Nuclear & Particle Physics Flashcards

Question

What is deep inelastic collisions?

Answer 1

* Quarks inside nucleons

Answer 2

* glue inside nucleons and nuclei

Answer 3

* defined from reaction rate * which is R_b = I_aNσ **(what's N and Ia?)** * σ = cross section = R_b/I_aN * effective area that quantifies the intrinsic likelihood of a scattering event when an incident beam strikes a target object, made of discrete particles * measured in units of area; most commonly a barn

Answer 4

* 1 barn = 10^-28 m² = 100 fm²

Answer 5

* quantifies the intrinsic ratte at which scatteed projectiles can be detected at a given angle * dσ/dΩ is not a differential, just a symbol * rate into a solid angle element is * dR_b = r(θ, φ) dΩ/4π * θ is the scattering angle measured between incident beam and scattered beam * φ is the azimuthal angle (usually scattering processes have azimuthal symmetry and therefore do not depend on φ) * differential cross section is: * dσ/dΩ = (dR_b/I_aN) r(θ, φ)/4π = r(θ, φ)/I_aN4π

Answer 6

* conservation of momentum, energy (elastic scattering) and angular mom. * relationship between scattering angle and impact parameter * b = d/2 cot(Θ/2) * b = impact parameter * d = distance of closest approach * Θ = scattering angle * scattering cross section * dσ/dΩ = (zZe²/4πε₀)(1/T_a)² 1/sin⁴(Θ/2) * (what's Ta?) (+ are we supposed to know this eqn?) * magnitude of change of momentum * |Δp| = |ћq| = 2mћv₀sin(Θ/2) * Rutherford cross section is * dσ/dΩ ~ 1/q⁴ * (What is the point of knowing this?)

Answer 7

* probability for transition between initial state and final state * λ = 2π/ћ ρ(E_f) |V_fi|² (what's the point of 2π here?) * V_fi is the matrix element: * V_fi = integral of ψ_f\*Vψ_i dv (volume integral?) * particle are plane waves * hence ψ_i= e^-ik_i.r = e^-ip_i/ћr * scattering transition probability is * |V_fi|² ~ |F|² where F is the form factor?

Answer 8

* V_fi = F = integral of e^i(p_f/ћ)**^r** V e^-i(p_i/ћ)**^r** dv * q defined as |Δp|/ћ * hence F(q) = integral of e^iq**^r** V dv * in spherical coordinates * F(q) = integral of e^iqrcosθ V r² dr sinθ dθ dφ * =

Answer 9

* e- scattered inside a +vely charged nucleus probes its interior * obtained by * measuring cross section vs scattering angle * deduce form factor * obtain charge density of nuclei

Answer 10

* nuclear radius * r = r₀ A^1/3 * r₀ = 1.25 fm * hence nuclear volume * V = r₀³ A * since nuclear size scales with number of constituents * like a drop of liquid

Answer 11

* from atomic transitions (point charge or uniformly charged sphere) - don't really know what this means * particularly for 1s where e- is inside nucleus * get isotope shifts

Answer 12

* where muon replaces e- * have smaller Bohr radius * hence greater sensitivity to size of nucleus

Answer 13

* Coulomb energy differences between mirror nuclei (e.g. 13-N(7,6) and 13-C(6,7)) * is there more to say about this?

Answer 14

* at short distances (higher energy) when incident α particle gets close enough to target nucleus that they interact via the nuclear force * (in addition to the Coulomb force that acts when they are far apart) * =\> projectile hits surface * point at which the formula breaks down gives a measure of the size of the nucleus

Answer 15

* at separations of less than 12.15 fm * corresponds to R₀ = 1.7 fm * greater than mean radius of 1.25 fm * but consistent with "skin thickness" of about 0.5 fm * which allows 2 nuclear distributions to overlap at these larger distances

Answer 16

* 1 amu = 1/12 of mass of ¹²C * **1 amu = 931.5 MeV** = 1.66 10-²⁷ kg

Answer 17

* B = Δmc² * where Δm = Zm_p + Nm_n - mx * Z = atomic number; m_p = mass of proton in amu * N = number of neutrons; m_n = mass of neutron in amu * m_x = mass of the particle in amu * binding energy for stable systems is +ve * assembled systems weighs less than sum of constituents

Answer 18

* **volume** (bulk) * a_vA =\> a_v = 15.5 MeV * **surface** reduces binding (based on strong force; surface tension in liquids) * a_s A^2/3 =\> a_s=1 6.8 MeV * **Coulomb** repulsion between protons * a_cZ(Z-1)/A^1/3 =\> a_c = 0.72 MeV * **(A)symmetry** energy (imbalance of no. of protons and neutrons for a given number of nucleons; Pauli exclusion principle) * a_A (A-2Z)²/A =\> a_A = 23 MeV * **Pairing** correction (spin coupling effects) * δ(A,Z) * = +δ₀;Z, N even (A even) * = 0; A odd * = -δ₀;Z, N odd (A even) * δ₀= a_p/A^1/2

Answer 19

* surface reduces binding for light nuclei * Coulomb reduces binding for heavy nuclei * Nuclear shell structure affects structure significantly * Upper limit in isotope chart set by fission * Superheavy nuclei has Z = 119

Answer 20

* dM/dZ = 0 * by beta decays * beta- = n -\> p + e^- + anti-v (hence for lower Z) * beta+ = p -\> n + e⁺ + v (hence for higher Z)

Answer 21

* existed in the universe at t \< 10^-5 - 10^-9 sec * what's t? * in the past * now the world is composed of hadrons * can it be recreated on Earth?

