Kinematics of collision processes
1)
Introduction
–
collision and decay processes
2) Ruthe
r
ford scattering (Ruthe
r
ford 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
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
摵楮i
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
扩b
瑲a湳景n浡瑩潮
潦
瑨e
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
…
)
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
.
Nuclear
decay
(radioactivity)
–
spontaneous
(not
always
–
induced
decay)
nuclear
transmutation
connected
with
particle
production
.
Elementary particle decay
-
the same for elementary particles
Ruthe
r
ford scattering
Target:
thin foil from
heavy nuclei (for example gold)
Beam: collimated low energy α particles with
velocity
v = v
0
<< c
,
after scattering
v = v
α
<< c
The interaction character and object structure are not
introduced
t
t
0
v
m
v
m
v
m
Momentum conservation law
:
..
(1.1)
and so:
…..
(1.1a)
square:
….. (1.1b)
2
t
2
t
t
t
2
2
0
v
m
m
v
v
m
m
2
v
v
Energy
conservation
law
:
(
1
.
2
a
)
2
t
t
2
2
0
v
m
2
1
v
m
2
1
v
m
2
1
and so:
.. (1.2b)
2
t
t
2
2
0
v
m
m
v
v
Using
comparison
of
equations
(
1
.
1
b)
and
(
1
.
2
b)
we
obtain
:
…………
(
1
.
3
)
t
t
2
t
v
v
2
m
m
1
v
For scalar product of two vectors it holds: so that we obtain:
cos
b
a
b
a
t
t
0
v
m
m
v
v
If
m
t
<<m
α
:
Left
side of equation
(1.3)
is
positive
→ from
right
side
results, that target and α particle are moving
to the
original direction
after scattering
→ only
small
deviation of α particle
If
m
t
>>m
α
:
Left
side
of
equation
(
1
.
3
)
is
negative
→
large
angle
between
α
particle
and
reflected
target
nucleus
results
from
right
side
→
large
scattering
angle
of
α
particle
Concrete
example
of
scattering
on
gold
atom
:
m
α
3
.
7
∙
10
3
MeV/c
2
,
m
e
0
.
㔱
敖⽣
2
a
m
Au
1
.
8
5
MeV/c
2
1)
If
m
t
=m
e
,
then
m
t
/m
α
1
.
-
4
:
We obtain from equation
(1.3)
:
v
e
= v
t
= 2v
α
cos
≤ 2v
α
We obtain from equation
(1.2b)
:
v
α
v
0
Then for magnitude of momentum i
t holds
:
m
e
v
e
= m
⡭
e
/m
⤠v
e
≤ m
∙1.4∙10
-
4
∙2v
α
2⸸뜱0
-
4
m
v
0
Maximal momentum transferred
to the electron is
≤ 2.8·10
-
4
of original momentum and
momentum of α particle decreases only for adequate (so negligible) part of momentum
.
cos
v
v
2
v
v
2
m
m
1
v
t
t
t
2
t
Reminder of equation
(1.3)
2
t
t
2
2
0
v
m
m
v
v
Reminder of equation
(1.2b):
Maximal angular deflection
α
of α particle
arise, if whole change of electron and α
momenta are
to the vertical direction. Then (
α
0
⤺
α
rad
瑡渠
α
= m
e
v
e
/m
v
0
≤ 2.8·10
-
4
α
≤ 0.016
o
2)
If
m
t
=m
Au
,
then
m
Au
/m
α
㐹
W攠扴a楮i晲m敱畡瑩潮e
⠱⸳(
㨠
v
Au
= v
t
= 2(m
α
/m
t
)v
α
cos
2⡭
α
v
α
)/m
t
We introduce this maximal possible velocity
v
t
in
(1.2b)
and we obtain:
v
α
v
0
Then for momentum is valid:
m
Au
v
Au
≤ 2m
v
α
㉭
v
0
Maximal momentum transferred
on Au nucleus
is double of original momentum
and α particle can
be backscattered with original magnitude of momentum (velocity).
Maximal angular deflection
α
of α particle
will be up to
180
o
.
Full agreement with Ruthe
r
ford experiment and atomic model:
1)
weakly scattered
獣s瑴敲楮e渠敬散n湳
2)
獣s瑴敲敤e瑯污牧攠a湧汥猠
–
scattering on massive nucleus
Attention remember!!:
we assumed that objects are point like and we didn't involve force character
.
