Modified Born-Lande Equation | Inorganic Chemistry PDF Download

 Page 1


Modified Born-Lande Equation to calculate Lattice 
Energy in a theoretical approach 
   
   
 
   
 
 
 
Abstract: 
Defects in ionic solid are very much common, which is increased with the rise in temperature. It 
causes the change in the value of many physical properties and varieties of physical parameters 
and the Lattice Energy is one such parameter to control the physical properties of the crystals. 
Considering the loss of ions from lattice points as random, the examination of each of the defects 
individually is going to be unpredictable, thus leading to almost nonattainment of the correct 
crystal structure with the theoretical calculations applying for available models. Here, in this 
present work, we have used some statistical methods and probabilistic approximation to introduce 
a novel idea of calculating the Madelung constant, and then Lattice Energy analytically. 
To make the understanding more lucid, we have taken one of the very common crystals, very 
popular in the crystallographic community, NaCl crystal having 6:6 co-ordination number, for 
which a significant number of Schottky defects are observed. 
During this study, we are bound to assume the random distribution of defects as Poisson 
distribution due to the fact that the number of defects is very less with respect to the total numbers 
of lattice points present in the crystal to calculate the Madelung Constant. 
 
Keywords: 
Madelung Constant, Lattice Energy, Schottky defects, Poisson distribution. 
Introduction: 
For undergraduate students in chemistry, solid-state chemistry solid-state, and structural 
chemistry is very important. It is because of many scientific inventions and due to the very 
many uses of solid-state devices from semiconductors to superconductors. But before that, we 
have to study the most important and basic, the ionic solid. To study properly the ionic-solids 
we must know about the lattice energy, Madelung constant, and many other parameters. 
Page 2


Modified Born-Lande Equation to calculate Lattice 
Energy in a theoretical approach 
   
   
 
   
 
 
 
Abstract: 
Defects in ionic solid are very much common, which is increased with the rise in temperature. It 
causes the change in the value of many physical properties and varieties of physical parameters 
and the Lattice Energy is one such parameter to control the physical properties of the crystals. 
Considering the loss of ions from lattice points as random, the examination of each of the defects 
individually is going to be unpredictable, thus leading to almost nonattainment of the correct 
crystal structure with the theoretical calculations applying for available models. Here, in this 
present work, we have used some statistical methods and probabilistic approximation to introduce 
a novel idea of calculating the Madelung constant, and then Lattice Energy analytically. 
To make the understanding more lucid, we have taken one of the very common crystals, very 
popular in the crystallographic community, NaCl crystal having 6:6 co-ordination number, for 
which a significant number of Schottky defects are observed. 
During this study, we are bound to assume the random distribution of defects as Poisson 
distribution due to the fact that the number of defects is very less with respect to the total numbers 
of lattice points present in the crystal to calculate the Madelung Constant. 
 
Keywords: 
Madelung Constant, Lattice Energy, Schottky defects, Poisson distribution. 
Introduction: 
For undergraduate students in chemistry, solid-state chemistry solid-state, and structural 
chemistry is very important. It is because of many scientific inventions and due to the very 
many uses of solid-state devices from semiconductors to superconductors. But before that, we 
have to study the most important and basic, the ionic solid. To study properly the ionic-solids 
we must know about the lattice energy, Madelung constant, and many other parameters. 
To analyze the various physical properties of an ionic rystal, Lattice energy is the most 
important parameter. It is very much useful in understanding the potential functions and 
binding forces which are responsible for binding in an ionic crystal. The lattice energy is 
experimentally determined using the Born Haber cycle. But sometimes all the experiments 
cannot be performed and so theoretical determination of lattice energy is very much 
important.
 [1-11] 
The lattice energy of a crystal is based on a model where ions are considered as the point 
charges placed in a fixed position in a regular array and the coulombic electrostatic force acting 
among the ions. The lattice energy is calculated theoretically using the Born-Lande equation in 
which another parameter Madelung constant is also related.  
The Born-Lande equation may appear in textbooks in the form      
                                              
