Sn

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Kinematics of collision processes
1) Introduction collision and decay
processes 2) Rutherford scattering (Rutherford
experiment from all sides). 3) Laws of
energy and momentum conservation. 4) Laboratory
and centre-of-mass frame. 5) Reaction energy,
decay energy. 6) Collision diagram of
momentum 7) Nonrelativistic, relativistic and
ultrarelativistic approach. 8) Relativistic
invariant kinematics variables. 9)
Ultrarelativistic approach rapidity 10)
Transformation of kinematic quantities and cross
sections from laboratory frame to
centre-of-mass and vice versa

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Introduction
Study of collisions and decays of nuclei and
elementary particles main method of microscopic
properties investigation.
Elastic scattering intrinsic state of motion of
participated particles is not changed ? during
scattering particles are not excited or deexcited
and their rest masses are not changed.
Inelastic scattering intrinsic state of motion
of particles changes (are excited), but particle
transmutation is missing.
Deep inelastic scattering very strong particle
excitation happens ? big transformation of the
kinetic energy to excitation one.
Nuclear reactions (reactions of elementary
particles) nuclear transmutation induced by
external action. Change of structure of
participated nuclei (particles) and also change
of state of motion. Nuclear reactions are also
scatterings. Nuclear reactions are possible to
divide according to different criteria
According to history ( fission nuclear reactions,
fusion reactions, nuclear transfer reactions )
According to collision participants (photonuclear
reactions, heavy ion reactions, proton induced
reactions, neutron production reactions )
According to reaction energy (exothermic,
endothermic reactions)
According to energy of impinging particles (low
energy, high energy, relativistic collision,
ultrarelativistic )
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Nuclear decay (radioactivity) spontaneous (not
always induced decay) nuclear transmutation
connected with particle production.
Elementary particle decay - the same for
elementary particles
Set of masses, energies and moments of objects
participating in the reaction or decay is named
as process kinematics . Not all kinematics
quantities are independent. Relations are
determined by conservation laws. Energy
conservation law and momentum conservation law
are the most important for kinematics.
Transformation between different coordinate
systems and quantities, which are conserved
during transformation (invariant variables) are
important for kinematics quantities determination.
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Rutherford scattering
Target thin foil from heavy nuclei (for example
gold)
Beam collimated low energy a particles with
velocity v v0 ltlt c, after scattering v
va ltlt c
The interaction character and object structure
are not introduced
Momentum conservation law
.. (1.1)
and so
.. (1.1a)
square
.. (1.1b)
Energy conservation law
(1.2a)
and so ..
(1.2b)
Using comparison of equations (1.1b) and (1.2b)
we obtain
(1.3)
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Reminder of equation (1.3)
If mtltltma
Left side of equation (1.3) is positive ?
from right side results, that target and a
particle are moving to the original direction
after scattering ? only small deviation of a
particle
If mtgtgtma
Left side of equation (1.3) is negative ? large
angle between a particle and reflected target
nucleus results from right side ? large
scattering angle of a particle
Concrete example of scattering on gold atom
ma ? 3.7103 MeV/c2 , me ? 0.51 MeV/c2 a mAu ?
1.8105 MeV/c2

1. If mt me , then mt/ma ? 1.410-4

Reminder of equation (1.2b)
We obtain from equation (1.3) ve vt 2vacos?
2va We obtain from equation (1.2b) va ? v0
Then for magnitude of momentum it holds meve
m?(me/m?) ve m?1.410-42va ? 2.810-4m?v0
Maximal momentum transferred to the electron is
2.810-4 of original momentum and momentum of
a particle decreases only for adequate (so
negligible) part of momentum .
Maximal angular deflection ?a of a particle
arise, if whole change of electron and a momenta
are to the vertical direction. Then (?a ? 0)
?a ?rad? ? tan ?a meve/m?v0 2.810-4 ? ?a
0.016o
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Reminder of equation (1.3)
Reminder of equation (1.2b)
2) If mt mAu , then mAu/ma ? 49
We obtain from equation (1.3) vAu vt
2(ma/mt)va cos? ? 2(mava)/mt We introduce this
maximal possible velocity vt in (1.2b) and we
obtain va ? v0
because
Then for momentum is valid mAuvAu 2m?va ?
