Kinematics of collision processes

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13 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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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



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
.


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





W攠扴a楮i晲m敱畡瑩潮e
⠱⸳(

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
(


㴠=e
⤠楳i

This force decreases quickly with distance and it acts along trajectory

L



A



琠㴠䰯⁶
0




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 →








o



a汳lv敲礠污牧攠獣s瑴敲楮ea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
.

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





pa牴捬e



a

桹灥牢污

楴i

湵捬敵n



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



瑨

湵n汥畳



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





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mm敮瑵m

扥景b攬e摵物湧a湤na晴敲獣s瑴敲楮ea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



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mm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




⠵(
:
































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





捯湮散敤

楴i

捯汬楳楯n

灡牡m整敲

b



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





瑨e

e摵捥

a湤



瑨e

瑯瑡l

ma獳敳



瑨e

獹獴sm
.



瑨e

a獥



敮e牡l

灯敮e楡l

m瑩潮

捡n



獰汩s



敷物瑴敮



敬e瑩癥

摩獴湣n

a湤

捥湴e



ma獳

捯牤楮瑥
:

CM
CM
r
v




䍥湴Ce



ma獳

浯瑩潮



畮景m

a湤

獴牡楧桴景牷a牤

捥瑲e



ma獳

mv敳



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



獰散楦楣

景牭



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



敬敡獥s

⡳(湴慮敯畳

摥捡ys



湵n汥i



灡牴i捬敳,

敡楯湳

ae

敡汩l敤

景

a湹

敮敲杹



a牲楶敤

灡牴楣汥
.



ae

瑡汫楮i

a扯畴

摥捡y

敮敲杹



獵捨

捡獥
.

䕬獴楣

獣s瑴敲楮e

Q

=

0

Endothermic

reactions

Q



0



敮敲杹

m畳



摥iv敲敤

⡲敡e瑩潮



湯n

灲捥敤

獰sna湥潵獬n,





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
,


††


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



灡瑩汥



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


+








+







+









=
<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



v敲e

畳敦畬



瑨

捡獥



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,



捡n

汯k



va物扬b







ma獳

獱畡e



數捨湧敤

灡瑩捬e

⡷楴i

敮敲杹

a湤

mm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ei瑹



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(
( )