8. Appl i c ati ons of N uc l ea r Sc i enc e
8.1 I n te rac tion of radiation with matte r
8 .1 .2 N eu tr o n S c a t ter in g a n d A b s o r p tio n
8 .1 .3 C h a r g ed p a r tic le in ter a c tio n
8 .1 .4 E lec t r o ma g n etic r a d ia tio n
W e h a v e n o w a clear er p ictu r e of th e n u clear s tr u ctu r e an d of th e r ad ioactiv e d eca y s, as w ell as th e for malis m –b as ed on q u an tu m mec h an ics an d q u an tu m fi eld th eor y – th at d es cr ib es th eir d y n amics . W e can tu r n to th e s tu d y of s ome ap p lication s of th es e id eas .
F ir s t, w e will s tu d y h o w r ad iation in ter acts with matter . Th is is fu n d amen tal b oth in or d er to k n o w wh at ar e th e eff ects of r ad iation emitted d u r in g n u clear p r o ces s es on th e mater ials ar ou n d (an d th e p eop le) an d in or d er to d ev is e d etector s th at can meas u r e th es e r ad iation s . A t th e s ame time th e k n o wled ge of h o w r ad iation in ter acts with matter lead s to man y imp or tan t ap p lication s in e.g. n u clear med icin e, for imagin g an d th er ap y , in mater ials s cien ce, for imagin g an d d iagn os tic, in agr icu ltu r e, ar c h eology etc. M os t of y ou migh t h a v e alr ead y s tu d ied th es e ap p lication s in 22.01 an d als o an aly zed th e p r o ces s es th at giv e r is e to th e in ter action s . Th u s w e will b e h er e on ly h a v e a q u ic k r ev iew, fo cu s in g mos tly on th e p h y s ical p r o ces s es .
Th en w e will s tu d y t w o n u clear r eaction s (fi s s ion an d fu s ion ) th at can b e u s ed as s ou r ces of en er gy (or in th e cas e of fu s ion , th at h old s th at p r omis e).
8. 1 Interac tion of radiation w ith matter
8.1.1 Cros s Sec ti on
0
≈
≈
≈
Clas s ically , th e cr os s s ection is th e ar ea on wh ic h a collid in g p r o jectile can imp act. Th u s for ex amp le th e cr os s s ection of a s p h er ical tar get of r ad iu s r is ju s t giv en b y π r 2 . Th e cr os s s ection h as th en u n its of an ar ea. Let’s con s id er for ex amp le a n u cleu s with mas s n u m b er A . Th e r ad iu s of th e n u cleu s is th en R R 0 A 1 / 3 = 1 . 25 A 1 / 3 fm an d th e clas s ical cr os s s ection w ou ld b e σ = π R 2 A 2 / 3 5 A 2 / 3 fm 2 . F or a t y p ical h ea v y n u cleu s , s u c h as gold , A = 197, w e h a v e σ 100fm 2 = 1b ar n (s y m b ol b , 1 b = 10 − 28 m 2 = 10 − 24 c m 2 = 100 f m 2 .
W h en s catter in g a p ar ticle off a tar get h o w ev er , wh at b ecomes imp or tan t is n ot th e h ead -on collis ion (as b et w een b alls ) b u t th e in ter action b et w een th e p ar ticle an d th e tar get (e.g. Cou lom b , n u clear in ter action , w eak in ter action etc.). F or macr os cop ic ob jects th e d etails of th es e in ter action s ar e lu mp ed togeth er an d h id d en . F or s in gle p ar ticles th is is n ot th e cas e, an d for ex amp le w e can as w ell h a v e a collis ion ev en if th e d is tan ce b et w een p r o jectile an d tar get is lar ger th an th e tar get r ad iu s . Th u s th e cr os s s ection tak es on a d iff er en t mean in g an d it is n o w d efi n ed as th e eff ectiv e ar ea or mor e p r ecis ely as a meas u r e of th e p r ob ab ilit y of a collis ion . Ev en in th e clas s ical an alogy , it is eas y to s ee wh y th e cr os s s ection h as th is s tatis tical mean in g, s in ce in a collis ion th er e is a cer tain (p r ob ab ilis tic) d is tr ib u tion of th e imp act d is tan ce.
Th e cr os s s ection als o d es cr ib es th e p r ob ab ilit y of a giv en (n u clear ) r eaction to o ccu r , a r eaction th at can b e gen er ally wr itten as :
a + X → X ′ + b or X ( a, b ) X ′
wh er e X is an h ea v y tar get an d a a s mall p r o jectile (s u c h as a n eu tr on , p r oton , alp h a...) wh ile X ′ an d b ar e th e r eaction p r o d u cts (again with b b ein g n u cleon s or ligh t n u cleu s , or in s ome cas es a gamma r a y ).
Th en let I a b e th e cu r r en t of in comin g p ar ticles , h ittin g on an h ea v y (h en ce s tation ar y ) tar get. Th e h ea v y p r o d u ct X ′ will als o b e almos t s tation ar y an d on ly b will es cap e th e mater ial an d b e meas u r ed . Th u s w e will ob s er v e th e b p r o d u cts ar r iv in g at a d etector at a r ate R b . If th er e ar e n tar get n u clei p er u n it ar ea, th e cr os s s ection can th en b e wr itten as
σ = R b
I a n
Th is q u an tit y d o n ot alw a y s agr ee with th e es timated cr os s s ection b as ed on th e n u cleu s r ad iu s . F or ex amp le, p r oton s catter in g x -s ection can b e h igh er th an n eu tr on s , b ecau s e of th e Cou lom b in ter action . Neu tr in os x -s ection th en will b e ev en s maller , b ecau s e th ey on ly in ter act v ia th e w eak in ter action .
A . Diff eren tia l cross section
Th e ou tgoin g p ar ticles ( b ) ar e s catter ed in all d ir ection s . Ho w ev er mos t of th e time th e d etector on ly o ccu p ies a s mall r egion of s p ace. Th u s w e can on ly meas u r e th e r ate R b at a p ar ticu lar lo cation , id en tifi ed b y th e an gles ϑ , ϕ . W h at w e ar e actu ally meas u r in g is th e r ate of s catter ed p ar ticles in th e s mall s olid an gle dΩ , r ( ϑ , ϕ ), an d th e r elev an t cr os s s ection is th e d iff er en tial cr os s s ection
d σ r ( ϑ , ϕ )
=
d Ω 4 π I a n
F r om th is q u an tit y , th e total cr os s s ection , d efi n ed ab o v e, can b e calcu lated as
σ = dΩ = s in ϑ dϑ dϕ
1 d σ
1 π 1 2 π d σ
4 π d Ω
(Notice th at h a v in g ad d ed th e factor 4 π giv es σ = 4 π d σ
0 0
for con s tan t
d Ω
d σ .)
