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The trip produced by these seeds is not unlike mushrooms, and can be quite intersting, but before one undertakes such an adventure, they must ask themselves if they are willing to deal with nausea the likes of which one never expects to experience from a small amount of seeds.
While waiting on line for Tool tickets, I consumed about 15-20 seeds in front of Tower Records. I had eaten a sandwhich minutes before, and thought that the meal would temper the nausea. After about fifteen minutes, I began to feel disconnected from my body, and the floor/ground began to move in waves. Like a bullet came the nausea, which became so severe that I knew, if my mouth were left open for more than threee seconds, I would have bathed the entire train car (as well as my friends) in vomit. Colors were brighter than usual, but I ws unable to appreciate this due to my sickness.
I tried to eat again, about a half hour after taking the seeds, but I had difficulty keeping food down. I sat in the park and rested my head on my lap, adjusting to the nausea and noticing an orgasmic buzz flowing throughout my body. I soon became exhausted, and knew that I would pass out if I did not lift my head. The colors were now ten times brighter, and everything had a green glaze to it. My mood and mindset remained unaffected.
At work (three hours later), I was hit by the second wave. My movements and perception became slowed, and I felt as if my every joint were having an orgasm. A sense of giddiness overtook me, and I would laugh childishly at seemingly unfunny things. This state lasted for fifteen minutes, and then I felt as I did prior to taking the seeds.
I ate more with a friend four hours later, and I experineced the same symptoms as before. the nausea was non-existant this time, and the trip more severe. I tried to purchase movie tickets, and found myself unable to interact with the woman at the ticket window. My thoughts were changing quickly and I could not fucus on what she said. My friend was beset with extreme nausea and pain. I soon noticed a sense of paranoia, and I was easilly frightened. I felt uncomfortable in my own body and with my own senses. She threw up five hours later, and the pain subsided. She had slight visual hallucinations, and went to sleep talking about the skulls in the room (?)
At best these seeds provide a dark-edged trip similar to mushrooms, and at worst leave raving and riddled with pain. Caveat Emptor.
Experience Reports are the writings and opinions of the individual authors who submit them.
Some of the activities described are dangerous and/or illegal and none are recommended by Erowid.


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--- layout: page title: Micromagnetism position: 2 --- {% raw %}
This chapter is in part taken from Abert (2013) .
| Model | Description | Length Scale |
|---|---|---|
| Atomic level theory | Quantum mechanical ab initio calculations | \(< 1\,\text{nm}\) |
| Micromagnetic theory | Continuous description of the magnetization | \(1-1000\,\text{nm}\) |
| Domain theory | Description of domain structure | \(1-1000\,\mu\text{m}\) |
| Phase theory | Description of ensembles of domains | \(> 0.1\,\text{mm}\) |
Ferromagnetic materials from the viewpoint of theoretical physics are most accurately described by the theory of quantum mechanics. In this theory the ferromagnet is described by a \(N\) -body problem whose complexity grows exponentially with the number of involved bodies \(N\) Thouless (1961) . Thus analytical calculations in this framework are restricted to very small systems.
Different models have been proposed in order to approximately describe ferromagnetic materials on a macroscopic scale. Depending on the simplifications introduced by a particular model it is able to describe the system accurately only under certain assumptions and on a certain length scale. Table 1 gives an overview of established models for the description of ferromagnets on different length scales.
For the description of ferromagnetism on the micron scale the theory of micromagnetism has proved to be a reliable tool. In contrast to domain theory it is able to resolve the inner structure of domain walls. On the other hand the micromagnetic equations can be solved numerically for relatively large system compared to atomistic approaches.
This chapter is organized as follows. In Sec. 1 the assumptions and simplifications of the micromagnetic model are discussed. Section 2 provides an overview over the different energy contributions of a ferromagnetic body. In Sec. 3 the Landau-Lifshitz-Gilbert equation is introduced, which describes the magnetization dynamics in micromagnetism. Section 4 and 5 discuss extensions, limits and solutions of the micromagnetic equations.
Magnetic matter basically consists of magnetic dipoles called elementary magnets. These dipoles can be identified with spins and orbital angular momentum of charges on the atomic level. Macroscopic magnetic properties are consequently derived from the interaction and superposition of these dipoles.
In the special case of ferromagnetic materials, electrons with overlapping wave functions favor a parallel spin alignment due to the so-called exchange interaction, see Sec. 2.1. The alignment of elementary magnets \(\boldsymbol{m}_i\) at places \(\boldsymbol{r}_i\) can thus be assumed to be locally almost parallel
\[\boldsymbol{m}_i \approx \boldsymbol{m}_j \quad \text{for} \quad |\boldsymbol{r}_i - \boldsymbol{r}_j| < \lambda \label{eq:neighbors_equal}\]
where \(\lambda\) is a measure for the range of the exchange interaction, called the exchange length. Further a homogeneous density of elementary magnets is assumed. Taking into account these assumptions the discrete distribution of magnetic moments \(\boldsymbol{m}_i\) can be well approximated by a continuous vector density \({\boldsymbol{M}_{}(\boldsymbol{r})}\) such that
\[\int_\Omega {\boldsymbol{M}_{}(\boldsymbol{r})} \,\text{d}\boldsymbol{r}\approx \sum_i \mathbb{1}_\Omega (\boldsymbol{r}_i) \boldsymbol{m}_i \label{eq:discrete_to_continuum}\]
holds approximately for volumes \(\Omega\) of the size \(\lambda^3\) and bigger. It is important to understand that the existence and the range of the exchange interaction is crucial for a good approximation in Eqn. \(\text{\ref{eq:discrete_to_continuum}}\) . The vector field \(\boldsymbol{M}\) is called magnetization. Due to the homogeneous density of elementary magnets it has a constant norm
\[{\boldsymbol{M}_{}(\boldsymbol{r})} = M_{\text{s}} \cdot {\boldsymbol{m}_{}(\boldsymbol{r})} \quad \text{with} \quad | {\boldsymbol{m}_{}(\boldsymbol{r})} | = 1. \label{eq:normalized_magnetization}\]
where \(M_{\text{s}}\) is called the saturation magnetization. In the following the unit vector field \(\boldsymbol{m}\) which should not be confused with the moments \(\boldsymbol{m}_i\) will often be used instead of \(\boldsymbol{M}\) .
The continuous magnetization field is a common parameter in classical electrodynamics Jackson (1999) . Micromagnetism extends the classical field theory by non classical effects such as the exchange interaction. These effects are expressed in the framework of a continuum theory, see Sec. 2. Moreover, micromagnetism describes the dynamics of the magnetization field by the Landau-Lifshitz-Gilbert equation, see Sec. 3. Due to this combination of classical field theory and quantum mechanics, micromagnetism is often referred to as semi-classical continuum theory.
The total energy of a ferromagnet with respect to the magnetization is influenced by a multitude of physical effects. While some of these effects have a classical description, like the demagnetization energy and the Zeeman energy, others have a quantum mechanical origin and have to be adapted to the continuum theory of micromagnetism, e.g. the exchange energy and the anisotropy energy. By finding local minima of the energy functional, stable magnetization configurations can be obtained. However, the total energy does also play an important role for the dynamics of a ferromagnet as will be seen in Sec. 3.
The characteristic feature of ferromagnetic materials is the existence of spontaneous magnetization. From classical electrodynamics it is known that neighboring spins energetically favor an antiparallel alignment Jackson (1999) . Consequently macroscopic magnetic bodies are expected to avoid a uniform magnetization, which in fact is the case for paramagnetic and diamagnetic materials.
However, the elementary magnets in ferromagnetic materials are subject to the so-called exchange interaction. This quantummechanical effect leads to an energetically favored parallel alignment of neighboring spins and thus to macroscopic uniform magnetization configurations.
Here we will use the classical Heisenberg Hamiltonian for two neighboring spins as starting point to derive a micromagnetic expression for the exchange energy
\[E_{i,j} = - J \, \boldsymbol{S}_i \cdot \boldsymbol{S}_j \label{eq:energetics_heisenberg}\]
where \(J\) is the so-called exchange integral and \(\boldsymbol{S}_i\) and \(\boldsymbol{S}_j\) are two neighboring classical spins. With the magnitude of the spins \(S = |\boldsymbol{S}_i| = |\boldsymbol{S}_j|\) und the unit vectors \(\boldsymbol{n}_i = \boldsymbol{S}_i / S\) and \(\boldsymbol{n}_j = \boldsymbol{S}_j / S\) this can be written as
\begin{align} E_{i,j} &= - J \, S^2 \, \boldsymbol{n}_i \cdot \boldsymbol{n}_j \label{eq:exchange_discrete} \\ &= - J \, S^2 \, [ 1 - \frac{1}{2} (\boldsymbol{n}_i - \boldsymbol{n}_j)^2 ]\end{align}
The exchange energy of a magnetic body is calculated by summing up
\[E = \sum_{i,j} - J_{ij} \, S^2 \, [ 1 - \frac{1}{2} (\boldsymbol{n}_i - \boldsymbol{n}_j)^2 ]\]
where \(J_{ij}\) is the exchange integral for spins \(i\) and \(j\) . That means \(J_{ij} \neq 0\) only for exchange coupled, usually neighboring, spins. In the theory of micromagnetism this expression has to be adapted to the continuous magnetization field \(\boldsymbol{m}\) . The scalar product in Eqn. \(\text{\ref{eq:exchange_discrete}}\) is analogously given by
\[{\boldsymbol{m}_{}(\boldsymbol{r})} \cdot \boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r}) = 1 - \frac{1}{2} [{\boldsymbol{m}_{}(\boldsymbol{r})} - \boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r})]^2 \label{eq:exchange_continuous_1}\]
where \(\Delta \boldsymbol{r}\) is chosen as distance vector between two exchange coupled magnetic moments. Expanding \(\boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r})\) in \(\Delta \boldsymbol{r}\) up to first order and inserting into Eqn. \(\text{\ref{eq:exchange_continuous_1}}\) yields
\[{\boldsymbol{m}_{}(\boldsymbol{r})} \cdot \boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r}) \approx 1 - \frac{1}{2} \sum_i (\Delta \boldsymbol{r} \cdot \boldsymbol{\nabla}m_i)^2. \label{eq:exchange_continuous_2}\]
The energy for a magnetic body is obtained by summing up contributions from different distance vectors \(\Delta \boldsymbol{r}_i\) depending on the crystal structure and integration over the magnetic body
\[E = \int_\Omega \sum_i A_i \, {\boldsymbol{m}_{}(\boldsymbol{r})} \cdot \boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r}_i) \,\text{d}\boldsymbol{r}\]
where \(A_i\) are called exchange constants and include the exchange integral and the magnitude of the spins involved. Inserting Eqn. \(\text{\ref{eq:exchange_continuous_2}}\) yields the general expression
\[E = C + \int_\Omega \sum_{i,j,k} A_{jk} \frac{\partial m_i}{\partial x_j} \frac{\partial m_i}{\partial x_k} \,\text{d}\boldsymbol{r}\label{eq:exchange_continuous_3}\]
for the exchange energy. The constant \(C\) results from the integration of the constant term of Eqn. \(\text{\ref{eq:exchange_continuous_2}}\) and can be omitted without changing the physics of the system. Here \(A_{jk}\) is a matrix of exchange constants. By rotation of the coordinate system this matrix can be diagonalized Döring (1966) which yields
\[E = \int_\Omega \sum_{i,j} A_{j} \left( \frac{\partial m_i}{\partial x_j} \right)^2 \,\text{d}\boldsymbol{r}. \label{eq:exchange_continuous_4}\]
In the case of cubic and isotropic materials the exchange constant does not depend on the spatial dimension und thus Eqn. \(\text{\ref{eq:exchange_continuous_4}}\) further reduces to
\[E = A \int_\Omega \sum_{i} (\boldsymbol{\nabla}m_i)^2 \,\text{d}\boldsymbol{r}= A \int_\Omega (\boldsymbol{\nabla}\boldsymbol{m})^2 \,\text{d}\boldsymbol{r}. \label{eq:exchange_continuous_final}\]
This expression turns out to accurately describe most materials and is usually used in micromagnetics. The exchange constant \(A\) is determined experimentally.
