Translational Symmetry

7 Key excerpts on "Translational Symmetry"

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  Photonic Crystals

  Molding the Flow of Light - Second Edition

  • John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, Robert D. Meade(Authors)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  Continuous Translational Symmetry Another symmetry that a system might have is continuous translation symmetry. Such a system is unchanged if we translate everything through the same distance in a certain direction. Given this information, we can determine the functional form of the system’s modes. A system with Translational Symmetry is unchanged by a translation through a displacement d. For each d, we can define a translation operator ˆ T d which, when 28 CHAPTER 3 operating on a function f(r), shifts the argument by d. Suppose our system is translationally invariant; then we have ˆ T d ε(r) = ε(r − d) = ε(r), or equivalently, [ ˆ T d , ˆ Θ] = 0. The modes of ˆ Θ can now be classified according to how they behave under ˆ T d . A system with continuous translation symmetry in the z direction is invariant under all of the ˆ T d ’s for that direction. What sort of function is an eigenfunction of all the ˆ T d ’s? We can prove that a mode with the functional form e ikz is an eigenfunction of any translation operator in the z direction: ˆ T d ˆ z e ikz = e ik(z−d) = (e −ikd )e ikz . (4) The corresponding eigenvalue is e −ikd . With a little more work, one can show the converse, too: any eigenfunction of ˆ T d for all d = d ˆ z must be proportional to e ikz for some k. 3 The modes of our system can be chosen to be eigenfunctions of all the ˆ T d ’s, so we therefore know they should have a z dependence of the functional form e ikz (the z dependence is separable). We can classify them by the particular values for k, the wave vector. (k must be a real number in an infinite system where we require the modes to have bounded amplitudes at infinity.) A system that has continuous Translational Symmetry in all three directions is a homogeneous medium: ε(r) is a constant ε (= 1 for free space). Following a line of argument similar to the one above, we can deduce that the modes must have the form H k (r) = H 0 e ik·r , (5) where H 0 is any constant vector.

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  Introduction to Solid State Physics for Materials Engineers

  • Emil Zolotoyabko(Author)
  • 2021(Publication Date)
  • Wiley-VCH

    (Publisher)

  Neumann's principle: the point group of the crystal is a sub-group of the group describing any of its physical properties. In simple words, the symmetry of physical property of the crystal cannot be lower than the symmetry of the crystal: it may be only equivalent or higher.

  In practical terms, it means that if physical property is measured along certain direction within the crystal and then the atomic network is transformed according any symmetry element of its point group and measurement repeats, we expect to obtain the measurable effect of the same magnitude and sign as before. Any deviation will contradict particular crystalline symmetry and, thus, the Neumann's principle. Using mathematical language, physical properties are, generally, described by tensors of different rank, for which the transformation rules under local symmetry operations are well-known. Tensor rank defines the number of independent tensor indices, i, k, l, m,…, each of them being run between 1 and 3, if the 3D space is considered. In most cases, physical property is the response to external field applied to the crystal. Note that external fields are also described by tensors, which are called field tensors to distinguish them from crystal (material) tensors.

  Figure 1.12 Illustration of the Biot–Savart law (Eq. (1.7) ).

  Tensors of zero rank are scalars. It means that they do not change at all under coordinate transformations related to symmetry operations. As an example of scalar characteristics, we can mention the mass density of a crystal. Tensor of rank one is a vector. It has one index i = 1,2,3, which enumerates vector projections on three mutually perpendicular coordinate axes within Cartesian (Descartes) coordinate system. It is easy to point out field vectors, for example, an applied electric field,

  ℰ

  i

  , or electric displacement field,

  D

  i

  . As crystal vector, existing with no external fields, one can recall the vector of spontaneous polarization, , in ferroelectric crystals (see Chapter 12 ). Spontaneous polarization, as well as polarization,

  P

  i

  , induced by external electric field, is defined as the sum of elementary dipole moments per unit volume. Note that polarization P is polar vector having three projections,

  P

  i

  , as e.g. radius-vector r (with projections,

  x

  i

  ). There exist also axial vectors (or pseudo-vectors), i.e. vector products (cross products) of polar vectors, which are used to describe magnetic fields and magnetic moments. In fact, magnetic field, ΔH, produced by the element

