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POSITRON AND ELECTRON INTERACTION WITH MATTER
DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE AWARD OF THE DEGREE OF
iWasiter of ^fjilosopljp
"^ *
IN
PHYSICS \yn ^n. ni
BY
A R U N PAWAR
• • i * . ;
m A^
UNDER THE SUPERVISION OF jx f ~ l ,
DR. T. S. GILL *• * -
DEPARTMENT OF PHYSICS ALIGARH MUSLIM UNIVERSITY
ALIGARH (INDIA)
2003
/v' v ' "' «-,
I % s .
/ '7 li ;v"-'=\3ri^
1 7 JAN 2011
DS3801
Dr, T. S. Gill
DEPARTMENT OF PHYSICS ALIGARH MUSLIM UNIVERSITY ALIGARH- 202002 (U. P.), INDIA Phone: +9] 571 2701001 (Off.)
+91 571 2601768 (Res.)
CERTIFICATE
This is 10 certify that the work reported in this dissertation entitled
"POSITRON AND ELECTRON INTERACTION WITH MATTER" is
original work of Mr. Aran Pawar, carried out under my supervision.
(Dr. T. S. Gill)
^^©oe^ir/j^
Ackno wledgemen ts
I am deeply indebted to Dr. T. S. Gill, Reader, Physics Department
Aligarh Muslim University, Aligarh, whose encouragement and guidance
has been a constant source of inspiration to me throughout this work.
I express my gratitude to Prof. A. K. Chaubey, Chairman, Physics
Department A.M.U. Aligarh, for their interest, and extending
departmental facilities during the period of this work.
I am thankful to all teachers, staff members and non-teaching staff of this
department.
I wish to express my thanks to Dr. Manoj Sharma (Young scientist) at the
same department and Dr. Pankaj Kumar Sharma with Dr. Avinash
Agrawal for their active help and cooperation.
I also wish to thanks to my younger brother Anuj and Mr. Sumit Pal Gill
for their help to make computer program for calculation.
I wish to special thanks to Mr. Nisar Ahmad to provide necessary help
and suggestion during the calculation,
I wish to thanks various of my colleagues, Ms. Unnati, Mr yatendra, Mr.
BK Sharma, Ms. Anjna Maheshwari, Mr. Ramprakash, Arafat Firdos,
and Mr. D. Singh for their cooperation during this work.
Finally I want to express my very special thanks to my parents with all
family members and intimate friends Dr. Sarita Verma and Ms. Nootan
for their support and constant encouragement and inspiration during the
period of work.
(ARUIV PAWAR)
CONTENTS
CHAPTER - ONE Introduction to different approaches in Positron and
Electron Penetration
1.1 Introduction 1
1.2 Range definition 4
1.3 Experimental Work 7
1.4 Theoretical developments 8
> C. S. D. A. Ranges 8 > Average penetration depth 9 > Projected Ranges. 11
References 15
CHAPTER- TWO Penetration of Positron and Electron Through different
materials
2.1 Introduction
2.2 lonisation and Excitation loss
2.3 Bremsstrahlung
> Classical theory of Bremsstrahlung > Quantum mechanical theory of
Bremsstrahlung > Comparison with classical theory
2.4 Empirical relations of total stopping power
of Electron and Positron
2.5 Multiple scattering theories
2.6 Straggling.
References
17
18
19
20
21 22
23
24
29
30
CHAPTER- THREE Theoretical approach for buildins transmission curves
3.1 Introduction
3.2 The present methods of calculation
3.3 Present methods for calculating
Ranges and Absorption coefficients.
References
31
32
36
39
CHAPTER- FOUR Results and Discussion of present theoretical approach
4.1 Introduction 40
4.2 Absorption coefficients 40
4.3 Comparison of present theoretical
and experimental Ranges. 41
References 44
T:L^Cr(RP^(FEJ^i:(RA'TlO!N'
y Introduction y (Rfluge {fefinition ^ ^JCperimentaCwor^ ^ lUeofeticaCdeveCopments
1.1 Introduction
The information about the interaction of electrons and positrons through
different materials is frequently required for the experimental methods
in nuclear, atomic and solid state physics.
The transmission and penetration of charged particles tlirough matter
has a great importance for experimental as well as theoretical Physics. In
the experimental study of the energy levels and transition of nuclei,
there are many cases, which require the measurement of the kinetic
energy of a charged particle by method, which depend upon absorption
and scattering phenomena.
The study of the passage of electrons and positrons through matter has
very useful applications in nuclear physics, semi conductor physics,
health physics and several other related fields. It is also useful for the
study of radiation damage, biological effect, and energy distribution at
different depth in matter.
The mechanism by which a charged particle loses its kinetic energy, or
is deflected from its original path, involve two principal types of
interaction,
(1) Inelastic interaction
(2) Elastic interaction
Further we can divide these two into following
(a) Inelastic collision with atomic electrons.
(b) Inelastic collision with a nucleus.
(c) Elastic collision with atomic electrons.
(d) Elastics collision with a nucleus.
In an absorbing material, a moving particle is slowed down and finally
brought to rest by the combined action of all four of these elastic and
inelastic processes. A particle whose initial kinetic energy is say 1 MeV
may have more than lO'* individual collision of each type.
Which type of interaction, if any will occur when a swift
particle passes a particular atom is described only by the laws of chance.
From collision theory, one can obtained the probabilities of any
particular type of collision, of any particular energy loss, and of any
particular change of direction of the motion of the incident particle.
After the first collision, these probabilities can be applied to a second
collision, then to a third, etc. This method is very complicated, but some
reasonable results have been obtained, notably at Los Alamos, using
electronic computing devices.
Electron bombardment of a sample is unique to microprobe analysis and
produces a large number of effects from the target materials. It is critical
to understand the volume of material from which these are produced and
the effects themselves that includes: X-rays (both continuum and
characteristic), backscattered and transmitted electrons, secondary
electrons Auger electrons etc. (fig. 1.1).
lonisation, bremsstrahlung radiation and annihilation are the three
mechanisms by which electron loses energy in medium.
Coulomb interactions between fast moving electrons and molecular
electrons excite and ionise the molecule producing ion pairs.
When a fast moving electron is accelerated or decelerated, a photon is
emitted, and such photons are called bremsstrahlung radiation (braking
radiation).
IiKideitf Ele( ti oil beam
.vecoiidJUT Elecliou
Aiigei' Elecri'Oii
back scartered Ekclioii;^
Tliiii s'peciineii
cliviiacreiisfic X - l a v s
Tiaiisinifed ElfCflOILS
bi-cin.sti'aldmt;s:
visible ligltr
If
absorbed ciuieiif
diffeiecfed election
Fig. 1.1 Effects produced by electron bombardment of a material
Although the positron is an anti particle of the electron, the cross section
(probability) for direct annihilation is much smaller than that for elastic
or inelastic scattering. For these latter processes, treating the positron as
a positive electron and using the usual theoretical methods developed for
electron scattering from atoms can carry out calculation. There are
several important differences between electron and positron scattering.
However since the positron is a distinct particle from the electron,
exchange interactions with the bound atomic electrons do not occur.
More interestingly, the formation of positronium (the bound state of an
electron and a positron) can and dose occur in positron-atom scattering.