Note that this result was derived from the classical Heisenberg model that assumes localized spins. In metallic ferromagnets the spins are not localized and the Heisenberg model does not apply. However, Eqn. \(\text{\ref{eq:exchange_continuous_final}}\) still describes the exchange interaction phenomenologically up to first order Hubert and Schäfer (1998) .
The demagnetization energy, also called magnetostatic energy or stray-field energy, is the energy of the magnetization in the magnetic field created by the magnetization itself. This means that this energy contribution accounts for the dipole–dipole interaction of the elementary magnets.
The demagnetization energy can be derived from classical electromagnetics. Maxwell’s equations for electrostatics, assuming a vanishing current \(\boldsymbol{J}\) , are given by
\begin{align} \boldsymbol{\nabla}\cdot \boldsymbol{B} &= 0 \label{eq:demag_zero_div} \\ \boldsymbol{\nabla}\times \boldsymbol{H} &= 0 \label{eq:demag_zero_curl}\end{align}
where the magnetic flux \(\boldsymbol{B}\) is connected to the magnetic field \(\boldsymbol{H}\) via the magnetization \(\boldsymbol{M}\)
\[\boldsymbol{B} = \mu_0 (\boldsymbol{H} + \boldsymbol{M}).\]
Equation \(\text{\ref{eq:demag_zero_curl}}\) is equivalent to the magnetic field \(\boldsymbol{H}\) being the gradient of a scalar potential \(u\) . Hence the demagnetization-field is the solution to the system
\begin{align} \Delta u &= \boldsymbol{\nabla}\cdot \boldsymbol{M} \label{eq:demag_poisson} \\ \boldsymbol{H} &= - \boldsymbol{\nabla}u. \label{eq:demag_scalar_potential}\end{align}
The boundary condition to this system is given in an asymptotical fashion by
\[{u_{}(\boldsymbol{r})} = \mathcal{O}(1 / |\boldsymbol{r}|) \text{ for } |\boldsymbol{r}| \rightarrow \infty. \label{eq:demag_open_boundary}\]
This condition states that the potential drops to zero at infinity. It is often referred to as open boundary condition. Equation \(\text{\ref{eq:demag_poisson}}\) is Poisson’s equation and can be solved with the well known Green’s function of the Laplacian, which naturally satisfies the boundary condition \(\text{\ref{eq:demag_open_boundary}}\) , see Jackson (1999)
\[{u_{}(\boldsymbol{r})} = - \frac{1}{4 \pi} \int \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' \label{eq:demag_whole_space}\]
where the integration is carried out over the whole space. In the case of an ideal magnetic body the magnetization is only defined on a finite region \(\Omega\)
\begin{align} |{\boldsymbol{M}_{}(\boldsymbol{r})}| = \left\{ \begin{array}{cl} M_{\text{s}} & \text{if } \boldsymbol{r} \in \Omega \\ 0 & \text{else} \end{array} \right.\end{align}
which leads to a discontinuity of \(\boldsymbol{M}\) on the boundary \(\partial \Omega\) .
Figure 1: Sketch of the limiting procedure used for the derivation of the boundary term in the integral solution of the scalar potential. The discontinuity of the magnetization across the sample boundary \(\partial \Omega\) is smoothed out in a region \(\Omega_{\text{d}}\) surrounding the sample.
In this case the solution of Eqn. \(\text{\ref{eq:demag_whole_space}}\) can be obtained by a limiting process. Consider a finite region \(\Omega_{\text{d}}\) that surrounds the magnetic body \(\Omega\) . Further we assume a smooth decay of the magnetization within \(\Omega_{\text{d}}\) , see Fig. 1. With the magnetization and its divergence vanishing in the outside region \(\mathbb{R}^3 \backslash \Omega \cup \Omega_{\text{d}}\) Eqn. \(\text{\ref{eq:demag_whole_space}}\) can be written as
\[{u_{}(\boldsymbol{r})} = - \frac{1}{4 \pi} \left[ \int_{\Omega} \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' + \int_{\Omega_{\text{d}}} \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' \right]. \label{eq:demag_split}\]
Applying Green’s theorem to the integral over \(\Omega_{\text{d}}\) yields
\[\int_{\Omega_{\text{d}}} \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' = \int_{\partial \Omega_{\text{d}}} \frac{{\boldsymbol{M}_{}(\boldsymbol{r}')} \cdot \boldsymbol{n}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{s}' - \int_{\Omega_{\text{d}}} {\boldsymbol{M}_{}(\boldsymbol{r}')} \cdot \boldsymbol{\nabla}' \frac{1}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' \label{eq:demag_green}\]
where \(\,\text{d}\boldsymbol{s}'\) is the area measure to \(\boldsymbol{r}'\) and \(\boldsymbol{n}\) is the unit outward normal. In the limit of a rapidly decaying magnetization \(\boldsymbol{M}\) the region \(\Omega_{\text{d}}\) can become infinitely small without changing the result of Eqn. \(\text{\ref{eq:demag_green}}\) . In this case the right-hand side of Eqn. \(\text{\ref{eq:demag_green}}\) reduces to the boundary integral since the volume integral has a finite integrand and is carried out over an infinitely small volume. The boundary \(\partial \Omega_{\text{d}}\) consists of an inner and an outer boundary. The integral over the outer boundary vanishes, because of a vanishing magnetization \(\boldsymbol{M}\) . The inner boundary coincides with the boundary of the magnetic body \(\partial \Omega\) except for the orientation. Thus the boundary integral can be replaced by an integral over \(\partial \Omega\) with opposite sign. Inserting into Eqn. \(\text{\ref{eq:demag_split}}\) results in
\[{u_{}(\boldsymbol{r})} = - \frac{1}{4 \pi} \left[ \int_\Omega \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' - \int_{\partial \Omega} \frac{{\boldsymbol{M}_{}(\boldsymbol{r}')} \cdot \boldsymbol{n}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{s}' \right]. \label{eq:demag_jackson}\]
The expressions \(\rho = - \boldsymbol{\nabla}\boldsymbol{M}\) and \(\sigma = \boldsymbol{M} \cdot \boldsymbol{n}\) are often referred to as magnetic volume charges and magnetic surfaces charges respectivly. An alternative expression to Eqn. \(\text{\ref{eq:demag_jackson}}\) can be obtained by applying Green’s theorem
\[{u_{}(\boldsymbol{r})} = \frac{1}{4 \pi} \int_\Omega {\boldsymbol{M}_{}(\boldsymbol{r}')} \cdot \boldsymbol{\nabla}' \frac{1}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}'. \label{eq:demag_potential_convolution}\]
The demagnetization field \(\boldsymbol{H}_{\text{demag}}\) is calculated as negative gradient of the potential \(u\) with respect to \(\boldsymbol{r}\)
\[{\boldsymbol{H}_{\text{demag}}(\boldsymbol{r})} = - \boldsymbol{\nabla}{u_{}(\boldsymbol{r})} = \int_\Omega \boldsymbol{\widetilde{N}}(\boldsymbol{r} - \boldsymbol{r}') {\boldsymbol{M}_{}(\boldsymbol{r}')} \,\text{d}\boldsymbol{r}' \label{eq:demag_field}\]
with the so-called demagnetization tensor \(\boldsymbol{\widetilde{N}}\) given by
\[\boldsymbol{\widetilde{N}}(\boldsymbol{r} - \boldsymbol{r}') = - \frac{1}{4 \pi} \boldsymbol{\nabla}\boldsymbol{\nabla}' \frac{1}{|\boldsymbol{r} - \boldsymbol{r}'|}.\]
According to classical electrodynamics the energy is given by
\begin{align} E &= - \frac{\mu_0}{2} \int_\Omega \boldsymbol{M} \cdot \boldsymbol{H}_{\text{demag}} \,\text{d}\boldsymbol{r}\label{eq:demag_energy}\\ &= - \frac{\mu_0}{2} \int \!\!\!\! \int_\Omega {\boldsymbol{M}_{}(\boldsymbol{r})} \boldsymbol{\widetilde{N}}(\boldsymbol{r} - \boldsymbol{r}') {\boldsymbol{M}_{}(\boldsymbol{r}')} \,\text{d}\boldsymbol{r}\,\text{d}\boldsymbol{r}'\end{align}
where the factor \(1/2\) accounts for the fact that the field is generated by the magnetization itself. Due to the integration over \(\boldsymbol{r}\) and \(\boldsymbol{r}'\) every dipole–dipole interaction between \({\boldsymbol{M}_{}(\boldsymbol{r})}\) and \({\boldsymbol{M}_{}(\boldsymbol{r}')}\) contributes twice to the result, which is corrected by this prefactor.
Depending on the crystal structure of a ferromagnetic material, it energetically favors the alignment of the magnetization parallel to certain axes. This energy contribution results from spin-orbit interactions Hubert and Schäfer (1998) and is referred to as anisotropy energy. The energetically favored axes are called easy axes. These axes are undirected which means that a local minimum of the energy at \(\boldsymbol{m}_{\text{min}}\) implies a local minimum at \(-\boldsymbol{m}_{\text{min}}\) with
\[E(\boldsymbol{m}_{\text{min}}) = E(-\boldsymbol{m}_{\text{min}}) \label{eq:aniso_symmetry}\]
Depending on the lattice structure, a material may have one or more of these easy axes. In the simplest case a material has a single easy axis. This uniaxial anisotropy energy is given by
\[E = - \int_\Omega [ K_{\text{u1}} ( \boldsymbol{m} \cdot \boldsymbol{e}_{\text{u}} )^2 + K_{\text{u2}} ( \boldsymbol{m} \cdot \boldsymbol{e}_{\text{u}} )^4 ] \,\text{d}\boldsymbol{r}\label{eq:aniso_uniaxial}\]
where \(\boldsymbol{e}_{\text{u}}\) is a unit vector pointing in the direction of the easy axis and \(K_{\text{u1}}\) and \(K_{\text{u2}}\) are called anisotropy constants. This phenomenological expression is the result of a Taylor expansion up to fourth order. Only even powers are considered in order to fullfill the symmetry condition \(\text{\ref{eq:aniso_symmetry}}\) . Uniaxial anisotropy occurs in materials with a hexagonal or tetragonal crystal structure, e.g. cobalt.