Answer 22

* no net density * =\> about equal amounts of matter and antimatter

Answer 23

* temperature predicted by theory (LQCD) is ~ 10¹²K ≈ 1000 billion degrees * obtainable by LHC (T = 250 MeV); RHIC (T = 180 MeV) * not really sure what the figure is showing? too much going on

Answer 24

* need symmetry breaking to give mass * Higgs boson is evidence of Brout-Englert-Higgs symmetry breaking mechanism (1964)

Answer 25

* Lagrangian: L = T-V * allows to determine equations of motion by sub. into Euler-Lagrange eqn * Principle of least action: * area under L vs t is the action * physical paths minimise the action

Answer 26

* baryons and mesons * baryons qqq or antibaryons q-q-q- (fermionic hadrons) * mesons qq- (bosonic hadrons)

Answer 27

* existence of Δ++ = uuu * to respect Pauli Principle * 3 values of colour (RGB - just representatives)

Answer 28

* M_p ≈ 938 MeV * M_n ≈ 939 MeV * ≈ 1 GeV * M_π = 140 MeV * M_e = 0.511 MeV * ћc = 197.32 MeVfm * fm/c = 0.3\*10^-23sec

Answer 29

* lepton number, charge, baryon number, CPT are conserved in all interactions * flavor and parity conserved in strong & EM, broken by weak * isospin conserved by strong, broken by EM (except I_z)

Answer 30

* SU(3) describing color and anticolor (R- = cyan, G- = magneta, B- = yellow) * g0 does not change colour hence only 8 * don't get why g7 or g8?

Answer 31

* **QED**: photon has no charge hence **no self interaction** * at large distances, **charge is screened** =\> α = 1/137 * **small distances, α increases** (since more charge seen) * **QCD**: gluon has colour charge =\> **self-interaction** (split and fuse) * at **smaller distances** (because of the **gluon cloud**), **α_s decreases** so less colour charge seen =\> **antiscreening** * gives rise to asymptotic freedom

Answer 32

* quark-quark potential vs distance (r) * at large distances, α_s increases =\> energy becomes infinite * no free quarks possible so always q and q- pairs * =\> confinement

Answer 33

* like water: T_HG = T_QGP, P_HG = P_QGP, μ_HG = μ_QGP * where P_HG = P_baryons + P_mesons ≈ P_mesons * P_HG ≈ P_π = g_π(π²/90)T⁴ ≈ 3(π²/90)T⁴ (g_π ≈ 3) * where P_QGP = P_quarks + P_gluons - B * P_QGP = (g_q + g_g)(π²/90)T⁴ - B ≈ 40(π²/90)T⁴ - B * MIT 'Bag Model' estimates Bag Constant, B ≈ 10²³ atm * from eqns for P, critical phase temp can be found * T(crit) ≈ 150 MeV * lattice QCD predicts T(crit) ≈ 175 MeV * E = kT =\> T ≈ 10¹² K

Answer 34

* e.g. at RHIC, proton with E = 100 GeV =\> γ = 100 GeV * at LHC, proton with E = 2750 to 7000 GeV =\> γ = 2750 to 7000

Answer 35

* set c = 1 * then E² - p² = m² which equals an invariant quantity, let's say s =\> E² - p² = m² ≡ s * e.g. for any system of 2 particles with (E₁, **p₁**) and (E₂, **p₂**) * s ≡ (E₁ + E₂)² - (**p₁** + **p₂**)² * s = E₁² + E₂² + 2E₁E₂ - **p₁**² - **p₂**² - 2**p₁**.**p₂** * s = m₁² + m₂² + 2E₁E₂ - 2**p₁**.**p₂** * 1) target at rest in lab frame: p2 = 0 hence E₂ = m₂ * s = m₁+ m₂ + 2E₁m₂ ≈ 2E₁m₂ * 2) collider in CM frame (CM frame so p=0 =\> **p₁**=**-p₂**) * s = (E₁ + E₂)² - (**p₁** + **p₂**)²= (2E)² * sqrt(s) = 2E

Answer 36

* tranverse variables are Lorentz invariant * p_t = sqrt(p_x² + p_y² ) [tranverse momentum] * m_t = sqrt (p_t² + m² ) [transverse mass] * longitudinal variables: rapidity is additive under Lorentz transformations from one frame of reference to another * y = 1/2 ln(E + p_z/E - p_z) * y = 1/2 ln(1 + βcosθ/1 - βcosθ) * β ≈ 0 =\> y ≈ β; β ≈ 1 =\> y ≈1/2 ln(2/1-β) * θ = 90 =\> y = 0 (midrapidity); θ = 0, y -\> infinity * E = m_t cosh y * pseudorapidity * η = ln cot(θ/2)

Answer 37

* because of colour confinement * thus gluons have colour and are not 'allowed' to move outside the nucleon

Answer 38

* screened Coulomb potential * nucleon-nucleon interaction is due to exchange of massie quanta (e.g. pions +,-, neutral) [not sure how this links in to what it is] * also is the eqn correct? One on wikipedia for classical potential of 2 fermions ineracting through a Yukawa potential is similar but g² * R = range of interaction * from uncertainty relation ΔEΔt ≈ ћ =\> mc²(R/c) ≈ ћ * R ≈ ћ/mc = ћc/mc²≈ 197.32/140 ≈ 1.5 fm * multiply by c/c to use ћc = 197.32

Answer 39

* pseudoscalar is a scalar with spin 0 but flips signs under parity transformation * pseudoscalar meson * e.g. pion (m = 140 MeV); J^P = 0 * a vector meson has spin 1 and -ve parity (J^P = 1-) * e.g. rho (m = 770 MeV) * omega (m = 782 MeV) * meson exchange underlies the nucleon-nucleon interaction

Answer 40

* Coulomb: form factor for extended density distribution (nuclei) * constant density =\> oscillating form factor * infinite range * Yukawa: form factor for potential mediated by heavy virtual particle

Answer 41

* of particle = +1 * of antiparticle = -1 * parity: P = (-1)^L+1 * C-parity: |qq_bar\> = C |q_barq\> * J^PC

Answer 42

* simplest bound nucleus * A=2, Z=1, hydrogen * only one bound state with E = -2.225 MeV * I^π = 1+ (what is this?) * R_ch= 2.14 fm (what is this?) * spin of neutron and proton = 1/2 each * I = 1

