2.2 RÉGIMEN JURÍDICO-NORMATIVO DEL TRABAJADOR
2.2.4 De la potestad disciplinaria o poder disciplinante del empleador
they are known as the complete elliptic integrals
E() = π/2 Z 0 dϕ (1− 2sin2ϕ)+12 , K() = π/2 Z 0 dϕ (1− 2sin2ϕ)−12
and D , R can be parameterized by D() = −4
E() , R() = 4 K() , ∈(0, 1) .
F.4
Phase transitions of the micromagnetic ground state
F.4.1 Numerical procedure
The procedure that is explained in F.2 can be used to identify the phase transitions of the micromagnetic model that are described in Chapter 5.2.4. In order to obtain Fig. 5.8 the magnetization is optimized and the energy E is calculated on a dense D-grid for each K-point. The crital points of the second-order transitions are detected by evaluating
d
dDE for fixed K and identifying the kinks in these curves (cf. Fig. F.2). In the vicinity of the first-order phase transitions it depends on the starting configuration whether the procedure of Chapter F.2 converges towards the global or just a local energy minimum, but the transition point can be identified by comparing the energies.
It is not advisable to localize the phase transitions by calculating order parameters like
R
dr|z| instead of ddDE , it requires very large numerical cutoffs to calculate these order parameters with sufficient accuracy.
The periodic states are calculated with periodic boundary conditions. The period length of the non-collinear states depends on K, D . Therefore, it is necessary to find the optimal period length for every fixed pair of K, D . In order to do this the energy is calculated in unit cells of different sizes and the optimal period length is determined in an iterative process. The size of the unit cell is varied by rescaling the parameters while the number of real-space lattice points is kept constant for better numerical stability.
F.4.2 Second-order transition from ferromagnetism to non-collinearity
For large |D| the ground state of the quasi one-dimensional micromagnetic model under- goes a phase-transition from the ferromagnetic to a non-collinear configuration. If the D-vector points parallel to the easy axis, this transition is second-order. An analytic expression for the corresponding critical point is deduced in this chapter.
In the vicinity of the critical point it is possible to describe the magnetization of the non-collinear state by its lowest order Taylor expansion around the ferromagnetic state. In this approximation it is straightforward to solve the Euler-Lagrange equations.
As in Chapter 5.2.4 the coordinate system is chosen such that D = D ez , Kx, Ky > Kz and as independent parameters are used
K = KKx−Kz y−Kz
, D = q −D
A (Ky−Kz) .
104 Appendix F. Details concerning the domain wall models (a) (b) (c) 1.50 1.54 1.58 0 −10 −20 0 −200 −400 E dE dD D 1.575 1.580 1.585 −380 −360 dE dD D 1.500 1.505 1.510 0.0 −0.1 −0.2 0.0 −20 −40 E dE dD D
Figure F.2: Behavior ofE and d
d DE when D is varied at fixed K. The zero point of the energy scale
corresponds to collinear magnetization parallel to the easy axis. The critical points are defined by the kinks in the (d E
d D)-curve. The transition from the truly 3-dimensional to the flat rotating
state can be localized easily (Fig. b). The transition from the collinear to the truly 3-dimensional state is less convenient from a numerical point of view: The period length of the non-collinear state depends onD and needs to be optimized for each data point. In this thesis this is done with an algorithm that is unstable in the ferromagnetic regime (where the period length is arbitrary), therefore there are no data points calculated for small D (Fig. a). But for every D there is a collinear solution of the Euler-Lagrange equations with E =d Dd E = 0 and Fig. c shows that E and
d E
d D at the same point connect continuously to this solution.
This figure is obtained forK = 0.25 . In Chapter F.4.2 it is shown that the critical point for the transition from the collinear to the non-collinear state is atD =√K+1=1.5 .
