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STATE UNIVERSITY OF NEW YORK AT STONY BROOK
DEPARTMENT OF PHYSICS AND ASTRONOMY
Part II.
Thursday, 2 September 1999 — Day 3
Comprehensive Examination in “Experiment” and “Breadth”
General instructions: Twelve problems are given. You should do any four, subject to the constraint
that you should answer no more than three from “experiment” and no more than three from
“breadth” (that is, not all four problems can be chosen from the same category). Each problem should
take about 34 hour and is worth twenty points. If a problem has subparts, each of these will be equally
weighted, unless indicated otherwise, with the sum totaling twenty points. Use one examination book
per problem and label it carefully with your name, the name of the problem’s author, and the date.
You may not use any materials other than this examination paper and the exam books supplied, a
calculator, and, with the proctor’s approval, a foreign language dictionary. None of these materials
may be shared between students.
“Experiment”
Experiment I. (Koch) [Note: problem continues onto next page.]
Apollo astronauts put a “corner cube” reflector array on the moon. They directed the array towards
earth so that light from pulsed lasers on earth could be aimed at it, retroreflected off it (i.e., antiparallel
to the incident direction), and sent back to earth for detection. That is what this problem is about.
Assume that the moon is L = 4 × 105 km from the earth. Consider on the moon one corner cube
retroreflector whose front face is a square = 0.1 m on a side. (Subject to the laws of (physical)
optics, the corner cube retroreflects any light entering it.) Let the laser on earth be a pulsed YAG
(yttrium-aluminum-garnet) laser, each pulse from which has a full-width-at-half-maximum duration
of ∆t1/2 = 5 ns and a total energy Ep = 1 J. The YAG laser wavelength is λ = 1.06 µm. Assume that
the output aperture of the YAG laser is a circle of diameter d = 0.1 m that is uniformly filled with
the laser light.
Useful constant: h = 6.63 × 10−34 Js.
a.
b.
c.
d.
e.
(3 points) How many photons are in one laser pulse?
(4 points) On earth, how would you measure the energy of the laser pulses? For this you have
available three different kinds of detectors: photomultiplier, photodiode, and thermopile. State
which one you would use and justify your answer. Draw a diagram of the apparatus you would
set up and use. Approximately what fractional precision (uncertainty in pulse energy divided
by pulse energy) would you expect to achieve with your apparatus?
(3 points) Make a numerical estimate (to a factor of 2 or better) of the the radius of the laser
beam when it arrives at the moon.
(3 points) Make a numerical estimate (to a factor of 2 or better) of the number of retroreflected
photons that arrive back at earth.
(3 points) Make a numerical estimate (to a factor of 2 or better) of the area of the “photon
receiver” (e.g., telescope) that is needed to get a detectable signal of, say, 10 photons, for each
laser pulse.
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f.
(4 points) On earth, how would you measure the “time of flight” for a laser pulse to travel from
the earth to the moon and back to the earth? What kind of photon detection system would
you use? Draw a diagram of the apparatus you would set up. Approximately what absolute
precision (in units of seconds) and what fractional precision (uncertainty in flight time divided
by the flight time) would you expect to achieve with your apparatus? Don’t just give numbers;
show how you did your calculation(s) and be sure to mention explicitly the important time scales
(e.g., rise- or fall-times) of all of the relevant devices/instruments in your diagram.
Experiment II. (Mihaly)
Nuclear magnetic resonance (NMR) is widely used in condensed matter physics, chemistry and
medicine. The measurement is performed on a collection of nuclear spins coupled to an external
magnetic field B by their magnetic moments µ. A resonant transition between the spin states can be
excited by electromagnetic radiation of angular frequency ω0 = 2µB/h̄ = γB , where γ is the ”gyromagnetic ratio”. The magnetization of the sample is M0 = µ∆n, where ∆n is the difference between
the number of spins parallel and antiparallel to the magnetic field, ∆n = nup − ndown .
