align:start position:0% in this demonstration I'm going to show align:start position:0% in this demonstration I'm going to show align:start position:0% in this demonstration I'm going to show you how to construct 2d brilliant zones align:start position:0% you how to construct 2d brilliant zones align:start position:0% you how to construct 2d brilliant zones and hopefully achieve a better align:start position:0% and hopefully achieve a better align:start position:0% and hopefully achieve a better understanding them and what they are so align:start position:0% understanding them and what they are so align:start position:0% understanding them and what they are so a brilliant zone is an important concept align:start position:0% a brilliant zone is an important concept align:start position:0% a brilliant zone is an important concept in material science and solid-state align:start position:0% in material science and solid-state align:start position:0% in material science and solid-state physics align:start position:0% physics align:start position:0% physics alike because it is used to describe the align:start position:0% alike because it is used to describe the align:start position:0% alike because it is used to describe the behavior of an electron in a perfect align:start position:0% behavior of an electron in a perfect align:start position:0% behavior of an electron in a perfect crystal system so what is a brilliant align:start position:0% crystal system so what is a brilliant align:start position:0% crystal system so what is a brilliant zone the Brillion zone is a particular align:start position:0% zone the Brillion zone is a particular align:start position:0% zone the Brillion zone is a particular choice of the unit cell of the align:start position:0% choice of the unit cell of the align:start position:0% choice of the unit cell of the reciprocal lattice is defined as the align:start position:0% reciprocal lattice is defined as the align:start position:0% reciprocal lattice is defined as the riegner Seitz cell of the reciprocal align:start position:0% riegner Seitz cell of the reciprocal align:start position:0% riegner Seitz cell of the reciprocal lattice it is constructed as the set of align:start position:0% lattice it is constructed as the set of align:start position:0% lattice it is constructed as the set of points and closed by the Bragg planes align:start position:0% points and closed by the Bragg planes align:start position:0% points and closed by the Bragg planes the planes perpendicular to a connection align:start position:0% the planes perpendicular to a connection align:start position:0% the planes perpendicular to a connection line from the origin of to each lattice align:start position:0% line from the origin of to each lattice align:start position:0% line from the origin of to each lattice point passing through the midpoint align:start position:0% point passing through the midpoint align:start position:0% point passing through the midpoint alternatively it is defined as the set align:start position:0% alternatively it is defined as the set align:start position:0% alternatively it is defined as the set of points closer to the origin than to align:start position:0% of points closer to the origin than to align:start position:0% of points closer to the origin than to any other reciprocal lattice point the align:start position:0% any other reciprocal lattice point the align:start position:0% any other reciprocal lattice point the whole reciprocal space may be covered align:start position:0% whole reciprocal space may be covered align:start position:0% whole reciprocal space may be covered without overlap with copies of such align:start position:0% without overlap with copies of such align:start position:0% without overlap with copies of such brilliant zone okay so that was a rather align:start position:0% brilliant zone okay so that was a rather align:start position:0% brilliant zone okay so that was a rather convoluted definition let's do it again align:start position:0% convoluted definition let's do it again align:start position:0% convoluted definition let's do it again let's brush up and be sure that we align:start position:0% let's brush up and be sure that we align:start position:0% let's brush up and be sure that we understand this definition we're going align:start position:0% understand this definition we're going align:start position:0% understand this definition we're going to define her the reciprocal space the align:start position:0% to define her the reciprocal space the align:start position:0% to define her the reciprocal space the Bragg planes as well but it's fine both align:start position:0% Bragg planes as well but it's fine both align:start position:0% Bragg planes as well but it's fine both of these will also do a quick revision align:start position:0% of these will also do a quick revision align:start position:0% of these will also do a quick revision of what is there a record lattice so align:start position:0% of what is there a record lattice so align:start position:0% of what is there a record lattice so that we can go from there to the align:start position:0% that we can go from there to the align:start position:0% that we can go from there to the reciprocal lattice the microscopic align:start position:0% reciprocal lattice the