Answer 43

* μ = AI * A = πr² * I = q/t * t = 2πr/v = 2πr/(p/m) [p=mv] * μ = qћl/2m * e-: μ = -eћl/2m_e ≡ -μ_Bl * nucleon: μ = eћl/2m_N ≡ μ_Nl * =\> μ ≡ g_llμ_B or μ ≡ g_llμ_N (g is the g-factor) * orbital e-: μ_l ≡ g_llμ_B; g_l = 1 * spin of e-: μs ≡ -gsμ_Bs; gs = 2 * expect g_s(p) = 2 & g_s(n) = 0 but in experiments, g_s(p) = 5.5856 & g_s(n) = -3.8261 * =\> nucleon has a substructure =\> quarks

Answer 44

* indicates proton and neutron spin are parallel * S = 1/2 + 1/2 = 1 * difficult to start from fundamental interaction =\> try effective model approach * quantum numbers * L= 0 * π = (-1)^L = +1 (parity) * I = 1 * I_z = -1, 0, +1 =\> deuteron is in TRIPLET state * =\> nucleon-nucleon interaction must be spin dependent (no singlet state exists)

Answer 45

* V₀ = 35 MeV (solved numerically & to first order) * by modelling the binding of the deuteron in a square well potential * an attractive central potential

Answer 46

* same way as bound problem of deutron but with E \> 0 * L = mvb * in QM: lћ = mvb * l = 0: l = mvb/ћ \<\< 1 * p = mv = sqrt(2mE) =\> E \<\< ћ²/2mb² ≈ 20 MeV * b = impact parameter * if neutron has low energy (E = keV or few MeV) then scattering is s-wave (l=0)

Answer 47

* to shift the phase of the scattered wave at points beyond the scattering regions, where the wave fn is that of a free particle * phase shift changes sign at high projectile energy * change in s-wave phase shift from +ve to -ve at about 300 MeV * =\> at these enegies, the incident nucleon is probing a repulsive core in the nucleon-nucleon interaction * repulsive at short distances * attractive in the well?

Answer 48

* number of particles per second * dσ = current into solid angle element dΩ/incident current = j_scattered r² dΩ/ j_incident * the differential cross section (cross section per unit solid angle) is dσ/dΩ = (sinδ)²/k² * and total cross section (L=0) integrated over all angles is * σ = 4π (sinδ)²/k²

Answer 49

* because nucleon-nucleon interaction is spin dependent

Answer 50

* not a length (unit is length though) but represents the strength of the interaction * bound state has +ve scattering length whereas an unbound has -ve

Answer 51

* central Coulomb potential: e-s experience attractive field of nucleus * atoms with many e-s =\> Hartree Fock approximation (effective charge) * atomic radius (nm) & ionisation energy (eV) vs Z shows smooth variations which correspond to gradual filing of an atomic shell * sudden jumps show transitions to the next shell

Answer 52

* particular stabilities at particle numbers 2, 8, 20, 28, 50, 82, 126 * suggests that atomic nucleus has a shell structure like the atom * =\> try to solve using central potential (though nuclear density profile suggests this will not work)

Answer 53

* Schrodinger eqn * wavefns are antisymmetric for fermionic systems * radial eqn * use infinite deep well & harmonic oscillator * which works OK for lowest states but not for higher states * large gaps occur between energy levels which are associated with closed shells

Answer 54

* since nucleon density is constant in nuclear interior (for r \< R) =\> can try potential with flat central part * state with angular mom. l has (2l+1) magnetic substates * m_l= -l, .., 0, ..., +1 * 2 spin states (-1/2, +1/2) * level degeneracy is 2(2l+1) * 1p tate is first state with l=1

Answer 55

* **s.l** term is important in nucleon-nucleon interaction * add V∝ **l.s** to W-S potential * calculate expectation value of V∝ **l.s** (spin orbit potential): * ΔE = energy difference between l states when they split up * spin orbit in nuclei has opposite sign(-) compared to atoms(+) * which sign are you talking about? * if **l** & **s** are no longer good quantum numbers =\> **j = l+s** is a good QN

Answer 56

* reproduces magic numbers * level degeneracy is 2j+1

Answer 57

* particle ⇔ antiparticle

Answer 58

* P (Parity): inversion of spatial coordinates (does not change spin and angular mom.) * C (Charge conjugation): inversion of all intrinsic QN (changes charge, lepton, baryon, strangeness etc. numbers) * C is Unitary (C² = 1) and Hermitian (C = C⁺) * C|ψ\>= η_c|ψ\> [what's η_c?] * strong and EM obey C symmetry; weak violates * e.g. neutrinos are left-handed & antineutrinos are right handed * T (Time): inversion of all derivatives e.g. mom. and angular mom. (leaves F=ma unchanged =\> forces are conserved) * CPT invariance absolutely true (Lorentz invariance =\> CPT invariance) * integer spin particles follow BE stats, 1/2 integer ones follow FD stats * q and q\_bar have identical mases and lifetimes * q\_bar has opposite QN of q * CP approximately conserved but partially broken in weak * e.g. the Kaon decays not equally prob. under CP * K₀^L -\> π+ + e- + anti-v_e * K₀^L -\> π- + e⁺ + v_e

Answer 59

* short range attractive nucleon-nucleon interaction bind nucleons into paris moving in time reverse orbits with **m_j=-j and m_j=+j** (magnetic moments; right hand rule) * even-even nuclei ground states: I^π = 0+ * odd-even nuclei ground states: filled proton shells do not contribute to structure * properties of ground state determined primarily by odd neutron =\> total spin and parity determined by valence nucleon or hole * excited states in odd-even nuclei: (don't know what the figure is showing?)