  Δl

  of a conducting wire carrying electric current, I

  c

  , is described by the Biot–Savart

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  Quantum Theory of Materials

  • Efthimios Kaxiras, John D. Joannopoulos(Authors)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Symmetries Beyond Translational Periodicity In the previous chapter we discussed in detail the effects of lattice periodicity on the single- particle wavefunctions and the energy eigenvalues. We also touched on the notion that a crystal can have symmetries beyond the translational periodicity, such as rotations around axes, reflections on planes, and combinations of these operations with translations by vectors that are not lattice vectors, called “non-primitive” translations. All these symmetry operations are useful in calculating and analyzing the physical properties of a crystal. There are two basic advantages to using the symmetry operations of a crystal in describing its properties. First, the volume in reciprocal space for which solutions need to be calculated is further reduced, usually to a small fraction of the first Brillouin zone, called the “irreducible” part; for example, in the FCC crystals with one atom per unit cell, the irreducible part is 1/48 of the full BZ. Second, certain selection rules and compatibility relations are dictated by symmetry alone, leading to a deeper understanding of the physical properties of the crystal as well as to simpler ways of calculating these properties in the context of the single-particle picture; for example, using symmetry arguments it is possible to identify the allowed optical transitions in a crystal, which involve excitation or de- excitation of electrons by absorption or emission of photons, thereby elucidating its optical properties. In this chapter we address the issue of crystal symmetries beyond translational periodic- ity in much more detail. To achieve this, we must make an investment in the mathematical ideas of group theory. We restrict ourselves to the bare minimum necessary to explore the effects of symmetry on the electronic properties of the crystal.

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  Qualitative Analysis of Physical Problems

  • M Gitterman(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  Such a group is called Abelian. In other cases, useful results can be obtained by considering Abelian subgroups of the system's symmetry group. For Abelian groups, it is found that all the irreducible representations are one dimensional. Thus, the symmetry of a function is represented by a series of numbers, as in Eq. (101), rather than by the matrices R of Eq. (104). This property of Abelian groups is associated with the well-known fact that commuting operators can be diagonalized simultaneously, since Eq. (101) can be regarded as an eigen-value equation for the operator Pj. We note that the following sets of opera-tions commute with each other: (i) two rotations about the same axis; (ii) rotations through 180° about mutually perpendicular axes; (iii) reflections in mutually perpendicular planes; (iv) a rotation and reflection in the plane perpendicular to the axis of rotation; (v) inversion and any rotation or reflection; (vi) any two translations. The Translational Symmetry of Crystals A crystal is distinguished from other states of matter by its Translational Symmetry. As we discussed in Section 3.1, for any crystal it is possible to choose three non-coplanar basis vectors, which we again denote by a l 5 a 2 , and a 3 , such that translations through the vectors η = n l 2i 1 + n 2 a 2 + rc 3 a 3 (3.107) 130 3 Symmetry connect equivalent sites in the crystal. Strictly speaking, such a definition is only accurate for an infinite system or for one without surfaces. Other-wise, any such translation will alter the distance of a point from the crystal's surfaces and so change its environment. However, at points far away from the surfaces, their influence should be negligible, and the sites will be equiva-lent for all effects and purposes. In view of this, it proves convenient to make them exactly equivalent mathematically by introducing cyclic boundary conditions, which eliminate the surfaces.

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  Introduction to Continuous Symmetries

  From Space-Time to Quantum Mechanics

  • Franck Laloë, Nicole Ostrowsky, Daniel Ostrowsky(Authors)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  These latter are symmetry transformations if the interactions within the system are electromagnetic (or strong), but no longer if weak interactions play a role (the product CPT nevertheless remains a symmetry operation). Note that the translation invariance of the evolution of an isolated system is a concept that would be hard to abandon completely as it is almost the definition of a so-called “isolated physical system”. As for the time translations, the basis of physics or even of the scientific method would be destroyed if they were not, at least approximately 2 , symmetry transformations for isolated systems: the same experiment performed today or tomorrow would yield different results. As it is considered at present that all the physical laws known or to be discovered must satisfy all these symmetries, these latter can be viewed as fun- damental “superlaws ” (Wigner), well worth studying. 1 Obviously, Galilean transformations must also be excluded since they only qualify as sym- metry transformations when they are approximations of the Lorentz transformations, in the “non-relativistic” limit (all the velocities ≪ light velocity, all the distances ∆x ≪ c∆t). 2 In certain cosmological theory, “fundamental” physical constants change (Dirac) as the Universe expands. This changes the group of transformations for which the physical laws are invariant. 3 CHAPTER I SYMMETRY TRANSFORMATIONS Comments: (i) If both T and T ′ are symmetry transformations, so is their product T ′ T (transformation obtained by applying T first, and then T ′ ): a group structure is thus expected for the set of symmetry transforma- tions. (ii) In the case of the time-reversal symmetry, one must actually change in figure 1 S ′ (t) into S ′ (−t) and invert the sense of the vertical arrow on the right, which represents the system evolution (cf. appendix and its figure 2). (iii) For an isolated physical system, all the translations and rotations are symmetry transformations.