microscopic align:start position:0% reciprocal lattice the microscopic perfect crystal is formed by adding align:start position:0% perfect crystal is formed by adding align:start position:0% perfect crystal is formed by adding identical building blocks so to say unit align:start position:0% identical building blocks so to say unit align:start position:0% identical building blocks so to say unit cells consisting of atoms and groups of align:start position:0% cells consisting of atoms and groups of align:start position:0% cells consisting of atoms and groups of atoms a unit cell is the smallest align:start position:0% atoms a unit cell is the smallest align:start position:0% atoms a unit cell is the smallest component of a crystal that one stacked align:start position:0% component of a crystal that one stacked align:start position:0% component of a crystal that one stacked together with pure translational align:start position:0% together with pure translational align:start position:0% together with pure translational repetition reproducing the whole crystal align:start position:0% repetition reproducing the whole crystal align:start position:0% repetition reproducing the whole crystal which essentially means that you can align:start position:0% which essentially means that you can align:start position:0% which essentially means that you can take the same thing over and over and align:start position:0% take the same thing over and over and align:start position:0% take the same thing over and over and over again and get the whole system done align:start position:0% over again and get the whole system done align:start position:0% over again and get the whole system done so and the groups of atoms these unit align:start position:0% so and the groups of atoms these unit align:start position:0% so and the groups of atoms these unit cells that form the microscopic crystal align:start position:0% cells that form the microscopic crystal align:start position:0% cells that form the microscopic crystal but infinite repetition is called the align:start position:0% but infinite repetition is called the align:start position:0% but infinite repetition is called the basis okay that seems quite clear and align:start position:0% basis okay that seems quite clear and align:start position:0% basis okay that seems quite clear and the basis is formed in such a way that align:start position:0% the basis is formed in such a way that align:start position:0% the basis is formed in such a way that it forms the lattice more commonly known align:start position:0% it forms the lattice more commonly known align:start position:0% it forms the lattice more commonly known as the Rivervale at every point of a align:start position:0% as the Rivervale at every point of a align:start position:0% as the Rivervale at every point of a breve lattice is equivalent to every align:start position:0% breve lattice is equivalent to every align:start position:0% breve lattice is equivalent to every other point which means that the align:start position:0% other point which means that the align:start position:0% other point which means that the arrangement atoms and the crystal is the align:start position:0% arrangement atoms and the crystal is the align:start position:0% arrangement atoms and the crystal is the same when viewed from different lattice align:start position:0% same when viewed from different lattice align:start position:0% same when viewed from different lattice points okay that also seems align:start position:0% points okay that also seems align:start position:0% points okay that also seems understandable and you should probably align:start position:0% understandable and you should probably align:start position:0% understandable and you should probably know that by now so any fundamental align:start position:0% know that by now so any fundamental align:start position:0% know that by now so any fundamental lattice must be definable by three align:start position:0% lattice must be definable by three align:start position:0% lattice must be definable by three primitive translational vectors a1 a2 align:start position:0% primitive translational vectors a1 a2 align:start position:0% primitive translational vectors a1 a2 and a3 the combination of these vectors align:start position:0% and a3 the combination of these vectors align:start position:0% and a3 the combination of these vectors is using to find the crystal align:start position:0% is using to find the crystal align:start position:0% is using to find the crystal translational vector R such that R is align:start position:0% translational vector R such that R is align:start position:0% translational vector R such that R is equal to a 1 and 1 plus a 2 and 2 plus a align:start position:0% equal to a 1 and 1 plus a 2 and 2 plus a align:start position:0% equal to a 1 and 1 plus a 2 and 2 plus a 3 and 3 where n are just arbitrary align:start position:0% 3 and 3 where n are just arbitrary align:start position:0% 3 and 3 where n are just arbitrary integers to show that Mark would show align:start position:0% integers to show that Mark would show align:start position:0% integers to show that Mark would show the size of our lattice so when we the align:start position:0% the size of our lattice so when we the align:start position:0% the size of our lattice so