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Two important BASIC CONCEPTS
•The “coupling” of a fermion (fundamental constituent of matter)
to a vector boson (the carrier or intermediary of interactions)
•Recognized symmetries are intimately related to CONSERVED quantities in nature which fix the QUANTUM numbers describing quantum states and help us characterize the basic, fundamental interactions between particles
Should the selected orientation of the x-axis matter?
As far as the form of the equations of motion? (all derivable from a Lagrangian)
As far as the predictions those equations make?Any calculable quantities/outcome/results?
Should the selected position of the coordinate origin matter?
If it “doesn’t matter” then we have a symmetry: the x-axis can be rotated through any direction of 3-dimensional space
orslid around to any arbitrary location
and the basic form of the equations…and, more importantly, all thepredictions of those equations are unaffected.
If a coordinate axis’ orientation or origin’s exact location “doesn’t matter” then it shouldn’t appear explicitly in the Lagrangian!
EXAMPLE: TRANSLATION
Moving every position (vector) in space by a fixed a(equivalent to “dropping the origin back” –a)
original descriptionof position
r
a
r' new descriptionof position
ar'r
iii qq
r'r
dq
rd
'
a
a
aˆ
i
iii
i dq
qrdq(qr
dq
rd )() a
dq
adq
i
i ˆˆ
or
under the newlyshifted basis qi
For a system of particles:
N
iirmT
1
2
21
acted on only by CENTAL FORCES: )()( rVrV function of separation
0
kk q
L
q
L
dt
d
no forces externalto the system
generalized momentum(for a system of particles,
this is just the ordinary momentum)
kk ppdt
d kk q
V
q
L
=for a system of particles
T may depend on q or r
but never explicitly on qi or ri
k
i
ii
k q
r
r
Vp
For a system of particles acted on only by CENTAL FORCES:
k
i
ii
k q
r
r
Vp
-Fi ai^
aFpi
ik ˆ
aFtotal ˆ
net force on a systemexperiencing only
internal forcesguaranteed
by the 3rd Lawto be
0 kp
Momentummust be conservedalong any direction
the Lagrangian is invariant totranslations in.
Particle properties/characteristicsspecifically their interactions
are often interpreted in terms ofCROSS SECTIONS.
Ei , pi
Ef , pf
EN , pN
recoilNfiEEE
,
recoilNfippp
,
The simple 2-body kinematics of scattering fixes the energy of particles scattered through .
For elastically scattered projectiles:The recoilingparticles areidentical to
the incomingparticles but
are in differentquantum states
The initialconditions
may bepreciselyknowable
onlyclassically!
Nuclear Reactions
Besides his famous scattering of particles off gold and lead foil, Rutherford observed the transmutation:
OHHeN 17
8
1
1
4
2
14
7
OpN 17
8
14
7 or, if you prefer
Whenever energetic particles(from a nuclear reactor or an accelerator)
irradiate matter there is the possibility of a nuclear reaction
Classification of Nuclear Reactions
• pickup reactionsincident projectile collects additional nucleons from the target O + d O + H (d, 3H)
Ca + He Ca + (3He,)
•inelastic scatteringindividual collisions between the incoming projectile and a single target nucleon; the incident particle emerges with reduced energy
2311
2412
Na + He Mg + d
16 8
15 8
31
4120
32
4020
32
9040
9140
Zr + d Zr + p (d,p)(3He,d)
•stripping reactionsincident projectile leaves one or more nucleons behind in the target
BB
BeC
LiN
O
HeO
pF
10
5
10
5
8
4
12
6
6
3
14
7
16
8
3
2
17
8
19
9
BB
BB
BeC
BeC
LiN
LiN
O
HeO
HF
dF
Ne
nNe
pF
11
5
9
5
10
5
10
5
9
4
11
6
8
4
12
6
7
3
13
7
6
3
14
7
16
8
3
2
17
8
3
1
17
9
18
9
20
10
19
10
19
9
2010[ Ne]*
The cross section is defined by the ratio
rate particles are scattered out of beamrate of particles focused onto target material/unit area
number of scattered particles/secincident particles/(unit area sec) target site density
a “counting” experiment
notice it yields a measure, in units of area
With a detector fixed to record data from a particular location , we measure the “differential” cross section: d/d.
