Suppose we are making the surface reciprocal lattice of Si(111) 1x1 ideal surface and Si(111) r3xr3-In, r7xr3-In, 4x1-In, r7xr7-In.

The reciprocal unit cell vector of Si(111) 1x1 is

a* = (r3/2 G, -1/2G) .....eq. (1)

b* = (0, G) .....eq. (2)

where G is the absolute value of the Si(111) 1x1 reciprocal unit cell vectors.

The r3xr3 reciprocal unit cell vectors can be expressed as

as* = 2/3 a* + 1/3 a* .....eq. (3)

bs* = -1/3 a* + 1/3 b* .....eq. (4)

The reciprocal unit cell vectors for all the other surface structures above can be found in the following comprehensive book.

"Surface phases on Silicon -preparations, Structures and Properties "

by V. G. Lifshits, A. A. Saranin, and A. V. Zotov, Wiley

Now, what we want to do now is

1. obtain the reciprocal lattice point set Gs(k, l) = k * as* + l * bs*

over k= -25 to 25 and l = -25 to 25 for the surface structures.

-25 to 25 can be a very big range for 1x1, but appropriate for big unit cells such as 7x7.

2. Express the point sets in terms of x and y.

3. Display the data point set on x-y graph.

Let Igor do that.

Copy and paste the following lines to the Igor command line.

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Make/N=(2601,2)/D indices,r7xr3A,r7xr3Axy, r7xr3Bxy,r7xr3Cxy,'4x1A','4x1Axy','4x1Bxy','4x1Cxy',r3xr3, r3xr3xy

Make/N=(2601,2)/D '1x1', '1x1xy', r7xr7A, r7xr7xy, r7xr7Bxy

indices[][0] = round((p-1300)/51) // set k = -25 to 25

indices[][1]= mod((p),51)-25 // set l = -25 to 25

// r3xr3------------------------

r3xr3[][0]= 2/3*indices[p][0] -1/3 * indices[p][1] //a* component of Gs(k, l) according to eqs. (3) and (4)

r3xr3[][1]= 1/3*indices[p][0] +1/3 * indices[p][1] //b* component of Gs(k, l) according to eqs. (3) and (4)

r3xr3xy[][0]= r3xr3[p][0]*sqrt(3)/2 //x component of Gs(k, l) according to eqs. (1) and (2)

r3xr3xy[][1]= -r3xr3[p][0]*1/2+r3xr3[p][1] //y component of Gs(k, l) according to eqs. (1) and (2)

// r7xr3------------------------

r7xr3A[][0]= 3/5*indices[p][0]-1/5*indices[p][1]

r7xr3A[][1]= 2/5*indices[p][0]+1/5*indices[p][1]

r7xr3Axy[][0]=r7xr3A[p][0]*sqrt(3)/2 //x component of Gs(k, l) for r7xr3 domain A

r7xr3Axy[][1]= -r7xr3A[p][0]*1/2+r7xr3A[p][1] //y component of Gs(k, l) for r7xr3 domain A

r7xr3Bxy[][0]= r7xr3A[p][0]*sqrt(3)/2-r7xr3A[p][1]*sqrt(3)/2 //x component of Gs(k, l) for r7xr3 domain B

r7xr3Bxy[][1]= r7xr3A[p][0]*1/2+r7xr3A[p][1]*1/2 //y component of Gs(k, l) for r7xr3 domain B

r7xr3Cxy[][0]= -r7xr3A[p][1]*sqrt(3)/2 //x component of Gs(k, l) for r7xr3 domain C

r7xr3Cxy[][1]= r7xr3A[p][0]-r7xr3A[p][1]/2 //y component of Gs(k, l) for r7xr3 domain C

// 4x1------------------------

'4x1A'[][0]= indices[p][0]

'4x1A'[][1]= 1/4*indices[p][1]

'4x1Axy'[][0]='4x1A'[p][0]*sqrt(3)/2

'4x1Axy'[][1]=-'4x1A'[p][0]*1/2+'4x1A'[p][1]

'4x1Bxy'[][0]= '4x1A'[p][0]*sqrt(3)/2-'4x1A'[p][1]*sqrt(3)/2

'4x1Bxy'[][1]= '4x1A'[p][0]*1/2+'4x1A'[p][1]*1/2

'4x1Cxy'[][0]= -'4x1A'[p][1]*sqrt(3)/2

'4x1Cxy'[][1]= '4x1A'[p][0]-'4x1A'[p][1]/2

//1x1------------------------

'1x1'[][0]= indices[p][0]

'1x1'[][1]= indices[p][1]

'1x1xy'[][0]= '1x1'[p][0]*sqrt(3)/2

'1x1xy'[][1]= -'1x1'[p][0]*1/2+'1x1'[p][1]

//r7xr7------------------------

r7xr7A[][0]= indices[p][0]*3/7-indices[p][1]*1/7

r7xr7A[][1]=indices[p][0]*1/7+indices[p][1]*2/7

r7xr7Axy[][0]= r7xr7A[p][0]*sqrt(3)/2

r7xr7Axy[][1]= -r7xr7A[p][0]*1/2+r7xr7A[p][1]

r7xr7Bxy[][0]= r7xr7A[p][0]*sqrt(3)/2

r7xr7Bxy[][1]= r7xr7A[p][0]*1/2-r7xr7A[p][1]

//Display------------------------

Display r3xr3xy[][1] vs r3xr3xy[][0]

ModifyGraph mode=3,marker=8

AppendToGraph r7xr3Axy[][1] vs r7xr3Axy[][0]

AppendToGraph r7xr7Bxy[][1] vs r7xr7Bxy[][0]

ModifyGraph mode=3,marker(r7xr7Bxy)=19,msize(r7xr7Bxy)=2,rgb(r7xr7Bxy)=(1,34817,52428)

//add the reciprocal lattices you are interested in. change color and size of the markers as you want.

//don' to forget to set the x axis and y axis have the same scale. Otherwise, the figure will be distorted.

ModifyGraph margin(left)=56,margin(bottom)=56

ModifyGraph margin(right)=14,margin(top)=14

ModifyGraph width=425.197,height=425.197

SetAxis left -2,2;DelayUpdate

SetAxis bottom -2,2

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