In chemistry and atomic physics, the electron affinity of an atom or molecule is defined as:
the change in energy (in kJ/mole) of a neutral atom or molecule (in the gaseous phase) when an electron is added to the atom to form a negative ion.
X + e– → X– + energy Affinity = – ∆H
In other words, it can be expressed as the neutral atom’s likelihood of gaining an electron. Note that ionization energies measure the tendency of a neutral atom to resist the loss of electrons. Electron affinities are more difficult to measure than ionization energies.
An atom of Carbon in the gas phase, for example, gives off energy when it gains an electron to form an ion of Carbon.
C + e– → C– – ∆H = Affinity = 153.9 kJ/mol
Electron affinity is one of the most important parameters that guide chemical reactivity. Molecules with high electron affinity form very stable negative ions which are important in the chemical and health industry as they purify the air, lift mood, and most importantly, act as strong oxidizing agents. To use electron affinities properly, it is essential to keep track of signs. When an electron is added to a neutral atom, energy is released. This affinity is known as the first electron affinity, and these energies are negative. By convention, the negative sign shows a release of energy. However, more energy is required to add an electron to a negative ion which overwhelms any release of energy from the electron attachment process. This affinity is known as the second electron affinity, and these energies are positive.
Halogens have the highest electron affinities among all elements. In fact, the electron affinity of Cl, 3.62 eV is the largest of all the elements. Superhalogens are molecules that have electron affinities (EA) greater than that of Cl, the element with the highest EA (3.62 eV).
It is well known that noble gases have closed electronic shell structure and hence have high ionization potentials and low electron affinities, due to which they are chemically inert and resistant to salt formation under most conditions.
Affinities of Nonmetals vs. Affinities of Metals
Metals: Metals like to lose valence electrons to form cations to have a fully stable shell. The electron affinity of metals is lower than that of nonmetals. Mercury most weakly attracts an extra electron.
Nonmetals: Generally, nonmetals have more positive electron affinity than metals. Nonmetals like to gain electrons to form anions to have a fully stable electron shell. Chlorine most strongly attracts extra electrons. The electron affinities of the noble gases have not been conclusively measured, so they may or may not have slightly negative values.
Electronegativity, symbol χ, is a chemical property that describes the tendency of an atom to attract electrons towards this atom. For this purpose, a dimensionless quantity, the Pauling scale, symbol χ, is the most commonly used.
The electronegativity of Carbon is:
χ = 2.55
In general, an atom’s electronegativity is affected by both its atomic number and the distance at which its valence electrons reside from the charged nucleus. The higher the associated electronegativity number, the more an element or compound attracts electrons towards it.
As it is usually calculated, electronegativity is not a property of an atom alone, but rather a property of an atom in a molecule. Even so, the electronegativity of an atom is strongly correlated with the first ionization energy, and negatively correlated with the electron affinity. Electrons with low ionization energies have low electronegativities because their nuclei do not exert a strong attractive force on electrons. Elements with high ionization energies have high electronegativities due to the strong pull exerted by the positive nucleus on the negative electrons. Therefore the electronegativity is greatest at the top-right of the periodic table and decreases toward the bottom-left.
Caesium is the least electronegative element (0.79); fluorine is the most (3.98).
Ionization energy, also called ionization potential, is the energy necessary to remove an electron from the neutral atom.
X + energy → X+ + e−
where X is any atom or molecule capable of being ionized, X+ is that atom or molecule with an electron removed (positive ion), and e− is the removed electron.
A Carbon atom, for example, requires the following ionization energy to remove the outermost electron.
C + IE → C+ + e− IE = 11.2603 eV
The ionization energy associated with removal of the first electron is most commonly used. The nth ionization energy refers to the amount of energy required to remove an electron from the species with a charge of (n-1).
1st ionization energy
X → X+ + e−
2nd ionization energy
X+ → X2+ + e−
3rd ionization energy
X2+ → X3+ + e−
Ionization Energy for different Elements
There is ionization energy for each successive electron removed. The electrons that circle the nucleus move in fairly well-defined orbits. Some of these electrons are more tightly bound in the atom than others. For example, only 7.38 eV is required to remove the outermost electron from a lead atom, while 88,000 eV is required to remove the innermost electron. Helps to understand the reactivity of elements (especially metals, which lose electrons).
In general, the ionization energy increases moving up a group and moving left to right across a period. Moreover:
Ionization energy is lowest for the alkali metals which have a single electron outside a closed shell.
Ionization energy increases across a row on the periodic maximum for the noble gases which have closed shells.
For example, sodium requires only 496 kJ/mol or 5.14 eV/atom to ionize it. On the other hand neon, the noble gas, immediately preceding it in the periodic table, requires 2081 kJ/mol or 21.56 eV/atom.
In chemistry and atomic physics, the electron affinity of an atom or molecule is defined as:
the change in energy (in kJ/mole) of a neutral atom or molecule (in the gaseous phase) when an electron is added to the atom to form a negative ion.
X + e– → X– + energy Affinity = – ∆H
In other words, it can be expressed as the neutral atom’s likelihood of gaining an electron. Note that ionization energies measure the tendency of a neutral atom to resist the loss of electrons. Electron affinities are more difficult to measure than ionization energies.
An atom of Boron in the gas phase, for example, gives off energy when it gains an electron to form an ion of Boron.
B + e– → B– – ∆H = Affinity = 26.7 kJ/mol
Electron affinity is one of the most important parameters that guide chemical reactivity. Molecules with high electron affinity form very stable negative ions which are important in the chemical and health industry as they purify the air, lift mood, and most importantly, act as strong oxidizing agents. To use electron affinities properly, it is essential to keep track of signs. When an electron is added to a neutral atom, energy is released. This affinity is known as the first electron affinity, and these energies are negative. By convention, the negative sign shows a release of energy. However, more energy is required to add an electron to a negative ion which overwhelms any release of energy from the electron attachment process. This affinity is known as the second electron affinity, and these energies are positive.
Halogens have the highest electron affinities among all elements. In fact, the electron affinity of Cl, 3.62 eV is the largest of all the elements. Superhalogens are molecules that have electron affinities (EA) greater than that of Cl, the element with the highest EA (3.62 eV).
It is well known that noble gases have closed electronic shell structure and hence have high ionization potentials and low electron affinities, due to which they are chemically inert and resistant to salt formation under most conditions.
Affinities of Nonmetals vs. Affinities of Metals
Metals: Metals like to lose valence electrons to form cations to have a fully stable shell. The electron affinity of metals is lower than that of nonmetals. Mercury most weakly attracts an extra electron.
Nonmetals: Generally, nonmetals have more positive electron affinity than metals. Nonmetals like to gain electrons to form anions to have a fully stable electron shell. Chlorine most strongly attracts extra electrons. The electron affinities of the noble gases have not been conclusively measured, so they may or may not have slightly negative values.
Electronegativity, symbol χ, is a chemical property that describes the tendency of an atom to attract electrons towards this atom. For this purpose, a dimensionless quantity, the Pauling scale, symbol χ, is the most commonly used.
The electronegativity of Boron is:
χ = 2.04
In general, an atom’s electronegativity is affected by both its atomic number and the distance at which its valence electrons reside from the charged nucleus. The higher the associated electronegativity number, the more an element or compound attracts electrons towards it.
As it is usually calculated, electronegativity is not a property of an atom alone, but rather a property of an atom in a molecule. Even so, the electronegativity of an atom is strongly correlated with the first ionization energy, and negatively correlated with the electron affinity. Electrons with low ionization energies have low electronegativities because their nuclei do not exert a strong attractive force on electrons. Elements with high ionization energies have high electronegativities due to the strong pull exerted by the positive nucleus on the negative electrons. Therefore the electronegativity is greatest at the top-right of the periodic table and decreases toward the bottom-left.
Caesium is the least electronegative element (0.79); fluorine is the most (3.98).
