Transcript Inside the proton

16.451 Lecture 4:
16/9/2003
1
Inside the Proton
• What we know about the proton’s internal structure comes from scattering experiments
• Experiments at SLAC (Stanford Linear Accelerator Centre) in the 1960’s and 70’s found
that the proton has an extended electric charge distribution:
 (r )  e o exp(  M r )
M  4.33 fm 1
 / e fm3
r2
1/ 2

12
 0.80 fm
M
• Nobel prize, 1990 to Friedman, Kendall and Taylor (Cdn!) for deep inelastic scattering
experiments that showed the existence of pointlike constituents inside the proton:
http://www.nobel.se/physics/laureates/1990/illpres/
Overview: Electron scattering experiments:
2
(Ref: Krane 3.1)
electron scatters from the proton’s electric charge distribution (r)
detector
Before
e

p
After
e

po

proton
Scattering rate is determined
by the cross-section:

q

d
d 
2
( ) 
 F (q )
d
d  o
point charge result (known)
1
  (r ) 



2 3
 e 

e

 iq.r
F (q 2 ) d 3 q

2
“Form factor” gives Fourier
transform of extended charge
distribution
3
What it looks like: proton electric form factor
4 – momentum transfer: Q2
Ref: Arnold et al., Phys. Rev. Lett. 57, 174 (1986)
“Dipole formula”
2


Q
p
2


GE (Q )  1 
2 

0.71 GeV 

2
(Inverse Fourier transform
gives charge density (r))
4
Details, Part I: Scattering Cross Section: , d/d
d( )  sin  d d
beam

target
• A beam particle will scatter from the target at any angle  if it approaches within a
(perpendicular) cross sectional area  centered on the target particle.
 Definition: total scattering cross section 
(units: area, eg. fm2)
• Scattering into a particular solid angle at (,) in 3d occurs if the beam particle
approaches within a (perpendicular) cross sectional area d/d centered on the target
 Definition: differential scattering cross section d/d

4
d ( ,  )
d  
d
(units: area/solid angle)
5
Units, etc: What is the right scale for ?
Cross-sectional area:
A   R  2.0 fm
2
proton, R ~ 0.8 fm
2
  A?
WRONG! Geometry has
nothing to do with the value
of . Scale is set by the
interaction and beam energy
Cross section unit:
“barn”
1 barn = 10-24 cm2 = 100 fm2
: b
d/d:
b/sr
Scale for proton-proton
scattering: ~ 0.01 b
Other reactions:
-p:
e-p:
 ~ 10-14 b
 < 10-9 b
but energy-scale dependent!
6
Connection to Experiment:
• Experimenters measure the scattering rate into a given solid angle  at (, ).
• Knowing target thickness, detector efficiency and solid angle yields d/d
dx
beam
nt = # of target nuclei
x
per unit volume
Interaction probability: dP  nt  dx
Transmission: T(x) = probability of getting to x without interacting = 1 – P(x)
T(x+dx) = T(x) [1 – dP] = T(x) [1 – nt  dx]
dT
  nt  dx
T
beam
x
0
dx
L
T ( x)  e
 nt  x
7
Targets: thick and thin!
A target is said to be “thin” if the transmission probability is close to 1.
Then, for target thickness x :
P( x)  ( 1  T ( x))  1  P( x)  nt  x
thin target:
1
x 
nt
Otherwise, attenuation in the target has to be accounted for explicitly via the
exponential relationship:
P( x)  1  e
nt  x
(always correct)
Thick target:
P(x)  1, essentially independent of x beyond a certain thickness.
8
connection to experiment continued....
target, L
beam
detector 
efficiency 
rate I()

rate Io
L
 d 
I ( )  I o P( , L)    I o nt L 
   ( small )
 d 
(thin target case!)
 d ( ) 
 I o nt L
 ( ) 
 sin  d d
 d 
det

9
Electron scattering apparatus at SLAC:
magnetic
spectrometer
shielded detector
package
Details, Part II: Statistical Accuracy in Experiments
(Krane, sec. 7.5)
10
Suppose we perform a scattering experiment for a certain time T. The differential
cross section is determined from the ratio of scattered to incident beam particles in
the same time period:
No  Io T ,
N ( )  I ( ) T

d ( )
N ( )
 const.
d
No
Scattering is a statistical, random process. Each beam particle will either scatter
at angle  or not, with probability P(). Individual scattering events are uncorrelated.
In this case, the statistical uncertainty in N() is said to follow “counting statistics”,
and the error in N() determines the statistical uncertainty in d/d:
N
N

