Applied Physics Department Fractional Domain Wall Motion Wesam Mustafa Al-Sharo'a Dr. Abdalla Obaidat May, 23, 07

Ferromagnetic materials are divided into a number of small regions called domains. Each domain is spontaneously magnetized to the saturation value, but the directions of magnetization of the various domains are different. Applying an external magnetic field convert the specimen from multi-domain state into one state in which it is a single domain

Domain walls are interfaces between regions in which the spontaneous magnetization has different directions, and then the magnetization must change direction by domain wall motion.

The equation of motion of the wall per unit area is: Where X describes the position of the wall Theory

The fractional calculus is defined by the derivative of order of a function The fractional equation of motion of one dimensional simple harmonic oscillator is: Where is a non integer 0 1

Case one: By using the Laplace transform relations, we get, Let with:

To simplified with the aid of the geometric series as : Then rewriting and as follows

Similarly After that

If and f (t) =0, the summation will vanish except if m=0 = The exact solution when is

-Case two: Following the same procedure as in case one, we get The solution is

-When, the above equation reduce to : -And when -

The equation of wall motion was solved by assuming, and then it was plotted for different values of ranging from 0.1 to 1 for and as shown in Figure (1) and Figure (2) respectively : Results and Discussions:

Figure (1): shows X as a function t of for and various value of ranging from 0.1 to 1, with, and

Figure (2): shows X as a function of t for and various value of ranging from 0.1 to 1

The equation of wall motion was solved Figure (3) and (4) has been plotted for different values of ranging from 0.1 to 1 for and

Figure (3): shows X as a function of t for and various value of Ranging from 0.1 to 1, with

Figure (4): shows X as a function t of for and various value of ranging from 0.1 to 1

The following two figures show the variation of the equation when an impulse driving force is applied : Figure (5) :Shows X(t) as a function of t for and = 0.1, 0.5, 0.7, 2 and 5

Figure (6) :Shows X(t) as a function of t for and = 0.1, 0.5, 0.7, 2 and 5

Conclusion It is concluded that the series solutions of the equation of motion of the wall is calculated by fractional analysis with a regular oscillatory behavior. The same results, when we affect two kinds of forced oscillator on the system, and the Figures show the series solution as a function of for =0.1, =0.5, and various of ranging from 0.1 to 1

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