Ionization energy, also called ionization potential, is the energy necessary to remove an electron from the neutral atom.
X + energy → X+ + e−
where X is any atom or molecule capable of being ionized, X+ is that atom or molecule with an electron removed (positive ion), and e− is the removed electron.
A Boron atom, for example, requires the following ionization energy to remove the outermost electron.
B + IE → B+ + e− IE = 8.298 eV
The ionization energy associated with removal of the first electron is most commonly used. The nth ionization energy refers to the amount of energy required to remove an electron from the species with a charge of (n-1).
1st ionization energy
X → X+ + e−
2nd ionization energy
X+ → X2+ + e−
3rd ionization energy
X2+ → X3+ + e−
Ionization Energy for different Elements
There is ionization energy for each successive electron removed. The electrons that circle the nucleus move in fairly well-defined orbits. Some of these electrons are more tightly bound in the atom than others. For example, only 7.38 eV is required to remove the outermost electron from a lead atom, while 88,000 eV is required to remove the innermost electron. Helps to understand the reactivity of elements (especially metals, which lose electrons).
In general, the ionization energy increases moving up a group and moving left to right across a period. Moreover:
Ionization energy is lowest for the alkali metals which have a single electron outside a closed shell.
Ionization energy increases across a row on the periodic maximum for the noble gases which have closed shells.
For example, sodium requires only 496 kJ/mol or 5.14 eV/atom to ionize it. On the other hand neon, the noble gas, immediately preceding it in the periodic table, requires 2081 kJ/mol or 21.56 eV/atom.
Electron Affinity and Electronegativity of Beryllium
Electron Affinity of Beryllium is — kJ/mol.
Electronegativity of Beryllium is 1.57.
First Ionization Energy of Beryllium is 9.3226 eV.
Electron Affinity
In chemistry and atomic physics, the electron affinity of an atom or molecule is defined as:
the change in energy (in kJ/mole) of a neutral atom or molecule (in the gaseous phase) when an electron is added to the atom to form a negative ion.
X + e– → X– + energy Affinity = – ∆H
In other words, it can be expressed as the neutral atom’s likelihood of gaining an electron. Note that ionization energies measure the tendency of a neutral atom to resist the loss of electrons. Electron affinities are more difficult to measure than ionization energies.
An atom of Beryllium in the gas phase, for example, gives off energy when it gains an electron to form an ion of Beryllium.
Be + e– → Be– – ∆H = Affinity = — kJ/mol
Electron affinity is one of the most important parameters that guide chemical reactivity. Molecules with high electron affinity form very stable negative ions which are important in the chemical and health industry as they purify the air, lift mood, and most importantly, act as strong oxidizing agents. To use electron affinities properly, it is essential to keep track of signs. When an electron is added to a neutral atom, energy is released. This affinity is known as the first electron affinity, and these energies are negative. By convention, the negative sign shows a release of energy. However, more energy is required to add an electron to a negative ion which overwhelms any release of energy from the electron attachment process. This affinity is known as the second electron affinity, and these energies are positive.
Halogens have the highest electron affinities among all elements. In fact, the electron affinity of Cl, 3.62 eV is the largest of all the elements. Superhalogens are molecules that have electron affinities (EA) greater than that of Cl, the element with the highest EA (3.62 eV).
It is well known that noble gases have closed electronic shell structure and hence have high ionization potentials and low electron affinities, due to which they are chemically inert and resistant to salt formation under most conditions.
Affinities of Nonmetals vs. Affinities of Metals
Metals: Metals like to lose valence electrons to form cations to have a fully stable shell. The electron affinity of metals is lower than that of nonmetals. Mercury most weakly attracts an extra electron.
Nonmetals: Generally, nonmetals have more positive electron affinity than metals. Nonmetals like to gain electrons to form anions to have a fully stable electron shell. Chlorine most strongly attracts extra electrons. The electron affinities of the noble gases have not been conclusively measured, so they may or may not have slightly negative values.
Electronegativity, symbol χ, is a chemical property that describes the tendency of an atom to attract electrons towards this atom. For this purpose, a dimensionless quantity, the Pauling scale, symbol χ, is the most commonly used.
The electronegativity of Beryllium is:
χ = 1.57
In general, an atom’s electronegativity is affected by both its atomic number and the distance at which its valence electrons reside from the charged nucleus. The higher the associated electronegativity number, the more an element or compound attracts electrons towards it.
As it is usually calculated, electronegativity is not a property of an atom alone, but rather a property of an atom in a molecule. Even so, the electronegativity of an atom is strongly correlated with the first ionization energy, and negatively correlated with the electron affinity. Electrons with low ionization energies have low electronegativities because their nuclei do not exert a strong attractive force on electrons. Elements with high ionization energies have high electronegativities due to the strong pull exerted by the positive nucleus on the negative electrons. Therefore the electronegativity is greatest at the top-right of the periodic table and decreases toward the bottom-left.
Caesium is the least electronegative element (0.79); fluorine is the most (3.98).
First Ionization Energy of Beryllium is 9.3226 eV.
Ionization energy, also called ionization potential, is the energy necessary to remove an electron from the neutral atom.
X + energy → X+ + e−
where X is any atom or molecule capable of being ionized, X+ is that atom or molecule with an electron removed (positive ion), and e− is the removed electron.
A Beryllium atom, for example, requires the following ionization energy to remove the outermost electron.
Be + IE → Be+ + e− IE = 9.3226 eV
The ionization energy associated with removal of the first electron is most commonly used. The nth ionization energy refers to the amount of energy required to remove an electron from the species with a charge of (n-1).
1st ionization energy
X → X+ + e−
2nd ionization energy
X+ → X2+ + e−
3rd ionization energy
X2+ → X3+ + e−
Ionization Energy for different Elements
There is ionization energy for each successive electron removed. The electrons that circle the nucleus move in fairly well-defined orbits. Some of these electrons are more tightly bound in the atom than others. For example, only 7.38 eV is required to remove the outermost electron from a lead atom, while 88,000 eV is required to remove the innermost electron. Helps to understand the reactivity of elements (especially metals, which lose electrons).
In general, the ionization energy increases moving up a group and moving left to right across a period. Moreover:
Ionization energy is lowest for the alkali metals which have a single electron outside a closed shell.
Ionization energy increases across a row on the periodic maximum for the noble gases which have closed shells.
For example, sodium requires only 496 kJ/mol or 5.14 eV/atom to ionize it. On the other hand neon, the noble gas, immediately preceding it in the periodic table, requires 2081 kJ/mol or 21.56 eV/atom.
Electron Affinity and Electronegativity of Lithium
Electron Affinity of Lithium is 59.6 kJ/mol.
Electronegativity of Lithium is 0.98.
First Ionization Energy of Lithium is 5.3917 eV.
Electron Affinity
In chemistry and atomic physics, the electron affinity of an atom or molecule is defined as:
the change in energy (in kJ/mole) of a neutral atom or molecule (in the gaseous phase) when an electron is added to the atom to form a negative ion.
X + e– → X– + energy Affinity = – ∆H
In other words, it can be expressed as the neutral atom’s likelihood of gaining an electron. Note that ionization energies measure the tendency of a neutral atom to resist the loss of electrons. Electron affinities are more difficult to measure than ionization energies.
An atom of Lithium in the gas phase, for example, gives off energy when it gains an electron to form an ion of Lithium.
Li + e– → Li– – ∆H = Affinity = 59.6 kJ/mol
Electron affinity is one of the most important parameters that guide chemical reactivity. Molecules with high electron affinity form very stable negative ions which are important in the chemical and health industry as they purify the air, lift mood, and most importantly, act as strong oxidizing agents. To use electron affinities properly, it is essential to keep track of signs. When an electron is added to a neutral atom, energy is released. This affinity is known as the first electron affinity, and these energies are negative. By convention, the negative sign shows a release of energy. However, more energy is required to add an electron to a negative ion which overwhelms any release of energy from the electron attachment process. This affinity is known as the second electron affinity, and these energies are positive.