Materials with a cube-symmetric lattice structure naturally feature three easy axes \(\boldsymbol{e}_i\) which are pairwise orthogonal
\[\boldsymbol{e}_i \cdot \boldsymbol{e}_j = \delta_{ij}.\]
The energy of such a cubic anisotropy is the result of an expansion in the magnetization components along the easy axes
\[E = \int_\Omega [ K_{\text{c1}} (m_1^2 m_2^2 + m_2^2 m_3^2 + m_3^2 m_1^2) + K_{\text{c2}} m_1^2 m_2^2 m_3^2 ] \,\text{d}\boldsymbol{r}\label{eq:aniso_cubic}\]
where \(m_i = \boldsymbol{e}_i\cdot\boldsymbol{m}\) is the magnetization component in direction of the anisotropy axis \(\boldsymbol{e}_i\) . Again terms which violate the symmetry condition \(\text{\ref{eq:aniso_symmetry}}\) are neglected. Moreover, only contributions which are constant under permutation of magnetization components \(m_i\) are considered in order to comply with the cubic symmetry. Cubic anisotropy occurs in materials such as iron which has a body-centered cubic structure or nickel which has a face-centered cubic structure.
Although the expressions for the anisotropy energy in Eqn. \(\text{\ref{eq:aniso_uniaxial}}\) and \(\text{\ref{eq:aniso_cubic}}\) have a pure phenomenological origin, they are able to describe anisotropy effects with a high accuracy. In practical applications the energy expressions are often reduced to the lowest order term.
The Zeeman energy of a ferromagnetic body is the energy of the magnetization in an external field \(\boldsymbol{H}_{\text{zeeman}}\) given by
\[E = - \mu_0 \int_\Omega \boldsymbol{M} \cdot \boldsymbol{H}_{\text{zeeman}} \,\text{d}\boldsymbol{r}.\]
The central equation in micromagnetism for the description of magnetization dynamics is the Landau-Lifshitz-Gilbert equation, which was originally proposed in Landau and Lifshitz (1935) . In this work from 1935 by Landau and Lifshitz the motion of the magnetization is described by a precessional term and a damping term. While the precessional term is physically derived in this work, the damping term is purely phenomenological. In 1955 Gilbert derived an equivalent expression for the Landau-Lifshitz-Gilbert equation from a Lagrangian formulation, where the damping is treated more strictly, see Gilbert (1955) and Gilbert (2004) .
In the following a classical Lagrangian method as well as a quantum mechanical approach are presented briefly to derive the Landau-Lifshitz-Gilbert equation. Both approaches give insight to different aspects of the micromagnetic model and are thus of interest for the application of this theory.
With an appropriate Lagrange functional the Landau-Lifshitz-Gilbert equation can be obtained by the Lagrangian formalism. Since the magnetization \(\boldsymbol{M}\) is assumed to be normalized, see Eqn. \(\text{\ref{eq:normalized_magnetization}}\) , its motion may locally be well described by the rotation of a rigid body. The angular velocity of a rigid body in terms of the Euler angles reads
\[\boldsymbol{\Omega} = \begin{pmatrix} \dot{\phi} \sin(\theta) \sin(\psi) + \dot{\theta} \cos(\psi) \\ \dot{\phi} \sin(\theta) \cos(\psi) + \dot{\theta} \sin(\psi) \\ \dot{\phi} \cos(\theta) + \dot{\psi} \end{pmatrix} \label{eq:llg_euler}\]
where the shorthand notation \(\dot{f} = \partial_t f\) is used. Note that the angular velocity \(\boldsymbol{\Omega}\) as well as the angles \(\theta\) , \(\phi\) and \(\psi\) have to depend on the location \(\boldsymbol{r}\) in order to describe the magnetization field \({\boldsymbol{m}_{}(\boldsymbol{r})}\) . Without loss of generality the rotation axes can be chosen such that the \(r_3\) -axis points into the direction of the magnetization in every point and thus
\[\boldsymbol{m} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.\]
In the rigid-body picture the magnetization is described by a rotationally symmetric stick. Since the angle \(\psi\) describes the rotation of the stick around its symmetry axis it may be set to \(\psi = 0\) without loss of generality Landau and Lifshitz (1969) . Thus Eqn. \(\text{\ref{eq:llg_euler}}\) reduces to
\[\boldsymbol{\Omega} = \begin{pmatrix} \dot{\theta} \\ \dot{\phi} \sin(\theta) \\ \dot{\phi} \cos(\theta) + \dot{\psi} \end{pmatrix}.\]
The Lagrangian of a dynamical system in general is given by
\[\mathcal{L} = T(\boldsymbol{q}, \dot{\boldsymbol{q}}) - V(\boldsymbol{q})\]
where \(T\) is the kinetic energy and \(V\) is the potential energy of the system and \(\boldsymbol{q}\) and \(\dot{\boldsymbol{q}}\) are the generalized coordinates and their time derivatives respectively. In micromagnetics the potential energy is given by the energy contributions discussed in Sec. 2. The kinetic energy however has no classical explanation since the magnetization does not have inertia in the classical sense. In Gilbert (1955) Gilbert proposed the Lagrangian
\[\mathcal{L} = - \frac{M_{\text{s}}}{\gamma} \dot{\phi} \cos(\theta) - U(\theta, \phi). \label{eq:llg_lagrangian}\]
As shown by Wegrowe et al. in Wegrowe and Ciornei (2012) this expression is equivalent to
\begin{align} \mathcal{L} &= \frac{1}{2} I [\dot{\phi} \cos(\theta) + \dot{\psi}]^2 - U(\theta, \phi) \\ &= \frac{1}{2} I \Omega_3^2 - U(\theta, \phi) \label{eq:llg_lagrangian_wegrowe}\end{align}
which describes a classical rotation energy with \(I\) being the moment of inertia. However, in this energy only the rotation around the symmetry axis of the magnetization is considered. From a classical point of view it is not clear why \(\Omega_1\) and \(\Omega_2\) may be neglected. In Wegrowe and Ciornei (2012) this choice of \(\mathcal{L}\) is discussed in detail.
In the following we will stick with the Lagrangian \(\text{\ref{eq:llg_lagrangian}}\) as proposed by Gilbert. According to the Lagrangian formulation, the equation of motion in terms of the generalized coordinates \(\boldsymbol{q} = (\theta, \phi)\) is given by
\[\frac{\text{d}}{\text{d}t} \frac{\delta \mathcal{L}}{\delta \dot{q_i}} - \frac{\delta \mathcal{L}}{\delta q_i} + \frac{\delta D}{\delta \dot{q_i}} = 0 \label{eq:llg_lagrange_formulation}\]
where \(\delta / \delta f\) denotes a functional derivative and \(D\) is an additional dissipative Rayleigh function that accounts for energy losses of the system. Inserting Eqn. \(\text{\ref{eq:llg_lagrangian}}\) into Eqn. \(\text{\ref{eq:llg_lagrange_formulation}}\) yields
\begin{align} \dot{\phi} \sin(\theta) &= \frac{\gamma}{M_{\text{s}}} \left[ \frac{\delta U}{\delta \theta} + \frac{\delta D}{\delta \dot{\theta}} \right] \label{eq:llg_lagrange_result_1} \\ \dot{\theta} &= - \frac{\gamma}{M_{\text{s}} \sin(\theta)} \left[ \frac{\delta U}{\delta \phi} + \frac{\delta D}{\delta \dot{\phi}} \right]. \label{eq:llg_lagrange_result_2}\end{align}
These equations of motion describe the dynamics of the magnetization by the Euler angles \({\phi_{}(\boldsymbol{r})}\) and \({\theta_{}(\boldsymbol{r})}\) . In order to obtain a description in cartesian coordinates, the time derivative of the magnetization in cartesian coordinates is considered
\[\partial_t \boldsymbol{m} = \boldsymbol{\Omega} \times \boldsymbol{m} = \begin{pmatrix} \dot{\phi} \sin(\theta) \\ - \dot{\theta} \\ 0 \end{pmatrix}. \label{eq:llg_rotating_frame}\]
Furthermore, since the magnetization is a unit vector field \(|\boldsymbol{m}| = 1\) , variations of the Euler angles \(\phi\) and \(\theta\) are given by
\begin{align} \delta m_1 &= \sin(\theta) \delta \phi \\ \delta m_2 &= \delta \theta.\end{align}
Hence Eqn. \(\text{\ref{eq:llg_lagrange_result_1}}\) and \(\text{\ref{eq:llg_lagrange_result_2}}\) can be written as
\begin{align} \partial_t m_1 &= \frac{\gamma}{M_{\text{s}}} \left[ \frac{\delta U}{\delta m_2} + \frac{\delta D}{\delta \dot{m_2}} \right] \\ \partial_t m_2 &= - \frac{\gamma}{M_{\text{s}}} \left[ \frac{\delta U}{\delta m_1} + \frac{\delta D}{\delta \dot{m_1}} \right]\end{align}
or equivalently
\[\partial_t \boldsymbol{m} = - \frac{\gamma}{M_{\text{s}}} \boldsymbol{m} \times \left( \frac{\delta U}{\delta \boldsymbol{m}} + \frac{\delta D}{\delta \dot{\boldsymbol{m}}} \right).\]
Choosing the reasonable dissipative function \(D = \alpha^\ast (\partial_t \boldsymbol{m})^2\) with \(\alpha^\ast \geq 0\) results in the Landau-Lifshitz-Gilbert equation
\[\partial_t \boldsymbol{m} = - \gamma (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}) + \alpha (\boldsymbol{m} \times \partial_t \boldsymbol{m}) \label{eq:llg_implicit}\]
where \(\boldsymbol{H}_{\text{eff}}\) is the so-called effective field given by
\[\boldsymbol{H}_{\text{eff}} = - \frac{1}{\mu_0 M_{\text{s}}} \frac{\delta U}{\delta \boldsymbol{m}}. \label{eq:llg_effective_field}\]
and \(\alpha \geq 0\) is a damping constant.
In Sec. 3.1 it is shown that the Landau-Lifshitz-Gilbert equation can be obtained by application of the classical Lagrange formalism. However, this method is based on a particular choice of the Lagrangian, which is not completely justifiable by classical theory.