In addition, the fact that the sign of the interaction between the positron
and the atom is the opposite of that for electrons leads to rather different
behaviour of the cross section.
As the electron passes through matter it loses its energy in ionising and
radiative collision. In each of these it may suffer significant deflections.
In addition there is a large number of deflection due to elastic scattering.
The net result is that the electron's path as it passes through the absorber
is very tortuous. In practice, one finds the path length to be from 1.2 to 4
times the thickness of the absorber traversed, the ratio being largest for
slow electrons in high Z materials.
The range R of a particle is an experimental concept, relating to the
thickness of an absorber, which the particle can just penetrate. Several
distinct definitions of range, which depend upon the method employed
to determine them, are in common uses.
1.2 Range Definitions
If a curve is plotted between the fraction of the incident Positrons and
electrons which pass through a given thickness and those thickness, It is
obtained that for high thickness of the absorber the curve passes into the
back-ground, which is due to cosmic and gamma rays. Schonland ' and
Flammersfield^ ', define the point at which the extension of the linear
portion of the transmission curve meets the background as the practical
range, the point where the tail of the transmission curve meets the back
ground is called the maximum range. The maximum range of electrons
and positrons is a very important quantity but it is very difficult to
explain theoretically, because the diffusion is a complicated
phenomenon, which sets in at the end of transmission curve. The
extrapolation of the linear portion of the transmission curve of the
monoenergetic electrons to the thickness axis known as practical
maximum range. A similar definition of range has given by some
workers like Ebert et al ^^\ Tabata et al ^'^\ They define the extrapolation
range Rex, as the point where the tangent at the steepest point on the
almost straight descending portion of the transmission curve meets the
thickness axis.
The practical range, the practical maximum range, and the extrapolation
range are very useful experimental quantity, including the straggling
effects hence can not be compared with theoretical straggling free range.
The range of electrons theoretically, can be given as the limiting
thickness of the absorber beyond which essentially none of the
originally incident electrons emerge. But due to the straggling effects
such a thickness practically dose not exist.
It is not possible to compare the experimental measurements with the
theoretical ranges.
c
o o
a.
CO c
6 z M
CO
UJ
00 u
H
\J
C3
* # "^ t> CO c C3 L_
0
fS
• »"
eJD
1
oi V CO 2 i— ^ c ^ E CO
E (U
x : • - 1
T3
§
^
.
o
o 31VU DNIlNnOO
The straggling free practical range Rfj-^ for electrons and positrons can
be define as the point where the extrapolation of the linear portion of the
transmission curve meets the straggling portion of the curve, when it is
extrapolated in the backward direction. All the ranges, which have
defined here, are shown in fig. (1.2) But extrapolated range R x is not
shown because it cannot be shown on semi log graph.
It has been proved experimentally that the range/J^^ is not sensitive to
the strength of the used source. But extrapolated range R x is very much
sensitive of the source strength and percentage of transmission reached.
For light and medium atomic number absorbers the tail of transmission
curve can be shown by a straight line this tail corresponds to straggling,
and error involved in its extrapolation in the backward direction is very
small. For high atomic number absorbers straggling part of the
transmission curve cannot be represented by a straight line because
some error is possible in its extrapolation in backward direction to find
R^fp. In the measurement that has been done by Gill ^^^ et al. this error is
found of the order of 2% for gold and lead.
Electron's total path length is a quantity, which is completely different
from its range. The distinction can be visualized from fig. (1.3) Where R
is the range.
The total path length has been observed in a few experiments, using
cloud chambers or photographic emulsions. Among these, one by E.J.
Williams ^^ provides a direct comparison of total path length S to range
R. In this experiment, monoenergetic electrons were produced in a cloud
chamber by the photoelectric absorption of monochromadc X-rays. The
Incident electron
End of path
6 ray or branch
Absorber
Fig, 1.3 _ ^ ^ ^
Tlie thickness of absorber is R + dR.
If the absorber had been of thickness R, the electron would have just
penetrated it and would be said to have a range R. The total path length S is
measured along the actual path of the electron and is always considerably
p-cater than R.
total path length and the maximum distance reached in the initial
direction were than measured.
Fig. (1.4) presents Williams observations on distribution of path length
S and range R in oxygen for 145 individual electrons whose kinetic
energy was 19.6 Kev. Several fundamental feature of the interaction of
electrons with matter can be visualised from a study of these curves in
this energy domain, radiative losses are negligible and curves represent
the effects of ionisation losses, elastic scattering, and straggling of
energy losses.
The curve S shows the fraction f of the electrons whose path length
exceeds the distance D cm. The slope of the curve ds/dD is the fraction
of electrons whose path lengths lie between D and D+dD. These slope
are sufficiently symmetric about f = 0.5, so that the mean path length S
can be taken as the path length at f = 0.5, when one half the electrons
have been stopped.
The broad distribution of path length is noteworthy and is due to the
statistical distribution of energy losses, or straggling for each electron.
Some have large losses, including one or more "branches" due to hard
collision, and have short path lengths. Other suffers smaller and fewer
losses per millimetre of path and have much longer path lengths. The
theoretical value of (dT/ds)ion relates to the average energy losses, and
thus its integral is S, corresponding to the average path length, or mean
path length.
The average path length of an electron of K.E. T will be
<n c o i -
oQ ^ A < co
^ Q G A ' - ' c c : «^^ O W) ^ . E c >
4 »
1.0
O.b
0
Distance D, cm of oxygen
Fig. (1.4)
The distribution of path lengths (S) and of range (R) for 19.6 Kev electrons
in O2 at 0° C and 1 - atm pressure:
R = 0.32 cm (mean range)
Ro =0.52 cm (extrapolated range)
S = 0.64 cm (mean free path length)
So = 0.82 cm (extrapolated path length)
^ = I * = l 7 : ^ "- ^ n
{dTldS)
where dS •
+ \(i^)ion \d^)rad
equation (1) will not be valid as the lower limit of T approach zero (Ti
1.3 Experimental Work
Some experimental work with electrons and positrons had been
reviewed by Katz and penfold ^ in Aluminium. They did experiment
only for electron interaction. Due to lack of positron sources no
attempt was made for positron transmission.
Seliger ^"^ and Gubemator ^ °-' ^ did some work on the transmission of
positron in the energy range of 180 Kev to 960 Kev, and 50 Kev to
160 Kev. Gubemater's experiment concludes that the ranges of
positron in Al are less than those of the electron in the energy region
below 160 Kev. These results are in qualitative agreements with the
similar measurements of Seliger. In 1959, Gubemator and
Flamammerfeld " mearured the ranges of 40 to 160 Kev electrons
and positrons in Cu, Ag and Au. They reported that in Cu the range of
positrons is less than that of electrons of the same energy while in Ag
and Au positron has large ranges than electrons.
Some work of on transmission of electrons and positrons has been
done by Gill ^^ et al. They studied experimentally the penetration of
electrons of energy E ax = 0.25 Mev, 0.77 Mev, 1.53 Mev, and 1.71
Mev through a large number of materials including some rare earth
materials. Also they experimentally investigated the penetration of
1.88 Mev positrons in various materials including rare earth materials
then they compared the results of electron and positron transmission.