2m?v0
Maximal momentum transferred on Au nucleus is
double of original momentum and a particle can be
backscattered with original magnitude of momentum
(velocity).
Maximal angular deflection ?a of a particle will
be up to 180o.
Full agreement with Rutherford experiment and
atomic model
1) weakly scattered ? - scattering on
electrons 2) ? scattered to large angles
scattering on massive nucleus
Attention remember!! we assumed that objects are
point like and we didn't involve force character.
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Inclusion of force character central repulsive
electric field
Thomson model positive charged cloud
with radius of atom RA
Thomson atomic model
Electrons
Electric field intensity outside
Electric field intensity inside
The strongest field is on cloud surface and
force acting on ? particle (Q ? 2e) is
Positive charged cloud
This force decreases quickly with distance and it
acts along trajectory L ? 2RA ? ?t L/ v0 ?
2RA/ v0 . Resulting change of particle ?
momentum given transversal impulse
Rutherford atomic model
Electrons
Maximal angle is
Positive charged nucleus
Substituting RA? 10-10m, v0 ? 107 m/s, Q 79e
(Thomson model) ???rad? ? tan ?? ? 2.710-4 ?
?? ? 0.015o only very small angles.
Estimation for Rutherford model
Substituting RA RJ ? 10-14m (only quantitative
estimation) tan ?? ? ? 2.7 ? ?? ? ? 70o ?
also very large scattering angles.
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Possibility of achievement of large deflections
by multiple scattering
Foil at experiment has 104 atomic layers. Let
assume

• Thomson model (scattering on electrons or on
  positive charged cloud)
• One scattering on every atomic layer
• Mean value of one deflection magnitude ?? ?
  0.01o. Either on electron or on positive
• charged nucleus

Mean value of whole magnitude of deflection after
N scatterings is (deflections are to all
directions, therefore we must use squares)
?
...
(1)
We deduce equation (1). Scattering takes place in
space, but for simplicity we will show problem
using two dimensional case
Deflections ?i are distributed both in positive
and negative directions statistically around
Gaussian normal distribution for studied case. So
that mean value of particle deflection from
original direction is equal zero
Multiple particle scattering
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the same type of scattering on each atomic layer
Then we can derive given relation (1)
Because it is valid for two inter-independent
random quantities a and b with Gaussian
distribution
And already showed relation is valid
We substitute N by mentioned 104 and mean value
of one deflection ? 0.01o. Mean value of
deflection magnitude after multiple scattering in
Geiger and Marsden experiment is around ? 1o.
This value is near to the real measured
experimental value.
Certain very small ratio of particles was
deflected more then 90o during experiment (one
particle from every 8000 particles). We determine
probability P(??), that deflection larger then ?
originates from multiple scattering.
If all deflections will be in the same direction
and will have mean value, final angle will be
100o (we accent assumption ? each scattering has
deflection value equal to the mean value).
Probability of this is P (1/2)N (1/2)10000
10-3010. Proper calculation will give
We substitute
Clear contradiction with experiment Thomson
model must be rejected
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Derivation of Rutherford equation for scattering
Assumptions
1) ? particle and atomic nucleus are point like
masses and charges. 2) Particle and nucleus
experience only electric repulsion force
dynamics is included. 3) Nucleus is very massive
comparable to the particle and it is not moving.
Acting force
Charged particle with the charge Ze produces a
Coulomb potential
Two charged particles with the charges Ze and Ze
and the distance
experience a Coulomb force giving rice to
a potential energy
Coulomb force is
1) Conservative force force is gradient of
potential energy
2) Central force
Magnitude of Coulomb force is
and force acts in the direction of
particle join.
Electrostatic force is thus proportional to 1/r2
? trajectory of ? particle is a hyperbola with
nucleus in its external focus.
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We define
Impact parameter b minimal distance on which ?
particle comes near to the nucleus in the case
without force acting.
Scattering angle ? - angle between asymptotic
directions of ? particle arrival and departure.
Geometry of Rutheford scattering.
Momenta in Rutheford scattering
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.....................(4)
We substitute (2) and (3) to (1)
We change integration variable from t to ?
