Now we analyze magnetization process in a film with the stripe domain structure (DS) neglecting the domain wall width, see Fig.2. The total volume energy density of DS is the sum of the domain wall, Zeeman and demagnetizing energies:
For periodical stripe domains with domain size D (means size of a domain magnetized opposite the applied field) the two first terms in Eq.8 can be easily obtained from a geometrical consideration. So, Eq.8 now becomes
with m= (D+-D–)/p. The demagnetizing energy is a rather complicated function and it can be represented as a series of its parameters (see the third term in Eq. A1, see the Appendix). The energy (Eq. 1.9) has two independent parameters describing domain geometry (e.g. m and p or D+ and D–) which can be used for EDS minimization. We choose m and p. So, the equilibrium normalized magnetization and DS period are defined by the two equations:
The explicit shape of Eqs. (1.10) and (1.11) one can find in the Appendix. Here we discuss the numerical solution of the system of equations (1.10) and (1.11). They are plotted in Fig. 1.3-1.5. In Figs. 1.3 and 4 we plot the film magnetization and inverse reduced domain period (p/d) vs H/4 π MS for different values of the ratio between the characteristic length and film thickness.
Using Eqs. 1.10,.11 one can obtain the useful, for experiment analysis, description of: (i) domain period p0 for H/4π M S <<1 and (ii) the initial susceptibility χ (=dM/dH H→0). In zero-field case the domain period is determined by the magnetostatic and domain wall energies competition and the equilibrium p0 can be easily calculated from the equation (see the Appendix):
where F(p0/d) is the function defined in the Appendix and plotted Fig. 1.5.
Knowing the material film parameters lc and d one can calculate the zero-field domain period with the help of the plot shown in Fig. 1.5, and contra versa. Solving numerically Eq.12 we find the equilibrium domain period as a function of the film thickness, see Fig. 1.6. It is important to note that the equilibrium domain period reaches its minimum at the certain film thickness which real meaning depends on the characteristic length of material. In the limiting case – large film thickness – the domain period grows proportionally to d1/2 as it is predicted by the above simplified theory (see Eg. 1.3). The Fig. 1.6 allows us compare the results obtained by rigorous calculations (Eq. 1.12) and the simple model (Eq. 1.3).
The initial magnetic susceptibility is given by χ = dM/dH =(1/4 π ) dm/dh and can be found by differentiating of Eq.(A1) with respect to h
Thus, Eq. 1.13 gives the initial susceptibility of stripe domain structure as a function of domain period (see Fig. 1.7). Passing in Eq.(1.13) to the limit p0→0 (which is a case of smallest domain wall energy and a dense domain structure, e.g. because of low anisotropy constant) one can arrive at χ0=1/4 π. The same result was previously obtained from Eq. 1.6 on the basis a simplified model. In the opposite limit, when p→∞ (or p>>d) Eg. 1.13 gives that χ→∞. Indeed, as it seen from Fig. 1.3 the susceptibility becomes infinity large before the saturation (the jumps on the m(H)-curves). Obviously, after saturation χ=0. Notice, the right side of Eq. 1.13 only includes geometrical parameters of the film and domain structure: d and p. In general, the magnetic susceptibility is affected by many factors, e.g. coercivity. The last may be a reason for a difference between values of χ measured from the virgin magnetization curve (or magnetization hysteresis loop) and determined from a DS image. Calculations of χ from analysis of domain structure images is more easy with the help of another option our experiment – registration of magnetic hysteresis loop .
One can use these figures to determine the characteristic length and the domain wall energy of a given film. Indeed, using these plots (Figs. 1.3 -1.6) and utilizing the measured domain period and domain sizes at different values of the magnetic filed it easy to determine with the help of the above given theory the material parameters: MS 