Reminder of equation
(1.3)
2
t
t
2
2
0
v
m
m
v
v
Reminder of equation
(1.2b):
cos
v
v
2
v
v
2
m
m
1
v
t
t
t
2
t
2
2
2
2
t
α
α
t
2
2
t
t
2
2
0
v
v
m
4m
v
m
v
2m
m
m
v
v
m
m
v
v
t
because:
Inclusion of force character
–
central repulsive electric field:
2
0
A
r
Q
4
1
)
R
E(r
Thomson model
–
positive charged cloud with
radius of atom R
A
:
Electric field intensity outside:
Electric field intensity inside:
3
A
0
A
R
Qr
4
1
)
R
E(r
2
A
0
A
MAX
R
4
Q
2
)
R
2eE(r
F
e
The strongest field is on cloud surface
and
force acting on
灡牴楣汥
(
儠
㴠=e
⤠楳i
This force decreases quickly with distance and it acts along trajectory
L
㉒
A
琠㴠䰯⁶
0
㉒
A
/ v
0
. Resulting change of particle
momentum =
given transversal impulse:
0
A
0
MAX
v
R
4
eQ
4
t
F
p
Maximal angle is:
2
0
A
0
v
m
R
4
eQ
4
/p
p
tan
Substituting
R
A
10
-
10
m
,
v
0
7
m/s
,
Q = 79e
(
Thomson model
):
rad
瑡渠
†
†
2⸷뜱.
-
4
→
†
0⸰.5
o
only very small angles.
Estimation for
Ruthe
r
ford model
:
Substituting
R
A
= R
J
-
14
m
(only quantitative estimation):
tan
†
†
2.7 →
㜰
o
a汳lv敲礠污牧攠獣s瑴敲楮ea湧汥l
.
Thomson atomic model
Electrons
Positive charged cloud
Electrons
Positive charged
nucleus
Rutherford
atomic model
Possibility of achievement of large deflections by multiple scattering
Foil at
experiment has 10
4
atomic layers. Let assume:
1)
Thomson model (scattering on electrons or on positive charged cloud)
2)
One scattering on every atomic layer
3)
Mean value of one deflection magnitude
0⸰.
o
. 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)
:
N
2
N
1
i
2
2
N
1
i
i
2
N
i
…..….
(1)
We
deduce
equation
(
1
)
.
Scattering
take
s
place
in
space,
but
for
simplicity
we
will
show
problem
using
two
dimensional
case
:
Multiple particle scattering
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:
0
N
1
i
i
N
1
i
i
the same type of scattering on each atomic layer:
i
2
2
i
2
N
1
i
2
i
1
1
1
1
2
N
1
i
1
N
1
i
N
1
i
j
j
i
2
2
N
1
i
i
N
2
2
N
i
N
i
j
j
i
N
i
i
i
Then we can derive given relation
(1):
ab
ab
M
N
1
b
a
M
N
1
b
M
1
a
N
1
b
a
M
N
1
k
k
M
1
j
j
N
1
i
i
M
1
j
j
N
1
i
i
Because it is valid for two
inter
-
independent
random quantities
a
and
b
with
Gaussian distribution:
And already showed relation is valid:
N
We
substitute
N
by
mentioned
10
4
and
mean
value
of
one
deflection
=
0
.
〱
o
.
Mean
value
of
deflection
magnitude
after
multiple
scattering
in
Geiger
and
Marsden
experiment
is
around
1
o
.
This
value
is
near
to
the
real
measured
experimental
value
.
Certain
very
small
ratio
of
particles
was
deflected
more
then
90
o
during
experiment
(
one
particle
from
every
8000
particles
)
.
We
determine
probability
P(
)
,
瑨a
e晬散瑩潮
污牧敲
瑨敮
originate
s
from
multiple
scattering
.
If all deflections will be in the same direction and will have mean value, final angle will be ~100
o
(we
accent assumption
敡捨獣s瑴敲楮e桡猠摥晬散s楯渠va汵攠敱畡氠瑯⁴桥m敡渠va汵攩l偲扡扩汩瑹b映
瑨楳楳i
‽
1⼲)
N
=(1/2)
10000
=
10
-
3010
.
Proper calculation will give:
2
e
P
We
substitute
:
3500
8100
2
1
90
10
90
e
e
P
o
o
o
Clear contradiction with
experiment
–
Thomson model must be rejected
Derivation
of
Ruthe
r
ford
equation
for
scattering
:
Assumptions
:
1)
灡牴楣汥
a湤瑯m楣i湵捬敵猠a攠灯楮琠汩l攠ma獳敳ea湤桡牧敳e
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
:
r
Ze
4
1
r
U
0
Two charged particles with the charges
Ze
and
Z‘e
and the distance
r
r
experience a Coulomb force giving rice to a
potential energy :
r
e
Z
Z
4
1
r
V
2
0
Coulomb
force
is
:
1
)
Conservative
force
–
force
is
gradient
of
potential
energy
:
r
V
r
F
2
)
Central
force
:
r
V
r
V
r
V
Magnitude
of
Coulomb
force
is
and
force
acts
in
the
direction
of
particle
join
.
