# A new method to find out the optimal neutron moderator size based on neutron
scattering instrument parameters

Petr Konik  Alexander Ioffe

###### Abstract

Simple and fast analytic method to find optimal combinations of sizes of
neutron moderator and optical system entrance, allowing for the full sample
illumination with minimum to none background, is developed. In the case of
employing low-dimensional para-hydrogen moderators with enhanced neutron beam
brilliance, the method allows to determine the minimum size of moderator,
which provides the highest sample flux while keeping the sample fully
illuminated.

This method can be used during the design of new neutron sources, upgrades of
neutron optical systems and moderator replacements.

###### keywords:

neutron optics , neutron cold moderators , para-hydrogen moderators , low-
dimensional moderators , brilliance transfer , neutron scattering instruments
, phase space

††journal: Nuclear Instruments and Methods A

[FZJ]organization=Jülich Centre for Neutron Science (JCNS) at Heinz Maier-
Leibnitz Zentrum (MLZ), Forschungszentrum Jülich GmbH,
addressline=Lichtenbergstr. 1, city=Garching, postcode=85748, country=Germany

##  1 Introduction

In the last decade low-dimensional neutron moderators filled with almost pure
para-hydrogen have been developed [1, 2]. Thanks to the large difference
between scattering cross-sections for thermal and cold neutrons, such tube-
like or disk-like moderators can provide significantly higher brilliance than
traditional voluminous cold moderators. Investigations carried out at ESS
showed potential gains of 2–3 times in neutron brilliance when reducing the
moderator size from 12 cm to 3 cm [1]. However, in many cases the use of such
small moderator results in the under-illumination of the neutron transport
system (NTS), i.e. moderator does not provide all neutrons with trajectories,
which could be accepted by NTS. Indeed, the full illumination of the NTS
entrance and the sample is required to reach the maximum sample flux.

However, if the neutron beam of a too large divergence or a too large size is
delivered to the sample, then the over-illumination occurs, that is
undesirable since such “useless” neutrons may lead to increase of background
and thus worsening experimental conditions. Hence, neutron scattering
instrument requirements for sample size and angular divergence impose
constraints on the parameters of the neutron beam leaving the NTS.

Therefore, the question arises, which combinations of moderator size and the
entrance size of NTS allow to obtain optimal sample illumination. Until the
present the straightforward approach to this problem has been used, involving
extensive and time-consuming Monte-Carlo simulations of both, moderator and
neutron optics [3, 4]. Few attempts have been made to constrain the
optimization process by reducing the number of free optimization parameters
[5, 6].

In this article we describe the analytic method allowing to quickly find
combinations of moderator and NTS entrance sizes that are optimal in the sense
of providing maximum sample flux with minimum background. While not fully
eliminating the need of Monte-Carlo simulations for the neutron optics
optimization, this method considerably decreases the number of free parameters
and allows to decouple the optimizations of moderator and neutron optics.

The rest of the paper is structured as follows.

In Sec. 2 the used notations and the phase space terminology are introduced.

In Sec. 3 we use phase space formalism to investigate connections between the
neutron scattering instrument requirements, NTS properties and moderator size.
The result is the simple expression allowing to calculate optimal moderator
size for any given instrument parameters.

In Sec. 4 we investigate the influence of deviations from the optimal
moderator size on the sample flux. Special attention is given to the para-
hydrogen moderators with size-dependent brilliance.

Finally, in Sec. 5 we give examples of practical applications of the developed
method. The cases of varying instrument parameters according to experimental
needs, as well as the optimization of a single moderator serving multiple
instruments are explored.

##  2 Notations

Since our aim is to investigate the sample illumination conditions (under- or
over-illumination), we consider neutron instrument holistically starting with
the neutron moderator, ending with the sample and disregarding the analyser-
detector part. We call the set of all elements situated between the moderator
and sample the neutron transport system (NTS), which may include neutron
guides [7, 8, 9], focusing mirrors [10, 11, 12], nested optics [13, 14, 15],
lenses [16, 17, 18], monochromators, entrance and exit slits, Soller
collimators, etc. The general layout of neutron instrument is shown in Fig. 1.
Here $D_{m}$ is the source size, $d_{s}$ is the sample size, $L_{in}$ is the
distance between the source and the NTS entrance, and $L_{out}$ is the
distance from the NTS exit to the sample. NTS entrance and exit sizes are
noted as $w_{in}$ and $w_{out}$, respectively. Finally, $\phi_{in}$ is the
neutron beam divergence that can be accepted by NTS, $\phi_{out}$ is the
neutron beam divergence at the NTS exit.

