Lecture delivered by Dr. Naseem Irfan (i) of xii

PAKISTAN INSTITUTE OF ENGINEERING AND APPLIED SCIENCES

FUNDAMENTALS OF NUCLEAR ENGINEERING

(Lecture Set 1 ) 31.01.2015

NEUTRONS, NEUTRON CROSS-SECTION AND ITS ENERGY DEPENDENCE

THE NEUTRON:

Neutrons play a fundamental role in initiating nuclear reactions. Having an uncharged nature, they remain unaffected by

the coulomb barrier and even a very low energy neutron can penetrate the nucleus and initiate nuclear reaction. But the

same nature makes their detection difficult as they produce little ionization when they pass through matter and they are

not deflected by electric or magnetic field. Due to these facts the discovery of neutron was an important milestone in the

nuclear sciences that afterward lead to the concept of nuclear reactors to harness the tremendous amount of nuclear

energy for power production.

In 1930, two German physicists, W. Bothe and H. Becker, observed that when beryllium, boron or lithium was

bombarded by alpha particles from radioactive polonium, the target material emitted a radiation that had much greater

penetrating power than the original alpha particle. A special reaction type studied by Bothe and Becker was

Bothe and Becker assumed this highly penetrating and non-ionizing product as high-energy gamma rays. Soon after

Curie and Joliot noticed that when these radiation fell on paraffin, an energetic proton was emitted. From the range of

these protons, they determined their energy to be 5.3 MeV. If radiation under study were indeed gamma rays, protons

could be knocked loose from paraffin by a Compton-like collision; from the Compton Scattering formula, they

computed that the energy of this Bothe and Becker termed gamma radiation would be at least 52 MeV to release such

protons. An emitted gamma ray of such energy seemed extremely unlikely. Experiments by James Chadwick in 1932

showed that the emitted particles were electrically neutral with mass approximately equal to that of proton. Thus in an

head-on collision, a 5.3 MeV neutral particle could transfer its energy entirely to struck proton. Chadwick christened

these neutral highly penetrating particles as neutrons and he is generally credited with being the discoverer of the

neutrons.

NEUTRON CATEGORIZATION BASED ON ENERGY CLASSIFICATION:

Neutrons of variety of energies are available in an operating nuclear reactor or from various nuclear sources. The

neutron in different energy groups behave in different manner. They may be categorized into three groups according to

their energies:

1. Thermal neutrons ( E 0.0253 eV)

2. Resonance neutrons ( from fraction of eV to thousands of eV)

3. Fast neutrons (ranging from neutrons E 100, 000 eV to even greater than 20,000,000 eV)

Polonium

alpha emitter

source

Beryllium

Target

High Energy

non-ionizing and

penetrating

radiation

Lecture delivered by Dr. Naseem Irfan (ii) of xii

THE NEUTRON SOURCES:

Beams of neutrons can be produced from a variety of nuclear reactions. Obviously neutrons cannot be accelerated like

charged particles, therefore, to start with, high-energy neutrons may be obtained and their energy can be reduced

through collisions with atoms of various materials. The process of slowing down fast neutrons is called moderation.

For practical applications, a neutron source must have a half-life of at least several years. It must also be intense (greater

than 105 n/s ) and free from other radiations such as beta particles and gamma rays.

Neutron Sources may be of (, n) type, spontaneous fission type, or photo-neutrons (, n) type. On the other hand they

may be produced by accelerators and nuclear reactors. Depending upon their composition, these neutron sources have

different neutron energy spectrum, different neutron yield and different half-lives. These may have some additional

gamma radiation of different energies and intensities ( emitter dependent).

The alpha neutron (, n) sources

The most familiar (, n) type sources are Ra-Be, Po-Be and Pu-Be. These all have similar spectrum with an average

energy of about 3-5 MeV, with a yield of approximately 1.5 x 107 n/sec per gram of radium whereas half life is about

19.4 years. This source has relatively more gamma rays emission associated with it. Po-Be source has the lowest

gamma emission, but its half life of 138 days is rather short. On the other hand Pu-Be source may not be used for

reactor start up as Pu-239 is a fissile material.

