Nucleosynthesis 8/21/12

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 distancein expansion of universe) To=13.7 billion years

2) Isotope geochemistry – 187Re 187Os, t1/2=40 billion years To=12-17 billion years238U decay, t1/2=4.5 billion years To=12.5-16 billion years

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 determinehow 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 -- appearance of space, time, energy,and superforce

10-43 s 1031 gravity separates10-35 s 1028 strong force and electro-weak force10-33 to 10-32 s 1027 inflation1 x 10-10 s 1015 electromagnetic and weak force3 x 10-10 to 5 x 10-6 s ~1013 stabilization of quarks, antiquarks6 x 10-6 1.4 x 1012 formation of protons and neutrons10s 3.9 x 109 stabilization of electrons and positrons3.8 m 9 x 108 formation of 2H, 3He, and 4He nuclei700,000 y 3000 electrons captured by nuclei

1992

2005

image microwaveradiation from 379,000 years after Big Bang

small temperaturedifferences (10-6 K)signify heterogeneousdistribution of matter

WMAP:Wilkinson Microwave Anisotropy Probe

age of universe =13.73 +/- 1%

http://map.gsfc.nasa.gov/

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

-process 24Mg, 28Si, 32S, 36Ar, 40Ca

e-process 56Fe & other transition

s-process up to mass 209

r-process up to mass 254

Nucleosynthesis Schematic

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

- 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=108K)

4He + 4He --> 8Be8Be + 4He --> 12C12C + 4He --> 16O and so on to 24Mg

- core contracts as He consumed, -process begins (T=109K)

20Ne --> 16O + 4He20Ne + 4He --> 24Mg and so on to 40Ca

For ‘small’ star, such as our Sun

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 + n21Ne + 4He --> 24Mg + nfollows 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 element- building 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)

Neutron #

Pro

ton

#Chart of the Nuclides, low mass

Entire chart of the nuclides

β decay

EC

α decay

The abundance of the elements - cosmic

- astronomers can detect different elements with spectroscopy (large telescopes equipped with high-resolution spectrometers)

Magic numbers: 2, 8, 20, 28, 50, 82,126

& even is always better than odd

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

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.

The abundance of the elements - our solar system

Unstable nuclides with half lives > 0.5Ma

Nuclear Physics & Radioactivity 8/21/12

What holds a nucleus together?

What drives radioactive decay?

What sets the timescale for radioactivedecay?

What happens during radioactive decay?

Lecture outline:1) nuclear physics

2) radioactive decay

3) secular equilibrium

4) counting statistics

particles in a cloud chamber

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, momentum4) symmetry

Binding energyLet’s revisit the fusion of four protons to form a 4He nucleus:

1 41 24( ) 1( ) 2 2

4(1.007277) 1(4.00150)

0.02761

eH He e E

m

m amu

*these masses comefrom the table of nuclides

We have calculated the mass deficit --> i.e. the whole is less than sum of the parts

The mass deficit is represented by a HUGE energy release, which can be calculatedusing Einstein’s famous equation, E=mc2, and is usually expressed in Mev

56F

e

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 better2) surface tension -- the less surface/volume the better (U better than He)3) spin pairing -- neutrons and protons have + and - spins, paired spins better4) 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 λ(units of y-1)

-the decay constant λ is time independent; the mean life is defined as τ=1/λ

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1000000

0 10000 20000 30000 40000 50000

Years

Nu

mb

er o

f 14

C a

tom

s

dNN

dtλ

0tN N e λ

t1/2 = 5730y

5730

N0

1/ 2

ln(2)

tλ

Activity calculations

- usually reported in dpm (disintegrations per minute), example: 14C activity = 13.56 dpm / gram C

Activity Nλ

0tA A e λ - because activity is linerarly proportional to number N,

then A can be substituted for N in the equation 0tN N e λ

Example calculation:

How many 14C disintegrations have occurred in a 1g wood sample formed in 1804AD?

T=208y

t1/2 = 5730y so λ = 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)*208y = 5.75e10 atoms

# decays = N0-N = 1.46e9 decays

Four types of radioactive decay

1) alpha () decay - 4He nucleus (2p + 2n) ejected2) beta (β) decay - change of nucleus charge, conserves mass3) gamma (γ) decay - photon emission, no change in A or Z4) spontaneous fission - for Z=92 and above, generates two smaller nuclei

decay

- involves strong and coloumbic forces- alpha particle and daughter nucleus have equal and opposite momentums

(i.e. daughter experiences “recoil”)

241 237 495 93 2Am Np He

β decay - three types

- converts one neutron into a proton and electron- no change of A, but different element- release of anti-neutrino (no charge, no mass)

1) β- decay

2) β+ decay

3) Electron capture

- converts one proton into a neutron and electron- no change of A, but different element- release of neutrino

- converts one proton into a neutron - no change of A, but different element- release of neutrino

γ decay

- 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

Fission tracks from 238U fission in old zircon

- heavy nuclides split into two daughtersand neutrons

- U most common (fission-track dating)

Decay chains and secular equilibrium

- three heavy elements feed large decay chains, where decay continues through radioactive daughters until a stable isotope is reached

238U --> radioactive daughters --> 206PbAlso 235U (t1/2)= 700MyAnd 232Th (t1/2)=10By

After ~10 half-lives, all nuclides in a decay chain will be in secular equilibrium, where

1 2( ) ( ) ( ) ...Activity P A D A D

234Th24d

Decay chains and secular equilibrium (cont)

Ex:

where λ1>>λ2

t/ τ1

λ1/ λ2=0.1

0.001

0.01

0.1

1

0 1 2 3 4 5

N/N

1o (

log

sca

le) N1

N2

N3

secular equilibriumλ1N

1=λ2N2

5τ2

2

N2o=0

N2o=N

1o

The approach to secular equilibrium is dictated by the intermediary, because the parent is always decaying, and the stable daughter is always accumulating.

Counting StatisticsRadioactive decay process behave according to binomial statistics.For large number of decays, binomial statistics approach a perfect Gaussian.

Observed # disintegrations

Num

ber

of O

bser

vatio

ns

Ex: 100 students measure 14C disintegrations in 1g of modern coral (A=13.56dpm)with perfect geiger counters, for 10 minutes

135.6

Ex

pe

cte

d v

alu

e (

N)

N+

sqrt

(N)

N-s

qrt

(N)

N+

2sq

rt(N

)

N-2

sqrt

(N)

N+

3sq

rt(N

)

N-3

sqrt

(N) 1σ=68.3%

2σ=95%3σ=99%

147.2124.0

Since the students only counted 135.6 disintegrations, they will only achieve a 1σ accuracyof ±sqrt(135.6)=±11.6 disintegrations …. Or in relative terms, 11.6d/135.6d = 8.5%

In other words, your 1σ relative error (in %) will be equal to (1/(sqrt(total counts)))*100