  A framework for understanding plane spin dynamics.

  Here do we need to do away with the notion of a ratio between up and down spins?
  How can we represent the notion of a spin wave when in general, the spin wave will me composed of multiple
  wavelengths after ANY interaction with a line of charges?  What is the "naked medium?"  For a particular
  overall wave vector imposed opon a medium of spins we require a particular ratio between the magnitudes of
  the two consistuient sets of spins so as to satisfy our requirement that the torque upon any individual
  spin be a constant of the motion as we time evolve.  This requirement is dependent upon the particular
  wave vector used, because for every wave vector that exists upon the spins we have a rotation from spin
  to spin to spin within the lattice, in multiple directions.  The effective torue felt by a particular spin
  depends only upon the relative angles of these two spins, and thus this torue will be dependent upon the
  wave vector existing upon the spins.  There are a few general trends to be aware of.  In every instance
  as we move towards long wavelength behavior the relative angle between spins decreases to zero.  This leads
  to the conclusion that we observe a return of symmetry to the system, whereby with no difference in angle,
  we can not have any distinction between a mode moving in either direction, and so the relative magnitudes of
  the spins must similarly approach the same value.  With this condition we see that the up and down spins
  have the same magnitude in this limit, as the wavelength goes to infinity.  There also exists a general trend
  whereby the relative magnitude becomes increasingly important as we move to shorter and shorter wavelengths.
  For the moment I am clearly ignoring the relative phases, which only come into play when we have additionally
  something that distinguishes the interaction with a grouping of spins to another grouping of spins.  In this
  case we clearly have a larger unit cell, and have more degrees of freedom to worry about, but the physical
  situation must additionally take the phase into account because we have another symmetry of the system which
  is disturbed by the presence of a higher level organizational pattern.  For the moment we should only consider
  general trends among spins within a medium that has the simple binary relationship spoken of.  For the moment
  I will entirely neglect these charge inhomogenieties however.  With a simple medium containing only up and
  down spins we observe that we must divide up the lattice into sets of up and down spins.  Each with a
  characteristic magnitude of tilting into the x and y plane, in which the spins precess.  We can satisfy the
  equations of motion of the system by then creating an equality between the spins in terms of their precessional
  frequency.  This is really the only constraint that we have upon the system.  The idea is that the individual
  parts of the system do not move RELATIVE to each other.  They move as a whole.  They are caught in lock-step
  destined to repeat their current behavior for all future time.  This makes the situation simple, and takes out
  any sort of time dependence.  All the spins are interacting with all of their neighbors individually, but it
  is almost as if it does not matter.  It poses no significant change to the system.  The system is self-preserving.
  This is of fundamental importance to our simple analysis of the system.  A problem, however, is that is does
  not leave room for consideration of time dependent interactions.  Indeed, it really doesnt allow any room
  for interaction whatsoever.  To take this type of thing into consideration we need to move to a new overall
  view of the system, which incorporates the new features of the system, somehow in the framework of the old
  system.  Or, we must change the fundamental nature of the system we have initially considered.  In this
  instance it seems that we do not need to entirely throw out our old system, but instead can expand upon it.

  We must first ask ourself what the purpose of the new system is, and whether our method of expanding the old
  system is rationalized.  What are the steps we can make to incorporate the new features?  Making giant leaps
  leads only to confusion and mistakes, and blurs our original motives.  Each new addition should correspond to
  something we MUST take into account in order to create the new system.  So, that being said, what can we do to
  modify the existing model?  Let us try to rationalize in this way.  Suppose the original system exists in a
  time independent state, whereby it is stuck within a particular mode.  The up spins have one magnitude and
  the down spins have another.  They have a relative phase which does not change from spin spin.  However, for
  each mode we could place upon the system there exists a DIFFERENT relative magnitude and DIFFERENT relative
  phase between spins.  I will now make the assumption, that if something perturbs the system, we can consider
  the new state of the system to be some superposition or addition, or conglomeration of the old modes of the
  system, which some minor modifications localized around the added disturbance.  We have a problem however,
  because now, in general, it seems that different parts of the waves which were indistinct with respect to
  any set of underlying spins (there is merely a net rotation for the local set of spins when comparing spins
  near a 'crest' and those near a 'trough', etc), may now have something different about their character for
  different 'parts' of the wave.  Is this a problem?  If the wave is interacting with a surface, are all parts
  of the wave to be treated equally?  Where is the preservation of this kind of symmetry?  What is the symmetry?
  In an optical situation we can assume that the phase does not matter in the idealized case of a light wave
  impingent upon a surface whereby it is reflected and refracted.  All parts of the wave have equal weight.
  Can we still make this assumption for spin waves incident on a line of charges?  For the moment let's remove
  the spins on the other side of the line of charges, as these introduce a change in the nature of symmetry
  as we move across pure AF region of the lattice, to moving across the line of charges to the next AF region
  in the lattice.  This makes the CONDITION that must be satisfied at the boundary complicated and obscures
  our ability to deciper the conditions on our set of solutions which will give an idea of what is going on.