. (5)
where d??dt is angular velocity of ? particle
motion around nucleus. Electrostatic action of
nucleus on particle is in direction of the join
vector ? ? force momentum do
not act ? angular momentum is not changing (its
original value is m?v0b) and it is connected with
angular velocity ?? d?/dt ? m???r2 const
m?r2 (d?/dt) m?v0b
then
we substitute dt/d? at (5)
................................ (6)
We substitute electrostatic force F (Z?2)
We obtain
because it is valid
We substitute to the relation (6)
Scattering angle ? is connected with collision
parameter b by relation
(7)
The smaller impact parameter b the larger
scattering angle ?.
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Energy and momentum conservation law
Just these conservation laws are very important.
They determine relations between kinematic
quantities. It is valid for isolated system
Conservation law of whole energy
Nonrelativistic approximation (m0c2 gtgt EKIN)
EKIN p2/(2m0)
Together it is valid for elastic scattering
Ultrarelativistic approximation (m0c2 ltlt EKIN) E
EKIN pc
Conservation law of whole momentum
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We obtain for elastic scattering
Using momentum conservation law
and
We obtain using cosine theorem
Nonrelativistic approximation
Using energy conservation law
We can eliminated two variables using these
equations. The energy of reflected target
particle EKIN 2 and reflection angle ? are
usually not measured. We obtain relation between
remaining kinematic variables using given
equations
Ultrarelativistic approximation
Using energy conservation law
We obtain using this relation and momentum
conservation law cos ? ? 1 and therefore ?
? 0
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Laboratory and centre-of-mass system
We are studying not only collisions of particle
with fixed centre. Also the description of more
complicated case can be simplified by separation
of the centre-of-mass motion in the case of
central potential. We solve problem using
advantageous coordinate system.
Laboratory system experiment is running in this
system, all kinematic quantities are measured in
this system. It is primary from the side of
experiment. Target particle is mostly in the rest
in this coordinate system (Experiments with
colliding beams are exception).
Centre-of-mass system centre-of-mass is in this
system in the rest and hence total momentum of
all particles is zero. Mostly we are interested
in relative motion of particles and no motion of
system as whole using of such coordinate system
is very useful.
Equations of motion can be written in form
(1)
where has in spherical coordinates form (
are appropriate unit vectors)
coordinate origin
i 1,2
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Potential energy depends only on relative
distance of particles.
Reminder of equations (1)
We define new coordinates
. (2)
Using relations (1) and (2) we obtain (it is
valid
)
where ? is the reduced and M the total masses of
the system. In the case of central potential
motion can be split by rewritten to relative
distance and centre-of-mass coordinates
The dynamics is completely contained in the
motion of a fictitious particle with the reduced
mass ? and coordinate r. In the centre-of-mass
system, the complete dynamics is described by the
motion of single particle, with the mass ?,
scattered by fixed central potential.
Kinetic energy splits into kinetic energy of
centre-of-mass and into the part corresponding to
relative particle motion (kinetic energy
in centre-of-mass system).
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Transformation relations between laboratory and
centre-of-mass system for kinematic quantities
We assume two particle scattering on fixed target
(v2p20) The centre-of-mass in the laboratory
system moves in the direction of arrived particle
motion with velocity
Particles are moving against themselves in the
centre-of-mass system with velocities
and then
(we see, that momenta have opposite directions
and they have the same magnitude)
and
Laboratory coordinate system
Centre-of mass coordinate system
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Laboratory coordinate system
Centre-of-mass coordinate system
Derivation of relation between scattering angles
in centre-of-mass and laboratory coordinate
systems
Relation between velocity components in direction
of beam particle motion is
Relation between velocity components
perpendicular to the direction of beam particle
motion
We divide these relations
................ (3)
It is valid for elastic scattering
We rewrite equation (3) to the form
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Reaction energy, decay energy
Up to now we studied only elastic scattering. To
extend our analysis on other reaction types
(decays, nuclear reactions or particle
creations), we introduce
Reaction energy Q is defined as difference of
sums of rest particle energies before reaction
and after reaction or as difference of sums
kinetic energies after reaction and before it
Value of Q is independent on coordinate system.
(Reminder m indicates rest mass)
Exothermic reactions Q ? 0 ? energy is released
(spontaneous decays of nuclei or particles,
reactions are realized for any energy of arrived
particle). We are talking about decay energy in
such case.