In pulsed NMR various (short) radiofrequency pulses are used to kick the spin system out of thermal
equilibrium. The spin dynamics is often described in a quasi-classical approximation. The time
development of the average magnetization of the spins has a relaxation term, and it also includes a
term responsible for precession, dM/dt = M × B. It is convenient to introduce a frame of reference
rotating with an angular frequency ω0 around an axis parallel to the magnetic field. In this frame of
reference the full equation of motion becomes
dMz /dt = (M0 − Mz )/T1
dMx /dt = γMy b0 − Mx /T2
dMy /dt = −γMxb0 − My /T2 ,
where b0 = B − ω0 /γ and the z axis is along the magnetic field.
The NMR instrument picks up the weak electromagnetic signal radiated by the precessing spins, and
it measures a quantity that is proportional to the magnitude of the magnetization component in the
x-y plane.
a. Describe a possible way of detecting the spin resonance signal. It is difficult to pick up the signal
right after the application of the radiofrequency pulse (“dead time” problem). Why? What
would you do to reduce the dead time?
b. Estimate the fraction ∆n/n = (nup − ndown )/(nup + ndown ) for hydrogen (protons) in a magnetic
field of 1 Tesla at room temperature.
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c. If one of your samples has a resistivity of 10 Ωcm, how accurately can you determine its dielectric
constant? (Valid reasoning is more important than precise numerics.)
d. Assume T1 = T2 = 2 sec. What is the time dependence of the NMR signal in a perfectly
homogeneous magnetic field?
−6 over the sample volume. Estimate
e. Our B = 1 T magnet has an inhomogeneity of δB/B = 10
the decay time of the NMR signal.
(The proton gyromagnetic ratio is γ = 2.76 × 108 rad T−1 sec−1 . Planck’s constant is h̄ = 1.05x10−34 Js.
Boltzmann’s constant is 1.38 × 10−23 J/K.)
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Experiment III. (Jacak)
a.
b.
How would you measure the distribution of charge inside an O16 nucleus? State what beam and
target you would use and identify the useful range of beam energy.
The form factor F (q) is given by
F (q) =
d3 reiq r ρ(r).
·
where ρ is the charge density and q is the magnitude of the vector q.
Show that the first two terms in the expansion of F (q) in powers of q give the total charge
and charge radius of the nucleus respectively. Assume the charge distribution is spherically
symmetric.
Experiment IV. (Hobbs)
There is a study underway at the Fermi national accelerator labaoratory to determine the long range
plans of the laboratory. One possibility is the construction of a muon collider, an accelerator in which
counter-rotating muon beams intersect at one point.
The machine might have the following characteristics:
Center of mass energy, ECM : 200—500 GeV
Muons/bunch: 2 × 1012
Number of bunches: 2 per beam
Luminosity: 1 × 1032 cm−2s−1, at ECM = 200 GeV
7 × 1032 cm−2s−1, at ECM = 500 GeV
a. Why build a muon collider with a center—of—mass energy lower than that of the existing Fermilab
proton/antiproton collider (which has ECM = 2000 GeV)?
b. This would be a new technology. Discuss why this machine would not use proven electronpositron collider techniques.
−6 s. For the
c. A disadvantage of a muon collider is that muons have a finite lifetime, τ = 2.2 × 10
parameters given above, what is the number of muon decays/sec?
d. Do the muon decays cause problems for experiments? If so, describe the problems and describe
methods for preventing them.
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Experiment V. (Walter)
You want to obtain a spectrum of a star with an R magnitude of 15. The R band is centered at 6400 Å
and has a flux λFλ = 1.44 × 10−5 erg cm−2 s−1 at R = 0.