when we the crystal lattice is repeated an infinite align:start position:0% crystal lattice is repeated an infinite align:start position:0% crystal lattice is repeated an infinite amount of times to create the perfect align:start position:0% amount of times to create the perfect align:start position:0% amount of times to create the perfect crystal structure and each of those align:start position:0% crystal structure and each of those align:start position:0% crystal structure and each of those lattices are translationally symmetric align:start position:0% lattices are translationally symmetric align:start position:0% lattices are translationally symmetric and the way I'll look at is is that one align:start position:0% and the way I'll look at is is that one align:start position:0% and the way I'll look at is is that one cannot tell their positional in crystal align:start position:0% cannot tell their positional in crystal align:start position:0% cannot tell their positional in crystal structure because every lattice looks align:start position:0% structure because every lattice looks align:start position:0% structure because every lattice looks this ok so now okay so that seems to align:start position:0% this ok so now okay so that seems to align:start position:0% this ok so now okay so that seems to make sense so now let's go through align:start position:0% make sense so now let's go through align:start position:0% make sense so now let's go through reciprocal space okay so every lattice align:start position:0% reciprocal space okay so every lattice align:start position:0% reciprocal space okay so every lattice has a reciprocal lattice associated align:start position:0% has a reciprocal lattice associated align:start position:0% has a reciprocal lattice associated through in crystallography terms the align:start position:0% through in crystallography terms the align:start position:0% through in crystallography terms the reciprocal lattices of the diffraction align:start position:0% reciprocal lattices of the diffraction align:start position:0% reciprocal lattices of the diffraction prior of a crystal or in quantum align:start position:0% prior of a crystal or in quantum align:start position:0% prior of a crystal or in quantum mechanics it's described this k space align:start position:0% mechanics it's described this k space align:start position:0% mechanics it's described this k space with K being 4k wave vectors in 3d align:start position:0% with K being 4k wave vectors in 3d align:start position:0% with K being 4k wave vectors in 3d lattice the vectors would be v1 v2 and align:start position:0% lattice the vectors would be v1 v2 and align:start position:0% lattice the vectors would be v1 v2 and v3 and they can be denoted it as well align:start position:0% v3 and they can be denoted it as well align:start position:0% v3 and they can be denoted it as well look at it just b1 b1 is equals to 2 pi align:start position:0% look at it just b1 b1 is equals to 2 pi align:start position:0% look at it just b1 b1 is equals to 2 pi times the cross product of the vectors align:start position:0% times the cross product of the vectors align:start position:0% times the cross product of the vectors a2 and a3 from our direct lattice align:start position:0% a2 and a3 from our direct lattice align:start position:0% a2 and a3 from our direct lattice divided by the triple cross scalar align:start position:0% divided by the triple cross scalar align:start position:0% divided by the triple cross scalar product of a 1 a 2 and a 3 in which case align:start position:0% product of a 1 a 2 and a 3 in which case align:start position:0% product of a 1 a 2 and a 3 in which case this cross product of a 2 and a 3 is the align:start position:0% this cross product of a 2 and a 3 is the align:start position:0% this cross product of a 2 and a 3 is the area of our vector of our two vectors in align:start position:0% area of our vector of our two vectors in align:start position:0% area of our vector of our two vectors in the triple scalar product is the volume align:start position:0% the triple scalar product is the volume align:start position:0% the triple scalar product is the volume of our system by simplifying it we can align:start position:0% of our system by simplifying it we can align:start position:0% of our system by simplifying it we can just get 2 pi over the height of our align:start position:0% just get 2 pi over the height of our align:start position:0% just get 2 pi over the height of our unit cell or we can put it in this way align:start position:0% unit cell or we can put it in this way align:start position:0% unit cell or we can put it in this way the larger our direct lattice gets the align:start position:0% the larger our direct lattice gets the align:start position:0% the larger our direct lattice gets the smaller and core person or reciprocal align:start position:0% smaller and core person or reciprocal align:start position:0% smaller and core person or reciprocal lattice becomes another observation that align:start position:0% lattice becomes another observation that align:start position:0% lattice becomes another observation that could actually be made by the reciprocal align:start position:0% could actually be made by the reciprocal align:start position:0% could actually be