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Physics - Paper – 203 ( Nuclear Physics)
Lecture Notes by Dr. N. R. Roy (Retd. Professor)
Guest Faculty, Post Graduate Department of Physics,
Ranchi University , Ranchi.
Students are advised to consult any standard text book for figures( Nuclear
Physics by S.N. Ghosal, by D.C.Tayal or any as per their convenience)
Unit – Nuclear detectors
IONIZATION CHAMBER :
A simple ionization chamber consists of a cylindrical metallic chamber with a thin conducting wire mounted along the axis. The arrangement is enclosed in a glass envelope which is filled with a gas such as air ,hydrogen ,carbon dioxide ,nitrogen ,argon. Methane at atmospheric or greater pressure.
When a charged particle enters the active volume of the chamber it produces a large number of ion pairs in the enclosed gas along its path. The central wire acts as the anode which is grounded through a resistance R. the cylindrical chamber acts as the cathode which is connected to a high voltage supply so as to have a negative potential say 𝑣𝑜 with respect to the anode.
A schematic diagram of a cylindrical ionization chamber is shown in the figure.
Figure
If n ion pairs are produced then a charge – ne get attracted towards the anode while a charge +ne (n positive ions) move slowly towards the cathode. A total charge a=2ne acts collected on the electrodes. If c be the capacity of the system electrodes the source e.m.f. generated in the cylinder by the passage of a charged particle is given by
𝐸 = 2 𝑛𝑒𝐶
The source e.m.f. produces a flow of current in the external circuit connected across the electrodes (anode & cathode) of R be the equivalent resistance of the circuit, the ionization current is given by
𝑖 = 𝐸𝑅 = (^2) 𝐶𝑅𝑛𝑒
The ionization current I is proportional to the 𝑧^2 of the particle and inversely to its velocity, this
allows us to distinguish between charged particle causing ionization such as electron, proton, ∝-
particle. Besides, it is possible, by measuring ionization current to detach the flux of such particles entering the chamber. A simple chamber has a limited application due to the fact that it delivers a very small ionization current of the order of 10 −^12 - 10 −^15 amp. Which is difficult to measure. For this reason it is usual to use d.c. amplifying devices to the external circuit.
The effectiveness of an ordinary chamber is improved by applying a large voltage across the electrodes. The ionization characteristic is shown in the figure
Figure
If we assume that three ionization sources with different primary ionization (𝑛1, 𝑛2, 𝑛 3 ion pairs) are placed near the chamber one after the other and the voltage applied across the electrodes is increased from zero then the ionization current increases along different paths for different primary sources. In different regions of these curves, the total ionization currents behave differently. In
region Ⅰ , the primary ions and electrons are partly collected by the electrodes and some pairs
recombine and lost due to collisions in the gas.
When the applied voltage is increased beyond ‘V 1 ’ (~20 volts) in region Ⅱ, almost all ion pairs are
collected by the electrodes and the ionization current become constant. In this region the chamber has application as ionization chamber for detection of different ‘Z’ number particles and for measurement of flux of charged particles.
As the applied voltage is further increased , the electrons produced in the primary ionization are accelerated to such high energy that they produce ‘Secondary ionization ‘ in collisions with neutral
atoms present in the gas. In region Ⅲ , the total ionization current is due to primary ions and the
large number of secondary ions produced. The total ionization current is seen to be increase linearly with voltage and proportional to the initial number of ions produced in the chamber. In this region output pulses are produced because one ion – pair on an average produces 10 to 10^5 secondary ion pairs during the processes of its collection and acceleration between the electrodes. An ionization chamber operated in this region of voltage is called proportional counter which is employed as pulse detector in proportional counting.
The proportional behavior of the counter is , however, lost as the applied voltage is increased beyond 500 volt when discharge takes place and avalanche production of ions takes place in the
chamber. In the region Ⅳ the avalanches are produced so rapidly that the ionization current
becomes completely independent of the primary ionization. This region is characteristic of the primary ionization and was used by ‘Geiger and Muller’ for charged particle counting. This region of voltage is called Geiger-Muller region.
Ionization chambers may be grouped into two categories depending up on the value of the time constant ‘RC’ of the system relative to the frequency of arrival of the ionizing events. If ‘RC’ is long compared to time interval between ionizing events , a steady state is reached and a direct current may be measured. If ‘RC’ is small , each pulse produced by an ionizing event may be detected separately. These two types are also known as integrating and non- integrating respectively and the respective ionization chambers are called current ionization chamber and pulse ionization chamber.
