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CHIRALITY, HANDEDNESS, AND PSEUDOVECTORS

CHIRALITY, HANDEDNESS, AND PSEUDOVECTORS. T. A. Kaplan, S.D. Mahanti Michigan State University Kyle Wardlow Iowa State University. APS March ‘08. INTRODUCTION

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CHIRALITY, HANDEDNESS, AND PSEUDOVECTORS

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  1. CHIRALITY, HANDEDNESS, AND PSEUDOVECTORS T. A. Kaplan, S.D. Mahanti Michigan State University Kyle Wardlow Iowa State University APS March ‘08

  2. INTRODUCTION Chirality: discovery of molecular handedness (Pasteur, 1860); term due to Kelvin ( 1890)—from Greek for hand. Definition, Kelvin: “An object is chiral if no mirror image of the object can be superimposed on itself.” In this context, Handedness Chirality. Important in biology; also applied to spin spirals. Another definition of handedness: That associated with the cross-product of 2 vectors (right-hand rule). (Also 19th century) Are these handednesses the same? This talk: Show the answer is “NO”. Proof by examples (E-M theory, spin spirals, multiferroics).

  3. Why worry? There has been misuse of the terms (our referee experience: people have claimed chirality same as handedness).

  4. Simplest examples 1. Coordinate system. Unit vectors (ordinary, i.e. polar) Handed: (right-handed) Chiral (e.g. reflection in z-x plane changes to left-handed). 2. Lorentz force Handed (Purcell): Achiral (not chiral) (because is a pseudovector, being polar)

  5. In summary, we have an example of an object that is chiral and handed and an example of an object that is achiral and handed. Therefore handedness cannot be equated to chirality in general. (Contradicts referees on paper about CoCr2O4, Choi et. al.) In fact this situation forces us to recognize two kinds of handedness: Handedness1 chirality. Handedness2 that associated with the cross product of two vectors.

  6. Then the picture of the x-y-z coordinate system shows both kinds of handedness: handedness1 (chirality, i.e. no reflection symmetry) handedness2 (vector product handedness). Whereas the picture of Lorentz force F, velocity v, and field B shows only the handedness2 (that of the vector product), because the picture is achiral.

  7. Some other examples. An elm tree. Chiral, but not handed2 (where are the vector products?) Commensurate spin spirals. 1D. (multiferroics) a) … …. (wavelength = 4) Handed2 (vector products ) and achiral: Reflection in plane of spins gives …. ….; translation by restores original, i.e. achiral.

  8. b) … (wavelength = 3) Handed2and chiral: Reflect in plane of spins: No translation will restore the original. In general: even: achiral; odd: chiral.

  9. SUMMARY • We’ve shown: • Chirality and handedness can not be equated in general. • There are historically two distinct kinds of handedness: • Handedness1 = chirality, • Handedness2 = that associated with vector cross products. • Logically should discard Handedness1 (redundant), • and put Handedness2=Handedness. But…

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