Passage of Charged Particles in matter
Submitted by: Sasmita Behera,
Advisor: Dr. Tania Moulik
School of Physical Sciences
NATIONAL INSTITUTE OF SCIENCE EDUCATION AND RESEARCH
Abstract:
In this report, I study the passage of charged particles in matter. When charged particles enter a material medium
they interact with the electrons and nuclei of the medium and lose energy as they penetrate into the medium.
The energy given off results in excitation or ionization of the atoms in the medium producing ion and electron
pairs. For some particles, the interaction also appears in the the form of electromagnetic radiation in a process
which is known as “bremsstrahlung”. In this report I concentrate on energy loss by heavy charged particles which
lose energy primarily by ionization of the medium through coulomb interaction. I have discussed and derived the
necessary formulae which describes the average energy loss and also discuss the fluctuation of energy loss and
the difference in behavior due to energy lost in thick and thin absorbers. Then, I have developed a code which
calculates the energy loss by classical treatment and by quantum treatment which is known as Bethe-Bloch
formula. In particular, I have chosen liquid hydrogen as the medium, although the formula applies equally to other
media and have studied energy loss by different particles.
1. Energy Transfer in a Coulumb Collision:
In this study ,we have considered the loss of kinetic energy of an incident charged particle due to
Coulombic interaction matter. We begin with an estimate of the energy loss suffered by an charged particle
when it interacts with a free electron. The electrons in the atom can be treated as free when the particle
velocity is much higher than the characteristic velocity of the electron in orbit.
Consider the incident particle having mass M, charge z1e and velocity v1 which interacts with a material
with mass m and charge z2e . Here, we consider small momentum transfers because the trajectory of
incident particle is not appreciably altered and the material particle only has small recoil. So, the net
momentum transferred to the electron as the particle moves from one end of cylinder to other end is
essentially entirely directed in the perpendicular direction along the negative y-axis, as
so the momentum along the horizontal direction vanishes. So we can write that
P=
(t)dt
changes sign
………………………(1)
So, we consider the interaction with a charged particle in the cylindrical shell a distance b from the axis. The
distance b is referred to as the impact parameter for the interaction.
The moving charge creates an electric and magnetic field at the location of the material particle. But its
magnetic interaction is not considered here due to the small velocity of the material particle. The net force
acting on the material particle is perpendicular to the cylinder. Then the transverse field in the rest frame of
the incident particle is
=
;………………………….(2)
The electric field observed in the lab frame changes with the time. If the incident particle reaches its
points of closest approach at t=0, then at time t the transverse electric field in the LAB frame is given by
the following :
;……………………………(3)
Equation (3) follows from the theory of special relativity which I discuss in a few paragraphs. To understand
how the electric field changes between the rest frame and the lab frame (one in which the frame is moving),
we should understand the basic postulates in Galilean transformation and then Lorentz transformation. Any
theory of 'relativity' is about relationships between different sets of coordinates against which physical
events can be measured.
1.1 Galilean Relativity:
For two references frames K, and
moving with a relative velocity V, the space and time coordinates in
the two frames, (x,y,z,t) and ( x',y',z',t) are related according to Galilean transformation as follows:
= x-Vt
=t;
So velocity of a particle in two frames will be related as (dx'/dt)=(dx/dt)-V;
v'=v-V;
In the Galilean transformation, time is an invariant in the two frames while position and velocity are
not (velocity is additive).
1.2 Lorentz Transformation:
The theory of special relativity arose from the fact that Maxwell's laws of electromagnetism must hold in all
inertial frames. However, there was an apparent contradition as the equations involved the velocity of the
electromagnetic wave which was exactly equal to the speed of light derived from other sources. This then
pointed to the fact that light is a wave and that the speed of light must be a constant, independant of the
frame of reference! the This constancy of the velocity of light, independent of the motion the source gives
rise to relations between space and time coordinates in different inertial frames which is called as Lorentz
transformation after the mathematician Lorentz who empirically derived the transformation equations for
space and time to explain the constancy of speed of light.
The equations for transformation in the case of Lorentz transformation are then given by :
=γ(
=γ(
-β
-β
)
)
(4)
(5)
=
(6)
=
(7)
Where
=
γ=
,
=|β|,
(8)
1.3 Inverse Lorentz Transformation:
It is basically Lorentz transformation by interchange of primed and unprimed variables along with a change
in sign of β. The coordinates perpendicular to the direction of relative motion are unchanged while parallel
coordinate and time are transformed. The equation for Lorentz transformation
=γ(
-β
)
(9)
=γ(
-β
);
(10)
(11)
(12)
1.4 Transformation of Electric Field from rest frame to a moving frame :
By following the equations of the Lorentz and inverve Lorentz transformation ,electric and magnetic can be
written in this form:
;
=γ(
-β
);……………………………(13)
=γ(
-β
);
=
;
=γ(
-β
);………………………………(14)
=γ(
-β
);
If we considered a point charge q moving in a straight line with a velocity v in K frame, but the charge is at
rest in the K’ frame. Suppose that the charge moves in the positive
direction and its closest distance of
approach to the observer is b. The observer is at the point P. At t=t’=0 the origins of two coordinate systems
coincide and the charge q is at closest distance to the observer.
