Transcript LiCAS - University of Oxford

Lecture 3
Using the SEMF and realising its
limitations
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
1
3.1 Overview

3.2 Stability of Nuclides




3.3 Decays






11 Nov 2004, Lecture 3
interpreting the table of nuclides
SEMF and the “valley of stability ”
SEMF and the “iron mountain ”
classification of decays
a-decay
b-decay
g-decay (off syllabus)
fission and the rest
end of lecture 3
3.4 Natural Radioactivity
Nuclear Physics Lectures, Dr. Armin Reichold
2
Z
Z
N
b stable
Neutron
longlived
9 yrs)
Magic
(>10
Numbers
155
Even
Odd
53
3
Even
N=Z
Odd
Odd
50
3
4
5
Odd
11
Z=92 (U)
N=160
Even A=const.
Even
Z=110
Proton
Magic
Numbers
• Even A stable nuclides
A=58
(Fe58, Ni58)
• Nuclides • SEMF total Ebind • SEMF Ebind/A
• N=Z • Ebind-contours
• Ebind/A-contours
-MeV
• A=const @ 58
• Z=92 (Uranium)
N
• Odd A stable nuclides
• odd-even summary
• Magic Proton Numbers
• Magic Neutron Numbers
3.2 The Valley of Stability
+1000 MeV
+500 MeV
0 Mev
-500 Mev
-1000 Mev
-1500 Mev
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
4
3.2 The Valley of Stability

Observation: stable nuclei not on a straight line in NZ plane. The SEMF predicts this:



Rich structure in location of stable elements




Coulomb term pulls them down (prefers Z<N) and …
… wins over Asymmetry term (prefers Z=N)
more stable isotopes of e-e then o-o nuclei (see b-decay)
No “life” beyond Z=92 (U) and a big gap from Z=82 to 92
(the region of natural radio activity)
Funny magic numbers for Z and N (see SEMF limitations)
But what about simple Ebind per nucleon
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
5
3.2 The Iron Mountain
-MeV
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
6
3.2 The Iron Mountain
Binding Energy vs. A for odd-A nuclei
Iron
Not smooth because Z
not smooth function of A
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
7
3.3 Classification of Decays
Protons
a-decay:
b-
b+
EC
•
•
•
•
emission of Helium nucleus
ZZ-2
NN-2
AA-4
b--decay
•
•
•
•
emission of e- and n
ZZ+1
NN-1
A=const
b+-decay
a
Neutrons
g-decay
• emission of g
• Z,N,A all const
• emission of e+ and n
• ZZ-1
• NN+1
• A=const
Electron Capture (EC)
• absorbtion of e- and emiss n
• ZZ-1
• NN+1
• A=const
8
3.3 b-decay
or
Into the valley of stability along the const. A direction


Q: How does nucleus “move” along constant A?
A: Via b-decay: nucleus emits e-,ne=(b-) or e+,ne=(b+)




DMnucl > me for bDMatom> 0 for b-
& DMnucl > me for b+
& DMatom>2me for b+
or via EC: like (b+) but swallow atomic e- instead instead of
Z
emitting e+


valley
DMnucl>-me
or
DMatom>0
unstable to
β+ decay
(or K capture)
Note: DMx = Mx(mother) – Mx(daughter)
Observe: e+- has continuous energy spectrum



unstable to
maximum of Ekin(e+-) = Qb-Erecoil(daughter) ≈ Qb
β- decay
1<Qb/MeV<15
ne carries the rest of Qb solving long standing puzzle of energy
conservation in b-decays
N
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
9
3.3 b-decay




Q: Where do e+- and ne (ne) come from?
A: Can’t be “in” the nucleus because nucleus is to
small a box for electrons of this energy
2 2
2
 Ebox=n h /8mea = 0.37 TeV @ n=1, a=1fm (i.e. n decay)
e and n produced during decay (particle physics)
Think of b-decay as n-decay inside the nucleus


n  p + e- + ne
Think of n-decay as quark decay inside the neutron


d-1/3  u+2/3 + Wfollowed by W-e- + ne
d
u
u
n
d
u
d
p
eW-
11 Nov 2004, Lecture 3
n( e)
10
3.3 b-decay and SEMF
av=15.56 MeV
ac=0.697 MeV
2
EBind = av A - as A - ac
3
Z
A
as=17.23 MeV
2
1
3
e=even
o=odd
+ 12 MeV (e-e)
ap= 0 MeV (o-e or e-o)
- 12 MeV (o-o)
(N - Z )
1
- aa
+ ap
A
A
2
aa=23.285 MeV
• Q: How do we find SEMF predictions for b-decay
• A: We need the optimum Z (max binding energy) at fixed A.
To make this easier lets consider A=odd i.e. ap=0 (even-odd or odd-even)
 dE bind 
= 4a A - Z
 dZ 