2
2
0
r
e
Z
Z
4
1
r
F
Electrostatic
force
is
thus
proportional
to
1
/r
2
牡橥捴潲
潦
pa牴捬e
楳
a
桹灥牢污
楴i
湵捬敵n
楮
楴i
數瑥牮慬
景捵
.
We
define
:
Impact
parameter
b
–
minimal
distance
on
which
灡瑩捬e
捯me
s
湥n
瑯
瑨
湵n汥畳
楮
瑨
捡se
楴桯畴
景捥
a捴楮
.
Scattering
angle
a湧汥
扥瑷敥e
a獹m灴潴楣
摩散瑩潮e
潦
particle
arrival
and
departure
.
Geometry of Rutheford scattering.
Momenta in
Rutheford scattering:
First
we
find
relation
between
b
and
:
Nucleus
gives
to
the
灡牴楣汥
imp畬獥
灡牴i捬
momentum
changes
from
original
value
p
0
to
final
value
p
:
dt
F
dt
F
p
p
p
0
…………. (1)
Using assumption about target fixation we obtain that
kinetic energy and magnitude of
灡牴楣汥mm敮瑵m
扥景b攬e摵物湧a湤na晴敲獣s瑴敲楮ea攠瑨攠獡m攺
p
0
= p
㴠=
v
0
=m
v
We
see
from
figure
:
2
sin
v
2m
p
2
sin
v
m
p
2
1
0
0
dt
cos
F
Because
impulse
is
in
the
same
direction
as
the
change
of
momentum,
it
is
valid
:
where
楳i牵湮楮a湧汥l扥瑷敥渠e††a湤n††a汯湧⁰慲瑩捬攠瑲慪散特.
F
p
……….. (2)
……………
(
3
)
We substitute
(2)
and
(3)
to
(1)
:
dt
cos
F
2
sin
v
m
2
0
0
…………
.....................
……
(
4
)
We
change
integration
variable
from
t
to
:
d
d
dt
cos
F
2
sin
v
m
2
2
1
2
1
-
0
…. (5)
where
d
摴
楳湧畬⁶敬捩瑹映
灡牴楣汥m瑩潮a畮搠湵捬敵献n䕬散瑲獴s瑩挠a捴楯渠映湵捬敵猠
渠灡牴楣汥n楳i楮i摩散瑩潮e映瑨攠橯楮jv散瑯e
†
†††††††
景捥mm敮瑵m摯潴捴
angular momentum
is not changing (its original value is
m
v
0
b
) and it is connected with angular
velocity
㴠
⽤琠
m
2
= const = m
2
(d
⽤琩㴠m
v
0
b
0
F
r
then:
b
v
r
d
dt
0
2
we substitute
dt/d
慴
⠵(
:
2
1
2
1
2
2
0
cos
Fr
2
sin
b
v
m
2
d
................................
…
(
6
)
2
2
0
r
2Ze
4
1
F
We
substitute
electrostatic
force
F
(
Z
=
2
)
:
We
obtain
:
2
cos
Ze
d
cos
4
2
d
cos
Fr
0
2
2
1
2
1
0
2
2
1
2
1
2
Ze
because
it
is
valid
:
2
cos
2
2
2
sin
2
sin
cos
2
1
2
1
2
1
2
1
d
We substitute to the relation
(6):
2
cos
Ze
2
sin
b
v
m
2
0
2
2
0
Scattering
angle
楳
捯湮散敤
楴i
捯汬楳楯n
灡牡m整敲
b
批
敬e瑩潮
:
b
Ze
E
4
Ze
b
v
m
2
2
cotg
2
KIN
0
2
2
0
0
… (7)
The
smaller
impact
parameter
b
the
larger
scattering
angle
.
Energy and momentum conservation law
Just
these
conservation
laws
are
very
important
.
T
hey
determine
relations
between
kinematic
quantities
.
It
i
s
valid
for
isolated
system
:
Conservation
law
of
whole
energy
:
Conservation
law
of
whole
momentum
:
i
f
n
1
j
j
n
1
k
k
p
p
i
f
n
1
j
j
n
1
k
k
E
E
j
f
n
1
j
j
KIN
2
0
n
1
k
k
KIN
2
0
E
c
m
E
c
m
j
i
f
f
n
1
j
j
KIN
n
1
j
j
2
0
n
1
k
n
1
k
k
KIN
k
2
0
E
c
m
E
c
m
i
KIN
2
i
0
f
KIN
2
f
0
E
c
M
E
c
M
Nonrelativistic
approximation
(
m
0
c
2
>>
E
KIN
)
:
E
KIN
=
p
2
/(
2
m
0
)
2
i
0
2
f
0
c
M
c
M
i
0
f
0
M
M
Together
it
is
valid
for
elastic
scattering
:
i
KIN
f
KIN
E
E
i
f
n
1
j
j
0
2
n
1
k
k
0
2
2m
p
2m
p
Ultrarelativistic
approximation
(
m
0
c
2
<<
E
KIN
)
:
E
≈
E
KIN
≈
pc
i
f
E
E
i
KIN
f
KIN
E
E
f
i
n
1
k
n
1
j
j
k
c
p
c
p
f
i
n
1
k
n
1
j
j
k
p
p
We
obtain
for
elastic
scattering
:
Using
momentum
conservation
law
:
sin
p
sin
p
0
2
1
and
cos
p
cos
p
p
2
1
1
We
obtain
using
cosine
theorem
:
cos
p
2p
p
p
p
1
1
2
1
2
1
2
2
Nonrelativistic
approximation
:
Using
energy
conservation
law
:
2
2
2
1
2
1
1
2
1
2m
p
2m
p
2m
p
We
can
eliminated
two
variables
using
these
equations
.