Halogens have the highest electron affinities among all elements. In fact, the electron affinity of Cl, 3.62 eV is the largest of all the elements. Superhalogens are molecules that have electron affinities (EA) greater than that of Cl, the element with the highest EA (3.62 eV).
It is well known that noble gases have closed electronic shell structure and hence have high ionization potentials and low electron affinities, due to which they are chemically inert and resistant to salt formation under most conditions.
Affinities of Nonmetals vs. Affinities of Metals
Metals: Metals like to lose valence electrons to form cations to have a fully stable shell. The electron affinity of metals is lower than that of nonmetals. Mercury most weakly attracts an extra electron.
Nonmetals: Generally, nonmetals have more positive electron affinity than metals. Nonmetals like to gain electrons to form anions to have a fully stable electron shell. Chlorine most strongly attracts extra electrons. The electron affinities of the noble gases have not been conclusively measured, so they may or may not have slightly negative values.
Electronegativity, symbol χ, is a chemical property that describes the tendency of an atom to attract electrons towards this atom. For this purpose, a dimensionless quantity, the Pauling scale, symbol χ, is the most commonly used.
The electronegativity of Lithium is:
χ = 0.98
In general, an atom’s electronegativity is affected by both its atomic number and the distance at which its valence electrons reside from the charged nucleus. The higher the associated electronegativity number, the more an element or compound attracts electrons towards it.
As it is usually calculated, electronegativity is not a property of an atom alone, but rather a property of an atom in a molecule. Even so, the electronegativity of an atom is strongly correlated with the first ionization energy, and negatively correlated with the electron affinity. Electrons with low ionization energies have low electronegativities because their nuclei do not exert a strong attractive force on electrons. Elements with high ionization energies have high electronegativities due to the strong pull exerted by the positive nucleus on the negative electrons. Therefore the electronegativity is greatest at the top-right of the periodic table and decreases toward the bottom-left.
Caesium is the least electronegative element (0.79); fluorine is the most (3.98).
Ionization energy, also called ionization potential, is the energy necessary to remove an electron from the neutral atom.
X + energy → X+ + e−
where X is any atom or molecule capable of being ionized, X+ is that atom or molecule with an electron removed (positive ion), and e− is the removed electron.
A Lithium atom, for example, requires the following ionization energy to remove the outermost electron.
Li + IE → Li+ + e− IE = 5.3917 eV
The ionization energy associated with removal of the first electron is most commonly used. The nth ionization energy refers to the amount of energy required to remove an electron from the species with a charge of (n-1).
1st ionization energy
X → X+ + e−
2nd ionization energy
X+ → X2+ + e−
3rd ionization energy
X2+ → X3+ + e−
Ionization Energy for different Elements
There is ionization energy for each successive electron removed. The electrons that circle the nucleus move in fairly well-defined orbits. Some of these electrons are more tightly bound in the atom than others. For example, only 7.38 eV is required to remove the outermost electron from a lead atom, while 88,000 eV is required to remove the innermost electron. Helps to understand the reactivity of elements (especially metals, which lose electrons).
In general, the ionization energy increases moving up a group and moving left to right across a period. Moreover:
Ionization energy is lowest for the alkali metals which have a single electron outside a closed shell.
Ionization energy increases across a row on the periodic maximum for the noble gases which have closed shells.
For example, sodium requires only 496 kJ/mol or 5.14 eV/atom to ionize it. On the other hand neon, the noble gas, immediately preceding it in the periodic table, requires 2081 kJ/mol or 21.56 eV/atom.
In chemistry and atomic physics, the electron affinity of an atom or molecule is defined as:
the change in energy (in kJ/mole) of a neutral atom or molecule (in the gaseous phase) when an electron is added to the atom to form a negative ion.
X + e– → X– + energy Affinity = – ∆H
In other words, it can be expressed as the neutral atom’s likelihood of gaining an electron. Note that ionization energies measure the tendency of a neutral atom to resist the loss of electrons. Electron affinities are more difficult to measure than ionization energies.
An atom of Helium in the gas phase, for example, gives off energy when it gains an electron to form an ion of Helium.
He + e– → He– – ∆H = Affinity = — kJ/mol
Electron affinity is one of the most important parameters that guide chemical reactivity. Molecules with high electron affinity form very stable negative ions which are important in the chemical and health industry as they purify the air, lift mood, and most importantly, act as strong oxidizing agents. To use electron affinities properly, it is essential to keep track of signs. When an electron is added to a neutral atom, energy is released. This affinity is known as the first electron affinity, and these energies are negative. By convention, the negative sign shows a release of energy. However, more energy is required to add an electron to a negative ion which overwhelms any release of energy from the electron attachment process. This affinity is known as the second electron affinity, and these energies are positive.
Halogens have the highest electron affinities among all elements. In fact, the electron affinity of Cl, 3.62 eV is the largest of all the elements. Superhalogens are molecules that have electron affinities (EA) greater than that of Cl, the element with the highest EA (3.62 eV).
It is well known that noble gases have closed electronic shell structure and hence have high ionization potentials and low electron affinities, due to which they are chemically inert and resistant to salt formation under most conditions.
Affinities of Nonmetals vs. Affinities of Metals
Metals: Metals like to lose valence electrons to form cations to have a fully stable shell. The electron affinity of metals is lower than that of nonmetals. Mercury most weakly attracts an extra electron.
Nonmetals: Generally, nonmetals have more positive electron affinity than metals. Nonmetals like to gain electrons to form anions to have a fully stable electron shell. Chlorine most strongly attracts extra electrons. The electron affinities of the noble gases have not been conclusively measured, so they may or may not have slightly negative values.
Electronegativity, symbol χ, is a chemical property that describes the tendency of an atom to attract electrons towards this atom. For this purpose, a dimensionless quantity, the Pauling scale, symbol χ, is the most commonly used.
The electronegativity of Helium is:
χ = —
In general, an atom’s electronegativity is affected by both its atomic number and the distance at which its valence electrons reside from the charged nucleus. The higher the associated electronegativity number, the more an element or compound attracts electrons towards it.
As it is usually calculated, electronegativity is not a property of an atom alone, but rather a property of an atom in a molecule. Even so, the electronegativity of an atom is strongly correlated with the first ionization energy, and negatively correlated with the electron affinity. Electrons with low ionization energies have low electronegativities because their nuclei do not exert a strong attractive force on electrons. Elements with high ionization energies have high electronegativities due to the strong pull exerted by the positive nucleus on the negative electrons. Therefore the electronegativity is greatest at the top-right of the periodic table and decreases toward the bottom-left.
Caesium is the least electronegative element (0.79); fluorine is the most (3.98).
Ionization energy, also called ionization potential, is the energy necessary to remove an electron from the neutral atom.
X + energy → X+ + e−
where X is any atom or molecule capable of being ionized, X+ is that atom or molecule with an electron removed (positive ion), and e− is the removed electron.
A Helium atom, for example, requires the following ionization energy to remove the outermost electron.
He + IE → He+ + e− IE = 24.5874 eV
The ionization energy associated with removal of the first electron is most commonly used. The nth ionization energy refers to the amount of energy required to remove an electron from the species with a charge of (n-1).
1st ionization energy
X → X+ + e−
2nd ionization energy
X+ → X2+ + e−
3rd ionization energy
X2+ → X3+ + e−
Ionization Energy for different Elements
There is ionization energy for each successive electron removed. The electrons that circle the nucleus move in fairly well-defined orbits. Some of these electrons are more tightly bound in the atom than others. For example, only 7.38 eV is required to remove the outermost electron from a lead atom, while 88,000 eV is required to remove the innermost electron. Helps to understand the reactivity of elements (especially metals, which lose electrons).
In general, the ionization energy increases moving up a group and moving left to right across a period. Moreover:
Ionization energy is lowest for the alkali metals which have a single electron outside a closed shell.
