Group Theoretical Methods in Physics by Robert Shar (Eds.)

By Robert Shar (Eds.)

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From a macroscopic to a B. From an averaged to a refined microscopic des- microscopic description cription (i) Basic symmetry (approximated) macroscopic crystal form point group K Q main reflexions only C 0(3) (or basic structure) average space group G (ii) C E(3) New admitted transformations space translations: Τ 3 d "internal" translations T^ requiring the extension to: E(3) = Τ (iii) Q 3 requiring the extension to: Λ 0(3) Pure new symmetries E(3) x E(d) (associated with the identity of the old ones) group of "primitive" (or group of "internal" lattice lattice) translations A ^ (iv) New symmetry group G microscopic symmetry group: 3-dim space group (v) translations refined symmetry group (3+d)dim superspace group Mathematical structure of G (group extension (in general non trivial)) O + A + G + K+ l 3 or A _ < G; G / A * Κ ο non-symmorphic case (vi) O + D + G + G + l d Ε D, < G ; G/D, Ξ Μ - G α d h modulated case C Relation with the basic (approximated) symmetry Κ C κ — ο Κ: point group of G = "homogeneous" parts of the space group G G C Ε — C G ο G : space group of £ G = "external" parts of the superspace group G 29 Nuclei, Atoms, Solids (vii) Physical properties (like systematic extinctions) (in general) - explained by G but - explained by G but not by Κ only - K q not by G ^ only symmetry elements - G easily recognized symmetry elements q easily recognized.

5) that / vw dx = X / w dx. 2 2 If w is normalized, we get r 2 2 λ = (X ,v) = J vw dx = (w , v ) . 6) v J X = w . u 2 To build Hamiltonian systems we need to choose some operator J. 7) J = D 7 was suggested by Clifford Gardner . metric and independent of u. 16) asso­ ciated with the Hamiltonians F and F 1 given in ( 3 . 8). 8) 2 2 Equation ( 3 . 8 ) 2 = JF- = Du = u 1 χ u u. = JF. 8) t 1 is linear and easily solved: u = u(x+t). Equation on the other hand is nonlinear and far from being easy to solve; it is the celebrated Korteweg-de Vries (KdV) equation.

Note that Σ , (Λ ,D,) and n+d η d 6 ^n+d^n'*\P ^ a v e t ^ i e s a m e v holohedry H. J For the corresponding representations Γ(Η) and f(Η) obtained by referring Η to bases Β and B, respectively, given by (1) (which are standard) one veri­ fies using (127) that Γ(Η) and f(H) are conjugated subgroups of Gl(n,d,Z). More generally the arbitrariness in the choice of bases (1) is as follows: (i) One may change the description of the main reflections by choosing another basis of Λ*: k (ii) * X % Ehk> h=l S Sje l(n,Z).

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