d Ω d Ω
B . Dou b ly d iff eren tia l cross section
W h en on e is als o in ter es ted in th e en er gy of th e ou tgoin g p ar ticles E b , b ecau s e th is can giv e in for mation e.g. on th e s tr u ctu r e of th e tar get or on th e c h ar acter is tic of th e p r o jectile-tar get in ter action , th e q u an tit y th at is meas u r ed is th e cr os s s ection as a fu n ction of en er gy . Th is can b e s imp ly
d σ d E b
if th e d etector is en er gy -s en s itiv e b u t collect p ar ticles in an y d ir ection , or th e d ou b ly d iff er en tial cr os s s ection
d 2 σ d Ω dE b
8.1.2 Neutron Sc atteri ng and A bs o rpti on
W h en n eu tr on s tr a v el in s id e a mater ial, th ey will u n d er go s catter in g (elas tic an d in elas tic) as w ell as oth er r eaction s , wh ile in ter actin g with th e n u clei v ia th e s tr on g, n u clear for ce. G iv en a b eam of n eu tr on with in ten s it y I 0 , wh en tr a v elin g th r ou gh matter it will in ter act with th e n u clei with a p r ob ab ilit y giv en b y th e total cr os s s ection σ T . A t h igh en er gies , r eaction s s u c h as (n ,p ), (n , α ) ar e p os s ib le, b u t at lo w er en er gy u s u ally wh at h ap p en s is th e cap tu r e of th e n eu tr on (n , γ ) with th e emis s ion of en er gy in th e for m of gamma r a y s . Th en , wh en cr os s in g a s mall r egion of s p ace dx th e b eam is r ed u ced b y an amou n t p r op or tion al to th e n u m b er of n u clei in th at r egion :
dI = − I 0 σ T ndx → I ( x ) = I 0 e − σ T nx
~
Th is for m u la, h o w ev er , is to o s imp lis tic: on on e s id e th e cr os s s ection d ep en d s on th e n eu tr on en er gy (th e cr os s s ection in cr eas es at lo w er v elo cit y as 1 /v an d at h igh er en er gies , th e cr os s s ection can p r es en t s ome r es on an ces – s ome p eak s ) an d n eu tr on s will los e p ar t of th eir en er gy wh ile tr a v elin g, th u s th e actu al cr os s s ection will d ep en d on th e p os ition . O n th e oth er s id e, n ot all r eaction s ar e ab s or p tion r eaction s , man y of th em will ”p r o d u ce” an oth er n eu tr on (i.e., th ey will on ly c h an ge th e en er gy of th e n eu tr on or its d ir ection , th u s n ot atten u atin g th e b eam). W e th en n eed a b etter d es cr ip tion of th e fate of a n eu tr on b eam in matter . F or ex amp le, wh en on e n eu tr on with en er gy 1M eV en ter s th e mater ial, it is fi r s t s lo w ed d o wn b y elas tic an d in elas tic collis ion s an d it is th en fi n ally ab s or b ed .
W e th en w an t to k n o w h o w man y collis ion s ar e n eces s ar y to s lo w d o wn a n eu tr on an d to calcu late th at, w e fi r s t n eed to k n o w h o w m u c h en er gy d o es th e n eu tr on lo os e in on e collis ion .
D iff er en t mater ials can h a v e d iff er en t cr os s s ection s , h o w ev er th e en er gy ex c h an ge in collis ion is m u c h h igh er th e ligh tes t th e tar get. Con s id er an elas tic collis ion with a n u cleu s of mas s M . In th e lab fr ame, th e n u cleu s is in itially at r es t an d th e n eu tr on h as en er gy E 0 an d momen tu m mv 0 . After th e s catter in g, th e n eu tr on en er gy is E 1 , s p eed v v 1 at an an gle ϕ with v v 0 , wh ile th e n u cleu s r ecoil giv es a momen tu m M V v at an an gle ψ (I will u s e th e n otation w for th e
HE-3 Cross Section (data from ENDF-VI.1 NJOY99)
total
absorption elastic
10 5
Cross section (barns)
4
10
10 3
10 2
10 1
10 0
10 -11 10 -10 10 -9 10 -8 10 -7 10 - 6 10 -5 10 -4 10 - 3 10 -2 10 - 1 10 0 10 1
Energy (MeV)
F ig . 4 7 : C r o s s s ec tio n σ ( E ) fo r t h e n eu tr o n -H e3 r ea c tio n s . T h e d a ta a n d p lo t c a n b e o b ta in ed o n lin e fr o m
http://t2.lanl.gov/data/ndviewer.html
U-235 cross sections FROM ENDF-VI.3
total absorption elastic gamma production
10 5
Cross section (barns)
4
10
10 3
10 2
10 1
10 0
10 -11 10 -10 10 - 9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1
Energy (MeV)
F ig . 4 8 : C r o s s s ec tio n σ ( E ) fo r t h e n eu tr o n -U 2 3 5 r ea c tio n s . N o tic e t h e 1 / v d ep en d en c e a t h ig h er en er g ies a n d r es o n a n c es a t lo w er en er g ies .
| |
magn itu d e of a v ector w v , w = w v ). Th e collis ion is b etter an aly zed in th e cen ter of mas s fr ame, wh er e th e con d ition of elas tic s catter in g imp lies th at th e r elativ e v elo cities on ly c h an ge th eir d ir ection b u t n ot th eir magn itu d e.
Th e cen ter of mas s v elo cit y is d efi n ed as v v
= m � v 0 + M V � 0 = m v v . Relativ e v elo cities in th e cen ter of mas s fr ame
C M m + M m + M 0
2
ar e d efi n ed as v u = v v − v v C M . W e can calcu late th e n eu tr on (k in etic) en er gy after th e collis ion fr om E 1 = 1 m | v v 1 | 2 . An
2
2
2
ex p r es s ion for | v v 1 | 2 is ob tain ed fr om th e CM s p eed :
2
| v v 1 |
= | v u 1 + v v C M |
= | v u 1 |
+ | v v C M |
+ 2 v u 1 · v v C M = u 2 + v 2 + 2 u 1 v C M cos ϑ
1 C M
wh er e w e d efi n ed ϑ as th e s catter in g an gle in th e cen ter of mas s fr ame (s ee fi gu r e 49 ).
Lab F r ame C en t er of M ass F r ame
m, v 1 , E 1
φ
m, u 1
e
m, v 0
, E 0
ʩ
M, V , E
m, u 0 M, U 0
M, U 1
F ig . 4 9 : N eu tr o n s c a t ter in g fr o m a n u c leu s . Left, la b fr a me. R ig h t, c e n ter o f ma s s fr a me
m + M m + M
G iv en th e as s u mp tion of elas tic s catter in g, w e h a v e | v u 1 | = | v u 0 | = u 0 , b u t u 0 = v 0 − v C M = v 0 C 1 − m r = M v 0 .
F in ally , w e can ex p r es s ev er y th in g in ter ms of v 0 :
2 M 2 2
m 2 2 M m
2 M 2 + m 2 + 2 mM cos ϑ
| v v 1 |
= ( m + M ) 2 v 0 + ( m + M ) 2 v 0 + 2 ( m + M ) v 0 ( m + M ) v 0 cos ϑ = v 0
( m + M ) 2
≈
W e n o w s imp lify th is ex p r es s ion b y mak in g th e ap p r o x imation M /m A , wh er e A is th e mas s n u m b er of th e n u cleu s . In ter ms of th e n eu tr on en er gy , w e fi n ally h a v e
E 1 = E 0
A 2 + 1 + 2 A cos ϑ
( A + 1) 2 ,
0
Th is mean s th at th e fi n al en er gy can b e eq u al to E 0 (th e in itial on e) if ϑ = 0 – cor r es p on d in g to n o collis ion – an d
r eac h es a min im u m v alu e of E 1
( A 1) 2
— = α E
= E
0 ( A + 1) 2
for ϑ = π (h er e α = ( A — 1) 2 ).
( A + 1) 2
Notice th at fr om th is ex p r es s ion it is clear th at th e n eu tr on lo os e mor e en er gy in th e imp act with ligh ter n u clei, in
p ar ticu lar all th e en er gy in th e imp act with p r oton :
A 2 + 2 A c os ϑ
• If A ≫ 1, E 1 ≈ E 0 A 2 ≈ E 0 , th at is , almos t n o en er gy is los t.
4
• If A = 1, E 1
= E 0 2+ 2 c o s ϑ = E 0
cos e ϑ ) 2 , an d for ϑ = π all th e en er gy is los t.
2
F or lo w en er gy , th e cr os s s ection is in d ep en d en t of ϑ th u s w e h a v e a fl at d is tr ib u tion of th e ou tgoin g en er gies : th e
2
p r ob ab ilit y to s catter in an y d ir ection is con s tan t, th u s P (cos ϑ ) = 1 . W h at is th e p r ob ab ilit y of a giv en en er gy E 1 ?