Now, in K’ frame the coordinates of the point P,where the fields are to be evaluated are
and P is at a distance r’=
away from q.
To express r' in coordinates in the K frame, we have to transform only t’ coordinate i.e t’=γ*tsince
=b,
=0 for the point P in the K frame.
For K’ frame the electric and magnetic fields are the following :
)
=γt,
=
;
;
=0
…………………….......(15)
=0
=0
=0……………………………… (16)
So in terms of the coordinates in the K Frame (substituting for r' and t' above), the non vanishing fields
components are :
=-
;
=
………………(17)
Then using the inverse Lorentz transformation, and using equations (13) and (14), the transverse fields in K
frame are:
=-
……………………………………(18)
=
……………………………………………(19)
while the other components are vanishing. We can now use equations (18), and the expression for E2
(transverse component) in equation (3) in section 1.1 and proceed with the derivation of the energy loss
formula.
1.5 Derivation of Energy Loss formula (Classical Derivation) :
So, transverse electric field from equation (18) is given by :
;
The momentum acquired by the bound particle is given by :
P=
;………………………………………………(20)
Now substituting the value of
P=
,we got previously
(t)dt……………………………………………………..(21)
=
=
;
eγb
=c
;
eγb=c;
;
=c
; put
=
=>dt=k
put
;put t= k
dӨ
=
; As t
;
t
=
=
=
;
=
; putting the value a,k and c in integration ,
And so we get,
=
=
∆p=
………………………………….(22)
The incident particle will have collisions with both and the nuclei and the electrons of the atoms. Since the
bound particle is assumed to have only a small velocity the energy transfer can be written :
∆E=
;…………………………………(23)
Here we can see that the energy transfer is inversely proportional to the square of the incident particle
velocity and the square of the impact parameter. Thus most of the the energy transfer is due to close
collision, so m=
;
.With Z electrons in an atom
and A=2Z;Here I have used z=1;and Z =1.Mass of the electron =0.510MeV,mass of the
proton=938.27MeV.So
=
So we see that atomic electrons are responsible for most of the energy loss in coulumbic interaction. So, in
proceeding sections we put m=
.
As most of energy lost by the incident particle is due to interactions with the atomic electrons. No of
electrons present in the cylindrical shell :
; but
=
Where
Integrating over the total energy transfer in each energy transfer in each b interval, total energy loss per unit
length is the following:
;……….............(24)
=
;……….................(25)
We have assumed the interaction is taking place between the electric field of the incident particle and a
free electron. But the electrons are actually bound to a atom. The interaction can only be considered to be
with a free electron if the collision time is short compared to characteristic orbital period of the electrons in
the atom. The transverse electric field in the LAB frame is very small except near t=0.So the full width at
half maximum of the E(t) distribution
;
Where
;
is times a constant of order 1.
; …………………………………………(26)
ω= characteristic orbital frequency
Substitute the values of equation (26) in the equation (25),we will get the final expression for energy
loss per unit length in the coulumbic interaction.
………………(27)
1.6 Quantum Treatment of Energy Loss Formula:
The semi classical treatment of the energy loss given in the proceeding section treats the quantum nature
of the particles in matter .A proper treatment must take into account two facts ,
(1) The energy transfers to the atomic electrons only occur in discrete amounts
(2) The wave nature of the particles
For very close collisions the classical specification of a particle as an object with a undefined position and
momentum conflicts with the uncertainty principle. So Bethe and Bloch treated the energy loss in the
matter in the frame work of their treatment.
He classifies atomic collisions according to the amount of momentum which is observable quantity in
contrast to the impact parameter which cannot be measured. He classifies two collision one is distant
collision however one can associate small momentum transfer processes with a large impact parameter and
close collision with large momentum transfer with small impact parameter.
The main features of Bethe Bloch Treatment :
(a) Both Excitation and ionization of electrons take place whose probabilities are calculated using first
order perturbation theory.
(b) Incident particles behave like plane wave.
(c) Spin and magnetic moments treated properly if Dirac functions are used.
(d) Perturbation used :Coulombic potential plus coupling to photon field.
Basing upon these features total energy loss is sum over all excitation energies weighted by cross
section,so
……………………………………………….(28)
Where
is limiting energy transfer
This above expression depends on the atomic properties through the mean ionization potential I.
For close collisions the interaction is with the free electron and not with atomic electron. So ,the
loss in close collision can be written as :
………………………………………….(29)
Where
=the cross section for an incident particle with energy E to loss an amount of
energy W in the collision with a free electron. The cross section depends on the type of incident
particle.
For spin 0 particles, heavier than electrons, the differential cross section is given by the following
expression :
………………………………………..(30)
While for heavy spin ½ particles, the differential cross section is:
………….(31)
When W <<
both cross sections reduce to that for coulomb scattering. Thus spin plays an
important role when W
.