The Photo-neutron (, n) sources

Photo-neutron sources (, n) again have a high gamma background, these are larger than (, n) sources and have

relatively low yield and half lives. There advantage lies in the production of mono-energetic neutrons with an energy

below 1 MeV. For low neutron energies Sb-Be ( , n) sources, which generate neutrons of 0.025 MeV, are often used in

calibration. The antimony beryllium carbide source is also commonly used for power plant reactor start up.

The spontaneous fission sources

More recently 252

Cf (Californium) has been suggested as a standard source. 252

Cf undergoes spontaneous fission with a

half life of about 2.55 years and has a fission spectrum similar to U-235. The neutron yield may be determined

indirectly from the activity of the fission product and the average number of neutrons per fission reaction (about 3.8) .

Because of the high energy yield (2.5 x 106 n/sec per microgram of

252Cf ) it is possible to fabricate point sources.

The accelerated charged partical (p,n) or (d,n) or (T,n) sources

Protons, deutrons, and tritium atoms can be accelerated to high energies in accelerator and fast neutrons can be obtained

in high source strength.

The reactor neutrons

The fission neutrons produced in a nuclear reactor have an energy distribution ranging from few keV to approximately

20 MeV, averaging about 2MeV.

Lecture delivered by Dr. Naseem Irfan (iii) of xii

THE CROSS SECTION FOR NEUTRON INTERACTION

The neutron interaction gives an idea that neutron goes toward a target atom rather towards its nucleus, collides with it,

either enter into it or is scattered. Now the bigger the nucleus of target atom will be, the more are the chances of neutron

to hit it. Thus chances of interaction of neutron with the target nuclei will be dependent upon the size of the nucleus in

terms of its cross sectional area.

By the word cross sectional area of a nucleus, the first thing that comes in mind is the actual size of the nucleus i.e. the

cross-sectional area of the three dimensional spherical nucleus projected on a two dimensional plane perpendicular to

the direction of propagation of incident neutron. In order to determine this actual area a thin sheet of a pure material

may be considered having a thickness dx such that only a single layer of atoms are present in this sheet and the nuclei

do not shadow each other [figure 2.1]. The material is assumed to have an atom density N atoms per unit volume of the

sheet.

Let each nucleus has an actual cross-sectional area .

Interaction probability =(net area of all the nuclei present in the area dA of the sheet of thicknessdx) .

(Area occupied by all the atoms in that target area where neutrons are incident i.e. dA)

Interaction probability = (number of atoms present in the area dA)( actual cross-sectional area of nucleus)

(Area occupied by all the atoms in that target area where neutrons are incident)

Interaction probability = [ (atom density)( volume occupied by incident area)] [ ]

[ dA ]

Interaction probability = [ (atoms present per unit volume)( volume occupied by incident area)] [ ]

[ incident area i.e. dA ]

Interaction probability = [(N ) (dA . dx)] [ ]

[ dA ]

Interaction probability = N . dx .

= Interaction probability

[ N . dx ]

Figure: 2.1 Determination of cross section

Lecture delivered by Dr. Naseem Irfan (iv) of xii

Which gives a mathematical definition of neutron cross section i.e. it is the interaction probability per unit atom density

per unit distance of neutron travel. The parameters on the right hand side are measurable i.e. interaction probability can

be determined by the ratio of number of neutron left uninteracted i.e. (I) to the number of neutron incident on the target

area i.e. (Io) [Figure 2.1] . Atom density and thickness of the sheet can also be determined and thus the neutron cross

section can be determined.

The observations show that there are many practical cases where the apparent area changes dramatically with the

incident neutron energy. These seemingly unphysical phenomena were attributed to the complex interaction of the

nuclear forces and particles. Nuclei can have an especially greater affinity for those neutrons that are capable of

exciting their discrete energy levels.

