<< 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 the meaninh of link
Aisha Reply
Yes wish to join am a science ⚗ student
Emmanuel Reply
What is acid
is agroup of substance that relaise hydrogen ion in the water solution.
how can someone understand chemistry vividly
Mercy Reply
Maybe by reading proofs or practical work and application in modern world.
what is isotopes
Samuel Reply
whats de shape of water
Amara Reply
water has no shape because it's liquid
water is a shapeless, odourless, colourless and tasteless substance that only takes the shape of its container.
i think they're referring to the molecular shape?
It has no shape but takes the shape of the container
what isthe maening pkw
Wilson Reply
what is the meaning of STP
introduction to chemistry for beginner
Lansana Reply
through out human history
marwan Reply
is euglena a unicellular organ
Agio Reply
is euglena a unicellular organism
how is hydrogen can be heated
Buwembo Reply
what is difference between atom and molecule
Aqeela Reply
Atom is the smallest part of matter; it consists of equal number of protons and electrons. It may have neutrons. A molecule is a compound made of atoms covalently bonded.
does amoeba has structure
Mercy Reply
what is the effect of green house on the earth 🌎
kolawole Reply
what is the effect of green house on the earth 🌎
what's an atom?
Davy Reply
it's the smallest unit of Matter
smallest part of an element
also, depending on its (atom's) structure, that is the amount of protons and neutrons and electrons, is the determining factors of what element it is.
is a smallast particals of an element
it is the smallest part of an element that can take part in a chemical reaction
is the smallest part of an element
An atom is the smallest indivisible part of a matter
Oy kl konsa test hay or kitna hay?
Faisal Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play

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?