Answer 60

* energy of nucleons can to good approximation be calculated from an effective central potential + spin-orbit coupling * energy levels show shell structure * magic numbers are understood * nucleons move indtly in this potential, unaffacted by others (extreme indt particle model) * works because Pauli blocking prevents nucleons inside the nucleus to scatter on each other (which would otherwise destroy the regular motion) * scatterings only allowed if final states are outside Fermi sphere |**p'_i**| \> |**p_F**|; |**p'**| \> |**p_F**|

Answer 61

* costs ≈ 2 MeV to break a pair * 2+ unexpected * 130-Sn

Answer 62

* new states corresponding to other excitations of the nucleus * not 'shell model states'

Answer 63

* involve all nucleus coherent motion * indt particle motion (single particle motion) =\> weak gamma transition; spherical shape * whereas correlated particle motion (collective rotation, vibration) =\> strong gamma transition; rugby football or pancake shape

Answer 64

* states are equidistant (compare Harmonic oscillator) * above 2 MeV, structure becomes complicated and no vibrational patterns can be seen

Answer 65

* vibrational quanta = phonons * dipole vibrations are shifts of CM: not allowed in absence of external forces * quadrupole vibration (λ = 2) is the lowest occuring mode * λ=2 phonon assigned parity (-1)^λ=+1 since it creates a shape with reflection symmetry ψ(**-r**)=ψ(**r**) * an even-even nucleus in GS has I^π = 0+ * =\> adding a λ=2 phonon create the first excited state I^π = 2+ * adding 2 λ=2 phonons gives the following possible states: 0+, 2+, 4+ * only total λ=λ1+λ2 values with μ=μ1+μ2 values from -λ to +λ are allowed * adding 3 λ=2 phonons gives possible states 0+, 2+, 3+, 4+, 6+

Answer 66

* using the quadrupole moment * spherical =\> Q = 0 * prolate =\> Q \> 0 * oblate =\> Q \< 0

Answer 67

* β ≈ 0.3 * they might be able to rotate

Answer 68

* fusion-evaporation heavy ion fusion reactions * beam nucleus hits target nucleus -\> fusion 10^-22 s -\> * either fast fission * or compond formation 10-¹⁹s; ~0.75 MeV; ~ 2\*10²⁰ Hz -\> rotation -\> cooling 10^-15s -\> ground state 10^-9s * p₁ = m₁v₁; p₂ = 0 * p_rec = m₁v₁;= (m₁+m₂)v_rec * E_cm = 1/2(m₁+m2)v_rec² * E₁-E_cm=E\* (what's E\*) * rotation increases with angular mom. and excitation energy * excited states can emit a photon to reduce rotation

Answer 69

* reflection symmetry ψ(**-r**) = ψ(**r**) * only states with +ve parity allowed * only even values of rotational angular momentum * from E vs I, can obtain moment of inertia J * e.g. get E in terms of 2+, 4+, 6+ * plot E vs I; curve goes through the points

Answer 70

* backbend is the breaking of a pair of nucleons due to the Coriolis force F_cor = 2m**v**×**ω** * Coriols force in a rotating nucleus can break a pair of nucleons moving in time reversed orbits * and align the single particle angular momenta along rotation axis * so **+m_j and -m_j** (+j and -j) becomes +j and +j

Answer 71

* uses a 3D deformed harmonic oscillator potential with axial symmetry, spin orbit and I² terms * then solve Schrodinger eqn to get nuclear energy levels * deformation means **J=l+s** is no longer a good QN =\> degeneracy of J states lifted * label single nucleon states by * N = n₃ + n_⊥ * Λ = component of I on symmetry axis * Ω = component of j on symmetry axis * f_7/2? =\> j=7/2; lowest energy (lies closest and will interact most strongly with the core) * Nilsson states: shell closures (magic numbers) for spherical nuclei (86 and 116, 82 and 126) * predict large deformation for N=86, e.g. nucleus 152-Dy

Answer 72

* gamma-rray of superdeformed band in 152-Dy (looks like a peanut) * different decay sequences ('bands') each correspond to a given shape =\> several shapes 'coexist' in the nucleus * prolate γ=0º, β=0.2; oblate γ=60º, β=-0.1; prolate superdeformed γ=0º, β=0.6 (angular momentum close to fission limit; excitation energy above GS) * superdeformed nuclei are exotic

Answer 73

* possibility of shell gaps at high N and Z numbers N=104 to N=118 * predicted island of stability at Z ≈ 110-120 and N ≈ 175-185 * Z = 109-118 * Z= 108 (Hassium) is longest lived isotope? =\> probably more stable can be found * Z = 115 recently announced (2013) * halo nuclei (weakly bound neutrons = \> neutron halo) * both superheavy and halo nuclei are exotic nuclei

Answer 74

* they can vibrate * lowest excitation is quadrupole vibration * states are equidistant * they have static deformations * they can rotate * energy states increases quadratically with angular mom. * moment of inertia can be measured from energies and sequence of gamma rays

Answer 75

* masses and Q-value uniquely determine α energy * Q = (3-10) MeV * Z = 82 shell * N = 126 shell

Answer 76

* Coulomb energy and Q value in MeV are at r=R in an potential barrier? * square well then exponential decay in V vs r

Answer 77

* λ = 1/Ta = P(formation)\*Frequency(knocking)\*P(barrier penetration) * probability to preform an α inside nucleus ≈ 0.01 -\> (nucleons near Fermi surface have similar momenta) * frequency of 'knocks' at barrier * t = R/v = 8/0.25c = 10^-22 s * v = 1/t ≈ 10²² s-1 * probability of barrier penetration * P= e^-2G where G = Gamow factor * exponential dependence of Gamow factor explains large variation in α-halflife as a consequence of variation of Q value relative to Coulomb barrier height * more probably at tips of a deformed nucleus (lower and thinner barrier)

Answer 78

* from relative abundance and half-life of * 235-U (0.72%; t₁ = 7.04\*10⁸ yrs) * and 238-U (99.27%; t₂ = 4.47\*10⁹ yrs) * N₁(t) = N₀₁e-^t/t₁ * if N₀(235-U) = N₀(238-U) =\> t = 5.9\*10⁹ yrs * if N₀(235-U) = 0.1N₀(238-U) =\> t = 3.14\*10⁹ yrs * compare nuclear synthesis models and Supernova models