                                             ?? ?? = -
???? ?? +
?? -
?? 2
?? ?? (1 -
1
?? ) ………………………………….(1) 
Where, U o is the Lattice energy N is the Avogrado Number, A is the Madelung Constant, z + is the   
charge of cation, z - is the charge of anion, e is the charge of an electron, R o is the equilibrium 
distance between the oppositely charged ions, n is the Born exponent.
[12-15] 
 
In the above equation, 1 is the expression of Lattice Energy where no defects or missing ions 
are taken into account. So, one can calculate the lattice energy of a perfect crystal where no 
ions are missing from their lattice points. But in reality, as temperature increases, the ions are 
displaced from their lattice position, and deects are formed in the ionic lattice. 
As defects are formed disorder is developed in the crystal which implies entropy of the system 
increases. But on the other hand formation of defects is an endothermic process. 
So, for this process,   
??? > 0 , ??? > 0 
The process of formation defects in a crystal lattice is entropically driven and so the defect 
concentration increases with an increase in temperature. Even at room temperature (25
o
C) in 
1cc of 6:6 NaCl crystal, there are about 10
6
 Schottky defects that cause a considerable change 
in Lattice energy. 
The total potential energy of a crystal will be the sum of both Coulombic attraction potential 
and short-range repulsive potential, which is referred to as Born repulsive potential. The 
repulsion potential is mainly due to the interpenetration of the electron clouds which is 
inversely proportional to Rn where R is the internuclear distance and n is the Born exponent.  
 
                                                         ?? ?????????????????? ?
1
?? ?? 
Page 3


Modified Born-Lande Equation to calculate Lattice 
Energy in a theoretical approach 
   
   
 
   
 
 
 
Abstract: 
Defects in ionic solid are very much common, which is increased with the rise in temperature. It 
causes the change in the value of many physical properties and varieties of physical parameters 
and the Lattice Energy is one such parameter to control the physical properties of the crystals. 
Considering the loss of ions from lattice points as random, the examination of each of the defects 
individually is going to be unpredictable, thus leading to almost nonattainment of the correct 
crystal structure with the theoretical calculations applying for available models. Here, in this 
present work, we have used some statistical methods and probabilistic approximation to introduce 
a novel idea of calculating the Madelung constant, and then Lattice Energy analytically. 
To make the understanding more lucid, we have taken one of the very common crystals, very 
popular in the crystallographic community, NaCl crystal having 6:6 co-ordination number, for 
which a significant number of Schottky defects are observed. 
During this study, we are bound to assume the random distribution of defects as Poisson 
distribution due to the fact that the number of defects is very less with respect to the total numbers 
of lattice points present in the crystal to calculate the Madelung Constant. 
 
Keywords: 
Madelung Constant, Lattice Energy, Schottky defects, Poisson distribution. 
Introduction: 
For undergraduate students in chemistry, solid-state chemistry solid-state, and structural 
chemistry is very important. It is because of many scientific inventions and due to the very 
many uses of solid-state devices from semiconductors to superconductors. But before that, we 
have to study the most important and basic, the ionic solid. To study properly the ionic-solids 
we must know about the lattice energy, Madelung constant, and many other parameters. 
To analyze the various physical properties of an ionic rystal, Lattice energy is the most 
important parameter. It is very much useful in understanding the potential functions and 
binding forces which are responsible for binding in an ionic crystal. The lattice energy is 
experimentally determined using the Born Haber cycle. But sometimes all the experiments 
cannot be performed and so theoretical determination of lattice energy is very much 
important.
 [1-11] 
The lattice energy of a crystal is based on a model where ions are considered as the point 
charges placed in a fixed position in a regular array and the coulombic electrostatic force acting 
among the ions. The lattice energy is calculated theoretically using the Born-Lande equation in 
which another parameter Madelung constant is also related.  
The Born-Lande equation may appear in textbooks in the form      
                                              
                                             ?? ?? = -
???? ?? +
?? -
?? 2
?? ?? (1 -
1
?? ) ………………………………….(1) 
Where, U o is the Lattice energy N is the Avogrado Number, A is the Madelung Constant, z + is the   
charge of cation, z - is the charge of anion, e is the charge of an electron, R o is the equilibrium 
distance between the oppositely charged ions, n is the Born exponent.
[12-15] 
 