In this section a quantum mechanical approach is discussed to derive the Landau-Lifshitz-Gilbert equation. As described in Sec. 1 the continuous magnetization field approximates a discrete distribution of spins. In quantum mechanics the components of the spin are described by the spin operators \(\hat{S}_i\) . In the Heisenberg picture the time development of the spin operators are given by
\[\frac{\text{d}\hat{S}_j}{\text{d}t} = \frac{1}{i \hbar} \left[ \hat{S}_j, \hat{H} \right]. \label{eq:llg_qm_heisenberg}\]
The contributions to the Hamiltonian in the context of micromagnetics will depend on the spin. Thus it is reasonable to expand the Hamiltonian in the spin operators
\begin{align} \left[ \hat{S}_j, \hat{H} \right] &= \sum_k \frac{\partial \hat{H}}{\partial \hat{S}_k} \left[ \hat{S}_k, \hat{S}_j \right] + \mathcal{O}(\hbar^2) \notag \\ &= i \hbar \sum_{k,l} \frac{\partial \hat{H}}{\partial \hat{S}_k} \epsilon_{jkl} \hat{S}_l + \mathcal{O}(\hbar^2).\end{align}
Inserting into Eqn. \(\text{\ref{eq:llg_qm_heisenberg}}\) and using the vector notation \(\hat{\boldsymbol{S}} = (\hat{S}_1, \hat{S}_2, \hat{S}_3)\) yields
\[\frac{\text{d}\hat{\boldsymbol{S}}}{\text{d}t} = - \hat{\boldsymbol{S}} \times \frac{\partial \hat{H}}{\partial \hat{\boldsymbol{S}}} + \mathcal{O}(\hbar).\]
In the classical limit the spin operators may be replaced by the continuous magnetization vector \(\boldsymbol{M} = M_{\text{s}} \boldsymbol{m}\) . Further the second term can be neglected since \(\hbar \rightarrow 0\) . Identifying \(\partial \hat{H} / \partial \hat{\boldsymbol{S}}\) with the effective field \(\boldsymbol{H}_{\text{eff}}\) in Eqn. \(\text{\ref{eq:llg_effective_field}}\) leads to
\[\partial_t \boldsymbol{m} = - \gamma' (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}).\]
This equation describes the precession of the magnetization around the effective field without any loss of energy. According to the original work of Landau and Lifshitz Landau and Lifshitz (1935) a phenomenological damping term is added in order to account for these losses. This damping term is constructed such that it is perpendicular to the precessional term. Further it should conserve the magnetization norm leading to
\[\partial_t \boldsymbol{m} = - \gamma' (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}) - \alpha' \boldsymbol{m} \times (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}). \label{eq:llg_explicit}\]
This is the explicit form of the Landau-Lifshitz-Gilbert equation. It can be shown that this form is equivalent to the implicit form in Eqn. \(\text{\ref{eq:llg_implicit}}\) . This is done by inserting Eqn. \(\text{\ref{eq:llg_implicit}}\) for \(\partial_t \boldsymbol{m}\) on the right-hand side of Eqn. \(\text{\ref{eq:llg_implicit}}\) and using the vector identity \(\boldsymbol{a} \times (\boldsymbol{b} \times \boldsymbol{c}) = (\boldsymbol{a} \cdot \boldsymbol{c}) \boldsymbol{b} - (\boldsymbol{a} \cdot \boldsymbol{b}) \boldsymbol{c}\) . The coefficients of the different versions of the Landau-Lifshitz-Gilbert equation satisfy the relations
\begin{align} \gamma' &= \gamma / (1 + \alpha^2) \\ \alpha' &= \alpha \gamma / (1 + \alpha^2).\end{align}
A full quantum mechanical description of a spin subject to exchange interaction, anisotropy and Zeeman field is given in Bode et al. (2012) where the Landau-Lifshitz-Gilbert equation is also obtained in a limit case.
Figure 2: Time evolution of a single magnetic moment as described by the Landau-Lifshitz-Gilbert equation. The motion can be divided into a precessional and a damping part. (a) Precessional motion around the effective field. (b) Damped motion. The magnetization relaxes towards the effective field. (c) Resulting motion including precession and damping.
The Landau-Lifshitz-Gilbert equation describes the motion of the magnetization in an effective field defined by Eqn. \(\text{\ref{eq:llg_effective_field}}\) . This motion can be described as the sum of a precessional term and a damping term. Figure 2 illustrates these two terms for the explicit formulation of the Landau-Lifshitz-Gilbert equation \(\text{\ref{eq:llg_explicit}}\) .
In the case of the Zeeman energy and the demagnetization energy, the effective field is an actual field in the classical sense. The Zeeman field is directly given by \(\boldsymbol{H}_{\text{zeeman}}\) , the demagnetization field is computed via Eqn. \(\text{\ref{eq:demag_field}}\) .
However, the exchange field as well as the anisotropy field have to be calculated via the variational derivative in Eqn. \(\text{\ref{eq:llg_effective_field}}\) . The exchange energy density is given by the integrand of Eqn. \(\text{\ref{eq:exchange_continuous_final}}\) . Applying the chain rule of variational calculus yields
\begin{align} {\boldsymbol{H}_{\text{ex}}(\boldsymbol{r})} &= - \frac{A}{\mu_0 M_{\text{s}}} \frac{\delta }{\delta \boldsymbol{m}} (\boldsymbol{\nabla}\boldsymbol{m})^2 \\ &= \frac{2 A}{\mu_0 M_{\text{s}}} \Delta \boldsymbol{m} \label{eq:effective_field_exchange}\end{align}
Analogously the effective fields for uniaxial and cubic anisotropy are obtained by computing the variational derivative of Eqn. \(\text{\ref{eq:aniso_uniaxial}}\) and \(\text{\ref{eq:aniso_cubic}}\) respectively
\begin{align} {\boldsymbol{H}_{\text{u}}(\boldsymbol{r})} &= \frac{2 K_{\text{u1}}}{\mu_0 M_{\text{s}}} \boldsymbol{e}_{\text{u}} (\boldsymbol{e}_{\text{u}} \cdot \boldsymbol{m}) + \frac{4 K_{\text{u2}}}{\mu_0 M_{\text{s}}} \boldsymbol{e}_{\text{u}} (\boldsymbol{e}_{\text{u}} \cdot \boldsymbol{m})^3 \\ {\boldsymbol{H}_{\text{c}}(\boldsymbol{r})} &= - \frac{2 K_{\text{c1}}}{\mu_0 M_{\text{s}}} \begin{pmatrix} m_1 m_2^2 + m_1 m_3^2 \\ m_2 m_3^2 + m_2 m_1^2 \\ m_3 m_1^2 + m_3 m_2^2 \end{pmatrix} - \frac{2 K_{\text{c2}}}{\mu_0 M_{\text{s}}} \begin{pmatrix} m_1 m_2^2 m_3^2 \\ m_1^2 m_2 m_3^2 \\ m_1^2 m_2^2 m_3 \end{pmatrix}.\end{align}
Figure 3: A so-called Gaussian pillbox \(\Omega_{\text{p}}\) on the outer boundary of the sample \(\Omega\). The discontinuity of the magnetization \(\boldsymbol{m}\) across the sample boundary \(\partial \Omega\) is smoothed out within the pillbox, as sketched on the right-hand side. Properties of the discontinuous system are obtained by considering the limit \(d \rightarrow 0\).
Due to the first derivate in time the Landau-Lifshitz-Gilbert equation is an initial value problem. In order to find a solution \(\boldsymbol{m}(\boldsymbol{r}, t)\) for \(t > t_0\) the initial magnetization \(\boldsymbol{m}(\boldsymbol{r}, t_0)\) has to be known.
Depending on the effective field, additional boundary conditions are required in order to find a unique solution of the Landau-Lifshitz-Gilbert equation. In fact, even with given boundary conditions the uniqueness of solutions to the Landau-Lifshitz-Gilbert equation could only be shown for special cases, see Carbou and Fabrie (2001a; Carbou and Fabrie 2001b; Alouges and Soyeur 1992) . From the effective field contribution discussed in Sec. 3.3, the exchange field is the only one that adds the need of boundary conditions due to its second order in space.
If this so-called exchange boundary condition is applied on the boundary of an ideal magnetic body, i.e. a boundary where the magnetization rapidly drops to zero, it is uniquely defined. Consider the Landau-Lifshitz-Gilbert equation without damping and with the exchange field as effective field.
\[\partial_t \boldsymbol{m} = - \gamma \boldsymbol{m} \times \Delta \boldsymbol{m}. \label{eq:llg_boundary_llg}\]
Since at the boundary of an ideal magnetic body the Laplacian of the magnetization \(\Delta \boldsymbol{m}\) is not defined, a limiting procedure is used to obtain the boundary condition. Like in Sec. 2.2 the magnetization is considered to decay continuously in a finite interval \(d\) . Equation \(\text{\ref{eq:llg_boundary_llg}}\) is integrated over a so-called Gaussian pillbox as sketched in Fig. 3. Application of Green’s theorem yields
\begin{align} \int_{\Omega_{\text{p}}} \partial_t m_i \,\text{d}\boldsymbol{r}&= \int_{\Omega_{\text{p}}} \epsilon_{ijk} m_j \Delta m_k \,\text{d}\boldsymbol{r}\label{eq:llg_boundary_green_1} \\ &= \int_{\partial \Omega_{\text{p}}} \epsilon_{ijk} m_j \frac{\partial m_k}{\partial \boldsymbol{n}} \,\text{d}\boldsymbol{r}- \int_{\Omega_{\text{p}}} \epsilon_{ijk} \boldsymbol{\nabla}m_j \cdot \boldsymbol{\nabla}m_k \,\text{d}\boldsymbol{r}. \label{eq:llg_boundary_green_2}\end{align}
The second integral on the right-hand side of Eqn. \(\text{\ref{eq:llg_boundary_green_2}}\) vanishes due to the skew-symmetric Levi-Civita tensor \(\epsilon\) . The boundary of the pillbox \(\partial \Omega_{\text{p}}\) coincides with the boundary of the magnetic body \(\Omega\) on one side except for the orientation. The other side of the pillbox is outside the magnetic body where \(|\boldsymbol{m}| = 0\) . Hence Eqn. \(\text{\ref{eq:llg_boundary_green_2}}\) reads
\[\int_{\Omega_{\text{p}}} \partial_t m \,\text{d}\boldsymbol{r}= - \int_{\partial_\Omega} \boldsymbol{m} \times \frac{\partial \boldsymbol{m}}{\partial \boldsymbol{n}} \,\text{d}\boldsymbol{r}\]
In the limit \(d \rightarrow 0\) the volume integral on the left-hand-side of Eqn. \(\text{\ref{eq:llg_boundary_green_1}}\) vanishes. Since the faces of the pill box can be chosen arbitrarily the boundary condition
\[\boldsymbol{m} \times \frac{\partial \boldsymbol{m}}{\partial \boldsymbol{n}} = 0\]
must hold in every boundary point. Further since \(|\boldsymbol{m}|=1\) the normal derivative of the magnetization is perpendicular to the magnetization in every point \(\partial \boldsymbol{m} / \partial \boldsymbol{n} \perp \boldsymbol{m}\) and thus
\[\frac{\partial \boldsymbol{m}}{\partial \boldsymbol{n}} = 0. \label{eq:exchange_boundary_condition}\]
This is the so-called exchange boundary condition, which is the right choice if the boundary of the computational domain \(\Omega\) coincides with the boundary of an ideal magnet as shown above. Equation \(\text{\ref{eq:exchange_boundary_condition}}\) was originally derived by Rado and Weertman in Rado and Weertman (1959) . Before, it was shown by Brown that the same boundary condition has to hold in energetic equilibrium Brown Jr. (1963) . Depending on further contributions to the effective field, such as surface anisotropy, this boundary condition changes accordingly, see Hubert and Schäfer (1998) .
As mentioned in Sec. 1 the magnetization \(\boldsymbol{m}\) is assumed to be normalized everywhere. This feature is preserved by the Landau-Lifshitz-Gilbert equation. Consider the time derivative of the squared magnetization
\[\partial_t |\boldsymbol{m}|^2 = \partial_t (\boldsymbol{m} \cdot \boldsymbol{m}) = 2 \partial_t \boldsymbol{m} \cdot \boldsymbol{m}.\]
Inserting Eqn. \(\text{\ref{eq:llg_implicit}}\) immediately yields \(\partial_t |\boldsymbol{m}|^2 = 0\) and thus also \(\partial_t |\boldsymbol{m}| = 0\) .
