o time-d ep en d en ce at all in th is p r ob lem (all th e s olu tion s ar e on ly a fu n ction of x , n ot of time). Th is is th e s ame s itu ation as s tation ar y w a v es for ex amp le in a r op e. Th e s tate of th e s y s tem is n ot ev olv in g. It is alw a y s (at an y time) d es cr ib ed b y th e s ame w a v es an d th u s at an y time w e will h a v e th e s ame ou tcomes an d p r ob ab ilit y ou tcomes for an y meas u r emen t.
C. Estima tes a n d sca lin g
In s tead of s olv in g ex actly th e p r ob lem for th e s econ d cas e, w e tr y to mak e s ome es timates in th e cas e th er e is a v er y s mall tu n n elin g p r ob ab ilit y . In th is cas e w e h a v e th e follo win g ap p r o x imation s for th e co efficien ts A, B , C an d D .
/
≪ ≈
- As s u min g T 1 w e ex p ect D 0 s in ce if th er e is a v er y s mall p r ob ab ilit y for th e p ar ticle to b e in r egion I I I, th e p r ob ab ilit y of comin g b ac k fr om it th r ou gh th e b ar r ier m u s t b e ev en s maller (in oth er w or d s , if D = 0 w e w ou ld h a v e an in cr eas in g p r ob ab ilit y to h a v e a w a v e comin g ou t of th e b ar r ier ).
- Als o, T ≪ 1 imp lies R ≈ 1. Th is mean s th at B / A ≈ 1 or B ≈ A .
- M atc h in g th e w a v efu n ction at x = 0, w e h a v e C = A + B ≈ 2 A .
- F in ally matc h in g th e w a v efu n ction at x = L w e ob tain :
ψ ( L ) = C e − κ L = 2 Ae − κ L = E e ik L
I I I
W e can th en calcu late th e tr an s mis s ion p r ob ab ilit y T fr om T = k | ψ R | 2 , with th es e as s ump tion s . W e ob tain
I
k | ψ R | 2
k | E | 2 | E | 2 4 | A | 2 e − 2 κ L
− 2 κ L
T = k | A | 2 = | A | 2 = | A | 2
→ T = 4 e
≈
≈ ≈
Th u s th e tr an s mis s ion p r ob ab ilit y d ep en d s on th e len gth of th e p oten tial b ar r ier (th e lon ger th e b ar r ier th e les s tr an s mis s ion w e h a v e, as it is in tu itiv e) an d on th e co efficien t κ . Notice th at κ d ep en d s on th e d iff er en ce b et w een th e p ar ticle en er gy an d th e p oten tial s tr en gth : If th e p ar ticle en er gy is n ear th e ed ge of th e p oten tial b ar r ier (th at is , ΔE 0) th en κ 0 an d th er e’s a h igh p r ob ab ilit y of tu n n elin g. Th is cas e is h o w ev er again s t ou r fi r s t as s u mp tion s of s mall tu n n elin g (th at’s wh y w e ob tain th e u n p h y s ical r es u lt th at T 4!!). Th e cas e w e ar e con s id er in g is in s tead wh er e th e p ar ticle en er gy is s mall comp ar ed to th e p oten tial, s o th at κ is lar ge, an d th e p ar ticle h as a v er y lo w p r ob ab ilit y of tu n n elin g th r ou gh .
3. 3 Alpha dec a y
∝
~ −
If w e go b ac k to th e b in d in g en er gy p er mas s n u m b er p lot ( B / A v s . A ) w e s ee th at th er e is a b u mp (a p eak ) for A 60 100. Th is mean s th at th er e is a cor r es p on d in g min im u m (or en er gy op tim u m) ar ou n d th es e n u m b er s . Th en th e h ea v ier n u clei will w an t to d eca y to w ar d th is ligh ter n u clid es , b y s h ed d in g s ome p r oton s an d n eu tr on s . M or e s p ecifi cally , th e d ecr eas e in b in d in g en er gy at h igh A is d u e to Cou lom b r ep u ls ion . Cou lom b r ep u ls ion gr o ws in fact as Z 2 , m u c h fas ter th an th e n u clear for ce wh ic h is A .
Th is cou ld b e th ou gh t as a s imilar p r o ces s to wh at h ap p en s in th e fi s s ion p r o ces s : fr om a p ar en t n u clid e, t w o d au gh ter n u clid es ar e cr eated . In th e α d eca y w e h a v e s p ecifi cally :
A X N − →
A − 4 X ′ + α
Z
2
wh er e α is th e n u cleu s of He-4: 4 He 2 .
