CHIRALITY HANDEDNESS AND PSEUDOVECTORS T A Kaplan S

INTRODUCTION Chirality: discovery of molecular handedness (Pasteur, 1860); term due to Kelvin ( 1890)—

Why worry? There has been misuse of the terms (our referee experience: people have

Simplest examples 1. Coordinate system. Unit vectors Handed: (ordinary, i. e. polar) (right-handed) Chiral

In summary, we have an example of an object that is chiral and handed

Then the picture of the x-y-z coordinate system shows both kinds of handedness: handedness

Some other examples. An elm tree. Chiral, but not handed 2 (where are the

b) … (wavelength = 3) Handed 2 and chiral: Reflect in plane of spins:

SUMMARY We’ve shown: • Chirality and handedness can not be equated in general. •

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

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 19 th century) Are these handednesses the same? This talk: Show the answer is “NO”. Proof by examples (E-M theory, spin spirals, multiferroics).

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

Simplest examples 1. Coordinate system. Unit vectors Handed: (ordinary, i. e. polar) (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, polar) being

In summary, we have an example of an object that is chiral and handed an example of an object that is achiral and handed. Therefore handedness cannot be equated to chirality in general. (Contradicts referees on paper about Co. Cr 2 O 4, Choi et. al. ) In fact this situation forces us to recognize two kinds of handedness: Handedness 1 chirality. Handedness 2 that associated with the cross product of two vectors.

Then the picture of the x-y-z coordinate system shows both kinds of handedness: handedness 1 (chirality, i. e. no reflection symmetry) handedness 2 (vector product handedness). Whereas the picture of Lorentz force F, velocity v, and field B shows only the handedness 2 (that of the vector product), because the picture is achiral.

Some other examples. An elm tree. Chiral, but not handed 2 (where are the vector products? ) Commensurate spin spirals. 1 D. (multiferroics) a) … …. (wavelength = 4) Handed 2 (vector products ) and achiral: Reflection in plane of spins gives …. ; translation by restores original, i. e. achiral.

b) … (wavelength = 3) Handed 2 and chiral: Reflect in plane of spins: No translation will restore the original. In general: even: achiral; odd: chiral.

SUMMARY We’ve shown: • Chirality and handedness can not be equated in general. • There are historically two distinct kinds of handedness: Handedness 1 = chirality, Handedness 2 = that associated with vector cross products. Logically should discard Handedness 1 (redundant), and put Handedness 2=Handedness. But…