Answer 79

* 14C prodn is constant over time * 14C absorption by living organisms is constant in time * 14C decay rate is constant (1 Bq per 4 g of C in 'fresh' organic matter) * expect 64/4 = 16 Bq * so sample has decayed by a factor 2/16 = 1/8 = 1/2³ * =\> time elapsed since absorption is 3\*5730 yrs = 17200 yrs * uncertainty on sample with N counts is sqrt(N)/N = 1/sqrt(N) * if 1% accuracy (Δt = 172 yrs) desire =\> need 10,000 counts = \> measure 5000 s for 64 g * if only 64 mg available =\> must measure for 5000\*1000 = 5\*10⁶ sec ≈ 58 days * so backgound radiation becomes important =\> shield

Answer 80

* β- (n-\>p+e-+antiv_e), β+ (p-\>n+e++v_e), e- capture(e-+p-\>n+v_e) * NB: free proton does not decay (p is lightest baryon) but is possible in nuclei if Q value \> 0 * continuous e- distribution from β decay until end point T_e = Q =\> 3 body decay =\> existence of neutrino * m_n-m_p = 1.3 MeV =\> free neutron decays with t_1/2 = 10.5 mn * neutrino capture * anti-v_e + p -\> n + e+ * v_e + n -\> p + e- * helicity of anti-v_e = +1 (right handed) * heclicity of v_e = -1 (left handed) * weak decay of quarks; weak interation mediate by W and Z bosons * involves matrix elements between initial and final nuclear states

Answer 81

* Fermi's Golden Rule * ρ(E_f) = density of final states = dn/dE_f * |V_fi|² = Matrix element * plane wave approximation for e and v (what's v?)

Answer 82

* E_f = E_e + E_v = E_e + qc * dq/dE_f = 1/c at fixed E_e * N(p) dp = C p2q2 dp * because N(p=0) =0 and for T_e=Q from figures of expected e- energy and momentum distributions

Answer 83

* F(Z',p) is Fermi fn which represents the Coulomb interaction between the daughter nucleus and the e- * Fermi integral is dimensionless * superallowed decays have shortest half-lives of log(ft) = 3-4 * log₁₀(ft) plotted due to large spread in half-lives * for 0+ -\> 0+: M_fi= √2 =\> g = 0.88\*10^-4 MeV fm3 (corresponds to the below for weak) * strong: 1 * EM: 10^-2 * Weak: 10^-5 * Gravity: 10^-39

Answer 84

* differences arrise from Coulomb interactions with daughter nucleus * e-: attraction * e+: repulsion

Answer 85

* neutrino mass * c²p dp = (T_e + m_ec²) dT_e * dN/dp = * -\> 0 if mv = 0 * -\> ∞ if mv \> 0 * at endpoint

Answer 86

* if neutrino and antineutrino were identical particles * but different helicity of neutrino and antineutrino suggest that they are different * Majorana particle (suggests that the neutrino is its own antiparticle) * if so then ₄₂Mo -\> ₄₄Ru

Answer 87

* neutron hits a uranium nucleus -\> nucleus spitting =\> two daughter nuclei + fast neutrons * spontaneous fission is due to tunneling through the fission barrier * overcome fission barrier in nuclear reactions by supplying an activation energy e.g. through neutron capture

Answer 88

* mass distribution fission fragments from fission of 235-U is quite symmetric * several neutrons are liberated because * heavy nuclei have large neutron excess (N\>Z) * lighter fission fragments have less neutron excess =\> neutrons are liberated during fission (prompt evaporation neutrons)

Answer 89

* governed by fission barrier height, nuclear viscosity and shell effects

Answer 90

* primordial: H, He, Li, B, Be * stellar: O, Fe, Th

Answer 91

* Era of Nuclei 10² sec * 10⁰ sec to 10^-4 sec to 10^-6 sec * Nucleosynthesis of Helium * Nucleosynthesis Era * disappearanceof positrons, antiprotons, antineutrons * Particle Era 10^-10 sec * confinement of quarks * formation of protons, neutrons, antiprotons, antineutrons * Era of Galaxies =\> Galaxy and Star Formation * 1 billion - 5 billion years

Answer 92

* H(WMAP) = 71 ± 4 * H(PLANCK) = 67 ± 1.2

Answer 93

* which implies S/V ∝ T³ * S constant =\> 1/R³ ∝ T³=\> T ∝ 1/R

Answer 94

* Time * Hubble relation * v = Hd * eqn of motion * H = v/d = 1/R dR/dt * Temp. (energy) * assume density of universe = density of radaition * Stefan-Boltzmann's law ς_rad = σT⁴ [do I need the other eqns?]

Answer 95

* microwave background (λ = 7.35 cm) =\> T = 2.7 K * n_γ = 4\*10⁸ photons/m³ * ς_γ= 2.5\*10⁵ ev/m³ (is this ev or eV?) * from luminous matter * n_baryon ≈ 0.4 baryons/m³ * n_baryon/n_γ = **η ≈ 10^-9**

Answer 96

* annihilation of quarks and antiquarks =\> photons * confinement of net quarks =\> hadrons * confinement in detail =\> Hadrons grouped together =\> nuclei * T = 175 MeV =\> t ≈ 10^-6 sec * after this time p and n are the only hadrons * mesons have decayed e.g. t_1/2(π) = 10^-8 sec

Answer 97

* weak interactions: mix n and p, but stops after 1 sec =\> fix the n to p ratio * neutrinos decouple at t ≈ 1 sec i.e. density of neutrinos is so small that neutrino capture no longer plays a role =\> ratio of n and p is fixed * in eqm, ratio of neutrons and protons given by Boltzmann factor * at 'freeze-out', neutron to proton ratio is about (t = 0.01 sec after Big Bang) * N_n/N_p = 0.88 for kT = 10 MeV * =\> starting condition for nucleus formation

Answer 98

* neutron + proton =\> deuteron * D + D =\> 3-He * first nuclei form: D, 3-He, 4-He, ..