P(E 1 ) ( A+1) 2
4AE 0
E 1
α E 0 E 0
F ig . 5 0 : P r o b a b ilit y d is t r ib u tio n o f th e o u t g o in g en er g y in n eu tr o n s ca tter in g fr o m a n u c leu s .
1 d E 2 E 0 A
2 d ϑ ( A + 1) 2
W e h a v e P ( E 1 ) dE 1 = − P (cos ϑ ) d (cos ϑ ) = − s in ϑ dϑ . Th en , s in ce = − s in ϑ , th e p r ob ab ilit y of a giv en
s catter in g en er gy is con s tan t, as ex p ected , an d eq u al to P ( E ) = ( A + 1) 2 . Notice th at th e p r ob ab ilit y is d iff er en t th an
1 4 E 0 A
2
zer o on ly for α E 0 ≤ E 1 ≤ E 0 . Th e a v er age s catter in g en er gy is th en ( E 1 ) = E 0 1+ α an d th e a v er age en er gy los t in a
2
( )
s catter in g ev en t is E l o ss = E 0 1 — α .
It s till r eq u ir es man y collis ion to los e en ou gh en er gy s o th at a fi n al cap tu r e is p r ob ab le. Ho w man y ?
Th e a v er age en er gy after on e collis ion is ( E 1 ) = E 0 1+ α . After t w o collis ion it can b e ap p r o x imated b y ( E 2 ) ≈
1+ α
( E ) = E
1+ α
2
. Th en , after n collis ion , w e h a v e ( E
) ≈ E
1+ α
n
= E
( E 1 ⟩
. Th u s , if w e w an t to k n o w
2
2
2
E 0
2
e ) e ) C r n
1
0
n
0
0
→ n log
= log
h o w man y collis ion s ar e n eed ed to r eac h an a v er age th er mal en er gy E th = ( E n ) w e n eed to calcu late n :
=
≈
→ n = log
/ log
E th ( E n ) ( E 1 ) n
E
0
E
0
E
0
( E 1 )
E
0
E th
E
0
E th
E
0
( E 1 )
E
0
Ho w ev er , th is calcu lation is n ot v er y p r ecis e, s in ce th e ap p r o x imation w e mad e, th at w e can calcu late th e a v er age en er gy after th e n th s catter in g ( E n ) con s id er in g on ly th e a v er age after th e ( n − 1) th s catter in g is n ot a go o d o n C e, s in c r e
th e en er gy d is tr ib u tion is n ot p eak ed ar ou n d its a v er age (b u t is q u ite fl at). Con s id er in s tead th e q u an tit y log
E n − 1
E n
an d tak e th e a v er age o v er th e p os s ib le fi n al en er gy (n ote th at th is is th e s ame as calcu latin g for th e fi r s t collis ion ):
log
= log
αE n − 1
f E n — 1 ) 1 E n − 1
E n
E n — 1
E
n
1 E n − 1
E n — 1 ( A + 1) 2 ( A − 1) 2
E
n
4 A E
n — 1
2 A
A − 1
A + 1
E n
P ( E n ) d E n = log
αE n − 1
dE n = 1 +
log
Th e ex p r es s ion ξ = ( log C E n − 1 r) d o es n ot d ep en d on th e en er gy , b u t on ly on th e mo d er atin g n u cleu s (it d ep en d s
Th en w e h a v e th at ( log C E r) = ( log C E n − 1 r ) or ( log ( E
on A ).
n
0
E n E n
n u m b er of collis ion s n eed ed to ar r iv e at a cer tain en er gy :
n ) ) = log ( E 0 ) − nξ , fr om wh ic h w e can calcu late th e
with ξ th e a v er age logar ith mic en er gy los s :
n ( E 0
→ E th
) = 1 log E th ξ E 0
—
( A 1) 2
ξ = 1 + log
A − 1
2 A A + 1
F or p r oton s ( 1 H), ξ = 1 an d it tak es 18 collis ion to mo d er ate n eu tr on s emitted in fi s s ion ( E = 2M eV) wh ile 2200 collis ion s ar e n eed ed in 238 U.
|
M ater ial |
A |
α |
ξ |
n |
|
H |
1 |
0 |
1 |
18.2 |
|
H 2 0 D He |
1& 16 2 4 |
– 0.111 0.360 |
0.920 0.725 0.425 |
19.8 25.1 42.8 |
|
Be C U |
9 12 238 |
0.640 0.716 0.983 |
0.207 0.158 0.0084 |
88.1 115 2172 |
8.1.3 Cha rged pa rti c l e i nterac ti on
Ch ar ged p ar ticles (s u c h as alp h a p ar ticles an d electr on s /p os itr on s ) goin g th r ou gh matter can in ter act b oth with th e n u clei –v ia th e n u clear in ter action an d th e cou lom b in ter action – an d with th e electr on clou d –v ia th e Cou lom b in ter action . Alth ou gh th e eff ects of a collis ion with th e ligh t electr on is goin g to aff ect th e collid in g p ar ticle m u c h les s th an an imp act with th e h ea v y n u cleu s , th e p r ob ab ilit y o f s u c h a collis ion is m u c h h igh er .
~
Th is can b e in tu itiv ely u n d er s to o d b y an aly zin g th e eff ectiv e s iz e of th e n u cleu s an d th e electr on ic clou d . W h ile th e n u cleu s h a v e a r ad iu s of ab ou t 8fm, th e atomic r ad iu s is on th e or d er of an gs tr oms (or 10 5 fm). Th en th e ar e a off er ed to th e in comin g p ar ticle is on th e or d er of π (8 f m ) 2 200 f m 2 = 2 b ar n s .
~
O n th e oth er s id e, th e electr on ic clou d p r es en t an ar ea of π (10 5 f m ) 2 π 10 8 b ar n s to th e in comin g p ar ticle. Alth ou gh th e cr os s s ection of th e r eaction (or th e p r ob ab ilit y of in ter action b et w een p ar ticles ) is n ot th e s ame as th e ar ea (as it is for clas s ical p ar ticles ) s till th es e r ou gh es timates giv e th e cor r ect or d er of magn itu d e for it.
Th u s th e in ter action s with th e electr on s in th e atom d omin ate th e o v er all c h ar ged p ar ticle/matter in ter action . Ho w ev er th e collis ion with th e n u cleu s giv es r is e to a p ecu liar an gu lar d is tr ib u tion , wh ic h is wh at lead to th e d is co v er y of th e n u cleu s its elf. W e will th u s s tu d y b oth t y p es of s catter in g for ligh t c h ar ged p r o jectiles s u c h as alp h a p ar ticles an d p r oton s .
A . A lp h a p a rticles collision with th e electron ic clou d
Let u s con s id er fi r s t th e s lo win g d o wn of alp h a p ar ticles in matter . W e fi r s t an aly ze th e collis ion of on e alp h a p ar ticle with on e electr on .
A lpha
B ef or e c ollision
v
A f t er c ollision v ’ v
v e v
F ig . 5 1 : Left: C h a r g ed p a r t ic les in ter a c t mo s tly w ith th e elec tr o n ic c lo u d . R ig h t: C o n s er v a tio n o f mo men tu m a n d kin etic en er g y in t h e c o llis io n o f a v er y h ea vy o b jec t w ith a v er y lig h t o n e
α α e
If th e collis ion is elas tic, momen tu m an d k in etic en er gy ar e con s er v ed (h er e w e con s id er a clas s ical, n on -r elativ is tic collis ion )
S olv in g for v a ′
an d v e w e fi n d :
m α v α = m α v α ′
+ m e v e , m α v 2 = m α v ′ 2 + m e v 2
v α ′
= v α
— 2 v
m e
e α
α m + m , v e
= 2 v
m α α m e + m α
≪ ≈
S in ce m e /m α 1, w e can ap p r o x imate th e electr on v elo cit y b y v e 2 v α . Th en th e c h an ge in en er gy for th e alp h a p ar ticle, giv en b y th e en er gy acq u ir ed b y th e electr on , is
ΔE = 1 m v 2 = 1 m (2 v ) 2 = 4 m e E
2 e e
2 e α
m α α
th u s th e alp h a p ar ticle lo os es a tin y fr action of its or igin al en er gy d u e to th e collis ion with a s in gle electr on :
1
ΔE α m e
E α ∼ m α ≪
Th e s mall fr action al en er gy los s y ield s th e c h ar acter is tics of alp h a s lo win g d o wn :
1. Th ou s an d s of ev en ts (collis ion s ) ar e n eed ed to eff ectiv ely s lo w d o wn an d s top th e alp h a p ar ticle
2. As th e alp h a p ar ticle momen tu m is b ar ely p er tu r b ed b y in d iv id u al collis ion s , th e p ar ticle tr a v els in a s tr aigh t lin e in s id e matter .
