LECTURE 19 REVIEW FOR MIDTERM1 PHY492 Nuclear and Elementary Particle Physics

LECTURE 19
REVIEW FOR MIDTERM1
PHY492 Nuclear and Elementary Particle Physics
Overview
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The midterm exam on Monday Feb 24 will cover the material in class up to Feb 21, MarCn Chs 1-­‐4, the quizzes and the homework sets. –  I intend to include: • 
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At least 1 problem from the quizzes. At least 1 problem from the homework sets At least 1 problem from MarCn These will not be idenCcal in wording/soluCon, but idenCcal in the concepts. This document gives you a sample of problems which I think are reasonable in Cme/
difficulty for the exam (individually, not in total). –  Do not assume that it gives you the total breadth to study. I didn’t cover every single concept in this review example. –  Do assume that I will expect that you at least understand the concepts in these problems, and that this understanding will benefit you greatly on the exam! • 
I do not expect you to memorize parCcle decays, parCcle properCes, details like ionizaCon energies, etc. But you should be able to make qualitaCve descripCons of the properCes based on the physics involved. • 
The exam will be closed book, closed notes, closed cell phone, closed laptop, etc. February 21, 2014 PHY492, Lecture 19 2 Recall
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Chapter 1: Intro – 
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Chapter 2: Nuclear Phenomenology – 
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RelaCvity Symmetries & ConservaCon Laws Feynman Diagrams Forces Cross secCons, Rutherford sca\ering Mass spectroscopy Nuclear shape & size Semi-­‐empirical mass formula Nuclear instability & decays Chapter 3: ParCcle Phenomenology –  Quarks & leptons –  Hadrons –  More on Feynman diagrams • 
Chapter 4: Experimental Methods – 
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Accelerators ParCcle interacCons with ma\er ParCcle detectors MulC-­‐component detector systems February 21, 2014 PHY492, Lecture 19 3 Hadron Example
February 21, 2014 PHY492, Lecture 19 4 Intro Example
Use the uncertainty principle to esCmate the minimum kineCc energy of an electron confined within a nucleus of width 10 fm. Assume the electron is fully relaCvisCc. February 21, 2014 PHY492, Lecture 19 5 Nuclear Example
What is the half-­‐life of the 0.17 eV resonance in 113Cd if it has a total width of 0.133 eV? Compare this with the collision Cme for a 0.17 eV neutron, which is approximately the Cme it takes for the neutron to pass through the nucleus. Take the nuclear radius to be R=1.2 A1/3 fm. February 21, 2014 PHY492, Lecture 19 6 Particle Example
What is the length of the longest drih tube in a linac, which, operaCng at a frequency of f=50 MHz, is capable of acceleraCng protons to a maximum energy of E=300 MeV? February 21, 2014 PHY492, Lecture 19 7 Particle Example
Show that the momentum p in GeV/c for a relaCvisCc parCcle of charge q moving in a circular orbit of radius R meters in a uniform magneCc field of B teslas is given by p = 0.3*q*B*R. February 21, 2014 PHY492, Lecture 19 8 Nuclear Example
Calculate the velocity of recoil electrons when light of wavelength 1 micron is Compton sca\ered through 90 degrees. Are the electrons relaCvisCc? February 21, 2014 PHY492, Lecture 19 9 Relativistic kinematics
