In the following an approximation for the resolution function of the SANS cam-era is obtained by assuming that the triangular wavelength and the trapeziodal beam divergence distributions can be described by gaussian distributions. Here the beam divergence will be assumed isotropic whereby the resolution function can be treated separately in the scattering plane and perpendicular to this. Furthermore the expression is only valid for scattering of neutrons without any energy transfer to the sample. The derivation is based on J.S. Pedersen et. al. [28], but extended to obtain the 3-dimensional resolution function.
Resolution function in the scattering plane
The momentum transfer κ of a neutron is given by the change of the wavevector kfrom the initial(i) to the final(f) state during a scattering process and it can be written in a complex form for the scattering plane shown on figure 59
κ = kf− ki
= 2π
λ eiθf − eiθi
(252) where λ is the neutron wavelength related to the wavevector by |k| = 2πλ and the real and imaginary part of κ is along the z axis and in the detector plane respectively. The assumption of scattering without energy transfer to the sample result in the condition |ki| = |kf|.
In the following the resolution function is formulated as the number of available neutrons at different momentum transfer vectors, which can be scattered into the same detector pixel specified by the angle θf as illustrated on figure 59.
The incoming beam is approximated by a gaussian distribution of the form
I(λ, θi) ∈ N(λ0, θi0, σλ2, σθi
2) (253)
which is centered around the wavelength λ0of the velocity selector setting and the direction θi = 0 of the z-axis of the camera. These values correspond to the nominal momentum transfer < κ > as marked by the black dots on figure 59.
Now we allow the incoming and scattered neutron direction to deviate from the nominal values < θi >= 0 and < θf >= θf0 and want to find the corresponding momentum transfer vector κ and the number of neutrons at that κ.
From the scattering angle θ = θf−θ2 i the direction of the momentum transfer θκ
is
θκ = θi+ θ +π 2
= θf+ θi
2 +π
2 (254)
and the nominal direction is < θκ >= θf02 +π2. Thus a more convenient de-scription of κ can be obtained by rotating the laboratory system into a coordinate system with the imaginary and real axis respectively along and perpendicular to the nominal < κ >.
κq = κ· e−iθf02
= 2π λ
ei(θf−θf02 )− ei(θi−θf02 )
(255)
Figure 59. Neutrons with different incoming angles θiand wavevector size |k| = 2πλ can be scattered into the same detector pixel at r0by different momentum transfer vectors κ = kf − ki. Blue and red arcs indicate changes of the direction of the incoming neutron at a short and long wavelength and the dots show the momentum transfer when the beam divergence is absent θi = 0. The scattering plane is spanned by the z axis and a radial detector direction given by φ in figure 20. θf and θκ are the angles of the scattered neutron and the momentum transfer vector κ.
We now have a transformation from a given wavelength and incoming neutron direction to the corresponding κq vector which can cause scattering into the de-tector position determined by θf. The number of neutrons at κq = f (λ, θi, θf) is given by the two dimensional gaussian intensity distribution as function of the wavelength λ and the angles θiand θf. However we seek the neutron distribution as function of κq and an approximation for this can be obtained by making a Taylor expansion of (255).
Taylor expansion of momentum transfer
The momentum transfer vector κq can be written as a first order Tailor expansion around the nominal < κq > with respect to the neutron wavelength deviation and the angle deviation of the incoming and scattered neutron wavevector giving
κq ≈ < κq > + ∂κq where k0 = 2πλ0. The advantage of the Tailor expansion is that the approxi-mate momentum transfer vector is given by a sum of normal distributed vari-ables, whereby the distribution of κq also will be normal distributed according to the convolution theorem : If a stochastic variable is given by Z = P
iaiXi with Xi∈ N(µi, σi2) being independent and normal distributed having a mean µi and variance σi2 then
Z ∈ N X Thus we can immediately write down the variance of the momentum transfer distribution from (256) and the distribution is given by
R(κq, < κq>) = 1 where the pre-factor ensured the area of the resolution function to be normal-ized.
What we learn from the distribution above is how the instrumental settings of the SANS camera will influence the number of available neutron at κq causing scattering into < κq >. The three contributions to the width σ2||along the momen-tum transfer is determined by the wavelength spread σλ of the velocity selector, the beam divergence σθi controlled by the collimation section, and the detector resolution σθf related to the pixel size of the detector. For the width perpendicular to the momentum transfer σ⊥2 the beam divergence term and detector resolution are found again, but the dependence on the wavelength spread does not appear.