The
energy
of
reflected
target
particle
E‘
KIN
2
and
reflection
angle
ψ
are
usually
not
measured
.
We
obtain
relation
between
remaining
kinematic
variables
using
given
equations
:
0
cos
p
p
m
m
2
m
m
1
p
m
m
1
p
1
1
2
1
2
1
2
1
2
1
2
1
0
cos
E
E
m
m
2
m
m
1
E
m
m
1
E
1
KIN
1
KIN
2
1
2
1
1
KIN
2
1
1
KIN
Ultrarelativistic
approximation
:
1
1
2
1
2
1
2
2
2
1
1
p
p
2
p
p
p
p
p
p
Using
energy
conservation
law
:
We obtain using this relation and momentum conservation law:
cos
1†
a湤⁴桥h敦潲攺
0
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
.
1
2
r
r
V
2
1
r
,
r
Remainder of centre
-
of
-
mass installation:
we assumed two particles (with masses
m
1
,
m
2
and
positions ) interacting mutually only by central potential
:
1
1
1
r
ˆ
,
r
r
2
2
2
r
ˆ
,
r
r
r
ˆ
r,
r
CM
CM
CM
r
ˆ
,
r
r
Equations of motion can be written in form:
1
2
1
1
1
r
r
V
r
m
1
2
2
2
2
r
r
V
r
m
…………(1)
where has in spherical coordinates form ( are
appropriate unit vectors):
i
i
i
ˆ
,
ˆ
,
ˆ
r
i
i
i
i
i
i
i
i
i
sin
r
ˆ
r
ˆ
r
r
ˆ
i
i = 1,2
coordinate
origin
Potential energy depends only on relative distance of particles.
We define new coordinates:
2
1
r
r
r
2
1
2
2
1
1
CM
m
m
r
m
r
m
r
………………………
.
(
2
)
1
2
1
1
1
r
r
V
r
m
1
2
2
2
2
r
r
V
r
m
Reminder of equations
(1):
Using
relations
(
1
)
and
(
2
)
we
obtain
(it
is
valid
)
:
r
V
r
V
r
V
1
2
r
ˆ
r
r
V
r
V
-
r
r
m
m
m
m
2
1
2
1
0
r
M
r
)
m
(m
CM
CM
2
1
r
ˆ
constant
r
CM
where
楳
瑨e
e摵捥
a湤
瑨e
瑯瑡l
ma獳敳
潦
瑨e
獹獴sm
.
䥮
瑨e
a獥
潦
敮e牡l
灯敮e楡l
m瑩潮
捡n
扥
獰汩s
批
敷物瑴敮
瑯
敬e瑩癥
摩獴湣n
a湤
捥湴e
潦
ma獳
捯牤楮瑥
:
CM
CM
r
v
䍥湴Ce
潦
ma獳
浯瑩潮
楳
畮景m
a湤
獴牡楧桴景牷a牤
捥瑲e
潦
ma獳
mv敳
楮
瑨e
污扯牡瑯特
楴i
a
捯湳瑡湴
v敬e捩瑹
†††††††††
楮摥灥湤敮瑬i
潦
獰散楦楣
景牭
潦
瑨
灯瑥湴楡l
.
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
,
獣s瑴敲敤e批楸敤e捥湴牡氠灯瑥湴楡氮
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)
.
Transformation relations
between laboratory and centre
-
of
-
mass system for kinematic quantities:
We assume two particle scattering on fixed target (
v
2
=p
2
=0
):
The centre
-
of
-
mass in the laboratory system moves in the direction of arrived particle motion with
velocity:
2
1
1
1
2
1
2
2
1
1
CM
CM
m
m
v
m
m
m
v
m
v
m
r
v
Particles
are
moving
against
themselves
in
the
centre
-
of
-
mass
system
with
velocities
:
2
1
1
2
CM
1
1
m
m
v
m
v
v
v
~
2
1
1
1
CM
2
2
m
m
v
m
v
-
v
v
~
1
1
1
1
v
v
~
m
p
~
1
2
2
2
v
v
~
m
p
~
and
then
:
(we
see,
that
momenta
have
opposite
directions
and
they
have
the
same
magnitude)
and
2
1
1
KIN
2
1
2
2
KIN
1
KIN
KIN
v
2
1
E
m
m
m
E
~
E
~
E
~
Laboratory
coordinate
system
Centre
-
of mass
coordinate
system
2
1
2
)
~
cos
2
1
(
~
cos
cos
We
rewrite
equation
(
3
)
to
the
form
:
It
is
valid
for
elastic
scattering
:
2
1
1
CM
m
m
v
~
v
1
CM
CM
1
1
v
~
v
~
cos
~
sin
v
~
cos
v
~
~
sin
v
~
tan
..............