Thus the concept of the nuclear cross section denoted by is now based on the idea that the effective size, a nucleus

show to a neutron, should be proportional to the probability that the incident neutron would react with it. Thus neutron

cross section may now be defined as:

The effective projected cross-sectional area surrounding an actual atomic nucleus, such that a neutron passing

within this area, will definitely interact with the nucleus.

This effective projected area may be smaller or larger [figure 2.2] than the actual physical cross-section of the nucleus

depending upon the relative probability for a reaction to occur.

Real or actual cross sectional area of the nucleus

Effective area would be more than actual area, if relative probability Effective area would be less than actual area, if relative probability of the neutron to interact with nucleus is more of the neutron to interact with nucleus is less

Effective cross sectional area of the nucleus

Figure: 2.2 Representation of cross section

This discussion suggested that the connotation of cross-sectional area may be dropped. Thus is merely a cross

section defined such that its product with the atom density N and distance dx is equal to the interaction probability

and hence instead of giving an exact physical sense of neutron cross sectional area it gives a measure of probability of

interaction of certain energy neutron with a specific nuclei. is more formally called microscopic neutron cross

section because it the word microscopic signifies application to describing the behavior on a nucleus-by-nucleus basis,

or actually submicroscopic basis.

The microscopic cross section depends upon factors like incident neutron energy and type of target nucleus for a

particular interaction. The defined unit is BARN which is equal to 1024

cm2. The cross section depends upon the

Lecture delivered by Dr. Naseem Irfan (v) of xii

nature of the target material and energy of the bombarding neutron and the nature of its interaction. Hence depending

upon the type of interaction of neutron with the nucleus the cross-sections are categorized in various types. The values

of neutron cross section of various materials range between 10-18

to 1027

cm2.It is applied in determination of reaction

rates, criticality studies, radiation shielding design, breeding or conversion ratio, thermal energy production and material

damage and life etc.

Figure 2.3 Examples of the dependence of cross-sections on neutron energy

Lecture delivered by Dr. Naseem Irfan (vi) of xii

Type of Cross Sections

Elastic Scattering Cross-sectiones.

In case of elastic scattering the probability of elastic scattering of neutrons is considered and the cross section is defined

as elastic scattering cross section es. In this case kinetic energy and momentum of both remain conserved and the

kinetic energy lost by the neutron would result in increase in kinetic energy of the hit nucleus. The targeted nucleus

remains in the ground state before and after the collision. This sort of interaction is usually more common for relatively

low energy neutrons and low mass number targets. e.g.

235U92 +

1n0

236U92

235U92 +

1n0

Inelastic Scattering Cross-sectioni.

For inelastic scattering the probability of inelastic scattering of neutrons is considered and the cross section is defined as

inelastic scattering cross section i. The kinetic energy does not remain conserved. The hit nucleus is left in excited

state. Compared to atomic excited states the energy of excited states of nuclei is considerably greater and instead of x-

rays, gamma ray emission takes place. Inelastic scattering cross section increases with increasing neutron energy and

increasing mass number of the target nuclei. Neutron slowing down is more rapid in inelastic scattering because neutron

energy is partially used up in excitation of target nucleus while remaining provides the kinetic energy to the struck

nucleus. e.g the same reaction above except the nucleus is left in excited state.

235U92 +

1n0 [

236U92]* [

235U92] * +

1n0

Capture Cross-sectionc.

The types of neutron interaction with a target nucleus, which involves capture of neutron such that the neutron remains

inside the nucleus as a part and a real absorption of neutron occurs and resulting excited nucleus with same atomic

number has a mass one unit higher than the previous number. The excited energy of the nucleus thus formed is

equivalent to one or more gamma rays that are emitted after wards and are called capture gamma rays The capture

gamma in this case has an energy of about 6MeV ( corresponding roughly to the binding energy per nucleon for an extra

nucleon). The example of such a reaction may be:

235U92 +

1n0 [

236U92]* [

236U92] +

00

Activation Cross-sectionact.