Z − 2 N − 2
Th e α d eca y s h ou ld b e comp etin g with oth er p r o ces s es , s u c h as th e fi s s ion in to eq u al d au gh ter n u clid es , or in to p air s in clu d in g 12 C or 16 O th at h a v e lar ger B / A th en α . Ho w ev er α d eca y is u s u ally fa v or ed . In or d er to u n d er s tan d th is ,
w e s tar t b y lo ok in g at th e en er getic of th e d eca y , b u t w e will n eed to s tu d y th e q u an tu m or igin of th e d eca y to ar r iv e at a fu ll ex p lan ation .
2
P article
4 He
Image by MIT OpenCourseWare.
F ig . 1 6 : A lp h a d ec a y s c h ema t ic s
3.3.1 Energeti c s
− −
In an aly zin g a r ad ioactiv e d eca y (or an y n u clear r eaction ) an imp or tan t q u an tit y is Q , th e n et en er gy r eleas ed in th e d eca y : Q = ( m X m X ′ m α ) c 2 . Th is is als o eq u al to th e total k in etic en er gy of th e fr agmen ts , h er e Q = T X ′ + T α (h er e as s u min g th at th e p ar en t n u clid e is at r es t).
W h en Q > 0 en er gy is r eleas ed in th e n u clear r eaction , wh ile for Q < 0 w e n eed to p r o v id e en er gy to mak e th e r eaction h ap p en . As in c h emis tr y , w e ex p ect th e fi r s t r eaction to b e a s p on tan eou s r eaction , wh ile th e s econ d on e d o es n ot h ap p en in n atu r e with ou t in ter v en tion . (Th e fi r s t r eaction is ex o-en er getic th e s econ d en d o-en er getic).
Q α
1
Notice th at it’s n o coin cid en ce th at it’s called Q . In p r actice giv en s ome r eagen ts an d p r o d u cts , Q giv e th e qu al ity of th e r eaction , i.e. h o w en er getically fa v or ab le, h en ce p r ob ab le, it is . F or ex amp le in th e alp h a-d eca y log ( t 1 / 2 ) ∝ √ ,
wh ic h is th e G eiger -Nu ttall r u le (1928).
Th e alp h a p ar ticle car r ies a w a y mos t of th e k in etic en er gy (s in ce it is m u c h ligh ter ) an d b y meas u r in g th is k in etic en er gy ex p er imen tally it is p os s ib le to k n o w th e mas s es of u n s tab le n u clid es .
W e can calcu late Q u s in g th e S EM F . Th en :
Z − 2
Z
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 )
W e can ap p r o x imate th e fi n ite d iff er en ce with th e r elev an t gr a d ien t:
∂ A ∂ Z
α
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 )
= 28 . 3 − 4 a v + a s A − 1 / 3 + 4 a c 1 − − 4 a sy m 1 − + 3 a p A − 7 / 4
8 Z Z 2 Z 2
3 3 A
A 1 / 3
A
S in ce w e ar e lo ok in g at h ea v y n u clei, w e k n o w th at Z ≈ 0 . 4 1 A (in s tead of Z ≈ A/ 2) an 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 econ d ter m comes fr om th e s u r face con tr ib u tion an d th e las t ter m is th e Cou lom b ter m (w e n eglect th e p air in g ter m, s in ce a p r ior i w e d o n ot k n o w if a p is zer o or n ot).
≥
Th en , th e Cou lom b ter m, alth ou gh s mall, mak es Q in cr eas e at lar ge A . W e fi n d th at Q 0 for A � 150, an d it is
≈ ≥
Q 6M eV for A = 200. Alth ou gh Q > 0, w e fi n d ex p er imen tally th at α d eca y on ly ar is e for A 200.
≈
87
81
F u r th er , tak e for ex amp le F r an ciu m-200 ( 200 F r 113 ). If w e calcu late Q α fr om th e ex p er imen tally fou n d mas s d iff er en ces w e ob tain Q α 7 . 6M eV (th e p r o d u ct is 196 A t). W e can d o th e s ame calcu lation for th e h y p oth etical d eca y in to a 12 C an d r emain in g fr agmen t ( 188 Tl 107 ):
Z
Z − 6
N − 6
Q 1 2 C = c 2 [ m ( A X N ) − m ( A − 12 X ′ ) − m ( 12 C )] ≈ 28 M eV
Th u s th is s econ d r eaction s eems to b e mor e en er getic, h en ce mor e fa v or ab le th an th e alp h a-d eca y , y et it d o es n ot o ccu r (s ome d eca y s in v olv in g C-12 h a v e b een ob s er v ed , b u t th eir b r an c h in g r atios ar e m u c h s maller ).