……………………
..
(
3
)
We
divide
these
relations
:
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
:
~
cos
v
~
v
cos
v
1
CM
1
CM
1
1
v
~
cos
v
~
cos
v
Relation between velocity components perpendicular to the direction of beam particle motion:
~
sin
v
~
sin
v
1
1
Laboratory
coordinate
system
Centre
-
of
-
mass
coordinate
system
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
:
Q
E
E
c
m
c
m
Q
i
f
f
i
n
1
j
KIN
j
n
1
k
KIN
k
f
n
1
k
2
k
i
n
1
j
2
j
Value
of
Q
is
independent
on
coordinate
system
.
(Reminder
:
m
indicates
rest
mass)
:
Exothermic
reactions
Q
0
敮敲杹
楳
敬敡獥s
⡳(湴慮敯畳
摥捡ys
潦
湵n汥i
潲
灡牴i捬敳,
敡楯湳
ae
敡汩l敤
景
a湹
敮敲杹
潦
a牲楶敤
灡牴楣汥
.
坥
ae
瑡汫楮i
a扯畴
摥捡y
敮敲杹
楮
獵捨
捡獥
.
䕬獴楣
獣s瑴敲楮e
Q
=
0
Endothermic
reactions
Q
0
敮敲杹
m畳
扥
摥iv敲敤
⡲敡e瑩潮
楳
湯n
灲捥敤
獰sna湥潵獬n,
楴
楳
necessary
certain
threshold
energy
of
arrived
particle
to
realize
reaction)
.
0
p
~
i
n
1
j
j
i
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
:
0
p
~
f
n
1
k
k
f
It
is
possible
case,
that
all
end
particles
have
zero
momentum
and
thus
also
their
individual
kinetic
energies
are
zero
:
0
E
~
f
n
1
k
KIN
k
f
Thus
threshold
energy
E
THR
in
centre
-
of
-
mass
coordinate
system
is
:
Q
Q
m
m
E
~
E
~
i
n
1
j
j
f
n
1
k
k
i
n
1
j
KIN
j
THR
i
f
i
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
m
1
a
m
2
.
The
target
particle
is
in
the
rest
in
the
laboratory
system
.
Centre
-
of
-
mass
is
moving
in
the
laboratory
system,
it
has
momentum
p
1
and
equivalent
kinetic
energy
:
)
m
2(m
p
E
~
2
1
2
1
KIN
Q
m
m
2
p
E
2
1
2
1
THR
this
energy
is
not
usable
for
reaction
.
That
means
threshold
energy
must
be
:
From definition
E
THR
is minimal
E
KIN 1
:
1
2
1
THR
2m
p
E
THR
1
2
1
E
2m
p
Substituting
p
1
2
into
previous
equation
:
2
1
THR
m
m
1
Q
E
Case
m
1
<<
m
2
leads
to
E
THR
=
|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):
cos
E
m
E
m
m
2
m
m
1
E
m
m
1
E
Q
KIN3
4
KIN1
3
1
4
1
KIN1
4
3
KIN3
We
often
need
relation
:
E
KIN
3
=f(E
KIN
1
,
⤬
††
睥
摥晩湥
†
x
√E‘
KIN
3
Solution
is
:
s
r
r
E
2
KIN3
where
cos
m
m
E
m
m
r
4
3
KIN1
3
1
4
3
1
4
KIN1
4
m
m
)
m
(m
E
Q
m
s
Inelastic
scattering
is
always
endothermic
(where
M
0
i
M
0
f
,
E
i
KIN
,
E
f
KIN
are
total
sums)
:
M
0
i
M
0
f
i
KIN
f
KIN
Q
0
a湤
䑥捡
潦
灡瑩汥
慴
e獴
:
Q
=
m
0
i
c
2
-
M
0
f
c
2
.
Momenta
of
particles
after
two
-
particle
decay
have
the
same
magnitude
but
opposite
direction
.
Isotropic
distribution
.
