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Nucleosynthesis
8/26/10
How did the various nuclides originate?
What determines their abundance?
When were the elements created?
Lecture outline:
1) The age of the universe
2)
The Big Bang
3)
Nucleosynthesis – initial + stellar
4)
Abundance of elements
900s exposure from Palomar
The Age of the Universe
Four methods of determining age of universe:
1)
Cosmological models – Ho (the Hubble constant – ratio of velocity to distance
in expansion of universe)
To=13.7 billion years
2)
Isotope geochemistry – 187Re  187Os, t1/2=40 billion years To=12-17 billion years
238U decay, t =4.5 billion years T =12.5-16 billion years
1/2
o
3)
Age of oldest star clusters -- measure luminosity
of brightest star, relies on
stellar evolutionary model,
To=11-13 billion years
4)
Oldest white dwarfs -- measure luminosity
of faint white dwarfs to determine
how long they have been cooling,
To=12-13 billion years
The Big Bang
- 1920’s: LeMaitre proposes on theoretical grounds that the
universe is expanding
- 1929: Hubble observed galaxies moving away from us with speeds
proportional to distance
- 1964: Penzias and Wilson detect ‘primordial static’ left over from
Big Bang
Time After Big Bang
Temperature (K)
Event
5.39 x 10-44 s
--
10-43 s
10-35 s
10-33 to 10-32 s
1 x 10-10 s
3 x 10-10 to 5 x 10-6 s
6 x 10-6
10s
3.8 m
700,000 y
1031
1028
1027
1015
~1013
1.4 x 1012
3.9 x 109
9 x 108
3000
appearance of space, time, energy,
and superforce
gravity separates
strong force and electro-weak force
inflation
electromagnetic and weak force
stabilization of quarks, antiquarks
formation of protons and neutrons
stabilization of electrons and positrons
formation of 2H, 3He, and 4He nuclei
electrons captured by nuclei
WMAP:
Wilkinson
Microwave
Anisotropy
Probe
1992
age of universe =
13.73 +/- 1%
image microwave
radiation from 379,000
years after Big Bang
2005
small temperature
differences (10-6 K)
signify heterogeneous
distribution of matter
http://map.gsfc.nasa.gov/
Nucleosynthesis Schematic
Nucleosynthetic process
Elements created
Big bang
1H, 4He, 2H, 3H
(Li, B?)
Main sequence stars:
Hydrogen burning
4He
Helium burning
12C, 4He, 24Mg, 16O, 20Ne
Carbon burning
24Mg, 23Na, 20Ne
CNO cycle
4He
x-process (spallation)
& supernova (?)
Li, Be, B
a-process
24Mg, 28Si, 32S, 36Ar, 40Ca
e-process
56Fe
s-process
up to mass 209
r-process
up to mass 254
& other transition
Nucleosynthesis during the Big Bang
- initially, protons (1H) and neutrons
combine to form 4He, 2H (D), and 3He
via exothermic fusion reactions.
- some uncertainty about whether
some B, Be, and Li were created at
this stage
- H & He comprise 99% of
mass of universe
Nucleosynthesis during small star evolution
For ‘small’ star, such as our Sun
- star must form from gravitational
accretion of ‘primordial’ H and He
- temperature ~ 107 after formation
- H-burning creates 4He from 1H,
longest stage of star (107 - 1010y)
- He-burning begins with formation
of Red Giant (T=108)
4He + 4He --> 8Be
8Be + 4He --> 12C
12C + 4He --> 16O and so on to 24Mg
- core contracts as He consumed,
 -process begins (T=109)
20Ne --> 16O + 4He
20Ne + 4He --> 24Mg and so on to 40Ca
Nucleosynthesis during small star evolution (cont)
For ‘small’ star, such as our Sun
- odd # masses created by proton bombardment
- slow neutron addition (s-process) during
late Red Dwarf:
13C + 4He --> 16O + n
21Ne + 4He --> 24Mg + n
follows Z/N stability up to mass 209
Nucleosynthesis during supernovae evolution
For massive stars
- same evolution as for small star,
up to Red Giant stage
- core contracts and heats at accelerating pace
- when T~3x109, several important elementbuilding processes occur:
- energetic equilibrium reactions between
n, p, and nuclei (e-process), builds up to 56Fe
- rapid addition of neutrons (r-process) builds
up to mass 254
Heavy element formation - the ‘s’ and ‘r’ processes
Neutron # (N)
The abundance of the elements - cosmic
- astronomers can detect different
elements with spectroscopy (large
telescopes equipped with high-resolution
spectrometers)
The abundance of the elements - cosmic
- the models of nucleosynthesis are driven
by the observed relative abundances of
the elements in this and other galaxies
Magic numbers: 2, 8, 20, 28, 50, 82,126
& even is always better than odd
The abundance of the elements - our solar system
Relative composition of heavy elements in sun very similar to “primordial”
crust (the carbonaceous chondrite), so we assume that solar system
was well-mixed prior to differentiation.
Nuclear Physics & Radioactivity
What holds a nucleus together?
What drives radioactive decay?
What sets the timescale for radioactive
decay?
What happens during radioactive decay?
Lecture outline:
1) nuclear physics
2)
radioactive decay
3)
secular equilibrium
4)
counting statistics
a particles in a cloud chamber
8/26/10
The Four Forces of Nature
Force
Strength
Range
Occurrence
Strong nuclear
1
<<1/r2 (finite, v. short)
inter-nucleon
Electromagnetic
10-2
1/r2 (infinite, but shielded
nucleus, atom
Weak nuclear
10-13
<<1/r2 (finite, v. short)
B-decay,
neutrinos
Gravity
10-39
1/r2 (infinite)
everywhere
Four Tenets of Nuclear Physics
1) mass-energy equivalence (E=mc2)
2) wave-particle duality (particles are waves, and waves are particles)
3) conservation of energy, mass, momentum
4) symmetry
Binding energy
Let’s revisit the fusion of four protons to form a 4He nucleus:
4( 11H )  1( 24 He)  2e  2 e  E
4(1.007277)  1(4.00150)  m
m  0.02761amu
*these masses come
from the table of nuclides
We have calculated the mass deficit --> i.e. the whole is less than sum of the parts
56Fe
The mass deficit is represented by a HUGE energy release, which can be calculated
using Einstein’s famous equation, E=mc2, and is usually expressed in Mev
Contributions to Binding Energy
EB = strong nuclear force binding -surface tension binding + spin pairing
+shell binding-Coulomb repulsion
1) strong nuclear force -- the more nucleons the better
2) surface tension -- the less surface/volume the better (U better than He)
3) spin pairing -- neutrons and protons have + and - spins, paired spins better
4) shell binding -- nucleus has quantized shells which prefer to be filled (magic numbers)
5) Coulomb repulsion -- packing more protons into nucleus comes at a cost (although
neutron addition will stabilize high Z nuclei)
Radioactive Decay
- a radioactive parent nuclide decays to a daughter nuclide
- the probability that a decay will occur in a unit time is defined as  (units of y-1)
- the decay constant  is time independent; the mean life is defined as =1/λ
dN
  N
dt
N0
N  N0e
ln(2)
t1/ 2

