NuclearNuclear andand ParticleParticle PhysicsPhysics 11 (PHY211/281)(PHY211/281) HerbstsemesterHerbstsemester 20192019 OlafOlaf SteinkampSteinkamp 36-J-05 [email protected] 044 63 55763 Units We are dealing with the extremely small, e.g. typical length scale is “femtometer” 1 fm = 10–15 m KT1 HS19 Basics (2/21) O. Steinkamp ElectronVolt (eV) SI units not very well suited, e.g. → Electron charge e = 1.6 × 10–19 C –27 → Proton rest mass mp = 1.67 × 10 kg 2 –10 → Proton rest energy Ep = mp c = 1.5 × 10 J Define “electronVolt” (eV) as our basic unit of energy: energy gained by an electron moved across a potential of 1 V 1 eV = 1.6 × 10–19 J → Binding energy of electrons in atoms few eV → Proton rest mass 938 MeV (MeV = 106 eV) KT1 HS19 Basics (3/21) O. Steinkamp Natural Units Derived units Observable Unit in SI units Energy eV 1.60 × 10–19 J Mass E = mc2 eV / c2 1.78 × 10–36 kg Momentum E = pc eV / c 5.34 × 10–28 kg m/s Time ΔE Δt ≥ ℏ ℏ / eV 6.58 × 10–16 s Distance Δx Δp ≥ ℏ ℏc / eV 1.97 × 10–7 m KT1 HS19 Basics (4/21) O. Steinkamp Natural Units Often use “Natural units”, where c ≡ 1 and ℏ = h ≡ 1 2 π Natural Observable Unit in SI units unit Energy eV eV 1.60 × 10–19 J Mass E = mc2 eV / c2 eV 1.78 × 10–36 kg Momentum E = pc eV / c eV 5.34 × 10–28 kg m/s Time ΔE Δt ≥ ℏ ℏ / eV 1 / eV 6.58 × 10–16 s Distance Δx Δp ≥ ℏ ℏc / eV 1 / eV 1.97 × 10–7 m → Energy, mass and momentum have the same unit → Time and distance have the same unit KT1 HS19 Basics (5/21) O. Steinkamp Natural Units Often use “Natural units”, where c ≡ 1 and ℏ = h ≡ 1 2 π Natural Observable Unit in SI units unit Energy eV eV 1.60 × 10–19 J Mass E = mc2 eV / c2 eV 1.78 × 10–36 kg Momentum E = pc eV / c eV 5.34 × 10–28 kg m/s Time ΔE Δt ≥ ℏ ℏ / eV 1 / eV 6.58 × 10–16 s Distance Δx Δp ≥ ℏ ℏc / eV 1 / eV 1.97 × 10–7 m → Express time and distance through units of energy, e.g. 1 fm ≈ 1 = 5 GeV−1 200MeV KT1 HS19 Basics (6/21) O. Steinkamp Natural Units Often use “Natural units”, where c ≡ 1 and ℏ = h ≡ 1 2 π Natural Observable Unit in SI units unit Energy eV eV 1.60 × 10–19 J Mass E = mc2 eV / c2 eV 1.78 × 10–36 kg Momentum E = pc eV / c eV 5.34 × 10–28 kg m/s Time ΔE Δt ≥ ℏ ℏ / eV 1 / eV 6.58 × 10–16 s Distance Δx Δp ≥ ℏ ℏc / eV 1 / eV 1.97 × 10–7 m → useful relation: ℏ c ≈ 200MeV⋅fm KT1 HS19 Basics (7/21) O. Steinkamp Electric charge Electric charge defined through Coulomb force e2 F C = 2 4 π ϵ0 r Use “Heaviside-Lorentz” units with ϵ0 ≡ 1 → Dimension of electric charge [q] = [F⋅r 2]1 /2 = [E⋅r ]1/2 = √ ℏ c → Dimensionless in natural units Fine-structure constant 2 2 α ≡ e = e = 1 4 π ℏ c 4 π 137 KT1 HS19 Basics (8/21) O. Steinkamp Some more numbers Binding energy of nucleons in nuclei few MeV (MeV = 106eV) Binding energy of quarks in nucleons few GeV (GeV = 109eV) Neutrino rest masses < 0.1 eV Electron rest mass 511 keV Rest mass of top quark 173 GeV Mass of 208Pb nucleus/atom 194 GeV Rest mass of W Boson 80 GeV Rest mass of Z Boson 91 GeV Rest mass of Higgs Boson 125 GeV Collision energy in LHC 13 TeV (TeV = 1012eV) Highest energy cosmic ray measured so far 300 EeV (EeV = 1018eV) KT1 HS19 Basics (9/21) O. Steinkamp KT1 HS19 KT1 Basics ( Basics Prefixes 10 / 21 ) O. Steinkamp O. [https://physics.nist.gov/cuu/Units/prefixes.html] Radioactive Decays Can happen when the sum of masses of produced particles is smaller than mass of decaying particle Some examples: Nuclear decays Decays of instable particles – – X* → X + γ π → μ νμ A A – π0 → γγ Z X → Z+1 Y + e + νe ρ0 → π+ π– A A + Z X → Z-1 Y + e + νe A A-4 4 Z X → Z-2 Y + He There are other rules besides energy conservation, we’ll discuss some of these in later