1
N
(Note: strictly speaking, N >> 1 for the Gaussian distribution to apply, but this is
the usual case in a scattering experiment anyway.)
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Interpretation of “counting statistics” error:
If we perform the same experiment many times, always counting for the same time T,
we will measure many different values of N, the number of scattered particles... the
distribution of values of N, call this Ni, will be a Gaussian or Normal distribution, with
the probability of observing a particular value given by:
P( N ) 
with standard deviation:
1
 N 2
N 
e
N
 ( N  N ) 2 / 2 N2
and mean value:
N
If we only do the measurement once, the best estimate of the statistical error comes
from assuming that the distribution of events follows counting statistics as above.
However, it is important to verify that this is the case!
(Electrical noise, faulty equipment, computer errors etc. can lead to distributions of
detected particles that do not follow counting statistics but in fact have much worse
behavior. This will never do!  ..... )
12
Example: checking on “counting statistics”
“Gamma Ray Asymmetry” histogram, Ph.D. thesis data (SAP)
A 
N  N
N  N
solid line:
fitted Gaussian function
 fit /  count  1.001
Sum of entries =
total number of
measurements, M
0.01 0.005
0.005
Mean value: A = - ( 0.9  0.9 ) x
0.01
10-5
 A 
 fit
M
-- very good agreement with counting statistics based on values of N+ and N-
13
continued...
Notes:
1. Time T required to achieve a given statistical accuracy:
  
T~ N ~ 

  
2
2. Beam time is expensive, so nobody can afford to waste it!
e.g. at Jefferson Lab: 34 weeks/year x 2 beams costs US$70M (lab budget)
 $625k Cdn/hour!
3. Efficient experiment design has statistical and systematic errors comparable,
counting rate optimized for “worst” data point (d/d smallest)
 see example, next slide...
14
Data from High Energy Electron-Proton Scattering at SLAC:
Note log scale!
10
0
cross section drop like
1/Q4 – most of the time is
spent at the highest Q2
data point!
-1
10
d/d (nb/sr)
-2
10
-3
10
-4
10
Q2 = 31 GeV2,
-5
10
N = 39 counts
Q2 = 12 GeV2,
N = 1779 counts
/ = 2%
/ = 8%
-6
10
-7
10
0
5
10
15
Q
(Statistical errors only shown)
2
20
25
30
35
2
[(GeV/c) ]
Ref: Sill et al., Phys. Rev. D 48, 29 (1993)
Tables from Sill et al. SLAC experiment, 1993
15
Independent systematic errors
added in quadrature to the statistical
error:
Beam energy and spectrometer angle
adjusted to vary the parameter of
interest, momentum transfer Q2
16.451
Homework Assignment #1
Due: Thursday Sept. 25th 2003
Note: I will be traveling on Sept. 25 and leaving my office by 3 pm. Assignments handed in after that time will count
as “late”, should be submitted to the department office for a date and time stamp, and will be collected Monday – SAP.
1. Penning trap problem: (Review your class notes from lectures 2 and 3.)
Use the equations ofmotion for trapped particles to work out expressions for the (modified) cyclotron frequency c’,
the axial frequency z and the magnetron frequency m .in terms of the fundamental particle and trap parameters.
(Note: assume that m << c and work this one out first in order to simplify the task of evaluating c’.) For each type
of motion, draw a sketch of the associated particle orbit and explain in simple terms how it comes about. Evaluate the
three frequencies numerically for a proton in a Penning trap with magnetic field B = 6.0 T, electrode potential V o = 100
V, and trap dimension d = 3 mm. What is the radius of the proton’s orbit if no axial motion is excited and the total
energy is 100 ħ c ?
2. Hyperfine splitting in hydrogen:
(Review lecture 3 and refer to Krane, sections 16.3-4.)
Use a semiclassical Bohr model of the hydrogen atom to estimate the hyperfine splitting between the F = 0 and F = 1
total angular momentum states. (Assume that the energy splitting is due to the interaction between the magnetic
moments of the proton and electron.) Draw a diagram to illustrate your approach to the calculation; verify that the
energy splitting should be proportional to the dot product of electron and proton spin vectors, and compare your
numerical value to the measured value for the hydrogen atom.
3. Counting Statistics: (Review class notes for lecture 4 and Krane, section 7.3)
a) Look closely at the graph from lecture 4 showing the gamma ray distribution that was analyzed to test for counting
statistics behavior and draw a hand sketch of what it looks like. Write down an expression for the Gaussian
distribution function that would have been fitted to the data to obtain the solid curve. Use the scales in the graph,
and this formula, to estimate the standard deviation  of the distribution, and the number of measurements M; from
these data, show that the stated error in the mean ( 0.9 x 10-5) is consistent with the distribution of measurements.
b) (Krane, problem 7.13) A certain radioactive source gives 3861 counts in a 10 minute counting period. When the
source is removed, the background alone gives 2648 counts in 30 minutes. Determine the net source counting rate
(counts per second) and its uncertainty, explaining your reasoning. (Treat the measurements of “background” and
“signal” events in a counting experiment as independent.)