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Theoretical background
Published in Guo-ping Zhang, Georg Lefkidis, Mitsuko Murakami, Wolfgang Hübner, Tomas F. George, Introduction to Ultrafast Phenomena from Femtosecond Magnetism to high-harmonic Generation, 2020
Guo-ping Zhang, Georg Lefkidis, Mitsuko Murakami, Wolfgang Hübner, Tomas F. George
Once we find a lattice, then we can figure out its symmetry by checking whether a particular symmetry operation leaves the lattice unchanged. These operations form a group. Since the lattice obeys the spatial translational symmetry (space) and additional symmetry operations, this symmetry group is called a space group.q A proper choice of lattice vectors is essential to this. For instance, if one chooses four atoms or two lattice points (see the dashed box on the right side of Fig. 3.3(a)), then this cell contains more than one lattice point and is not a primitive cell. Since two lattice vectors have different lengths, a 60° rotation symmetry is artificially lost. Figure 3.3(b) has two primitive lattice vectors, a1 = (a, 0, 0) and a2 = (a2,−3a2,0), with the same length |a1| = |a2|, so this choice keeps the rotation symmetry of the crystal. Here a is the lattice constantr which is the side length of the Bravais lattice.
Chapter 11: Group Methods for ODES
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
What is of interest is the case where, among all symmetry groups admitted by the Euler-Lagrange equation (11.3.3.9), there are also groups admitted by the Lagrangian L(x,y,yx′,…,yx(n)) $ L(x, y, y_{x}^{'} , \ldots , y_{x}^{(n)} ) $ . Such symmetry groups are known as variational or Nötherian; they play a major role in physics and mathematics, since they are closely related to conservation laws. Obviously, the order of equation (11.3.3.9) is 2n.
Principles of Calculation of Symmetry Properties of High-Rank Material Tensors
Published in S. V. Popov, P Svirko Yu, N. I. Zheludev, Susceptibility Tensors for Nonlinear Optics, 2017
S. V. Popov, P Svirko Yu, N. I. Zheludev
Crystallography shows that all in all there are 32 classes of crystal symmetry (see e.g. Juretschke 1974, Sirotin and Shaskolskaya 1979, Pelliott and Dawber 1990, Vainshtein 1981 and Bhagavantam 1966). This means that all crystals are divided into 32 groups; within each group, any physical properties of crystals are invariant under a certain set of symmetry operations, known as the crystal symmetry group. The number of symmetry operations in a group is the group order (we will denote it by the letter m). All symmetry groups are listed in table 3.1 where the so-called international notation is used to designate classes (second column). In this table, the 32 crystal classes are complemented by the so-called Curie groups (textures), which describe media with infinite-fold symmetry axes. Classes are subsumed into systems according to the main symmetry attributes common to all classes of the system.
Investigations on the enhancement of thermomagnetic properties in Fe2.4Ga0.6O4
Published in Phase Transitions, 2023
K. Rećko, M. Orzechowska, W. Olszewski, A. Beskrovnyy, M. Biernacka, U. Klekotka, A. Miaskowski, K. Szymański
A space group is formed when combining a point symmetry group with a set of lattice translation vectors. The space group Fdm requires four transition vectors: (1) – corresponding to the vertex translation and three related to the centering of unit cells on the face (the first numerical row in the second column of Table 1). According to Wyckoff notation, atomic coordinates, as a result of the symmetry operation, guarantee the invariance of the sublattice. The atomic coordinates together with selective transition vectors allow tracking variants of the spin order in sublattices with mixed occupancy and variable magnetic modes. Analyzing the occupancy of the A-sublattice by gallium cations, one can simulate all possible configurations of ion arrangements in the spinel structure (Figure 3(a–f)). Considering the equivalent combinations of two of the four translation vectors (row 2 of Table 1), it is possible to analyze the distribution of cations in the B sublattice, in which only iron ions are found. Depending on the implemented translation vectors, for example, (1) and (2) (Figure 3(a)) or (1) and (3) (Figure 3(b)) the magnetic ordering propagates differently at (16d) positions. Two of the six configurations favor the formation of homogeneous chains of Fe2+ ions along the diagonal of the base separated by non-magnetic gallium chains and correspondingly perpendicular chains formed by clearly weaker magnetic moments of Fe3+ (see Figure 3(b,e)). It is easy to predict that, under the action of an alternating magnetic field, such systems will require longer relaxation times and will be associated with slower superparamagnetic fluctuations. The other four configurations will be related with shorter relaxation times and faster superparamagnetic fluctuations. Obviously, the configuration distribution can change regardless of the increasing gallium content in the layout or the cation distribution in the sublattices in favor of random configurations. The other source leading to the creation of non-equivalent configurations is non-stoichiometric sample composition.