 A =odd
11 Nov 2004, Lecture 3
 2ac 8a A 
 1 +
 = 0
A 
A 3
Nuclear Physics Lectures, Dr. Armin Reichold
11
3.3 b-decay and SEMF


Aodd  ac 2 3
Aodd  Aodd3
yielding: Z =
Aodd + 1  
+ 1


2  4aa
2  133.63

2
-1


evaluate:



-1

for small Aodd: A2/3<< 133  Z≈A/2≈N
for large Aodd: A=105  Z=3/4 N (Z=45; N=60):
Quite close to reality. The nearest nuclei are:




A=103; Z=45; N=58: 10345Rh ,even-odd, stable
A=106; Z=46; N=60: 10646Pd ,even-even, stable
A=105; Z=46; N=59: 10546Pd ,odd-even, stable
A=105; Z=45; N=60: 10545Rh ,odd-even, meta-stable,
decays via b- to 10546Pd in 38h
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
12
3.3 b-decay and SEMF

Odd A. A=135
Odd A b-decays:




Single parabola
even-odd and odd-even
single parabolic minimum
only one b-stable nucleus for
each odd A
nearly exclusively single bdecays occur in nature
double b-decay is 2nd order
weak process and very rare
β-
ββ+
ββ54
52
Te
11 Nov 2004, Lecture 3
I
Nuclear Physics Lectures, Dr. Armin Reichold
Xe
EC
56
Cs
Ba
58
La
Ce
Pr
13
3.3 b-decay and SEMF






Even A:
two parabolae
one for o-o & one for e-e
lowest o-o nucleus often has
two decay modes
since double b-decay is
extremely weak most e-e nuclei
have two stable isotopes
there are nearly no stable o-o
nuclei in nature because these
can nearly all b-decay to an e-e
nucleus
11 Nov 2004, Lecture 3
Even A. A=102
Two parabolae separated by 2δ,
odd-odd and even-even
odd-odd
β+
ββ+
β-
ββ+
44
42
Mo
Tc
Nuclear Physics Lectures, Dr. Armin Reichold
Ru
Rh
even-even
46
Pd
Ag
48
Cd
14
3.3 b-decay and SEMF

Consequence: 2 or more even A, 1 or no odd A
11 Nov 2004, Lecture 3
15
3.3 a-decay
Observation:




emits a with Ekin≈4 MeV
RTh≈1.2*2321/3 fm = 7.36 fm
a has Epot(RTh)=24 MeV
a has negative kinetic energy
up to R=8*RTh
232
90Th
Geiger-Nuttal Plot
Conclusion:


a must tunnel out of the
nucleus
half lifes should have exp(Ekin)
dependence (true over 24
orders, see Geiger-Nuttal plot)
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
16
3.3 a-decay
DEsep≈6MeV per nucleon for heavy nuclei
DEbind(42a)=28.3 MeV > 4*6MeV
Neutrons
11 Nov 2004, Lecture 3
Protons
Alphas
Nuclear Physics Lectures, Dr. Armin Reichold
17
3.3 a-decay
(energetics)

What can SEMF say about a-decay?
Decay is possible if Mnucl(N,Z)-Mnucl(N-2,Z-2)>M(a)
 SEMF as function of A only (dA=dN+dZ & dN=dZ) and
Ebindterm
/A
ignoring pairing
(odd A only)
[MeV]
a
E bind = E bind  Z=2,A=4  E bind (Z , A) - E bind (Z - 2, A - 4)

dE bind
dE bind
 d E bind A  E bind
a
E bind = 28.3 MeV  4
4
= 4 A
d (E bind+ / A )
dA dZ =dN
dA
dA=
A
slope