Momenta
of
products
:
f
02
f
01
f
02
f
01
f
2
f
1
m
m
Q
m
2m
p
p
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
:
CM
1
1
v
v
~
v
1
2
1
1
1
1
p
m
m
m
p
~
p
CM
2
2
v
v
~
v
1
2
1
2
1
2
p
m
m
m
p
~
p
(We obtain law of momentum conservation for studied case by sum of these equations
: )
2
1
1
p
p
p
Such
relations
are
initial
equations
for
construction
of
vector
diagram
of
momenta
:
1
p
AC
1
)
Momentum
of
impinging particle we represent by oriented abscissa
.
AC
2
1
m
:
m
OC
:
AO
2)
We divide abscissa
to two parts in the proportion
1
2
1
2
1
p
m
m
m
p
~
3)
We describe circle around the point
O
passing through the point
C
. The circle
radius
is
equal to magnitude of momenta
p
1
in the centre
-
of
-
mass system
. The circle
geometrical place of vertexes
B
of vector triangle of momenta
ABC
(represents law of
momentum conversation), which sides and represent possible momenta of particles
after collision in the laboratory system.
AB
BC
m
1
< m
2
:
m
1
= m
2
:
m
1
> m
2
:
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
⸠䅬汯A敤eva汵敳l
of scattering angle
楮i瑨攠污扯牡瑯特獹獴敭a湤敦汥e瑩潮a湧汥l
楮⁴桥
污扯牡瑯特獹獴敭a攠楮i
瑨
瑡扬攺
~
m
1
m
2
m
1
= m
2
m
1
m
2
v
1
> v
CM
v
1
= v
CM
v
1
< v
CM
+
⼲
+
㴠
⼲
+
⼲
=
<0,
=
<0,
⼲
=
<0,
MAX
>
=
<0,
⼲
=
<0,
⼲
=
<0,
⼲
m
1
< m
2
:
m
1
= m
2
:
m
1
> m
2
:
In
the
laboratory
system
:
m
1
< m
2
業灩湧楮i灡牴楣汥猠a攠獣e瑴敲敤e瑯瑨桥h楳灨敲敳
m
1
= m
2
業灩湧楮i灡牴楣汥猠a攠獣e瑴敲敤
to front hemisphere
m
1
> m
2
業灩湧楮i灡牴楣汥猠a攠獣e瑴敲敤
to front hemisphere to cone
with top angle
2
MAX
(direction of impinging particles is axe of cone):
sin
MAX
=m
2
/m
1
Relation
between
scattering
angle
and
reflection
angle
in
the
laboratory
and
the
centre
-
of
-
mass
system
(remainder
of
elastic
scattering
assumption)
:
2
~
2
1
m
m
~
cos
~
sin
tg
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.
Relativistic description
–
nonrelativistic and ultrarelativistic approximations
Total
energy
is
connected
with
momentum
by
relativistic
relation
:
4
2
0
2
2
c
m
c
p
E
We
label
rest
mass
m
m
0
.
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)
:
4
2
2
2
2
c
m
c
p
E
It
is
valid
not
only
for
single
particle
but
also
for
particle
system
in
the
given
time
:
2
n
1
i
2
i
2
2
n
1
i
i
2
n
1
i
i
c
m
c
p
E
2
2
2
2
2
c
m
c
E
p
We
express
kinetic
energy
and
momentum
:
2
KIN
mc
E
E
... (1)
2
f
n
1
j
2
j
2
2
1
2
2
2
1
f
c
m
c
p
)
c
m
(E
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
p
2
:
2
f
n
1
j
2
j
4
2
1
2
1
2
2
2
1
f
c
m
c
m
E
c
m
E
2
f
n
1
j
2
j
4
2
2
4
2
1
2
2
1
f
c
m
c
m
c
m
c
m
E
2
and
E
KIN
1
:
2
f
n
1
j
2
j
4
2
2
4
2
1
4
2
1
2
2
1
KIN
f
c
m
c
m
c
m
c
m
m
2
c
m
E
2
We
obtain
:
2
2
n
1
j
2
2
2
1
2
j
n
1
j
2
2
2
1
2
j
KIN1
THR
c
2m
c
m
c
m
c
m
c
m
c
m
c
m
E
E
f
f
Because
:
f
n
1
j
2
2
2
1
2
j
c
m
c
m
c
m
Q
we
obtain
2
2
2
1
2
2
2
2
2
1
THR
c
m
Q
m
m
1
Q
c
2m
Q
c
2m
c
2m
Q
E
2
1
THR
m
m
1
Q
E
In
n
o
relativistic
approximation
(
Q<<m
2
c
2
)
we
obtain
known
relation
.