how tightly focused or intense the beam is density of nucleartargets
v t
d
Incident mono-energetic beamscattered particles
A
N = number density in beam (particles per unit volume)
N number of scattering centers in targetintercepted by beamspot
Solid angle d representsdetector counting the dN particles per unit time that
scatter through into d
FLUX = # of particles crossing through unit cross section per sec = Nv t A / t A = Nv
Notice: qNv we call current, I, measured in Coulombs.
dN N F d dN = N F d dN = N F d
dN = FN d N F d
the “differential” cross section
R
R
R
R
R
the differential solid angle d for integration is sin dd
R
R
Rsin
RsindRd
RsindRd
ddR
ddRd sin
sin2
2
Symmetry arguments allow us to immediately integrate out
Rsind
R
RR
R
and consider rings definedby alone
Integrated over all solid angles Nscattered = N F TOTAL
Nscattered = N F TOTALThe scattering rate
per unit time
Particles IN (per unit time) = FArea(of beam spot)
Particles scattered OUT (per unit time) = F N TOTAL
AvogadroN
A
N
Earth Moon
Earth Moon
In a solid•interatomic spacing: 15 Å (15 10-10 m)•nuclear radii: 1.5 5 f (1.55 10-15 m)
for some sense of spacing consider the ratio
orbital diameterscentral body diameter
~ 10s for moons/planets
~100s for planets orbiting sun
the ratio orbital diameterscentral body diameter
~ 66,666 for atomic electronorbitals to their own nucleus
Carbon 6COxygen 8OAluminum 13AlIron 26FeCopper 29CuLead 82Pb
A solid sheet of lead offers how much of a (cross sectional) physical target (and how much empty space) to a subatomic projectile?
82Pb207
Number density, n: number of individual atoms (or scattering centers!) per unit volume
n= NA / A where NA = Avogadro’s Number
A = atomic weight (g) = density (g/cc)
w
n= (11.3 g/cc)(6.021023/mole)/(207.2 g/mole)
= 3.28 1022/cm3
82Pb207w
For a thin enough layer
n(Volume) (atomic cross section)= n(surface areaw)(r2)
as a fraction of the target’s area: = n(w)13cm)2
For 1 mm sheet of lead: 0.00257 1 cm sheet of lead: 0.0257
Actually a projectile “sees”
nw nuclei per unit area
but Znw electrons per unit area!
that general description of cross section
let’s augmented with the specific example of
Coulomb scattering
q2
Recoil oftarget
BOTH target and projectile will move in response to
the forces between them. q1
q1
20
21202
tanbmv
bmv
K
But here we areinterested onlyin the scattered
projectile
impact parameter, b
d
q2
b
A beam of N incident particles strike a (thin foil) target.The beam spot (cross section of the beam) illuminates n scattering centers.
If dN counts the average number of particles scattered between and d
dN/N = n dusing
dx
du
uu
dx
d2cos
1tan
20
21
2tan
mbv
becomes:
dbvmb
qqd 2
02
21
2
2cos2
1
d = 2 b db
d
q2
b
20
21
2tan
mbv
2tan2
0
21
mv
qqb
dbvmb
qqd 2
02
212 2cos2
1
and
dqq
vmbdb
2cos2 221
20
2
so
d
mv
mvd
3
20
212
21
20
2tan2cos
d
mv
mvd
3
20
212
21
20
2tan2cos
sin
2sin2
2cos2sin
2sin
2cos
2sin
2cos
44
3
32
d
mv
qqd
2sin2
sin4
2
20
21
d
q2
b