If the energy functional of a magnetic system does not explicitly depend on the time, i.e. if the external field is constant in time, the time derivative of the energy density may be written as
\begin{align} \partial_t U &= \frac{\delta U}{\delta \boldsymbol{m}} \cdot \partial_t \boldsymbol{m} \\ &= - \mu_0 M_{\text{s}} \boldsymbol{H}_{\text{eff}} \cdot \partial_t \boldsymbol{m}.\end{align}
Replacing \(\partial_t \boldsymbol{m}\) with Eqn. \(\text{\ref{eq:llg_explicit}}\) and spatial integration yields
\begin{align} \partial_t E &= \mu_0 M_{\text{s}} \int_\Omega \boldsymbol{H}_{\text{eff}} [ \gamma' \boldsymbol{m} \times \boldsymbol{H}_{\text{eff}} + \alpha' \cdot \boldsymbol{m} \times (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}) ] \,\text{d}\boldsymbol{r}\\ &= - \mu_0 M_{\text{s}} \alpha' \int_\Omega | \boldsymbol{m} \times \boldsymbol{H} |^2 \,\text{d}\boldsymbol{r}\label{eq:llg_energy_loss}\\ &\leq 0.\end{align}
This means that the energy of a magnetic system is always a non increasing function in time. The Landau-Lifshitz-Gilbert equation is said to have Lyapunov structure d’Aquino, Serpico, and Miano (2005; Cimr á k 2007) .
In the special case of no damping \(\alpha = 0\) the right-hand side of Eqn. \(\text{\ref{eq:llg_energy_loss}}\) vanishes
\[\partial_t E = 0.\]
In this case the energy of the system ist preserved and the Landau-Lifshitz-Gilbert equation has Hamiltonian structure.
As discussed in Sec. 1 the micromagnetic model introduces a set of simplifications to the quantum mechanical description of magnetism. While justified for a broad range of applications, these simplifications may be inappropriate in some cases. The successfull application of micromagnetism to physical problems requires a solid understanding of the origin of this model. It is crucial to consider its underlying assumptions and simplifications in order to predict the validity of simulation results for specific physical problems. Without claiming to be complete, this section discusses a number of limits and extension to the basic micromagnetic model as introduced in the preceding sections.
The central assumption in micromagnetics is the homogeneous saturation magnetization \(M_{\text{s}}\) . This assumption is justified by the fact that ferromagnetic materials are subject to exchange coupling which leads to locally almost perfectly aligned magnetic moments.
However, certain magnetic processes involve the creation of magnetic singularities called Bloch points. At these Bloch points the magnetization changes rapidly in space, which is inconsistent with the basic assumption of a homogeneous \(M_{\text{s}}\) in micromagnetics. Despite this fact it was shown in Thiaville et al. (2003) that micromagnetic simulations involving the creation of Bloch points are able to describe the corresponding processes in accordance with experiments, although the energy density at the Bloch point is underestimated.
Another example for the violation of the micromagnetic assumption of a locally homogeneous magnetization is given by thermal effects. Thermal effects are most naturally reflected by local perturbation of magnetic moments. Perturbation of a single magnetic moment, however, obviously breaks the homogeneity of the magnetization. In the framework of classical micromagnetics a possible approach for consideration of finite temperature is the reduction of the saturation magnetization according to a mean-field approximation, see Kittel and McEuen (1996) . Another approach is to add a fluctuating field to the effective field, which converts the Landau-Lifshitz-Gilbert equation into a stochastic differential equation, see Lyberatos, Berkov, and Chantrell (1993) . Both of these techniques do not account for local changes in the saturation magnetization and as a result both methods fail to describe the magnetization dynamics correctly when approaching the Curie temperature. This deficiency is overcome by the Landau-Lifshitz-Bloch equation, which extends the Landau-Lifshitz-Gilbert equation not only by a fluctuating field, but also by a term that allows the change of the magnetization modulus Garanin (1997) .
Other successfull extensions to the micromagnetic model include the description of spin polarized currents and its interactions with the magnetization configuration Zhang and Li (2004) . A famous application for this interaction is the magnetic racetrack memory proposed by Parkin et al. in Parkin, Hayashi, and Thomas (2008) .
The micromagnetic model was designed to describe the energetics and dynamics in a homogeneous material. Thus the description of interfaces between different materials gives rise to another limit of the model. The magnetic coupling at these interfaces is often not entirely clear and can be subject to quantum mechanical effects that are not covered by the micromagnetic model. These effects include the Ruderman-Kittel-Kasuya-Yosida interaction (RKKY) Parkin and Mauri (1991) and the Dzyaloshinskii-Moriya Interaction (DMI) Sergienko and Dagotto (2006) . Work has been done on the integration of these effects into micromagnetic theory and computations, see Heide, Bihlmayer, and Bl ü gel (2008; Lopez-Diaz et al. 2012; Thiaville et al. 2012) . A common simplification for the description of thin multilayer materials however is the assumption of a bulk material with efficient material constants. These efficient material constants are determined by experiments.
The Landau-Lifshitz-Gilbert equation with effective field contributions as introduced in the preceding sections is a non-linear partial differential equation in space and time. Apart from some simplified edge cases the arising system of equations cannot be solved analytically. This work is dedicated to the numerical solution of the micromagnetic equations. In the following chapters the discrete solution of the different subproblems is discussed in detail. An overview over existing methods is given and novel methods are introduced. Finally the open-source finite-element micromagnetic simulation code magnum.fe is presented.
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Roberto Firmino opens up on Liverpool position under Jurgen Klopp
Liverpool forward Roberto Firmino has been vital to the success of their powerful front three under Jurgen Klopp
by Irishmirror.ieLiverpool striker Roberto Firmino has emerged as a key piece of Jurgen Klopp's attacking puzzle - and he insists he is happy to continue being a team player.
Firmino has often been hailed as the man who knits together the Reds' powerful front three to bring out the best in Mohamed Salah and Sadio Mane.
He has now opened up on his exact role in the side and states he will play wherever it takes to win games.
It has certainly done the trick for the Brazilian so far, with the Champions League winners sitting top of the Premier League table after last weekend's win over title rivals Manchester City .
Speaking to Premier League Productions, Firmino said of his eight assists this campaign: “That is just me, my personality, to try to be a good and humble guy.
“To try to help my team in the best way with assists and goals. I like that, I like to win all the time, and for that I will always be like this.
“I am a guy who respects my position on the pitch. Where the team needs me, I will play. As a No.9, but sometimes I play as a No.10, and I try to give my best.
“As I mentioned before, I always want to help my team as best as I can and to always be chasing the win, always winning games. That is what matters the most."
Firmino also believes he has simply gone from strength to strength since moving to the Premier League in 2015.
“I think I improved a lot in every way: physically, tactically, mentally, I learned a bit of every aspect.
“But I do not want to stop here. I want to keep improving - I always want more, I have big goals, I want to keep winning titles with this club and that is it.”


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Jennifer Dornan murder trial collapses
The jury presiding over a murder trial arising from the death of a west Belfast mother-of-three was discharged by a senior judge.
As the trial at Belfast Crown Court was due to enter its second day, a legal issue arose which prompted a defence application to discharge the six men and women on the jury and which resulted in the trial being halted.
The case against Raymond O'Neill was opened by a prosecuting barrister on Monday, when a jury was told it was the Crown's case that he murdered Jennifer Dornan (30) in her west Belfast home in the early hours of Sunday August 2, 2015.
The 41-year old, from Amcomri Street in Belfast, was also charged with arson with intent to endanger life, and denied both this and the murder charge.
As the hearing was due to enter its second day, the application was made to Mr Justice Horner, who granted it and discharged the jury.
Expressing an apology to all those involved in the hearing for the delay, Mr Justice Horner then re-listed the trial to start again in the New Year.


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Tribal Air Monitoring Tools and Procedures
| Download | File Name | Section | File Type |
|---|---|---|---|
| Download | Admin DM Flowchart | Admin and Tools | |
| Download | Admin DM Flowchart | Admin and Tools | xlsx |
| Download | Admin Sample Contract | Admin and Tools | docx |
| Download | Admin Sample Contract 2 | Admin and Tools | docx |
| Download | Admin SelectingaLaboratory | Admin and Tools | |
| Download | Admin TechGuidanceForDevelopingTribalAirPrograms | Admin and Tools | |
| Download | AQS and PM-cont-Reporting-Tech-Note-053106 | AQS | |
| Download | AQS data upload SOP | AQS | |
| Download | AQS pipe delimited example | AQS | pptx |
| Download | AQS QA data format SOP NNEPA | AQS | |
| Download | AQS Quick reference Guide v2 | AQS | ppt |
| Download | AQS-user data coding manual feb 2018 0 | AQS | |
| Download | AQS Certification Tribal Question and Answer Session | AQS | pptx |
| Download | AQS Creating a Monitor via Online Maintenance | AQS | pptx |
| Download | AQS Creating a Site via Online Maintenance | AQS | pptx |
| Download | AQS CreatingASiteTribal Question and Answer Session | AQS | pptx |
| Download | AQS Data Certification CNEP | AQS | |
| Download | AQS QA Transaction Formats | AQS | |
| Download | AQS SubmittingAQS Report | AQS | docx |
| Download | AQS UploadingDataIntoAQS | AQS | docx |
| Download | aqs user guide 2018 2 | AQS | |
| Download | Establishing a Monitor in AQS | AQS | |
| Download | Establishing a Site in AQS | AQS | |
| Download | IndigenousDataSovereignty | Data Sovereignty | |
