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 19th 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 xˆ , yˆ , zˆ (ordinary, i.e. polar)
Handed:
ẑ
zˆ  xˆ  yˆ (right-handed)
Chiral (e.g. reflection in z-x plane
ŷ
changes to left-handed).
x̂
2. Lorentz force
F
B
v
q
Handed (Purcell): F = v  B
c
Achiral (not chiral)
(because B is a pseudovector, F, v being
polar)
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:
 chirality.
Handedness2  that associated with the cross
Handedness1
product of two vectors.
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.
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 Sn  Sn1) and achiral:
Reflection in plane of spins gives
….
….; translation by  / 2
restores original, i.e. achiral.
b) …
(wavelength = 3)
Handed2 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:
Handedness1 = chirality,
Handedness2 = that associated with vector cross
products.
Logically should discard Handedness1 (redundant),
and put Handedness2=Handedness. But…