SCINTILLATION COUNTER :
One of the earliest methods of detecting nuclear radiation was by the ‘Luminescence’ or ‘Scintillations’ (light flashes) they produced on striking certain substances. In 1903 , ‘Crookes’ in
England and ‘Elster and Geitel’ in Germany independently reported that 𝛼- particle impinging on ZnS
end windows of mica, cellophane or glass about 1 to 2 𝑚𝑔/𝑐𝑚^2 thick allow
𝛼 𝛽 particles to enter into the counter.
For detecting ‘ 𝛽’ particles thin walled glass counters (the cathode is often a thin layer of colloidal
carbon deposited on the inner wall of glass envelope) which allow the entry of 𝛽-rays from all
directions, can be detection of 𝛾-rays is mainly through the ejection of photoelectrons from the
cathode cylinder into the gas. The probability of this process varies as 𝑍^4 and hence counters having high ‘Z’ (Bismuth or Lead) cathode are used for the detection of 𝛾 – rays.
There are two important characteristic curves exhibited by Geiger Counters. one is called the ‘ Plateau curve’. It is plot of counting rate ( current pulses per minute ) against the voltage applied between the anode and the cathode. The G.M. counter is connected with a device capable of recording only relatively large pulses. Till the voltage applied reaches the value called the starting potential , the pulses are too small to be detected.
Figure.
As the voltage increases , the amplification increases and the number of pulses per time increases to attain a practically constant value. The flat portion of the curve is called the ‘Plateau’. This is the Geiger tube region for which the count rate is nearly independent of the potential difference across the electrodes.
Beyond plateau , the electric field becomes so large that a continuous discharge takes place and count rate increases rapidly. It may be noted that the shape of the knee of plateau curve is partly a property of the associated recording circuit.
The plateau has a width of a few hundred volts. A good G.M. tube has plateau length of 100 to 200 volts or more with a slope of 5% counts per 100 volts applied.
Another type of characteristic curve is that of the starting potential as a function of the pressure of the gas in the counter. From these curves it is seen that the starting potential increases practically linearly with the pressure of the gas filling the tube.
The efficiency of the counter is defined as the ratio of the observed counts per second to the number of ionizing particles entering the counter per second. Counting efficiency is defined as the ability of its counting if at -least one pair is produced in it .It is given by
𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = 1 − 𝑒𝑠^ 𝑙^ 𝑝 where, s is specific ionization at one atmosphere ,
p is the pressure in atmosphere ,and
𝑙 is the path length of the ionizing particle in the counter.
Main features of a G.M.Counter :
- Constant output pulse size , independent of initial ionization.
A long insensitive time ( paralysis time ) to allow entry of each particle 3) Infinite life
Can detect and count 𝛼 , 𝛽 , 𝛾 and cosmic rays.
G.M. counter has proved to be a very sensitive device for detecting and counting charged particles. It produces large pulses requiring no amplification.
It however suffers from the disadvantages of
i) being quite slow
ii) being incapable of providing information about the type of the particle which may have produced a count
iii)being incapable of providing information about the energy of the particle counted.
QUENCHING :
The methods used to prevent continuous series of pulses to take place in a G.M. counter , are called ‘Quenching’. There are three types of quenching involved in the operation of G.M. counter (a) Quenching of photons in the initial avalanche (b) electrostatic quenching off the avalanche by the positive ion space charge , and (c) quenching of the secondary emission ,when the positive ions reach the cathode. Quenching of the discharge improves the resolving power of the counter.
RESOLVING TIME :
Let the counting system with a resolving time ‘ 𝜏’ responds at a rate ‘𝑛’ counts per unit time when
exposed to ‘𝑁’ initiating ionization events per unit time. Clearly in unit time the total insensitive time becomes ‘𝑛 𝜏’ and the number of counts missed per unit time becomes ‘𝑁 𝑛 𝜏 ’. Since the number of counts missed is the error in counting ,we get,
𝑁 𝑛 𝜏 = 𝑁 − 𝑛
Clearly the actual count rate is
𝑁 = 𝑛 ( 1 − 𝑛𝜏) ??
From a knowledge of ‘ 𝜏 ’ , the actual count rate can be determined.