1.4 Theoretical Developments
(a.) C.S.D.A. Ranges
The rate of energy loss of electrons and positrons is always subject
to statistical fluctuations. For the simplified evaluation it is assume
that during slowing down process, the rate of energy loss along the
entire path is always equal to the mean rate of energy loss. This is
called C.S.D.A. ranges, (CSDA means, continuous - slowing - down
- approximation.). The CSDA range is the path length, which a
particle travels in the course of slowing down, in a homogenous
medium. Berger and Seltzer ' ^ calculated these CSDA ranges with
energy loss due to ionisation and excitation ^^^\ and also
bremsstrahlung process ''*l
CSDA ranges are calculated by integrating the reciprocal of total
stopping power: -
T\
1 fdE^' dT + R'-iT,) (1.3)
where Ti is some lower limit of energy below which the calculation can
not be possible
Normally T,= 1 KeV
For intermediate and high-energy electrons R (Ti) is considered
negligible. And we know:-
UE^"-
p V dx ;,„,„/ p
dE_
\dxy
1 ^dE^ +
coll P
±
(1.4)
where + ve and - ve signs are for positron and electron respectively
and p is the density of materials.
If we compare the CSDA ranges of electron and positron in any material
with the conesponding experimental values, it is observed that CSDA
range are always greater than the measured values.
(b.) Average penetration depth
Rohrlich and Carlson ' ^ calculated average penetration depth, Zj*.
Which is define as the amount of absorber thickness which when placed
in the path of the beam of positrons and electrons, such that particles
lose completely their initial orientation.
Mathematically the average penetration depth is given by the condition
< C o s e )average = l /c (1.5)
where 6 is the angle of multiple scattering.
This is the condition when the particle losses its memory of initial
direction at average energy
Ed = Ydmc' (1.6)
The average penetration depth corresponding to yd" is given by,
10
2 * = j ^ *(/«,/; dy
where ^M/o'7) = (Co5^>',, = G{r),
And G*(/) = 7 + 1
7^ Y
y-ip
(1.7)
(1.8)
(1.9)
The constants a~ and b* were considered to be approximately of the
same order of magnitude for small and large values of atomic numbers.
Using this expression Rohrlich and Carlson obtained the average
penetration depths of positrons and electrons for energy ranges 0.1 to
2.04 MeV for Aluminium and lead. They found that the valueZj/z;
first increase with energy upto 1.0 MeV and there after decrease with
energy. While in the case of lead, this value of Zljl'^ first increase
with energy upto 1 MeV and then becomes constant. But they did not
explain the reason of this behaviour.
Some other limitations of this method are as follows:
1. The constant a" and b* had been designed forZ^/z; in almunium
and lead only. It is very difficult to give seprate constant for every
element.
2. There is no indication has been given for the percentage error, in
the final result.
3. In the expression for ( CosO ) and Zd^ the rate the energy loss that
is dT/ds is found. They used energy loss due to collision only; the
energy loss due to radiation has been not taken into account. The
estimated contribution due to radiation loss of 2 MeV electron in
the case of lead is about 21% of total energy loss. And for energy
more than 2 MeV the radiation loss increase and it should be
taken into account.
Actually the average penetration depth given by the equation (1.5) is
not experimental quantity. There is no known evidence, which could
support some kind of relationship between the estimated average
penetration depth and the measured ranges. Tomlin ' ^ have
reinvestigated the problem of electron penetration using lewis theory ' ^
of multiple scattering and reported relatively simple expression for the
first and second moments of the electron distribution in depth.
(c.) Projected ranges
Rohrlich and Carlson ''' considered only the elastic scattering of
electrons and positrons to calculate the value of Z / .
While the CSDA ranges ' ^ were based on purely inelastic
consideration. Both the approaches do not interpret the measured ranges.
When electrons and positron passing through an absorber both elastic
and inelastic interaction are possible.
Both the process have been taken into account for calculating the
projected range Rp* for positrons and electrons, by Batra and Sehgal. ' " 20]
The mean projected range can be define as the mean projection of the
path of these particles on the direction of incident in the absorber. They
considered that the inelastic scattering is statistically independent of
energy loss fluctuations. Near the end of the range multiple scattering is
large due to the small energy. At the large multiple scattering angle the
electrons diffuse randomly this is called straggling. They assumed that
the electrons first undergo a straight motion and their interaction with
matter is only through inelastic process.
12
When electrons traverse a small thickness x of the absorber the mean
square angle of multiple scattering ( 0 >x~ is required. The total stopping
is needed as input for calculating < 0 >, power — P dx • - '" V ^ Jiolal
A simple empirical relation of total stopping power found by Batra and
Sehgal ' ' ° this expression can easily be integrable.
In order to take into account the random motion of electrons and
positrons by multiple scattering they given a definition of transport
mean free path tr, this transport mean free path can be define as the
average distance a particle traverse before being scattered through an
angle > n/2. L 9 - 1 .
If Tr' is the energy and ( 0 >x~ is the mean square projected angle
corresponding to the instant when the motion of the particle become
random, the projected range is given by:
Ki^)=RL{T)-R!jT^^) • (1.10)
These values of projected ranges ' " ° are comparable with the
experimental values given by Gill et al ^^\ For 6 < Z > 13 materials the
agreement is good. For intermediate and heavy elements the calculations
by Batra and sehgal ' " ° give lower values of range as compared to the
experimental values by Dr. Gill ^^\ For intermediate Z values the
difference is small but goes on increasing with increasing value of Z. At
Z = 82, the theoretical values are off by about 25% for particle energies
of 1 MeV.
The difference in the theoretical and experimental values may be due to
following reasons:
13
1. Batra and Sehgal ' ' ^ had used Mott's expression ^ ' for elastic
scattering cross section, tliis expression is in the form of a power
series in aZ, where a = 1/137. They used only the first term of
this series and leaved the higher order terms. For small value of Z
the term {aZf and (aZ)^ and so on can be ignored but for high Z
values these term can not be ignored.
2. In these calculations ' "^°\ the range coming from the diffusion
part was not taken into account.
It is found from their reference 20, figs 5-6, that as the incident kinetic
energy of electrons and positrons decreases and also with the increase of
atomic number Z of the absorber, the fraction of energy left with these
particle increases.
If the diffusion part of the ranges is taken into account, the calculated
values of straggling free practical ranges in very good agreement with
experimental values.
Gill et al ^^ have developed a theoretical model for obtaining
transmission curves of different energies electrons and positrons using
the energy loss expression '''" ^ and Mott's scattering ^ ' cross section
for the scattering of these particles. The values of ranges and absorption
coefficients thus obtained have been compared with the experimental
data.
If a comparison is made between the experimental values of Dr. Gill
and the theoretical values of Batra and Sehgal, ' "' ° the values are in
good agreement for low Z- materials except the case of carbon. For the
intermediate Z the difference is about 16% and is about 30% in high Z
materials. If higher order term is taken into account in Mott's
expression, or if the correction factor is used the values of range is
14
improved by about 6 to 12% in low Z and 10 to 18% in high Z
materials.
If the contribution of diffusion part is also taken into account with
correction factor or higher order term, the difference in experimental
and theoretical values are - 0.01 % to 6% in low Z for low energy and
about 6 to 9.5%) for high Z at low energy. Similarly this difference is
about - 0.50%) to 5.3%) in low Z for the energy 1.53 MeV and it is upto
8.4% in high Z, at 1.53 MeV energy.