The interaction similar to capture except that the reaction occurs in materials other than the fuel is known as activation

and the probability of such a reaction may be deduce by considering the activation cross section. The entire field of

activation analysis is based on producing artificial radionuclides by neutron bombardment of an unknown sample. Then

by using gamma ray information to determine the identity and quantity of each species, the composition of initial

material can be determined, often to a high degree of accuracy. The example may be taken as that of sodium:

23Na11 +

1n0 [

24Na11]* [

24Na11] +

00

Multiple Neutron Cross-sectionxn.

An incident neutron on capture in certain nuclei could result in emission of more than one neutron i.e. instead of

emitting gamma ray the compound nucleus may excite by emitting more than one neutron. The multiple neutron

reactions are generally endoergic. In reaction with uranium-235, a neutron above the threshold energy of about 6MeV

evolves a pair of neutrons.

235

U92 + 1n0 [

236U92]* [

234U92] + 2[

1n0]

or 233

U92 + 1n0 [

234U92]* [

232U92] + 2[

1n0]

Lecture delivered by Dr. Naseem Irfan (vii) of xii

Fission Cross-sectionf.

Neutron after absorption in certain nuclei causes the nucleus to break up in two parts, the process is known as fission

and one of the source of energy. The fission cross section is thus a measure of the fission of the nuclei in this interaction.

The typical fission reaction of U-235 yields two fission-fragment nuclei plus several gamma and neutrons. The reaction

may be given as

233U92 +

1n0 [

234U92]* F1 + F2 + x[

1n0] +

00

Charged Particle Cross-section(n,p) or (n,d), or (n,).

Neutron on absorption in some nuclei may result in emission of various charged particles like proton, deuteron or alpha

particle etc. Although it is not common to U-235, there are many neutron-initiated reactions that produce light charged

particles. An important example is the reaction with boron in which boron-10 is converted to lithium-7 plus an alpha

particle. On the basis of this reaction, boron is often used as a poison or reactor control material for removing neutron

when it is desired to shut down the fission chain reaction.

10B5 +

1n0 [

11B5]*

7Li3 +

42

Other neutron induced reaction may yield protons or deuterons. Multiple charged particle emissions, possibly

accompanied by neutrons(s) and/or gamma(s), have also been observed.

Absorption Cross-sectiona.

The overall cross-section which is the summation of all above mentioned capture or activation cross section, charged

particle, multiple neutron production, as well as fission cross section except the scattering ones are altogether termed as

absorption cross section.

a = [ c + (n,p) or (n,d), or (n,) + (n,xn) ] + [f ] Total Cross-sectiont.

The summation of all possible interactions of neutron is known as total cross section i.e. all types of absorption cross

sections and all scattering cross section are considered in it

t = [ a] + [s ]

A subset-tree diagram is given in figure 2.4 and a schematic categorization of various types of cross section is given in

figure 2.5.

Figure 2.4 Different types of neutron cross sections (Knief, pp-49)

Lecture delivered by Dr. Naseem Irfan (viii) of xii

Figure 2.5 Different types of neutron interactions

ENERGY DEPENDENCE OF NEUTRON CROSS SECTION

As mentioned previously (and indicated in figure 2.3), the cross section depends upon main factors like the energy of

incident particle and the nature of target nuclei. The cross section vs energy plot for different material may be entirely

different which indicated the dependence of cross section on the type and nature of target material. However, some

features of these -vs-E plots are common. For instance the case of uranium-235 may be considered in detail as it is the

most important material in reactor physics. The close study of energy dependence of its absorption cross section reveals

the existence of three major regions [Figure 2.6].

1. Thermal region (low energy or 1/V region)

2. Resonance region (epithermal region)

3. Fast neutron region

In general in each three neutron energy ranges a certain pattern exists that may be used to estimate the value of cross

sections.