You propose to use the KPNO (Kitt Peak National Observatory) 4-m echelle spectrograph. It has a
resolution ∆λλ =40,000. The efficiency of the slit plus spectrograph plus detector is 4% (that is, 4%
of the photons hitting the primary mirror are recorded). The slit width corresponds to 0.5 arcsec on
the sky. The slit length is 10 arcsec, and the seeing is 1 arcsec. The CCD (charge-coupled-device)
detector has 2 pixels per resolution element in the dispersion direction, and 3 pixels per arcsecond in
the other direction. For convenience of calculation, assume that any profile is a top-hat (e.g., take the
stellar image to be a uniformly lit square with 1 arcsecond sides, the profile of an unresolved line to
be constant within a width of one resolution element and zero outside, etc.).
a. (10 points) You want to detect an unresolved absorption line of equivalent width 0.1 Å. How
long do you need to integrate to detect that line with a S/N (signal-to-noise) ratio of 10? Assume
no sky background and a CCD read noise of 3 electrons rms.
b. (6 points) If you can measure the position of the line to 0.25 resolution elements, with what
accuracy can you determine the radial velocity and proper motion?
c. (4 points) How long do you need to integrate to measure the continuum with S/N=30 per
resolution element? Assume a (continuum) sky background of 20 mag/sq. arcsec and 3 electrons
read noise. Use the flux from the ends of the slit for sky subtraction.
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Experiment VI. (Peterson)
The Space Interferometry Mission (SIM) proposes to launch a Michelson interferometer, basically a
Young’s two slit experiment. In this case the apertures (mirrors) are 0.3 m in diameter, separated
by 10 m. The detectors are CCDs (charge-coupled-devices) whose sensitivities (plus the effect of 12
reflections off silvered surfaces) limit the spectral range effectively to 500 — 1000 nm. With these
parameters, answer the following:
a. (5 points) The Airy disk for the individual mirrors defines an angular region on the sky that is
viewed at any instance. What is the size (diameter) of this region at 500 nm? At 1000 nm?
b. (5 points) The interference of the beams from the two mirrors creates a fringe pattern modulating
the Airy disk. At 500 nm what is the fringe width (peak to peak)? How many fringes are there
across the Airy pattern? Answer these two questions for 1000 nm.
SIM is designed to make exquisitely accurate measurements that will provide positions accurate to a
few µarcsec (1 µarcsec ≈ 5 × 10−12 radians) around the sky. This will allow distances to be measured
by triangulation to essentially any star in the Milky Way. Tricks may allow its reach to be extended
well beyond that – one such trick involves nearby spiral galaxies.
Normal spiral galaxies, including the Milky Way, are observed to exhibit sensibly circular rotation.
Moreover, the rotation velocity rises rapidly close to the center and then becomes nearly constant over
a range of many kiloparsecs, at least to the visible edge of the galaxy. SIM proposes to use that fact to
determine the distance to a number of nearby galaxies. The following outlines roughly the situation
for M31, the Andromeda galaxy.
c.
d.
(5 points) M31 is expected to show measurable angular velocities of individual stars due to
rotation. The (relatively flattened) system is observed with a relatively large inclination, approaching 80◦ . Bright stars, observable along the apparent minor axis, will show transverse
angular velocities of about 73 µarcsec/yr (microarcseconds per year – note that 1 km s−1 at a
distance of 1 pc gives 0.21094 arcsec/yr). Those stars at the end of the major axis will show
16.4 µarcsec/yr and radial velocities of 260 kms−1 . What is the total circular velocity for the
system in these outer regions, the actual inclination, and the distance (in Mpc)?
(5 points) If this flat rotation law is seen to hold along the major axis out to 1◦ from the center,
what is the system’s approximate mass (ignore corrections for flattening)? How would that
change if we discovered a star at twice that distance from the center showing the same motion?
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“Breadth”
Breadth I. (Bergeman) *******PUT IN FIGURE LABEL THE CURVES a,...,h****
The plot shows as a function of a parameter |χ|, which is proportional to a static magnetic field B,
the energy levels of an atomic 2S1/2 state (in L-S coupling notation). The labels a,. . . ,h shown are
non-physical, i.e., arbitrary.
a. In the limit of B = 0, what are the “best” quantum numbers for each of the levels shown?
b. What quantum numbers best describe each of the levels in the limit of large B? (Be sure that
you describe what “large” B means physically.)
c. For the atom in this problem, what is I, the nuclear spin quantum number?
d. Write down and explain the physical meaning of the Hamiltonian (perturbation) interaction that
is responsible for the splitting |∆W | at B= 0.
e. Magnetostatic atom traps have a mininum of the magnetic field at their center. What simple
coil configuration can be used to produce such a minimum? Atoms in which of the above states
can be confined in such a trap?