This is because a change in λ will only cause a change in the length of κq, but no rotation since all derivatives with respect to λ have no real part, Re∂nκq
∂λn
= 0.
Thus the effect of the wavelength spread only enters in the cross terms of the Tailor expansion These cross terms are not easy to include in the widths, since the convolution theorem is only valid for independent variables. An estimate of the effect of these terms can be made if the wavelength spread (λ−λλ00) is assumed to be a constant, whereby it is seen that an extra term similar to the beam divergence must be added to σ2⊥
σ2⊥=< κ >2(σi
2)2(1 +(λ − λ0) λ0
)2 (262)
However the σ⊥2 will be dominated by the beam divergence since (λ−λλ 0)
0 =
0.1 − 0.2 in most cases.
Detector resolution
The detector consists of pixels of a finite size ∆r and neutrons scattered in a small angular range around θf will be detected in the same pixel at r0as shown on figure
59. By representing the pixel as a gaussian distribution centered at r0 and with a width
σr= 1 2√
2ln2∆r (263)
the corresponding width of the angle θf is found by Taylor expanding θf = arctan (rz). In the small angle limit r << z one gets
σθf =σr
z (264)
The relation between standard deviation σ and Full Width Half Maximum
∆F W HM of a gaussian has been used above
∆F W HM = 2√
2ln2σ (265)
Resolution function perpendicular to the scattering plane
In order to include the resolution perpendicular to the scattering plane we need a 3 dimensional transformation from wavelength, in- and out of plane beam divergence into the momentum transfer. Figure (60) show the projection of the incoming and scattered neutron onto the scattering plane spanned by the z axis and the radial direction eR of the detector plane. When the nominal values of the angles are inserted the nominal momentum transfer plane is obtained.
< θi >= 0 < φi>= 0
< θf >= θf0 < φf >= 0 (266)
Figure 60. The momentum transfer can be decomposed into components in the scattering plane described by (kip, kf p) and an out-of-plane component depending on the angles (φi, φf) that the incoming and scattered neutron are tilted with respect to the scattering plane.
The 3-dimensional momentum transfer is decomposed into the scattering plane and a direction eφ perpendicular to the scattering plane
κ = κp+ κφeφ
= kf p− kip+ (kf φ− kiφ)eφ
= |kf| cos (φf)eiθf − |ki| cos (φi)eiθi + {|kf| sin (φf) − |ki| sin (φi)} eφ
= 2π
λ cos (φf)eiθf − cos (φi)eiθi + 2π
λ {sin (φf) − sin (φi)} eφ (267)
The in-plane component has been written in the complex notation used in the previous section and the rotation from the laboratory coordinate system into a system following the nominal momentum transfer can be done again.
κq = κe−iθf02
Once again the Tailor expansion can be made to obtain an approximation for the κq distribution, but it is easily seen that the out-of-plane angles are decoupled from the in-plane momentum transfer whereby the in-plane distribution of the previous section is found again. The out-of-plane component however gives
κφ ≈ < κφ> + ∂κφ
and the out-of-plane width σφ2 is
σ2φ= k02
(σ2φf + σφ2i) (270)
where the two terms show the influence of the out-of-plane beam divergence and the out-of-plane detector resolution respectively.
Combined resolution function
The combined resolution function is then given by
R(κq, < κq>) = 1 where the width σ||along the momentum transfer vector is denoted the longitu-dinal resolution, σ⊥in the plane is called the transverse resolution and the out-of plane width σφis denoted the azimuthal resolution related to the azimuthal angle φ.
The widths may now be expressed from the instrument settings derived in sec-tion 4 and 264
σλ = λ0
2√ 2ln2
∆λ
λ0 (275)
σθi = 1 2√
2ln2∆θi (276)
σθf = 1 2√
2ln2
∆r
z (277)
Figure (61) show the combined resolution function for scattering into a detector pixel corresponding to the nominel < κ > at the center of the cigar shaped gaussian distribution. Thus the resolution function gives the intensity scattered into the pixel as a reciprocal lattice vector τ is moved around in space by changing the angle ω.
Figure 61. Gaussian resolution function for scattering into the detector pixel cor-responding to the nominal momentum transfer < κ > at the center of the dis-tribution. The shape of the distribution is given by the longitudinal width σ||, the transverse width σ⊥ and the azimuthal width σφ.