In the above equation, 1 is the expression of Lattice Energy where no defects or missing ions 
are taken into account. So, one can calculate the lattice energy of a perfect crystal where no 
ions are missing from their lattice points. But in reality, as temperature increases, the ions are 
displaced from their lattice position, and deects are formed in the ionic lattice. 
As defects are formed disorder is developed in the crystal which implies entropy of the system 
increases. But on the other hand formation of defects is an endothermic process. 
So, for this process,   
??? > 0 , ??? > 0 
The process of formation defects in a crystal lattice is entropically driven and so the defect 
concentration increases with an increase in temperature. Even at room temperature (25
o
C) in 
1cc of 6:6 NaCl crystal, there are about 10
6
 Schottky defects that cause a considerable change 
in Lattice energy. 
The total potential energy of a crystal will be the sum of both Coulombic attraction potential 
and short-range repulsive potential, which is referred to as Born repulsive potential. The 
repulsion potential is mainly due to the interpenetration of the electron clouds which is 
inversely proportional to Rn where R is the internuclear distance and n is the Born exponent.  
 
                                                         ?? ?????????????????? ?
1
?? ?? 
                                                   or, ?? ?????????????????? =
?? ?? ??    [B is the proportionality constant] 
 
    If an ion is surrounded by c numbers of oppositely charged ions then B can be written in 
terms of c and repulsive coefficient (b) according to the following relation: 
                                                         ?? = ???? [16] 
c is the first order coordination number concerning our reference ion. For example, if we 
consider NaCl (6:6) ionic crystal then c=6, and if CsCl (8:8) ionic crystal then c=8. 
If a particular crystal contains a considerable numbers of defects then the following parameters 
related to the crystal used in calculate lattice energy according to the Born-Lande equation will 
change: 
1) Madelung Constant (A changes to A
*
) 
2) Equilibrium distance (R o changes to ?? ?? *
) 
3) B will change (B to B
*
). 
 
Now we aim to find the expression of Lattice Energy in terms of new Madelung constant (A*), 
and new equilibrium distance () using the new value of proportionality constant B* and also the 
new value or the expression of A* and ?? ?? *
. So, we define a new parameter ?? which is the 
average number of missing ions from a particular distance at a particular temperature for a 
particular crystal. 
The Madelung Constant represents all the electrostatic interaction among all the ions in a solid 
crystal lattice. It is a dimensionless quantity related to the crystal which is invariant for a 
specific crystal. Madelung constant depends on the number of ions but also the location of the 
ions. The Madelung constant is widely used because it does not depend on the crystal unit cell 
and it has the same value for the crystal having identical invariant symmetry. [17]
 
                       
The dimensionless Madelung constant at ?? ?? h
 site is defined by, 
              
                                                      A=?
?? ?? ?? ????
?? 0  
/
??       
here ?? ?? is the charge at the ?? ?? h
 site, 
?? ????
=|?? ?? - ?? ?? | which is the distance between the ?? ?? h 
and ?? ?? h
 site and ?? 0
 is a chosen reference 
distance.  
From the definition of the Madelung constant, it can be concluded that the Madelung Constant 
not only depends on the number of missing ions but also the location of the voids caused by the 
Page 4


Modified Born-Lande Equation to calculate Lattice 
Energy in a theoretical approach 
   
   
 
   
 
 
 
Abstract: 
Defects in ionic solid are very much common, which is increased with the rise in temperature. It 
causes the change in the value of many physical properties and varieties of physical parameters 
and the Lattice Energy is one such parameter to control the physical properties of the crystals. 
Considering the loss of ions from lattice points as random, the examination of each of the defects 
individually is going to be unpredictable, thus leading to almost nonattainment of the correct 
crystal structure with the theoretical calculations applying for available models. Here, in this 
present work, we have used some statistical methods and probabilistic approximation to introduce 
a novel idea of calculating the Madelung constant, and then Lattice Energy analytically. 
To make the understanding more lucid, we have taken one of the very common crystals, very 
popular in the crystallographic community, NaCl crystal having 6:6 co-ordination number, for 
which a significant number of Schottky defects are observed. 
During this study, we are bound to assume the random distribution of defects as Poisson 
distribution due to the fact that the number of defects is very less with respect to the total numbers 
of lattice points present in the crystal to calculate the Madelung Constant. 
 