| Download | Intellectual Property Rights Data Sovereignty | Data Sovereignty | |
| Download | TribalAuthorityRuleData Sovereignty | Data Sovereignty | |
| Download | TribalAuthorityRuleSummary Data Sovereignty | Data Sovereignty | |
| Download | UN Indigenous Data Governance Data Sovereignty | Data Sovereignty | |
| Download | Best Practices for Review and Validation of Ambient Air Monitoring Data - Final (August 2021) | DataAnalysis | |
| Download | Data 1-VideoDemoDataAnalysis RawData | DataAnalysis | xlsx |
| Download | Data 2-VideoDemoDataAnalysisNoBlanks | DataAnalysis | xlsx |
| Download | Data 3-VideoDemoDataWoErrors | DataAnalysis | xlsx |
| Download | Data 4-VideoDemoDataWPivotTable | DataAnalysis | xlsx |
| Download | Data 5-VideoDemoGroupedPivotTable | DataAnalysis | xlsx |
| Download | Data 6-VideoDemoPivotChartLineO3 | DataAnalysis | xlsx |
| Download | Data 6-VideoDemoPivotChartPrettyO3 | DataAnalysis | xlsx |
| Download | Data 7-VideoDemoPivotChart O3 PM25 Column | DataAnalysis | xlsx |
| Download | Data 8-VideoDemoPivotChart O3 PM25 Column | DataAnalysis | xlsx |
| Download | Data 8 VideoDemoPivotChartLineO3 PM25 2ndaryAxis | DataAnalysis | xlsx |
| Download | Data Analyses WaterExamples | DataAnalysis | |
| Download | Data DataFinalization | DataAnalysis | doc |
| Download | Data ReviewAndDocumentation | DataAnalysis | doc |
| Download | Data SOP & ChecklistForPM25DataReview | DataAnalysis | doc |
| Download | Data Validation Water 29 Palms | DataAnalysis | |
| Download | Data ValidationReportExample | DataAnalysis | docx |
| Download | Data VideoDemoDataAnalysisW ReferenceToVideos | DataAnalysis | docx |
| Download | Data Water 29 Palms DataValidation | DataAnalysis | |
| Download | Data WaterAnalysesLabQA andReferences | DataAnalysis | |
| Download | EPA Audits-Options Available for Tribes to Meet Independent Performance Evaluation Requirements Final 9 07 | EPA | docx |
| Download | EPA Data Quality Assessment Statistical Methods for Practitioners | EPA | |
| Download | EPA Graded Approach to QAPPs | EPA | docx |
| Download | EPA Guidance and Policy for Implementation of Tribal Air Monitoring Programs | EPA | |
| Download | EPA Guidance for Data Quality Assessment Data Analysis | EPA | |
| Download | EPA Guidance on Environmental Data Verification and Data Validation | EPA | |
| Download | EPA Index to EPA Test Methods | EPA | |
| Download | EPA QA Guidance for Conducting Brownfields Site Assessments | EPA | |
| Download | EPA QA Guidance G5-final | EPA | |
| Download | EPA QA Handbook for Air Pollution Measurement Systems Meteorological Measurements | EPA | |
| Download | EPA QA Handbook for Air Pollution Measurement Systems | EPA | |
| Download | PEP-Definitions and Responsibilities | EPA | ppt |
| Download | PEP-Technical Documents | EPA | pptx |
| Download | QC 40CFRAppA BiasPrecision | EPA | pptx |
| Download | QC HowEPAcalcsConfLevelsAndDrift DASCmod2020 | EPA | xlsx |
| Download | TSA ChecklistAppendixOnly | EPA | docx |
| Download | TSA QAGD Final with Appendices 11-30-17 | EPA | |
| Download | TSA QAGD Final part1 | EPA | docx |
| Download | Equipment Air Monitoring Equipment Operation & Maintenance Tutorial Videos | Equipment | |
| Download | Equipment MaintenanceCalibrationCertificationFirmwareSchedulesRev5 | Equipment | docx |
| Download | Equipment OzoneTransferStandardGuidance | Equipment | |
| Download | Equipment StationOperationLogbookEntriesGuide | Equipment | docx |
| Download | InstrumentationPapp-Gaseous Training part 1 and 2 2 09 | Equipment | ppt |
| Download | IAQ Healthy Homes Diaqnostic Assessment | Indoor Air | |
| Download | IAQ AmprobeCO QSGuide | Indoor Air | docx |
| Download | IAQ EXTECH-RH300 QSGuide | Indoor Air | docx |
| Download | IAQ Fluke922Micromanometer QSGuide | Indoor Air | docx |
| Download | IAQ FlukeThermalImagerTIR105 QSGuide | Indoor Air | docx |
| Download | IAQ FlukeThermalImagerTIR105 QSGuide 000 | Indoor Air | docx |
| Download | IAQ J-Lite QSGuide | Indoor Air | docx |
| Download | IAQ J-Lite QSGuide 000 | Indoor Air | docx |
| Download | IAQ J-Lite QSGuide 001 | Indoor Air | docx |
| Download | IAQ TramexMoisture QSGuide | Indoor Air | docx |
| Download | IAQ TSIModel7515 CO2 QSGuide | Indoor Air | docx |
| Download | IAQ TSIModel7515 CO2 QSGuide 000 | Indoor Air | docx |
| Download | IAQ TSIModel7515 CO2 QSGuide 001 | Indoor Air | docx |
| Download | 5-Roland Level I SOP 021313 | QAQC | doc |
| Download | bam1020 service zero test template | QAQC | xls |
| Download | Cherokee Nation Criteria QAPP - revised sections 2020 | QAQC | |
| Download | Cherokee Nation NCore May 27-31 2019 | QAQC | xls |
| Download | Cherokee Nation NCore Multipoint Second Quarter 2019 | QAQC | xls |
| Download | QC BAM 1020 Audit Sheet | QAQC | doc |
| Download | QC BAM 1020 QA Audit Form | QAQC | XLS |
| Download | QC BAM 1020 Verification Sheet | QAQC | docx |
| Download | QC COC Region 1 QAPP resource | QAQC | |
| Download | QC DronePlanDraft | QAQC | docx |
| Download | QC GPS 29 Palms GPS Data Collection-Garmin - Ver 1 | QAQC | |
| Download | QC GPS 29 Palms GPS Data Collection-Trimble - Ver 1 | QAQC | |
| Download | QC Ozone Monitor Performance Audit Form | QAQC | xls |
| Download | QC Radon Test Field Deployment and Retrieval Log | QAQC | xlsx |
| Download | QC Region 1 Document ControlQAPP resource | QAQC | |
| Download | QC Region1 Sample TrackingQAPP resource | QAQC | |
| Download | QC 2-Stilwell Level I SOP 032013 | QAQC | doc |
| Download | QC 5-Roland Level I SOP 021313 | QAQC | doc |
| Download | QC BAM1020 SinglePoint Clark county | QAQC | xlsx |
| Download | QC BAM MN | QAQC | xls |
| Download | QC BAM PM10 PM25 AuditSheet CTCR | QAQC | doc |
| Download | QC BAM PM10 PM25 Sheet CTCR | QAQC | doc |
| Download | QC BAM Sheet PM CTCR | QAQC | docx |
| Download | QC Calibration PM10 BAM-1020 | QAQC | xls |
| Download | QC Cherokee Nation Criteria QAPP - revised sections 2020 | QAQC | |
| Download | QC Cherokee Nation Criteria QAPP 2019 (2) | QAQC | |
| Download | QC Cherokee Nation Criteria QAPP 2019 | QAQC | |
| Download | QC Cherokee Nation NCore May 27-31 2019 | QAQC | xls |
| Download | QC Cherokee Nation NCore Multipoint Second Quarter 2019 | QAQC | xls |
| Download | QC Cherokee-Stilwell Level I SOP 032013 | QAQC | doc |
| Download | QC Continuous PM Thermo Beta 5014i Clark county | QAQC | xlsx |
| Download | QC CO Multipoint Clark county | QAQC | xlsx |
| Download | QC DASC DataAssessmentStatisticalCalculatorForNAAQS | QAQC | xls |
| Download | QC FDL Radon QAPP GAP | QAQC | doc |
| Download | QC Gaseous NCore Multipoint Calibration | QAQC | xls |
| Download | QC Gas PM10 PM25 NCore Multipoint Calibration | QAQC | xls |
| Download | QC Met | QAQC | xlsx |
| Download | QC Met Audit | QAQC | xls |
| Download | QC Nephelometor Form CTCR | QAQC | xls |
| Download | QC NO2 Multipoint Clark county | QAQC | xlsx |
| Download | QC NOX Multipoint Clark county | QAQC | xlsx |
| Download | QC O3 Multipoint Clark county | QAQC | xlsx |
| Download | QC Ozone Audit form CNEP | QAQC | xls |
| Download | QC Ozone Audit form MN | QAQC | xls |
| Download | QC Partisol Dichot 2025i Clark county | QAQC | xlsx |
| Download | QC Pb QAPP AZ DEQ | QAQC | doc |
| Download | QC PM25 2025 FR Maintenance Calibration Log | QAQC | xlsx |
| Download | QC QAPP Airsheds of the Colville Reservation | QAQC | |
| Download | QC QAPP BAM PM25 CTCR | QAQC | |
| Download | QC Region 1 Document ControlQAPP resource | QAQC | |
| Download | QC Region8 QAPP checklist | QAQC | doc |
| Download | QC Sampling Line Residence Calculation Clark county | QAQC | xlsx |
| Download | QC SelectingaLaboratory | QAQC | |
| Download | QC SO2 Monitor Performance Audit Form | QAQC | xlsx |
| Download | QC SO2 Multipoint Clark county | QAQC | xlsx |
| Download | QC SO2 Performance Audit | QAQC | xlsx |
| Download | QC Sonic Anemometer Field Clark county | QAQC | xlsx |
| Download | QC Technical Systems Audit Checklist with Appendices | QAQC | |
| Download | QC Temperature Verification | QAQC | xlsx |
| Download | QC URG-3000 Carbon Sampler Clark county | QAQC | xlsx |
| Download | QC Zero Air Verification | QAQC | xlsx |
| Download | QC Zero Gas Verification Clark county | QAQC | xlsx |
| Download | QREST 12-2020 Level 1 and 2 Report YPT PM10a | QREST | xlsx |
| Download | QREST-ImportMonitorsFromAQS | QREST | xlsx |
| Download | QREST-ImportSitesFromAQS | QREST | xlsx |
| Download | QREST-Level 2 Data Review | QREST | xlsx |
| Download | QREST-Level0ValidationCriteria | QREST | xlsx |
| Download | QREST-Manual Importing | QREST | xlsx |
| Download | QREST-Raw Data QuickLook | QREST | xlsx |
| Download | QREST-Sending to AQS | QREST | xlsx |
| Download | QREST-SettingImportConfig | QREST | xlsx |
| Download | QREST-SupportingDocumentation | QREST | xlsx |
| Download | QREST-UsersAndAccounts | QREST | xlsx |
| Download | RadiationRadon U-238 & Th-232 Decay Chain Summary | Radiation | doc |
| Download | Radiation Files U-238 & Th-232 Decay Chain Summary | Radiation | doc |
| Download | RadonCRM CalibrationsAndUncertainties | Radiation | xlsx |
| Download | RadonUnitsConversionFactors | Radiation | |
| Download | Report Gross Alpha and Gross Beta Radioactivity in Drinking Water | Radiation | |
| Download | Report BogenGoldinPopulationExposuresToNaturalRadiationIntheUS | Radiation | |
| Download | Report Crosswalk and Required QAPP Elements for Graded Approach | Reports | docx |
| Download | Report EIIP Residential Wood Combustion Coordination Project | Reports | |
| Download | Report Flowchart for Tribal Air Monitoring Continuous Data Management | Reports | xls |
| Download | Report Guidelines on the Meaning and Use of Precision and Bias Data Required by 40 CFR | Reports | |
| Download | Report AQS DataCertificationPackage | Reports | |
| Download | Report AuditAndQtrlyReport Cherokee2016 | Reports | doc |
| Download | Report BAM1020 Data Validation Checklist | Reports | xls |
| Download | Report DataCorrectiveActionReport GilaRiver | Reports | docx |
| Download | Report ITEC Quarterly Report to EPA CNEP | Reports | doc |
| Download | Report Precipitation Calibration | Reports | |
| Download | How to Download and Interpret Purple Air Data | Sensors | |
| Download | PurpleAirSensorTimeStampCorrectionRev2022 | Sensors | xlsx |
| Download | PurpleAirSensorTimeStampCorrectionSampleDataRev2022 | Sensors | xlsx |
| Download | Site EPA Example Site Form | Site | doc |
| Download | Site RoutineSiteCheckList | Site | xlsx |
| Download | QC GPS 29 Palms GPS Data Collection-Garmin - Ver 1 | SOPs | |
