Lecture 3: Tools of the Trade

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Lecture 3 Tools of the Trade

• High Energy Units
• 4-Vectors
• Cross-Sections
• Mean Free Path

Useful Sections in Martin Shaw
Section 1.5, Appendix A, Appendix B
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High Energy Units
High Energy Units (''natural" units)
''c ? 1"
but really its just LYING!
Say that youre talking about mass, momentum,
time and length but actually convert everything
to units of energy using fundamental constants
and pretend not to notice
1 eV/c2 1.782x10-36 kg
m ? E L ? 1/E p ? E t ? 1/E
E ? mc2
E ?c/?
? ?c/L
?c 197 MeV fm (1 fm 10-15m)
E?pc
1 eV/c 5.346 ??10-28 kg m/s
E??
? ?/t
? 6.58 ??10-22 MeV s
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Dealing with High Energy Units
However
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Conversion Example
Convert the following to MKS units of
acceleration
2.18x10-34 GeV
want to multiply by units LT-2E-1
? Ea Ta LbT-b
?acb
Ea Ta-b Lb
comparing ? a -1 b 1
4.5x1032 m/s2/GeV
(2.18x10-34 GeV)(4.5x1032 m/s2/GeV)
9.8 m/s2
g
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Tips 1
Steves Tips for Becoming a Particle Physicist
1) Be Lazy
2) Start Lying
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4-Momenta
4-Momenta
But (luckily) the basics are pretty
straight-forward
Define the 4-component vector P ? (E, px,
py, pz) (E, p)
(''natural" units, otherwise E ? E/c)
Define the scalar product of 2 such beasts as
P1 ? P2 E1E2 - p1 ??p2 Note that
this means P ? P P2 E2 -
p2 m2
? relativistic invariant !!
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Basic Recipe Idea
1) Represent the reaction in terms of 4-momenta
(statement of energy and momentum conservation)
2) Algebraically manipulate 4-vectors to simplify
and insure the result will be couched in
terms of relevant angles, energies etc.
3) Square both sides of the equation, choosing
convenient reference frames for each side
4) Let cool for 15 minutes and serve warm with
plenty of custard and a nice, hot mug of tea
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Example beam collisions
2 particle beams cross with angle ?. Find the
total CM energy in the limit Em. What is this
for a head-on collision?
( )2 2
P1 P2 PT
P12 P22 2P1 P2 PCM2
2E1E2 (1 cos?)
? ECM2 2E1E2 (1 - cos?)
so, for a head-on collision ECM 2(E1E2)1/2
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Example threshold production
1 2 3 4 5 6
(fixed)
P1 P2 PF where PF
(E3E4E5E6 , p3p4p5p6)
(4mp , 0) in CM at threshold

• mp2 mp2 2E1E2 2p1p2 (4mp)2 - (0)2

(P1 P2)2 PF2
(in lab, p2 0 , so E2mp)
2E1mp 2mp2 16mp2
E1 7mp
T2 E - mp
gmp - mp mp(g - 1)
? g1.03 b 0.253
T1? E? - mp 4.4mp
E? g ( E - b p )
1.03 (E1- 0.253p1)
5.4mp
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Example beta decay
Pn Pp Pe Pn??
( )2 ( )2
Pn - Pn Pp Pe
Pn2 Pn2 - 2(EnEn - pnpncosq) Pp2 Pe2
2(EpEe - pppecosa)
mn2 0 - 2En(En - pncosq) mp2 me2
2(EpEe - pppecosa)
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Lorentz Transformations
Keeping with the vector idea, we can also write
the Lorentz transformation as
where ? velocity of moving frame
? ( 1 - ?2 ) -1/2 p component
of p parallel to ?
Note this transformation matrix also
applies for other types of
similarly defined 4-vectors as well!
Basic 4-Vectors X ? ( t, x ), P ? ( E, p )
Relativistic Doppler Shift
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More 4-Vectors
More fun with 4-Vectors!
Start with X ? ( t, x ), P ? ( E, p )
Take time-derivative of X
...but which time?
d/dt X (dt/dt d/dt) t, (dt/dt d/dt) x
( ?, ? v ) ? V
Similary, d/dt P (? dE/dt, ? f ) ? F
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Pions can decay via the reaction p m nm .
Find the energy of the neutrino in the rest
frame of the pion.
Pp Pm Pn
mm2 mp2 0 2(EpEn pppn)
mp2 - mm2 2(En Ep pppn)
En (mp2 - mm2 )/2mp
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Derive an expression relating the emission angle
of the muon or neutrino with respect to the beam
in the CM to that in the lab frame.
Transverse momentum plabsinq pcmsina
Longitudinal momentum plabcosq g (pcmcosa
bEcm)
divide tanq sina g (cosa bEcm/pcm)-1
sina g (cosa b/bp)-1
Are there any limits to the lab directions of the
muon or neutrino?
To find maximum angle, set dq/da 0
Thus, a solution is well defined if bp/b lt 1, or
if b gt bp
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A cloud of matter is ejected with high velocity
from a distant galactic nucleus. The cloud moves
from A to B in the diagram, emitting photons
seen, over a period of several years, by an
observer on Earth. In the Earths frame of
reference, the clouds velocity has constant
magnitude v and is at an angle ? with the line of
sight.
A
The points B, C represent the space coordinates
of space-time points on the paths of photons
seen at different times by the observer. Derive
an expression, in terms of v and for the
apparent transverse velocity of the cloud which
would be deduced by the observer from the
difference between the arrival times of photons
with the paths shown and from the measured
spatial separation of B and C.
L
?
v
B
C
Time for matter to get from A to B L/v
Time for light to get from A to C L cos?/c
Distance from C to B L sin?
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Cross-Sections
Cross-Sections etc.
(a quick review of useful definitions and
relations)
Interaction Probability Fraction of area
occupied by targets
? NT / A Interactions Nint ? NT
NB /A
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Alternative expressions
Alternate expression to eliminate A
NT (/Volume) x Volume ?? L A
Alternate expression to connect ? with theory
Rate Nint/t
vB NB NT ? / V
vBNBNT? / (LA)
Prob/Time Rate/(NB NT)
W vB ? / V
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Transmission Experiment
Relate the interaction cross-section to easily
measurable target properties and the fraction of
the beam which survives intact after passing
through the target
Example Transmission experiment
Beam Counters ?
NB
Transmission Counter ? NTR
Nint NB - NTR
(1-fTR)/(??L)
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Mean Free Path
Mean Free Path
(Interaction Length)
?? volume per interaction V/N
1/???????????
????????????????
(Poisson) P ?n e-? / n! In this case n 0
and ? N V/(??) x/?