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  Thinking about Physics

  • Roger G. Newton(Author)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  112 SYMMETRY IN PHYSICS autonomous and no longer translationally invariant. The same ap-plies to invariance under time translation: unless the equations contain explicitly time-dependent parameters, the equations are invariant under translations in time. Thus the results of experi-ments on autonomous systems depend neither on the location of the laboratory nor on the time at which they are performed. These three symmetries or invariances—with respect to rotations, spa-tial translations, and time translations—embodied in the laws of physics are also sometimes expressed by saying that the universe is isotropic and uniform, both spatially and temporally. Since Einstein's introduction of the special theory of relativity, an additional prerequisite has been added for any theory to be considered acceptable: it must be form invariant under Lorentz transformations. Since the latter are formally quite analogous to spatial rotations, this symmetry translates itself into a requirement similar to rotational invariance. The first step toward enforcing it is to classify all quantities according to their transformation proper-ties under such transformations. However, spatial coordinates and time transform together, so that if the location and time of an event as seen in one laboratory are given by x 11 , (JL = 0,1, 2, 3, where x° = ct, the same event observed in another laboratory moving with respect to the first with a uniform velocity v has the four space-time coordinates 9 which is such that (x 0 ) 2 - (x 1 ) 2 - (x 2 ) 2 - (x 3 ) 2 = (x 0 ) 2 - (x 1 ) 2 - (x 2 ) 2 -(x 3 ) 2 . A scalar, therefore, is a quantity that is invariant under Lorentz transformations, and the 4 quantities A^'-itL , as measured in the first laboratory, are the components of a four-tensor of rank n if in the second they have the values p-... V 9 This transformation may combine both a boost and a spatial rotation. 113

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  Invitation To Contemporary Physics

  • Harry Chi-sing Lam, Narendra Kumar, Ho-kim Quang(Authors)
  • 1991(Publication Date)
  • World Scientific

    (Publisher)

  But this will take us very far afield. We will be content with just mentioning it. In addition to these continuous symmetries, there are discrete space-time symmetries too. One of them is the symmetry under space reflection, also called the mirror symmetry or parity. This produces enigmatic effects in the ordinary laboratory physics and chemistry as also in the extra-ordinary processes involving elementary particles. We will take this up next. 3. REFLECTION SYMMETRY We have spoken of objects having bilateral symmetry, also called the left-right symmetry. A butterfly with outstretched wings or a maple leaf for example. When an object is reflected in a mirror, the left and the right sides of it get interchanged. Thus, an object having bilateral symmetry is by definition superposable on its mirror image. The mirror is just an optical device that enables us to visualize the result of reflection of objects in space through a plane. For these reasons the terms bilateral symmetry, left-right symmetry, mirror symmetry and the symmetry under space reflection are all used interchangeably. In physics, handedness is often referred to as chirality. There is something that sets this symmetry apart from the rest that we have discussed so far. As noted above it is a discrete symmetry unlike the continuous symmetry of rotation or translation, say. Changes caused by continuous symmetry operations can be made arbitrarily small. Not so with discrete ones. You reflect or you don't. Nothing in between. Also, unlike these, it is a non-performable symmetry operation. Space reflection involves turning the object inside out laterally, an operation we can hardly perform continuously. But we can and we do visualize it by the optical trick of reflect-ing it in a mirror. Having visualized it so, nothing prevents us from making a physical copy of the image, using silly putty, say, which can then be tested for superposability on our object.

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