made by the reciprocal lattices that the reciprocal lattice of align:start position:0% lattices that the reciprocal lattice of align:start position:0% lattices that the reciprocal lattice of the reciprocal lattice is the direct align:start position:0% the reciprocal lattice is the direct align:start position:0% the reciprocal lattice is the direct lattice bit okay for simplicity's sake align:start position:0% lattice bit okay for simplicity's sake align:start position:0% lattice bit okay for simplicity's sake let's look at a transformation from 2d align:start position:0% let's look at a transformation from 2d align:start position:0% let's look at a transformation from 2d lattice to a reciprocal lattice okay so align:start position:0% lattice to a reciprocal lattice okay so align:start position:0% lattice to a reciprocal lattice okay so we have a visualization here where we align:start position:0% we have a visualization here where we align:start position:0% we have a visualization here where we can change align:start position:0% can change align:start position:0% can change the length of our X vector indirect align:start position:0% the length of our X vector indirect align:start position:0% the length of our X vector indirect space and the length of our vibrator align:start position:0% space and the length of our vibrator align:start position:0% space and the length of our vibrator indirect space and we can change whether align:start position:0% indirect space and we can change whether align:start position:0% indirect space and we can change whether or not we're seeing this as a direct align:start position:0% or not we're seeing this as a direct align:start position:0% or not we're seeing this as a direct lattice or by pressing this button as a align:start position:0% lattice or by pressing this button as a align:start position:0% lattice or by pressing this button as a reciprocal lattice so as we can see but align:start position:0% reciprocal lattice so as we can see but align:start position:0% reciprocal lattice so as we can see but increasing the length of our direct align:start position:0% increasing the length of our direct align:start position:0% increasing the length of our direct direct vector we change the sizes of our align:start position:0% direct vector we change the sizes of our align:start position:0% direct vector we change the sizes of our reciprocal lattice vectors and the other align:start position:0% reciprocal lattice vectors and the other align:start position:0% reciprocal lattice vectors and the other way around align:start position:0% align:start position:0% okay so now we had a definition of the align:start position:0% okay so now we had a definition of the align:start position:0% okay so now we had a definition of the reciprocal space in the rect space let's align:start position:0% reciprocal space in the rect space let's align:start position:0% reciprocal space in the rect space let's go back to our definition so the first align:start position:0% go back to our definition so the first align:start position:0% go back to our definition so the first brilliant zone can be defined as a set align:start position:0% brilliant zone can be defined as a set align:start position:0% brilliant zone can be defined as a set of points in reciprocal space that can align:start position:0% of points in reciprocal space that can align:start position:0% of points in reciprocal space that can be reached from a specific point of align:start position:0% be reached from a specific point of align:start position:0% be reached from a specific point of origin without crossing any breakpoints align:start position:0% origin without crossing any breakpoints align:start position:0% origin without crossing any breakpoints came so what our Bragg planes a Bragg align:start position:0% came so what our Bragg planes a Bragg align:start position:0% came so what our Bragg planes a Bragg plane or in this case a Bragg line is a align:start position:0% plane or in this case a Bragg line is a align:start position:0% plane or in this case a Bragg line is a Bragg line with perpendicular bisector align:start position:0% Bragg line with perpendicular bisector align:start position:0% Bragg line with perpendicular bisector reciprocal lattice vector a vector which align:start position:0% reciprocal lattice vector a vector which align:start position:0% reciprocal lattice vector a vector which connects two lattice points and the align:start position:0% connects two lattice points and the align:start position:0% connects two lattice points and the closest Bragg planes are essentially align:start position:0% closest Bragg planes are essentially align:start position:0% closest Bragg planes are essentially accosting the Brillion zone now can show align:start position:0% accosting the Brillion zone now can show align:start position:0% accosting the Brillion zone now can show that the Bragg planes for the closest align:start position:0% that the Bragg planes for the closest align:start position:0% that the Bragg planes for the closest neighbors with this being our original align:start position:0% neighbors with this being our original align:start position:0% neighbors with this being our original lattice point these are the four closest align:start position:0% lattice point these are the four