Lecture delivered by Dr. Naseem Irfan (ix) of xii

Figure: 2.6 A sketch of microscopic fission/absorption cross section and total cross section for fissile 235

U92 .

Thermal region:

The plot shows a straight line on a log-log scale with a slop of 0.5 in the range of 0.0001eV to about 1 eV. Using the

straight line equation:

y = m x + c

ln (a ) = -0.5 ln(E) + ln ( C )

ln (a ) = ln ( C ) - 0.5 ln(E)

ln (a ) = ln ( C ) - ln (E)0.5

ln (a ) = ln [( C ) / (E)0.5

]

(a ) = [( C ) / (E)0.5

]

a 1 / (E)0.5

Thus in the low energy range, the microscopic absorption cross section is inversely proportional to the square root of the

neutron energy. As temperature is proportional to the kinetic energy therefore same relation also exist for temperature.

Based on the expression of kinetic energy, it can be observed that energy is proportional to square of the velocity there

for there would be a simple inverse relationship of cross section with velocity. Greater (a) at lower (E) indicates that

at low energy or velocity there would be a greater probability of absorption of neutrons. This may be explained by the

fact that among neutrons passing close to a given nucleus, the slower ones spend more time near the nucleus and

experience the nuclear forces for a greater period of time. The absorption probability, then, tends to vary inversely as the

neutron speed and, thus, give ruse to the 1/v behavior. Hence we can write a continuity type equation of cross section for

energy, temperature and velocity as below:

THERMAL REGION RESONANCE REGION FAST REGION

Lecture delivered by Dr. Naseem Irfan (x) of xii

ENERGY TEMPERATURE VELOCITY

The above inverse relation hides in it a very important fact regarding an in built and inherent safety feature of this

thermal region concerning the operation of a power reactor. It indicates that if due to some reason power level increases

in an uncontrolled fashion it results in an increase in temperature which in turn increases the thermal vibrations and thus

the energy of the system. As the above inverse relations imply that due to this increase in energy or temperature of the

system, the cross section will decrease resulting in decreased probability of fission (f contributes the major part of

absorption cross section (a ) for U-235) will decrease and rate or nuclear reaction is resulting in decreased energy

production which is a safety feature. This is why approximately all the power reactors operating today are thermal

reactors, working in low energy (1/v) by slowing down the fast fission neutrons to this thermal energy region. This is

done by the process of moderation i.e. when neutrons are emitted, they have an energy of 2MeV and are in fast energy

range. When a moderator medium is provided between the fuel rods then due to multiple collisions with moderator

nuclei the fast neutrons loose the energy and thus they are brought to energy range of 1/v region causing more fission.

Resonance region:

The absorption cross section rises sharply to fairly high values for certain neutron energies resulting in sharp peaks. The

very high, narrow peaks in this region are a result of the nucleuss affinity for neutrons whose energies closely match its

discrete, quantum energy levels. Due to this matching of neutron energy and quantum energy levels, these absorption

peaks are called resonance peaks.

According to nuclear shell model, an atomic nucleus can be stable only when its

energy corresponds to that of a particular quantum state. Every nucleus has

several such states, the lowest being the stable or ground state, whereas the

others are various excited quantum states or energy levels. For small excitation

energies, as are of interest here, the quantum levels are fairly widely spaced and

distinct; at high energies the spacing becomes very much closer and the quantum

states are virtually continuos. When a nucleus captures a neutron and the energy

of the resulting compound nucleus is equal or very close to one of the quantum

states of the nucleus, the probability of capture is exceptionally high which

explains the resonance absorption with its associated large capture cross section.