Keywords: 
Madelung Constant, Lattice Energy, Schottky defects, Poisson distribution. 
Introduction: 
For undergraduate students in chemistry, solid-state chemistry solid-state, and structural 
chemistry is very important. It is because of many scientific inventions and due to the very 
many uses of solid-state devices from semiconductors to superconductors. But before that, we 
have to study the most important and basic, the ionic solid. To study properly the ionic-solids 
we must know about the lattice energy, Madelung constant, and many other parameters. 
To analyze the various physical properties of an ionic rystal, Lattice energy is the most 
important parameter. It is very much useful in understanding the potential functions and 
binding forces which are responsible for binding in an ionic crystal. The lattice energy is 
experimentally determined using the Born Haber cycle. But sometimes all the experiments 
cannot be performed and so theoretical determination of lattice energy is very much 
important.
 [1-11] 
The lattice energy of a crystal is based on a model where ions are considered as the point 
charges placed in a fixed position in a regular array and the coulombic electrostatic force acting 
among the ions. The lattice energy is calculated theoretically using the Born-Lande equation in 
which another parameter Madelung constant is also related.  
The Born-Lande equation may appear in textbooks in the form      
                                              
                                             ?? ?? = -
???? ?? +
?? -
?? 2
?? ?? (1 -
1
?? ) ………………………………….(1) 
Where, U o is the Lattice energy N is the Avogrado Number, A is the Madelung Constant, z + is the   
charge of cation, z - is the charge of anion, e is the charge of an electron, R o is the equilibrium 
distance between the oppositely charged ions, n is the Born exponent.
[12-15] 
 
In the above equation, 1 is the expression of Lattice Energy where no defects or missing ions 
are taken into account. So, one can calculate the lattice energy of a perfect crystal where no 
ions are missing from their lattice points. But in reality, as temperature increases, the ions are 
displaced from their lattice position, and deects are formed in the ionic lattice. 
As defects are formed disorder is developed in the crystal which implies entropy of the system 
increases. But on the other hand formation of defects is an endothermic process. 
So, for this process,   
??? > 0 , ??? > 0 
The process of formation defects in a crystal lattice is entropically driven and so the defect 
concentration increases with an increase in temperature. Even at room temperature (25
o
C) in 
1cc of 6:6 NaCl crystal, there are about 10
6
 Schottky defects that cause a considerable change 
in Lattice energy. 
The total potential energy of a crystal will be the sum of both Coulombic attraction potential 
and short-range repulsive potential, which is referred to as Born repulsive potential. The 
repulsion potential is mainly due to the interpenetration of the electron clouds which is 
inversely proportional to Rn where R is the internuclear distance and n is the Born exponent.  
 
                                                         ?? ?????????????????? ?
1
?? ?? 
                                                   or, ?? ?????????????????? =
?? ?? ??    [B is the proportionality constant] 
 
    If an ion is surrounded by c numbers of oppositely charged ions then B can be written in 
terms of c and repulsive coefficient (b) according to the following relation: 
                                                         ?? = ???? [16] 
c is the first order coordination number concerning our reference ion. For example, if we 
consider NaCl (6:6) ionic crystal then c=6, and if CsCl (8:8) ionic crystal then c=8. 
If a particular crystal contains a considerable numbers of defects then the following parameters 
related to the crystal used in calculate lattice energy according to the Born-Lande equation will 
change: 
1) Madelung Constant (A changes to A
*
) 
2) Equilibrium distance (R o changes to ?? ?? *
) 
3) B will change (B to B
*
). 
 