Figure 1: Layout of a neutron instrument from the optical point of view.

For a newly designed instrument only $d_{s}$, $L_{in}$ and $L_{out}$ are
constrained based on instrument’s science case, potential radiation or thermal
damage to neutron optics and requirements for sample environment size or
angular resolution, respectively. All other parameters are free to vary:
moderator size $D_{m}$ has to be adapted to instrument’s needs; guide shape,
coating, $w_{in}$ and $w_{out}$ have to provide the highest NTS performance,
etc.

To evaluate performance of NTS the phase space diagram technique can be
employed [19, 20, 6]. The phase space (PS) is a space, where each point
corresponds to the unique state of the system — a single neutron trajectory in
the neutron beam in our case. This 5-dimensional PS is built upon two space
coordinates (i.e. the beam cross-section), two angles defining the direction
of beam propagation, and wavelength. The ensemble of all points (neutron
trajectories) in PS is called the phase space volume $V$ and constitutes the
whole neutron beam.

The brilliance $b$ of the neutron beam is the the PS density defined as the
total number $\Phi$ of neutron trajectories divided by the PS volume they
occupy:

$b=\dfrac{\Phi}{V}.$ (1)

Usually the neutron wavelength is not changed during the beam propagation
through NTS, allowing to exclude this coordinate from further considerations.
Assuming rectangular cross-section of NTS, we can study separately two PS
projections, horizontal and vertical, each built with one space and one
angular coordinates. From here on we use the term “phase space” referring to
one of those projections. Phase space volumes corresponding to each of these
projection are 2-dimensional, so that brilliance is measured in
$\dfrac{\text{n}}{\text{s}\cdot\text{cm}\cdot\text{rad}}$, rather than in
$\dfrac{\text{n}}{\text{s}\cdot\text{cm}^{2}\cdot\text{sr}}$ as usual. Note,
the method derived in this paper can be easily expanded to account for the
5-dimensional PS as a whole, however corresponding formulas become cumbersome.

According to Liouville theorem brilliance $b_{out}$ at the NTS exit cannot be
larger than brilliance $b_{in}$ at its entrance :

$b_{out}\leq b_{in}.$ (2)

For optical system with no transmission losses (ideal brilliance transfer)
$b_{out}=b_{in}$ and

$V_{out}=V_{in},$ (3)

where $V_{out}$ and $V_{in}$ are the PS volumes of the beam at the NTS exit
and entrance, respectively. We use this assumption for the rest of the paper,
except for Sec. 4.2, where transmission losses are taken into account.

##  3 Choice of optimal sizes of moderator and NTS entrance

###  3.1 Instrument requirements

To achieve the expected instrument performance each point of the sample with
size $d_{s}$ must be illuminated by a neutron beam with divergence
$2\alpha_{s}$, where $\alpha_{s}$ is required geometric resolution defined by
momentum transfer or energy resolution.

Corresponding PS volume $V_{s}$ has a shape of parallelogram and is equal to

$V_{s}=2d_{s}\alpha_{s}=d_{s}(\alpha_{s}^{\prime}+\alpha_{s}^{\prime\prime}),$
(4)

$\alpha_{s}=\dfrac{\alpha_{s}^{\prime}+\alpha_{s}^{\prime\prime}}{2},$ (5)

where $\alpha_{s}^{\prime}$ and $\alpha_{s}^{\prime\prime}$ define positions
of parallelogram corners (Fig. 2).

Here we introduce factor $n\geq 1$ describing the inclination of the
parallelogram:

$\alpha_{s}^{\prime}=n\alpha_{s}.$ (6)

Taking into account (5) we can write

$\alpha_{s}^{\prime\prime}=(2-n)\alpha_{s}.$ (7)

Figure 2: Different PS volumes $V_{s}$, which can be required by instrument.

Practically $\alpha_{s}$ is defined by the collimation system of the
instrument. In the case of the double-slit collimation, the first slit $s_{1}$
is the exit aperture of NTS, so that $s_{1}=w_{out}$. If the second slit is
placed at the sample $s_{2}=d_{s}$, then $\alpha_{s}^{\prime}$ and
$\alpha_{s}^{\prime\prime}$ correspond to the angles shown in Fig. 3a and can
be calculated as follows:

$\alpha_{s}^{\prime}=\dfrac{s_{1}+s_{2}}{2L_{out}},$ (8)
$\alpha_{s}^{\prime\prime}=\dfrac{s_{1}-s_{2}}{2L_{out}}.$ (9)

From here on all equations are given in the small angle approximation, which
is usually appropriate for neutron optics. Note that
$\alpha_{s}^{\prime\prime}$ can be negative, corresponding to the case of the
sample being larger than the exit of NTS.