References
1. B. F. Schonland, Proc. Roy Soc. (London) A135, 429-58 (1925).
2. A. Flammersfield, Z. Nature-forsch 2a, 370 (1947).
3. P. J. Elbert, A. F. Lauzon E. M. Lent, Phys. Rev. 183, 422-30
(1969).
4. T. Tabata, R. Ito, S. Okabe and Y. Fujita, J. Appl. Phys. 42, 3361
(1971).
5. Dr. T. S. Gill, Ph.D. Thesis 1988, A. M. U. Aligarh
6. E. J. Williams Proc. Roy. Soc. (London) ABO, 310 (1931).
7. L. Katz and A.S. Penfold, Rev. of Mod. Phys. 24, 28-44 (1952).
8. H. H. Seliger, Phys. Rev. 88, 408 (1952).
9. H. H. Seliger, Phys. Rev. 100, 1029 (1955).
lO.K. Gubemator, Zeit Fur Physik, 152, 183 (1958).
U.K. Gubemator and A. Flammerfeld, Z. Phys. Vol 156, No.2, 179
(1959).
12.M. J. Berger And S. M. Seltzer, Studies in penetration of charged
Particles Through matter. National Research council publication
No. 1133, P. 205-268(1964).
13.F. Rohrilish and B. C. Corlson, Phys, Rev. 93, 38 (1954).
14. W. Heither, the quantum theory of radiation (Oxford Univ. press
London, 1954) third edition, P. 242.
15.S. G. Tomlin, Proc. Phys. Soc. 99, 805 (1966).
16.H. W. Lewis, Phys. Rev. 78, 526 (1950).
17.R. K. Batra and M. L. Sehgal, Nuclear Physics A156, 314- 20
(1970).
18.R. K. Batra, Penetration of electrons and positrons in matter,
Ph.D. thesis (1971).
19.R. K. Batra and M. L. Sehgal, Phys. Rev. B, 23, No. 9, 4448
(1981).
20.R. K. Batra and M. L. Sehgal, Nucl. Inst. Methods 109, 565
(1973).
21.N. F. Mott. Proc. Roy. Soc. (London) A124, 425, (1929),and
A135, 429 (1932).
^ Introduction y Ionization andej(citation foss y (BremsstrafiCung y Empirical reCations of totaCstopping power of electron and
positrons y MuCtipCe scattering tfieories > StraggCing
17
2.1 Introduction
When a beam of electrons or positrons of some kinetic energy interact
on a target foil, they penetrate through the foil by undergoing two major
kinds of interaction: the slowing down process and the elastic scattering.
These processes depend upon the energy of the particle and atomic
number of materials traversed. A moving electron by virtue of its
moving electric field creates disturbances in the electronic structure of
the atoms and molecules of the medium through which it moves. These
disturbances is due to transfer of energy from the moving electron to the
surrounding atoms and molecules, raising them to excited states and
frequently producing ionisation and the rapture of the molecular bonds.
Most of the energy of the incident electrons or positrons is dissipated by
the collisions between the incident electrons and orbital electrons of
material. The energy loss due to collisions between the incident
electrons and the nuclei is negligible small because of the large
difference in their masses.
Fast electrons or positrons also lose energy due to the emission of
electromagnetic radiation in the coulomb fields of nuclei and the orbital
electrons. This is known as Bremsstrahlung loss. This loss is very small
due to the coulomb field of orbital electrons in comparison to that of
nuclear coulomb field.
The total energy loss per unit path length is given by:
\ dE^' ( j_^Y f_i^Y p dx J lolal
+ P'^-'') colli V Pdx)
(2.1) rud.
2.2 lonisation and excitation loss
Bathe and Bloch ' gave the expression for the average energy loss due
to the coHisions between the incident electrons and atomic electrons,
using Moeller's differential cross section ^^\ But the expression given by
Bathe and Bloch cannot distinguish between the energy loss of electrons
and positrons. Rohrlich and Carlson ^^ had given the formula of average
energy loss due to the collisions between positrons and electrons with
orbital electrons.
The Bathe Bloch formula "' of average energy loss by collision is
obtained for electrons under the assumption that above a certain
fractional energy transfer E, the atomic electrons are treated as free.
Under this assumption, Moller's cross section ^ •' for scattering of free
electrons by free electrons at rest in Bom's approximation is applicable.
For low energy transfers i.e. 0< E < Ei, an explicit summation over the
various excitation probabilities can be carried out. The average energy
loss per unit path length in a medium for electrons and positrons is given
by
^ 1 ^ p dx
\ ±
/colli
Ine'NZ
Am^c^j^'
^ r y + O I' 2
+r{/)-^ (2.2)
where /"(/) = i-/?
and /"( /)-21n2-4
2 2 / - 1
r in 2 + -
8 r J for electrons
12
^^ 14 10 4 23 + + ^ + • cr + 1) ir+if (r+i)'
(2.3)
(2.4)
for positrons
Where
e = electronic charge
iTioC = rest mass energy of electron
Z = atomic number of the absorber
P = v/c
T = Kineticenergyofelectron in units of its rest mass energy
y = total energy of electron expressed in units of rest mass
energy
I = average ionisation potential
A = atomic weight
p = Density of medium
5 = Density effect con-ection factor
it is clear by this equation that the difference in the average energy loss
of positrons and electrons is due to the function f * (y). Rohrlich and
Carlson ^^^ plotted these functions versus y. They conclude that slow
positrons lose energy at a faster rate than slow electrons and vice-versa
occurs in the case of fast electrons and positrons. At energy of about 350
Kev the rate of energy loss positrons is the same as that of electrons ^^\
2.3 Bremsstrahlung
Besides the energy loss due to excitation and ionization the energy loss
is also possible due to emission of bremsstrahlung. This loss occurs
when the electrons accelerated in coulomb field of nucleus.
20
Bremsstrahlung is the continuous spectra associated with the deflection
of incident charged particle by the coulomb field of nuclei.
(a) Classical theory of Bremsstrahlung
According to classical theory whenever an incident charged particle is
deflected from its path or has its velocity changed it should emit
electromagnetic radiation whose amplitude is proportional to the
acceleration. The acceleration produced by a nucleus of charge Ze on a
particle of charge ze and mass M, is proportional to Zze / M. (M is the
mass of the incident particle).
We know by coulomb force F oc qi q2 / r
Or M.a oc Ze.ze
Or a oc Zze^/M (2.5)
Thus the intensity, which is proportional to the square of the product of
the amplitude and the charge ze, will very as
Amplitude oc acceleration
Intensity oc (amplitude x ze)
oc Zze
•ze
Or intensity oc (Z^z''e^)/M^ (2.6)
Thus the total bremsstrahlung per atom varies as the square of the
atomic number of the absorbing material. We also see that the total
bremsstrahlung varies inversely with the square of the mass of incident
particle.
Therefore Proton and alpha particle will produce about one millionth of
bremsstrahlung of an electron of same velocity. Because of this strong
mass dependence bremsstrahlung is almost completely negligible for all
21
swift particles otlier than electrons. The incident particle can radiate any
amount of energy from zero to its kinetic energy 'T'.