Now we aim to find the expression of Lattice Energy in terms of new Madelung constant (A*), 
and new equilibrium distance () using the new value of proportionality constant B* and also the 
new value or the expression of A* and ?? ?? *
. So, we define a new parameter ?? which is the 
average number of missing ions from a particular distance at a particular temperature for a 
particular crystal. 
The Madelung Constant represents all the electrostatic interaction among all the ions in a solid 
crystal lattice. It is a dimensionless quantity related to the crystal which is invariant for a 
specific crystal. Madelung constant depends on the number of ions but also the location of the 
ions. The Madelung constant is widely used because it does not depend on the crystal unit cell 
and it has the same value for the crystal having identical invariant symmetry. [17]
 
                       
The dimensionless Madelung constant at ?? ?? h
 site is defined by, 
              
                                                      A=?
?? ?? ?? ????
?? 0  
/
??       
here ?? ?? is the charge at the ?? ?? h
 site, 
?? ????
=|?? ?? - ?? ?? | which is the distance between the ?? ?? h 
and ?? ?? h
 site and ?? 0
 is a chosen reference 
distance.  
From the definition of the Madelung constant, it can be concluded that the Madelung Constant 
not only depends on the number of missing ions but also the location of the voids caused by the 
removal of the ions. And, in an ionic solid it is not straightforward rather difficult to study the 
position as well as the number of defects. Moreover, the impossibility of examining the defects 
separately led us to open a new methodology for this study.  
From the structural point of view, in a crystalline substance, the number of ions missing from a 
particular position (w.r.t the reference ion) is random and so, a probability distribution of the 
random variable can help us to calculate the Madelung Constant easily. 
As the numbers of defects with respect to the total numbers of lattice points are very small and 
so the probability of finding defects is very small in a specific position so maybe it follows the 
Poisson distribution. 
A discrete random variable X is said to have a Poisson distribution with parameter ?>0 if for k= 
0, 1, 2, 3, …., the probability mass function (p.m.f.) of X is given by, 
                                           ?? (?? = ?? ) = 
?? -?? ?? ?? ?? !
          [18,19] 
Where, 
•    e is Euler's number (e = 2.71828...) 
•    ?? is the mean or average of the variable x. (to be edited by DJ) 
 
 
The Poisson distribution can be applied to systems with a large number of possible events, each 
of which is rare. The number of such events that occur during a fixed time interval is, under the 
right circumstances, a random number with a Poisson distribution. 
Hence the probability of missing an ion from a specific distance is very less it may follow the 
Poisson distribution. Using the distribution the change of the Madelung constant can be 
determined. Then it can be used to estimate other parameters. 
 
Expression of Lattice Energy in terms of A
*
, B
*
 and ?? ?? *
:  
For one mole of any crystal the modified potential energy can be written as, 
?? = -
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? +
?? *
?? ?? ?? 
                                                             (
????
????
)
?? =?? ?? *
= +
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? ?? * 2
-
?? ?? *
?? ?? ?? * ?? +1
   
At equilibrium distance the potential energy is minimum. So, if ?? ?? *
be the equilibrium distance 
then we can say that, 
(
????
?? ?? )
?? =?? ?? *
= 0 
So, 
Page 5


Modified Born-Lande Equation to calculate Lattice 
Energy in a theoretical approach 
   
   
 
   
 
 
 
Abstract: 
Defects in ionic solid are very much common, which is increased with the rise in temperature. It 
causes the change in the value of many physical properties and varieties of physical parameters 
and the Lattice Energy is one such parameter to control the physical properties of the crystals. 
Considering the loss of ions from lattice points as random, the examination of each of the defects 
individually is going to be unpredictable, thus leading to almost nonattainment of the correct 
crystal structure with the theoretical calculations applying for available models. Here, in this 
present work, we have used some statistical methods and probabilistic approximation to introduce 
a novel idea of calculating the Madelung constant, and then Lattice Energy analytically. 
To make the understanding more lucid, we have taken one of the very common crystals, very 
popular in the crystallographic community, NaCl crystal having 6:6 co-ordination number, for 
which a significant number of Schottky defects are observed. 
During this study, we are bound to assume the random distribution of defects as Poisson 
distribution due to the fact that the number of defects is very less with respect to the total numbers 
of lattice points present in the crystal to calculate the Madelung Constant. 
 