Figure 3:  (a) Instrument with double-slit collimation. (b) Instrument with
Soller collimation.

The case of double-slit collimation demonstrates the physical meaning of $n$
as an indicator of the relative size of collimation slits. If the first slit
is $x$ times larger than the second one, then

$\alpha_{s}^{\prime}=\dfrac{(x+1)s_{2}}{2L_{out}},$ (10)
$\alpha_{s}^{\prime\prime}=\dfrac{(x-1)s_{2}}{2L_{out}},$ (11)
$\alpha_{s}=\dfrac{(x+1)s_{2}+(x-1)s_{2}}{4L_{out}}=\dfrac{xs_{2}}{2L_{out}}.$
(12)

Taking into account Eq. (6) we get

$n(x)=\dfrac{x+1}{x}.$ (13)

In reflectometry slits of equal sizes are mostly used, meaning $x=1$ and
$\alpha_{s}^{\prime\prime}=0$, what corresponds to $n(1)=2$ (see Fig. 2a). For
small-angle scattering experiments $x=2$ is considered to be optimal what
corresponds to $n(2)=1.5$ (Fig. 2b).

In case of double-slit collimation $L_{out}$ is the distance between the slits
and is not constrained by instrument requirements. Instead, each particular
scattering technique requires a certain value $n$. Based on geometric
resolution requirement $\alpha_{s}$, one can calculate collimation base using
Eqs. (6,8):

$\alpha_{s}^{\prime}=n\alpha_{s}=\dfrac{x+1}{x}\alpha_{s},$ (14)

$\alpha_{s}^{\prime}=\dfrac{s_{1}+s_{2}}{2L_{out}}=\dfrac{(x+1)d_{s}}{2L_{out}},$
(15)

$L_{out}=\dfrac{d_{s}}{2\alpha_{s}(n-1)}.$ (16)

Note that in some cases it is not possible to place the second collimation
slit directly at the sample. If that happens, in the following considerations
we will consider this slit as the “sample”, since optimally the sample should
accept all neutrons going through the second slit. The size of this slit
should be chosen to provide the full illumination of the real sample.

An alternative way to collimate the neutron beam is to use Soller collimator
(Fig. 3b). In this case divergence at any point of the sample is equal to the
collimation angle and PS volume $V_{s}$ has the rectangular shape:
$\alpha_{s}^{\prime\prime}=\alpha_{s}^{\prime}=\alpha_{s}$ and $n=1$ (see Fig.
2c). $L_{out}$ is constrained by geometrical restrictions around the sample,
e.g. the size of bulky sample environment.

If neutron instrument does not use any collimation device, then geometric
resolution is defined by the natural divergence of neutron beam leaving the
NTS (e.g. determined by the critical angle of mirror coating). Since Soller
collimator is an integral part of NTS, there is no difference when compared to
the previous case: PS volume $V_{s}$ still has the rectangular shape and $n=1$
(Fig. 2c).

For instruments using the beam focusing on the detector (see e.g. [10]) the
detector pixel can be considered as the “sample”, while the real sample can be
placed anywhere between the detector and the NTS exit.

###  3.2 Sample illumination

Let us consider the shape of PS volume of the neutron beam $V_{out}$ at the
NTS exit (Fig. 4). Two extreme cases can be distinguished:

  1. 1.

Phase space non-focusing (PS NF) exit of NTS, when there is no correlation
between angle and position of each neutron trajectory (Fig. 4a);

  2. 2.

Phase space focusing (PS F) exit of NTS: PS volume $V_{out}^{F}$ at the NTS
exit has a special shape (position-angle correlation), so that after
propagation to the sample position the shape of $V_{out}^{F}$ matches $V_{s}$
(Figs. 4d,f).

Figure 4: Evolution of the shape of beam PS volume between NTS exit and sample
position for non-focusing case (upper row (a–c)) and focusing case (lower row
(d–f)). Dark blue arrows in panels (a) and (d) show the direction where points
move when the beam propagates, length of arrows corresponds to the speed of
this movement. The PS volume required by the instrument (shown with red line)
is similar to one depicted in Fig. 2b.

Note that we speak of focusing in phase space, not in real one. Focusing here
and further on refers to matching of two PS volumes shapes.

