Topic 3 - The University of Sheffield/file/Topic3.pdf · included (by Gamow) as a joke ! Proposed...
Transcript of Topic 3 - The University of Sheffield/file/Topic3.pdf · included (by Gamow) as a joke ! Proposed...
Topic 3
Primordial nucleosynthesis
Evidence for the Big Bang ! Back in the 1920s it was
generally thought that the Universe was infinite
! However a number of experimental observations started to question this, namely: • Red shift and Hubble’s Law • Olber’s Paradox • Radio sources • Existence of CMBR
Red shift and Hubble’s Law ! We have already discussed red shift in the
context of spectral lines (Topic 2) ! Crucially Hubble discovered that the
recessional velocity (and hence red shift) of galaxies increases linearly with their distance from us according to the famous Hubble Law
V = H0d where
H0 = 69.3 ±0.8 (km/s)/Mpc and 1/H0 = Age of Universe
Olbers’ paradox ! Steady state Universe is:
infinite, isotropic or uniform (sky looks the same in all directions), homogeneous (our location in the Universe isn’t special) and is not expanding
! Therefore an observer choosing to look in any direction should eventually see a star
! This would lead to a night sky that is uniformly bright (as a star’s surface)
! This is not the case and so the assumption that the Universe is infinite must be flawed
Radio sources
! Based on observations of radio sources of different strengths (so-called 2C and 3C surveys)
! The number of radio sources versus source strength concludes that the Universe has evolved from a denser place in the past
! This again appears to rule out the so-called Steady State Universe and gives support for the Big Bang Theory
Cosmic Microwave Background ! CMBR was predicted as early as 1949 by Alpher and Herman
(Gamow group) as a “remnant heat” left over from the very hot and dense initial Universe
! They predicted that after the Big Bang the Universe should “glow” in the gamma ray part of the spectrum
! This will subsequently cool as the Universe expands shifting the wavelength of this “last light” to a temperature of ~5K
! Eventually observed in 1965 by Penzias and Wilson ! The CMBR is now a very
powerful tool for cosmologists ! Recent experiments such as
COBE and WMAP have measured the CMBR anisotropies at the 10-5 level
! Gives us information on Big Bang, Dark Matter, etc.
! Subsequently they proposed a single process for all elemental abundances in the Universe - that of neutron capture
! Protons via β-decay: n → p + e- + νe
! First step: p + n → 2H + γ
αβγ theory (Origin of Chemical Elements) ! Actually Alpher & Gamow: Bethe
included (by Gamow) as a joke ! Proposed an early Universe that was
hot and dense ! Assumed that the Early Universe
consisted only of neutrons ! As the temperature fell neutron decay
to protons was possible
αβγ theory
νe
νe
αβγ theory - abundances ! Successive neutron
capture creates heavier elements
! At each step the progress controlled by the balance between the rate of production and the rate of destruction
! By setting up and solving a sequence of differential equations of this type, a distribution could be produced in reasonable agreement with the trend of the observed abundances
dNA/dt = F(S,T)[σ A-1NA-1 - σANA] F is collision frequency (function of thermodynamic state variables) NA is the no. of atoms with atomic no. A σA is the neutron capture cross-section
For these calculations
capture cross-sections measured at Los Alamos
during World War II were used
(1 MeV neutrons =1010K)
Cross-sections (quick revision) ! Consider the simple case in which a
beam of particles is incident on nuclei of some type, then the cross-section is the probability of a particular process occurring per target nucleus, per incident particle
! The total area “blocked out” is the (number of nuclei per unit volume) x (the volume) x (σ). Thus the fraction of the beam which is removed by the reaction is:
! In neutron capture the rate at which the reaction is occurring depends upon the relative velocity v of the particles and target nuclei and is given by the product of particle density, the relative velocity, the cross section and the total number of target nuclei.