The range ofK can be restricted to K ∈(0, 1] , (cf. Chapter 5.2.4). In Cartesian coordinates the energy functional (5.3) has the form
E =
Z
dr ( ˙x2+ ˙y2+ ˙z2− D (x ˙y − y ˙x) + K x2+ y2)
where the condition x2+ y2+ z2 = 1 reduces the degrees of freedom .
ForK > 0 the D-vector points parallel to the easy axis and in the ferromagnetic state there holds x = y = 0 , z = 1 . In the ferromagnetic limit x, y can be used as independent variables. With
z =q1− x2− y2 , ˙z = p− x ˙x − y ˙y 1− x2− y2
the integrand of the energy functional can be written and expanded around x = y = 0 as L(x, y, ˙x, ˙y) = ˙x2+ ˙y2+(x ˙x + y ˙y)
2
1− x2− y2 − D (x ˙y − y ˙x) + K x 2+ y2
= ˙x2+ ˙y2− D (x ˙y − y ˙x) + K x2+ y2 +O(x + y + ˙x + ˙y)4 and the Euler-Lagrange equations become
1 2Lx = − D ˙y + K x − ¨x + O (x + y + ˙x + ˙y)3 = 0 ,! 1 2Ly = +D ˙x + y − ¨y + O (x + y + ˙x + ˙y)3 = 0 .!
If the higher-order terms are neglected these equations have the general periodic solution x = αx cos( ω r + px) , y = αy cos( ω r + py)
F.4. Phase transitions of the micromagnetic ground state 105
where the phases can be chosen such that
x = αx cos(ω r) , y = αy sin(ω r) . In order to determine α = αy
αx and ω these expressions can be inserted into the Euler-
Lagrange equations 1 2x−1Lx= − ω α D + K + ω2 = 0 1 2y−1Ly =− ω α−1D + 1 + ω2= 0 ⇒ α = ω 2+K ωD 2 ω2 =D2− K − 1 ±q(D2− K − 1)2− 4 K .
The next step is to find the area in the (D, K)-space where the last expression has a real solution for ω. This area is defined by the conditions that the radicand is positive and that 2 ω2 is positive. It is useful to split this analysis in the casesD2>K+1 and D2<K+1 .
• case D2 >K + 1 :
At first the sign of the radicand is examined:
(D2− K − 1)2− 4 K > 0 ⇔ D2>K + 2√K + 1 =√K + 12 .
Under the assumption that the radicand is positive the last expression for 2 ω2 is positive as well, i.e.
D2− K − 1 ±q(D2− K − 1)2− 4 K > 0 is fulfilled for all K >0 , D2>K+1 .
• case D2<K + 1 :
If the radicand is positive and D2<K+1 then the condition D2− K − 1 ±q(D2− K − 1)2− 4 K > 0
cannot be fulfilled if “±” is replaced with “−”, if “±” is replaced with “+” then one gets
D2− K − 1 > −q(D2− K − 1)2− 4 K ⇔ (D2− K − 1)2 < (D2− K − 1)2− 4 K . The last equation cannot be fulfilled forK >0 .
Thus it turns out that the lowest-order expansion of the Euler-Lagrange equations has a non-collinear solution in the caseK >0 if and only if
|D| > Dc= √
K + 1 .
This inequality defines the critical point for the transition from the ferromagnetic to the non-collinear state, if the non-collinear state is lower in energy than the ferromagnetic state: If |D| is smaller than Dc=
√
K+1 there cannot be any continuous transition into a non-collinear state, if|D| is just larger than Dcthe system can lower its energy by changing to the non-collinear solution. The lengthy analytical evaluation of the sign of E =Rdr L in the limit |D| & Dcis skipped since the numerical analysis shows that the non-collinear state lowers the energy. Of course one cannot completely rule out the possibility of a first-order phase transition at |D| ≤ Dcthat would prevent the system from reaching the critical point, but this has not been observed in the numerical simulations.
106 Appendix F. Details concerning the domain wall models