Elastic scattering Q 0
Endothermic reactions Q ? 0 ? energy must be
delivered (reaction is not proceed spontaneously,
it is necessary certain threshold energy of
arrived particle to realize reaction).
Threshold energy in centre-of-mass coordinate
system
Using definition of centre-of-mass system we
obtain for beginning state
We obtain from momentum conservation law
It is possible case, that all end particles have
zero momentum and thus also their individual
kinetic energies are zero
Thus threshold energy ETHR in centre-of-mass
coordinate system is
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Threshold energy in laboratory system
Usually we need to know reaction threshold in
laboratory system. We assume nonrelativistic
reaction of two particles with rest masses m1 a
m2. The target particle is in the rest in the
laboratory system. Centre-of-mass is moving in
the laboratory system, it has momentum p1 and
equivalent kinetic energy
this energy is not usable for reaction. That
means threshold energy must be
From definition ETHR is minimal EKIN 1
Substituting p12 into previous equation
Case m1 ltlt m2 leads to ETHR Q
Relation between reaction energy and kinematic
variables of arrived and scattered particle can
be written (we use the same procedure as for
similar relation for elastic scattering)
We often need relation
EKIN 3 f(EKIN 1,?), we define x ? vEKIN 3
Solution is
where
and
Inelastic scattering is always endothermic
(where M0i M0f, EiKIN, EfKIN are total sums)
M0i ? M0f ? EiKIN ? EfKIN ? Q ? 0
Decay of particle at rest Q m0ic2 -M0fc2.
Momenta of particles after two-particle decay
have the same magnitude but opposite direction.
Isotropic distribution. Momenta of products
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Collision momentum diagram
We assumed again, that target nucleus is in the
rest and no relativistic approximation. We write
relations between momenta of particles before and
after collision
(We obtain law of momentum conservation for
studied case by sum of these equations
)
Such relations are initial equations for
construction of vector diagram of momenta
1) Momentum of impinging particle we
represent by oriented abscissa .
2) We divide abscissa to two parts in the
proportion
m1 lt m2
m1 m2
m1 gt m2

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The point A can be inside given circle, on it or
outside dependent on ratio of particle masses. A
scattering angle in centre-of-mass system can
take all possible values from 0 do ?.
Allowed values of scattering angle ? in the
laboratory system and reflection angle ? in
the laboratory system are in the table
m1 ? m2 m1 m2 m1 ? m2
v1 gt vCM v1 vCM v1 lt vCM
?? ? ?/2 ?? ?/2 ?? ? ?/2
? lt0,?gt ? lt0,?/2gt ? lt0,?MAX gt
? lt0,?/2gt ? lt0,?/2gt ? lt0,?/2gt
m1 lt m2
m1 m2
m1 gt m2
m1 lt m2 ? impinging
particles are scattered to both hemispheres
m1 m2 ? impinging particles
are scattered to front hemisphere
m1 gt m2 ? impinging particles are
scattered to front hemisphere to cone with top
angle 2?MAX (direction of impinging particles is
axe of cone) sin?MAX m2/m1
In the laboratory system
Relation between scattering angle and reflection
angle in the laboratory and the centre-of-mass
system (remainder of elastic scattering
assumption)
Vector momentum diagram provides full
information given by conservation laws of energy
and momentum. It shows possible variants of
particle fly away but it has no information about
probabilities of realisation of particular
possible variants.
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Relativistic description nonrelativistic and
ultrarelativistic approximations
Total energy is connected with momentum by
relativistic relation
We label rest mass m ? m0. Rest masses and rest
energies are invariant under Lorentz
transformation (they are the same in all inertial
coordinate systems) and then invariant is also
quantity (optimal coordinate system can be chose
for its calculation)
It is valid not only for single particle but also
for particle system in the given time
... (1)
We express kinetic energy and momentum
Threshold energy in centre-of-mass system leads
to zero sum of kinetic energies of system in
ending state. We express invariant (1) for
beginning state of system in laboratory and for
ending state in centre-of-mass systems
We substitute p2
and EKIN 1
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We express EKIN1
In no relativistic approximation (Qltltm2c2) we
obtain known relation.