! We shall discuss neutron capture further in understanding the production of elements heavier than Iron
dN/N = - nσ dxwhere n = number density x beam area
Integration yields N = N0 exp(- nσx)
or N = N0 exp(- x /λ )where λ is the mean free path
αβγ theory - success and failure ! Abundance for He agrees well with observation ! By splitting the elements into 15 “groups” by atomic weight
and using an average cross-section for each group gives a reasonable fit to abundance data
! BUT predicted abundances for heavier elements were incorrect
! Problem getting past A=4 due to lack of stable elements with A=5, 8
! Results carved the way for calculations of thermonuclear fusion
! Discussion is relevant to neutron capture topic later
This is an extract from the “Chart of nuclides”
Big Bang: Underlying principles I
! Universe expanded some 14 billion years ago from a singularity
! At extremely high temperatures elementary particles can simply be created from thermal energy kT = mc2 (essentially E = mc2)
! After the BB the Universe expands and cools ! As temperatures fall below the threshold
temperature for particle production then annilihilation rate > creation rate
Big Bang; Underlying Principles II
! Normal physics laws (including standard model of particle physics)
! Small matter-antimatter asymmetry ! Gravitation described by General Relativity ! Cosmological principal (Universe is
homeogeneous and isotropic) Robertson-Walker metric
! Expansion of the Universe is governed by field equations of GR
The Big Bang
Time
Space
Key events after Big Bang Time Temp/Energy Event 10-43 s kT = 1019 eV Planck era, quantum gravity, prior
to this all forces one, gravity first to decouple, many exotic particles
10-35 s kT = 1015 eV Inflation starts, Strong nuclear force decouples
10-10 s -10-4 s
T = 1015 K – 1012 K
Free electrons, quarks, photons, neutrinos all strongly interacting
10-4 s -101 s
T = 1012 K – 1010 K
Free electrons, protons, neutrons, photons, neutrinos all strongly interacting
Key events after Big Bang Time Temp/Energy Event 101 s T = 1010 K Neutrinos “decouple” from the
cosmic plasma (cross-section falls dramatically)
102 s T = 7.5-6x109 K Pair production of e+e- ceases
102 s kT = 0.8 MeV Proton:neutron ratio is frozen Next 300 s
Thermal energy still high enough to photodissociate atoms Neutron decay continues, n:p ratio changing
Next 103 s
Primordial nucleosynthesis starts Note ions not atoms due to mean thermal energy
Key events after Big Bang Time Temp/Energy Event
~ 103 s to 400,000 years
T ~ 108 or 9 K to T = 3000 K
“Dark ages”: Universe is a sea of free nuclei, electrons and photons. Photons Thomson scatter off electrons so Universe remains opaque to photons. Physics in this period is less well-established.
380,000 years
T = 3000 K Photons can no longer ionize, photons decouple, “last scattering surface”. Origin of CMBR.
Fundamental forces
Cosmic Microwave Background
Cosmic Microwave Background
Very close to a perfect thermal (Black Body) spectrum with a temperature
of 2.7K
The neutron:proton ratio ! The main 3 reactions involved in determining
the number of protons and neutrons in the early Universe are:
(i) n + e+ � p + νe (+ 1.8 MeV) (ii) p + e- (+0.8MeV) � n + νe
(iii) n � p + e- + νe (+ 0.8 MeV) ! Note that reaction (ii) is endothermic in a left-
right direction i.e. requires energy into the system (KE of incoming particles) in order to proceed
The neutron:proton ratio ! At T > 1010 K, kT > 1 MeV, t < 1 s, reactions (i) and (ii)
maintain protons and neutrons in thermal equilibrium • When kT >> mn – mp = Δm, protons and neutrons are nearly equal in
number • When Δm becomes significant compared to kT, the neutron-proton
ratio is given by the Boltzmann factor exp(−Δmc2/kT)
! At T ~ 1010 K, kT ~ 0.8 MeV, t ~ 1 s, the reaction rates for (i) and (ii) become slow compared to the expansion rate of the universe • neutrinos decouple (weak interaction rate slow compared to
expansion rate) • e+e− pair creation suppressed (γ energies drop below 0.511 MeV) • neutron:proton ratio “freezes out”
! Below this temperature only reaction (iii) continues
The neutron:proton ratio ! We use the Boltzmann distribution to estimate the
n:p ratio at this point
! hence
! where kT = 0.8 MeV and (mn - mp) = 1.3 MeV/c2
This yields a value of Nn:Np ~ 0.2
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N ∝m32 exp −mc
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# $ $
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Primordial nucleosynthesis ! At this point kT is too high
for primordial nucleosynthesis to start (formation of nuclei) due to dissociation
! Therefore reaction (iii) continues in the left-right direction – this is neutron decay
! After a further 300 seconds primordial nucleosynthesis starts
p + n ⇔ 2H + γ2H + 2H ⇔ 3He + n 2H + 2H ⇔ 3H + p
3H + 2H ⇔ 4He + n 3He + 2H ⇔ 4He + p
2H + 2H ⇔ 4He 3He + 4He ⇔ 7Be + γ
3H + 4He ⇔ 7Li + γ 7Be + n ⇔ 7Li + p
7Li + p ⇔ 24He Note: ions not atoms
Solved problem ! If the neutron:proton ratio starts at 0.2 and the neutron continues to decay
for a further 300 seconds what is the neutron:proton ratio at the end of this period given that the neutron’s lifetime is 890 seconds?
! The neutron’s lifetime is 890 seconds therefore in 300 seconds:
! Therefore the fraction of neutrons that have decayed = 0.286 ! Next we write
where = 0.2 and d=0.286 to give = 0.135
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NN0
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300890
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=Nn (1− d)Np + dNn
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Nn
Np
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Nn
Np
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& ' ' t= 300s
Abundances vs time Note that a
neutron:proton ratio of 0.135:1
is equivalent to 12:88
Assuming that the 12 neutrons go to forming
4He we would expect
76% Hydrogen (1H) and
24% Helium (4He) - in excellent agreement
with observation
Modern day abundances ! Comparison of modern
day elemental abundances from primordial nucleosynthesis can also give important cosmological information such as the baryon density or the baryon to photon ratio
! Concordance with CMB is important check on theory
Summary ! Big Bang Nucleosynthesis (BBNS)
successfully predicts the production of light elements shortly after the Big Bang
! The thermal history of the early Universe and nuclear physics are used to explain the sequence of events
! Light element abundances can be accurately predicted and related to cosmological parameters