Thus the maximum quantum energy (hvmax) at the short wavelength of
the continuous X- rays spectrum is
hVmax = T. This relation is called Duane and Hunt's law. *^
(b) Quantum Mechanical Theory of Bremsstrahlung
The deflection of a swift electron of velocity v = pc, rest mass mo, by a
nucleus of charge Ze fall in the domain of Z/137 p « 1, if Z is not too
large.
In a quantum mechanical treatment we know that the first approximation
of Bom's method calls for neglecting Z/137 p compared with unity.
Born's first approximation is therefore applicable to the problem of
bremsstrahlung, except for initial or final electrons of very low velocity.
Bethe and Heitler and others, using Dirac's relativistic theory of the
electron and the first approximation of Bom, have developed the
quamtum mechanical theory for the bremsstrahlung of relativistic
electrons. The non-relativistic theory has been developed by
Sommerfield. Using exact wave functions.
The total bremsstrahlung energy (in MeV per incident electron of kinefic
energy Eg in MeV) is given by
I = KZ Ee^ for E, > 2.5 MeV, where K ~ 7 x lO"""
In quantum mechanical theory a plane wave representing the electron
enters the nuclear field, is scattered and has a small finite chance of
emitting a photon. The electron is acted on by the electromagnetic field
of the emitted photon as well as by the coulomb field of nucleus. The
intermediate states of the system involve the negadve energy state,
22
which characterized the Dirac electron theory. The theory of
bremsstrahlung is intimately related to the theory of electron pair
production by energetic photons in the field of a nucleus. Because the
radioactive process involves the coupling of the electron with the
electromagnetic field of the emitted photon the cross sections for
radiation are the order of 1/137 times the cross sections for elastic
scattering.
(c) Comparison with Classical Theory
The classical theory of bremsstrahlung gives incorrectly the emission of
radiation in every collision in which an electron is deflected. Yet for the
averages over-all collisions, the classical and the quantum- mechanical
cross section are of the same order of magnitude,
Z 2 f ^2 \
rad J 3 ^ K'^hc' J
cm^ I nucleus (2.7)
In the quantum mechanical model there is a small but finite probability
that a photon will be emitted each time a particle suffers a deflection
however this probability is so small that usually no photon is emitted. In
the few collisions, which are accompanied by photon emission, a
relatively large amount of energy is radiated. In this way the quantum
theory replaces the multiple of small classical energy losses by a much
smaller number of large energy losses the average being about the same
in the two theories. Of course, the spectral distributions are very
different in the two models. All experimental are in agreement with the
quantum mechanical model.
23
2.4 Empirical relations of total stopping power of electrons
and positrons
The expression of the total stopping power of electrons and positrons
has been given by Batra and Sehgal ^'^\
For the energy range T < 500 Kev, total stopping power is given by
' 1 dE^'
p dx = {mz^c)r[r) (2.8)
/ iiiiul
.2.4
where ^(Y) = — — for positrons (2.9)
,2.56
and f (y) = —J— for electrons (2.10)
and Y = 1— (2.11)
and for the energy between 0.5 Mev to 5.0 Mev the expression is given
by
^ 1 dEY v' =imz + c)^— (2.12)
The value of constant a , b , m, z, are given in table
Table-1 forT<500Kev
24
S.No.
1.
2.
3.
Z- values
1 < Z < 1 0
10<Z<38
3 8 < Z < 9 2
—• - 2
m (Mev.cm /gm)
- 0.0300
- 0.00595
- 0.00285
C (Mev.cm /gm)
1.1700
0.9282
0.8100
Table -2 for T lying between 0.5 to 5.0 Mev
S.No.
1.
2.
3.
Z- values
1 < Z < 10
10<Z<36
3 6 < Z < 9 2
m (Mev.cm /gm)
- 0.0330
- 0.0097
- 0.0048
C (Mev.cm^/gm)
1.3230
1.0911
0.9156
= -0.0038 = - 0.0040
= 1.8402 = 1.8160
2.5 Multiple scattering theories
If the thickness of absorbing materials or foil is such that a single
particle can suffer a large number of scattering collisions, then statistical
method become applicable for the estimation of mean deflections. These
25
phenomena are called multiple scattering. The determination of the
ranges from the electron transmission curve is very difficult due to the
statistical fluctuations of path lengths of the electrons in the material
traversed. So many attempts have been made to develop the theory of
multiple scattering. Several review papers on small angle multiple
scattering of fast charged particle are available in the literature ^ ' l
The single scattering of charged particle by the coulomb field of a
nucleus was first given by Rutherford. The differential cross section of
elastic scattering of a non-relativistic particle of charged ze by a nucleus
of charged Ze within the angle 9 to 0 + d9 is given by:
cT[e)i7T%\\\ed0^—~Y^,—-.inimede (2.13) 16£' sin [OH]
where E is the K.E. of the particle .
Mott ' ^ derived an exact expression for the scattering of fast electrons
by a bare nucleus. He used the second order wave equation for Dirac
electrons thus the spin and the relativistic effects were taken into
account. Mott's formula is valid when Ze /hv « 1.
This formula is in the form of a series in Legendre polynomials and its
expansion in power of aZ is equivalent to the solution in Bom
approximation. Here a = Z/137, fine structure constant. The resulting
expression contains only first order tenn of the series in aZ is given by:
^'^^'^^ = ^ | " S : 1 —JrZiT\ [l->^'sin^^/2 + H^^sin^/2(l-sin^/2)] ( Y 1
_sii/(^/2)_
(2.14)
26
0 T 9 9
where YO "= eVmoC" and y = (T + nioC )/moC
P= v/c, nio = rest mass of electron.
The upper sign stand for positrons and lower for electrons,
The difference between the cross section of electrons and positrons is
that the sign of the last term is reversed. This is because the last term
proportional to the first power of the interaction potential. It is also clear
from the equation (2.14) positron scattering is always less than the
electron scattering under identical conditions. The last term gives the
difference between positrons and electrons scattering and is very
important for heavy nuclei. For heavy element the power series in aZ
converge too slowly. The theory of multiple scattering of fast electrons
given by Willams ^ "'° is based on the assumption that the small angle
multiple scattering will yield a Gaussian distribution. The projection of
the angular distribution of the scattering on a plane containing the
incident beam was also considered. The effects of the shielding of the
nucleus by its orbital electrons and the finite size of the nucleus on the
scattering probability were also included. The effects of the finite size of
the nucleus and its shielding by orbital electrons were also considered.
In order to derive the expression for scattering cross section ' "" ^ he used
the potential
V{r) = —e'''" (2.15) / •
where a = —^ and ao = radius of the first bohr orbit of the Hydrogen
atom. The expression of the scattering cross section is written as
4 sm' 012 ( l - ^ 1 l + t \
2asm0l2 (2.16)
27
the scattering for 9 » 9min the shielding thus little affects,
where Qm\n ~ ^/a, and it reduces for 8 « Qm\n •
the scattering through an angle 0 depends on the field at distances from
the nucleus of the order of %/Q, this the consequence of bom
approximation.