Relativistic kinematics :
relativistic
momentum
non-relativistic
P
(total) energy E
kinetic energy T
E Useful relation
February 21, 2014 PHY492, Lecture 19 10 Relativity
Lorentz transformation:
Consider a particle of rest mass m with its co-ordinates
(ct,x,y,z) in a frame S. If a second frame S’ is moving
with speed v in the z direction, its co-ordinates
(ct’,x’,y’,z’) in the S’ frame become;
S’ y
S v
z
Lorentz Invariance:
February 21, 2014 PHY492, Lecture 19 11 x
Relativity (2)
Lorentz transformation for energy and momentum :
A four-vector for the momentum P and the energy E
is defined as (E/c, Px, Py, Pz)
cf. (ct,x,y,z)
Lorentz Invariance:
The following values are invariant constants
- the rest frame mass Mc2 (= [E2 – P2c2]1/2)
- the invariant mass “s” (squared) for 2 particle system
(Ea/c, Pa), (Eb/c, Pb)
s = [(Ea+Eb)2 – (Pac+Pbc)2]/c4
February 21, 2014 PHY492, Lecture 19 12 Ch 1 review
Consider a parCcle of rest mass m in a frame S, with its co-­‐ordinates (t,r)=(ct,x,y,z) and its speed u along the z direcCon. In a second frame S’, its co-­‐ordinates are (t’,r’)=(ct’,x’,y’,z’) and its speed is u’. S and S’ coincide at t=0 and S’ is moving with uniform speed v in the posiCve z direcCon with respect to S. (a) Write the Lorentz transformaCon between (t,r) and (t’,r’) (b) Show that the parCcle’s speed u’ in the S’ frame is related to its speed u in the S frame by Note that u is given by dz/dt (for u’, dz’/dt’). y
S S’ v
x
z
February 21, 2014 PHY492, Lecture 19 13 Partity transformation - example
Spherical polar co-ordinate (x,y,z) → (r,θ,φ)
rcosθ X = r sinθ cosφ
Y = r sinθ sinφ
Z = r cosθ
r →
θ →
φ →
r
π - θ
π + φ
r θ In this co-ordinate, r → -­‐ r means; Z rsinθcosφ φ rsinθ rsinθsinφ X Y It can be shown for sperical harmonics function that
^ P Ylm(θ,φ) = (-1)l Ylm(θ,φ)
February 21, 2014 PHY492, Lecture 19 14 Ch 1 review
Prove the following parity transformaCon for spherical harmonic funcCons Yl,m(θ,φ), The funcCon Yl,m(θ,φ) (l,m: integer) is defined as;
February 21, 2014 PHY492, Lecture 19 15 Rutherford Scattering (3)
differential cross section dσ
dΩ
J 2πb |db|
J
(flux) dσ
2π sinθ dθ
dΩ
dσ db b
dΩ
= sinθ
dθ
= J
b
2πb·db
zZe2
1
b = 8πε0 Ekin cot(θ/2) zZe2
θ
useful relation (fine structure constant)
2πsinθdθ
e2
1
α=
(SI)
db 2(θ/2)
4πε0hc
=
cosec
dθ
(text book) 16πε0 Ekin
=
e2
hc
(cgs) =
1
137
zZe2
1
2
dσ
4(θ/2) =
(
)
cosec
dΩ
16πε0 Ekin
February 21, 2014 PHY492, Lecture 19 16 Ch 1 review
February 21, 2014 PHY492, Lecture 19 17 Ch 1 review
February 21, 2014 PHY492, Lecture 19 18 Ch 1 review
February 21, 2014 PHY492, Lecture 19 19 Ch 2 review
February 21, 2014 PHY492, Lecture 19 20 HW2-2 review
February 21, 2014 PHY492, Lecture 19 21 Activity
Intensity of decays (activity)
dN
A = = λN ;
dt dN
= (- λ )dt N N number of nuclei
λ decay constant
N (t) = N0 exp( - λt ) Decay probability λ is constant in time and equal for all mother (initial) nuclei. Units
Bq (becquerel) = 1 decay per second
Ci (Curie)
= 3.7 × 1010 Bq February 21, 2014 PHY492, Lecture 19 22 Ch2 review
February 21, 2014 PHY492, Lecture 19 23 Ch 2 review
February 21, 2014 PHY492, Lecture 19 24 Mass Formula
Semi-empirical Mass Formula is given by ;
M(Z,A) = Z · (Mp + me) + N · Mn
corresponds
to “ – B/c2 “ -15.56 A + 17.23
A2/3 +
