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Difference between Primitive cell, unit cell and a wigner-seitz cell.

by shayaan_musta
Tags: cell, difference, primitive, unit, wignerseitz
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shayaan_musta
#1
Aug11-11, 10:15 AM
P: 184
Hi to all experts!
I know individually about primitive cell("It has lattice point at corners only") and unit cell("It has lattice point at corners as well as at center if bcc or at faces if fcc or at bases if it is base centered")
Are these right?
But I don't know what is Wigner-Seitz cell.

But I know that Primitive cell is a special case of unit cell and Wigner Seitz cell is a special case of primitive cell.
Am I right?

Here is space lattice pic. Please identify primitive cell and a unit cell and also a Wigner-Seitz cell in it.

Kindly help me in easiest way.

Thanks for your contribution.
Attached Thumbnails
preview_html_880098a.png  
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Foder
#2
Aug12-11, 03:46 AM
P: 8
Hello, here is the answer:

Unit cell: volume formed by the arbitrary chosen basis vectors (normally within the 14 Bravais systems). They fill the complete space by translational symmetry.

Primitive cell: smallest possible unit cell.

Wigner-Seitz cell: smallest possible primitive cell, which consist of one lattice point and all the sorrounding space closer to it than to any other point. The construction of the W-S cell in the reciprocal lattice delivers the first Brillouin zone (important for diffraction).

These concepts (or rather the way I put them) may help you to understand when you already have an idea about them, like you seem to do.

In your simple cubic system, the unit and primitive cell (both the same here) woud be one of those cubes. The W-Z cell would be each single lattice point and sorroundings. That example doesn't help you to visualize the differences properly, I recommend you to analyze it for a face centered cubic case.
shayaan_musta
#3
Aug12-11, 09:02 AM
P: 184
Hi Foder!

As you said "Wigner-Seitz cell: smallest possible primitive cell, which consist of one lattice point and all the sorrounding space closer to it than to any other point"

Will you please explain me how W-Z could be the smallest primitive cell while primitive cell is smallest one with 8 lattice points at corners only.

Here is face centered cubic figure, could you differentiate three definitions with the help of this figure now please.

Also kindly tell me how to find lattice points per unit cell in a face centered cubic(may be I would be doing wrong calculations)

Thanks again expert.
Attached Thumbnails
index.jpeg  

Foder
#4
Aug12-11, 09:20 AM
P: 8
Difference between Primitive cell, unit cell and a wigner-seitz cell.

Quote Quote by shayaan_musta View Post
Hi Foder!

As you said "Wigner-Seitz cell: smallest possible primitive cell, which consist of one lattice point and all the sorrounding space closer to it than to any other point"

Will you please explain me how W-Z could be the smallest primitive cell while primitive cell is smallest one with 8 lattice points at corners only.

Here is face centered cubic figure, could you differentiate three definitions with the help of this figure now please.

Also kindly tell me how to find lattice points per unit cell in a face centered cubic(may be I would be doing wrong calculations)

Thanks again expert.
WZ is the smallest possible primitive cell, but that doesnt mean that it is always appliable/representative of the crystal. For example, in a system with two different elements you can build the WZ but it won't be a primitive cell, in the sense that with only one kind of atom you cannot reproduce the whole lattice by translational symmetry.

Number of lattice point per unit cell for fcc = 1/8 * 8 + 1/2 * 6 = 4. Then, the 4 points basis is (0,0,0,);(1/2,1/2,0);(1/2,0,1/2);(0,1/2,1/2). In the attachment you may visualize the difference between the fcc and its primitive.

The WZ is easier to visualize in 2D, through its construction procedure, which is very simple (can read it at Wiki)
Attached Thumbnails
fcc.JPG   200px-Wigner-Seitz-Zelle.png  
shayaan_musta
#5
Aug12-11, 12:34 PM
P: 184
Quote Quote by Foder View Post
WZ is the smallest possible primitive cell, but that doesnt mean that it is always appliable/representative of the crystal. For example, in a system with two different elements you can build the WZ but it won't be a primitive cell, in the sense that with only one kind of atom you cannot reproduce the whole lattice by translational symmetry.
I got above justification. Thanks for understanding me & thanks also for helping me.


[/QUOTE]Number of lattice point per unit cell for fcc = 1/8 * 8 + 1/2 * 6 = 4. Then, the 4 points basis is (0,0,0,);(1/2,1/2,0);(1/2,0,1/2);(0,1/2,1/2). In the attachment you may visualize the difference between the fcc and its primitive.[/QUOTE]

When you are calculating lattice point per unit cell then please clearify the following:
what is 1/8?
what is 8?
what is 1/2?
what is 6?
In general, please provide general formula for calculating lattice points per unit cell for any Braivis lattice.

Thanks Again expert.
Foder
#6
Aug12-11, 04:00 PM
P: 8
Each corner point of a cubic cell is shared by 8 identical contiguous cubic cells hence, for each one of these cells corresponds 1/8 atom (or molecule or whatever). There are 8 corners per cubic cell, then we got the 1/8 * 8.

