1. Bond Length and Measurements of Radius
AND
Radius Ratio and Co-Ordination Polyhedra

2. CONTENTS
- Introduction
- Bond Length
- Measurements of Radius
- Radius Ratio
- Coordination Polyhedra
- Conclusions
- References

3. INTRODUCTION
Distances between center of bonded ions are
called bond length, or bond distances. Bond length
vary depending on many factors, but in general,
they are very consistent. Of course the bond orders
affect bond length, but bond length of the same
order for the same pair of atoms in various
molecules are very consistent.

4. BOND LENGTH
Bond length is related to bond order, when
more electrons participate in bond formation the
bond will get shorter. Bond length is also inversely
related to bond strength and the bond dissociation
energy, as (all other things being equal) a stronger
bond will be shorter. A comparison of bond lengths
is shown in the following table.

5. BOND LENGTH (contd)
Therefore, the bond length of a :
triple bond < double bond < single bond.

6. BOND LENGTH (contd)
The length of the bond is determined by the
number of bonded electrons (the bond order). The
higher the bond order, the stronger the pull
between the two atoms and the shorter the bond
length. Generally, the length of the bond between
two atoms is approximately the sum of the radii of
the two atoms, X + Y as shown in the following
figure. Bond length is given in picometers (pm) or
angstroms (A).

7. In a bond between two identical atoms half the bond distance is equal to
the Metallic radius, Covalent radius and Ionic radius respectively.
X YX Y X Y
Vander Waals radius is defined as half of the inter-nucleus separation of
two non-bonded atoms of the same element on their closest possible
approach.
BOND LENGTH (contd)
In a bond between two identical atoms half the bond distance is equal to
the Metallic radius, Covalent radius and Ionic radius respectively.

8. BOND LENGTH (contd)
It should be noted that the terms 'Covalent bond
distance (X + Y)' and 'Bond Length' are often used
interchangeably.
The two terms have essentially the same definition
with the difference only being that Covalent bond
distance can only refer to covalently bonded atoms
while bond length could theoretically refer to two
atoms bonded in any way.

9. MEASUREMENTS OF RADIUS
Bond lengths are measured:
by X-ray diffraction of solids,
by Electron diffraction.

10. MEASUREMENTS OF RADIUS
X-ray diffraction of solids
An X-ray which reflects from the surface of a
substance has travelled less distance than
an X-ray which reflects from a plane of
atoms inside the crystal. The penetrating
X-ray travels down to the internal layer,
reflects, and travels back over the same
distance before being back at the surface as
shown in the figure below:

11. nλ = 2dsinθ
MEASUREMENTS OF RADIUS
X-ray diffraction of solids (contd)
The distance travelled depends on the separation of
the layers and the angle at which the X-ray entered
the material. For this wave to be in phase with the
wave which reflected from the surface it needs to
have travelled a whole number of wavelengths
while inside the material. Bragg expressed this in an
equation now known as Bragg's Law:

12. λ = 2dsinθ λ / 2sinθ = d
MEASUREMENTS OF RADIUS
X-ray diffraction of solids (contd)
Where:
λ is the wavelength of the rays
θ is the angle between the incident rays and the surface of the crystal
d is the spacing between layers of atoms (distance)
and constructive interference occurs when n is an integer (whole
number).
When n is an integer (1, 2, 3 etc.) the reflected waves from different
layers are perfectly in phase with each other and produce a bright point
on a piece of photographic film. In this case, we use n=1 to get the
shortest spacing between layers of atoms. We can rewrite Bragg’s
equation as following:

13. d
Half of distance “d” is the radius of each atom.
d
d
MEASUREMENTS OF RADIUS
X-ray diffraction of solids (contd)

14. MEASUREMENTS OF RADIUS
Electron diffraction
Electron diffraction refers to the wave
nature of electrons. However, from a
technical or practical point of view, it may
be regarded as a technique used to study
matter by firing electrons at a sample and
observing the resulting interference
pattern.
This phenomenon is commonly known as
the wave-particle duality, which states
that the behaviour of a particle of matter
(in this case the incident electron) can be
described by a wave. For this reason, an
electron can be regarded as a wave much
like sound or water waves. This technique
is similar to X-ray and neutron diffraction.

