1. Inside metals, electrons are weakly bound to atoms and move freely through the metal. 2. These free electrons move through a periodic potential created by the positive ions and other electrons. 3. The potential energy of the electrons is periodic inside the metal but rises suddenly at the boundaries, as shown in the figures. Electrons with energy below a binding energy Eb are tightly bound to atoms, while higher energy electrons between Eb and the Fermi energy Ef can move through the metal.

4. SUBMITTED TO :-
Mr. Gyanrao Dhote

5. ELECTRONS
IN PERIODIC
ELEMENTS

7. Each substance is composed
of atoms. In a metal ,the
outermost electrons of its
constituent atoms are most
weakly wound with the
atoms , so they get separated
from their atoms and move
freely inside the entire

8. Substances. These electrons are
called the free electrons .
Each free electron inside a
metal moves in the electric field
of positive ions and of other free
electrons. Since the crystal
structure itself is periodic,
therefore the potential energy of

10. electron also changes perioical-
ly with the position co–ordinate
i.e. the motion of electron inside
a metal is in a periodic potential
well.
We know that inside a metal
(atomic number z ), the
potential energy of a free
electron at a distance x in the

11. potential field of an atom i is
given as
Ux = -Ze2/4 x
Hence a graph plotted for Ux
versus x is a rectangular
hyperbola as shown in fig.4.1.
From the graph , it is clear that
for different values of x,the

13. value of Ux is negative and at
x=∞, the value of Ux is zero.
But inside the metal ,
atoms are arranged in a
definite order, therefore the
potential energy of electrons
in combined field of atoms i&i
can be represented by the
complete curve as shown in

14. Fig.4.2. Remember that in
fig.4.2 , the dotted curves
represent the potential energy
of electron in the field of atom
i&i separetely , while the
complete curve represents the
resultant potential energy .
In a metal, there are
number of atoms arranged in

15. a definite order in any
direction , hence in the length
L of the metal, the resultant
potential energy of electron in
the fields of i, j, k, l,……….etc
can be represented as shown
in fig .4.3. From fig.4.3 ,it is
clear that at the boundaries of
metal (i.e., at its free surfaces)

17. the potential energy suddenly
rises and tends to become
zero, while inside the metal,
the potential well is not of
uniform depth everywhere,
but it is periodic.
Fig.4.3. also shows the
different energy levels inside
the metal. All the electrons

18. from the lowest energy to Eb
are bound with their atoms
and they can vibrate about
their main positions only with
a very small amplitude, but
they cannot leave their atoms.
The electrons of energy higher
than this ,with energy in
between Eb& Ef can move

19. anywhere within the metal, but
on reaching at the boundary,
they have to face a surface
barrier & hence they cannot
emerge out of the metal
surface. Here Ef is the Fermi
energy level (i.e., the level of
maximum kinetic energy of
electrons ). The depth of

20. Potential well is Es & Es-Ef=θ is
the work function of the metal.
The separation between the two
consecutive atoms (i.e.,lattice
constant) is a
Thus , according to band model
1. Each electron in a metal is in
the electric field produced due to
charge distribution of positive

21. ions and remaining electrons.
2. Electron is associated with the
entire crystal, & not only with
an atom .
3. Electron moves in a periodic
potential produced by the ion
cores & other electron inside the
crystals.(fig.4.3.). This
periodicity vanishes at the free

22. surface of the crystal .The motion
of electron inside the crystal is
like the elastic waves in a
continous medium.
To obtain the different energy
states of electron in the periodic
potential well , we will have to
find the solution of wave
equation corresponding to the

23. wave equation ψ associated
with the electron . But it is not
so easy to definite the periodic
potential . Hence to explain the
behavior of electrons in the
periodic potential, Kronig &
Penny gave a simple one
dimensional model which is
called the Kronig-Penney

24. model.