## The PhotoElectron Angular Distribution

Photoionization of randomly oriented molecules with polarized (or un-polarized) light produces a photoelectron angular distribution (PAD) that takes a well known form:

`I`(θ)=1 +

`βP`

_{2}(cos θ).

`P`

_{2}is the 2

^{nd}Legendre Polynomial function and provides a non-isotropic part to the overall distribution. The parameter, β ( −1 ≤ β ≤ +2 ) specifies the magnitude of the anisotropy which ranges from a pure cos

^{2}θ to sin

^{2}θ form, but which therefore remains symmetrical around the direction θ = 90°.

Less widely recognised, is that this equation is just a special (though common) sub-case of the more general expression:

`I`(θ) = 1 +

_{p}`b`

_{1}

^{p}

`P`

_{1}(cosθ) +

`b`

_{2}

^{p}

`P`

_{2}(cosθ)

`b`are determined by the photoionization dynamics and depend on the light polarization —

_{n}^{p}`p`= 0 for linear,

`p`= ±1 for left, right circular polarization (CPL) states. Coupling of the electron and photon angular momenta introduces certain symmetry properties and constraints. For the

`P`

_{2}terms

`b`

_{2}

^{0}= − ½ ×

`b`

_{2}

^{±1}≡ β . More particularly,

`b`

_{1}

^{0}= 0 while

`b`

_{1}

^{±1}also vanishes for

**achiral**molecules; in such circumstances this general expression simply reduces to the well-known first form.

But, for the specific case of a chiral molecule ionized with
CPL, these `P`_{1}
(1^{st} Legendre Polynomial) coefficients
are found to be antisymmetric for exchange of handedness of
either the molecular enantiomer or the light polarization:
`b`_{1}^{+1}
= − `b`_{1}^{−1}
.

## Asymmetry in the PAD

The
additional `P`_{1}
term in the general PAD expression,opposite, is a simple cosθ
function. For a chiral molecule ionized by CPL it therefore
introduces a
**forward-backward asymmetry** into the PAD –
θ measured from the light beam propagation direction –
which reverses with exchange of either the enantiomer or CPL
state. Using the symmetry properties of
the `b _{n}^{p}`
coefficients, a quantitative measure of this PAD asymmetry for
a

*given*enantiomer/CPL pair is written:

`G`

_{AD}=

`I`(0°) −

_{p}`I`(180°) = 2

_{p}`b`

_{1}

^{p}

The alternative asymmetry
factors `G`_{AD} and
Γ provide essentially identical measures, and either allow
experemental measurements to be related to the value of the
`b`_{1}^{p}
coefficient which can be calculated using appropriate quantum
treatments.

## Angle Resolved Circular Dichroism

For a given enantiomer, one can define the photoelectron
circular dichroism (**PECD**) as the difference
seen, at a fixed detection angle θ, between left and
right CPL:

_{0}=

`I`

_{+1}(0°) −

`I`

_{−1}(0°) = [

`I`

_{+1}(θ) −

`I`

_{−1}(θ)] ⁄ cosθ = 2

`b`

_{1}

^{+1}

`b`

_{1}

^{±1}coefficients. Γ therefore measures the dichroism in the angle-resolved photoelectron spectrum. For the other enantiomer,

`b`

_{1}

^{+1}will have opposite sign, so that the PECD, Γ, can also be expected to change sign.

## Giant Chiral Asymmetry

The PECD chiral asymmetry discussed on this page is
obtained from pure electric dipole interactions, unlike
normal CD that requires much weaker electric quadrupole and
magnetic dipole terms. Typical
`G`_{AD} or
Γ_{0} values
(10^{-2}–
2×10^{-1}) consequently exceed
more conventional CD by 2–3 orders of
magnitude. Indeed PECD is the strongest chiroptical
asymmetry that can be observed in randomly oriented, dilute
systems in the absence of any cooperative enhancement
effects.

## More About ...

## Recording PECD

This picture shows a 3D slice, mathematically inverted from a raw experimental 2D image, of valence photoelectrons from pure R-enantiomers of camphor. The HOMO photoelectron band appears as the reddish outer ring structure and the forward–backward (here it appears up–down) chiral asymmetry can be seen in this false colour representation. Also visible is a similar asymmetry, but with the opposite direction, from the HOMO-1 orbital ionization (the inner ring structure).

For sensitivity, however, we prefer to record PECD as a difference image, switching between left and right CPL. The asymmetry in such difference images appears as a positive signal in one direction, a negative in the other — as can be seen in other illustrations on this page.

**Notice that**

`G`_{AD}and Γ are essentially equivalent measures of the dynamical parameter`b`_{1}^{p}