15. λ = h / mv (1)
MEASUREMENTS OF RADIUS
Electron diffraction (contd)
Electron diffraction is most frequently used in
solid state physics and chemistry to study the
crystal structure of solids. However, in this, we
use this method to find out the distance
between spacing layers of atom.
In 1924, de Broglie suggested that subatomic
particles such as electrons, neutrons or protons
might have the wave characteristics with the
associated wavelength:
Where h is the Planck’s constant, m the mass of
the particle and v its velocity.

16. Ve = ½(mv2
)
λ = h / (2emV)½
= 1.23 / (V)½
nm
λ = dsinθ ≈ dθ = d(D/2L)
(2)
(3)
(4)
MEASUREMENTS OF RADIUS
Electron diffraction (contd)
The velocity v can be obtained from the
classical expression:
and substituted into the de Broglie
equation:
The Bragg condition for diffraction for
small angles is

17. λ = d(D/2L) (5)
d(D/2L) = 1.23 / (V)½
nm (6)
MEASUREMENTS OF RADIUS
Electron diffraction (contd)
From the latest equation, we can rewrite
as following:
where d = the inter-atomic spacing, D is
the ring diameter, and L is the path length
from the target at the gun aperture to the
luminescent screen. By combining this (5)
equation with the (3) equation, we get:
where V = potential difference (Voltage)

18. MEASUREMENTS OF RADIUS
Electron diffraction (contd)
APPARATUS is the instrument helping us
to find out the value of D, L and V.
The electron diffraction tube comprises a
'gun' which emits a narrow converging
beam of electrons within an evacuated
clear glass bulb on the surface of which is
deposited a luminescent screen. Across
the exit aperture of the 'gun' lies a micro-
mesh nickel grid onto which has been
vaporized a thin layer of target.
The beam penetrates through this target
to become diffracted into two rings.

19. Target
D
L
(V)
MEASUREMENTS OF RADIUS
Electron diffraction (contd)
The electron diffraction tube is shown in the following figure:

20. RC
RA
RADIUS RATIO
The cation-anion radius ratio is the ratio of
the ionic radius of the cation to the ionic
radius of the anion in a cation-anion
compound as shown the figure below. This
is simply given by RC/RA.
The allowed size of the cation is determined
by the critical radius ratio. If the cation is
too small, then it will attract the anions into
each other and they will collide hence the
compound would collapse, this occurs when
the radius ratio drops below 0.155.

21. The figure above shows the radius ratio related to stability of atoms
RADIUS RATIO (contd)
At the stability limit the cation is touching
all the anions and the anions are just
touching at their edges (radius ratio =
0.155). Beyond this stability limit (radius
ratio > 0.155) the compound will be stable.

22. COORDINATION POLYHEDRA
A coordinated polyhedron of anions is formed about
each cation, the cation-anion distance determined by
the sum of ionic radii and the coordination number
(C.N.) by the radius ratio.
The numbers in the table are mathematically derived
minimum radius ratios for that geometry. An
octahedron may form with a radius ratio greater than
or equal to .414, but as the ratio rises above .732, a
cubic conformation becomes more stable. These
mathematically derived ratios are deviated from in
practice; thus, octahedral salt structures with a radius
ratio of less than .414 or more than .732 have been
observed. If the radius ratio falls below the minimum
calculated for ions modelled as spherical balls, it is
presumed that the ions are compressed into oblong
balls that are not perfect spheres.

23. COORDINATION POLYHEDRA
The below table gives the relation
between radius ratio and coordination
number

24. CONCLUSIONS
 Distances between center of bonded ions are
called bond length, or bond distances.
 Bond length vary depending on many factors,
however, they are very coherent.
 Bond length is also inversely related to bond
strength and the bond dissociation energy, as (all
other things being equal) a stronger bond will be
shorter.
 Measurements of Radius have various methods to
be used but X-Ray diffraction and Electrons diffraction
are frequently used.
 Radius Ratio is used to show the stability of atoms.
 Radius Ratio is also used to indicate coordination
numbers, as a result, its ratio can tell type of
polyhedra.

25. REFERENCES
 A Basic Course in Crystallography by JAK
Tareen and TRN Kutty
 Basic Elements of Crystallography by
Teresa Szwacka, Nevill Gonzalez Szwacki
 The Basics of Crystallography and
Diffraction: Third Edition by C. Hammond
 http://en.wikipedia.org/wiki/Radius_ratio

http://www.phy.davidson.edu/ModernPhysicsLabs/elecdff.html

http://www-outreach.phy.cam.ac.uk/camphy/xraydiffraction/xraydi