If the finite size of nucleus is not considered William ^ " ^ used the
potential:
V{r). (Ze'^
V ^ J
[\-e-'"') (2.17)
where b = nuclear dimensions,
and the scattering cross section is given by,
a\ {0). zV (l-/?^ sin'012
1 + ^bsm0l2^''
-2
(2.18)
It can be seen that the nucleus size correction reduces the scattering,
for 9 > ?C/b, but the shielding reduces it for 9 < ^/a, where a = atomic
dimension.
Thus the correction apply to regions which do not overlap, hence the
general solution of a(9) is given by:
Am'V \s\n 012)1 4a ' \ \ / /
(2.19)
28
Elton " reported on the effect of the finite size of the nucleus on the
elastic scattering cross section. In 1948, Moliere ' ^ proposed a useful fit
to the Thomson- Fermi function for heavy atoms. Later, Bethe ' ^
proved that multiple scattering of electrons given by Moilere^ ' can be
obtained from the exact theory of Goudsmith and Saundersons. The
problem of scattering through the angles for which sin0 cannot be
replaces by 0 was studied by Goudsmith and saunderson ^^"^K Lawis ' ^
gave an approximate solution of the integro -differential diffusion
equation of the multiple scattering problem in an infinite, homogeneous
medium without using small angle approximation. He also included the
energy loss by considering the energy of a particle as a function of its
residual range, the effect of straggling in energy loss assumed to be
small. On the other side so many workers ^^^'^^^ have obtained the r.m.s.
angle of multiple scattering experimentally by measuring the tracks of
scattered electrons and positrons by cloud chamber technique or nuclear
emulsions. It was reported that the experimental curve between r.m.s.
angle of the normal part of multiple scattering distribution versus energy
agrees well with the theory of Willams^ '' ^ and Molier' ^ l
The systematic calculations of the penetration of electrons and positrons
through matter require the correct and convenient form of multiple
scattering cross sections as input data. The results from experiments ' "
^'K indicate that at energies greater than 1 MeV, for electron there is
some agreement with Moilier's ' ^ theory. Below 1 MeV the
comparison between the Moilier's theory and the various experiments
on multiple scattering shows disagreement with decreasing energy. The
experimental evidence for difference in multiple scattering of electron
and positrons is contradictory.
29
2.6 Straggling
Identical charged particles all having the same initial velocity do not all
have the same range. It is found that the ranges are distributed over a
small interval about a mean range. This phenomenon is called range
straggling and is due to several effects, Since the No. of collision is
large and are independent to each other, hence the distribution may be
expected roughly to be Gaussian.
We have discussed in last article of multiple scattering that the
effects of multiple scattering preclude exact calculations of the range of
electrons. Such calculations are further complicated by the statistical
fluctuations, or straggling of energy losses. Straggling like scattering is
much more pronounced in the case of electrons than for heavy particles.
This is because heavy particles lose most of their energy in ionizing
collisions with atomic electrons, where conservation of momentum
permits fractional energy transfers of the order of the ratio of the masses
(~ mo /m), therefore each collision results in the transfer of only a small
fraction of the energy of a heavy particle. On the other hand, electron
can lose up to one half of their energy in an ionizing collision. In
addition to this, the electron may also lose any fraction of its energy in a
radiative collision.
Among the most instructive observations on electron absorption
are the classic straggling and absorption measurements, made by white
and Millington ''' ^ with the aid of a beta ray magnetic spectrograph, on
seven conversion electron groups from Ra (B + C), covering an energy
range from 0.155 to 1.33 Mev.
Fig. 2.1 shows the effect on a homogeneous number- energy
distribution of 0.2065 - MeV electrons after passing through thin
mica foils of varying thickness.
0.18 0.19 0.20 Electron energy, Mev
0.21
Fig. 2.1
Transmission through mica of monoenergelic electrons The position of peaks gives the most probable energ>' loss.
30
References
1. H. A. Bethe, and buch fur Physik ( Julius Springer, Berline 1933),
Vol 24/2, P. 273.
2. C. Moller, Ann. Physik, 14, 531 (1932).
3. E. Rohrlich and B. C. Carlson, Phys. Rev. 93, 38 (1954).
4. R. K. Batra and M. L. Sehgal, Nucl. Inst. Meth. 109, 565 (1973).
5. R. D. Birkoff, passage of fast electrons Through Matter, Hand
buch Der, Physik Vol. 34,115 (1958).
6. M. I. Berger, Monte carlo Calculation of the penetration and
diffusion of fast charged particles, "methods of computational
physics", Vol. 1, 135 (1963), academic press.
7. T. Scott Willams, Rev. Mod. Phys. Vol. 35/2, 231 (1963).
8. N. F. Mott and H. S. W. Massey, The theory of atomic collisions,
Oxford University presses. London (1949).
9. E. J. Williams, Proc. Roy. Soc. (London) A169, 531 (1957).
lO.E. J. Williams, Phys. Rev. 58, 292 (1940).
1 l.L. R. B. Elton, Proc. Phy. Soc. A63,115 (1950).
12.G. Moliere, Z. Naturforch 24, 133 (1947) and 34, 78 (1948).
13.H. A. Bethe, Phys. Rev, 89, 1256 (1953).
M.S. Goundersmith and J. L. Saunderson, Phys. Rev. 57, 24 (1940)
andPhys. Rev. 58, 36(1940).
15.H. W. Lewis, Phys. Rev. 78, 526 (1950).
16.G. Groetzinger, M. J. Berger and F. L. Ribe, Phys. Rev. 77, 584
(1950).
17.F. F. Heymann and, W. F. Williams, Phil. Mag. 1, 212 (1956).
18.W. Duane and F. L. Hunt, Phy, Rev. 6, 166 (1915).
19.P. White and G. Millington, Proc. Roy. Soc. (London) A 120, 701
(1928).
^ Introduction ^ Hie present methods of caCcuCation y Tfie present methods for calcuCating ranges andaSsorption coefficient
31
3.1 Introduction With the help of theoretical developments, ^ '" which has already been
discussed in chapter 1, it is not possible to get any useful information
about the absorption coefficients, ranges, etc of electrons and positrons.
Because it is very difficult to make transmission curves of these
particles by these theories. '"
If the initial energies of electrons and positrons are same no difference
can be found out in the characteristics of transmission with the help of
these theories, because by these theories no transmission curve can be
constructed. ^ ""
The transmission curves for electron and positron to get information
about the ranges and absorption coefficient have been made
theoretically by Dr, Gill ^^^ in large number of materials for Z numbers
between 6 to 82, including some rare earth materials like Neodymium,
Yttrium, Ytterbium, and Holmium.
It is of great interest to study the penetration of electrons and positrons^
through rare earth materials, since no information is available for the
electron positron penetration through these materials. The properties of
these materials have been a subject of many latest investigations. ^^'^'^^\
The theoretical approach given by Dr. Gill ^^ was the first such attempt
to reproduced transmission curves of positrons and electrons almost in
the same way as the experimental transmission curves. In this method
the absorber was considered to consist of a large number of extremely
thin slices and Mott's "'^ quantum mechanical expression for the
scattering cross section had been used for computing the scattered flux
at each individual slice of absorber. Inelastic scattering and nuclear
screening had also been taken into account.
32
The calculation done by Dr. Gill ^^ has the agreement with experimental
values of ranges and absorption coefficients, when they used single
scattering formula ''' ^ multiplied by an empirically determined
correction factor. The energy and Z- dependence of that correction
factor is also calculated.