| Download | QC GPS 29 Palms GPS Data Collection-Trimble - Ver 1 | SOPs | |
| Download | QC Region1 Sample TrackingQAPP resource | SOPs | |
| Download | SOP Continuous Measurement of Particulate Matter | SOPs | |
| Download | SOP for BAM 1020 PM2.5 Monitors | SOPs | |
| Download | SOP for Continuous Measurement of Particulate Matter Minnesota Pollution Control Agency | SOPs | |
| Download | SOP for Meteorological Sensors CTCR | SOPs | |
| Download | SOP & ChecklistForPM25DataReview | SOPs | doc |
| Download | SOP 1-point O3 QC check | SOPs | docx |
| Download | SOP BAM BKGD Test | SOPs | doc |
| Download | SOP BAM ErrorRetrieval | SOPs | doc |
| Download | SOP BAM FilterTape | SOPs | doc |
| Download | SOP BAM FlowCalibration | SOPs | doc |
| Download | SOP BAM FlowCheck v.3 | SOPs | doc |
| Download | SOP BAM InletTubeCleaning | SOPs | doc |
| Download | SOP BAM LeakCheck | SOPs | doc |
| Download | SOP BAM Nozzle Vane Cleaning | SOPs | doc |
| Download | SOP BAM PM25 | SOPs | |
| Download | SOP BAM PM25 PM10 MetOne MN | SOPs | |
| Download | SOP BAM PumpMufflerCleaning | SOPs | doc |
| Download | SOP CreatingBAM ValidatedExcelFileFromTextFile | SOPs | docx |
| Download | SOP CreatingNewImportFile | SOPs | doc |
| Download | SOP CTCR BAM SOP 2012 | SOPs | |
| Download | SOP CTCR Meteorological SOP 2012 | SOPs | |
| Download | SOP DataReviewAndDocumentation | SOPs | doc |
| Download | SOP Data ComparingDataToARB | SOPs | doc |
| Download | SOP Gaseous Analyzers Clark county | SOPs | |
| Download | SOP Gravimetric Laboratory Filter Handling Guide | SOPs | docx |
| Download | SOP Gravimetric Laboratory Operations Clark county | SOPs | |
| Download | SOP Independent Audit O3 CO SO2 NO met | SOPs | doc |
| Download | SOP Instrument Acceptance Testing-Setup-Operating Guide | SOPs | docx |
| Download | SOP LeadInTotalSuspendedParticulates | SOPs | docx |
| Download | SOP Level I Check NO NO2 NOx PM10 PM2.5 | SOPs | doc |
| Download | SOP Meteorological Equipment Clark county | SOPs | |
| Download | SOP Meteorological MN | SOPs | docx |
| Download | SOP Partisol 2025 | SOPs | docx |
| Download | SOP PM2.5 BAM 1020 DataRetrieval | SOPs | docx |
| Download | SOP PM Instruments Clark county | SOPs | |
| Download | SOP QA Audits and Corrective Action Clark county | SOPs | |
| Download | SOP QC DronePlanDraft | SOPs | docx |
| Download | SOP Scheduling AQS SubmissionOpenNode2 | SOPs | docx |
| Download | SOP Shelter Temperature Calibration Guide | SOPs | docx |
| Download | SOP Standards Operations Clark county | SOPs | |
| Download | SOP Volt Meter Verification Guide | SOPs | docx |
| Download | SOP Wind Speed Sensor | SOPs | docx |
| Download | SOP Zero Air Generator Verification Guide | SOPs | docx |
| Download | QC NationalLakesAssessmentQAPP | Water | |
| Download | QC NewRiversWater QAPP | Water | |
| Download | QC Water FishConsumptionLimits | Water | |
| Download | QC Water FishSampling | Water | |
| Download | QC Water Region 1 Fish samples by Electrofishing in Streams | Water | |
| Download | QC Water Region 9 Surface Water Sampling for Metals | Water | |
| Download | QC Water Region 9 Surface Water Sampling General | Water | |
| Download | QC Water SOP 2013 Surface Water Sampling | Water | |
| Download | QC Water SOP 29 Palms Dissolved Oxygen | Water | |
| Download | QC Water SOP 29 Palms Field pH - Ver 2 | Water | |
| Download | QC Water SOP 29 Palms Field Turbidity - Ver 1 | Water | |
| Download | QC Water SOP 29 Palms Ground Water Sampling Ver 14 | Water | |
| Download | QC Water SOP 29 Palms Monitoring Well Pressure Transducer Maintenance | Water | |
| Download | QC Water SOP 29 Palms Sample Handling & Preservation Table | Water | |
| Download | QC Water SOP 29 Palms Static Water Level Determination Ver 12 | Water | |
| Download | QC Water SOP 29PalmsFieldConductivity | Water | |
| Download | QC Water SOP 29PalmsStreamGauging | Water | |
| Download | QC Water SOP ECASOP-YSI Sondes Rev7 | Water | |
| Download | QC Water SOP ERTSOP2001 Gen Field Sampling Guidelines | Water | |
| Download | QC Water SOP ERTSOP2006 Equip Decon | Water | |
| Download | QC Water SOP ERTSOP2007 Groundwater Sampling | Water | |
| Download | QC Water SOP ERTSOP2012 Soil Sampling | Water | |
| Download | QC Water SOP ERTSOP2016 Sediment Sampling | Water | |
| Download | QC Water SOP ERTSOP2043 Water Level Measurement | Water | |
| Download | QC Water SOP ERTSOP2044 Well Development | Water | |
| Download | QC Water SOP ERTSOP2050 Geoprobe Operation | Water | |
| Download | QC Water SOP FLCAL4SOP | Water | |
| Download | QC Water SOP forSurfaceWaterSamplingExample29Palms | Water | |
| Download | QC Water SOP Region 1 Ambient Water Sampling | Water | |
| Download | QC Water SOP Region 1 EIASOP-Soil Sed Solid | Water | |
| Download | QC Water SOP Region 1 Groundwater Monitor Well Install via Geoprobe | Water | |
| Download | QC Water SOP Region 1 Groundwater Sampling | Water | |
| Download | QC Water SOP Region 1 Low Level Metals Ambient Water Sampling | Water | |
| Download | QC Water SOP Region 1 lowflow8 | Water | |
| Download | QC Water SOP Region 1Transport & Operation of Boats for Sampling | Water | |
| Download | QC Water SOP Region 6 Canopy Cover and Slope Gradient Estimation | Water | |
| Download | QC Water SOP Region 6 Flow Measurement SOP updated 01-31-03 | Water | |
| Download | QC Water SOP Region 6 pH & DO Instrument Calibration SOP | Water | |
| Download | QC Water SOP Region 6 Sample Handling and Preservation Tables | Water | |
| Download | QC Water SOP Region 6 Water Sampling for Dissolved Metals-in-Water | Water | |
| Download | QC Water SOP Region 6 Water Sampling for Dissolved Metals | Water | |
| Download | QC Water SOP Region 9 Groundwater GUI | Water | |
| Download | QC Water SOP Region 9 Sedimentsample GUI | Water | |
| Download | QC Water SOP Region 9 Soilsample GUI | Water | |
| Download | QC Water SOP Region 9 VOCsoilsample GUI | Water | |
| Download | QC Water SOP Sample Preservation Specs | Water | |
| Download | QC Water SOP Sample Preservation SpecsA | Water | |
| Download | QC Water SOP Sample Preservation SpecsB | Water | |
| Download | QC Water SOP SAMPLING EQUIPMENT DECONTAMINATION | Water | |
| Download | QC water qapp-TN-106 | Water | |
| Download | QC ExampleSurface Water QAPP | Water | |
| Download | QC NC waterQAPP | Water | |
| Download | SOP Water MaineStreamSampling | Water | |
| Download | SOP WaterBasicParameters | Water | |
| Download | SOP WaterCaliforniaStreamBioassessmentProcedure2003 | Water | |
| Download | SOP WaterCommunityBioAssessmentEPA Guidance | Water | |
| Download | SOP WaterVolunteerStreamSamplingMethodsManual | Water | |
| Download | Water ExampleSurface Water QAPP | Water | |
| Download | Water FishConsumptionLimits | Water | |
| Download | Water FishSampling | Water | |
| Download | Water Region 1 Fish samples by Electrofishing in Streams | Water | |
| Download | Water Region 9 Surface Water Sampling for Metals | Water | |
| Download | Water Region 9 Surface Water Sampling General | Water | |
| Download | Water SOP 2013 Surface Water Sampling | Water | |
| Download | Water SOP 29 Palms Dissolved Oxygen | Water | |
| Download | Water SOP 29 Palms Field pH - Ver 2 | Water | |
| Download | Water SOP 29 Palms Field Turbidity - Ver 1 | Water | |
| Download | Water SOP 29 Palms Ground Water Sampling Ver 14 | Water | |
| Download | Water SOP 29 Palms Monitoring Well Pressure Transducer Maintenance | Water | |
| Download | Water SOP 29 Palms Sample Handling & Preservation Table | Water | |
| Download | Water SOP 29 Palms Static Water Level Determination Ver 12 | Water | |
| Download | Water SOP 29PalmsFieldConductivity | Water | |
| Download | Water SOP 29PalmsStreamGauging | Water | |
| Download | Water SOP ECASOP-YSI Sondes Rev7 | Water | |
| Download | Water SOP ERTSOP2001 Gen Field Sampling Guidelines | Water | |
| Download | Water SOP ERTSOP2006 Equip Decon | Water | |
| Download | Water SOP ERTSOP2007 Groundwater Sampling | Water | |
| Download | Water SOP ERTSOP2012 Soil Sampling | Water | |
| Download | Water SOP ERTSOP2016 Sediment Sampling | Water | |
| Download | Water SOP ERTSOP2043 Water Level Measurement | Water | |
| Download | Water SOP ERTSOP2044 Well Development | Water | |
| Download | Water SOP ERTSOP2050 Geoprobe Operation | Water | |
| Download | Water SOP FLCAL4SOP | Water | |
| Download | Water SOP forSurfaceWaterSamplingExample29Palms | Water | |
| Download | Water SOP Region 1 Ambient Water Sampling | Water | |
| Download | Water SOP Region 1 EIASOP-Soil Sed Solid | Water | |
| Download | Water SOP Region 1 Groundwater Monitor Well Install via Geoprobe | Water | |
| Download | Water SOP Region 1 Groundwater Sampling | Water | |
| Download | Water SOP Region 1 Low Level Metals Ambient Water Sampling | Water | |
| Download | Water SOP Region 1 lowflow8 | Water | |
| Download | Water SOP Region 1Transport & Operation of Boats for Sampling | Water | |
| Download | Water SOP Region 6 Canopy Cover and Slope Gradient Estimation | Water | |
| Download | Water SOP Region 6 Flow Measurement SOP updated 01-31-03 | Water | |
| Download | Water SOP Region 6 pH & DO Instrument Calibration SOP | Water | |
| Download | Water SOP Region 6 Sample Handling and Preservation Tables | Water | |
| Download | Water SOP Region 6 Water Sampling for Dissolved Metals-in-Water | Water | |
| Download | Water SOP Region 6 Water Sampling for Dissolved Metals | Water | |
| Download | Water SOP Region 9 Groundwater GUI | Water | |
| Download | Water SOP Region 9 Sedimentsample GUI | Water | |
| Download | Water SOP Region 9 Soilsample GUI | Water | |
| Download | Water SOP Region 9 VOCsoilsample GUI | Water | |
| Download | Water SOP Sample Preservation Specs | Water | |
| Download | Water SOP Sample Preservation SpecsA | Water | |
| Download | Water SOP Sample Preservation SpecsB | Water | |
| Download | Water SOP SAMPLING EQUIPMENT DECONTAMINATION | Water | |
| Download | WaterBasicParameters | Water | |
| Download | WaterCommunityBioAssessmentEPA Guidance | Water | |
| Download | WaterNationalLakesAssessmentQAPP | Water | |
| Download | WaterNewRiversWater QAPP | Water | |
| Download | WaterVolunteerStreamSamplingMethodsManual | Water | |
| Download | Water NC waterQAPP | Water | |
| Download | water qapp-TN-106 | Water |


object Valence : TuneableTrackAttribute<Float>
A measure from 0.0 to 1.0 describing the musical positiveness conveyed by a track. Tracks with high valence sound more positive (e.g. happy, cheerful, euphoric), while tracks with low valence sound more negative (e.g. sad, depressed, angry).