 t
14
Number of C atoms
1000000
900000
800000
t1/2 = 5730y
700000
1/2
1/2
1/2
N0 
 N0 / 2 
 N0 / 4 
 N0 / 8
t
600000
500000
400000
t
t
300000
200000
100000
0
0
5730
10000
20000
30000
Years
40000
50000
Activity calculations
Activity   N
A  A0e  t
- usually reported in dpm (disintegrations per minute),
example: 14C activity = 13.56 dpm / gram C
- because activity is linerarly proportional to number N,
 t
then A can be substituted for N in the equation N  N 0 e
Example calculation:
How many 14C disintegrations have occurred in a 1g wood sample formed in 1804AD?
T=206y
t1/2 = 5730y so  = 0.693/5730y = 1.209e-4 y-1
N0=A0/λ
so N0=(13.56dpm*60m/hr*24hr/day*365days/y) /1.209e-4= 5.90e10 atoms
N(14C)=N(14C)0*e-(1.209e-4/y)*206y = 5.75e10 atoms
# decays = N0-N = 1.5e9 decays
Four types of radioactive decay
1) alpha ( ) decay - 4He nucleus (2p + 2n) ejected
2) beta () decay - change of nucleus charge, conserves mass
3) gamma () decay - photon emission, no change in A or Z
4) spontaneous fission - for Z=92 and above, generates two smaller nuclei
 decay
241
95
a
4
Am 
 237
Np

93
2 He
- involves strong and coloumbic forces
- alpha particle and daughter nucleus have equal and opposite momentums
(i.e. daughter experiences “recoil”)
decay - three types
1) β- decay
3
1
H 
 23 He  e  e
- converts one neutron into a proton and electron
- no change of A, but different element
- release of anti-neutrino (no charge, no mass)
2) β+ decay

C  115 B  e   e
11
6
- converts one proton into a neutron and electron
- no change of A, but different element
- release of neutrino
3) Electron capture
7
4
Be  e EC
 37 B  e
decay
3
2
He* 
 23 He  
- conversion of strong to coulombic E
- no change of A or Z (element)
- release of photon
- usually occurs in conjunction with other decay
Spontaneous fission
256
100
112
Fm sf
 140
Xe

Pd  4n
54
46
- heavy nuclides split into two daughters
and neutrons
- U most common (fission-track dating)
Fission tracks from 238U fission in old zircon
Decay chains and secular equilibrium
- three heavy elements feed large decay chains,
where decay continues through radioactive
daughters until a stable isotope is reached
234Th
24d
238U
--> radioactive daughters --> 206Pb
Also 235U (t1/2)= 700My
And 232Th (t1/2)=10By
After ~10 half-lives, all nuclides
in a decay chain will be in secular
equilibrium, where
Activity( P)  A( D1 )  A( D2 )  ...
Decay chains and secular equilibrium (cont)

1
Ex: N 
 N2
1
where 1>>2
2

2

 N3
N3
1
1 / 2 =0.1
N2o=0
N2o=N1 o
scale)
N1
N/ N1 o ( log
0.1
sec u
la r e
q u i l ib
1 N
ri u m
=
1  N
2
N2
0.01
2
0.001
0
5 2
1
2
t/ 1
3
4
5
The approach to secular equilibrium is dictated by the intermediary,
because the parent is always decaying, and the stable daughter is
always accumulating.
Counting Statistics
Radioactive decay process behave according to binomial statistics.
For large number of decays, binomial statistics approach a perfect Gaussian.
N+3sqrt(N)
N+2sqrt(N)
N+sqrt(N)
Expected value (N)
N-sqrt(N)
N-2sqrt(N)
N-3sqrt(N)
Number of Observations
Ex: 100 students measure 14C disintegrations in 1g of modern coral (A=13.56dpm)
with perfect geiger counters, for 10 minutes
1=68.3%
2=95%
3=99%
124.0 135.6 147.2
Observed # disintegrations
Since the students only counted 135.6 disintegrations, they will only achieve a 1 accuracy
of ±sqrt(135.6)=±11.6 disintegrations …. Or in relative terms, 11.6d/135.6d = 8.5%
In other words, your 1 relative error (in %) will be equal to (1/(sqrt(total counts)))*100