DEAD TIME and RECOVERY TIME :
The variation of the output signal with time is shown in the figure.
Figure
The paralysis time may belong as high as few millisecond.
Unit- Nuclear reactor Theory:
NUCLEAR FISSION CHAIN REACTION: FOUR FACTOR FORMULA
These neutrons are slowed down by collisions with the other materials ( moderating materials). However, during the slowing down process some of them are captured by U^238 to form U^239 which decays to Np^239 and finally to Pu^239 , particularly at energies in the region of strong resonance absorption.
[reaction:- 92 238𝑈 + 0 1𝑛 → (92 239 𝑈 )∗^ → 92 239𝑈 + 𝛾 → 93 239𝑁𝑝 + −1 0𝑒 → 94 239𝑃𝑢 + −1 0𝑒 ]
To account for the above ,we introduce a quantity ‘p’ called the resonance escape probability. Clearly , a fraction 𝜈𝜖 1 − 𝑙𝑓 p of neutrons which escapes without being captured and the fraction , 𝜈𝜖 ( 1 − 𝑙𝑓 ) ( 1-p), get captured forming Pu^239 nuclei. Hence , the number of neutrons which are slowed down to thermal energies is 𝜈𝜖 ( 1 − 𝑙𝑓 ) p.
Some of the thermal neutrons leak out of the assembly. We define a quantity ‘𝑙𝑡 ’ called the thermal neutron leakage factor. Clearly , the number of thermal neutrons that remain in the core is,
𝜈𝜖 1 − 𝑙𝑓 (1 − 𝑙𝑡 ) p. (2)
A fraction of these thermal neutrons is absorbed by Uranium and the remaining by other materials .The fraction of thermal neutrons absorbed by U^235 causing fission as compared to all thermal neutron absorption in the assembly is called the thermal utilization factor ,usually denoted by the symbol ‘f’. Hence , the number of thermal neutrons that are absorbed by U^235 is thus
= 𝜈𝜖 1 − 𝑙𝑓 1 − 𝑙𝑡 p f (3)
Not all of these neutrons absorbed by U^235 will cause fission of U^235.
If ‘ 𝜍𝑡 ’ be the cross section for thermal neutron absorption in the assembly and ‘ 𝜍𝑓 ’be the
cross section of thermal neutron absorption by U^235 causing fission then we get the number of second generation neutrons absorbed by U^235 to be
𝜈𝜖 1 − 𝑙𝑓 1 − 𝑙𝑡 𝑝 𝑓 (^) 𝜍𝜍 (^) 𝑡𝑓 (4)
The above quantity usually denoted by ‘𝑅’ is called the reproduction factor or multiplication factor as,
𝑅 = 𝜈𝜖 1 − 𝑙𝑓 1 − 𝑙𝑡 𝑝 𝑓 (^) 𝜍𝜍 (^) 𝑡𝑓 (5)
The quantity , 𝜈 𝜍 (^) 𝑓
𝜍 𝑡^ ,usually denoted by ‘^ 𝜂’ represents the number of fast fission neutrons
produced by each thermal neutrons absorption by U^235 causing fission. We can hence, write the multiplication factor as,
𝑅 = 𝜖 1 − 𝑙𝑓 𝑝 1 − 𝑙𝑡 𝑓 𝜂 (6)
In the special case when the assembly size is of infinite dimension both ‘𝑙𝑓 ’ and ‘𝑙𝑡 ’ reduce to zero. Thus for an infinite core the multiplication factor can be written as ,
𝑅 = 𝜖 𝑝 𝑓 𝜂 (7)
The above relation is referred to as ‘ the Four Factor Formula’
In the infinite system the condition for self- sustaining chain reaction is that each neutron generation just replaces the previous one , i.e. , 𝑅∞ = 1. The system is then said to be ‘ critical’.
For a finite system because some neutrons are lost from the walls of the assembly the critical condition is that ‘𝑅 ’must be greater than 1. The exact number by which 𝑅 should be greater than 1 depends upon the shape and the actual size of the system and the arrangement of materials in the system.
If for an infinite system 𝑅 > 1,then the number of thermal neutrons increases steadily. This is known as the ‘ Super-critical state’. On the other hand if for the system 𝑅 < 1 , the number of neutrons steadily decreases to zero and this state is known as ‘the Sub-critical state’.