Th u s , lo ok in g on ly at th e en er getic of th e d eca y d o es n ot ex p lain s ome q u es tion s th at s u r r ou n d 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 ome of th is tigh tly b ou n d n u clid es , e.g O -16 etc.)
- W h y th er e’s n o s p on tan eou s fi s s ion in to eq u al d au gh ter s ?
- W h y th er e’s alp h a d eca y on ly for A ≥ 200?
Q α
- W h at is th e ex p lan ation of G eiger -Nu ttall r u le? log t 1 / 2 ∝ √ 1
3.3.2 Quantum mec hani c s des c ri pti on of al pha dec a y
W e will u s e a s emi-clas s ical mo d el (th at is , com b in in g q u an tu m mec h an ics with clas s ical p h y s ics ) to an s w er th e q u es tion s ab o v e.
In or d er to s tu d y th e q u an tu m mec h an ical p r o ces s u n d er ly in g alp h a d eca y , w e con s id er th e in ter action b et w een th e d au gh ter n u clid e an d th e alp h a p ar ticle. J u s t p r ior to s ep ar ation , w e can con s id er th is p air to b e alr ead y p r es en t ins ide th e p ar en t n u clid e, in a b ou n d s tate. W e will d es cr ib e th is p air of p ar ticles in th eir c enter of m as s co or d in ate fr ames : th u s w e ar e in ter es ted in th e r elativ e motion (an d k in etic en er gy ) of th e t w o p ar ticles . As often d on e in th es e s itu ation s , w e can d es cr ib e th e r elativ e motion of t w o p ar ticles as th e motion of a s in gle p ar ticle of r ed u ced mas s
µ =
m α m ′ m α + m ′
(wh er e m ′ is th e mas s of th e d au gh ter n u clid e).
→
Con s id er for ex amp le th e r eaction 238 U 234 Th + α . W h at is th e in ter action b et w een th e Th an d alp h a p ar ticle in
th e b ou n d s tate?
- A t s h or t d is tan ce w e h a v e th e n u clear for ce b in d in g th e 238 U.
- A t lon g d is tan ces , th e cou lom b in ter action p r ed omin ates
Th e n u clear for ce is a v er y s tr on g, attr activ e for ce, wh ile th e Cou lom b for ce amon g p r oton s is r ep u ls iv e an d will ten d to ex p el th e alp h a p ar ticle.
S in ce th e fi n al s tate is k n o wn to h a v e an en er gy Q α = 4 . 3M eV, w e will tak e th is en er gy to b e as w ell th e in itial en er gy of th e t w o p ar ticles in th e p oten tial w ell (w e as s u me th at Q α = E s in ce Q is th e k in etic en er gy wh ile th e p oten tial en er gy is zer o). Th e s ize of th e p oten tial w ell can b e calcu lated as th e s u m of th e d au gh ter n u clid e ( 234 Th ) an d alp h a r ad ii:
R = R ′ + R α = R 0 ((234) 1 / 3 + 4 1 / 3 ) = 9 . 3fm .
≫
≫ −
O n th e oth er s id e, th e Cou lom b en er gy at th is s ep ar ation is V C o ul = e 2 Z ′ Z α /R = 28 M eV Q α (h er e Z ′ = Z 2 ). Th en , th e p ar ticles ar e in s id e a w ell, with a h igh b ar r ier (as V C o ul Q ) b u t th er e is s ome p r ob ab ilit y of tu n n elin g, s in ce Q > 0 an d th e s tate is n ot s tab ly b ou n d .
Th u s , if th e p ar en t n u clid e, 238 U, w as r eally comp os ed of an alp h a-p ar ticle an d of th e d au gh ter n u clid e, 234 Th ,
| |
th en with s ome p r ob ab ilit y th e s y s tem w ou ld b e in a b ou n d s tate an d with s ome p r ob ab ilit y in a d eca y ed s tate, with th e alp h a p ar ticle ou ts id e th e p oten tial b ar r ier . Th is las t p r ob ab ilit y can b e calcu lated fr om th e tu n n elin g p r ob ab ilit y P T w e s tu d ied in th e p r ev iou s s ection , giv en b y th e amp litu d e s q u ar e of th e w a v efu n ction ou ts id e th e b ar r ier , P T = ψ ( R o ut ) 2 .
