| Topic: |
Science > Physics |
| User: |
"idyllic" |
| Date: |
09 Oct 2004 08:49:33 AM |
| Object: |
Lattice plane |
Dear all,
I'm confused by calculating the distance of lattice planes (hkl). My
trouble is: why the neighboring plane of hx+ky+lz=0 is hx+ky+lz=1 ?
This is a mathematical problem, but I can't conquer it. Any help will
be appreciated.
Sincerely Idyllic
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| User: "idyllic" |
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| Title: Re: Lattice plane |
09 Oct 2004 11:13:17 PM |
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(idyllic) wrote in message news:<f9b8e54f.0410090549.1243a037@posting.google.com>...
Dear all,
I'm confused by calculating the distance of lattice planes (hkl). My
trouble is: why the neighboring plane of hx+ky+lz=0 is hx+ky+lz=1 ?
This is a mathematical problem, but I can't conquer it. Any help will
be appreciated.
Sincerely Idyllic
aha...I'm so stupid.
Considering the integer solutions of equations of three variables,
naturally, the neighboring planes should be:
hx+ky+lz=...-3,-2,-1,0,1,2,3...
(hx+ky+lz=fraction number has no integer solution, i.e. lattice point)
and each equation as hx+ky+lz=integer exist integer solution which has
been prooved by Euler---material of senior high school math.
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| User: "Edward Green" |
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| Title: Re: Lattice plane |
10 Oct 2004 08:02:08 AM |
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(idyllic) wrote in message news:<f9b8e54f.0410092013.54c78c1f@posting.google.com>...
(idyllic) wrote in message news:<f9b8e54f.0410090549.1243a037@posting.google.com>...
Dear all,
I'm confused by calculating the distance of lattice planes (hkl). My
trouble is: why the neighboring plane of hx+ky+lz=0 is hx+ky+lz=1 ?
This is a mathematical problem, but I can't conquer it. Any help will
be appreciated.
Sincerely Idyllic
aha...I'm so stupid.
Considering the integer solutions of equations of three variables,
naturally, the neighboring planes should be:
hx+ky+lz=...-3,-2,-1,0,1,2,3...
(hx+ky+lz=fraction number has no integer solution, i.e. lattice point)
and each equation as hx+ky+lz=integer exist integer solution which has
been prooved by Euler---material of senior high school math.
Hmm... well, just so you have company, I was thinking to myself "but
how do you know that hx+ky+lz = n in general _has_ an integer solution
-- until I read your next line. Now I'm wondering how to prove it.
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| User: "idyllic" |
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| Title: Re: Lattice plane |
03 Nov 2004 12:36:46 PM |
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(Edward Green) wrote in message news:<eca320d0.0410100502.5134fe58@posting.google.com>...
idyllic.math@gmail.com (idyllic) wrote in message news:<f9b8e54f.0410092013.54c78c1f@posting.google.com>...
idyllic.math@gmail.com (idyllic) wrote in message news:<f9b8e54f.0410090549.1243a037@posting.google.com>...
Dear all,
I'm confused by calculating the distance of lattice planes (hkl). My
trouble is: why the neighboring plane of hx+ky+lz=0 is hx+ky+lz=1 ?
This is a mathematical problem, but I can't conquer it. Any help will
be appreciated.
Sincerely Idyllic
aha...I'm so stupid.
Considering the integer solutions of equations of three variables,
naturally, the neighboring planes should be:
hx+ky+lz=...-3,-2,-1,0,1,2,3...
(hx+ky+lz=fraction number has no integer solution, i.e. lattice point)
and each equation as hx+ky+lz=integer exist integer solution which has
been prooved by Euler---material of senior high school math.
Hmm... well, just so you have company, I was thinking to myself "but
how do you know that hx+ky+lz = n in general _has_ an integer solution
-- until I read your next line. Now I'm wondering how to prove it.
If (h,k,l)|n then this equation has integer solution.
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| User: "zigoteau" |
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| Title: Re: Lattice plane |
04 Nov 2004 03:35:41 AM |
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(idyllic) wrote in message news:<f9b8e54f.0410090549.1243a037@posting.google.com>...
Hi, Idyllic,
I'm confused by calculating the distance of lattice planes (hkl). My
trouble is: why the neighboring plane of hx+ky+lz=0 is hx+ky+lz=1 ?
This is a mathematical problem, but I can't conquer it. Any help will
be appreciated.
Your question suggests that your knowledge is very basic. To help me
just the level of my reply, could you answer the following two
questions:
1. What is the general equation of a plane?
2. In Euclidian geometry, it takes three points to determine a plane.
How would you go about determining the equation of a plane through
three points whose coordinates are given?