dA
-3
7.7x10-3 MeV
 Slope in Ebind/A (A≥120) is 7.7*10
MeV
E 

28.3 MeV  4  A  7.7  10 -3 MeV + bind  
A 

E bind
 7.075 MeV + A  7.7  10 -3 MeV  Aacritical  151
A

11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold





18
3.3 a-decay
(energetics-but)


but the world is full of isotopes with A>151
and only 7 natural a-emitters observed with
A<206 because …



barrier penetration has t~exp(-Ea)
energies are too low to get t << age of earth
(4*109 years)
Note: Shell effects O(1 MeV) make the life
times of a-emitters deviate by several orders
of magnitude from SEMF predictions
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
19
3.3 a-decay


SEMF says they should not exist
It is a shell effect, off syllabus
(the 3-odd ones out)
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
20
3.3 a-decay
(the fine print)

To compute decay rates one needs

a lecture from Dr. Weidberg …
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
21
3.3 g-decay






Very similar to atomic physics transitions
Q: When do nuclear g-decays happen?
A: When there is not enough E to emit a
strongly interacting particle (Nucleon), often
after other nuclear decays
Egatomic<100 keV ; Egnuclear<O(1 MeV)
But: heavy nuclear rotational states can have
Egnuclear, rot<O(10 keV)
Note: Not on syllabus
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
22
3.3 Fission and the Rest







Fission in the liquid drop model:
Yet another tunneling
process
Complicated dynamics
Coulomb repulsion
fights surface term
Call it surface barrier
Theoretical limit:
Z2/A>18 (9842Mo) could
decay …
… but does not
because …
11 Nov 2004, Lecture 3
23
3.3 Fission and the Rest

It would take forever
15
Fission is mainly asymmetric
10
log10(t/1 year)

5
0
-5
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
Z2/A
24
3.3 Fission and the Rest



Fission barrier changes
with Z2/A (and via
SEMF this is a change
with A)
Thus the huge lifetime
variation observed
Beyond Z2/A=43
(which does not exist)
there would be no
fission barrier
11 Nov 2004, Lecture 3
Epot [MeV]
Nuclear Physics Lectures, Dr. Armin Reichold
25
3.3 Fission and the Rest





t=0
Fission products:
too rich in neutrons (valley is
curved ) emit neutrons
(needed for reactors)
highly excited  g-decay
still away from valley of
stability  b-decay
tunneling: tfis~exp(-Efis)
excited nuclei (n-capture)
decay much faster via
fission (reactors)
11 Nov 2004, Lecture 3
t≈10-14 s
t>10-10 s
Nuclear Physics Lectures, Dr. Armin Reichold
26
3.3 Others



Best to emit something with very large
binding energy  12C has been observed
Anything else is just asymmetric fission
And then there is fusion (separate chapter)
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
27
3.4 Natural Radioactivity
• Three “chains” of natural radioactivity parents:
232Th, 235U, 238U (made by last super nova, t>age
of earth)
• 40K (odd-odd, Z=19, N=21, t=1.3*1019 years, b- or
EC)
• short-lived but naturally regenerated radioactive
nuclei, eg 14C (radio-carbon)
• natural life times O(1s)<t<age-of-universe
• all types of decays present
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
28
Protons
93
α
89
α
87
α
85
α
206Pb
83
β-
α
91
210Po
238U
234U
ββ-
α
β-
β-
214Po
α
β-
α
β-
230Th
α
234Th
226Ra
222Rn
218Po
214Pb
81
210Tl
79
126
128
130
132
134
238U
11 Nov 2004, Lecture 3
136
138
series
Nuclear Physics Lectures, Dr. Armin Reichold
140
142
144
146
Neutrons
29
End of Lecture 3
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
30
Notes:


In the following I reproduce some slides that
have animated overlays and can not be read
completely with the overlays turned on. The
number of the slide they refer to is indicated
in the top right corner.
There is one additional slide on g-decays (off
syllabus)
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
31
Z slide 3a
Neutron
Magic
Numbers
Z=92 (U)
N=160
A=const.
Z=110
N=Z
Proton
Magic
Numbers
A=58
(Fe58, Ni58)
• Nuclides • SEMF total Ebind
• N=Z
• Ebind-contours
• A=const @ 58
• Z=92 (Uranium)
N
• Magic Proton Numbers
• Magic Neutron Numbers
Z
longlived
Z
N
b stable
Even
Even
155
11
Even
Odd
53
3
Odd
Even
50
3
Odd
Odd
4
(>109 yrs)
slide 3b
5
-MeV
• Even A stable nuclides
• Nuclides
• N=Z
• SEMF Ebind/A
• Ebind/A-contours
N
• Odd A stable nuclides
• odd-even summary
3.3 b-decay
slide 9a
or
Into the valley of stability along the const. A direction