In
ultrarelativistic
approximation
(Q>>m
1
c
2
a
Q>>m
2
c
2
)
:
E
THR
=
Q
2
2
2
4
2
2
4
2
1
4
2
1
2
f
n
1
j
2
j
1
KIN
c
2m
c
m
c
m
c
m
m
2
c
m
E
f
We express
E
KIN1
:
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)
:
2
CM
2
CM
x
x
c
v
1
c
E
~
v
p
~
p
y
y
p
~
p
2
CM
x
CM
c
v
1
p
~
v
E
~
E
We use polar coordinate system:
pcos
p
x
psin
p
y
~
cos
p
~
p
~
x
~
sin
p
~
p
~
y
and
We
derived
relation
for
angle
:
CM
2
CM
2
CM
2
CM
2
CM
x
2
CM
y
x
y
v
~
cos
v
~
c
v
1
~
sin
v
~
c
E
~
v
~
cos
p
~
c
v
1
~
sin
p
~
c
E
~
v
p
~
c
v
1
p
~
p
p
tan
In
nonrelativistic
approximation
,
where
v
CM
<<
c
we
obtain
known
relation
,
which
we
already
derived
.
Relativistic relation between scattering angle in centre
-
of
-
mass and laboratory system
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
:
2
2
2
1
KIN1
1
CM
CM
c
m
c
m
E
c
p
c
v
We
use
relation
between
kinetic
energy
and
momentum
:
2
1
2
2
1
4
2
1
KIN1
c
m
c
p
c
m
E
2
1
KIN1
2
KIN1
1
c
m
2E
E
c
p
We
obtain
:
2
2
2
1
KIN1
2
1
KIN1
2
KIN1
CM
c
m
c
m
E
c
m
2E
E
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
π/
2
:
2
1
KIN1
2
KIN1
2
2
KIN1
4
2
2
1
CM
2
CM
c
m
2E
E
c
m
2E
c
m
m
c
v
~
v
1
v
~
tan
In
ultrarelativistic
approximation
(
E
KIN
1
>>
m
1
c
2
and
E
KIN
1
>>
m
2
c
2
)
we
obtain
:
0
E
c
2m
c
v
~
tan
KIN1
2
2
In
the
laboratory
system,
particles
are
produced
to
the
very
small
angle
.
Relativistic invariant variables
We
can
obtain
velocity
of
centre
-
of
-
mass
during
scattering
of
two
particles
with
the
rest
masses
m
1
a
nd
m
2
by
total
relativistic
momentum
and
total
relativistic
energy
:
2
1
2
1
CM
CM
E
E
c
)
p
p
(
c
v
(1
.
a)
m
1
refers
to the projectile mass and
m
2
to target mass. We use laboratory
k
inematic variables and
we obtain:
2
2
4
2
1
2
2
1
1
2
2
1
1
c
m
c
m
c
p
c
p
c
m
E
c
p
CM
…………… (1
.
b)
Nonrelativistic
approximation
(
m
1
c
2
p
1
c
)
:
)c
m
(m
v
m
c
m
c
m
c
v
m
2
1
1
1
2
2
2
1
1
1
CM
…………….. (1.c)
Ultrarelativistic
approximation
(
m
1
c
2
p
1
c
a
m
2
c
2
p
1
c
)
:
2
1
2
2
2
1
2
1
1
2
2
2
1
2
1
1
2
2
2
1
2
1
CM
CM
)
c)
p
(
)
c
m
(
(
c))
p
(
)
c
m
(
(
1
c)
(p
)
c
(m
c))
(p
)
c
(m
(
1
)
c
p
(
)
c
(m
))
c
(p
)
c
m
(
(
1
1
2
1
1
1
2
1
2
2
2
1
2
1
p
c
m
2
1
p
c
m
1
)
c
p
(
)
c
m
(
)
c)
p
(
)
c
m
(
(
1
)
p
c
m
1
(
1
2
CM
For
m
1
m
2
:
and
c
2m
p
p
c
m
2
1
)
1
(
)
1
(
2
1
2
/
1
1
2
2
/
1
CM
CM
2
/
1
2
CM
CM
We
obtain
general
relativistic
relation
for
CM
:
using
equation
(
1
.
b)
:
2
2
2
1
2
2
1
2
c
m
E
c
p
CM
2
2
2
1
2
2
1
4
2
2
4
2
1
2
2
2
1
2
2
1
4
2
2
2
2
1
2
1
2
CM
)
c
m
(E
c
m
2E
c
m
c
m
)
c
m
(E
c
p
c
m
c
m
2E
E
1
So
that
(
m
1
2
c
4
=
E
1
2
-
p
1
2
c
2
)
:
and we obtain
1/2
2
2
1
4
2
2
4
2
1
2
2
1
2
/
1
2
CM
CM
)
c
m
2E
c
m
c
(m
c
m
E
)
1
(
… (2)
Equation
is
reduced
for
limits
E
1
p
1
c
m
1
c
2
and
p
1
c
m
2
c
2
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
(
p
2
=
0
)
:
2
2
1
2
2
2
1
2
2
2
1
2
2
1
c
p
)
c
m
(E
c
)
p
p
(
)
E
(E
s
2
2
1
4
2
2
4
2
1
2
2
1
2
2
1
4
2
2
2
1
c
m
2E
c
m
c
m
c
p
c
m
2E
c
m
E
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)
:
2
2
2
1
2
2
1
2
2
1
4
2
2
4
2
1
c
)
p
~
p
~
(
)
E
~
E
~
(
c
m
2E
c
m
c
m
s
2
TOT
2
2
1
E
~
)
E
~
E
~
(
and
s
is square of total energy accessible in centre
-
of
-
mass system. Then
TOT
TOT
TOT
2
2
1
CM
E
~
E
E
~
c
m
E
Invariant
variable
s
is
often
used
for
description
of
high
-
energy
collisions
.