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Breadth II. (Mendez) ********************PUT IN DIAGRAM*****************
a.
b.
c.
d.
e.
Derive the classical formula for the Hall resistance where, as shown in the figure, a slab-shaped
conducting sample carries a current I parallel to its long axis. The slab’s thickness and width
are t and w, respectively, and the magnetic field B is applied perpendicular to the sample. The
conductor contains a volume density N of free carriers with charge −e. Show that transverse
to the current there exists a voltage of magnitude VH = IB/N te, giving a Hall “resistance”
RH = B/N te.
Taking the view that a two-dimensional (2D) electron gas is really just a thin three-dimensional
one, obtain an expression for the 2D classical Hall resistance in terms of the 2D electron density
NS (= “areal density”).
In a quantum-mechanical treatment of the 2D gas, the electrons are quantized in energy levels
(Landau levels). The degeneracy of each level is equal to the number of flux quanta per unit
area. Express the level degeneracy in terms of B, e, and h. (Hint: if you do not remember
the definition of flux quantum, use dimensional analysis to write it in terms of the fundamental
constants e and h.)
Let us consider a 2D electron gas with areal density NS under a magnetic field B at which
there are exactly n Landau levels fully occupied. Expressing NS in terms of n, B , e, and h, and
using the result obtained in part b., write an expression for RH in terms of n and fundamental
quantities. It is observed experimentally that for such a magnetic field the resistance is quantized.
(Quantum Hall effect)
Make a qualitative plot of the experimental observation in a very high mobility 2D electron gas
of VH versus B, from B = 0 up to the quantum limit, that is, fields at which the the number of
occupied Landau levels is less than n = 1.
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Breadth III. (Kuo)
The nuclear shell model is a very successful model for nuclear structure. This model assumes that the
nucleons in a nucleus are independent particles moving in a common harmonic oscillator well, with a
small spin-orbit splitting.
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17 (Z=8) and Ca41 (Z=20) especially simple? Write down the shell
a. Why are He (Z=2), O
model wave functions for the ground states of these nuclei, giving the spin (J), orbital angular
momentum (l), and parity (P) for each of them ?
The shell model Hamiltonian is written as
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H = C + 2pm + 21m r2ω2 + δ(l · s)
where C and δ are constants; l and s are orbital angular momentum and spin angular momentum,
respectively; and m is the nucleon mass. For the following questions, please show your derivations in
detail in order to get full credit. (Use mc2 = 938 MeV and h̄c = 197.3 MeV·fm.)
b.
c.
Suppose the 3/2-5/2 spin-orbit splitting of O17 is 5 MeV. What value for δ one should use in
order to reproduce this splitting?
The operator for magnetic moment is given as µ = g l + g s. For the proton, g = 1 and g = 5.58;
for the neutron, g = 0 and g = −3.82. Calculate the magnetic moments of O17 and F17 (Z=9).
Suppose the rms (root-mean-square) radius of O16 at its ground state is 3.1 fm. To reproduce
this radius, what value of ω should one use for the above shell model Hamiltonian? (Hint: use
the virial theorem, which for the oscillator potential V reads < V >=< E > /2.)
l
l
d.
s
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s
l
s
Breadth IV. (Grannis)
The Standard Model unification of the weak and electromagnetic interactions employs Yang-Mills
fields for massless bosons
(V + , V 0 , V − ) and B 0
that transform as a triplet under SU(2) and singlet under U(1), respectively. However, the short range
of the weak force implies that the actual carriers of the weak interaction are massive. Indeed, in 1983
the W and Z bosons were discovered at CERN with the masses 80 and 91 GeV, respectively.