Consider case 1 in more details. PS volume at the PS non-focusing exit of the
NTS has rectangular shape (Fig. 4a) and can be calculated as

$V_{out}^{NF}=2w_{out}\phi_{out}.$ (17)

In other words neutron trajectories going through a given point at the NTS
exit may have any angle with the NTS axis within the divergence $\phi_{out}$.
Figs. 4a–c show the transformation of PS volume shape between the NTS exit and
sample. As neutrons are moving along trajectories with a given angle, their
space coordinates are changed proportionally to this angle (see dark blue
arrows) and the length of their trajectory, so that the rectangular shaped PS
volume is transformed into the parallelogram shaped one. In agreement with
Liouville theorem the volume itself is conserved, only its shape is
transformed.

The PS volume $V_{s}$ required by the instrument is shown in Fig. 4c by red
dashed line (see also Fig. 2). Optimal instrument performance is reached if
$V_{s}$ is fully inscribed in the transformed $V_{out}^{NF}$, and the excess
volume $V_{out}^{NF}-V_{s}$ is minimal. In other words, these conditions
correspond to the full sample illumination and minimum of background and using
(6) and (7) we can write:

$\phi_{out}=\alpha_{s}^{\prime}=n\alpha_{s},$ (18)

$w_{out}=2L_{out}\alpha_{s}^{\prime\prime}+d_{s}=2L_{out}(2-n)\alpha_{s}+d_{s}.$
(19)

These two conditions are important for further discussion and we will refer to
them as Optimal and Full Sample Illumination (OFSI) conditions. Practically
these conditions allow one to minimise the over-illumination of the sample.
The over-illumination can lead to following consequences:

  1. 1.

Neutrons with too high angles hit the sample, that worsens the resolution and
violates basic instrument requirements. This is prevented by condition (18).

  2. 2.

Some neutrons reach the sample position outside of the sample. They can be
scattered at the sample holder that results in unwanted background at the
detector and reduce the signal-to-background ratio. The number of such
neutrons is minimised by condition (19), while still providing full sample
illumination.

A non-optimal situation, where both OFSI conditions are violated, is shown in
Fig. 4c. The sample is under-illuminated since
$\phi_{out}<\alpha_{s}^{\prime}$, that provides better than required
resolution however with decreased sample flux. Simultaneously $w_{out}$ is
larger than required, leading to a high proportion of “useless” neutrons at
the sample position.

If both OFSI conditions (18) and (19) are met, we can write

$V_{out}^{NF}=2(2L_{out}(2-n)\alpha_{s}+d_{s})n\alpha_{s}$ (20)

Thus, the optimal instrument performance can be achieved, only if this
equation for PS volume at the NTS exit holds true.

Another extreme case of the NTS exit is the PS focusing one (PS F), which
brings to the sample the exact PS volume, required by the instrument, and this
volume depends neither on $L_{out}$ nor on $n$:

$V_{out}^{F}=V_{s}=2d_{s}\alpha_{s}.$ (21)

In this case the OFSI conditions (18) and (19) are also held, despite they
were initially derived for the case of PS non-focusing case.

Figs. 4d–f show the evolution of the shape of PS volume provided by such NTS
exit and allow to reconstruct what type of position-angle correlation is
required at the NTS exit.

The great diversity of modern NTSs does not allow for an immediate answer
which of them have PS focusing or non-focusing exits. As an obvious example of
NTS with phase space NF exit one can consider a straight neutron guide. An
optimized elliptic focusing guide could be NTS with phase space F exit. One
should check the PS volume of the beam at the sample position to determine
precisely the NTS exit type. Certainly, the shape of PS volume from a
realistic NTS is constrained by two extreme cases described above in Eqs.
(20,21).

###  3.3 NTS entrance illumination

NTS entrance illumination scheme is shown in Fig. 5a. A homogeneous and
isotropic neutron source of size $D_{m}$ is separated from the NTS entrance of
size $w_{in}$ by the distance $L_{in}$. While the moderator emits neutrons in
all directions we are only concerned with those hitting the NTS entrance.
Geometry shown in Fig. 5a defines the collimation of the incident neutron
beam. At the distance $L_{in}$ from the moderator surface the PS volume
$V_{m}$, which can be potentially accepted by the NTS entrance, has a shape of
parallelogram, as shown in Fig. 5b.

(a)

(b)

Figure 5: (a) NTS entrance illumination scheme. (b) PS volume $V_{m}$ provided
by the moderator given the distance $L_{in}$ and NTS entrance size $w_{in}$.