3. Th e c ol l is ions ar e d u e to Cou lom b in ter action , wh ic h is an in fi n ite-r an ge in ter action . Th en , th e alp h a p ar ticle in ter acts s im u ltan eou s ly with man y electr on s , y ield in g a con tin u ou s s lo win g d o wn u n til th e p ar ticle is s top p ed an d a cer tain s top p in g r an ge.
4. Th e electr on s wh ic h ar e th e collis ion tar gets get ion ized , th u s th ey lead to a v is ib le tr ail in th e alp h a p ar ticle p ath (e.g. in clou d c h am b er s )
D
: St oppi ng p o w er W e calcu lated th e en er gy los t b y th e alp h a p ar ticle in th e collis ion with on e electr on . A mor e imp or tan t q u an tit y is th e a v er age en er gy los s of th e p ar ticle p er u n it p ath len gth , wh ic h is called th e s top p in g p o w er . W e con s id er an alp h a p ar ticle tr a v elin g alon g th e x d ir ection an d in ter actin g with an electr on at th e or igin of th e x -ax is an d at a d is tan ce b fr om it. It is n atu r al to as s u me cy lin d r ical co or d in ates for th is p r ob lem.
elec tr on
v
r
b
x z
alpha
r
F y
b
x
z
db
y
F ig . 5 2 : Geo metr y fo r th e a lp h a / elec tr o n c o llis io n . Left: I mp a c t p a r a met er b a n d c ylin d r ic a l c o o rd in a tes ( x, b ). R ig h t: C o u lo m b fo r c e p a r a llel to th e mo men tu m c h a n g e (in th e y d ir ec tio n ) .
Th e c h an ge in momen tu m of th e electr on is giv en b y th e Cou lom b for ce, in tegr ated o v er th e in ter action time. Th e Cou lom b in ter action is giv en b y F v = e Q r ˆ , wh er e v r = r r ˆ is th e v ector join in g th e alp h a to th e electr on . O n ly th e
4 π ǫ 0 | � r | 2
( x 2 + b 2 ) 1 / 2
comp on en t of th e for ce in th e “r ad ial” ( y ) d ir ection giv es r is e to a c h an ge in momen tu m (th e lon gitu d in al for ce wh en in tegr ated h as a zer o n et con tr ib u tion ), s o w e calcu late F v · y ˆ = | F | r ˆ · y ˆ . F r om th e fi gu r e ab o v e w e h a v e r ˆ · y ˆ = b
4 π ǫ 0 ( x 2 + b 2 ) 3 / 2
an d fi n ally th e for ce F y = e Q b . Th e c h an ge in momen tu m is th en
Δp = F y dt = v
0 — ∞
1 ∞ 1 ∞ dx e 2 Z α b
α
4 π ǫ
0
( x 2 + b 2 ) 3 / 2
d t
wh er e w e u s ed th e r elation d x = v α b et w een th e alp h a p ar ticle v elo cit y (wh ic h is con s tan t with time u n d er ou r
as s u mp tion s ) an d Q = Z α e = 2 e . By con s id er in g th e electr on in itially at r es t w e h a v e
Δp = p e = 4 π ǫ v b (1 + ξ 2 ) 3 / 2 = 2 4 π ǫ v
e 2 Z 1
0
d ξ e 2 Z α
0
α
b
wh er e w e u s ed ξ = x /b . Th en , th e en er gy los t b y th e alp h a p ar ticle d u e to on e electr on is
e
p 2
ΔE =
4 2
e Z
= 2 α
2 m (4 π ǫ 0 ) 2 m e v 2 b 2
db
b
x
alpha
dx
F ig . 5 3 : T o fi n d th e s to p p in g p o w er w e in t eg r a te o v er a ll imp a c t p a r a met er s b , in a s ma ll th ic kn es s d x .
W e n o w s u m o v er all electr on s in th e mater ial. Th e n u m b er of electr on s in an in fi n ites imal cy lin d er is dN e =
A
n e 2 π bdbdx , wh er e n e is th e electr on ’s n u m b er d en s it y (wh ic h can b e e.g. calcu late fr om n e = N A Z ρ , with N A
Av ogad r o’s n u m b er an d ρ th e mas s d en s it y of th e mater ial).
Th en
1 d E 1
4 π e 4 Z 2 n e
1 db
α
− dE = 2 π dx n e ΔE bdb → d x = − 2 π n e ΔE ( b ) bdb = − (4 π ǫ ) 2 m v 2 b
0 e α
∞
α
Th e in tegr al s h ou ld b e ev alu ated b et w een 0 an d . Ho w ev er th is is n ot math ematically p os s ib le (s in ce it d iv er ges ) an d it is als o p h y s ically u n s ou n d . W e ex p ect in fact to h a v e a d is tan ce of clos es t ap p r oac h s u c h th at th e max im u m en er gy ex c h an ge (as in th e h ar d -on collis ion s tu d ied p r ev iou s ly ) is ac h iev ed . W e h ad ob tain ed E e = 2 m e v 2 . Th en w e
s et th is en er gy eq u al to th e electr on ’s Cou lom b p oten tial en er gy : E e ≈
1 e 2
1 e 2
4 π ǫ 0 b m in
fr om wh ic h w e ob tain
b m in ∼ 4 π ǫ
2 m v 2
0 e α
Th e max im u m b is giv en b y ap p r o x imately th e Boh r r ad iu s (or th e atom’s r ad iu s ). Th is can b e calcu lated b y s ettin g
1 e
2
4 π ǫ 0 b m a x ∼ E I wh er e E I is th e mean ex citation en er gy of th e atomic electr on s . Th en wh at w e ar e s tatin g is th at th e
max im u m imp act p ar ameter is th e on e at wh ic h th e min im u m en er gy ex c h an ge h ap p en , an d th is min im u m en er gy
~
—
is th e min im u m en er gy r eq u ir ed to ex cite (k n o c k off ) an electr on ou t of th e atom. Alth ou gh th e mean ex citation en er gy of th e atomic electr on s is a con cep t r elated to th e ion ization en er gy (wh ic h is on th e or d er of 4 15eV) h er e E I is tak en as an emp ir ical p ar ameter , wh ic h h as b een fou n d to b e w ell ap p r o x imated b y E I 10 Z eV (with Z th e atomic n u m b er of th e tar get). F in ally w e h a v e
b m ax
2 m e v 2
α
=
b m in Z α E I
an d th e s top p in g p o w er is
α
α
b m ax
4 π e 4 Z 2 n e
= (4 π ǫ ) 2 m v 2 ln Λ
with Λ called th e Cou lom b logar ith m.