Ionization energy increases across a row on the periodic maximum for the noble gases which have closed shells.
For example, sodium requires only 496 kJ/mol or 5.14 eV/atom to ionize it. On the other hand neon, the noble gas, immediately preceding it in the periodic table, requires 2081 kJ/mol or 21.56 eV/atom.
Electron Affinity and Electronegativity of Hydrogen
Electron Affinity of Hydrogen is 72.8 kJ/mol.
Electronegativity of Hydrogen is 2.2.
First Ionization Energy of Hydrogen is 13.5984 eV.
Electron Affinity
In chemistry and atomic physics, the electron affinity of an atom or molecule is defined as:
the change in energy (in kJ/mole) of a neutral atom or molecule (in the gaseous phase) when an electron is added to the atom to form a negative ion.
X + e– → X– + energy Affinity = – ∆H
In other words, it can be expressed as the neutral atom’s likelihood of gaining an electron. Note that ionization energies measure the tendency of a neutral atom to resist the loss of electrons. Electron affinities are more difficult to measure than ionization energies.
An atom of Hydrogen in the gas phase, for example, gives off energy when it gains an electron to form an ion of Hydrogen.
H + e– → H– – ∆H = Affinity = 72.8 kJ/mol
Electron affinity is one of the most important parameters that guide chemical reactivity. Molecules with high electron affinity form very stable negative ions which are important in the chemical and health industry as they purify the air, lift mood, and most importantly, act as strong oxidizing agents. To use electron affinities properly, it is essential to keep track of signs. When an electron is added to a neutral atom, energy is released. This affinity is known as the first electron affinity, and these energies are negative. By convention, the negative sign shows a release of energy. However, more energy is required to add an electron to a negative ion which overwhelms any release of energy from the electron attachment process. This affinity is known as the second electron affinity, and these energies are positive.
Halogens have the highest electron affinities among all elements. In fact, the electron affinity of Cl, 3.62 eV is the largest of all the elements. Superhalogens are molecules that have electron affinities (EA) greater than that of Cl, the element with the highest EA (3.62 eV).
It is well known that noble gases have closed electronic shell structure and hence have high ionization potentials and low electron affinities, due to which they are chemically inert and resistant to salt formation under most conditions.
Affinities of Nonmetals vs. Affinities of Metals
Metals: Metals like to lose valence electrons to form cations to have a fully stable shell. The electron affinity of metals is lower than that of nonmetals. Mercury most weakly attracts an extra electron.
Nonmetals: Generally, nonmetals have more positive electron affinity than metals. Nonmetals like to gain electrons to form anions to have a fully stable electron shell. Chlorine most strongly attracts extra electrons. The electron affinities of the noble gases have not been conclusively measured, so they may or may not have slightly negative values.
Electronegativity, symbol χ, is a chemical property that describes the tendency of an atom to attract electrons towards this atom. For this purpose, a dimensionless quantity, the Pauling scale, symbol χ, is the most commonly used.
The electronegativity of Hydrogen is:
χ = 2.2
In general, an atom’s electronegativity is affected by both its atomic number and the distance at which its valence electrons reside from the charged nucleus. The higher the associated electronegativity number, the more an element or compound attracts electrons towards it.
As it is usually calculated, electronegativity is not a property of an atom alone, but rather a property of an atom in a molecule. Even so, the electronegativity of an atom is strongly correlated with the first ionization energy, and negatively correlated with the electron affinity. Electrons with low ionization energies have low electronegativities because their nuclei do not exert a strong attractive force on electrons. Elements with high ionization energies have high electronegativities due to the strong pull exerted by the positive nucleus on the negative electrons. Therefore the electronegativity is greatest at the top-right of the periodic table and decreases toward the bottom-left.
Caesium is the least electronegative element (0.79); fluorine is the most (3.98).
First Ionization Energy of Hydrogen is 13.5984 eV.
Ionization energy, also called ionization potential, is the energy necessary to remove an electron from the neutral atom.
X + energy → X+ + e−
where X is any atom or molecule capable of being ionized, X+ is that atom or molecule with an electron removed (positive ion), and e− is the removed electron.
A Hydrogen atom, for example, requires the following ionization energy to remove the outermost electron.
H + IE → H+ + e− IE = 13.5984 eV
The ionization energy associated with removal of the first electron is most commonly used. The nth ionization energy refers to the amount of energy required to remove an electron from the species with a charge of (n-1).
1st ionization energy
X → X+ + e−
2nd ionization energy
X+ → X2+ + e−
3rd ionization energy
X2+ → X3+ + e−
Ionization Energy for different Elements
There is ionization energy for each successive electron removed. The electrons that circle the nucleus move in fairly well-defined orbits. Some of these electrons are more tightly bound in the atom than others. For example, only 7.38 eV is required to remove the outermost electron from a lead atom, while 88,000 eV is required to remove the innermost electron. Helps to understand the reactivity of elements (especially metals, which lose electrons).
In general, the ionization energy increases moving up a group and moving left to right across a period. Moreover:
Ionization energy is lowest for the alkali metals which have a single electron outside a closed shell.
Ionization energy increases across a row on the periodic maximum for the noble gases which have closed shells.
For example, sodium requires only 496 kJ/mol or 5.14 eV/atom to ionize it. On the other hand neon, the noble gas, immediately preceding it in the periodic table, requires 2081 kJ/mol or 21.56 eV/atom.
Reactor coolant pumps (RCPs) are used to pump primary coolant around the primary circuit. The purpose of the reactor coolant pump is to provide forced primary coolant flow to remove and transfer the amount of heat generated in the reactor core.
Nuclear power plants rely on cooling systems to ensure safe, continuous operation of the nuclear reactor. Cooling systems naturally ensure a heat transfer from a reactor core to steam generators, which is the main purpose of the cooling systems. Because of the large amount of heat generated in the reactor core by the fission reaction, the cooling systems demand a large volumetric flow of water (~80000 m3/hr) to ensure a sufficient and safe heat transfer. The cooling water is usually supplied by two or more large centrifugal pumps called reactor coolant pumps (RCPs). RCPs are not usually “safety system”, as defined. After the loss of RCPs the reactor must be shutdown immediately. Sufficient and safe residual heat removal is then ensured by a natural circulation flow through the reactor. However, natural circulation is not sufficient to remove the heat being generated when the reactor is at power.
Reactor coolant pumps (RCPs) are used to pump primary coolant around the primary circuit. The purpose of the reactor coolant pump is to provide forced primary coolant flow to remove and transfer the amount of heat generated in the reactor core. There are many designs of these pumps and there are many designs of primary coolant loops. There are significant differences between pumps for different reactor types. This article is focused on RCPs for pressurized water reactors. Most of PWRs use four RCPs in two or four loops design.
Generally reactor coolant pumps are powerful, they can consume up to 6 MW each and therefore they can be used for heating the primary coolant before a reactor startup.
Most of RCPs are vertical installed on a cold leg of a primary loop, but also a direct connection to a steam generator is possible. The reactor coolant enters the suction side of the pump at high pressure and temperature (~16MPa; 290°C; 554°F). The water is increased in velocity by the pump impeller. This increase in velocity is converted to pressure in the discharge volute. At the discharge of the reactor coolant pump, the reactor coolant pressure will be approximately 0,5MPa higher than the inlet pressure. After the coolant leaves the discharge side of the pump, it will enter the cold leg and continue to the reactor. The coolant will then pass through the nuclear core and through the fuel, where collects heat and is sent back to the steam generators.
The main components of a reactor coolant pump
Electric motor. The motor is a large, air or water (seal-less RCPs) cooled, induction motor.
Impeller. Impeller is a rotor used to increase the pressure and flow of a coolant.
Shaft (Rotor). Shaft is a mechanical component for transmitting torque from the motor to the impeller.
Shaft seal package. Shaft seal package is used to prevent any water from leaking up the shaft into the containment.