The
quantity
s
楳
v敲e
畳敦畬
楮
瑨
捡獥
潦
捯汬楤敲l
.
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)
:
2
2
i
1
f
1
2
i
1
f
1
c
)
p
p
(
)
E
(E
t
……………………. (3a)
c
2m
p
2
1
CM
Energy
and
momentum
conservation
laws
are
valid
and
we
can
express
t
also
in
target
variables
:
2
2
i
2
f
2
2
i
2
f
2
c
)
p
p
(
)
E
(E
t
………………
..
(
3
b)
variable
t
is
invariant
and
it
can
be
calculated
in
arbitrary
coordinate
system
.
We
add
yet
variable
u
:
2
2
i
1
f
2
2
i
1
f
2
c
)
p
p
(
)
E
(E
u
2
2
i
2
f
1
2
i
2
f
1
c
)
p
p
(
)
E
(E
u
or
Variables
t
,
u
and
s
are
named
as
Lorentz
invariant
Mandelstam
variables,
which
sum
generally
satisfy
equation
:
f
4
2
2
4
2
1
i
4
2
2
4
2
1
c
m
c
m
c
m
c
m
u
t
s
p
~
p
~
p
~
p
~
f
i
f
i
E
~
E
~
In the case of elastic scattering in the centre
-
of
-
mass system
(for both particles and
)
)
~
cos
1
(
c
p
~
2
c
p
~
p
~
2
p
~
p
~
t
2
2
2
i
1
f
1
2
i
1
2
f
1
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
q
2
(
q
2
c
2
=
-
t
),
which
is
equal
to
square
of
momentum
transferred
to
target
nucleus
q
2
⡭
2
v
2
)
2
in
no
relativistic
approximation
.
Feynman
graph
:
Because
–
1
≤
cos
≤
1
it
is
valid
t
0
.
啳楮U
(
3
aⱢ,
睥
捡n
汯k
潮
va物扬b
慳
潮
†
ma獳
獱畡e
†
潦
數捨湧敤
灡瑩捬e
⡷楴i
敮敲杹
a湤
mm敮瑵m
)
.
Imag楮i特
ma獳
v楲瑵慬
灡牴楣汥
.
i
2
f
2
E
E
i
2
f
2
p
p
Ultrarelativistic approximation
-
rapidity
In
high
-
energy
physics
(ultrarelativistic
collisions
v敬ei瑹
潦
b敡m
灡牴捬敳
v
)
湥
k楮ima楣
va物扬b
–
rapidity
–
is
useful
to
introduce
(usually
we
have
c=
1
,
m
is
total
mass)
:
We
choose
beam
direction
as
axe
z
,
thus
we
can
write
total
energy
and
momentum
of
particle
as
:
E = m
T
c
2
cosh y, p
x
, p
y
a p
z
= m
T
c sinh y
2
e
e
cosh(y)
y
y
1
e
1
e
e
e
e
e
tanh(y)
2y
2y
y
y
y
y
Reminder
:
2
e
e
sinh(y)
y
y
2
y
2
x
2
2
2
2
T
p
p
c
m
c
m
We
introduced
transversal
mass
m
T
:
and
rapidity
y
:
z
z
p
c
E
p
c
E
ln
2
1
y
and
thus
:
cos
1
cos
1
ln
2
1
mvcos
mc
mvcos
mc
ln
2
1
y
For
nonrelativistic
limit
(
β
→
0
)
:
y
=
β
For
ultrarelativistic
limit
(
β
→
1
)
:
y
→
∞
Rapidity
using
leads
to
very
simple
transformation
from
one
coordinate
system
to
another
:
21
1
2
y
y
y
where
y
21
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
:
CM
y
y
y
~
Examples
:
GSI Darmstadt (
E
LAB
= 1GeV/A
y=0.458 β=0.875
)
SPS CERN (
E
LAB
= 200GeV/A y=6.0 β=1.000
)
LHC CERN (
E
LAB
=3500+3500GeV/A y=17.8 β=1.000
)
Relation between transversal
component of velocity and
rapidity
cos
2
1
ln
2
1
cos
1
cos
2
cos
1
cos
1
ln
2
1
)
0
y(
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