Using Kotlin for Android Development
Kotlin is a great fit for developing Android applications, bringing all of the advantages of a modern language to the Android platform without introducing any new restrictions:
- Compatibility: Kotlin is fully compatible with JDK 6, ensuring that Kotlin applications can run on older Android devices with no issues. The Kotlin tooling is fully supported in Android Studio and compatible with the Android build system.
- Performance: A Kotlin application runs as fast as an equivalent Java one, thanks to very similar bytecode structure. With Kotlin's support for inline functions, code using lambdas often runs even faster than the same code written in Java.
- Interoperability: Kotlin is 100% interoperable with Java, allowing to use all existing Android libraries in a Kotlin application. This includes annotation processing, so databinding and Dagger work too.
- Footprint: Kotlin has a very compact runtime library, which can be further reduced through the use of ProGuard. In a real application, the Kotlin runtime adds only a few hundred methods and less than 100K to the size of the .apk file.
- Compilation Time: Kotlin supports efficient incremental compilation, so while there's some additional overhead for clean builds, incremental builds are usually as fast or faster than with Java.
- Learning Curve: For a Java developer, getting started with Kotlin is very easy. The automated Java to Kotlin converter included in the Kotlin plugin helps with the first steps. Kotlin Koans offer a guide through the key features of the language with a series of interactive exercises.
Kotlin for Android Case Studies
Kotlin has been successfully adopted by major companies, and a few of them have shared their experiences:
- Pinterest has successfully introduced Kotlin into their application, used by 150M people every month.
- Basecamp's Android app is 100% Kotlin code, and they report a huge difference in programmer happiness and great improvements in work quality and speed.
- Keepsafe's App Lock app has also been converted to 100% Kotlin, leading to a 30% decrease in source line count and 10% decrease in method count.
Tools for Android Development
The Kotlin team offers a set of tools for Android development that goes beyond the standard language features:
- Kotlin Android Extensions is a compiler extension
that allows you to get rid of
findViewById()
calls in your code and to replace them with synthetic compiler-generated properties. - Anko is a library providing a set of Kotlin-friendly wrappers around the Android APIs, as well as a DSL that lets you replace your layout .xml files with Kotlin code.
Next Steps
- Download and install Android Studio 3.0, which includes Kotlin support out of the box.
- Follow the Getting Started with Android and Kotlin tutorial to create your first Kotlin application.
- For a more in-depth introduction, check out the reference documentation on this site and Kotlin Koans.
- Another great resource is Kotlin for Android Developers, a book that guides you step by step through the process of creating a real Android application in Kotlin.
- Check out Google's sample projects written in Kotlin.


codecademy
Businessmodel of Codecademy
Customer Segments
Codecademy is targeted at users that wish to learn coding and programming skills, catering to a broad range of abilities and interests, primarily those with no or little knowledge of coding.
These could be jobseekers, those seeking to change their career path, hobbyists, people who want to set up their own website and people who wish to expand their skillset.
In 2011, Sims stated that the majority of the Company’s users were aged between 15 and 30. In the intervening years, this may have changed, with the Company now boasting more than 24 million users.
The US is its largest market, accounting for around 30% of the site’s web traffic, followed by the UK with just under 7%, India with around 6%, Canada with 3% and China with just under 3%.
Value Propositions
Codecademy’s most immediate value to users is that its platform and resources are accessible free of charge. The Company provides a broad range of resources and learning content, including a number of courses contributed by leading tech companies, that make coding more accessible to all demographics, including students, professionals and non-technical users.
Codecademy’s resources are available in multiple languages allowing users around the world to utilise the platform. Users are also provided technical and community support from the Company and its user base.
Channels
Codecademy’s learning platform can be accessed through its mobile and desktop websites at www.codecademy.com. The Company also provides an iOS app named Codecademy: Hour of Code, that allows users to start learning programming skills on the go.
Customer Relationships
Codecademy is available to users on a self-service basis. Customers can register via the Company’s homepage and start learning at their own pace. Upgrading to the platform’s premium subscription can also be done via the website with no interaction with Codecademy representatives necessary.
Codecademy provides a range of support resources to its users – including FAQs and guides – through its Knowledge Centre and enables users and Codecademy representatives to discuss and collaborate on issues through its community forum.
Premium customers are provided instant support from one of Codecademy’s technical advisors.
Customers can also interact with the Company through its blog; its YouTube, Facebook, Twitter, Medium, LinkedIn and Instagram accounts; and through its own community-led sub-reddit.
Key Activities
Codecademy is an online learning company. It develops and operates a platform that allows users to learn coding and programming skills free of charge. The platform provides a range of courses, including coding classes for ten different languages, as well as for specific tasks such as designing a website or developing an API.
Key Partners
Codecademy partners with a range of companies, including tech firms, software developers, content providers, and other online learning companies such as Galvanize. This includes partners in the US as well as across the UK, France, Estonia, Argentina and Brazil. Codeacademy provides lessons for building APIs contributed by high-profile tech firms including Twitter, Evernote, Box, Microsoft SkyDrive, Github, MailChimp and Gilt. The Company has also partnered with Periscope to provide courses designed to teach SQL.
The Company has also partnered with non-profits and government bodies on various coding initiatives. It collaborated with the Obama Administration as part of the TechHire initiative to bring coding education to underserved groups, partnering in with Libraries Without Borders to help 600 minority students participate in Codecademy Meets. Codecademy has also partnered with Google and DonorsChoose in a scheme to encourage high school girls to study Computer Science.
Codecademy does not currently generate significant revenue. As such, its financial investors remain key partners.
Key Resources
Codecademy’s key resources are its software platform, its IT infrastructure, its learning resources, and its personnel. The Company’s partnerships, in particular its content partners and financial investors, are key to the success of the Codecademy platform. Codecademy does not have any patent applications filed with the US Patent and Trademark Office.
Cost Structure
Codecademy incurs costs relating to the development of its platform, maintenance of its IT infrastructure, the creation of its content, the management of its partnerships and the retention of its personnel. On its website the Company lists a team of 27 staff members working out of a single office in New York, representing costs in the form of salaries, benefits, rent and utilities.
Revenue Streams
Codecademy generates revenue using a freemium business model. The Company provides access to its learning platform free of charge but also offers a more advanced, more feature-laden premium package for which users pay a monthly subscription. The Premium package is currently priced at $19.99 per month or $199.99 for a full year.
Codecademy only relatively recently began to monetise its offerings after a great deal of speculation as to what the Company’s business model would be. Further revenue streams are expected to be added; however, revenue generation is not currently a priority of Codecademy Chief Executive Officer Sims, according to reports.


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SURVIVING PICASSO A film review by Scott Renshaw Copyright 1996 Scott Renshaw
(Warner Bros.) Starring: Anthony Hopkins, Natascha McElhone, Julianne Moore. Screenplay: Ruth Prawer Jhabvala. Producers: Ismail Merchant, David L. Wolper. Director: James Ivory. MPAA Rating: R (brief nudity, adult themes) Running Time: 125 minutes. Reviewed by Scott Renshaw.
I'm not sure what it is that inspired the highly respected Merchant Ivory production team to turn from fiction to biography, but I wish they would drop it already. For years, director James Ivory, producer Ismail Merchant and screenwriter Ruth Prawer Jhabvala turned out exceptionally well-crafted adaptations of novels commonly held to be too difficult to film: A ROOM WITH A VIEW, HOWARDS END, THE REMAINS OF THE DAY. Then last year they made JEFFERSON IN PARIS, a study of Thomas Jefferson in which Nick Nolte looked as bored in the title role as the audience must have been watching him. SURVIVING PICASSO is not nearly as paralyzingly dull, but it demonstrates once again that real people seem to elude Ivory and company.
Frequent Merchant Ivory collaborator Anthony Hopkins stars as celebrated artist Pablo Picasso, who is a living legend in his 60s when the film opens in occupied Paris in 1943. Picasso is all too aware of his own celebrity, and uses it to seduce impressionable young women like aspiring artist Francoise Gilot (Natascha McElhone).  Francoise believes it will be different with her, but eventually discovers that Picasso uses far more than he loves.
SURVIVING PICASSO is a particularly frustrating piece of work because you can see what it is Ivory was trying to do. Francoise is introduced early as a strong-willed woman who risks the wrath of her domineering father (Bob Peck) when she decides to pursue her artistic career, setting up a conflict between Francoise and Picasso in which he has greater difficulty reducing her to yet another of his weeping conquests. McElhone, a newcomer with piercing eyes, brings out the intelligence in Francoise, but she becomes a distressingly passive character, even more so when she becomes a narrator for flashbacks. Her unique personality has little chance to emerge, and even seems to be contradicted directly by a voice-over near the end in which Francoise notes that Picasso "made me strong." It is a baffling conclusion, since she seems infinitely stronger before their relationship begins.
Picasso himself is similarly enigmatic, played by Anthony Hopkins as an egomanical bon vivant, turning his fame into a method for making people feel insignificant, and honored merely to be in his presence. Ivory and Jhabvala do a fine job of showing us the effect of Picasso's behavior on those around him, but they are much less effective at showing us the man himself. It does not help that we are never able to see Picasso's own work to reflect his personality (his estate did not allow them to be used), making his art little more than a footnote to SURVIVING PICASSO. The film-makers remain distant, showing us bits and pieces of Picasso's life, like a speech to a Communist Party conference, without connecting them to the man he became. We are never allowed inside the passionate artist; Hopkins is technically proficient, wandering accent notwithstanding, but his is a role which seems designed to be observed rather than understood.
You can always count on a Merchant Ivory film to look splendid, and there is superb work once again from cinematographer Tony Pierce-Roberts and production designer Lucianna Arrighi. It is a beautiful environment they have created for this story, but the characters in that environment are disappointing. There are several moments in SURVIVING PICASSO which inspire a curiosity about this fascinating man, notably his rivalry with contemporary Henri Matisse (Joss Ackland), but there is always the nagging sense, as there was in JEFFERSON IN PARIS, that Ivory doesn't really have a point of view. His biographies are safe, almost documentary detailings of events in an individual's life which do not add up to a story. Picasso and Jefferson were both fascinating historical figures with private and public lives full of contradictions, and they should have made for great drama. Ivory simply seems too respectful of his subjects, content to lay out events like they might appear in a history text without much in the way of interpretation. The irony is that he has been able to take far more internalized fictional characters, like THE REMAINS OF THE DAY's James Stevens, and make them extraordinarily vivid. He has not shown the same facility with bringing life to characters who actually lived.
On the Renshaw scale of 0 to 10 abstract artists: 5.
-- Receive Scott Renshaw's reviews directly via email from Marquee! Visit http://www.marquee.com/marquee-mailinglists.html for details. Visit Scott Renshaw's MoviePage http://www-leland.stanford.edu/~srenshaw
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