Download Electron momentum calculations

Electron momentum calculations
Geometry of the SX-ARPES experiment at the ADRESS beamline is shown on the figure. The
horizontal incident photon beam and analyser axis form the vertical measurement plane (MP). The
analyser axis is inclined by α = 20o. The
manipulator axis is horizontal and perpendicular
to the MP. The parallel momentum k//x is varied
through the primary manipulator rotation θ M,
e
and k//y through the tilt ϕM. The analyser slit can
θA
be oriented either in the MP or perpendicular to
it, θA and ϕA being the angles along the analyzer
ϕA
k z // n
slit in these two cases. The origin and sign
x
θM
k //
convention for θ M, ϕM, θA and ϕA is indicated on
α
the figure. Note that with this convention the
increase of each of these coordinates increases
manip axis
the corresponding k//.
The photon momentum is calculated as
p ph =
k
2p
hv
12400
y
//
ϕM
hv
Hereinafter the momenta are measured in Å-1 and energies in eV.
With our sign convention, the two parallel momentum components of the initial (photohole) state
can be found as
k x = 0.5124 hv − eφ + EB sin(ϑ A + ϑM ) − 2π ⋅ hv ⋅ cos(α + ϑM ) / 12400
k y = 0.5124 hv − eφ + EB sin(ϕ A + ϕ M ) + 2π ⋅ hv ⋅ sin(α + ϑM ) sin ϕ M / 12400 ,
where eφ >0 is the workfunction, EB < 0 binding energy, ϕA = 0 if the slit is oriented in the MP and
θA = 0 if perpendicular to it, and ϑN is the surface normal angle relative to the analyzer axis (in the
shown case ϑN < 0). Note that (1) the second term accounts for pph and does not depend on the θA/
ϕA angles along the slit; (2) due to rather grazing light incidence in our case, the photon momentum
correction to k //y is small near the normal emission (at hv = 1000 eV, θM = 0 and ϕM = 10o, for
example, the correction is only ~0.03 Å-1 ) but increases at larger θM and ϕM.
For the perpendicular momentum,
(
)
k z = 0.5124 hv + EB + V000 − 3.81 (k xf ) 2 + (k yf ) 2 + 2π ⋅ hv ⋅ sin(α + ϑM ) / 12400 ,
where V000 > 0 is the inner potential relative to EF, and k xf = 0.5124 hv − ef + EB sin(ϑ A + ϑM ) and
k yf = 0.5124 hv − ef + EB sin(ϕ A + ϕ M ) are the final state (without the pph-correction) parallel
momentum components.