3. Suppose I have a unit cubic lattice, i.e. atoms at all points with
three integer coordinates (a,b,c). Could you describe in words what
distinguishes a 'lattice plane' from any other sort of plane?
Cheers,
Zigoteau.
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| User: "idyllic" |
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| Title: Re: Lattice plane |
04 Nov 2004 08:52:11 PM |
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(zigoteau) wrote in message news:<a836cacf.0411040135.947ce8d@posting.google.com>...
idyllic.math@gmail.com (idyllic) wrote in message news:<f9b8e54f.0410090549.1243a037@posting.google.com>...
Hi, Idyllic,
I'm confused by calculating the distance of lattice planes (hkl). My
trouble is: why the neighboring plane of hx+ky+lz=0 is hx+ky+lz=1 ?
This is a mathematical problem, but I can't conquer it. Any help will
be appreciated.
Your question suggests that your knowledge is very basic. To help me
just the level of my reply, could you answer the following two
questions:
1. What is the general equation of a plane?
With any oblique axes, the equation is the same: ax+by+cz=k
2. In Euclidian geometry, it takes three points to determine a plane.
How would you go about determining the equation of a plane through
three points whose coordinates are given?
In general, you can solve the linear equations to determine the
coefficients of the above equation of plane. Note that, in oblique
axes, vector (a,b,c) is no longer the "normal vector."
3. Suppose I have a unit cubic lattice, i.e. atoms at all points with
three integer coordinates (a,b,c). Could you describe in words what
distinguishes a 'lattice plane' from any other sort of plane?
I don't understand your question 3.
But my original problem is just an integer solution in an Oblique
axes. All lattice plane can be described by "ax+by+cz=integer" because
every point is an integer solution to that corresponding equation and
each that equation exist a corresponding integer solution.
Cheers,
Zigoteau.
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| User: "zigoteau" |
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| Title: Re: Lattice plane |
06 Nov 2004 07:48:13 AM |
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(idyllic) wrote in message news:<f9b8e54f.0411041852.231f503e@posting.google.com>...
Hi, Idyllic,
I'm confused by calculating the distance of lattice planes (hkl). My
trouble is: why the neighboring plane of hx+ky+lz=0 is hx+ky+lz=1 ?
This is a mathematical problem, but I can't conquer it. Any help will
be appreciated.
1. What is the general equation of a plane?
With any oblique axes, the equation is the same: ax+by+cz=k
Fine.
2. In Euclidian geometry, it takes three points to determine a plane.
How would you go about determining the equation of a plane through
three points whose coordinates are given?
In general, you can solve the linear equations to determine the
coefficients of the above equation of plane. Note that, in oblique
axes, vector (a,b,c) is no longer the "normal vector."
Sounds OK. In fact you can write down the solution as a determinant:
| x x_1 x_2 x_3 |
| y y_1 y_2 y_3 | = 0
| z z_1 z_2 z_3 |
| 1 1 1 1 |
3. Suppose I have a unit cubic lattice, i.e. atoms at all points with
three integer coordinates (a,b,c). Could you describe in words what
distinguishes a 'lattice plane' from any other sort of plane?
I don't understand your question 3.
A lattice plane is one which goes through three lattice points. In
fact, if it goes through three, then it goes through infinitely many.
But my original problem is just an integer solution in an Oblique
axes. All lattice plane can be described by "ax+by+cz=integer" because
every point is an integer solution to that corresponding equation and
each that equation exist a corresponding integer solution.
Well then, I'm not sure why you have a problem. From the determinant
form given above, you can see that, in the coordinate system defined
by the unit cell vectors, the cofactors of each term in the first
column are all integers.
Cheers,
Zigoteau.
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| User: "Theo Wollenleben" |
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| Title: Re: Lattice plane |
09 Oct 2004 01:50:43 PM |
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idyllic wrote:
I'm confused by calculating the distance of lattice planes (hkl). My
trouble is: why the neighboring plane of hx+ky+lz=0 is hx+ky+lz=1 ?
This is a mathematical problem, but I can't conquer it. Any help will
be appreciated.
Let a_x,a_y,a_z be the basis of the lattice and b_h,b_k,b_l the basis of
the reciprocal lattice. A reciprocal lattice vector b=h*b_h+k*b_k+l*b_l
defines lattice planes {r=x*a_x+y*a_y+z*a_z | b.r=h*x+k*y+l*z=constant}.
The distance of neighboring planes is 1/|b|.
Let's calculate the distance of the planes b.r=0 and b.r=1. The distance
of a point r=x*a_x+y*a_y+z*a_z on b.r=1 from b.r=0 is b.r/|b|=1/|b|.
Therefore b.r=1 must be the neighboring plane of b.r=0.
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