3.2 The present methods of calculation
The absorber has been considered to consist of a large numbers of thin
slices. The electrons and positrons of same energy were allowed to fall
perpendicularly on the plane of the absorbers. Both elastics and inelastic
scattering of electrons and positrons occur in each slice. It is also
considered that the numbers of particles, which get scattered through an
angle > 90° are supposed to be removed from incident beam.
For the energies between 0.5 to 5.0 MeV the following expression is
used to find out the energy dissipation in each slice.
UdE^
P dx \UX J luial / ^
Where upper + ve sign is for positrons and lower - ve sign is for
electrons
Y is the total energy of electrons and positrons in the unit of rest mass of
electron.
a~ and b~ are the constants and their values are given by
a = -0.0038, b^ = 1.8402 ,
a" = -0.0040, b" = 1.8160
the values of constants m and c for different Z are shown in table.
33
Atomic Nos.
1 < Z < 1 0
11 < Z < 3 6
3 7 < Z < 9 2
m (Mev-cm^-g"')
- 0.0330
- 0.0097
- 0.0048
c (Mev-cm^-g"')
1.3230
1.0911
0.9156
for the energies less than 0.5 Mev following expression is used
P {mz\c)F^{y)
total
(3.2)
where F" (y)
and F"(Y)
r 2.4
/"-I
7 2.56
f-\
for positrons
for electrons
(3.3)
(3.4)
Since the particles loose their energy in each slice, so that it is clear that
the incident energy on the next slice is less than the energy lost in
previous slice. So we can say as the number of slices increase, the
percentage transmission goes on falling.
The following Mott's scattering formula was used by Dr. Gill et al. ^^^
{ 2 \ 2
d(/ --NTZ^Z^-X) V^c J
Z+lYl-/ • -4
sm (^^
—
^2J - sin(<^/2)cos^/2)
[1 - / sin'(5^/2) ± a/ll-sm6>/2)sm6^/2]d0 (3.5)
34
Where -ve sign stands for electrons and +ve sign for positrons
N
Z
e =
a =
P Where y is
7
number of atoms/cm
atomic numtjer of absorber
charge on electron
Z/137
(y'-i)/y'
given in term of the incident kinetic energy T by 9 9
(T + moC ) / nioC
Where moc^ is the rest mass energy of the electron
In the above expression the term Z (Z+1) replaces ^^ the term Z in the
original Mott's ""'^ formula, and is a satisfactory approximation to
account for screening of the nucleus by the orbital electrons. Also for
the inelastic deflection ^^ another term (Z+1) / Z is used. But for the
present calculations we used the Mott's scattering formula "'^ up to fifth
order of aZ, the final expression is given below:
da =N72.{Z+X)- • ^ —^' yin^c^ J
sin- l - |sin(<9/2)cos^/2)
[\-/fsm\0l2)±7raJl\-sm0l2)sm0l2+(/R^+(//^R^Mc^ I /^R
{a' I0')R, +a'R,+a'^R\^a' I/f)R, + ( ^ lj3)R,+(^/^\ ]d0
35
Where symbols has their usual meanings, and 'R' are the complicated
expression and involve gamma function of complex nature. In the
present calculations all the higher order terms in the Mott's scattering
expression are used, that is the expression given by R. M. Curr ^^ up to
fifth power of aZ, has been used for calculating the transmission of
electrons and positrons.
The values of R's for the angle 90° has been used, the values are given
below,
Ri
0.650
R:
1.477
^ 2
0.259
R3
0.744
K
0.789
R4
0.955
^ 4
2.658
K
0.151
Rs
0.891
R's
0.958
K
l . l f
For the present calculations the values of R's has been used for the angle
9 = n/2, because for computing the transmission of electrons and
positrons through a slice, the numbers of particles scattered through an
angle > 90° have been considered to be removed from the incident beam
for the successive slice. In the calculation done by Dr. Gill ^^ the
scattering formula '"'" ^ multiplied by an empirically determined
correction factor was used, but no correction factor is used for the
present calculations.
To calculate the transmission by the final expression (equation 3.6) it is
considered that all the incident particles scattered through an angle > 90°
have been removed or absorbed from the incident beam while traversing
an individual slice of the absorber. The choice of thickness of slices is
36
arbitrary and it is taken to be extremely small. The thickness of each
slice is taken to be 22 x 10' x p gm/cm^ and 11 x 10" x p gm / cm^
where p is the density of the material. The fraction of incident particles
removed from the beam or the removed flux is given by:
]dJ]da (3.7) !rll I 0
In the Mott's expression (equation 3.6) the term p = (y - 1)/ Y goes on
changing at each subsequent slice. For the first slice the total incident
energy of the incident particles is measured by y = (T + moC^) / moC^,
where T is the incident kinetic energy and moC" is the rest mass energy
of electron or positron. The beam looses its energy in each slice, the
energy loss of the particles in each slice is computed by using the
formula "'^ for stopping power (equations. 3.1, 3.2) for different Z
values and energy ranges. For every slice the incident energy will be
equal to the incident energy of previous slice less the energy lost in it.
3.3 Present methods for calculating Ranges and Absorption
coefficients
When positrons and electrons pass through the matter elastic and
inelastic collision with the atoms of absorber take place. Due to the
multiple scattering the path of the incident particles gets distorted and
the kinetic energy of incident particle goes on decreasing the distortion
process become more rapid. At the low energy the multiple scattering is
enhanced. After a level at low energy the motion of these particles
become random. This random motion is the diffusion of these particles.
37
The lower energy below, which the diffusion starts, depends upon the
kinetic energy of incident particle and the nature of the absorber.
The incident positrons and electrons can travel an average distance even
after the diffusion process starts. This average distance is a substantial
fraction of the practical range in the absorber. By using this fact the
straggling free practical ranges of electrons and positrons can be
calculated. Both parts that is before and after diffusion process must be
taken into account.
In the theory of electron penetration, by Bethe ' ^ et al the diffusion of
electrons is taken into account. The assumption of this theory '"' make
the present approach of calculations of the straggling free practical
ranges simpler, and bring close to experimental values.
In the beginning of the journey in absorbing material electrons and
positrons travel along a straight path due to very little multiple
scattering. As the energy goes on decreasing the multiple scattering
starts and when the motion of these particles become random the
diffusion sets in. For the calculation of critical energy Tr~, at which the
diffusion process starts the multiple scattering during their slowing
down in target material is taken into account. However, both multiple
scattering and slowing down cannot be easily treated simultaneously. So
this situation can be simplified by assuming a direct transition from a
straight motion into diffusion. On the basis of transport mean free path
Bethe ' ^ studied this transition
That is
< C o s 0 ) average = l / e ( 3 . 8 )
Where, 0 is the angle between the direction of motion of the particle in
absorber and direction of initial beam.