Figure 2.7 Illustrates the resonance effect. The lines at the right indicate schematically the (virtual) quantum levels of

the compound nucleus. The lines marked Eo, at the left, represents the energy of the compound nucleus formed when the

target nucleus absorbs a neutron of zero kinetic energy Ek. This would be the difference of mass of neutron outside

the nucleus i.e. (M) and the mass inside the nucleus after capture (M/) i .e. Eo = M - M

/. The excitation energy of the

compound nucleus Eo is then just (numerically) equal to the binding energy of the neutron. An examination of the figure

2.7 shows that the energy does not corresponds to any of the quantum levels of the compound nucleus. However, if the

a 1 / (E)0.5

[a ] = Constant / (E)0.5

[a ] (E)0.5

= Constant

[a ]1 (E

1)

0.5=[a ]

2 (E

2)

0.5

Similarly for

a 1 / (T)0.5

we get

[a ]1 (T

1)

0.5=[a ]

2 (T

2)

0.5

same for velocity

a 1 / (v)

we get

[a ]1 (v

1)

=[a ]

2 (v

2)

Figure 2.7 Formation of excited compound

nucleus and quantum energy levels

Lecture delivered by Dr. Naseem Irfan (xi) of xii

neutron has a certain amount of kinetic energy EK, sufficient to bring the energy of the compound nucleus up to E1, the

nucleus will be in one of its quantum states. Thus, when the neutron has kinetic energy Ek=E1-Eo, resonance absorption

occurs, and the nuclear cross section is exceptionally large. There will, similarly, be resonance absorption for neutrons

of kinetic energy E2-Eo so that the total energy E2 is equal to that of another quantum level of the compound nucleus, as

seen in the figure 2.7.

As a massive nuclei has greater number of quantum energy levels compared to smaller mass nuclei, specific energies for

which an absorption peak may occur depends considerably on type and size of the target nuclei. Most of the elements

having low mass numbers do not exhibit resonance absorption peaks. This does not mean that they are not important as

regard to be used as reactor material. There may be the cases that there total cross section is then contributed by s

instead of a and still they may be used as reflector or moderator.

The resonance peaks are much more pronounced in case of heavier nuclei e.g. 238

U92 which has 8 to 9 peaks of very

heavy energy [figure 2.8]. The greater number of peaks in this case may be attributed to more number of nucleon in

238U92 as compared to

235U92 and thus greater number of shell and quantum level states are available.

Figure: 2.8 A sketch of total microscopic cross section for 238

U92 .

An idea about the precision or the sharpness of the resonance peaks can be expressed by the resonance peak width.

Usually it is taken in keV at a cross-section equal to one half of the peak value. It has been observed that with increasing

temperature resonance width increases and peak height decreases. This effect is also termed as Doppler Broadening.

The widening of resonance peak would contribute more towards absorption of neutron in this region. As maximum

absorption chances are present in this resonance region therefore this region is the most suitable reason for reactors

which are meant for production of fissile material from fertile material. Hence the converters or production reactors are

operated in this region.

Fast region:

In this region absorption cross-section decreases as energy increases beyond the resonance range. Theoretically cross-

section at any energy is difficult to predict because here the cross-section decreases and increases abruptly and

haphazardly.

Fast breeder reactors are operated in this region. When the fissile nuclide produced is identical with that used to

maintain fission chain, the reactor is called breeder reactor. A fast breeder reactor more usually utilizes Pu-239 as the

Lecture delivered by Dr. Naseem Irfan (xii) of xii

fuel and U-238 as the fertile species. These reactors can generate power at the same time producing more Pu-239 than is

consumed. The fertile species is employed in the form of blanket, which surrounds the core, and so large proportions of

the neutrons escaping from the core are captured in the blanket. In most breeder reactor design, especially for fast

reactors, some fertile material, (e.g. U-238) is included in the core. Fissile nuclei are then formed both in the core and in

the blanket. The fissile material produced in the core is useful as fuel at the place of formation. That generated in the

blanket, however, must be returned to the core in some manner if it is to be utilized efficiently.

The cross sections in fast region are usually low, being less than 10 barns in most cases and becoming even smaller for

energies of the order of 0.1 MeV or more. The absorption cross-section at high energies of an order of 1MeV are then

similar in magnitude to the geometric cross section of the nucleus, i.e. 2 to 3 barns.