0 e α
m in
0 e α
S in ce th e s top p in g p o w er , or en er gy los t p er u n it len gth , is giv en b y th e en er gy los t in on e collis ion (or ΔE ) times th e n u m b er of collis ion (giv en b y th e n u m b er of electr on p er u n it v olu me times th e p r ob ab ilit y of on e electr on collis ion , giv en b y th e cr os s s ection ) w e h a v e th e r elation :
d E
— d x
= σ c n e ΔE
fr om wh ic h w e can ob tain th e cr os s s ection its elf. S in ce ΔE = 2 m e v 2 , w e h a v e
α
2 π e 4 Z 2
σ c = ln Λ
e α
Th is can als o b e r ewr itten in ter ms of mor e gen er al con s tan ts . W e d efi n e th e clas s ical electr on r ad iu s as
e
1 e 2
0
r e = 4 π ǫ
m c 2 ∼ 2 . 8 f m,
c
wh ic h is th e d is tan ce at wh ic h th e Cou lom b en er gy is eq u al to th e r es t mas s . Alth ou gh th is is n ot clos e to th e r eal s ize of an electr on (as for ex amp le w e w ou ld ex p ect th e electr on r ad iu s –if it cou ld b e w ell d efi n ed – to b e m u c h s maller th an th e n u cleu s r ad iu s ) it giv es th e cor r ect or d er of magn itu d e of th e eff ectiv e ar ea in th e collis ion b y c h ar ged p ar ticles . Als o w e wr ite β = v , s o th at
Z
2
σ c = 2 π r 2 α ln Λ
e β 4
S in ce β is u s u ally q u ite s mall for alp h a p ar ticles , th e cr os s s ection can b e q u ite lar ge. F or ex amp le for a t y p ical alp h a
2
c 2
en er gy of E α = 4M eV, an d its r es t mas s m α c 2 ∼ 4000M eV, w e h a v e v ∼ 2 × 10 — 3 . Th e Cou lom b logar ith m is on
th e or d er of ln Λ ∼ 5 − 15, wh ile 2 π r 2 ∼ 1 b ar n . Th en th e cros s s ection is σ c ∼ 1 4 · 10 6 / 4 · 10 b = 5 × 10 6 b .
e 2
D : St oppi ng Lengt h Th is is d efi n ed b y
2
1 d E
1 /l α = − E d x .
Th en w e can wr ite an ex p on en tial d eca y for th e en er gy as a fu n ction of d is tan ce tr a v eled in s id e a mater ial: E ( x ) =
E 0 ex p ( − x /l α ). Th u s th e s top p in g len gth als o giv es th e d is tan ce at wh ic h th e en er gy h as b een r ed u ced b y 1 /e ( ≈ 63%).
1000
100
10
1
100
10
1
0.1
Stopping power[MeV cm 2 /g]
Stopping power[MeV cm 2 /g]
0.01 0.1 1 10 0.01 0.1 1 10
Ener gy [MeV] Ener gy [MeV]
F ig . 5 4 : S to p p in g p o w er fo r a lp h a p a rt ic les (left ) a n d p r o to n s (r ig h t ) in g r a p h ite. x-a xis : E n er g y in M eV . y-a xis : S to p p in g p o w er (M eV c m 2 / g ). T h e r ed c u r v e is th e to ta l s to p p in g p o w er , g iv en b y th e C o u lo m b s t o p p in g p o w er fr o m c o llis io n w ith th e elec tr o n s (b lu e) a n d th e R u th er fo r d s to p p in g p o w er (b la c k) fr o m c o llis io n w ith th e n u c lei. T h e d a ta is ta k en fr o m N I S T .
In ter ms of th e cr os s s ection th e s top p in g len gth is :
1 /l
= 4 m 2 σ Z n,
α m α c
wh er e n , th e atomic n u m b er d en s it y can b e ex p r es s ed in ter ms of th e mas s d en s it y an d th e Av ogad r o n u m b er ,
A
n = ρ N A .
× ×
Ex amp le S top p in g len gth for lead : 1 / l α = 4 10 4 c m — 1 or l α = 2 . 5 10 — 5 cm. Th e r an ge of th e p ar ticle in th e mater ial is h o w ev er man y s top p in g len gth s (on th e or d er of 10), th u s th e r an ge in lead is ar ou n d 2 . 5 µ m.
D
: R ange. Th e r an ge is mor e p r ecis ely d efi n ed as th e d is tan ce a p ar ticle tr a v els b efor e comin g to r es t. Th en , th e r an ge for a p ar ticle of in itial k in etic en er gy E α is d efi n ed as
R ( E α ) =
dx = −
dE
dx
1 r ( E = 0) 1 E α dE — 1
r ( E α )
0
Notice th at th es e is a s tr on g d ep en d en ce of th e s top p in g p o w er on th e mas s d en s it y of th e mater ial (a lin ear d ep en d en ce) s u c h th at h ea v ier mater ials ar e b etter at s top p in g c h ar ged p ar ticles .
Ho w ev er , for alp h a p ar ticles , it d o es n ’t tak e a lot to b e s top p ed . F or ex amp le, th ey ar e s top p ed in 5 mm of air . Bragg Cu rv e – Th e Br agg cu r v e d es cr ib es th e S top p in g p o w er as a fu n ction of th e d is tan ce tr a v eled in s id e matter . As th e s top p in g p o w er (an d th e cr os s s ection ) in cr eas e at lo w er en er gies , to w ar d th e en d of th e tr a jector y th er e is an in cr eas e in en er gy los t p er u n it len gth . Th is giv es r is e to a c h ar acter is tic B r agg p e ak in th e cu r v e. Th is featu r e is
1.2
1.0
0.8
0.6
0.4
0.2
Stopping power
10 20 30 40
Range (mm)
F ig . 5 5 : B r a g g c u r v e fo r p r o to n s (d is ta n c e in mm)
ex p loited for ex amp le for r ad iation th er ap y , s in ce it allo ws a mor e p r ecis e s p atial d eliv er y of th e d os e at th e d es ir ed lo cation .
B . R u th erfo rd - Cou lomb sca tterin g
Elas tic Cou lom b s catter in g is called Ru th er for d s catter in g b ecau s e of th e ex p er imen ts car r ied ou t in Ru th er for d lab in 1911-1913 th at lead to th e d is co v er y of th e n u cleu s . Th e ex p er imen ts in v olv ed s catter in g alp h a p ar ticles off a th in la y er of gold an d ob s er v in g th e s catter in g an gle (as a fu n ction of th e gold la y er th ic k n es s ).
Th e in ter action is giv en as b efor e b y th e Cou lom b in ter action , b u t th is time b et w een th e alp h a an d th e p r oton s in th e n u cleu s . Th u s w e h a v e s ome d iff er en ce with r es p ect to th e p r ev iou s cas e. F ir s t, th e in ter action is r ep u ls iv e (as b oth p ar ticle h a v e p os itiv e c h ar ges ). Th en mor e imp or tan tly , th e p r o jectile is n o w th e s maller p ar ticle, th u s lo os in g con s id er ab le en er gy an d momen tu m in th e in ter action .
d Ω
W h at w e w an t to calcu late in th is in ter action is th e d iff er en tial cr os s s ection d σ . Th e d iff er en tial (in fi n ites imal) cr os s s ection can b e calcu lated (in a clas s ical p ictu r e) b y con s id er in g th e imp act p ar ameter b an d th e s mall an n u lar r egion b et w een b an d b + db :
dσ = 2 π bdb
→
Th en th e d iff er en tial cr os s -s ection , calcu lated fr om th e s olid an gle dΩ = dϕ s in ϑ dϑ 2 π s in ϑ dϑ (giv en th e s y mmetr y ab ou t ϕ ), is :
d σ 2 π bdb b d b
= =
d Ω 2 π s in ϑ dϑ s in ϑ d ϑ
W h at w e n eed is th en a r elation s h ip b et w een th e imp act p ar ameter an d th e s catter ed an gle ϑ (s ee fi gu r e).
p
alpha
v
b
r min
d
Nucleus
x
p=m v 0
F ig . 5 6 : R u th er fo r d s c a tter in g a n d mo men t u m c h a n g e
In or d er to fi n d b ( ϑ ) w e s tu d y th e v ar iation of en er gy , momen tu m an d an gu lar momen tu m. Con s er v ation of en er gy s tates th at:
1 2 1 2 z Z e 2
0
2 mv 0 = 2 mv + 4 π ǫ r
wh ic h giv es th e min im u m d is tan ce (or d is tan ce of clos es t ap p r oac h ) for zer o imp act p ar ameter b = 0, th at h ap p en s
wh en th e p ar ticle s top s an d gets d efl ected b ac k : 1 mv 2 =
z Z e 2 .