Bearings. Bearings constrain relative motion of the shaft (rotor) and reduce friction between the rotating shaft and the stator. RCPs usually use a combination of fluid dynamic bearings and hydrostatic bearings in the radial bearing assembly (water lubricated; close to the primary coolant) and oil lubricated bearings used in the thrust (axial) bearing assembly (in the motor section).
Flywheel. The flywheel provides flow coastdown in case of loss of power.
Auxilliary systems. Oil lubrication system, oil lift system, seal leakoff system, seal cooling system etc.
Use of shaft seals.
The seal package is located on the shaft between the electric motor and the impeller and prevents any primary coolant from leaking up the shaft into the containment. Any coolant that does leak up the shaft is collected and routed to the seal leakoff system.
RCPs with shaft seals. In order to maintain pump pressure and restrict water volume loss, the pumps typically utilize a multi-stage mechanical face seal system. This configuration allows use of flywheel, which provides rotating inertia to ensure a slow decrease in coolant flow in order to prevent fuel damage as a result of a loss of power to the pump motors. An integrity of the pressure boundary in the event of a postulated flywheel failure has to be proved.
Seal-less RCPs . This pumps do not have any shaft seals and any large flywheels. Entire RCP systems (motor, impeller, shaft, fluid bearings) are sealed at high pressure side of primary circuit. Such RCPs (Canned motor pumps) do not require so much external systems (no lube oil system, seal injection and leak-off system ) as the RCPs with shaft seals. These pumps can be more reliable, but sufficient rotating inertia (internal flywheels) to provide flow coastdown has to be proved.
A pressurizer is a component of a pressurized water reactor. Pressure in the primary circuit of PWRs is maintained by a pressurizer, a separate vessel that is connected to the primary circuit (hot leg) and partially filled with water which is heated to the saturation temperature (boiling point) for the desired pressure by submerged electrical heaters.
A pressurizer is a key component of PWRs.
Temperature in the pressurizer can be maintained at 350 °C (662 °F), which gives a subcooling margin (the difference between the pressurizer temperature and the highest temperature in the reactor core) of 30 °C. Subcooling margin is very important safety parameter of PWRs, since the boiling in the reactor core must be excluded. The basic design of the pressurized water reactor includes such requirement that the coolant (water) in the reactor coolant system must not boil. To achieve this, the coolant in the reactor coolant system is maintained at a pressure sufficiently high that boiling does not occur at the coolant temperatures experienced while the plant is operating or in an analyzed transient.
Gas Compression in Pressurizer
Pressure in the primary circuit of PWRs is maintained by a pressurizer, a separate vessel that is connected to the primary circuit (hot leg) and partially filled with water which is heated to the saturation temperature (boiling point) for the desired pressure by submerged electrical heaters. During the plant heatup the pressurizer can be filled by nitrogen instead of saturated steam.
Assume that a pressurizer contains 12 m3 of nitrogen at 20°C and 15 bar. The temperature is raised to 35°C, and the volume is reduced to 8.5 m3. What is the final pressure of the gas inside the pressurizer? Assume that the gas is ideal.
Solution:
Since the gas is ideal,we can use the ideal gas law to relate its parameters, both in the initial state i and in the final state f. Therefore:
pinitVinit = nRTinit
and
pfinalVfinal = nRTfinal
Dividing the second equation by the first equation and solving for pf we obtain:
pfinal = pinitTfinalVinit / TinitVfinal
Note that we cannot convert units of volume and pressure to basic SI units, because they cancel out each other. On the other hand we have to use Kelvins instead of degrees of Celsius. Therefore Tinit = 293 K and Tfinal = 308 K.
It follows, the resulting pressure in the final state will be:
pfinal= (15 bar) x (308 K) x (12 m3) / (293 K) x (8.5 m3) = 22 bar
Enthalpy of Water - 0.1 MPa, 3 MPa, 16 MPa
Latent heat of vaporization – water at 0.1 MPa (atmospheric pressure)
hlg = 2257 kJ/kg
Latent heat of vaporization – water at 3 MPa (pressure inside a steam generator)
hlg = 1795 kJ/kg
Latent heat of vaporization – water at 16 MPa (pressure inside a pressurizer)
The heat of vaporization diminishes with increasing pressure, while the boiling point increases. It vanishes completely at a certain point called the critical point.
Functions
Extensive and intensive properties of medium in the pressurizer.
Pressure in the pressurizer is controlled by varying the temperature of the coolant in the pressurizer. For these purposes two systems are installed. Water spray system and electrical heaters system. Volume of the pressurizer (tens of cubic meters) is filled with water on saturation parameters and steam. The water spray system (relatively cool water – from cold leg) can decrease the pressure in the vessel by condensing the steam on water droplets sprayed in the vessel. On the other hand the submerged electrical heaters are designed to increase the pressure by evaporation the water in the vessel. Water pressure in a closed system tracks water temperature directly; as the temperature goes up, pressure goes up.
Over-pressure relief system
Part of the pressurizer system is an over-pressure relief system. In the event that pressurizer pressure exceeds a certain maximum, there is a relief valve called the pilot-operated relief valve (PORV) on top of the pressurizer which opens to allow steam from the steam bubble to leave the pressurizer in order to reduce the pressure in the pressurizer, thus leads to reduction of pressure in the whole system. This steam is routed to a large relief tank in the reactor containment building where it is cooled and condensed back into liquid and stored for later disposition. There is a finite volume to these tanks and if events deteriorate to the point where the tanks fill up, a secondary pressure relief device on the tank(s), often a rupture disc, allows the condensed reactor coolant to spill out onto the floor of the reactor containment building where it pools in sumps for later disposition.
The pressurizer is equipped also with safety valves system (“safety system”), which are also routed to the relief tank. The safety valves system is used to emergency pressure reduction during emergency conditions.
Water level monitoring
Since the reactor coolant system is completely flooded during normal operations, there is no point in monitoring coolant level in any of the other vessels. But early awareness of a reduction of coolant level (or a loss of coolant) is very important to the safety of the reactor core. The pressurizer is deliberately located high in the reactor containment building such that, if the pressurizer has sufficient coolant in it, one can be reasonably certain that all the other vessels of the reactor coolant system (which are below it) are fully flooded with coolant. There is therefore, a coolant level monitoring system on the pressurizer and it is the one reactor coolant system vessel that is normally not full of coolant.
Main components of pressurizer
Pressure vessel
Relief system. Consist of relief and safety valves and pressurized relief tank. The relief system is used to emergency pressure reduction during emergency conditions and the system.
Water spray system. Consist of spray nozzles and spray line. The water spray system is used to pressure reduction during normal conditions and the system is used to mixing the coolant inside the pressurizer.
Electrical heaters. The submerged electrical heaters are used to pressure increase during normal conditions.
Surge line nozzle. Connects the pressurizer with the primary circuit.
Nuclear reactor and primary coolant system of WWER-1200. Source: http://www.bellona.ru/
Steam generators are heat exchangers used to convert feedwater into steam from heat produced in a nuclear reactor core. The steam produced drives the turbine. They are used in the most nuclear power plants, but there are many types according to the reactor type. The boiling water reactor does not require steam generators since the water boils directly in the reactor core. In other types of reactors, such as the pressurised heavy water reactors of the CANDU design, the primary fluid is heavy water. Liquid metal cooled reactors such as the Russian BN-600 reactor also use heat exchangers between a secondary sodium circuit and a tertiary water circuit.
Design of Steam Generator
To increase the amount of heat transferred and the power generated, the heat exchange surface must be maximalized. This is obtained by using tubes. Each steam generator can contain anywhere from 3,000 to 16,000 tubes, each about 19mm diameter.While the secondary fluid is always water, the reactor coolant (carbon dioxide, sodium, helium) depends on the reactor type. Where the coolant is pressurized water, two solutions have been adopted. In the first of these, the secondary water flows through straight tubes welded to tubesheets at both ends. This is the “once-through” type of steam generator. To eliminate the loads exerted on the tubesheets by differential thermal expansion between outside shell and the tubes, a second solutions is often employed. This alternative gives acope for thermal expansion by using U-tubes welded to a single tubesheet. The tubes carry the pressurized primary coolant and are surrounded by the secondary water, which is turned into steam.