38
By knowing ( Cos 0 ) average, the desired range can be calculated. Batra
and Sehgal ^^'^^ calculated the critical kinetic energy Tr~ for electrons and
positrons and plotted against incident kinetic energy T for different
materials. If the critical kinetic energy Tr" for any particular initial
kinetic energy and absorber is calculated, the parts of range before and
after diffusion process starts can also be calculated. The straight path
covered by electrons and positrons or the projected range ^ ' ^ is given in
continuous slowing down approximation as
Rp-(T) = R'cSDA (T) - R'cSDA(Tr-) (3.9)
Where R"CSDA (T) = CSDA range of positrons and electrons for kinetic
energy T,
And R CSDA (Tr) - CSDA ranges of electron and positron at kinetic
energy Tr" (or at critical energy)
And can be obtained by using Berger and Saltzer ^^ table.
39
References
1. E. J. Williams, phy. Rev. 58, 292 (1940).
2. L. V. Spencer, phys. Rev. 98, 1597 (1955).
3. Dr. T. S. Gill, PhD thesis, A. M. U. Aligarh (1988).
4. N. F. Mott, Proc Roy Soc. (London) A124, 425 (1929) and A135, 429
(1932).
5. N. F. Mott and H. S. W. Massey, The theory of atomic collision
Oxford University press, London (1949).
6. R. K. Batra & M. L. Sehgal, Nuclear Physics A156, 314-320, (1970).
7. R. K. Batra & M. L. Sehgal, Nucl. Inst. Methods (109), 565 (1973)
8. R. K. Batra & M. L. Sehgal, Phys. Rev. B23, No.9, 4448 (1981).
9. M. J. Berger and S. M. Seltzer, NAS - NRC, Publ. No. 1133 (1964)
and NASA SP-3012 (1964).
10. F. Rohrlich and B. C. Carlson, Phys. Rev. 93, 38(1954).
11. P.S. Takhar and M. L. Sehgal, Nuovo Cimento (Italy) 47B, 31
(1978).
12. R. M. Curr, Proc Roy Soc. A68, 156 (1955).
13. H. A. Bethe, M. E. Rose, Proc Am. Phil. Soc. 78, 573 (1938).
14. F. H. Spedding and A. H. Danne, the rare earths (John Wiley and
sons, Inc.) New York (1961).
15. K. A. Gschneider, Jr. Rare Earths, the Fraternal fifteen, published by
united states Atomic energy commission, Division of technical
information (1966).
y IntrocCuction y Jl6soTption coefficients y Comparison of present tfieoreticaCamf^ExperimentaC (Ranges y transmission curves plbtteif 6y present theory y Curves for JiBsorption coefficients y Curves for ^n^es
40
4.1 Introduction
In this chapter the transmission curves are build to view the
percentage of transmitted beam intensity theoretically as a flmction of
the thickness of the absorber, through which the incident beam is
passing. Tlie transmission curves for rare earth materials and a large
number of absorber is plotted. The shape of these curves is the same as
that of experimental curves. Some curves are shown from figs. 4.1 to
4.12. By using these curves the values of Rsfp and the absorption
coefficients are determined. The shape of these curves for other
absorber is the same.
Figs. 4.13 to 4.16 shows that how we have measured the values of Rsfp
the extrapolation on the curves for measurement of Rs^ is shown.
4.2 Absorption coefficients
For measuring the absorption coefficients we have measured the slope
of curves between 33% to 67% of transmission. These absorption
coefficients are plotted against the Z number of the absorber. Figs 4.17
to 4.20 shows the comparison of absorption coefficients obtained from
present theoretical approach with experimental values of Dr. Gill. '^
The con^arison of the absorption coefficient of positrons and
electrons at the equal energy Emax is the right way of comparison, this
type of comparison is shown in fig. 4.21.
Rohrlich and Carlson * studied the differences in positrons and
electrons scattering. They determined the energy loss and multiple
4)
scattering of positrons and electrons in AJuminum and Lead. The
energy loss of positrons is greater than that of electrons at very low
energy and for low Z region, the multiple scattering of electrons is
slightly greater than that of positrons. Hence the transmission of
electrons is to be higher than that of positrons at very low energy and
for low Z region. Fig 4,20 shows dependence of absorption coefficient
for positrons and electrons at energy 1.88 MeV. It shows that as Z
increases the absorption also increases. It is clear from these figs. That
there is a difference between the transmission behavior of positrons
and electrons of the same initial kinetic energy.
4.3 Comparison of present theoretical and experimental
Ranges
In the figs 4.22 to 4.24, the values of ranges obtained by present
theoretical approach with experimental values of Dr. Gill '^ are
shown. It is clear from these figures. There is a very small difference
between the present theoretical curves and experimental curves. Or we
can say the values obtained by present theoretical approach are very
close to the experimental values.
The good agreement is because in the present calculation we used the
Mott's scattering expression ^^ including with higher order terms. The
total expression up to fifth order term in ocZ has already given in
chapter 3 (equation 3.6).
This expression is given by R. M Curr ^^ and can be written in the
following simplest form.
42
<y{e,r) = z' e \2m^c'^ 2 o2
1 - / 3 2 ^
ysin'e 2
1 - ^ ^ i n ^ ^ / + a /3 /? ,+a^ /?^+a^^^ /? ,+^ + a^ /?3 +
Where a = Z/137, p = v/c, G = angle of scattering,
for electrons and positrons only the appropriate sign is to be changed.
'R' is the complicated expression and involves gamma fiinction of
complex nature, which has already discussed in third chapter.
ITie values of theoretically calculated ranges by Batra and Sehgal ^ ' ^
and experimental values '"^ agree for lower Z materials with the
exception of carbon. This difference is of the order of 16% in
intermediate Z values and is the order of 30% in high Z region.
But the difference between the experimental ranges and the ranges
obtained by present theoretical approach after including the higher
order terms in Mott ^^ and Massay ' ^ expression is improved by about
6 to 12% in intermediate Z region and 10 to 18 % in the high Z region.
If the contribution due to diffusion part is also taken into account the
difference in experimental and theoretical values becomes only 0.01 %
to 6 % in low and intermediate Z region at energy 0.77 MeV and only
6 to 9% for high Z region. Similarly at energy 1.53 Mev this
difference becomes only 0.5% to 5.3% in low and intermediate Z and
up to 8.4% in high Z materials. This difference is shown in figs 4.25 to
4.27.
43
At low energies and for high Z materials the difference between
experimental and present theoretical values is larger than the
difference at high energy and low Z materials.
This difference in experimental values and the values given by present
theoretical approach may be due to the errors in the total stopping
power expression ' " l These expressions are used as input data for the
present calculation. There can be an error of about 4 % in the input
data for stopping power. Also the experimental data has some errors.
Hence the net effects of these errors produce the difference in
experimental and present theoretical values.
References ""
1. R. K. Batra & M. L. Sehgal, Nuclear Physic
(1970).
2. R. K. Batra & M. L. Sehgal, Nucl. Inst. Methods (109), 565
(1973)
3. R. K. Batra & M. L. Sehgal, Phys. Rev. B23, No.9, 4448
(1981).
4. Dr. T. S. Gill, PhD thesis, A. M. U. Aligarh (1988).
5. N. F. Mott, Proc Roy Soc. (London) A124, 425 (1929) and
A135,429 (1932).
6. N. F. Mott and H. S. W. Massey, The theory of atomic collision
Oxford University press, London (1949).
7. R. M. Curr, Proc Roy Soc. A68, 156 (1955).
8. F. Rohrlich and B. C. Carlson,Phys. Rev. 93, 38(1954).
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