2 0 4 π ǫ 0 d
Th e momen tu m c h an ges d u e to th e Cou lom b for ce, as s een in th e cas e of in ter action with electr on s . Her e h o w ev er
2 2
th e n u cleu s almos t d o es n ot acq u ir e an y momen tu m at all, s o th at on ly th e momen tu m d ir ection is c h an ged , b u t n ot its ab s olu te v alu e: in itially th e momen tu m is p 0 = mv 0 alon g th e in comin g (x ) d ir ection , an d at th e en d of th e in ter action it is s till mv 0 b u t alon g th e ϑ d ir ection . Th en th e c h an ge in momen tu m is Δp = 2 p 0 s in ϑ = 2 mv 0 s in ϑ
—
2
(s ee F ig. ab o v e). Th is momen tu m d iff er en ce is alon g th e d ir ection δ p ˆ , wh ic h is at an an gle π ϑ with x . W e th en
{ } | |
s witc h to a r efer en ce fr ame wh er e v r = r , γ , with r th e d is tan ce v r an d γ th e an gle b et w een th e p ar ticle p os ition an d δ p ˆ .
Th e momen tu m c h an ge is b r ou gh t ab ou t b y th e for ce in th at d ir ection :
·
Δp = F δ p ˆ dt =
0
dt
1 ∞ v
z Z e 2 1 ∞ r ˆ · δ p ˆ z Z e 2 1 ∞ co s γ
4 π ǫ
0
0
4 π ǫ
0
0
r 2
2 2
Notice th at at t = 0, γ = − π — ϑ (as v r is almos t align ed with x ) an d at t = ∞ , γ = π — ϑ (F ig). Ho w d o es γ c h an ges
with time?
Th e an gu lar momen tu m con s er v ation (wh ic h is alw a y s s atis fi ed in cen tr al p oten tial) p r o v id es th e an s w er . A t t = 0, th e an gu lar momen tu m is s imp ly L = mv 0 b . A t an y later time, w e h a v e L = m v r × v v . In th e co or d in ate s y s tem v r = { r , γ } th e v elo cit y h as a r ad ial an d an an gu lar comp on en t:
v v = r ˙ r ˆ + r γ ˙ v γ
an d on ly th is las t on e con tr ib u tes to th e an gu lar momen tu m (th e oth er b ein g p ar allel):
L = mr 2 d γ
1 γ ˙
→ =
d t r 2 v 0 b
p=m v 0
p
p
r
d Nucleus x
F ig . 5 7 : M o men tu m c h a n g e a n d c o o r d in a te s ys tem ( { r , γ } ) fo r R u t h er fo r d s c a tter in g .
In s er tin g in to th e in tegr al w e h a v e :
z Z e 2 1 ∞ cos γ γ ˙ z Z e 2 1 π + ϑ cos γ z Z e 2 ϑ
2
Δp = dt = dγ = 2 cos
4 π ǫ 0
0 v 0 b
4 π ǫ 0 π − ϑ
2
v 0 b
4 π ǫ 0 v 0 b 2
By eq u atin g th e t w o ex p r es s ion s for Δp , w e h a v e th e d es ir ed r elation s h ip b et w een b an d ϑ :
2 mv 0 s in 2 = 4 π ǫ v b 2 cos 2 → b = 4 π ǫ mv 2 cot 2
0 0 0 0
F in ally th e cr os s s ection is :
= (4 T a ) — 2 s in — 4
ϑ
d Ω 4 π ǫ 0 2
2
2
∝ ≈ × ≥
(wh er e T a = 1 mv 0 is th e in cid en t –alp h a– p ar ticle k in etic en er gy ). In p ar ticu lar , th e Z 2 , T — 2 an d s in — 4 d ep en d en ce ar e in ex cellen t agr eemen t with th e ex p er imen ts . Th e las t d ep en d en ce is c h ar acter is tic of s in gle s catter in g ev en ts an d ob s er v in g p ar ticles at lar ge an gles (alth ou gh les s p r ob ab le) con fi r m th e p r es en ce of a mas s iv e n u cleu s . Con s id er gold foil of th ic k n es s ζ = 2 µ m an d an in cid en t b eam of 8M eV alp h a p ar ticles . Th e imp act p ar ameter th at giv es a s catter in g an gle of 90 d egr ees or mor e is b ≤ d = 14fm. Th en th e fr action of p ar ticles with th at imp act p ar ameter is π b 2 , th u s w e h a v e ζ nπ b 2 7 . 5 10 — 5 p ar ticles s catter in g at an an gle 90 ◦ (with n th e tar get d en s it y ). Alth ou gh
th is is a s mall n u m b er , it is q u ite lar ge comp ar ed to th e s catter in g fr om a u n ifor mly d en s e tar get.
C. Electron stop p in g in ma tter
Electr on s in ter act with matter main ly d u e to th e Cou lom b in ter action . Ho w ev er , th er e ar e d iff er en ces in th e in ter action eff ects with r es p ect to h ea v ier p ar ticles . Th e d iff er en ces b et w een th e alp h a p ar ticle an d electr on b eh a v ior in matter is d u e to th eir v er y d iff er en t mas s :
1. Electr on -electr on collis ion s c h an ge th e momen tu m of th e in comin g electr on , th u s d efl ectin g it. Th en th e p ath of th e electr on is n ot s tr aigh t an y mor e.
2. Th e s top p in g p o w er is m u c h les s , s o th at e.g. th e r an ge is 1cm in lead . (r emem b er th at th e r atio of th e en er gy los t to th e in itial en er gy for alp h a p ar ticles w as s mall, s in ce it w as p r op or tion al to th e r atio of mas s es -electr on to alp h a. Her e th e mas s es r atio is 1, an d w e ex p ect a lar ge c h an ge in en er gy ).
3. Electr on s h a v e mor e often a r elativ is tic s p eed . F or ex amp le, electr on s emitted in th e b eta d eca y tr a v el at r elativ is tic s p eed .
4. Th er e is a s econ d mec h an is m for d eceler ation . S in ce th e electr on s can u n d er go r ap id c h an ges of v elo cit y d u e to th e collis ion , it is con s tan tly acceler atin g (or d eceler atin g) an d th u s it r ad iates . Th is r ad iation is called Bremss trah l u n g , or b r ak in g r ad iation (in G er man ).
A
Th e s top p in g p o w er d u e to th e Cou lom b in ter action can b e calcu lated in a v er y s imilar w a y to wh at d on e for th e alp h a p ar ticle. W e ob tain :
d E
− = 4 π
e 2 2 Z ρ N 1
ln Λ ′
�
He r e Λ ′ is n o w a d iff er en t r atio th an th e on e ob tain ed for th e alp h a p ar ticles , b u t with a s imilar mean in g: Λ ′ =
2 m v 2 E
T ( T + m c 2 ) , wh er e again w e can r ecogn ize th e r atio of th e electr on en er gy (d eter min in g th e min im u m d is tan ce)
e I
an d th e mean ex citation en er gy E I (wh ic h s ets th e max im u m d is tan ce) as w ell as a cor r ection d u e to r elativ is tic eff ects .