In the special case of a reactor in which the fuel contains only U^235 and not U^238 then both the factors 𝜖 & 𝑝 are reduced to unity. The multiplication factor in this case thus becomes 𝑅 = 𝑓 𝜂.
For a finite system the effective multiplication factor ‘𝑅𝑒𝑓𝑓 ’ becomes
𝑅𝑒𝑓𝑓 = 1 − 𝑙𝑓 1 − 𝑙𝑡 𝑓 𝜂 (8)
and clearly we have , 𝑅𝑒𝑓𝑓 < 𝑅∞.
The magnitude of 𝑅𝑒𝑓𝑓 determines the speed with which the number of thermal neutrons build up and the rate at which fission occurs and the energy is released.
If 𝑅𝑒𝑓𝑓 > 1, for some assembly we can decrease it by progressively reducing the reactor size, causing increase in the neutron loss through leakage. The size of the assembly for which 𝑅𝑒𝑓𝑓 is equal to 1 is called the critical size.
CLASSIFICATION OF NEUTRONS :
Neutrons are divided in different classes according to their kinetic energy , such as ,
- SLOW NEUTRONS :-
Neutrons having energy in the range 0 to 1000 eV are slow neutrons. Slow neutrons can be further classified as :
a) Thermal neutrons : Neutrons having energy of about 1/40 eV =0.025 eV.
b) Cold neutrons : Neutrons having energy less than that of thermal neutrons.
c) Epithermal neutrons : Neutrons having energy greater than or equal to 1eV.
- INTERMEDIATE NEUTRONS :-
Neutrons having energy in the range 1000 eV to 0.5 MeV are called intermediate neutrons.
- FAST NEUTRONS :-
using the principle of conservation of linear momentum and conservation of energy we find the following relation to hold ,
𝐸 1 𝐸 0 =^
𝐴 2 + 1 + 2 𝐴 𝑐𝑜𝑠𝑐𝑜𝑠 𝜙 𝐴 + 1 2 (9)
For ,𝜙 = 0 , 𝐸 1 = 𝐸 0 (Grazing collision)
For ,𝜙 = 𝜋 , 𝐸 1 = 𝛼𝐸 0 (Head-on collision)
where, 𝛼 = (𝐴^ −^ 1 )^
2 (𝐴 + 1 ) 2 (10) Clearly , maximum energy is transferred from neutron to the moderator molecule under the condition of head – on collision. Thus , in single collision maximum possible energy loss of the neutron is given by
( 𝛥𝐸 ) (^) 𝑚𝑎𝑥 = 𝐸 0 − ( 𝐸 1 )𝑚𝑖𝑛
= 𝐸 0 − 𝛼 𝐸 0 = 𝐸 0 (1 − 𝛼 ). (11)
The maximum fractional energy loss of the neutron is, thus , (𝛥𝐸 ) (^) 𝑚𝑎𝑥 𝐸 0 =^1 –^ 𝛼^ (12)
From the above we find that maximum loss of K.E depends upon the atomic weight of the nucleus encountered by the neutron.
For Hydrogen, ‘A’ =1 and hence 𝛼 = 0. Thus the loss of K.E is maximum and is proportional to
For large values of ‘A’ , we get ,
𝛼 = (𝐴−1)^
2 (𝐴+1) 2 =^
1 − (^) 𝐴^12 1 + (^1) 𝐴^2
= (1 − 𝐴^1 ) 2 1 + 1 𝐴^ −^2
= 1 + 𝐴^4 2 − 4 𝐴 ≈ 1 − 𝐴^4.
The maximum fractional energy loss in such a case becomes equal to ,
1 − 𝛼 = 1 − 1 + (^4) 𝐴 = 4/𝐴. (13)
We thus find that light nuclei are more effective moderators than heavy nuclei and the loss of energy per collision is proportional to the initial energy of the neutron.