Ho w d o w e r elate th is p r ob ab ilit y to th e d eca y r ate?
W e n eed to m u ltip ly th e p r ob ab ilit y of tu n n elin g P T b y th e fr eq u en cy f at wh ic h 238 U cou ld actu ally b e fou n d as b ein g in t w o fr agmen ts 234 Th + α (alth ou gh s till b ou n d togeth er in s id e th e p oten tial b ar r ier ). Th e d eca y r ate is th en giv en b y λ α = f P T .
T o es timate th e fr eq u en cy f , w e eq u ate it with th e fr eq u en cy at wh ic h th e comp ou n d p ar ticle in th e cen ter of mas s fr ame is at th e w ell b ou n d ar y : f = v in /R , wh er e v in is th e v elo cit y of th e p ar ticles wh en th ey ar e in s id e th e w ell
10
M eV
V Z
8
6
Q
4
2
R
R c
0
10
...
60
70
80
-2
Nucleus- separ a tion (fm)
F ig . 1 7 : P o t en tia l w ell fo r a lp h a d ec a y tu n n elin g . T h e in n er r a d iu s is R w h ile th e in ters ec tio n o f Q α w ith th e p o ten tia l is R c
(n o t t o s c a le).
Lab fr ame C en t er of mass fr ame
par en t
daugh t er
C ompound
R
R
F ig . 1 8 : P o s it io n s o f d a u g h ter a n d a lp h a p a r tic les in th e n u c leu s , a s s een in (left) th e la b o r a to r y fr a me a n d (r ig h t) in th e c en ter o f ma s s fr a me. Wh en t h e r ela tiv e d is t a n c e is z er o , th is c o r r esp o n d to a u n d ivid ed (p a r en t) n u c lid e. Wh en th e r ela tiv e d is t a n c e is R , it c o r r es p o n d s to a s ep a ra te a lp h a a n d d a u g h ter n u c lid e in s id e th e n u c leu s .
2 in
(s ee car to on in fi gu r e 18 ). W e h a v e 1 mv 2 = Q α + V 0 ≈ 40M eV, fr om wh ic h w e h a v e v in ≈ 4 × 10 22 fm/s . Th en th e
fr eq u en cy is f ≈ 4 . 3 × 10 21 .
2
| |
Th e p r ob ab ilit y of tu n n elin g is giv en b y th e amp litu d e s q u ar e of th e w a v efu n ction ju s t ou ts id e th e b ar r ier , P T = ψ ( R c ) , wh er e R c is th e co or d in ate at wh ic h V C o ul ( R c ) = Q α , s u c h th at th e p ar ticle h as again a p os itiv e k in etic en er gy :
R c =
e 2 Z α Z ′
Q α
≈ 63fm
~
Recall th at in th e cas e of a s q u ar e b ar r ier , w e ex p r es s ed th e w a v efu n ction in s id e a b ar r ier (in th e clas s ically for b id d en r egion ) as a p lan e w a v e with imagin ar y momen tu m, h en ce a d eca y in g ex p on en tial ψ in ( r ) e − κ r . W h at is th e r elev an t momen tu m 1 κ h er e? S in ce th e p oten tial is n o lon ger a s q u ar e b ar r ier , w e ex p ect th e momen tu m (an d k in etic en er gy ) to b e a fu n ction of p os ition .
Th e total en er gy is giv en b y E = Q α an d is th e s u m of th e p oten tial (Cou lom b ) an d k in etic en er gy . As w e’v e s een th at th e Cou lom b en er gy is h igh er th an Q , w e k n o w th at th e k in etic en er gy is n egativ e:
1 2 k 2 Z α Z ′ e 2
with µ th e r ed u ced mas s
Q α = T + V C o ul = 2 µ + r
m α m ′
µ =
m α + m ′
an d k 2 = − κ 2 (with κ ∈ R ). Th is eq u ation is v alid at an y p os ition in s id e th e b ar r ier :
κ ( r ) =
J 2 µ [ V ( r ) − Q ] =
2 µ Z α Z ′ e 2
1 2 r
— Q α
1 2 C o ul α
T k T
L
If w e w er e to con s id er a s mall s lice of th e b ar r ier , fr om r to r + dr , th en th e p r ob ab ilit y to p as s th r ou gh th is b ar r ier w ou ld b e dP T ( r ) = e − 2 κ ( r ) d r . If w e d iv id e th en th e total b ar r ier r an ge in to s mall s lices , th e fi n al p r ob ab ilit y is th e p r o d u ct of th e p r ob ab ilities dP k of p as s in g th r ou gh all of th e s lices . Th en log ( P T ) = log ( dP k ) an d tak in g th e
con tin u ou s limit log ( P T ) = J log [ dP ( r )] = − 2 J κ ( r ) dr .
