<< Chapter < Page Chapter >> Page >
Two images are shown. The first image shows a cube with black dots at each corner and a red dot in the center of each face of the cube. This cube is stacked with seven others that are not colored to form a larger cube. The second image is composed of eight small green spheres that form the corners of a cube with six other small green spheres located in the faces of the cube. Eight larger green spheres are spaced inside the cube and all of the spheres are connect to one another by lines. The name under this image reads “C a F, subscript 2, face-centered unit cell.”
Calcium fluoride, CaF 2 , forms an FCC unit cell with calcium ions (green) at the lattice points and fluoride ions (red) occupying all of the tetrahedral sites between them.

Calculation of ionic radii

If we know the edge length of a unit cell of an ionic compound and the position of the ions in the cell, we can calculate ionic radii for the ions in the compound if we make assumptions about individual ionic shapes and contacts.

Calculation of ionic radii

The edge length of the unit cell of LiCl (NaCl-like structure, FCC) is 0.514 nm or 5.14 Å. Assuming that the lithium ion is small enough so that the chloride ions are in contact, as in [link] , calculate the ionic radius for the chloride ion.

Note: The length unit angstrom, Å, is often used to represent atomic-scale dimensions and is equivalent to 10 −10 m.


On the face of a LiCl unit cell, chloride ions contact each other across the diagonal of the face:

Three images are shown. The first shows a cube of alternating green and purple spheres. A smaller cube within that cube is outlined and a larger version of it appears next. This figure is a grey cube that appears to be made up of spheres. There are small spaces between each sphere. There is a right triangle outlined in this cube and a larger version of it appears next. This right triangle has two sides labeled “a,” and the hypotenuse, which spans two half-circles and one full one is labeled, “r, 2 r, and r.”

Drawing a right triangle on the face of the unit cell, we see that the length of the diagonal is equal to four chloride radii (one radius from each corner chloride and one diameter—which equals two radii—from the chloride ion in the center of the face), so d = 4 r . From the Pythagorean theorem, we have:

a 2 + a 2 = d 2

which yields:

( 0.514 nm ) 2 + ( 0.514 nm ) 2 = ( 4 r ) 2 = 16 r 2

Solving this gives:

r = ( 0.514 nm ) 2 + ( 0.514 nm ) 2 16 = 0.182 nm ( 1.82 Å ) for a Cl radius .

Check your learning

The edge length of the unit cell of KCl (NaCl-like structure, FCC) is 6.28 Å. Assuming anion-cation contact along the cell edge, calculate the radius of the potassium ion. The radius of the chloride ion is 1.82 Å.


The radius of the potassium ion is 1.33 Å.

Got questions? Get instant answers now!

It is important to realize that values for ionic radii calculated from the edge lengths of unit cells depend on numerous assumptions, such as a perfect spherical shape for ions, which are approximations at best. Hence, such calculated values are themselves approximate and comparisons cannot be pushed too far. Nevertheless, this method has proved useful for calculating ionic radii from experimental measurements such as X-ray crystallographic determinations.

X-ray crystallography

The size of the unit cell and the arrangement of atoms in a crystal may be determined from measurements of the diffraction of X-rays by the crystal, termed X-ray crystallography    . Diffraction is the change in the direction of travel experienced by an electromagnetic wave when it encounters a physical barrier whose dimensions are comparable to those of the wavelength of the light. X-rays are electromagnetic radiation with wavelengths about as long as the distance between neighboring atoms in crystals (on the order of a few Å).

When a beam of monochromatic X-rays strikes a crystal, its rays are scattered in all directions by the atoms within the crystal. When scattered waves traveling in the same direction encounter one another, they undergo interference , a process by which the waves combine to yield either an increase or a decrease in amplitude (intensity) depending upon the extent to which the combining waves’ maxima are separated (see [link] ).

Questions & Answers

What is whizatron?
Frendick Reply
What is stoichometry
ngwuebo Reply
what is atom
yinka Reply
An indivisible part of an element
the smallest particle of an element which is indivisible is called an atom
An atom is the smallest indivisible particle of an element that can take part in chemical reaction
is carbonates soluble
Ebuka Reply
what is the difference between light and electricity
Joshua Reply
What is atom? atom can be defined as the smallest particles
what is the difference between Anode and nodes?
What's the net equations for the three steps of dissociation of phosphoric acid?
Lisa Reply
what is chemistry
Prince Reply
the study of matter
what did the first law of thermodynamics say
Starr Reply
energy can neither be created or distroyed it can only be transferred or converted from one form to another
Graham's law of Diffusion
Ayo Reply
what is melting vaporization
Anieke Reply
melting and boiling point explain in term of molecular motion and Brownian movement
Scientific notation for 150.9433962
Steve Reply
what is aromaticity
Usman Reply
aromaticity is a conjugated pi system specific to organic rings like benzene, which have an odd number of electron pairs within the system that allows for exceptional molecular stability
what is caustic soda
Ogbonna Reply
sodium hydroxide (NaOH)
what is distilled water
is simply means a condensed water vapour
advantage and disadvantage of water to human and industry
Abdulrahman Reply
a hydrocarbon contains 7.7 percent by mass of hydrogen and 92.3 percent by mass of carbon
Timothy Reply

Get the best Chemistry course in your pocket!

Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Chemistry' conversation and receive update notifications?