10
1
0.1
0.01
Stopping power[MeV cm 2 /g]
Stopping power[MeV cm 2 /g]
0.1 1 10 100 1000
0.1 1 10 100 1000
100
10
1
0.1
Ener gy [MeV] Ener gy [MeV]
F ig . 5 8 : S to p p in g p o w er fo r elec tr o n s in g r a p h ite (left ) a n d Lea d (r ig h t). x-a xis : E n er g y in M eV . y-a xis : S to p p in g p o w er (M eV c m 2 / g ). T h e r ed c u r v e is th e t o ta l s t o p p in g p o w er , g iv en b y t h e C o u lo m b s to p p in g p o w er (b lu e) a n d th e r a d ia tiv e s t o p p in g p o w er (b la c k). N o te th e d iffer en t c o n tr ib u tio n s o f t h e t w o t yp es o f p r o c es s es fo r t h e t w o n u c lid es : th e B r ems s t r a h lu n g is m u c h h ig h er fo r h ea vier elemen ts s u c h a s Lea d . T h e d a ta is ta k en fr o m N I S T .
A q u an tu m mec h an ical calcu lation giv es s ome cor r ection s (for th e alp h a as w ell) th at b ecome imp or tan t at r elativ is tic en er gies (s ee K r an e):
d E
− = 4 π
e 2 2 Z ρ N 1
[ln Λ ′ + r elativ is tic cor r ection s ]
A
d x 4 π ǫ 0 A m e c 2 β 2
T o th is s top p in g p o w er , w e m u s t ad d th e eff ects d u e to th e “b r ak in g r ad iation ”. In s tead of calcu latin g th e ex act con tr ib u tion (s ee K r an e), w e ju s t w an t to es timate th e r elativ e con tr ib u tion of Br ems s tr ah lu n g to th e Comp ton s catter in g. Th e r atio b et w een th e r ad iation s top p in g p o w er an d th e cou lom b s top p in g p o w er is giv en b y
— d x
/ − d x
= l c f
d E d E
e 2 Z T + m e c 2
T + m e c 2 Z
r c
c
m e c 2
m e c 2
1600
≈
wh er e f c
2
e
~ 10 − 12 is a factor th at tak es in to accou n t r elativ is tic cor r ection s an d r emem b er k c
≈ 137
. Th en th e
≫
1
r ad iation s top p in g p o w er is imp or tan t on ly if T m e c 2 an d for lar ge Z. Th is ex p r es s ion is v alid on ly for r elativ is tic en er gies ; b elo w 1M eV th e r ad iation los s es ar e n egligib le. Th en th e total s top p in g p o w er is giv en b y th e s u m of th e t w o con tr ib u tion s :
d E d E d E
= +
d x d x c d x r
S in ce th e electr on d o not h a v e a lin ear p ath in th e mater ials (b u t a r an d om p ath with man y collis ion s ) it b ecomes mor e d ifficu lt to calcu late r an ges fr om fi r s t p r in cip les (in p r actice, w e can n ot ju s t tak e dt = dx /v as d on e in th e calcu lation s for alp h as ). Th e r an ges ar e th en calcu lated emp ir ically fr om ex p er imen ts in wh ic h th e en er gy of mon o en er getic electr on b eams is v ar ied to calcu late R ( E ). NIS T p r o v id es d atab as es of s top p in g p o w er an d r an ges for electr on s (as w ell as for alp h a p ar ticles an d p r oton s , s ee th e S T AR d atab as e at h ttp ://www.n is t.go v /p ml/d ata/s tar /in d ex .cfm . Sin ce th e v ar iation with th e mater ial c h ar acter is tics (on ce n or malized b y th e d en s it y ) is n ot lar ge, th e r an ge meas u r ed for on e mater ial can b e u s ed to es timate r an ges for oth er mater ials .
8.1.4 El ec tromagneti c radi ati on
Th e in ter action of th e electr omagn etic r ad iation with matter d ep en d s on th e en er gy (th u s fr eq u en cy ) of th e e.m. r ad iation its elf. W e s tu d ied th e or igin of th e gamma r ad iation , s in ce it d er iv es fr om n u clear r eaction s . Ho w ev er , it is in ter es tin g to als o s tu d y th e b eh a v ior of les s en er getic r ad iation s in matter .
In or d er of in cr eas in g p h oton en er gy , th e in ter action of matter with e.m. r ad iation can b e clas s ifi ed as :
Ra y leigh S catter ing l ω < E I
∼ eV
Ph oto electr ic Ab s or p tion l ω ≥ E I
∼ k eV
Comp ton S catter in g l ω ∼ m e c 2
∼ M eV
P air Pr o d u ction l ω > 2 m e c 2
≥ M eV
Vis ib le X-r a y s γ -r a y s h ar d γ -r a y s
Her e E I i s th e ion ization en er gy for th e giv en tar get atom.
A clas s ical p ictu r e is en ou gh to giv e s ome s calin g for th e s catter in g cr os s s ection . W e con s id er th e eff ects of th e in ter action of th e e.m. w a v e with an os cillatin g d ip ole (as cr eated b y an atomic electr on ).
0
~ −
—
Th e electr on can b e s een as b ein g attac h ed to th e atom b y a ”s p r in g”, an d os cillatin g ar ou n d its r es t p os ition with fr eq u en cy ω 0 . W h en th e e.m. is in cid en t on th e electr on , it ex er ts an ad d ition al for ce. Th e for ce actin g on th e electr on is F = eE ( t ), with E ( t ) = E 0 s in ( ω t ) th e os cillatin g electr ic fi eld . Th is os cillatin g d r iv in g for ce is in ad d ition to th e attr action of th e electr on to th e atom k x e , wh er e k (giv en b y th e Cou lom b in ter action s tr en gth an d r elated to th e b in d in g en er gy E I ) is lin k ed to th e electr on ’s os cillatin g fr eq u en cy b y ω 2 = k /m e . Th e eq u ation of motion for
e
th e electr on is th en
2
e
m e x ¨ e = − k x e − eE ( t ) → x ¨ e + ω 0 x e = − m
E ( t )
W e s eek a s olu tion of th e for m x e ( t ) = A s in ( ω t ), th en w e h a v e th e eq u ation
( − ω 2 + ω 2 ) A = − e E
1 e
0
→ A = E
0
0 m e 0
ω 2 − ω 2 m e
W e h a v e alr ead y s een th at an acceler ated c h ar ge (or an os cillatin g d ip ole) r ad iates , with a p o w er
2 e 2 2
P = 3 c 3 a
wh er e th e accelar ation a is h er e a = − ω 2 A s in ( ω t ), giv in g a mean s q u ar e acceler ation
� �
2
ω 2 e
2 1
0 e
Th e r ad iated p o w er is th en
a = ω 2 − ω 2 m E 0 2
2
P = 1 e
2 ω 4
c E 2
0
0
3 m e c 2 ( ω 2 − ω 2 ) 2
10
0.1
0.001
Range [g/cm 2 ]
0.01 0.1 1 10 100 1000
Ener gy [MeV]
F ig . 5 9 : R a n g e fo r a lp h a p a r tic les (b la c k) a n d elec tr o n s (r ed ) in Lea d ( s o lid c u r v es ) a n d a ir (d a s h ed ). x-a xis : E n er g y in M eV . y-a xis : R a n g e (g / c m 2 ). N o t e th e m u c h lo n g er r a n g e fo r elec t r o n s th a n fo r a lp h a p a rtic les . On ly a t v ery h ig h en er g y , fo r lea d , th e r a n g e is s h o r ter fo r elec tr o n s , th a n ks to th e c o n tr ib u tio n fr o m B r ems s tr a h lu n g . T h e d a ta is ta k en fr o m N I S T .
Th e r ad iation in ten s it y is giv en b y I
= c E 2
(r ecall th at th e e.m. en er gy d en s it y is giv en b y u =
1 E 2 an d th e
0
~ ×
0 8 π 2
in ten s it y , or p o w er p er u n it ar ea, is th en I c u ). Th en w e can ex p r es s th e r ad iated p o w er as cr os s -s ection r ad iation
in ten s it y :
P = σ I 0
σ =
8 π
3 m e c 2 ω − ω
2
e 2
ω 2
0
2
2
2
Th is y ield s th e cr os s s ection for th e in ter action of e.m. r ad iation with atoms :
or in S I u n its :
σ = = 4 π r 2
8 π e 2
2 ω 2 2
2 ω 2 2
e
3 4 π ǫ 0 m e c 2 ω 2 − ω 2 3 ω 2 − ω 2
0 0
wh er e w e u s ed th e clas s ical electr on r ad iu s r e .