closest align:start position:0% lattice point these are the four closest neighbors in a simple I'd say cubic but align:start position:0% neighbors in a simple I'd say cubic but align:start position:0% neighbors in a simple I'd say cubic but it's actually just the square lattice align:start position:0% it's actually just the square lattice align:start position:0% it's actually just the square lattice because it's in 2d and if we add the align:start position:0% because it's in 2d and if we add the align:start position:0% because it's in 2d and if we add the second large vectors before the second align:start position:0% second large vectors before the second align:start position:0% second large vectors before the second closest neighbors you can see these ones align:start position:0% closest neighbors you can see these ones align:start position:0% closest neighbors you can see these ones and essentially conveys the same align:start position:0% and essentially conveys the same align:start position:0% and essentially conveys the same information when we go to a higher order align:start position:0% information when we go to a higher order align:start position:0% information when we go to a higher order of closest neighbors we can see that the align:start position:0% of closest neighbors we can see that the align:start position:0% of closest neighbors we can see that the system gets a lot more complex okay so align:start position:0% system gets a lot more complex okay so align:start position:0% system gets a lot more complex okay so now we've seen what our back planes can align:start position:0% now we've seen what our back planes can align:start position:0% now we've seen what our back planes can go towards brilliant zones so this is align:start position:0% go towards brilliant zones so this is align:start position:0% go towards brilliant zones so this is the first brilliant result it is what it align:start position:0% the first brilliant result it is what it align:start position:0% the first brilliant result it is what it seems it is as you can imagine the Bragg align:start position:0% seems it is as you can imagine the Bragg align:start position:0% seems it is as you can imagine the Bragg planes just go here and the brilliant align:start position:0% planes just go here and the brilliant align:start position:0% planes just go here and the brilliant zone shows us the area in reciprocal align:start position:0% zone shows us the area in reciprocal align:start position:0% zone shows us the area in reciprocal space that is closer to our lattice align:start position:0% space that is closer to our lattice align:start position:0% space that is closer to our lattice point than any other lattice point which align:start position:0% point than any other lattice point which align:start position:0% point than any other lattice point which are essentially the Bragg planes as well align:start position:0% are essentially the Bragg planes as well align:start position:0% are essentially the Bragg planes as well as we can see if our reciprocal lattice align:start position:0% as we can see if our reciprocal lattice align:start position:0% as we can see if our reciprocal lattice origin point isn't here this the square align:start position:0% origin point isn't here this the square align:start position:0% origin point isn't here this the square is closer to this point than to any align:start position:0% is closer to this point than to any align:start position:0% is closer to this point than to any other point and after these lines it align:start position:0% other point and after these lines it align:start position:0% other point and after these lines it gets the other way around align:start position:0% gets the other way around align:start position:0% gets the other way around so we can move on forward to a higher align:start position:0% so we can move on forward to a higher align:start position:0% so we can move on forward to a higher order of brilliant zones align:start position:0% align:start position:0% this is the Brillion zone for the second align:start position:0% this is the Brillion zone for the second align:start position:0% this is the Brillion zone for the second closest neighbor as you can see it it is align:start position:0% closest neighbor as you can see it it is align:start position:0% closest neighbor as you can see it it is rather similar it just takes the second align:start position:0% rather similar it just takes the second align:start position:0% rather similar it just takes the second closest neighbors and essentially draws align:start position:0% closest neighbors and essentially draws align:start position:0% closest neighbors and essentially draws another square it but it's a bit tilted align:start position:0% another square it but it's a bit tilted align:start position:0% another square it but it's a bit tilted to this okay so now let's look at the align:start position:0% to this okay so now let's look at the align:start position:0% to this okay so now let's look at the third one for the third brilliant zone align:start position:0% third one for the third brilliant zone align:start position:0% third one for the third brilliant zone that gets a bit more complex because if align:start position:0% that gets a bit more complex because if align:start position:0% that