Remembering that in the C-M system in a scattering process all scattering angles are equally probable , we find that the probability that a neutron of energy ‘𝐸 0 ’ will have its K.E lying in the interval E & E+dE , after the scattering , is given by
𝑑𝑊 = − 𝐸𝑑^ 𝐸
0 (1−𝛼)^
{the – ive sign is introduced to indicate that energy is lost in the scattering process.}
Since , the above expression is independent of ‘E’ we can say that all values of ‘E’ are equally probable after a collision. This probability per unit energy loss is given by ,
𝑃 𝐸 = − 𝑑𝑊𝑑𝐸 = (^) 𝐸^1 0 1 −^ 𝛼^
Thus , average energy loss of the scattered neutrons after one collision is ,
𝐸 0 𝛼 𝐸 0 𝐸^0 −𝐸^ 𝑃 𝐸 𝑑𝐸 𝐸 0 𝛼 𝐸 0 𝑃 𝐸 𝑑𝐸
=
𝐸 0 𝛼𝐸 0 𝐸^0 − 𝐸^ 𝑃^ 𝐸^ 𝑑𝐸
=
𝐸 0 𝛼𝐸 0 𝐸^0 − 𝐸^
1 𝐸 0 (1− 𝛼) 𝑑𝐸
= (^) (1−𝛼^1 ) 𝛼𝐸^ 𝐸^00 𝑑𝐸 − (^) 𝛼𝐸𝐸^00 𝐸^ 𝐸 0
= 𝐸 0 − (1 +^ 𝛼 2 )𝐸^0
= 𝐸 0 ( 1^ − 2 𝛼^ ) (16)
Therefore average energy loss is ( 1^ − 2 𝛼^ ).
We thus find that the same fraction of neutron energy is transferred to the moderating molecule in each successive collision.
TRANSPORT MEAN FREE PATH ( 𝜆𝑡𝑟 ) :-
The scattering angles in the Lab. -system are smaller than the scattering angles in the C.M- system. This means that in the Lab. system the neutrons show a preferential forward scattering although the scattering is spherically symmetric i.e. , isotropic in the C.M- system. The deviation from spherical symmetry is expressed in terms of the average of cosines of all possible scattering
angles i.e. , by 𝑐𝑜𝑠 𝑐𝑜𝑠 𝜃. It is possible to show that 𝑐𝑜𝑠 𝑐𝑜𝑠 𝜃 = 2/3𝐴.
We find that for large ‘A’ , 𝑐𝑜𝑠 𝑐𝑜𝑠 𝜃 is small and hence the average forward scattering is
small making the scattering almost isotropic. The forward scattering is predominant in scattering by only light nuclei. Predominant forward scattering in the L.-system affect the mean free path ( the average distance traversed between two successive scattering ) in the sense that the average distance travelled by the neutron before it is scattered through a maximum angle of 90^0 is greater than the corresponding average distance for isotopic scattering. If the increased effective mean free
path for non- isotropic scattering is called the ‘Transport mean free path’ ( 𝜆𝑡𝑟 ) and is related to
isotropic scattering mean free path ( 𝜆𝑠 ) according to the formula
iv)Low absorption cross-section.
The first three quantities ‘𝜉 , 𝜍𝑠 & 𝑁 ’ jointly determine the slowing down ability of a moderator and their product is called the slowing down power.
SDP = 𝜉 𝜍𝑠 𝑁. (23)
We may write the above equation as
SDP = 𝜉 𝑠 (24)
where 𝑠 = 𝜍𝑠 𝑁 , is the macroscopic cross-section.
We may further express ‘SDP’ as ,
SDP = (^) 𝜆𝜉 𝑠
where 𝜆𝑠 is the scattering mean free path.
From the above we find that a good moderator is one which has large value of ‘ 𝜉’ and large
value of macroscopic scattering cross section.
MODERATING RATIO ( mr):
The ratio of ‘SDP’ to macroscopic absorption cross section is called the moderating ratio.
𝑚 𝑟 = 𝑆𝐷𝑃𝑎
= 𝜉^ 𝑎𝑠. (26)
The quantity ‘mr’ decides the properties of good moderator.
SLOWING DOWN DENSITY :
The rate at which the neutrons are slowed down past a given energy ‘E’ per unit volume of a moderator is called the slowing down density and is usually denoted as ‘q(E)’.
Because of the discontinuous nature of slowing down process and presence of absorption resonances for particular value of neutron energy , the variation of ‘q( E)’ with ‘E’ is quite complex. However, we may treat the slowing down process as a virtually continuous process by using sufficient number of moderator nuclei of large mass number for which average energy loss per collision is very small.
If the rate of production of neutrons with an initial high energy ‘𝐸 0 ’ is a constant ,say, ‘Q’ neutrons per unit volume per second and no neutrons are lost due to escape or leakage or due to absorption before they have slowed down to the energy ‘E’, then slowing down density ‘q( e)’remains constant and is equal to Q. This is the goal of an ideal moderator.