  However, I must first get back to one point.  Why is is rationalized that our original set of solutions can
  act as a BASIS for expansion of our new solution.  In general the new solution may be heavily time dependent.
  But, for any particular INSTANT in time, the system will need to satisfy our conditions on solutions of the
  system.  This may be too heavy of an assumtion, but it seems to be the only constructive one we can make at
  the moment.  Additionally, this simplifies our analysis, because we can now look at individual snapshots of
  the system subject to some conditions that lead to a net solution that should satisfy the system for future
  times.  We really have two conditions here.  That the system satisfy our conditions for any instant of time,
  and that if it satisfies the system for one instant, it will for the next instant.  Hopefully, in a way which
  is periodically repeating for all time.  The infinitesimal and the whole.

  The other type of interaction we can look at is that of a one time 'collision' of a spin wave with the line
  of charges.  This is the type of interaction that would more likely occur in my mind.  As it has the character
  of not really bringing infinite quantities into play.  The spin waves we have so far considered are infinite
  in extent, exist for all time, and are of a nature that requires we look at all spins within a particular
  set of spins in the same manner.  I don't like this notion, because it seems unphysical and unlikely.  Instead,
  I would rather consider a sort of 'packet' of spin excitation which 'strikes' the line of charges and creates
  some sort of reaction.  Currently I have no place to consider the actual interaction of the spin wave with the
  line of charges, since I don't know what would govern the dynamics.  But, some part of the wave will react as
  if the charges weren't there, and some part will interact with them.  Initially, we can therefore at least look
  at what would be the interaction of the spin wave with a changing value of J.  This presents an interesting
  problem in and of itself, for it illustrates the basic notion of how spin waves can be affected by changes in
  the underlying medium on which the waves are propogating.  In addition, we will expect to observe some
  properties which are analogous to optical refraction and reflection at an interface.  As we will see, they
  are quite similar.  Spin waves also automatically behave as bosons.  So I ask, if we have a spin wave trapped
  in an AF medium, between two lines of charge, and it acts like a boson, don't we have a spin wave laser?
  And further, if the action of this spin wave laser is to create spin wave coherence, and these spin waves
  interact with the walls of the compartment, where we have other electrons, free to move, and delocalized,
  can these electrons not further interact by way of bosonic behavior, linked across the sample, from line to
  line, by spin waves?  Its not that complicated, and seems quite beautiful.  Lets hope I can sort this out.

  The nonlinear behavior and the addition of two waves are one in the same.  The waves do not add as simple
  linear superpositions, rather, they are of a more complicated nature.  A single wave obeys a scalar
  multiplication property, as we can assume from the differential equations governing the motion of the system.
  However, I am not sure how precise this statement is, or how valid it is to assume in the given situation.
  Our solutions are such that the torques are confined to xy planes within which the motion of the spins occurs.
  This is a classical view of the situation, however, the spins really do transform quantum mechanically as
  their classical counterparts, so our assumption that they behave as VECTORS is valid.  Once we have made this
  assumption, we can perform vector operations on the spins and transform them in this manner to obtain a
  description of the system.  The spins individually are then seen as moving on cones of varying radius at the
  'base'.  This value is the sigma I have used for the spins.  One value of sigma is related to other values
  by a ratio, which is found as an exact solution to our initial condition (boundary) that the spins all
  precess in the xy plane, with a constant angular frequency.  This is our only constraint.  It additionally
  means that the spins must satisfy certain properties.  We find three things as a result of constraint.  Before
  delving into what these are, I would like to clarify the constraint, and identify the precise structure.
  First and foremost, we require that the spins move in the xy plane.  In spherical coordinates, which are a
  much more natural coordinate system for the problem, we find that this means the spins (which are also of
  fixed magnitude 1) the projection in the xy plane is of fixed magnitude.  Therefore the polar angle is fixed.
  Then, we have a statement from the torques, which is an equivalent ratio among the spins, stating that for
  every spin, the torque per unit xy plane projection is fixed.  AND, the torque is in the xy plane, so the
  z component of the torque is fixed to zero.