gets a bit more complex because if we scroll a bit backwards we can see align:start position:0% we scroll a bit backwards we can see align:start position:0% we scroll a bit backwards we can see that the third bread the breakfast for align:start position:0% that the third bread the breakfast for align:start position:0% that the third bread the breakfast for the thirtieth closest neighbors are align:start position:0% the thirtieth closest neighbors are align:start position:0% the thirtieth closest neighbors are these ones so we might think that this align:start position:0% these ones so we might think that this align:start position:0% these ones so we might think that this whole thing would be that brilliant the align:start position:0% whole thing would be that brilliant the align:start position:0% whole thing would be that brilliant the third brilliant look but it's actually align:start position:0% third brilliant look but it's actually align:start position:0% third brilliant look but it's actually not because with every next system it align:start position:0% not because with every next system it align:start position:0% not because with every next system it gets a bit more complex and it's align:start position:0% gets a bit more complex and it's align:start position:0% gets a bit more complex and it's actually taking account both the third align:start position:0% actually taking account both the third align:start position:0% actually taking account both the third right planes and the first ones okay so align:start position:0% right planes and the first ones okay so align:start position:0% right planes and the first ones okay so now let's do another thing let's turn align:start position:0% now let's do another thing let's turn align:start position:0% now let's do another thing let's turn off on that we can see all the Bragg align:start position:0% off on that we can see all the Bragg align:start position:0% off on that we can see all the Bragg planes and turn up another notch okay so align:start position:0% planes and turn up another notch okay so align:start position:0% planes and turn up another notch okay so here we can see that the fourth Brillion align:start position:0% here we can see that the fourth Brillion align:start position:0% here we can see that the fourth Brillion zone gets a lot more complex now we can align:start position:0% zone gets a lot more complex now we can align:start position:0% zone gets a lot more complex now we can see all the Bragg planes for the quite align:start position:0% see all the Bragg planes for the quite align:start position:0% see all the Bragg planes for the quite for the closest neighbors and these align:start position:0% for the closest neighbors and these align:start position:0% for the closest neighbors and these lines could get quite difficult to align:start position:0% lines could get quite difficult to align:start position:0% lines could get quite difficult to understand by drawing themselves but we align:start position:0% understand by drawing themselves but we align:start position:0% understand by drawing themselves but we can help ourselves out with this align:start position:0% can help ourselves out with this align:start position:0% can help ourselves out with this visualization so yeah we can see that align:start position:0% visualization so yeah we can see that align:start position:0% visualization so yeah we can see that they start to interact with each other align:start position:0% they start to interact with each other align:start position:0% they start to interact with each other and thus make a more difficult structure align:start position:0% and thus make a more difficult structure align:start position:0% and thus make a more difficult structure and if we go to the fifth one and see align:start position:0% and if we go to the fifth one and see align:start position:0% and if we go to the fifth one and see show that we can see all the brilliant align:start position:0% show that we can see all the brilliant align:start position:0% show that we can see all the brilliant zones okay so here we can see that it align:start position:0% zones okay so here we can see that it align:start position:0% zones okay so here we can see that it gets becomes quite a nice drawing if we align:start position:0% gets becomes quite a nice drawing if we align:start position:0% gets becomes quite a nice drawing if we were to keep adding them let's just it align:start position:0% were to keep adding them let's just it align:start position:0% were to keep adding them let's just it let's head to the ninth one okay so see align:start position:0% let's head to the ninth one okay so see align:start position:0% let's head to the ninth one okay so see here we can see a high order brilliant align:start position:0% here we can see a high order brilliant align:start position:0% here we can see a high order brilliant zones and it actually looked kind of align:start position:0% zones and it actually looked kind of align:start position:0% zones and it actually looked kind of nice which is also interesting because align:start position:0% nice which is also interesting because align:start position:0% nice which is also interesting because it's also an art form drawing high align:start position:0% it's also an art form drawing high align:start