Keywords: 
Madelung Constant, Lattice Energy, Schottky defects, Poisson distribution. 
Introduction: 
For undergraduate students in chemistry, solid-state chemistry solid-state, and structural 
chemistry is very important. It is because of many scientific inventions and due to the very 
many uses of solid-state devices from semiconductors to superconductors. But before that, we 
have to study the most important and basic, the ionic solid. To study properly the ionic-solids 
we must know about the lattice energy, Madelung constant, and many other parameters. 
To analyze the various physical properties of an ionic rystal, Lattice energy is the most 
important parameter. It is very much useful in understanding the potential functions and 
binding forces which are responsible for binding in an ionic crystal. The lattice energy is 
experimentally determined using the Born Haber cycle. But sometimes all the experiments 
cannot be performed and so theoretical determination of lattice energy is very much 
important.
 [1-11] 
The lattice energy of a crystal is based on a model where ions are considered as the point 
charges placed in a fixed position in a regular array and the coulombic electrostatic force acting 
among the ions. The lattice energy is calculated theoretically using the Born-Lande equation in 
which another parameter Madelung constant is also related.  
The Born-Lande equation may appear in textbooks in the form      
                                              
                                             ?? ?? = -
???? ?? +
?? -
?? 2
?? ?? (1 -
1
?? ) ………………………………….(1) 
Where, U o is the Lattice energy N is the Avogrado Number, A is the Madelung Constant, z + is the   
charge of cation, z - is the charge of anion, e is the charge of an electron, R o is the equilibrium 
distance between the oppositely charged ions, n is the Born exponent.
[12-15] 
 
In the above equation, 1 is the expression of Lattice Energy where no defects or missing ions 
are taken into account. So, one can calculate the lattice energy of a perfect crystal where no 
ions are missing from their lattice points. But in reality, as temperature increases, the ions are 
displaced from their lattice position, and deects are formed in the ionic lattice. 
As defects are formed disorder is developed in the crystal which implies entropy of the system 
increases. But on the other hand formation of defects is an endothermic process. 
So, for this process,   
??? > 0 , ??? > 0 
The process of formation defects in a crystal lattice is entropically driven and so the defect 
concentration increases with an increase in temperature. Even at room temperature (25
o
C) in 
1cc of 6:6 NaCl crystal, there are about 10
6
 Schottky defects that cause a considerable change 
in Lattice energy. 
The total potential energy of a crystal will be the sum of both Coulombic attraction potential 
and short-range repulsive potential, which is referred to as Born repulsive potential. The 
repulsion potential is mainly due to the interpenetration of the electron clouds which is 
inversely proportional to Rn where R is the internuclear distance and n is the Born exponent.  
 
                                                         ?? ?????????????????? ?
1
?? ?? 
                                                   or, ?? ?????????????????? =
?? ?? ??    [B is the proportionality constant] 
 
    If an ion is surrounded by c numbers of oppositely charged ions then B can be written in 
terms of c and repulsive coefficient (b) according to the following relation: 
                                                         ?? = ???? [16] 
c is the first order coordination number concerning our reference ion. For example, if we 
consider NaCl (6:6) ionic crystal then c=6, and if CsCl (8:8) ionic crystal then c=8. 
If a particular crystal contains a considerable numbers of defects then the following parameters 
related to the crystal used in calculate lattice energy according to the Born-Lande equation will 
change: 
1) Madelung Constant (A changes to A
*
) 
2) Equilibrium distance (R o changes to ?? ?? *
) 
3) B will change (B to B
*
). 
 
Now we aim to find the expression of Lattice Energy in terms of new Madelung constant (A*), 
and new equilibrium distance () using the new value of proportionality constant B* and also the 
new value or the expression of A* and ?? ?? *
. So, we define a new parameter ?? which is the 
average number of missing ions from a particular distance at a particular temperature for a 
particular crystal. 
The Madelung Constant represents all the electrostatic interaction among all the ions in a solid 
crystal lattice. It is a dimensionless quantity related to the crystal which is invariant for a 
specific crystal. Madelung constant depends on the number of ions but also the location of the 
ions. The Madelung constant is widely used because it does not depend on the crystal unit cell 
and it has the same value for the crystal having identical invariant symmetry. [17]
 
