delta functions and convolutions
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Some additional math concepts
Complex numbers
The complex number a + ib should be viewed simply as the point in the xy-
plane having Cartesian coordinates (a, b), or equivalently as the vector
connecting the origin to that point.
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Eulers formula:
Important relations:
1
1It follows that 1 if n is integer
Then: if n is integer
1 1
0
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Let us recall the scalar product, of two vectors, one of the direct lattice, one of the
reciprocal lattice:
X
X = integer number
1
Then:
This property will be extremely useful in the future
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The Dirac delta function
The Dirac delta function is defi
ned by theproperties:
0 for 0
for 0and
1
Area under the box=1
x=0
a
1/a
The delta function as a limit of box function
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The delta function as a limit of a gaussian function
1 2
x
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The Dirac delta function in 2D...
x=0
x=x0x=0
0 for for
The Dirac delta function in 3D...
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Lattice functions
where and is an integer
Delta functions can be used to represent lattice functions. For example, in a
one-dimensional space, a lattice with period a may be represented by:
vanishes everywhere except at the lattice points where it becomes infinite
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Lattice functions
,,
,,
By the same fashion, a three-dimensional lattice defined by the basis vectorsa, b and c may be represented by:
vanishes everywhere except at the lattice points where it becomes infinite
Where
,, and u, v, w are integers
,,
,,
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Convolutions
A convolution between two functions and is denoted by and is defined by:
A convolution is in essence an integral that expresses the amount of
overlap of one function as it is shifted over another function. It
therefore "blends" one function with another.
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One example of convolution
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Another example of convolution
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Other examples of convolution
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Convolutions with delta functions
1. Flip :
0
0
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Convolutions with delta functions
2. Translate by :
0
0
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Convolutions with delta functions
3. Move from left toright ( runs from to +)
0
0
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Convolutions with delta functions
3. Move from left toright ( runs from to +)
0
0
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Convolutions with delta functions
3. Move from left toright ( runs from to +)
0
0
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Convolutions with delta functions
0
0
4. At some point starts overlapping with
Because is infinetly narrow at , and the area under is equal to 1, in theonly point of overlap, the area under is equal to
starts appearing
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Convolutions with delta functions
0
0
The area under is always equal to
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Convolutions with delta functions
0
0
The area under is always equal to
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Convolutions with delta functions
0
0
The area under is always equal to
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Convolutions with delta functions
0
0
The area under is always equal to
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Convolutions with delta functions
0
0
We are clearly flipping horizontally as advances and thefunction is filtered through
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Convolutions with delta functions
0
0
We are clearly flipping horizontally as advances and thefunction is filtered through
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Convolutions with delta functions
0
0
We are clearly flipping horizontally as advances and thefunction is filtered through
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Convolutions with delta functions
0
0
We are clearly flipping horizontally as advances and thefunction is filtered through
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Convolutions with delta functions
0
0
We are clearly flipping horizontally as advances and thefunction is filtered through
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Summarizing:
The convolution of with has the effect of translating from theorigin to
0
0
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Convolutions
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We come now to an important conclusion:
A crystal can be described mathematically as the convolution of a function which describes the content of the unit cell (for example the electronic density
distribution) with the lattice function
,,
,,