The Standard Model says that a spontaneous symmetry breaking causes the transformation of (V + , V 0 , V − )
and B 0 into W + , W − , Z 0 , γ through the introduction of scalar Higgs bosons.
a. Give an example of spontaneous symmetry breaking in another branch of physics.
b. What is the minimum number of Higgs bosons that must be put into the electroweak interaction?
How many survive the spontaneous symmetry breaking to be candidates for real, observable,
new particles?
c. What experimental results have been obtained to limit the possible mass of the Higgs boson?
(One such experiment or set of results is sufficient.)
d. Give an example of a phenomenon that should be observed in the unified electroweak interaction
that differs from what we would expect without electroweak unification.
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Breadth V. (Solomon)
We investigate the strength and excitation of a molecular line in an interstellar cloud. Note that
−27 erg·s, k = 1.38 × 10−16 erg/K, and mH = 1.67 × 10−24 g.
h = 6.63 × 10
a. (6 points) The term critical density is used to describe the physical condition necessary in an
interstellar molecular cloud for the thermal excitation (and presence of a strong thermal emission
line) of a rotational transition from a particular molecule. Define critical density in terms of
molecular parameters and parameters of the cloud for a transition with energy E = hν at
kinetic temperature Tkin . (Be specific and define all terms; for simplicity assume that there are
only 2 levels.)
b. (5 points) Estimate the critical density for a 2-level molecule (say, with J = 0, 1 rotation levels)
with a spontaneous transition rate A10 = 10−7 s−1 in a cloud with a kinetic temperature of 20
K. (Make any assumptions necessary about other molecular parameters.)
c. (4 points) The critical density may in reality be different if the above spectral line has high
optical depth. What happens at high optical depth? Give an approximate expression for the
critical density in the escape probablility approximation for an optical depth τ01 1.
11 Hz.
d. (5 points) Assume that the frequency of the transition is exactly 100,000 MHz or 10
Calculate the optical depth τ through a cloud with a total velocity width of 1 km/s, a
density of absorbing molecules of 0.1 cm−3 , and a path length of 3 pc (1019 cm). Assume that
the line profile is rectangular, the levels J = 0 and J = 1 are the only levels, and the excitation
temperature is 20 K. A10 = 10−7 s−1 . (You need to determine the absorption cross section and
allow for stimulated emission to calculate the optical depth.)
ij
c3 A , where B must be multiplied by the total
Remember that the Einstein B coefficient B = 8πhν
3 ji
c2 A if the
energy density per unit frequency interval to give the transition probability, or Bji = 2hν
3 ji
specific intensity of radiation is used.
ji
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Breadth VI. (Lanzetta)
We investigate “damped Lyman-α” absorption systems, which are detected as neutral hydrogen absorption lines in spectra of quasi-stellar objects (QSOs). They are of particular interest because they
contain most of the mass density of neutral hydrogen in the Universe. We will determine their number
and mass density from the observed number of times our lines of sight to distant QSOs go through
them.
It is generally believed that the absorption systems are produced by the gas disks of normal spiral
galaxies, of characteristic radius r = 15 h−1 kpc (where h is the dimensionless Hubble constant,
defined by H0 = 100 h km s−1 Mpc−1 ). At redshift z = 3, the number density per line of sight of the
absorbers is n ≈ 0.2 per unit redshift interval, and the characteristic neutral hydrogen column density
of the absorbers is N ≈ 1021 cm−2 .
Adopt a standard Friedmann cosmological model with deceleration parameter q0 = 0.5 (i.e. a flat
universe with zero cosmological constant), for which cosmic time is related to redshift by
2
−3/2. Assume that the objects causing the damped Lyman-α lines are fixed in comoving
t =
3H0 (1+ z )
coordinates. Then calculate
a. (5 points) the proper and comoving number densities of the damped Lyman-α systems at z = 3,
b. (5 points) the proper and comoving number densities of the damped Lyman-α systems at z = 0,
c. (5 points) the fraction of the sky covered by the damped Lyman-α systems between z = 3 and
3.1, and
d. (5 points) the comoving mass density of neutral hydrogen contained in the damped Lyman- α
systems.
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