Q: How does nucleus “move” along constant A?
A: Via b-decay: nucleus emits e-,ne=(b-) or e+,ne=(b+)




& DMnucl > me for b+
& DMatom>2me for b+
or via EC: like (b+) but swallow atomic e- instead instead of
emitting e+


DMnucl > me for bDMatom> 0 for b-
DMnucl>-me
or
DMatom>0
Note: DMx = Mx(mother) – Mx(daughter)
Observe: e+- has continuous energy spectrum



maximum of Ekin(e+-) = Qb-Erecoil(daughter) ≈ Qb
1<Qb/MeV<15
ne carries the rest of Qb solving long standing puzzle of energy
conservation in b-decays
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
34
3.3 b-decay
slide 9b
or
Into the valley of stability along the const. A direction


Q: How does nucleus “move” along constant A?
A: Via b-decay: nucleus emits e-,ne=(b-) or e+,ne=(b+)




& DMnucl > me for b+
& DMatom>2me for b+
or via EC: like (b+) but swallow atomic e- instead instead of
Z
emitting e+


DMnucl > me for bDMatom> 0 for b-
DMnucl>-me
or
DMatom>0
unstable to
β+ decay
(or K capture)
Note: DMx = Mx(mother) – Mx(daughter)
Observe: e+- has continuous energy spectrum



unstable to
maximum of Ekin(e+-) = Qb-Erecoil(daughter) ≈ Qb
β- decay
1<Qb/MeV<15
ne carries the rest of Qb solving long standing puzzle of energy
conservation in b-decays
N
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
35
slide 18
3.3 a-decay
(energetics)

What can SEMF say about a-decay?
Decay is possible if Mnucl(N,Z)-Mnucl(N-2,Z-2)>M(a)
 SEMF as function of A only (dA=dN+dZ & dN=dZ) and
ignoring pairing term (odd A only)
a
E bind
= E bind  Z=2,A=4  E bind (Z , A) - E bind (Z - 2, A - 4)

dE bind
dE bind
 d E bind A  E bind
a
E bind = 28.3 MeV  4
4
= 4 A
+
dA dZ =dN
dA
dA
A


 Slope in Ebind/A (A≥120) is 7.7*10-3 MeV
E 

28.3 MeV  4  A  7.7  10 -3 MeV + bind  
A 

E bind
 7.075 MeV + A  7.7  10 -3 MeV  Aacritical  151
A

11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold





36
3.3 a-decay
slide 18
(energetics)

What can SEMF say about a-decay?


Decay is possible if Mnucl(N,Z)-Mnucl(N-2,Z-2)>M(a)
SEMF as function of A only (dA=dN+dZ & dN=dZ) and ignoring pairing
term (odd A only)
Ebind/A
[MeV]
d (E bind / A )
dA
7.7x10-3 MeV
slope =
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
37
slide 20a
3.3 a-decay


SEMF says they should not exist
It is a shell effect, off syllabus
(the 3-odd ones out)
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
38
slide 20b
3.3 a-decay


SEMF says they should not exist
It is a shell effect, off syllabus
(the 3-odd ones out)
11 Nov 2004, Lecture 3
Nuclear Physics Lectures, Dr. Armin Reichold
39
additional information
off syllabus


3.3 g-decay
Q: What if J=0 nucleus needs to loose Energy
A: It can’t loose it via g

it could loose it via pair-creation if E>2me (virtual g
does not have to have S=1 and converts to pair in
e
1
emitted electron
g*
J=0 S0 state)
nucl.

e+ emitted positron
if E<2me could do internal conversion (a’la Auger
- emitted electron
e
in atomic)
g*
nucl.
11 Nov 2004, Lecture 3
e- absorbed atomic electron
Nuclear Physics Lectures, Dr. Armin Reichold
40