Check LLG Solution and Approximation

1. Check LLG Solution and Approximation Huanlong

2. Equations Integration LLG: Exact solution: Our Approximation : Sun Approximate : Japanese Approximation :

3. Results I: integration from LLG E: Exact solution A1: Our approximate solution A2: Japanese approximation AS: J. Sun’s approximation

4. Results

5. Results

6. Conclusion • Exact solution is the same as numerical integration of LLG equation. • Japanese approximation is always better(almost the same as the exact solution for these small initial angles) than our approximation, since it keeps more terms. • Compare J. Sun’s approximation with ours.

7. J. Sun’s approximation vs. ours • Both approximations ignore the second term, which yields a largerτ. • Sun’s approximation also change tan(θ/2) to θ/2 • tan(x) < x for all 0 < x < π / 2 • the difference between tan(x) and x increases as x increases. Which will reduce the value of τ. • Need to compare the increase of τ(both) with the decrease of τ(Sun). τ Ours tan(x) –>x Sun’s no 2nd term Exact solution Approximations tan(x) –>x Sun’s

8. Compare Approximations Time(time difference between the exact solution and approximations) to reach the equator as a function of initial angles Small overdrive: Sun’s is better than ours

9. Compare Approximations Time(time difference between the exact solution and approximations) to reach the equator as a function of initial angles large overdrive: Ours is better than Sun’s

10. Compare Approximations • τE: switching time from the exact solution. • τO: switching time from our approximation. • τS: switching time from Sun’s approximation. • Sun’s approximation is better for i < 6 • Ours approximation is better for i > 6. • In experimental condition, Sun’s approximation is better. Doesn’t change much with initial angles Sun’s is better Ours is better

11. Results

12. Results

13. Results