There are two designes for U-tubes steam generators. Design with tube bundle arranged vertically and design with tube bundle arranged horizontally. Horizontal steam generators are used in the VVER type reactors. In commercial power plants, there are 2 to 6 steam generators per reactor; each steam generator (vertical design) can measure up to 70 feet (~21m) in height and weigh as much as 800 tons.
The materials that make up the steam generators and tubes are specially made and specifically designed to withstand the heat, high pressure and radiation. The water tubes also have to be able to resist corrosion from water for an extended period of time.
Steam Generator – vertical
Operating conditions
The hot primary coolant (water 330°C; 626°F; 16MPa) is pumped into the steam generator through primary inlet. High pressure of primary coolant is used to keep the water in the liquid state. Boiling of the primary coolant shall not occur. The liquid water flows through hundreds or thousands of tubes (usually 1.9 cm in diameter) inside the steam generator. The feedwater (secondary circuit) is heated from ~260°C 500°F to the boiling point of that fluid (280°C; 536°F; 6,5MPa). Heat is transferred through the walls of these tubes to the lower pressure secondary coolant located on the secondary side of the exchanger where the coolant evaporates to pressurized steam (saturated steam 280°C; 536°F; 6,5 MPa). The pressurized steam leaves the steam generator through a steam outlet and continues to the steam turbine. The transfer of heat is accomplished without mixing the two fluids to prevent the secondary coolant from becoming radioactive. The primary coolant leaves (water 295°C; 563°F; 16MPa) the steam generator through primary outlet and continues through a cold leg to a reactor coolant pump and then into the reactor.
Evaporation of water at high pressure - Energy balance in a steam generator
Steam Generator – vertical
Calculate the amount of primary coolant, which is required to evaporate 1 kg of feedwater in a typical steam generator. Assume that there are no energy losses, this is only idealized example.
Balance of the primary circuit
The hot primary coolant (water 330°C; 626°F; 16MPa) is pumped into the steam generator through primary inlet. The primary coolant leaves (water 295°C; 563°F; 16MPa) the steam generator through primary outlet.
hI, inlet = 1516 kJ/kg
=> ΔhI = -206 kJ/kg
hI, outlet = 1310 kJ/kg
Balance of the feedwater
Temperature gradients in typical PWR steam generator.
The feedwater (water 230°C; 446°F; 6,5MPa) is pumped into the steam generator through the feedwater inlet. The feedwater (secondary circuit) is heated from ~230°C 446°F to the boiling point of that fluid (280°C; 536°F; 6,5MPa). Feedwater is then evaporated and the pressurized steam (saturated steam 280°C; 536°F; 6,5 MPa) leaves the steam generator through steam outlet and continues to the steam turbine.
hII, inlet = 991 kJ/kg
=> ΔhII = 1789 kJ/kg
hII, outlet = 2780 kJ/kg
Balance of the steam generator
Since the difference in specific enthalpies is less for primary coolant than for feedwater, it is obvious that the amount of primary coolant will be higher than 1kg. To produce of 1 kg of saturated steam from feedwater, about 1789/206 x 1 kg = 8.68 kg of primary coolant is required.
Core inlet temperature
Core inlet temperature. Core inlet temperature is directly given by system parameters in steam generators. When steam generators are operated at approximately 6.0MPa, it means the saturation temperature is equal to 275.6 °C. Since there must be always ΔT (~15°C) between the primary circuit and the secondary circuit, the reactor coolant (in the cold leg)have about 290.6°C (at HFP) at the inlet of the core. As the system pressure increases, the core inlet temperature must also increase. This increase causes slight increase in fuel temperature.
Moisture separation
A moisture separation is important to maintain the moisture content of the steam as low as possible to prevent damage to the turbine blading. The vertical steam generators must use multiple stage moisture separation. The horizontal separators can use the moisture separation, but it is not necessary, since the steam releases the two phase fluid much more slowly and the produced steam is generally without moisture.
The vertical steam generators use usually two stages of moisture separation. One stage causes the mixture to spin, which slings the water to the outside. The water is then drained back to be used to make more steam. The drier steam is routed to the second stage of separation. In this stage, the mixture is forced to make rapid changes in direction. Because of the steam’s ability to change direction and the water’s inability to change, the steam exits the steam generator, and the water is drained back for reuse. The two stage process of moisture removal is so efficient at removing the water that for every 100 pounds of steam that exits the steam generator, the water content is less than 0.25 pounds.
Nuclear reactor and primary coolant system of WWER-1200. Source: http://www.bellona.ru/
In general, the neutron cross-section is an effective area that quantifies the likelihood of certain interaction between an incident neutron and a target object. It must be noted this likelihood does not depend on real target dimensions, because we are not describing geometrical cross-section. In the vicinity of target nucleus, neutron is subjected to strong nuclear forces of target nucleons. The interaction strongly depends on many variables, such as type of target nucleus and the neutron energy. For example, the likelihood that a thermal neutron will be absorbed by xenon-135 is about a million times higher than it will be scattered.
At this point we have to distinguish between two basic types of nuclear or neutron cross-sections.
Microscopic Cross-section. The effective target area in m2 presented by a single nucleus to an incident neutron beam is denoted the microscopic cross section, σ. The microscopic cross-sections characterize interactions with single isotopes and are a part of data libraries, such as ENDF/B-VII.1.
Macroscopic Cross-section. The macroscopic cross-section represents the effective target area of all of the nuclei contained in the volume of the material (such as fuel pellet). The units are given in cm-1. It is the probability of neutron-nucleus interaction per centimeter of neutron travel. These data are commonly used by codes for reactor core analyses and design. These codes are based on pre-computed assembly homogenized macroscopic cross-sections.
Various microscopic cross-section for uranium-235 and incident thermal neutron.
Barn – Unit of Cross-section
The cross-section is typically denoted σ and measured in units of area [m2]. But a square meter (or centimeter) is tremendously large in comparison to the effective area of a nucleus, and it has been suggested that a physicist once referred to the measure of a square meter as being “as big as a barn” when applied to nuclear processes. The name has persisted and microscopic cross sections are expressed in terms of barns. The standard unit for measuring a nuclear cross section is the barn, which is equal to 10−28 m² or 10−24 cm². It can be seen the concept of a nuclear cross section can be quantified physically in terms of “characteristic target area” where a larger area means a larger probability of interaction.
Typical Values of Microscopic Cross-sections
Uranium 235 is a fissile isotope and its fission cross-section for thermal neutrons is about 585 barns (for 0.0253 eV neutron). For fast neutrons its fission cross-section is on the order of barns.
Boron is commonly used as a neutron absorber due to the high neutron cross-section of isotope 10B. Its (n,alpha) reaction cross-section for thermal neutrons is about 3840 barns (for 0.025 eV neutron).
Gadolinium is commonly used as a neutron absorber due to very high neutron absorbtion cross-section of two isotopes 155Gd and 157Gd. 155Gd has 61 000 barns for thermal neutrons (for 0.025 eV neutron) and 157Gd has even 254 000 barns.
The extent to which neutrons interact with nuclei is described in terms of quantities known as cross-sections. Cross-sections are used to express the likelihood of particular interaction between an incident neutron and a target nucleus. It must be noted this likelihood does not depend on real target dimensions. In conjunction with the neutron flux, it enables the calculation of the reaction rate, for example to derive the thermal power of a nuclear power plant. The standard unit for measuring the microscopic cross-section (σ-sigma) is the barn, which is equal to 10-28 m2. This unit is very small, therefore barns (abbreviated as “b”) are commonly used.