           ==> torques (global restrictions - all spins equivalent)
               z magnitdue of torque fixed to zero
               xy magnitude of torque / xy magnitude of spin = constant for all spins      **MOST IMPORTANT
           therefore torque magnitudes are fixed,

           ==> spins (individual restrictions - sets of spins equivalent)
               z magnitude of spin fixed
               xy magnitude of spin fixed
           therefore spin magnitudes are fixed

  What about phases of the spins?  These come out as a RESULT of our conditions on the system in our process
  of solving the system, however, it seems to be inconsistent that we only seem to be able to take the
  'magnitude-centric' perspective of solving the system.  In reality, we should for example be able to choose
  phases as a starting point for our constraint.  For if they are implied by our constraint on the magnitudes
  then there is a mapping in some sense between the sets of constraints, we should be able to start with a
  constraint on the phases and move to that of the magnitudes.  I feel I am fooling myself however, because
  there is another piece of information here, and that is the mapping between the constraints.  Ask, for instance
  what makes these two hypothetical constraints (which must come out of the problem) equivalent?  There seems
  to be a hidden constraint.  And yet, we have shown we can solve the system for at least 2 spins with differing
  values of Ja and Jb.  Is this the symmetry of the system, and if so, where does this constraint come from,
  and how can we understand it.

  The above seems to be a compatible set of restrictions, in that, if we were to relax the condition on the xy
  magnitude for the spins, we would find ourselves immediately having to relax the condition on the z magnitude
  of the spins, and then also the xy magnitude of the torques.  Given the torque equation is from a cross product,

  torque = det[ ^i    ^j    ^k
                x1    y1    z1
                x...  y...  z... ]

  This would then require that all x, y, and z components of spins would be variable.  This could be additionally
  solvable, but increases the complexity of the problem dramatically.  Our current conditions side step this
  complication but without introducing any approximation along the way.  They are a way for us to look at only
  the xy plane behavior of spins...  Which happen to themselves only be interacting in an xy plane.  There seems
  to be a natural progression then from the behavior we are examining on the smallest scale to the larger scale.

  What would a superposition of these waves look like for something like a guassian packet?  This is a kind
  of fourier transform, but since solutions aren't superpositions of each other I don't believe we can simply ADD
  given preexisting solutions to produce new ones... (Need to prove this not just say it.)  And, to have something
  like a Fourier decomposition we must have a set of solutions from which to construct new solutions.  This is 
  really the overarching idea.  Compose new solutions from old solutions.  But how do we add the old solutions.
  Erica Carlson suggested that I add the spin sigma values with phase information, as if they were vectors.  We can
  afterwards solve to find the z components which keep the spin an overall magnitude of 1. This clearly works for
  consideration of one solution.  Adding a given solution to itself will still be a solution.  This is the scalar
  property of solutions mentioned before, and will certainly hold true.  Is there anything wrong with this method
  for additions of different solutions, however?  And if there is, how can we show it?  

  Perhaps we can take one trivial example as a demonstration.  We can make three types of combination.  Waves of 
  equal magnitude wavelength in different directions, waves of different magnitude wavelength in the same direction,
  and the general case, waves of different wavelength in different directions.  If a wave carries a certain
  momentum one would expect that the addition of two waves would conserve the momentum.  This momentum is in fact
  defined by the wavelenth (k vector) of the wave, and is in proportion to it (linearity).  So, we expect for two
  waves in the positive x direction with k1 and k2, the sum of the waves should simply have k = k1 + k2.  We now
  want to see if this is consistent with solutions to the equations for our system.  




  There will still however be a certain type of trend.  For a given medium as we increase wavelength
  the condition on the ratio between the magnitudes of the spins (just up and down, here, and don't need to
  consider phases?) changes.  The trend is that for longer wavelength, the ratio is more towards 1.  For shorter
  wavelengths the ratio is closer to zero or infinity.  We will take the convention that all ratios we talk about
  must be between 0 and 1 by taking all sigmas as compared with the largest magnitude sigma given the value of 1,
  to avoid infinity. 




