Clearly , an ideal modulator should reduce the neutron escape to negligible proportions which can be achieved by using moderators having thick walls .It may however be noted that absorption of neutrons can never be reduced to zero.
Let the number of neutrons per unit volume having energy lying between ‘𝐸 & 𝐸 + 𝛥𝐸’ be 𝐸 𝛥𝐸. If 𝛥𝑡 be the time needed by the neutrons to transit through the energy level 𝛥 𝐸 then we have ,
𝑛 𝐸 𝛥 𝐸 = − 𝑞 𝐸 𝛥𝑡 (27)
The – ive sign arises because energy decreases as time increases.
The number of collisions suffered by this group of neutrons per second is equal to , 𝑣 𝜆 =^ 𝑣^ 𝑠^ (28) where ‘𝑣 ’ is the average velocity of the group of neutrons under consideration. Hence, the neutron density that crosses the energy ‘E’ per sec. is given by
𝑞 𝐸 = 𝑛 𝐸 𝛥𝐸 𝑣 𝑠 (29)
In the event of small energy change per collision ,we may write 𝛥 𝐸 𝐸 =^ 𝛥(𝑙𝑜𝑔 𝑙𝑜𝑔^ 𝐸) =^ 𝜉
Thus , 𝑞 𝐸 = 𝑛 𝐸 𝜉 𝐸 𝑣 𝑠 (30)
In the absence of neutron absorption , the neutron flux per unit energy range is given by ,
𝑞 (𝐸) 𝜉 𝐸 𝑠 =^
𝑄 𝜉 𝐸 𝑠.^ (31)
The above expression shows that the slowing down neutron flux per unit energy range is inversely proportional to the energy ‘E’.
We define a quantity ‘F(E)’ called the ‘ collision density’ as
𝐹 𝐸 = 𝜙 𝐸 𝑠 = (^) 𝜉𝑄 𝐸^ 𝑠𝑠 = (^) (𝜉𝑄 𝐸). (32)
The above relation shows that the scattering loss ( i.e., no. of neutrons scattered out of the energy interval 𝛥𝐸 per sec. per unit vol.) is equal to the neutron in-flux gain ( no. of neutrons scattered into the energy interval 𝛥𝐸 per sec. per unit vol.).
NEUTRON DIFFUSION :
Diffusion is the process of motion of particles of one kind between particles of another kind.
It is often necessary to know the spatial distribution of neutrons during slowing down process that takes place due to collision of neutron with moderator nuclei. This can be obtained using the theory of diffusion.
𝐽𝑥 = 𝐽𝑥^ +^ − 𝐽𝑥^ −^ (37)
Similarly , we may write ,
𝐽𝑦 = 𝐽𝑦^ +^ − 𝐽𝑦^ −^ (37a)
and, 𝐽𝑧 = 𝐽𝑧+^ − 𝐽𝑧−^. (37b)
It can be shown that current density components can be related to neutron flux and macroscopic scattering cross section as ,
𝐽𝑥^ +^ = 𝜙 4 − (^6 1) 𝑠^ 𝜕𝜙𝜕 𝑥
𝐽𝑥^ −^ = 𝜙 4 + 6 1 𝑠 𝜕^ 𝜕𝜙 𝑥
So as to get ,
𝐽𝑥 = − (^3 1) 𝑠^ 𝜕𝜙𝜕𝑥 = − 𝜆 3 𝑡𝑟^ 𝜕𝜙𝜕𝑥 (38a)
(where, 𝜆𝑡𝑟 is the transport mean free path of neutron )
Similarly, 𝐽𝑦 = − 𝜆 3 𝑡𝑟^ 𝜕𝜙𝜕𝑦 (38b)
𝐽𝑧 = − 𝜆 3 𝑡𝑟^ 𝜕𝜙𝜕𝑧 (38c)
using the above results we get,
𝐽 = − 𝜆^3 𝑡𝑟 𝑖 𝜕𝜙𝜕𝑥 + 𝑗 𝜕𝜙𝜕𝑦 + 𝑘 𝜕𝜙𝜕𝑧
we have , 𝐽 = − 𝐷 0 𝛻𝜙. (39)
comparing we obtain ,
𝐷 0 = + 𝜆^3 𝑡𝑟. (40)
NEUTRON LEAKAGE RATE :
Let us now consider an elementary rectangular parallelopiped of volume
𝑑𝑉 = 𝑑𝑥 𝑑𝑦 𝑑𝑧, about a point (x,y,z) within the volume under consideration. The no. of neutrons that enter the volume through the face area ‘dy dz’ along the x-axis per sec.