The cross-section σ can be interpreted as the effective ‘target area’ that a nucleus interacts with an incident neutron. The larger the effective area, the greater the probability for reaction. This cross-section is usually known as the microscopic cross-section.
The concept of the microscopic cross-section is therefore introduced to represent the probability of a neutron-nucleus reaction. Suppose that a thin ‘film’ of atoms (one atomic layer thick) with Na atoms/cm2 is placed in a monodirectional beam of intensity I0. Then the number of interactions C per cm2 per second will be proportional to the intensity I0 and the atom density Na. We define the proportionality factor as the microscopic cross-section σ:
σt = C/Na.I0
In order to be able to determine the microscopic cross section, transmission measurements are performed on plates of materials. Assume that if a neutron collides with a nucleus it will either be scattered into a different direction or be absorbed (without fission absorption). Assume that there are N (nuclei/cm3) of the material and there will then be N.dx per cm2 in the layer dx.
Only the neutrons that have not interacted will remain traveling in the x direction. This causes the intensity of the uncollided beam will be attenuated as it penetrates deeper into the material.
Then, according to the definition of the microscopic cross section, the reaction rate per unit area is Nσ Ι(x)dx. This is equal to the decrease of the beam intensity, so that:
-dI = N.σ.Ι(x).dx
and
Ι(x) = Ι0e-N.σ.x
It can be seen that whether a neutron will interact with a certain volume of material depends not only on the microscopic cross-section of the individual nuclei but also on the density of nuclei within that volume. It depends on the N.σ factor. This factor is therefore widely defined and it is known as the macroscopic cross section.
The difference between the microscopic and macroscopic cross sections is extremely important. The microscopic cross section represents the effective target area of a single nucleus, while the macroscopic cross section represents the effective target area of all of the nuclei contained in certain volume.
Nuclear Radius
Typical nuclear radii are of the order 10−14 m. Assuming spherical shape, nuclear radii can be calculated according to following formula:
r = r0 . A1/3
where r0 = 1.2 x 10-15 m = 1.2 fm
If we use this approximation, we therefore expect the geometrical cross-sections of nuclei to be of the order of πr2 or 4.5 x 10−30 m² for hydrogen nuclei or 1.74 x 10−28 m² for 238U nuclei.
Since there are many nuclear reaction from the incident particle point of view, but, in nuclear reactor physics, neutron-nuclear reactions are of particular interest. In this case the neutron cross-section must be defined.
Total Cross-section
In general, nuclear cross-sections can be measured for all possible interaction processes together, in this case they are called total cross-sections (σt). The total cross section is the sum of all the partial cross sections such as:
The total cross-section measures the probability that an interaction of any type will occur when neutron interacts with a target.
Microscopic cross-sections constitute a key parameters of nuclear fuel. In general, neutron cross-sections are essential for the reactor core calculations and are a part of data libraries, such as ENDF/B-VII.1.
The neutron cross-section is variable and depends on:
Target nucleus (hydrogen, boron, uranium, etc.). Each isotop has its own set of cross-sections.
Type of the reaction (capture, fission, etc.). Cross-sections are different for each nuclear reaction.
Neutron energy(thermal neutron, resonance neutron, fast neutron). For a given target and reaction type, the cross-section is strongly dependent on the neutron energy. In the common case, the cross section is usually much larger at low energies than at high energies. This is why most nuclear reactors use a neutron moderator to reduce the energy of the neutron and thus increase the probability of fission, essential to produce energy and sustain the chain reaction.
Target energy (temperature of target material – Doppler broadening). This dependency is not so significant, but the target energy strongly influences inherent safety of nuclear reactors due to a Doppler broadening of resonances.
Microscopic cross-section varies with incident neutron energy. Some nuclear reactions exhibit very specific dependency on incident neutron energy. This dependency will be described on the example of the radiative capture reaction. The likelihood of a neutron radiative capture is represented by the radiative capture cross section as σγ. The following dependency is typical for radiative capture, it definitely does not mean, that it is typical for other types of reactions (see elastic scattering cross-section or (n,alpha) reaction cross-section).
The capture cross-section can be divided into three regions according to the incident neutron energy. These regions will be discussed separately.
1/v Region
Resonance Region
Fast Neutrons Region
Charts of Cross-sections
Uranium 238. Comparison of cross-sections.
Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data LibraryGadolinium 155 and 157. Comparison of radiative capture cross-sections.
Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library
For thermal neutrons (in 1/v region), absorption cross sections increases as the velocity (kinetic energy) of the neutron decreases. Source: JANIS 4.0
1/v Region
In the common case, the cross section is usually much larger at low energies than at high energies. For thermal neutrons (in 1/v region), also radiative capture cross-sections increase as the velocity (kinetic energy) of the neutron decreases. Therefore the 1/v Law can be used to determine shift in capture cross-section, if the neutron is in equilibrium with a surrounding medium. This phenomenon is due to the fact the nuclear force between the target nucleus and the neutron has a longer time to interact.
This law is aplicable only for absorbtion cross-section and only in the 1/v region.
Example of cross- sections in 1/v region:
The absorbtion cross-section for 238U at 20°C = 293K (~0.0253 eV) is:
.
The absorbtion cross-section for 238U at 1000°C = 1273K is equal to:
This cross-section reduction is caused only due to the shift of temperature of surrounding medium.
Resonance Region
Energy levels of compound state. For neutron absorption reaction on 238U the first resonance E1 corresponds to the excitation energy of 6.67eV. E0 is a base state of 239U.
The largest cross-sections are usually at neutron energies, that lead to long-lived states of the compound nucleus. The compound nuclei of these certain energies are referred to as nuclear resonances and its formation is typical in the resonance region. The widths of the resonances increase in general with increasing energies. At higher energies the widths may reach the order of the distances between resonances and then no resonances can be observed. The narrowest resonances are usually compound states of heavy nuclei (such as fissionable nuclei).
Since the mode of decay of the compound nucleus does not depend on the way the compound nucleus was formed, the nucleus sometimes emits a gamma ray (radiative capture) or sometimes emits a neutron (scattering). In order to understand the way, how a nucleus will stabilize itself, we have to understand the behaviour of compound nucleus.
The position of the energy levels during the formation of a compound nucleus. Ground state and energy states.
The compound nucleus emits a neutron only after one neutron obtains an energy in collision with other nucleon greater than its binding energy in the nucleus. It have some delay, because the excitation energy of the compound nucleus is divided among several nucleons. It is obvious the average time that elapses before a neutron can be emitted is much longer for nuclei with large number of nucleons than when only a few nucleons are involved. It is a consequence of sharing the excitation energy among a large number of nucleons.
This is the reason the radiative capture is comparatively unimportant in light nuclei but becomes increasingly important in the heavier nuclei.
It is obvious the compound states (resonances) are observed at low excitation energies. This is due to the fact, the energy gap between the states is large. At high excitation energy, the gap between two compound states is very small and the widths of resonances may reach the order of the distances between resonances. Therefore at high energies no resonances can be observed and the cross section in this energy region is continuous and smooth.
The lifetime of a compound nucleus is inversely proportional to its total width. Narrow resonances therefore correspond to capture while the wider resonances are due to scattering.
The radiative capture cross-section at energies above the resonance region drops rapidly to very small values. This rapid drop is caused by the compound nucleus, which is formed in more highly-excited states. In these highly-excited states it is more likely that one neutron obtains an energy in collision with other nucleon greater than its binding energy in the nucleus. The neutron emission becomes dominant and gamma decay becomes less important. Moreover, at high energies, the inelastic scattering and (n,2n) reaction are highly probable at the expense of both elastic scattering and radiative capture.
Macroscopic Cross-section
The difference between the microscopic cross-section and macroscopic cross-section is very important and is restated for clarity. The microscopic cross section represents the effective target area of a single target nucleus for an incident particle. The units are given in barns or cm2.
While the macroscopic cross-section represents the effective target area of all of the nuclei contained in the volume of the material. The units are given in cm-1.
