The physics angles θ^{∗} and ϕ^{∗} in RSS with Maple V

The physics system for asymmetry calculation is defined as z = along incident electron momentum k⃗, i.e. along beam y = along cross product k⃗ × k⃗′ (points down in lab for k⃗′ horizontal) x = normal to y, z (points to HMS in Hall C) First, rotate counterclockwise about z by the azimuthal scattering angle ϕ

= ϕ_{e}:

Rzphi:=linalg[matrix]([[cos(phi),sin(phi),0], [-sin(phi),cos(phi),0],[0,0,1]])

Rzphi := [ \begin{matrix}cos(ϕ) & sin(ϕ) & 0 \\ −sin(ϕ) & cos(ϕ) & 0 \\ 0 & 0 & 1 \end{matrix} ]

(There are no subscripts in Maple V, so we use plain theta, phi for θ ≡ θ_{q}, ϕ ≡ ϕ_{e}).

Then, rotate about the physics y by the angle θ_{q} between k⃗ and q⃗ = k⃗ − k⃗′ from k⃗ to q⃗.

This is a CLOCKWISE rotation, since q⃗ is to the left of k⃗ and y is down, so sines are *(-1).

> Ryq:=linalg[matrix]([[cos(theta),0,sin(theta) ],[0,1,0],[-sin(theta),0,cos(theta)]]);

Ryq := [ \begin{matrix}cos(θ) & 0 & sin(θ) \\ 0 & 1 & 0 \\ −sin(θ) & 0 & cos(θ) \end{matrix} ]

Final rotation, about the rotated z^{∗} axis

Rzg:=linalg[matrix]([[cos(gamma),sin(gamma),0 ],[-sin(gamma),cos(gamma),0],[0,0,1]])

Rzg := [ \begin{matrix}cos(γ) & sin(γ) & 0 \\ −sin(γ) & cos(γ) & 0 \\ 0 & 0 & 1 \end{matrix} ]

> with(linalg):
> Rstar:=multiply(Rzg,Ryq,Rzphi);

Rstar := [cos(γ) cos(θ) cos(ϕ) − sin(γ) sin(ϕ) , cos(γ) cos(θ) sin(ϕ) + sin(γ) cos(ϕ) , cos(γ) sin(θ)] [−sin(γ) cos(θ) cos(ϕ) − cos(γ) sin(ϕ) , −sin(γ) cos(θ) sin(ϕ) + cos(γ) cos(ϕ) , −sin(γ) sin(θ) ] [−sin(θ) cos(ϕ) , −sin(θ) sin(ϕ) , cos(θ)] Since there is no third rotation, i.e. γ=0,

Rzg1:=linalg[matrix](3,3,[1,0,0,0,1,0,0,0,1])

Rzg1 := [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ]

> Rstar1:=evalm(Rzg1&*Ryq&*Rzphi);

Rstar1 := [ \begin{matrix}cos(θ) cos(ϕ) & cos(θ) sin(ϕ) & sin(θ) \\ −sin(ϕ) & cos(ϕ) & 0 \\ −sin(θ) cos(ϕ) & −sin(θ) sin(ϕ) & cos(θ) \end{matrix} ]

PARA: B points along -z in all systems

> Bpara:=linalg[matrix](3,1,[0,0,-Bz]);

Bpara := [ \begin{matrix}0 \\ 0 \\ −Bz \end{matrix} ]

> Bstarpara:=evalm(Rstar1 &* Bpara);

Bstarpara := [ \begin{matrix}−sin(θ) Bz \\ 0 \\ −cos(θ) Bz \end{matrix} ]

> cosstar:=Bstarpara[3,1]/Bz;

cosstar := −cos(θ)

> phistar:=arctan(Bstarpara[2,1]/Bstarpara[1,1] );

phistar := 0

ϕ^{∗} = 0 or 180, depending on the sign of the denominator. Here denominator is negative, so ϕ^{∗} = 180

PERP: B points along -x in physics system

> Bperp:=linalg[matrix](3,1,[-Bx,0,0]);

Bperp := [ \begin{matrix}−Bx \\ 0 \\ 0 \end{matrix} ]

> Bstarperp:=evalm(Rstar1 &* Bperp);

Bstarperp := [ \begin{matrix}−cos(θ) cos(ϕ) Bx \\ sin(ϕ) Bx \\ sin(θ) cos(ϕ) Bx \end{matrix} ]

> cossstar:= Bstarperp[3,1]/Bx;

cossstar := sin(θ) cos(ϕ)

> phistar:=arctan(Bstarperp[2,1]/Bstarperp[1,1] );

phistar := −arctan(\frac{sin(ϕ)}{cos(θ) cos(ϕ)})

So cos θ^{∗} = sin θ_{q} cos ϕ_{e}, and tan ϕ^{∗} is in second quadrant (+ sine, − cosine),

so ϕ^{∗} = 180 + arctan(\frac{tan ϕ_{e}}{− cos θ_{q}} )