1 2 ((Z-N)/2)2
+ 93.14
A
4 Z2
0.697
A1/3
+Δ
3 5 1 Volume term; each nucleon feels the effect of the nucleons surrounding it 2 Surface term; Volume term overestimates the effect at the surface 3 Coulomb term; Repulsive force among the charged particles (protons) 4 Asymmetry term;
5 Pairing term;
February 21, 2014 Nuclei tend to have N = Z - 12A-1/2 for even-even nuclei, ; + 12A-1/2 for odd-odd nuclei,
0 for others PHY492, Lecture 19 25 Ch 2 review
February 21, 2014 PHY492, Lecture 19 26 Leptons and Quarks in the Standard model
Leptons :
Quarks :
elementary particles
in the standard model
All have the spin of 1/2 strongly interacting particles
fundamental constituents of matter,
but cannot be detected directly
(hadron : meson, baryon) GeneraCons (flavors) 1 2 3 parCcles ( e- ) ( µ- ) ( τ- ) ( ) ( ) ( ) νe
νµ
ντ
u
d c
s t
b u
d c
s t
b ( e+ ) ( µ+ ) ( τ+ ) ( ) ( ) ( ) AnC parCcles νe
February 21, 2014 νµ
ντ
PHY492, Lecture 19 27 Feynman Diagram
Rules for Feynman Diagrams
1. Initial state (left) and final state (right)
2. Fermions (electrons) are drawn as solid lines
3. Bosons (photons) are represented by wiggly (broken) lines
4. Arrowhead pointing to the right (left) indicate particles (anti particles)
does not indicate the particle’s direction of motion
5. Lines that end at boundaries are free particles
6. Energy (or momentum) is conserved at a vertex
e- e- 7. Charge is conserved at a vertex
γ e- February 21, 2014 PHY492, Lecture 19 e- 28 Feasibility of reactions
Check the following conservation laws;
- Q (charge) must be conserved
- Lepton numbers Leµτ must be conserved
- Quark number Nq (Baryon number B) must be conserved
- Each Nf (f=u,d,s,c,b,t) must be conserved
for strong, electromagnetic interactions
- Mass, energy, momentum conservation. (a) Σ0 → Λ + γ
(b) p + p → Σ+ + n + Κ0 + π+
(c) Ξ- → Λ + π-
(d) Δ+ → p + π0
February 21, 2014 PHY492, Lecture 19 29 Ch 3 review
February 21, 2014 PHY492, Lecture 19 30 Ch 3 review
February 21, 2014 PHY492, Lecture 19 31 Charge independence of nuclear forces
Flavor independence of the strong forces
between u and d quarks leads directly the
charge independence of nuclear forces.
charge symmetry p equals to
p
n n
Structure of
Structure of
AZN
is similar to
11
5 B6
ANZ
11C5
6 (mirror nuclei) February 21, 2014 PHY492, Lecture 19 32 Ch 3 review
February 21, 2014 PHY492, Lecture 19 33 Lifetime
Lifetime of hadrons has a close relation to the interaction
involved in the decay. interaction
K*+(us) → K0(ds) + π+(ud) π0(uu,dd) → γ + γ
strong interaction
1.3×10-23 s electromagnetic
interaction
0.8×10-16 s π+(ud) → µ+ + νµ
weak interaction
n(udd) → p(uud) + e- + νe weak interaction
February 21, 2014 lifetime PHY492, Lecture 19 2.6×10-8 s 103 s 34 Ch 3 review
February 21, 2014 PHY492, Lecture 19 35 Ch 3 review
February 21, 2014 PHY492, Lecture 19 36 Ch 4 review
A charged parCcle loses a kineCc energy of about 1 MeV while traversing a counter. The energy loss W required to produce the electron-­‐hole pair depends on the type of detector used. Typical values of W are 100-­‐300eV for scinCllators, 10-­‐30eV for gas detectors, and 3eV for semiconductor detectors, respecCvely. (a) Calculate the number of electron-­‐ion pairs for each detector. (b) Discuss the expected energy resoluCon for each detector. February 21, 2014 PHY492, Lecture 19 37