                       
The dimensionless Madelung constant at ?? ?? h
 site is defined by, 
              
                                                      A=?
?? ?? ?? ????
?? 0  
/
??       
here ?? ?? is the charge at the ?? ?? h
 site, 
?? ????
=|?? ?? - ?? ?? | which is the distance between the ?? ?? h 
and ?? ?? h
 site and ?? 0
 is a chosen reference 
distance.  
From the definition of the Madelung constant, it can be concluded that the Madelung Constant 
not only depends on the number of missing ions but also the location of the voids caused by the 
removal of the ions. And, in an ionic solid it is not straightforward rather difficult to study the 
position as well as the number of defects. Moreover, the impossibility of examining the defects 
separately led us to open a new methodology for this study.  
From the structural point of view, in a crystalline substance, the number of ions missing from a 
particular position (w.r.t the reference ion) is random and so, a probability distribution of the 
random variable can help us to calculate the Madelung Constant easily. 
As the numbers of defects with respect to the total numbers of lattice points are very small and 
so the probability of finding defects is very small in a specific position so maybe it follows the 
Poisson distribution. 
A discrete random variable X is said to have a Poisson distribution with parameter ?>0 if for k= 
0, 1, 2, 3, …., the probability mass function (p.m.f.) of X is given by, 
                                           ?? (?? = ?? ) = 
?? -?? ?? ?? ?? !
          [18,19] 
Where, 
•    e is Euler's number (e = 2.71828...) 
•    ?? is the mean or average of the variable x. (to be edited by DJ) 
 
 
The Poisson distribution can be applied to systems with a large number of possible events, each 
of which is rare. The number of such events that occur during a fixed time interval is, under the 
right circumstances, a random number with a Poisson distribution. 
Hence the probability of missing an ion from a specific distance is very less it may follow the 
Poisson distribution. Using the distribution the change of the Madelung constant can be 
determined. Then it can be used to estimate other parameters. 
 
Expression of Lattice Energy in terms of A
*
, B
*
 and ?? ?? *
:  
For one mole of any crystal the modified potential energy can be written as, 
?? = -
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? +
?? *
?? ?? ?? 
                                                             (
????
????
)
?? =?? ?? *
= +
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? ?? * 2
-
?? ?? *
?? ?? ?? * ?? +1
   
At equilibrium distance the potential energy is minimum. So, if ?? ?? *
be the equilibrium distance 
then we can say that, 
(
????
?? ?? )
?? =?? ?? *
= 0 
So, 
0 = +
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? ?? * 2
-
?? ?? *
?? ?? ?? * ?? +1
 
Or,  
?? ?? *
?? ?? ?? * ?? +1
= +
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? ?? * 2
 
 
Or,  ?? *
= +
?? *
?? +
?? -
?? 2
(4?? ?? ?? )?? ?? ?? * 2
× ?? ?? * ?? +1
 
Or,  ?? *
= +
?? *
?? +
?? -
?? 2
(4?? ?? ?? )?? ?? ?? * ?? -1
 
If we replace the expression of B
*
 in the expression of modified potential Energy (U) then we 
will get the expression of Lattice Energy when the interionic distance is ?? ?? *
. The Modified Lattice 
energy is denoted as ?? ?? *
 and its expression will be, 
 
?? ?? *
= -
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? ?? *
+
?? *
?? +
?? -
?? 2
?? (4?? ?? ?? )?? ?? ?? *?? × ?? ?? *?? -1
 
???? , ?? ?? *
= -
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? ?? *
+
?? *
?? +
?? -
?? 2
?? (4?? ?? ?? )?? ?? ?? *
 
???? , ?? ?? *
= -
?? *
?? ?? +
?? -
?? 2
(4?? ?? ?? )?? ?? *
(1 -
1
?? ) 
 
The modified equation to calculate Lattice Energy contains the term A
*
 and R
*
. These are the 
value of the Madelung constant and equilibrium distance when the defects in the ionic solid is 
taken into account. So, we aim to find the new Madelung constant(A
*
) and equilibrium distance 
(?? ?? *
). 
Calculation of A
*
: 
The general expression of Madelung constant of an ionic crystal is defined by the following 
expression: 
?? = ?
?? ?? (
?? ????
?? ?? / )
8
?? =1
 
here ?? ?? is the charge at the ?? ?? h
 site, 
?? ????
=|?? ?? - ?? ?? | which is the distance between the ?? ?? h 
and ?? ?? h
 site and ?? 0
 is a chosen reference 
distance.