| align:start position:0% |
| |
| in this demonstration I'm going to show |
|
|
| align:start position:0% |
| in this demonstration I'm going to show |
| |
|
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| align:start position:0% |
| in this demonstration I'm going to show |
| you how to construct 2d brilliant zones |
|
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| 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 |
|
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| align:start position:0% |
| and hopefully achieve a better |
| |
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| align:start position:0% |
| and hopefully achieve a better |
| understanding them and what they are so |
|
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| align:start position:0% |
| understanding them and what they are so |
| |
|
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| align:start position:0% |
| understanding them and what they are so |
| a brilliant zone is an important concept |
|
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| align:start position:0% |
| a brilliant zone is an important concept |
| |
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| align:start position:0% |
| a brilliant zone is an important concept |
| in material science and solid-state |
|
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| align:start position:0% |
| in material science and solid-state |
| |
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| align:start position:0% |
| in material science and solid-state |
| physics |
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| align:start position:0% |
| physics |
| |
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| align:start position:0% |
| physics |
| alike because it is used to describe the |
|
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| align:start position:0% |
| alike because it is used to describe the |
| |
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| align:start position:0% |
| alike because it is used to describe the |
| behavior of an electron in a perfect |
|
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| align:start position:0% |
| behavior of an electron in a perfect |
| |
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| align:start position:0% |
| behavior of an electron in a perfect |
| crystal system so what is a brilliant |
|
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| align:start position:0% |
| crystal system so what is a brilliant |
| |
|
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| align:start position:0% |
| crystal system so what is a brilliant |
| zone the Brillion zone is a particular |
|
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| align:start position:0% |
| zone the Brillion zone is a particular |
| |
|
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| align:start position:0% |
| zone the Brillion zone is a particular |
| choice of the unit cell of the |
|
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| align:start position:0% |
| choice of the unit cell of the |
| |
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| align:start position:0% |
| choice of the unit cell of the |
| reciprocal lattice is defined as the |
|
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| 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 |
|
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| 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 |
|
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| align:start position:0% |
| lattice it is constructed as the set of |
| |
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| align:start position:0% |
| lattice it is constructed as the set of |
| points and closed by the Bragg planes |
|
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| align:start position:0% |
| points and closed by the Bragg planes |
| |
|
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| 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 |
|
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| align:start position:0% |
| point passing through the midpoint |
| |
|
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| align:start position:0% |
| point passing through the midpoint |
| alternatively it is defined as the set |
|
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| align:start position:0% |
| alternatively it is defined as the set |
| |
|
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| align:start position:0% |
| alternatively it is defined as the set |
| of points closer to the origin than to |
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| align:start position:0% |
| of points closer to the origin than to |
| |
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| align:start position:0% |
| of points closer to the origin than to |
| any other reciprocal lattice point the |
|
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| align:start position:0% |
| any other reciprocal lattice point the |
| |
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| align:start position:0% |
| any other reciprocal lattice point the |
| whole reciprocal space may be covered |
|
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| align:start position:0% |
| whole reciprocal space may be covered |
| |
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| align:start position:0% |
| whole reciprocal space may be covered |
| without overlap with copies of such |
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| align:start position:0% |
| without overlap with copies of such |
| |
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| align:start position:0% |
| without overlap with copies of such |
| brilliant zone okay so that was a rather |
|
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| align:start position:0% |
| brilliant zone okay so that was a rather |
| |
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| align:start position:0% |
| brilliant zone okay so that was a rather |
| convoluted definition let's do it again |
|
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| align:start position:0% |
| convoluted definition let's do it again |
| |
|
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| align:start position:0% |
| convoluted definition let's do it again |
| let's brush up and be sure that we |
|
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| align:start position:0% |
| let's brush up and be sure that we |
| |
|
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| align:start position:0% |
| let's brush up and be sure that we |
| understand this definition we're going |
|
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| align:start position:0% |
| understand this definition we're going |
| |
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| align:start position:0% |
| understand this definition we're going |
| to define her the reciprocal space the |
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| align:start position:0% |
| to define her the reciprocal space the |
| |
|
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| align:start position:0% |
| to define her the reciprocal space the |
| Bragg planes as well but it's fine both |
|
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| align:start position:0% |
| Bragg planes as well but it's fine both |
| |
|
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| align:start position:0% |
| Bragg planes as well but it's fine both |
| of these will also do a quick revision |
|
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| align:start position:0% |
| of these will also do a quick revision |
| |
|
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| align:start position:0% |
| of these will also do a quick revision |
| of what is there a record lattice so |
|
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| align:start position:0% |
| of what is there a record lattice so |
| |
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| align:start position:0% |
| of what is there a record lattice so |
| that we can go from there to the |
|
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| align:start position:0% |
| that we can go from there to the |
| |
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| align:start position:0% |
| that we can go from there to the |
| reciprocal lattice the microscopic |
|
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| align:start position:0% |
| reciprocal lattice the microscopic |
| |
|
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| align:start position:0% |
| reciprocal lattice the microscopic |
| perfect crystal is formed by adding |
|
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| align:start position:0% |
| perfect crystal is formed by adding |
| |
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| align:start position:0% |
| perfect crystal is formed by adding |
| identical building blocks so to say unit |
|
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| align:start position:0% |
| identical building blocks so to say unit |
| |
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| align:start position:0% |
| identical building blocks so to say unit |
| cells consisting of atoms and groups of |
|
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| align:start position:0% |
| cells consisting of atoms and groups of |
| |
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| align:start position:0% |
| cells consisting of atoms and groups of |
| atoms a unit cell is the smallest |
|
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| 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 |
|
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| align:start position:0% |
| component of a crystal that one stacked |
| |
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| align:start position:0% |
| component of a crystal that one stacked |
| together with pure translational |
|
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| 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 |
| |
|
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| 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 |
|
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| 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 |
|
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| 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 |
| |
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| align:start position:0% |
| Brillion zone or on its boundary |
| characterized States in a system with |
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| align:start position:0% |
| characterized States in a system with |
| |
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| align:start position:0% |
| characterized States in a system with |
| lattice period periodicity for example |
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| align:start position:0% |
| lattice period periodicity for example |
| |
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| align:start position:0% |
| lattice period periodicity for example |
| phone out or or electron state but for |
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| align:start position:0% |
| phone out or or electron state but for |
| |
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| align:start position:0% |
| phone out or or electron state but for |
| that the whole of the video thanks and |
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| align:start position:0% |
| that the whole of the video thanks and |
| |
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| align:start position:0% |
| that the whole of the video thanks and |
| this code for this demonstration was |
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| align:start position:0% |
| this code for this demonstration was |
| |
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| align:start position:0% |
| this code for this demonstration was |
| taken and editors from Mathematica |
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| align:start position:0% |
| taken and editors from Mathematica |
| |
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| align:start position:0% |
| taken and editors from Mathematica |
| demonstration since made by their slope |
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| align:start position:0% |
| demonstration since made by their slope |
| |
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| align:start position:0% |
| demonstration since made by their slope |
| cloth |