= 𝐽𝑥 𝑑𝑦𝑑𝑧.
The no. of neutrons that leave the volume through the opposite face per sec.
= 𝐽𝑥 + 𝜕 𝜕^ 𝐽𝑥𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑧.
Thus the net no. of neutrons that leak through the volume in 1 sec. through the faces perpendicular to the x-axis is
= 𝜕 𝜕^ 𝐽𝑥𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑧.
Considering similar leakage of neutrons through the faces perpendicular to the y- & z-axes ,we obtain ,the net leakage rate of neutrons from the elementary volume ‘dV’,
= 𝜕 𝜕^ 𝐽𝑥𝑥 + 𝜕 𝜕^ 𝐽𝑦𝑦 + 𝜕 𝜕^ 𝐽𝑧𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧.
(^2) 𝜙 𝜕𝑥 2 +^
𝜕^2 𝜙 𝜕 𝑦 2 +^
𝜕^2 𝜙 𝜕 𝑧^2 𝑑𝑉 = − 𝐷 0 𝛻^2 𝜙 𝑑𝑉
= −𝐷 𝛻^2 𝑛 𝑑𝑉. (41)
THERMAL NEUTRON DIFFUSION :
Consider a mono-energetic group of thermal neutrons. When these neutrons collide with moderator nuclei , the net energy change is zero.
Let 𝑛 ( 𝑟) be the neutron density at the point 𝑟(𝑥 , 𝑦, 𝑧). The value of 𝑛(𝑟) depends upon :
i) the rate of production of thermal neutrons per unit volume i.e. , on ‘Q’
ii) the rate of absorption of thermal neutrons per unit volume i.e. , on 𝑛 𝑟 𝑣 𝑎.
iii) the rate of diffusion or leakage per unit volume i.e. , on −𝐷 𝛻^2 𝑛.
Considering the above we get , 𝜕 𝑛 𝑟 𝜕 𝑡 =^ 𝑄 − 𝑛 𝑟^ 𝑣^ 𝑎^ −^ (^ −𝐷^ 𝛻
= 𝑄 − 𝑛 𝑟𝜆𝑎^ 𝑣 + 𝐷 𝛻^2 𝑛
= 𝑄 − 𝑛 𝑟𝜆^ 𝑎𝑣 + 𝜆^ 𝑡𝑟^ 𝑣^ 𝛻
(^2) 𝑛 3.^ (42)
{ 𝜆𝑎 = 1 𝑎 is the absorption mean free path for thermal neutrons }
The above equation is referred to as the general diffusion equation. Under steady state condition ,
𝜕 𝑛 𝑟 𝜕 𝑡 = 0.^ (43) Since , production of thermal neutrons is mainly due to slowing down of fast neutrons , we may replace ‘Q’ by slowing down density ‘q’. Hence , under the steady state condition we obtain ,
𝑞 − 𝑛 𝑟𝜆^ 𝑎𝑣 + 𝜆^ 𝑡𝑟^ 𝑣^ 𝛻
(^2) 𝑛 3 =^0
In terms of ‘𝑛’ , the rate of diffusion is given by 𝜕 𝑛 𝜕 𝑡 =^ −𝐷^ 𝛻
2 𝑛 = − 𝜆^ 𝑡𝑟
3 𝑣^ 𝛻
The no. of neutrons that slow down into the energy interval ‘∆𝐸 ’ and remain with energy ‘E’ per sec. are,
= 𝑞 𝐸 + ∆ 𝐸 − 𝑞 𝐸
= (^) 𝜕𝜕 𝐸𝑞 ∆ 𝐸. (50)
we then obtain ,
− 13 𝜆 (^) 𝑡𝑟 𝑣 𝛻^2 𝑛 = (^) 𝜕𝜕 𝐸𝑞 ∆ 𝐸 (51)
we have, 𝑞 𝐸 = 𝑛 𝐸 𝐸 𝜉 𝑣 𝑠
Differentiating the above we obtain ,
𝛻^2 𝑞 = 𝐸 𝜉 𝑣 𝑠 𝛻^2 𝑛
or, 𝛻^2 𝑛 = 𝛻
(^2) 𝑞 𝐸 𝜉 𝑣 𝑠 (52)
Substituting the above we obtain ,
