Sherman function

File:Spin-orbit coupling potential.png

When an electron beam is polarized, an unbalance between spin-up, $n\_\{up\}$, and spin-down electrons, $n\_\{down\}$, exists. The unbalance can be evaluated through the polarization $P$ {{cite journal |last1=Mayne |first1=K. I. |title=Polarized electron beams |journal=Contemporary Physics |date=July 1969 |volume=10 |issue=4 |pages=387–412 |doi=10.1080/00107516908204794|bibcode=1969ConPh..10..387M }} defined as

:$P\; =\; \backslash frac\{n\_\{up\}-n\_\{down\}\}\{n\_\{up\}+n\_\{down\}\}$.

It is known that, when an electron collides against a nucleus, the scattering event is governed by Coulomb interaction. This is the leading term in the Hamiltonian, but a correction due to spin-orbit coupling can be taken into account and the effect on the Hamiltonian can be evaluated with the perturbation theory. Spin orbit interaction can be evaluated, in the rest reference frame of the electron, as the result of the interaction of the spin magnetic moment of the electron

:$\backslash boldsymbol\{\backslash mu\}\_S\; =\; -g\_\backslash text\{s\}\; \backslash mu\_\backslash text\{B\}\; \backslash frac\{\backslash mathbf\{S\}\}\{\backslash hbar\},$

with the magnetic field that the electron sees, due to its orbital motion around the nucleus, whose expression in the non-relativistic limit is:

:$\backslash mathbf\{B\}\; =\; \backslash frac\{1\}\{m\_\backslash text\{e\}\; ec^2\}\; \backslash frac\{1\}\{r\}\; \backslash frac\{\backslash partial\; U(r)\}\{\backslash partial\; r\}\; \backslash mathbf\{L\}.$

In these expressions $\backslash mathbf\{S\}$ is the spin angular-momentum, $\backslash mu\_\backslash text\{B\}$ is the Bohr magneton, $g\_\backslash text\{s\}$ is the g-factor, $\backslash hbar$ is the reduced Planck constant, $m\_\backslash text\{e\}$ is the electron mass, $e$ is the elementary charge, $c$ is the speed of light, $U\; =\; eV$ is the potential energy of the electron and $\backslash mathbf\{L\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{p\}$ is the angular momentum.

Due to spin orbit coupling, a new term will appear in the Hamiltonian, whose expression is{{cite book |last1=Griffiths |first1=Davis J. |title=Introduction to quantum mechanics |date=2005 |publisher=Pearson Prentice Hall |isbn=0131118927 |edition=2nd}}{{page needed|date=July 2020}}

:$V\_\backslash text\{SO\}\; =\; -\backslash boldsymbol\{\backslash mu\_s\}\backslash cdot\backslash mathbf\{B\}$.

Due to this effect, electrons will be scattered with different probabilities at different angles. Since the spin-orbit coupling is enhanced when the involved nuclei possess a high atomic number Z, the target is usually made of heavy metals, such as mercury, gold{{cite book |last1=Ciullo |first1=Giuseppe |last2=Contalbrigo |first2=Marco |last3=Lenisa |first3=Paolo |title=Polarized Sources, Targets and Polarimetry : Proceedings of the 13th International Workshop. |date=2009 |publisher=World Scientific Publishing Co Pte Ltd |isbn=9781283148580 |pages=337}} and thorium.{{cite journal |last1=Berti |first1=G. |last2=Calloni |first2=A. |last3=Brambilla |first3=A. |last4=Bussetti |first4=G. |last5=Duò |first5=L. |last6=Ciccacci |first6=F. |title=Direct observation of spin-resolved full and empty electron states in ferromagnetic surfaces |journal=Review of Scientific Instruments |date=July 2014 |volume=85 |issue=7 |pages=073901 |doi=10.1063/1.4885447|pmid=25085146 |bibcode=2014RScI...85g3901B |hdl=11311/825526 |s2cid=38096215 |hdl-access=free }}

If we place two detectors at the same angle from the target, one on the right and one on the left, they will generally measure a different number of electrons $n\_\{R\}$ and $n\_\{L\}$. Consequently it is possible to define the asymmetry $A$, as

:$A\; =\; \backslash frac\{n\_\{R\}-n\_\{L\}\}\{n\_\{R\}+n\_\{L\}\}$.

