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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2. Translate by :

0

0


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3. Move from left toright ( runs from to +)

0

0


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0

0


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0

0


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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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0

0

The area under is always equal to


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0

0



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0

0



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0

0



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0

0

We are clearly flipping horizontally as advances and thefunction is filtered through


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0

0



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0

0



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0

0



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0

0



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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

,,

,,

Delta Functions and Convolutions

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