- #1
sokrates
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Hi everybody:
I have been working on a "SIMPLE" derivation (for a long time now) and I 'll go insane if I don't get this right. I am trying to do a similar research problem - but before I attempt to change anything I'd like to make clear that I can do this right. The formulas look complicated but this is a simple perturbative analysis - and I am just not clear on some of the "definitions" the author uses. Maybe you could help me on clarifying what he means
If you have PRB access the paper is J.Z. Sun - "Spin-Current Interaction with a monodomain magnetic body : A model study" -(2001)
However, I'll repeat all the crucial steps here and present my confusion over some overly simplified derivations by Sun.
Up until Equation (11) in the paper, he writes down the LLG equation in spherical coordinates with subject to relevant anisotropic forces and the spin current.
He arrives at the following ODE system:
<br /> \frac{d \phi}{dt}=-1-{\it hp}\, \left( \cos \left( \phi \right) -\alpha\,\sin \left( <br /> \phi \right) \right) \cos \left( \phi \right) -h+{\it hs}\,\alpha <br /><br /> \frac{d\theta}{dt}= \theta \left( -\alpha-{\it hp}\,\cos \left( \phi \right) \sin \left( \phi<br /> \right) -{\it hp}\, \left( \cos \left( \phi \right) \right) ^{2}<br /> \alpha-h\alpha-{\it hs} \right)<br />
Then he solves these sets of equations for hs=0, alpha=0 to find the "unperturbed solution".
He gives these unperturbed solutions as:
<br /> \phi(t)= arctan \left[\left(\frac{\epsilon+1}{\epsilon}\right)^{1/2} cot(w_p t)\right]<br />
<br /> \theta(t)= \theta_0\left(\frac{2 \epsilon + 1 + cos(2 w_p t)}{2\epsilon + 1}\right)^{1/2}<br />
where he defines:
<br /> \epsilon = \frac{1+h}{hp} \ \ and \ \ w_p = hp \sqrt{\epsilon (1+\epsilon)}<br />
There's a redundant relation here -- but maybe useful:
<br /> \theta (t) = \frac{\theta_0^2 \epsilon}{\epsilon + cos^2(\phi(t))}<br />in implicit form. (theta_0 is the initial condition)
To apply the perturbative analysis and see how the system reacts to the perturbation, he uses the energy function which is defined as:
<br /> U(\theta,\phi)={\theta}^{2}+{\it hp}\,{\theta}^{2} \left( \cos \left( \phi \right) <br /> \right) ^{2}-2\,h<br />
Now; treating hs and alpha terms as perturbations to the original equation, uses all equations above to SOMEHOW get to the following: (which is a mystery to me!)
<br /> \left<\frac{dU}{dt}\right> = -(2 hp \epsilon \theta_0^2) hs - (2 hp \epsilon \theta_0^2) \alpha hp \left[-\epsilon (1+\epsilon) A + (1+ 2\epsilon) + \frac{hs}{hp^2} B \right]<br />
where A and B are:
<br /> A = \left<\frac{1}{\epsilon + cos^2(\phi)}\right> = \frac{2 \epsilon}{2\epsilon(1+\epsilon)}<br />
<br /> B = \left<\frac{sin(\phi)cos(\phi)}{\epsilon+cos^2(\phi}\right> = 0<br />
--> How is the averaging over U done here? He says nothing about it in the paper apart from saying it's an average. If it's a time-average over, say one, period - then it becomes extremely messy because of the way theta and phi depend on time?
In other words, how are A and B evaluated?
--> How can you arrive at the dU/dt term based on what I wrote above? I tried the obvious: take derivative of U wrt time and substitute all the terms using the theta, phi relations but it got very messy and I came nowhere close to what he has...
But since I have no idea on how he takes the average, maybe I am missing something obvious?... Even the slightest hint would be greatly appreciated!
I have been working on a "SIMPLE" derivation (for a long time now) and I 'll go insane if I don't get this right. I am trying to do a similar research problem - but before I attempt to change anything I'd like to make clear that I can do this right. The formulas look complicated but this is a simple perturbative analysis - and I am just not clear on some of the "definitions" the author uses. Maybe you could help me on clarifying what he means
If you have PRB access the paper is J.Z. Sun - "Spin-Current Interaction with a monodomain magnetic body : A model study" -(2001)
However, I'll repeat all the crucial steps here and present my confusion over some overly simplified derivations by Sun.
Up until Equation (11) in the paper, he writes down the LLG equation in spherical coordinates with subject to relevant anisotropic forces and the spin current.
He arrives at the following ODE system:
<br /> \frac{d \phi}{dt}=-1-{\it hp}\, \left( \cos \left( \phi \right) -\alpha\,\sin \left( <br /> \phi \right) \right) \cos \left( \phi \right) -h+{\it hs}\,\alpha <br /><br /> \frac{d\theta}{dt}= \theta \left( -\alpha-{\it hp}\,\cos \left( \phi \right) \sin \left( \phi<br /> \right) -{\it hp}\, \left( \cos \left( \phi \right) \right) ^{2}<br /> \alpha-h\alpha-{\it hs} \right)<br />
Then he solves these sets of equations for hs=0, alpha=0 to find the "unperturbed solution".
He gives these unperturbed solutions as:
<br /> \phi(t)= arctan \left[\left(\frac{\epsilon+1}{\epsilon}\right)^{1/2} cot(w_p t)\right]<br />
<br /> \theta(t)= \theta_0\left(\frac{2 \epsilon + 1 + cos(2 w_p t)}{2\epsilon + 1}\right)^{1/2}<br />
where he defines:
<br /> \epsilon = \frac{1+h}{hp} \ \ and \ \ w_p = hp \sqrt{\epsilon (1+\epsilon)}<br />
There's a redundant relation here -- but maybe useful:
<br /> \theta (t) = \frac{\theta_0^2 \epsilon}{\epsilon + cos^2(\phi(t))}<br />in implicit form. (theta_0 is the initial condition)
To apply the perturbative analysis and see how the system reacts to the perturbation, he uses the energy function which is defined as:
<br /> U(\theta,\phi)={\theta}^{2}+{\it hp}\,{\theta}^{2} \left( \cos \left( \phi \right) <br /> \right) ^{2}-2\,h<br />
Now; treating hs and alpha terms as perturbations to the original equation, uses all equations above to SOMEHOW get to the following: (which is a mystery to me!)
<br /> \left<\frac{dU}{dt}\right> = -(2 hp \epsilon \theta_0^2) hs - (2 hp \epsilon \theta_0^2) \alpha hp \left[-\epsilon (1+\epsilon) A + (1+ 2\epsilon) + \frac{hs}{hp^2} B \right]<br />
where A and B are:
<br /> A = \left<\frac{1}{\epsilon + cos^2(\phi)}\right> = \frac{2 \epsilon}{2\epsilon(1+\epsilon)}<br />
<br /> B = \left<\frac{sin(\phi)cos(\phi)}{\epsilon+cos^2(\phi}\right> = 0<br />
--> How is the averaging over U done here? He says nothing about it in the paper apart from saying it's an average. If it's a time-average over, say one, period - then it becomes extremely messy because of the way theta and phi depend on time?
In other words, how are A and B evaluated?
--> How can you arrive at the dU/dt term based on what I wrote above? I tried the obvious: take derivative of U wrt time and substitute all the terms using the theta, phi relations but it got very messy and I came nowhere close to what he has...
But since I have no idea on how he takes the average, maybe I am missing something obvious?... Even the slightest hint would be greatly appreciated!
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