File:Sherman function.png

The Sherman function $S(\backslash theta)$ is a measure of the probability of a spin-up electron to be scattered, at a specific angle $\backslash theta$, to the right or to the left of the target, due to spin-orbit coupling.{{cite book |last1=Chao |first1=Alexander W. |last2=Mess |first2=Karl H. |title=Handbook of accelerator physics and engineering |date=2013 |publisher=World scientific |isbn=978-9814415859 |pages=756–757 |edition=Second}}{{cite book |last1=Joachim |first1=Kessler |title=Polarized electrons |date=1976 |publisher=Springer-Verlag |isbn=978-3-662-12721-6 |page=49}} It can assume values ranging from -1 (spin-up electron is scattered with 100% probability to the left of the target) to +1 (spin-up electron is scattered with 100% probability to the right of the target). The value of the Sherman function depends on the energy of the incoming electron, evaluated via the parameter $\backslash beta\; =\; \backslash frac\{v\}\{c\}$. When $S(\backslash theta)=0$, spin-up electrons will be scattered with the same probability to the right and to the left of the target.

Then it is possible to write

:$n\_\{R\}\; =\; \backslash frac\{n\_\{up\}[1+S(\backslash theta)]+n\_\{down\}[1-S(\backslash theta)]\}\{2\}$

:$n\_\{L\}\; =\; \backslash frac\{n\_\{up\}[1-S(\backslash theta)]+n\_\{down\}[1+S(\backslash theta)]\}\{2\}.$

Plugging these formulas inside the definition of asymmetry, it is possible to obtain a simple expression for the evaluation of the asymmetry at a specific angle $\backslash theta$,{{cite journal |last1=Sherman |first1=Noah |last2=Nelson |first2=Donald F. |title=Determination of Electron Polarization by Means of Mott Scattering |journal=Physical Review |date=15 June 1959 |volume=114 |issue=6 |pages=1541–1542 |doi=10.1103/PhysRev.114.1541|bibcode=1959PhRv..114.1541S }} i.e.:

:$A\; =\; PS(\backslash theta)$.

Theoretical calculations are available for different atomic targets{{cite journal |last1=Czyżewski |first1=Zbigniew |last2=MacCallum |first2=Danny O’Neill |last3=Romig |first3=Alton |last4=Joy |first4=David C. |title=Calculations of Mott scattering cross section |journal=Journal of Applied Physics |date=October 1990 |volume=68 |issue=7 |pages=3066–3072 |doi=10.1063/1.346400|bibcode=1990JAP....68.3066C }} and for a specific target, as a function of the angle.

File:Mott scattering of an electron beam with S(θ) = 0.5.png

To measure the polarization of an electron beam, a Mott detector is required.{{cite journal |last1=Nelson |first1=D. F. |last2=Pidd |first2=R. W. |title=Measurement of the Mott Asymmetry in Double Scattering of Electrons |journal=Physical Review |date=1 May 1959 |volume=114 |issue=3 |pages=728–735 |doi=10.1103/PhysRev.114.728|bibcode=1959PhRv..114..728N |hdl=2027.42/6796 |hdl-access=free }} In order to maximize the spin-orbit coupling, it is necessary that the electrons arrive near to the nuclei of the target. To achieve this condition, a system of electron optics is usually present, in order to accelerate the beam up to keV{{cite journal |last1=Petrov |first1=V. N. |last2=Landolt |first2=M. |last3=Galaktionov |first3=M. S. |last4=Yushenkov |first4=B. V. |title=A new compact 60 kV Mott polarimeter for spin polarized electron spectroscopy |journal=Review of Scientific Instruments |date=December 1997 |volume=68 |issue=12 |pages=4385–4389 |doi=10.1063/1.1148400|bibcode=1997RScI...68.4385P }} or to MeV{{cite web |last1=Steigerwald |first1=M. |title=MeV Mott Polarimetry at Jefferson Lab |url=https://www.jlab.org/accel/inj_group/mott/mott.pdf |accessdate=25 June 2020}} energies. Since standard electron detectors count electrons being insensitive to their spin,{{cite journal |last1=Ladislas Wiza |first1=Joseph |title=Microchannel plate detectors |journal=Nuclear Instruments and Methods |date=June 1979 |volume=162 |issue=1–3 |pages=587–601 |doi=10.1016/0029-554X(79)90734-1|bibcode=1979NucIM.162..587L }} after the scattering with the target any information about the original polarization of the beam is lost. Nevertheless, by measuring the difference in the counts of the two detectors, the asymmetry can be evaluated and, if the Sherman function is known from previous calibration, the polarization can be calculated by inverting the last formula.

In order to characterize completely the in-plane polarization, setups are available, with four channeltrons, two devoted to the left-right measure and two devoted to the up-right measure.

In the panel it is shown an example of the working principle of a Mott detector, supposing a value for $S(\backslash theta)=0.5$. If an electron beam with a 3:1 ratio of spin-up over spin-down electrons collide with the target, it will be splitted with a ratio 5:3, according to previous equation, with an asymmetry of 25%.

• Spin–orbit interaction
• Mott scattering
• Photoemission spectroscopy

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Category:Electron beam

Category:Foundational quantum physics

Category:Scattering