In ultrarelativistic approximation (Qgtgtm1c2 a
Qgtgtm2c2) ETHR Q2
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Relativistic relation between scattering angle in
centre-of-mass and laboratory system
Lorentz transformation of momenta and energy
from centre-of-mass system to laboratory system
is (centre-of-mass moves to the direction of axis
y)
We derived relation for angle ?
In nonrelativistic approximation, where vCM ltlt c
we obtain known relation, which we already
derived.
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In common practice, kinetic energy of impinging
particle is used instead centre-of-mass velocity
Centre-of-mass velocity in the laboratory system
is given by ratio between total momentum and
total energy of the system in the laboratory
system
We use relation between kinetic energy and
momentum
We obtain
This relation can be substitute to the relation
for scattering angle. We will show special case,
when scattering angle in the centre-of-mass
system is p/2
In ultrarelativistic approximation (EKIN 1 gtgt
m1c2 and EKIN 1 gtgt m2c2) we obtain
In the laboratory system, particles are produced
to the very small angle.
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Relativistic invariant variables
We can obtain velocity of centre-of-mass during
scattering of two particles with the rest masses
m1 and m2 by total relativistic momentum and
total relativistic energy
(1.a)
m1 refers to the projectile mass and m2 to target
mass. We use laboratory kinematic variables and
we obtain
(1.b)
Nonrelativistic approximation (m1c2 ??p1c)
.. (1.c)
Ultrarelativistic approximation (m1c2 ?? p1c a
m2c2 ?? p1c)
For m1 ? m2
and
We obtain general relativistic relation for ?CM
using equation (1.b)
So that (m12c4 E12-p12c2)
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and we obtain
(2)
Equation is reduced for limits E1 ? p1c ?? m1c2
and p1c ?? m2c2 to formerly given
ultrarelativistic limit
Quantity in the divisor (2) is invariant scalar.
We prove this using the square of following
four-vector in the laboratory frame (p2 0)
This scalar has the same value in arbitrary
reference frame. It has simple interpretation in
the centre-of-mass reference frame (total
momentum in this reference frame is zero)
and s is square of total energy accessible in
centre-of-mass system. Then
Invariant variable s is often used for
description of high-energy collisions. The
quantity ?s is very useful in the case of
colliders.
Invariant variable t is also often used square
of the four-momentum transfer in a collision
(square of the difference in the energy-momentum
four-vectors of the projectile before and after
scattering)
. (3a)
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Energy and momentum conservation laws are valid
and we can express t also in target variables
.. (3b)
variable t is invariant and it can be calculated
in arbitrary coordinate system.
We add yet variable u
or
Variables t, u and s are named as Lorentz
invariant Mandelstam variables, which sum
generally satisfy equation
In the case of elastic scattering in the
centre-of-mass system (for both particles
and
)
Because 1 cos? 1 it is valid t ? 0. Using
(3a,b) we can look on variable t as on
mass-square of exchanged particle (with energy
and momentum ).
Imaginary mass ? virtual particle.
Such diagrams were pioneered by R. Feynman in the
calculation of scattering amplitudes in QED and
they are referred to as Feynman graphs. Let us
define a variable q2 ( q2c2 -t ), which is
equal to square of momentum transferred to target
nucleus q2 ? (m2v2)2 in no relativistic
approximation.
Feynman graph
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Ultrarelativistic approximation -rapidity
In high-energy physics (ultrarelativistic
collisions ? velocity of beam particles v ? c)
new kinematic variable rapidity is useful to
introduce (usually we have c1, m is total mass)
We choose beam direction as axe z, thus we can
write total energy and momentum of particle as
E mTc2cosh y, px, py a pz mTc sinh y
Reminder
We introduced transversal mass mT
and rapidity y
and thus
For nonrelativistic limit (ß ? 0) y ß
For ultrarelativistic limit (ß ? 1) y ? 8
Rapidity using leads to very simple
transformation from one coordinate system to
another
where y21 is rapidity of the coordinate system 2
in the system 1. Thus we write for
transformation from the laboratory to the
centre-of-mass systems
Examples GSI Darmstadt ( ELAB 1GeV/A
y0.458 ß0.875 ) SPS
CERN ( ELAB 200GeV/A y6.0
ß1.000 ) LHC CERN (
ELAB35003500GeV/A y17.8 ß1.000 )
Relation between transversal component of
velocity and rapidity