The total energy loss is sum of contribution from close and distant collisions. Hence, the energy loss for spin 0
particles is:
Substituting equation (30) in equation (29) we obtain equation below:
………………(32)
Note that we did not use the impact parameter at all in this derivation.
To separate factors relating to the incident particle, the material medium, and the intrinsic properties of the
electron,we define classical radius of the electron is:
=
; ………………..(33)
;
Putting the values of
,
in the equation (32) we will get
…………………..(34)
From equation (34) we have seen that energy loss depends upon the charge and velocity of
incident particle, but not mass. The energy loss depends on the material linearly through the
electron density factor
and logarithmically through the mean ionization potential I.
As the velocity of the particle increases from zero ,
falls due to the
factor. As
continues to increase and the velocity of the incident charged particle is equal to the speed of the light,
then
factor begins to dominate and
starts to increase which as the region relativistic rise.
Hence, the energy loss reaches a minimum at a certain value of the momentum. Also, the relativistic rise
does not continue indefinite .We have discussed previously that interaction of charged particle with an
isolated atom. But physically for dense material inter atomic spacing is so small that dense matter makes
interaction with other surrounding particles like nucleus and proton ,so it can cause a screening of the
projectile’s electric field. This screening effect explains the reduction of energy loss for distant collision
which is basically “density effect” of the material. It causes the energy loss in region of relativistic rise which
increase by
instead of
and the energy loss is constant at large γ.
Taking density effect into account ,energy loss per unit length is of the following :::
…………………………………..(35)
2. Landau Fluctuations in Energy Loss:
The amount of energy lost by the charged particle that lay transversed a fixed thickness of absorber will vary
due to statistical nature of its interaction with individual atom in the material.
The fluctuation of energy loss by ionization of a charged particle in a thin layer of matter was first
described by Landau. This gives to a universal asymmetric probability density function characterized by a
narrow peak with along tail towards the positive values.1 It is possible due to the small number of individual
collision, each with a small probability of transferring comparatively large amounts of energy, producing
high energy recoil electrons, typically called delta rays.
The number of delta rays produced is characterized by a parameter which is the energy above which
there will be on an “average” one delta ray produced. Based on this parameter and another parameter
 which represents the low energy cutoff of possible energy losses, the most probable energy loss can
be given by
ξ(ln +0.198Where
………………………………………….(36)
= density effect correction
The full width at half maximum is given by FWHM = 4.02 ξ .
The landau function is a complex integral and is difficult to calculate. Hence, we use a simplified form
which reproduces the gross characteristics of the landau distribution given by Moyal
1 http://pdg.lbl.gov/2009/reviews/rpp2009-rev-passage-particles-matter.pdf
f( )=
………………………….(37)
with = (E-Ep)/FWHM, Ep = most probable energy loss.
For a thick absorber energy loss the landau distribution approaches a Gaussian because of increase in the
number of delta rays produced thus averaging out the behavior. I calculate the most probable energy loss
and obtain the landau distribution as given in Fig 4.
Results and Discussions:
In order to study the behavior of charged particles like pions,muons,alpha and kaon particle in
classical and quantum mechanical treatment, the plots of energy loss per unit length as a function of
momentum (both in MEVs) are studied. Whereas Figure 1 shows the energy loss per unit length versus
momentum plot in classical treatment, Figure 2 shows the same using Bethe –Bloch formula in
quantum treatment as well as classical treatment and figure 3 shows the same using the Bethe –Bloch
formula in quantum treatment only without considering the density effects. All these figures shows an
exponential decrease in the value of energy loss per unit length with increase in momentum. This can be
attributed to the fact that most of the energy loss occurs near the end of the path where the velocity is
smallest. All the
curve show the
drop for small momentum and a region of minimum
ionization for higher momentum. In figures 1 and 3 the behavior of charged particles in classical and
quantum mechanical treatment are studied which indicates that when β increases then lnγ2 factor
dominate and dE/dX starts to increase .But the energy loss is found to be increasing by lnγ instead of
lnγ2 and it is constant at large γ when density effect is taken into consideration. However when density
effect has not been considered (Figure 3),
curve decreases as ln
factor instead of lnγ and
starts increasing at momentum 400 MeV/c. In Figure 4, the Landau distribution of the energy
loss of the particles indicates that for thin absorbers small energy transfers are more likely than the
large energy transfer. However for thick absorbers the energy loss does not follow the same pattern
rather is distributed all over the range of energy transfers.
Fig. 1 : dE/dx vs Momentum of heavy charged particles using the equation as obtained in the classical
treatment
Fig 2: dE/dx vs Momentum of heavy charged particles using Bethe Bloch and classical treatment.
Fig 3 : dE/dx vs momentum using Bethe-Bloch formula (without density correction)
Figure 4:-Landau distribution in energy loss
Acknowledgement:
I would like to thank my advisor, Dr.Tania Moulik under whose guidance I am able to complete my project.
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