R
R
R c R c
T
F in ally th e p r ob ab ilit y of tu n n elin g is giv en b y P T = e − 2 G , wh er e G is calcu lated fr om th e in tegr al
∫ ∫
R C R C
G = dr κ ( r ) = dr
R R
2 µ Z α Z ′ e 2
1 2 r
— Q α
W e can s olv e th e in tegr al an aly tically , b y lettin g r = R y = y Z α Z ′ e 2 , th en
Z α Z 0 e 2 2 µc 2 ∫ 1 J 1
c Q α
α
wh ic h y ield s
G = 1 c Q
dy
R / R C
y − 1
G =
ar ccos
c
—
R
c
1 − R
Z α Z ′ e 2 2 µc 2 � � J R � J R J
1 c Q α
R
R � Z α Z ′ e 2 2 µc 2 π
� J R �
R
c
c
=
1 c Q
α
g
2
wh er e to s imp lify th e n otation w e u s ed th e fu n ction
π
g ( x ) = 2 ( ar ccos ( x ) − x J 1 − x 2 ) .
F in ally th e d eca y r ate is giv en b y
λ α = R e
v in − 2 G
wh er e G is th e s o-called G amo w factor .
� J � ≈ � − J �
In or d er to get s ome in s igh t on th e b eh a v ior of G w e con s id er th e ap p r o x imation R ≪ R c :
G = 1 E G g R 1 E G 1 4 R
2 Q α R c 2 Q α π R c
E G =
2 π Z Z e µ c
α
2
1 c
2
2
2
wh er e E G is th e G amo w en er gy :
R c
F or ex amp le for th e 238 U d eca y s tu d ied E G = 122 , 000M eV (h u ge!) s o th at J E G /Q α = 171 wh ile g ( V R ) ≈ 0 . 5 18.
Th e ex p on en t is th u s a lar ge n u m b er , giv in g a v er y lo w tu n n elin g p r ob ab ily : e − 2 G λ α = 1 . 6 × 10 − 17 s or t 1 / 2 = 4 . 5 × 10 9 y ear s , clos e to wh at ob s er v ed .
= e − 89
= 4 × 10 − 39 . Th en ,
Th es e r es u lts fi n ally giv e an an s w er to th e q u es tion s w e h ad r egar d in g alp h a d eca y . Th e d eca y p r ob ab ilit y h as a v er y s tr on g d ep en d en ce on n ot on ly Q α b u t als o on Z 1 Z 2 (wh er e Z i ar e th e n u m b er of p r oton s in th e t w o d au gh ter s ). Th is lead s to th e follo win g ob s er v ation s :
- O th er t y p es of d eca y ar e les s lik ely , b ecau s e th e Cou lom b en e r gy w ou ld in cr eas e con s id er ab ly , th u s th e b ar r ier b ecomes to o h igh to b e o v er come.
- Th e s ame is tr u e for s p on tan eou s fi s s ion , d es p ite th e fact th at Q is m u c h h igh er ( ∼ 200M eV).
≥
- W e th u s fi n d th at alp h a d eca y is th e op timal mec h an is m. S till, it can h ap p en on ly for A 200 ex actly b ecau s e oth er wis e th e tu n n elin g p r ob ab ilit y is v er y s mall.
- Th e G eiger -Nu ttall la w is a d ir ect con s eq u en ce of th e q u an tu m tu n n elin g th eor y . Als o, th e lar ge v ar iation s of th e d eca y r ates with Q ar e a con s eq u en ce of th e ex p on en tial d ep en d en ce on Q .
A fi n al w or d of cau tion ab ou t th e mo d el: th e s emi-clas s ical mo d el u s ed to d es cr ib e th e alp h a d eca y giv es q u ite accu r ate p r ed iction s of th e d eca y r ates o v er man y or d er of magn itu d es . Ho w ev er it is n ot to b e tak en as an in d ication th at th e p ar en t n u cleu s is r eally alr ead y con tain in g an alp h a p ar ticle an d a d au gh ter n u cleu s (on ly , it b eh a v es as if it w er e, as lon g as w e calcu late th e alp h a d eca y r ates ).
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22.02 Introduction to Applied Nuclear Physics
Spring 2012
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