Each atom located at the faces of a cubic cell is shared by 2 contiguous cells, then, to each one corresponds 1/2 of atom. There are 6 faces in a cubic cell, then we obtain the 1/2 * 6.

Atoms corresponding to a fcc: 1/8 * 8 + 1/2 * 6 = 4.

I hope it is clear now.
shayaan_musta
#7
Aug13-11, 05:58 AM
P: 184
Ok Thanks. It brings me a lot of help.
I am a student of electronic engineering and studying solid state as a subject. And I have a test of solid state on 19th august.
Today I will start a new thread "Bravais lattice in two dimensions". So kindly contribute your experience in it with me.
Thanks for your whole contribution.
Take care a lot.
solas99
#8
Aug16-11, 08:29 AM
P: 69
how do i construct/ sketch a lattice and reciprocal lattice for 2D surface lattice from vectors
a=i+4j
b=3i

what is the way of working this out, is there a procedure where i can follow.I have an exam soon its the only problem im stuck upon.
tnks
shayaan_musta
#9
Aug17-11, 02:40 PM
P: 184
Quote Quote by solas99 View Post
how do i construct/ sketch a lattice and reciprocal lattice for 2D surface lattice from vectors
a=i+4j
b=3i

what is the way of working this out, is there a procedure where i can follow.I have an exam soon its the only problem im stuck upon.
tnks
Hello solas99!

As you are studying solid state physics, means you are visualize physics in your mind.
I can tell you how to construct a lattice but not reciprocal because I don't know so much about reciprocal.

a=i+4j
=> 1unit on x-axis and 4units on y-axis and 0 on z-axis. Imagine a vector in your mind with this configuration.
b=3i
=> 3units on x-axis, 0 on y-axis and 0 on z-axis. Imagine this one too.

Now translate it and you have a lattice in 2D space. Please visualize.
solas99
#10
Aug18-11, 08:45 AM
P: 69
thanks for quick reply, done the translation on graph paper, tried to visualise as well..
Ive got a line from the origin going up to point i+4j, then i drew another line from the origin using vector b=3i.
do i get a triangular lattice? (only 5 possible types of crystal lattice in 2D)
shayaan_musta
#11
Aug18-11, 11:54 AM
P: 184
Quote Quote by solas99 View Post
do i get a triangular lattice? (only 5 possible types of crystal lattice in 2D)
You are welcome.

And you will not get triangular lattice. you must get a rectangular lattice.
This shows that you didn't translate your vector.

As you can see in the image given below.

In fig#1 two vectors a and b are sketched. And in fig#2 they are translated in order to make lattice.
Hope you got my point.
Attached Thumbnails
untitled.JPG  
solas99
#12
Aug18-11, 04:54 PM
P: 69
hey,
Yeh i kinda figured it out after so many times. :) thanks

still cant figure out how to sketch the reciprocal lattice :(

any ideas,
merci
shayaan_musta
#13
Aug19-11, 09:57 AM
P: 184
You have four vectors, "a", "b", "reciprocal a" and "reciprocal b".

"Reciprocal a" is orthogonal to "b", "reciprocal b" is orthogonal to "a".

The shadow that makes "reciprocal a" on "a" (or vice versa) must be 1.

The shadow that makes "reciprocal b" on "b" (or vice versa) must be 1.

Mathematically

"Reciprocal a" dot "b" equal to zero
"Reciprocal b" dot "a" equal to zero

"Reciprocal a" dot "a" equal to one
"Reciprocal b" dot "b" equal to one

The reciprocal and the original vector is a dot product i.e. a.a'=0
where,
a=original vector
a'=reciprocal vector

similarly,
b.b'=0

Does this help you?
solas99
#14
Aug19-11, 12:12 PM
P: 69
yes, but i still dont get how im supposed to sketch it :S ive got some books etc..but im still dont understand..
shayaan_musta
#15
Aug20-11, 01:35 PM
P: 184
Quote Quote by solas99 View Post
yes, but i still dont get how im supposed to sketch it :S ive got some books etc..but im still dont understand..
Hello solas99!

Use formula for reciprocal lattice.

don't you know the vector formula for reciprocal lattice???
rida
#16
Apr25-13, 10:56 AM
P: 13
can any one tell me the physical significance of weigner seitz cell
nasu
#17
Apr25-13, 11:05 AM
P: 1,970
Quote Quote by Foder View Post
Wigner-Seitz cell: smallest possible primitive cell, which consist of one lattice point and all the sorrounding space closer to it than to any other point.
This is misleading.
All primitive cells in a given latiice have the same volume.
The WS cell is one of the possible primitive cells.
What does it mean for the WS cell to be the "smallest"?
It has the same volume as any other primitive cell.
DrDu
#18
Apr25-13, 11:29 AM
Sci Advisor
P: 3,593
Quote Quote by rida View Post
can any one tell me the physical significance of weigner seitz cell
The Wigner Seitz cell is often near to spherical. This is used in the Wigner Seitz method to calculate the binding energy of primitive monovalent metallic lattices: The WS cell is replaced by a sphere of the same volume and special boundary conditions (vanishing of the derivative of the wavefunction) are applied to calculate the modified atomic orbitals.


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