position:0% it's also an art form drawing high drawing high order billion zones align:start position:0% drawing high order billion zones align:start position:0% drawing high order billion zones the higher you go the more complex and align:start position:0% the higher you go the more complex and align:start position:0% the higher you go the more complex and more discrete the system gets in align:start position:0% more discrete the system gets in align:start position:0% more discrete the system gets in essentially it looks nicer okay so after align:start position:0% essentially it looks nicer okay so after align:start position:0% essentially it looks nicer okay so after looking at this we can get a short align:start position:0% looking at this we can get a short align:start position:0% looking at this we can get a short definition of how to construct to the align:start position:0% definition of how to construct to the align:start position:0% definition of how to construct to the brilliant zones so enth brilliant zone align:start position:0% brilliant zones so enth brilliant zone align:start position:0% brilliant zones so enth brilliant zone can be defined as the area or volume if align:start position:0% can be defined as the area or volume if align:start position:0% can be defined as the area or volume if we look in 3d in reciprocal space that align:start position:0% we look in 3d in reciprocal space that align:start position:0% we look in 3d in reciprocal space that can be reached from the origin but align:start position:0% can be reached from the origin but align:start position:0% can be reached from the origin but crossing exactly n minus one Bragg align:start position:0% crossing exactly n minus one Bragg align:start position:0% crossing exactly n minus one Bragg planes align:start position:0% planes align:start position:0% planes so we can look also add a brilliant zone align:start position:0% so we can look also add a brilliant zone align:start position:0% so we can look also add a brilliant zone for a system where the atoms are not align:start position:0% for a system where the atoms are not align:start position:0% for a system where the atoms are not perfectly in the perfect square lattice align:start position:0% perfectly in the perfect square lattice align:start position:0% perfectly in the perfect square lattice but are offset a bit making sort of say align:start position:0% but are offset a bit making sort of say align:start position:0% but are offset a bit making sort of say a triangular lattice and as we can see align:start position:0% a triangular lattice and as we can see align:start position:0% a triangular lattice and as we can see here the first billion zone is it such align:start position:0% here the first billion zone is it such align:start position:0% here the first billion zone is it such such a hexagon and the second one align:start position:0% such a hexagon and the second one align:start position:0% such a hexagon and the second one already gets a bit more difficult by align:start position:0% already gets a bit more difficult by align:start position:0% already gets a bit more difficult by forming a star shape and the third one align:start position:0% forming a star shape and the third one align:start position:0% forming a star shape and the third one is again so to say FX agon and it goes align:start position:0% is again so to say FX agon and it goes align:start position:0% is again so to say FX agon and it goes on like that align:start position:0% on like that align:start position:0% on like that okay so and then so what information is align:start position:0% okay so and then so what information is align:start position:0% okay so and then so what information is the brilliance don't hold on what does align:start position:0% the brilliance don't hold on what does align:start position:0% the brilliance don't hold on what does it give us the short vectors in the align:start position:0% it give us the short vectors in the align:start position:0% it give us the short vectors in the Brillion zone or on its boundary align:start position:0% Brillion zone or on its boundary align:start position:0% Brillion zone or on its boundary characterized States in a system with align:start position:0% characterized States in a system with align:start position:0% characterized States in a system with lattice period periodicity for example align:start position:0% lattice period periodicity for example align:start position:0% lattice period periodicity for example phone out or or electron state but for align:start position:0% phone out or or electron state but for align:start position:0% phone out or or electron state but for that the whole of the video thanks and align:start position:0% that the whole of the video thanks and align:start position:0% that the whole of the video thanks and this code for this demonstration was align:start position:0% this code for this demonstration was align:start position:0% this code for this demonstration was taken and editors from Mathematica align:start position:0% taken and editors from Mathematica align:start position:0% taken and editors from Mathematica demonstration since made by their slope align:start position:0% demonstration since made by their slope align:start position:0% demonstration since made by their slope cloth