lectures ... KT1 HS19 Basics (11/21) O. Steinkamp Radioactive Decays Constant probability λ for a single nucleus/particle to decay For an ensemble of N particles: d N (t ) = −λ N (t ) dt N(t ) = N (t=0) e−λ t Mean lifetime Halflife τ = 1/ λ t 1/2 = τ ⋅ln2 Intensity of a given source = mean number of decays per time dN (t ) I(t) = − = N (t=0) λ e−λ t = I (t=0) e−λ t dt KT1 HS19 Basics (12/21) O. Steinkamp Radioactive Decays The number of decays measured in a fixed time interval Δt is a random number that follows a … Poisson distribution μ n P(n ∣μ) = e−μ n! with expectation value μ = λ ⋅Δt Standard deviation σ of the Poission distribution (= uncertainty on the measurement) σ = √ μ ≈ √ n KT1 HS19 Basics (13/21) O. Steinkamp Lifetime and Decay Width Stable particles have a well-defined mass m , e.g. me = 511 keV But for short-lived particles with finite lifetime τ ΔE Δt ≥ ℏ implies that the mass of the particle has an uncertainty Γ = ℏ / τ The probability distribution for the mass is described by a relativistic Breit-Wigner distribution f (m) ∝ 1 short-lived particles (m2−M 2)2 + M2 Γ2 are also called ”resonances” M is the mean mass of the particle Γ is called the decay width KT1 HS19 Basics (14/21) O. Steinkamp Lifetime and Decay Width Plays a role only for very short-lived particles, e.g. for Z boson: for muon: τ ≈ 3 × 10–25 sec τ ≈ 2.2 × 10–6 sec Γ ≈ 2.5 GeV Γ ≈ 3 × 10–19 GeV ( Γ / m ≈ 2 % ) ( Γ / m ≈ 3 × 10–18 ) Usually quote decay width for very short-lived particles and quote lifetime for longer-lived particles, e.g. from Particle Data Group: KT1 HS19 Basics (15/21) O. Steinkamp Z boson decay width Determination of the number of neutrino species at LEP: Measure the width of the Z mass distribution → width depends on lifetime → lifetime depends on the number of different possible decay modes → number of decay modes depends on the number of neutrino species Measured width agrees well with three types of neutrinos, i.e. three generations of elementary particles KT1 HS19 Basics (16/21) O. Steinkamp Special Relativity: Four-Vectors (Contravariant) four-vector aμ ≡ (a0 ; a1 , a2 , a3 ) Example: space-time xμ = (t ; x , y , z) = (t ; x⃗ ) Example: energy-momentum μ p = (E ; p x , p y , p z) = (E ; p⃗ ) ( using natural units with c ≡ 1 ) KT1 HS19 Basics (17/21) O. Steinkamp Special Relativity: Four-Vectors Covariant four-vector ν 0 1 2 3 aμ ≡ (a0 ; a1 , a2 , a3 ) = gμ ν a = (a ;−a ,−a ,−a ) with the metric tensor 1 0 0 0 g ≡ 0 −1 0 0 μ ν 0 0 −1 0 ( 0 0 0 −1 ) Using “Einstein notation”, summation over repeated indices is implied 3 ν ν gμ ν a ≡ ∑ gμ ν a ν=0 KT1 HS19 Basics (18/21) O. Steinkamp Special Relativity: Four-Vectors Scalar product of two four-vectors μ μ ν 0 0 1 1 2 2 3 3 a bμ = a gμ ν b = (a b −a b −a b −a b ) μ μ aμ b = a bμ Norm of a four-vector 2 μ μ 0 0 1 1 2 2 3 3 | a| = a aμ = aμ a = (a a −a a −a a −a a ) KT1 HS19 Basics (19/21) O. Steinkamp Special Relativity: Derivative Contravariant derivative μ ∂ ≡ ∂ = ∂ ; − ∂ , − ∂ , − ∂ = ∂ ; −∇⃗ ∂ x μ ( ∂ t ∂ x ∂ y ∂ z ) ( ∂ t ) Covariant derivative ∂ ∂ ∂ ∂ ∂ ∂ ⃗ ∂μ ≡ = ; + , + , + = ; ∇ ∂ x μ ( ∂ t ∂ x ∂ y ∂ z ) ( ∂ t ) Derivative of a four-vector aμ 0 μ ∂ a ∂ a = + ∇⃗⋅a⃗ μ ∂ t In particular: 2 2 2 2 2 μ ∂ ∂ ∂ ∂ ∂ ⃗ 2 μ ∂μ ∂ = − − − = −∇ = ∂ ∂μ ( ∂ t 2 ∂ x 2 ∂ y 2 ∂ z2 ) ∂ t 2 KT1 HS19 Basics (20/21) O. Steinkamp Lorentz Transformation Consider a reference frame that moves at relative speed β = v/c along the x-axis Four-vectors transform from the rest frame to this moving reference frame as μ μ ν x ' = Λ ν x with γ −β γ 0 0 Λμ ≡ −β γ γ 0 0 ν 0 0 1 0 ( 0 0 0 1 ) and γ = 1 √ 1−β2 KT1 HS19 Basics (21/21) O. Steinkamp.