Answer 99

* n + p \<-\> D + γ (2.225 MeV) * deuterium only survives if number of gamma rays with Eγ \> 2.225 MeV is sufficiently small * integrate tail of black body radiation * N(Eγ \> E₀) = integral of n(E) dE from E₀ to ∞ * plotting fraction f of photons of energies above E₀ gives T and time of formation of deuteron

Answer 100

* formed between 3 mins and 15 mins after * limited by neutron decay time at 15 mins

Answer 101

* formation of Li and Be limited by Coulomb barrier (MeV) * temp now \< 0.1 MeV (10² to 10⁵ s after) * all other nuclei have higher binding energy than deuterium * hence more stable * nuclei up to 7-Li and 7-Be can be formed * but reactions cannot go beyond A=7 * because ⁸Be = \> ⁴He + ⁴He (t_1/2 = 10^-16 sec)

Answer 102

* original n to p ratio * hence number of neutrino (lepton) families * all neutrons end up in ⁴He =\> limiting factor * N_He/N_p = 0.081 (calculated from N_n/N_p & corrected for β decay from t = 3-220 s) * M_He/M_p ≈ 0.24 * measurements predict that there are 3 families of leptons * more lepton families =\> mix n and p better =\> higher n/p ratio * hence more He * (is there a 4th family of neutrinos - sterile neutrinos?)

Answer 103

* when energy of light can longer ionise H (E \< 13.6 eV) * light decouples (photons seen in CMB are from this period) * at this point, the Universe is about 1 million years old * stars birth rate peaks at ~0.5 - 1 billion years * early galaxy formation at t = 3 billion years * Solar System forms at ~10 billion years * 23% dark matter = WIMPs? Neutralinos? Axions? =\> LHC? * 73% dark energy

Answer 104

* general relativity predicts * (a"/a') α - (ρ + 3ρ) * =\> something out there with p \< - ρ/3 * =\> dark energy; what is it?

Answer 105

* p+p -\> α in stars (alpha capture for A \< 60) * fusion (tunneling model) is sub-barrier * height of Coulomb barrier limits fusion to lighter elements in smaller stars (low internal temp.) * initial conditions from primordial nucleosynthesis M_He/M_p ≈ 0.24 * make heavier nuclei by * ⁴He + ⁴He -\> ⁸Be -\> ⁴He + ⁴He: * t(breakup) ≈ 10-16 s; Q = 91.9 keV * Q = 7.45 MeV for ⁴He + ⁸Be -\> ¹²C * =\> requires very large rate * only possible through resonant reactions * barrier penetration strongly enhanced if final nucleus has state that matches the incoming energy

Answer 106

* for a + X -\> Y * reaction rate: R ∝ n(E) σ(E) * n(E) ∝ e-^E/kT√E * σ(E) ∝ (1/√E) e^-2G * =\> R ∝ e^-(E/kT)-2G * curves for reaction ¹²C + ¹²C at temp. corresponding to kT = 0.1 MeV * should be studied near the peak of the effective cross section

Answer 107

* due to overlap with final states of compound nucleus * reaction rate: R ∝ n(E) σ(E) * n(E) ∝ e^-E/kT√E σ(E) ∝ (1/√E) e^-2G -\> σ(E) ∝ (1/√E) e^-2G S(E) =\> R ∝ e^-(E/kT)-2G

Answer 108

* up to ⁵⁶Fe * in heavy stars when He is used up * ¹²C + ¹²C -\> ²⁰Ne + ⁴He * relative abundance of C, O, Ne, Fe higher than the rest * end at ⁵⁶Ni, ⁵⁶Co, ⁵⁶Fe (max. of binding energy curve)

Answer 109

* stellar nucleosynthesis for A \> 60 * build up of heavier elements by neutron capture (no Coulomb barrier) * cross section for formation is determined by * neutron flux * element lifetime vs. Beta decay * slow (S) process * all neutron flux, requires long half-lives * rapid (R) process * large neutron flux wins over short half-lives

Answer 110

* red giant star: T = (1-2)\*10⁸ K; nⁿ = 10¹⁴ m^-3 * rate corresponds to one capture per approx. 20 years * =\> nuclear reactions proceed via the slow process * i.e. determined by beta half-lives of elements produced in the chain

Answer 111

* blue = slow * rapid = red * green = superheavies (SHE)?

Answer 112

* large neutron flux generated in supernova explosions * heavy elements on Earth are thought to be due to mass accretion from debris from a local (in our Galaxy) supernova explosion

Answer 113

* knowledge of 'exotic nuclei' i.e. unstable nuclei occuring in S and R processes necessary for astro calculations * these nuclei cannot be made using stable projectiles and target nuclei * have to use unstable isotopes as projectiles * called Radioactive Isotope Beams or Rare Isotope Beams

Answer 114

* Δp small =\> one wavelength * Δp medium =\> wave packet made of several waves * Δp large =\> wave packet made of lots of waves

Answer 115

* Einstein's theory of special relativity brings space and time at the same level * based on 2 postulates * speed of light is the same for all observers, no matter what their relative speeds are * laws of physics are the same in any inertial frame of reference (Lorentz invariant) * m₀ = rest mas of a particle * number of particles not conserved e.g. in a p-p collision, many particles are produced

Answer 116

* describes time evolution of ψ * works for spin 0 particles * operators (observables) follow special relativity * respects Lorentz invariance * space & time derivatives to the same power

Answer 117

* ρ may take on -ve values * -ve prob. density = probably not conserved? * considering Klein-Gordon eqn is second order in ∇