− 13 𝜆 (^) 𝑡𝑟 𝑣 𝛻
(^2) 𝑞 𝐸 𝜉 𝑣 𝑠 =^
𝜕 𝑞 𝜕 𝐸 ∆^ 𝐸
or, 𝛻^2 𝑞 = − 𝜆 𝑠 𝜆𝜕 𝑡𝑟^ 𝑞
3 𝜉
𝜕 𝐸 𝐸
Let us introduce a new variable ‘ 𝜏 ’ according to
𝑑𝜏 = − 𝜆^ 𝑠 3 𝜆𝜉^ 𝑡𝑟^ 𝜕𝐸^ 𝐸 (54)
so that we have ,
𝜏 = (^) 𝐸𝐸 0 𝑑𝜏
= (^) 𝐸𝐸 0 − 𝜆^ 𝑠 3 𝜆𝜉^ 𝑡𝑟^ 𝜕𝐸^ 𝐸
= 𝜆^ 𝑠 3 𝜆𝜉^ 𝑡𝑟 𝑙𝑛 𝑙𝑛 𝐸 𝐸^0 (55)
Using the above we obtain ,the equation ,
𝛻^2 𝑞 − 𝜕𝜕^ 𝜏𝑞 = 0 (56)
The above is the general equation of diffusion for neutrons of all energies and is referred to as the ‘general Fermi Age Equation’.
Variable ‘ 𝜏 ’defined above is called the ‘ Fermi Age or Neutron Age’ .It may be noted that the
dimensions of 𝜏 is not of time but is that of area or square of length. The quantity 𝜏 plays that
role in slowing down process as does time in heat conduction equation.
1 𝜉 𝑙𝑛 𝑙𝑛^
𝐸 0 𝐸 represents the average number of collisions a neutron undergoes with
moderator nuclei when its energy changes from ‘𝐸 0 𝑡𝑜 𝐸 ’. Writing , (^) 𝜉^1 𝑙𝑛 𝑙𝑛 𝐸 𝐸^0 = 𝐶 ,
we get,
𝜏 = 𝜆^ 𝑠^3 𝜆^ 𝑡𝑟𝐶 = 𝜆 3 𝑡𝑟 𝛬𝑠 (57)
where ‘ 𝛬𝑠 ’ represents the total path length a neutron covers between the moment the
slowing down begins and the moment it attains thermal energy. WE see that ‘ 𝛬𝑠 ’ is analogous to
‘𝜆𝑎 ’ i.e., the thermal diffusion length. We can thus define a quantity
𝜏 0 = 13 𝜆 𝑡𝑟 𝜆𝑎 = 𝐿 𝑓^2. (58)
The quantity defined in this manner is called ‘ Fast Diffusion Length ’. CRITICAL SIZE OF A REACTOR :
The size of a reactor that operates under the condition of exact balance between thermal neutron production and neutron loss is known as the critical size and under the above condition the reactor is said to be ‘ Critical’. For such a reactor a relation between its geometric properties and the material properties of the assembly is known as the ‘ Critical equation’.
For all types of reactors we have the following relationship between the neutron flux and the slowing down density (q) ;
𝛻^2 𝜙 − 𝜆^3 𝑡𝑟^ 𝜙 𝜆𝑎 + 𝜆^3 𝑡𝑟^ 𝑞 = 0 (1)
‘𝑞 ’ is a function of position co-ordinate (𝑟 ) and the Fermi age or neutron age ( 𝜏 ) i.e. ,
The Fermi age equation can thus be written as ,
𝛻^2 𝑞 𝑟 , 𝜏 = 𝜕𝑞^ 𝜕(^ 𝑟𝜏^ , 𝜏) (3)
Solution of equation (3) can be expressed in the form ,
𝑞 = 𝑞 𝑟 , 𝜏 = 𝑅 𝑟 𝑇 (𝜏 ) (4)
we get from eq. (4)
𝛻^2 𝑞 𝑟 , 𝜏 = 𝛻^2 𝑅 𝑟 𝑇 𝜏