A macroscopic cross-section is derived from microscopic cross-section and the atomic number density:
Σ=σ.N
Here σ, which has units of m2, is the microscopic cross-section. Since the units of N (nuclei density) are nuclei/m3, the macroscopic cross-section Σ have units of m-1, thus in fact is an incorrect name, because it is not a correct unit of cross-sections. In terms of Σt (the total cross-section), the equation for the intensity of a neutron beam can be written as
-dI = N.σ.Σt.dx
Dividing this expression by I(x) gives
-dΙ(x)/I(x) = Σt.dx
Since dI(x) is the number of neutrons that collide in dx, the quantity –dΙ(x)/I(x) represents the probability that a neutron that has survived without colliding until x, will collide in the next layer dx. It follows that the probability P(x) that a neutron will travel a distance x without any interaction in the material, which is characterized by Σt, is:
P(x) = e-Σt.x
From this equation, we can derive the probability that a neutron will make its first collision in dx. It will be the quantity P(x)dx. If the probability of the first collision in dx is independent of its past history, the required result will be equal to the probability that a neutron survives up to layer x without any interaction (~Σtdx) times the probability that the neutron will interact in the additional layer dx (i.e., ~e-Σt.x).
P(x)dx = Σtdx . e-Σt.x = Σt e-Σt.x dx
Mean Free Path
From the equation for the probability of the first collision in dx we can calculate the mean free path that is traveled by a neutron between two collisions. This quantity is usually designated by the symbol λ and it is equal to the average value of x, the distance traveled by a neutron without any interaction, over the interaction probability distribution.
whereby one can distinguish λs, λa, λf, etc. This quantity is also known as the relaxation length, because it is the distance in which the intensity of the neutrons that have not caused a reaction has decreased with a factor e.
For materials with high absorption cross-section, the mean free path is very short and neutron absorption occurs mostly on the surface of the material. This surface absorption is called self-shielding because the outer layers of atoms shield the inner layers.
Macroscopic Cross-section of Mixtures and Molecules
Most materials are composed of several chemical elements and compounds. Most of chemical elements contains several isotopes of these elements (e.g., gadolinium with its six stable isotopes). For this reason most materials involve many cross-sections. Therefore, to include all the isotopes within a given material, it is necessary to determine the macroscopic cross section for each isotope and then sum all the individual macroscopic cross-sections.
In this section both factors (different atomic densities and different cross-sections) will be considered in the calculation of the macroscopic cross-section of mixtures.
First, consider the Avogadro’s number N0 = 6.022 x 1023, is the number of particles (molecules, atoms) that is contained in the amount of substance given by one mole. Thus if M is the molecular weight, the ratio N0/M equals to the number of molecules in 1g of the mixture. The number of molecules per cm3 in the material of density ρ and the macroscopic cross-section for mixtures are given by following equations:
Ni = ρi.N0 / Mi
Scattering of slow neutrons by molecules is greater than by free nuclei.
Note that, in some cases, the cross-section of the molecule is not equal to the sum of cross-sections of its individual nuclei. For example the cross-section of neutron elastic scattering of water exhibits anomalies for thermal neutrons. It occurs, because the kinetic energy of an incident neutron is of the order or less than the chemical binding energy and therefore the scattering of slow neutrons by water (H2O) is greater than by free nuclei (2H + O).
Example - Macroscopic cross-section for boron carbide in control rods
A control rod usually contains solid boron carbide with natural boron. Natural boron consists primarily of two stable isotopes,11B (80.1%) and10B(19.9%). Boron carbide has a density of 2.52 g/cm3.
Determine the total macroscopic cross-section and the mean free path.
Density:
MB = 10.8
MC = 12
MMixture = 4 x 10.8 + 1×12 g/mol
NB4C = ρ . Na / MMixture
= (2.52 g/cm3)x(6.02×1023 nuclei/mol)/ (4 x 10.8 + 1×12 g/mol)
= 2.75×1022 molecules of B4C/cm3
NB = 4 x 2.75×1022 atoms of boron/cm3
NC = 1 x 2.75×1022 atoms of carbon/cm3
NB10 = 0.199 x 4 x 2.75×1022 = 2.18×1022 atoms of 10B/cm3
NB11 = 0.801 x 4 x 2.75×1022 = 8.80×1022 atoms of 11B/cm3
NC = 2.75×1022 atoms of 12C/cm3
the microscopic cross-sections
σt10B = 3843 b of which σ(n,alpha)10B = 3840 b
σt11B = 5.07 b
σt12C = 5.01 b
the macroscopic cross-section
ΣtB4C= 3843×10-24 x 2.18×1022 + 5.07×10-24 x 8.80×1022 + 5.01×10-24 x 2.75×1022
= 83.7 + 0.45 + 0.14 = 84.3 cm-1
the mean free path
λt= 1/ΣtB4C = 0.012 cm = 0.12 mm (compare with B4C pellets diameter in control rods which may be around 7mm) λa ≈ 0.12 mm
Example - Atomic number density of 235U in uranium fuel
It was written the macroscopic cross-section is derived from microscopic cross-section and the atomic number density (N):
Σ=σ.N
In this equation, the atomic number density plays the crucial role as the microscopic cross-section, because in the reactor core the atomic number density of certain materials (e.g., water as the moderator) can be simply changed leading into certain reactivity changes. In order to understand the nature of these reactivity changes, we must understand the term the atomic number density.
Most of PWRs use the uranium fuel, which is in the form of uranium dioxide (UO2). Typically, the fuel have enrichment of ω235 = 4% [grams of 235U per gram of uranium] of isotope 235U.
Calculate the atomic number density of 235U (N235U), when:
the molecular weight of the enriched uranium MUO2 = 237.9 + 32 = 269.9 g/mol
the uranium density ⍴UO2 = 10.5 g/cm3
NUO2 = ⍴UO2 . NA / MUO2
NUO2 = (10.5 g/cm3) x (6.02×1023 nuclei/mol)/ 269.9 NUO2 = 2.34 x 1022 molecules of UO2/cm3
NU = 1 x 2.34×1022 atoms of uranium/cm3
NO = 2 x 2.34×1022 atoms of oxide/cm3
N235U = ω235.NA.⍴UO2/M235U x (MU/MUO2)
N235U = 0.04 x 6.02×1023 x 10.5 / 235 x 237.9 / 269.9 =9.48 x 1020 atoms of 235U/cm3
Doppler Broadening of Resonances
In general, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of kinetic energies of molecules or atoms. In reactor physics a particular case of this phenomenon is the thermal Doppler broadening of the resonance capture cross sections of the fertile material (e.g.,238U or240Pu) caused by thermal motion of target nuclei in the nuclear fuel.
Doppler effect improves reactor stability. Broadened resonance (heating of a fuel) results in a higher probability of absorbtion, thus causes negative reactivity insertion (reduction of reactor power).
The Doppler broadening of resonances is very important phenomenon, which improves reactor stability, because it accounts for the dominant part of the fuel temperature coefficient (the change in reactivity per degree change in fuel temperature) in thermal reactors and makes a substantial contribution in fast reactors as well. This coefficient is also called the prompt temperature coefficient because it causes an immediate response on changes in fuel temperature. The prompt temperature coefficient of most thermal reactors is negative.
It was written, in some cases the amount of absorption reactions is dramatically reduced despite the unchanged microscopic cross-section of the material. This phenomena is commonly known as the resonance self-shielding and also contributes to to the reactor stability. There are two types of self-shielding.
Energy Self-shielding.
Spatial Self-shielding.
See also: Resonance Self-shieldingAn increase in temperature from T1 to T2 causes the broadening of spectral lines of resonances. Although the area under the resonance remains the same, the broadening of spectral lines causes an increase in neutron flux in the fuel φf(E), which in turn increases the absorption as the temperature increases.
References:
Nuclear and Reactor Physics:
J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