Answer 118

* γ depends on Pauli spin matrices * Lorentz invariant * includes spin (all fermions have spin) * no problem with -ve prob. density anymore * ψ is not a complex number anymore, is a 4-component spinor BUT * still problem of energy solns \< 0 * solved by antiparticles * still problem of conserved number of particles

Answer 119

* interpret a -ve energy soln as a -ve energy particle which propagates backwards in time * or +ve energy anti-particle which progates forwards in time

Answer 120

* fields for matter and for forces * continuous quantities over space * not wave fns * particles are excited states of an underlying physical field * excited states have arbitrarily large number of particles * providing QFT systems with effectively an infinite number of degrees of freedom * particles are called field quanta * Lagrangian formalism L = T-V; [L] = energy

Answer 121

* normally, vacuum is defined as state with no particles (classical) * in quantum field, balls and springs are never stationary * always moving even when not enough energy has been added to the field to create a particle * "virtual particles" can briefly and spontaneously appear from vacuum and disappear again even if insufficient energy * vacuum itself has random and indelible fluctuations * sometimes their influence can be felt by the way they kick around real particles

Answer 122

* light disturbances of the field does not work * field only accepts energies above a certain threshold * if the field is tapped hard enough * a particle is created * particle can propagate stably through the field

Answer 123

* when (self or not) interacting fields are introduced, the number of particles is not necessarily conserved

Answer 124

* can be used to derive the dynamics of any physical system * when applied to a quantum system with Lagrangian expressed in terms of space-time fields, principle can determine the dynamics of elementary particles in QFT

Answer 125

* interaction terms between corresponding underlying fields * free field theory neglects interaction with other particles (fields) * unrealistic because even for one free particle, it interacts with the field it generates itself * in QFT approach, a vibration in the e- field induces a vibration in the photon field * photon field vibration transports energy and momentum to another e- vibration and is absorbed * this idea of fields and vibrations explains how the universe works at a deep and fundamental level * fields span all space

Answer 126

* symmetry; any transformations of the generalised coordintates q, of the associated velocities dq/dt (possibly of the time variable t * that leaves value of the Lagrangian unaffected * continous symmetry; symmetry with a continuous constant parameter, typically infiniteimal that can be dialled and that measures how far from the identity the transformation is bringing us * measures the size of the transformation

Answer 127

* e.g. Lagrangian for a free particle: L = T-V = q²/2m * examples of L invariant under infinitesimal translation * in space =\> total momentum conserved * in time =\> total energy is conserved * rotation =\> total angular momentum conserved

Answer 128

* classical transformations treat time and space differently * CM fails because it is not Lorentz invariant * special relativity treats time as a dimension (same footing as spatial dimension) and not just as a parameter * conservation of 4 vector total momentum

Answer 129

* charge conjugation * changing particles into anti-particles * parity inversion * reversing direction of each of the 3 spatial coordinates * time reversal * changing the direction of time not obvious whether laws of nature should look the same for any of these changes

Answer 130

Changes sign of * helicity * posn * velocity, momentum * accn, force * E field Leaves unchanged * time * spin * mass * energy * B field, magnetisation

Answer 131

* P(particle) = +1 * P(antiparticle) = -1 * P(meson) = P(q).P(anti q).(-1)^L = (-1)^L+1 * comes from P property of spherical harmonics * P(baryons) = P(q)(-1)^L1+L2+L3? = (-1)^L1+L2+L3 * P(photons) = -1

Answer 132

* that reactions governed by EM and strong interactions are invariant under P transform * parity conserved if Hamiltonian is invariant under parity operation (reversal of sign of all the coordinates; true for strong and EM but not weak) * binding of quarks into hadrons is dominated by strong with influence from EM and a vanishingly small effect from weak * hence parity is another QN useful in defining a particle * these 2 particles have same mass and lifetime, initially thought to be distinct ones since P was supposed to be conserved in all interactions * θ+ -\> π+π0 =\> P = (-1)²=+1 * τ+ -\> π+π0π0 and π+π-π+ =\> P = (-1)³ = -1 * but Yang and Lee (1956) suggested that it is the same particle that can decay in states of different parity =\> hence P not conserved in weak * Wu experiment * are A and B equally probable (can I count as many e- up as down (given direction for spin - direction of H; I don't know what this means)? * No; neutrinos are mainly left handed and antineutrinos are right handed * parity not conversed in weak interactions

Answer 133

* P(s) = s * e.g. Higgs boson * P(p) = -p * P(**v**) = -**v** * P(**a**) = -**a**

Answer 134

* C(particle) = antiparticle =\> C²(particle) = particle * C² = 1 =\> C = ±1 * some neutral particles can satisfy C(particle)= antiparticle = particle * e.g. π0 (C=1), γ (C=-1) not neutron, neutrino * Changes sign of: * electric charge, magnetic moment * baryon number * strangeness * charm * beauty * top * other internal quantum numbers * Leaves unchanged * spin * mass * energy * momentum * conserved in strong and EM, not weak

Answer 135

* decay of polarised muons (anti-muons) * observe angular distributions of e-s or e+s from decay in rest frame of decaying particle * if decay is * C-parity invariant =\> expect Γ+ = Γ- and ξ+ = ξ- * P-parity invariant =\> expect ξ± = 0 * measured Γ+ = Γ- and ξ+ = -ξ- * hence both C and P are violated in muon decay

Answer 136

* No since in kaon, charm and beauty mesons systems CP is violated in weak interactions at level of 10^-3 * CP is good symmetry only for EM and strong interactions

Answer 137

* momentum and spin direction

Answer 138

* CPT must be conserved in all interactions * e.g. particles and antiparticles have same mass and lifetime (CPT invariance theorem) * proving wrong by experiment =\> Nobel Prize * principle of detailed balance * A + B -\> C + D * C + D -\> A + B (time T reverseal) * happen at the same rate * would expect T to be violated in weak since CP is but CPT is not (though it is hard to measure)

Answer 139

* P and T violation due to EDM of the neutron * expect D to be 10^-15 * since neutron has radius of about 10^-13 * include charge and other effects * experiment shows neutron EDM \< 0.29\*10^-25 e-cm