A . R a yleigh S ca tterin g
≪
W e fi r s t con s id er th e limit in wh ic h th e e.m. r ad iation h as v er y lo w en er gy : ω ω 0 . In th is limit th e electr on is in itially b ou n d to th e atom an d th e e.m. is n ot goin g to c h an ge th at (an d b r eak th e b ou n d ). W e can s imp lify th e
fr eq u en cy factor in th e s catter in g cr os s -s ection b y
ω 2 ω 2
2 2 2
ω — ω ≈ ω , th en w e ha v e:
0 0
σ R = 3
8 π
e 2
2
ω
4
4 π ǫ m c 2
0 e
ω 4
0
Th e Ra y leigh s catter in g h as a v er y s tr on g d ep en d en ce on th e w a v elen gth of th e e.m. w a v e. Th is is wh at giv es th e b lu e color to th e s k y (an d th e r ed color to th e s u n s ets ).
B . Th omson S ca tterin g
Th oms on s catter in g is s catter in g of e.m. r ad iation th at is en er getic en ou gh th at th e electr on ap p ear s to b e in itially u n b ou n d fr om th e atom (or a fr ee electr on ) b u t n ot en er getic en ou gh to imp ar t a r elativ is tic s p eed to th e electr on . (If th e electr on is a fr ee electr on , th e fi n al fr eq u en cy of th e electr on will b e th e e.m. fr eq u en cy ).
W e ar e th en con s id er in g th e limit:
l ω 0 ≪ l ω ≪ m e c 2
ω 2 — ω 2 ( ω 0 / ω ) 2 — 1
2
wh er e th e fi r s t in eq u alities tells u s th at th e b in d in g en er gy is m u c h s maller th an th e e.m. en er gy (h en ce fr ee electr on ) wh ile th e s econ d tells u s th at th e electr on will n ot gain en ou gh en er gy to b ecome r elativ is tic.
0
Th en w e can s imp lify th e factor
ω = 1 ≈ − 1 an d th e cr os s s ection is s imp ly
σ T = 3
8 π
e 2
4 π ǫ m c 2
0 e
2
2
3
with σ T ∼ b ar n . Notice th at con tr as tin g with th e Ra y leigh s catter in g, Th oms on s catter in g cr os s -s ection is com
p letely in d ep en d en t of th e fr eq u en cy of th e in cid en t e.m. r ad iation (as lon g as th is is in th e giv en r an ge). Both th es e
≪
t w o t y p es of s catter in g ar e elas tic s catter in g, mean in g th at th e atom is left in th e s ame s tate as it w as in itially (s o con s er v ation of en er gy is s atis fi ed with ou t an y ad d ition al en er gy comin g fr om th e in ter n al atomic en er gy ). Ev en in Th oms on s catter in g w e n eglect th e r ecoil of th e electr on (as s tated b y th e in eq u alit y l ω m e c 2 ). Th is mean s th at th e electr on is n ot c h an ged b y th is s catter in g ev en t (th e atom is n ot ion ized ) ev en if in its in ter action with th e e.m. fi eld it b eh a v es as a fr ee electr on .
3 e
Notice th at th e cr os s s ection is p r op or tion al to th e clas s ical electr on r ad iu s s q u ar e: σ T = 8 π r 2 .
C. P h oto electric Eff ect
≈
A t r es on an ce ω ω 0 th e cr os s -s ection b ecomes (math ematically ) in fi n ite. Th e r es on an ce con d ition mean s th at th e
e.m. en er gy is eq u al to th e ion ization en er gy E I of th e electr on . Th u s , wh at it r eally h ap p en s is th at th e electr on gets ejected fr om th e atom. Th en ou r s imp le mo d el, fr om wh ic h w e calcu lated th e cr os s s ection , is n o lon ger v alid (h en ce th e in fi n ite cr os s s ection ) an d w e n eed Q M to fu lly calcu late th e cr os s -s ection . Th is is th e p h oto electr ic eff ect. Its cr os s s ection is s tr on gly d ep en d en t on th e atomic n u m b er (as σ pe ∝ Z 5 )
D. Comp ton S ca tterin g
Comp ton s catter in g is th e s catter in g of h igh ly en er getic p h oton s fr om electr on s in atoms . In th e p r o ces s th e electr on acq u ir e an en er gy h igh en ou gh to b ecome r elativ is tic an d es cap e th e atom (th at gets ion ized ). Th u s th e s catter in g is n o w in elas tic (comp ar ed to th e p r ev iou s t w o s catter in g) an d th e s catter in g is an eff ectiv e w a y for e.m. r ad iation to los e en er gy in matter . A t lo w er en er gies , w e w ou ld h a v e th e p h oto electr ic eff ect, in wh ic h th e p h oton is ab s or b ed b y th e atom. Th e eff ect is imp or tan t b ecau s e it d emon s tr ates th at ligh t can n ot b e ex p lain ed p u r ely as a w a v e p h en omen on .
'
F ig . 6 0 : P h o to n / E lec tr o n c o llis io n in C o mp to n s c a tter in g .
F r om con s er v ation of en er gy an d momen tu m, w e can calcu late th e en er gy of th e s catter ed p h oton .
{
E γ + E e = E γ ′ + E e ′ → l ω + m e c 2 = l ω ′ + � | p | 2 c 2 + m 2 c 4
l v k = l v k ′ + p v →
l k = l k ′ cos ϑ + p cos ϕ
l k ′ s in ϑ = p s in ϕ
′ [ 2 ] �
c 2
F r om th es e eq u ation s w e fi n d p 2 = ( ω — ω ) l ( ω ′ − ω ) − 2 mc an d cos ϕ = 1 − l 2 k ′ 2 s in 2 ϑ /p 2 . S olv in g for th e
k
c h an ge in th e w a v elen gth λ = 2 π w e fi n d (with ω = k c ):
Δλ = (1 − cos ϑ )
2 π l
m e c
or for th e fr eq u en cy :
[ l ω
] — 1
e
l ω ′ = l ω 1 + m c 2 (1 − cos ϑ )
Th e cr os s s ection n eed s to b e calcu lated fr om a fu ll Q M th eor y . Th e r es u lt is th at
m e c 2 σ C ≈ σ T l ω
th u s Comp ton s catter in g d ecr eas es at lar ger en er gies .
E. P a ir P ro d u ction
P air p r o d u ction is th e cr eation of an electr on an d a p os itr on p air wh en a h igh -en er gy p h oton in ter acts in th e v icin it y of a n u cleu s . In or d er n ot to v iolate th e con s er v ation of momen tu m, th e momen tu m of th e in itial p h oton m u s t b e ab s or b ed b y s ometh in g. Th u s , p air p r o d u ction can n ot o ccu r in emp t y s p ace ou t of a s in gle p h oton ; th e n u cleu s (or an oth er p h oton ) is n eed ed to con s er v e b oth momen tu m an d en er gy .
Ph oton -n u cleu s p air p r o d u ction can on ly o ccu r if th e p h oton s h a v e an en er gy ex ceed in g t wice th e r es t mas s ( m e c 2 ) of an electr on (1.022 M eV):
l ω = T e − + m e − c 2 + T e + + m e + c 2 ≥ 2 m e c 2 = 1 . 022 M eV
P air p r o d u ction b ecomes imp or tan t after th e Comp ton s catter in g falls off (s in ce its cr os s -s ection is ∝ 1 /ω ).
MIT OpenCourseWare http://ocw.mit.edu
22.02 Introduction to Applied Nuclear Physics
Spring 2012
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