Answer 140

* occur at about the same rate * only exception is if kinematically not allowed (energy not conserved etc.) * trivial, the theoretical calculations (probability to happen) are the same (e.g. revert arrows in Feynman diagrams and respect conservation of lepton, baryon etc.) * e.g. anti-v p -\> n e+ OK then * v n -\> p e- OK * anti-v n -\> p e- DOES NOT EXIST

Answer 141

* hadrons, B is defined to be * B = 1/3 [N(q) - N(anti-q)] * baryons have B = 1 * antibaryons have B = -1 * mesons have B = 0 * there is no operator (transformation) that implies the existence of conserved quantities lepton and baryon number * but no evidence is yet seen for total baryon or lepton number violation * e.g. in a process A + B + C ... -\> F + G + H ... * total L and total B are conserved, and in all interactions that have been tested * only exception is in the neutrino sector (since neutrinos are seen to oscillate)

Answer 142

* (b) * happens in the atmosphere when neutrinos propagate down to Earth and while this occurs, v_μ -\> v_e * only observed case of single lepton flavour number hanging

Answer 143

* p -\> e+ γ (does not happen) * matter-antimatter asymmetry & neutron oscillations

Answer 144

* p and n are 2 states of the same particle (nucleon); differ by isospin * pion is the strong interaction particle * flavour symmetry in strong interactions = hadrons in patterns can be obtained by rotating in flavour space (all different flavour states of the same particle) * =\> isospin is conserved in strong interactions * strong interaction between us and ds is the same * consider total angular momentum of bond state (including spin) and small quark mass differences * can explain masses and magnetic moments of hadrons quite well * isospin symmetry is not an exact symmetry of strong interactions (p and n are not identical except for their charge; what? they don't have the same charge) but a good approximate one

Answer 145

* Y = B + S hypercharge * I₃ = Q - Y/2 isospin * Q = electric charge * see if one can organise and find pattern =\> Eightfold way * why these patterns =\> quantum chromodynamics (QFT)

Answer 146

* strangeness (S) = **-** [N(s) - N(anti-s)] * charmness (C) = N(c) - N(anti-c) * beauty (B) = **-** [N(b) - N(anti-b)] * top (T) = N(t) - N(anti-t) * upness (U) = N(u) - N(anti-u) * downness (D) = **-** [N(d) - N(anti-d)]

Answer 147

* systematic method for expanding in the coupling constant * when Lagrangian is the sum of free particle and interaction terms * each interaction has a multiplicative factor ig (or iα) = coupling constant * expand interaction-field theory around free-field theory as a power series in ig * interaction conceived as a small perturbation on free theory =\> interaction can be calculated as a Taylor expansion in ig

Answer 148

* α^N/2 * or M² of order α^N

Answer 149

* ig_eγ_μ * γ_μ is same matrix seen in Dirac's eqn * g_e = √α_EM

Answer 150

* can measure α(Q) in ee -\> μμ scattering * α(Q) increases with Q (i.e. closer to bare charge; large R test charge sees screened e- charge)

Answer 151

* expressed in terms of g-factor μ = gSe/2m * where S = spin & g = 2 from first order diagram * a = anomalous magnetic moment = (g-2)/2 is predicte dto be different from 0 due to higher order diagrams

Answer 152

* only interaction changing flavour of quarks (i.e. changing one type of quark into another) * sees 3 families of falvour whereas photon interacts individually interacts with each charged particle * propagated by carrier particles (W, Z) that have significant masses, feature explained in Standard Model by Higgs mechanism * only interaction that violates P and CP symmetry (recall experimental evidence lecture 2, p11 and 15)

Answer 153

* projection of spin along direction of flight of a particle * not Lorentz invariant (can always find a reference system where LH looks RH) =\> only L. inv. if mass = 0 * helicity is commuting with H =\> a set of eigenstates for helicity is also a set of eigenstates for H * Dirac soln in base u1, u2 (v1, v2 for anti-particle) =\> Diract soln in base u↑, u↓ (v↑, v↓)

Answer 154

* interactions of e- and its associated neutrino should be identical to those of muon and its neutrino, or tau and its neutrino =\> meaning strength is the same or α_weak at W lepton neutrino vertex is identical

Answer 155

* Cabibbo mixing matrix couldn't explain why the measured decay rate of us du was much smaller

Answer 156

* only left-handed particles and right-handed antiparticles * helicity of * left-handed particles = -1 * right-handed particles = +1

Answer 157

* when QM calculation gives larger flu of W bosons than incoming flux of e-s/e+s

Answer 158

* electroweak symmetry * if we know sinθ_W from ratio of strength of W and Z interactions at low energy (G_Z/G_F) =\> predict Mw and Mz * Higgs particle * something to do with cross section and differential cross section

Answer 159

* LEP collider

Answer 160

* AFB = σ_F - σ_B/σ_F + σ_B * result of the fact Z couples to LH and RH chiral fermions differently

Answer 161

* 18 (but 19 apparently) * 9 fermion masses * 3 CKM mixing angles + 1 phase * 1 EM coupling constant * 1 strong coupling constant * 1 weak coupling constant * G_F = 1.16637\*10^-5 GeV^-2 * 1 Z⁰ mass * m_Z = 91.187621 GeV/c² * 1 Higgs mass

Answer 162

* global symmetry transformation; when sphere is simply rotated about its axis, shapes of lines are not changed * local symmetry transformation; lines of longitude and lattitude are twisted * shape of ballon does not change; remains invariant or is conserved in these transformations * Gauge principle is the requirement that U(1) phase invariance should be local * generated interaction between Dirac fermion and gauge field A; nothing else than the vertex of GED

Answer 163

* way to keep Lagrangian gauge symmetric (all particles apparently massless), and yet add mass terms for fermions and bosons * requires adding a new complex field - Higgs field (spin 0, I3W = 1/2)