Angles $\alpha_{m}^{\prime}$ and $\alpha_{m}^{\prime\prime}$ are defined as
follows:

$\alpha_{m}^{\prime}=\dfrac{D_{m}+w_{in}}{2L_{in}},$ (22)

$\alpha_{m}^{\prime\prime}=\dfrac{D_{m}-w_{in}}{2L_{in}}.$ (23)

Then PS volume $V_{m}$ can be calculated as

$V_{m}=w_{in}(\alpha_{m}^{\prime}+\alpha_{m}^{\prime\prime})=w_{in}\dfrac{D_{m}}{L_{in}}.$
(24)

Depending on e.g. the shape and coating of the neutron guide the NTS can
accept this PS volume either fully or partially. As in the previous section,
one can consider two extreme cases: with or without PS focusing. The latter
corresponds to the NTS entrance, which accepts rectangular PS volume without
any position-angle correlations, while the former corresponds to a specific
NTS entrance with the acceptance exactly matching PS volume shown in Fig. 5b.

Figure 6: Evolution of the PS volume $V_{m}$ shape between the moderator (a,d)
and the NTS entrance (c,f). First row a–c shows non-focusing case, second row
d–f shows focusing case. Shown with red dashed line is the PS volume accepted
by the NTS.

The evolution of the PS volume $V_{m}$ between the moderator and the NTS
entrance is shown in Fig. 6. The shape of the PS volume delivered to the NTS
entrance is the same for both PS NF and PS F cases, however the shape of
accepted PS volumes is different as shown by red dashed lines in Fig. 6c and
Fig. 6f, respectively.

PS non-focusing NTS entrance only accepts PS volume (see Fig. 6c):

$V_{in}^{NF}=2w_{in}\phi_{in}.$ (25)

If there is excessive incident PS volume, which comprises neutron trajectories
entering the NTS under angles exceeding the critical angle of the guide walls
coating for corresponding wavelengths, it will be absorbed in the coating of
the neutron guide walls creating a high-energy gamma-background, so that quite
bulky and rather expensive radioactive shielding around the NTS will be
required. This background can be minimized by imposing the condition (26) that
still allows for the full illumination of the NTS entrance (see Fig. 6c):

$\phi_{in}=\alpha_{m}^{\prime\prime}.$ (26)

Then from Eq. (23), we can obtain the optimal moderator size $D_{opt}$:

$D_{opt}=w_{in}+2L_{in}\phi_{in}.$ (27)

If the actual source size deviates from this value, then either the NTS
entrance is over-illuminated leading to an additional background along the
transport system, or it is under-illuminated leading to reduced instrument
performance.

Now it is possible to rewrite the expression (25) for PS volume:

$V_{in}^{NF}=w_{in}\dfrac{D_{opt}-w_{in}}{L_{in}}.$ (28)

Note that in the case of PS non-focusing NTS entrance the optimal moderator
size $D_{opt}$ is always larger than the NTS entrance size $w_{in}$.

Another extreme case is that of the PS focusing NTS entrance (Fig. 6f), for
which

$V_{in}^{F}=V_{m}.$ (29)

The optimal moderator size then can be calculated from Eq. (24). In this case
optimal moderator can be of any size relative to the NTS entrance, including
smaller than that.

In practice, all known to us modern NTSs have the PS non-focusing entrance,
where $\phi_{in}$ is determined by critical momentum transfer of guide walls.
At the moment, we don’t have any suggestions for the construction of optical
system with such position–angle correlation at its finite size entrance as
shown in Fig. 6f.

###  3.4 Moderator size for optimal and full sample illumination

A NTS with PS focusing properties at the entrance does not necessarily possess
them at the exit. One can distinguish four extreme cases that are the
combinations of PS focusing/non-focusing properties at the entrance and exit
of the NTS. We notate them with two symbols, referring to the entrance and the
exit, respectively: NF–F, NF–NF, F–NF and F–F. For example, the NTS of the
NF–F type possess PS non-focusing entrance and PS focusing exit.

In case of ideal transport $V_{in}=V_{out}$ (see Eq. (3)). Then we can tie
together instrument parameters and the optimal moderator size. Collecting
corresponding expressions from previous subsections we obtain:

  1. a)

for NF–NF type NTS

$w_{in}\dfrac{D_{opt}-w_{in}}{L_{in}}=V_{in}^{NF}=V_{out}^{NF}=2n\alpha_{s}(2L_{out}\alpha_{s}(2-n)+d_{s});$
(30)

  2. b)

for F–NF

$w_{in}\dfrac{D_{opt}}{L_{in}}=V_{m}=V_{in}^{F}=V_{out}^{NF}=2n\alpha_{s}(2L_{out}\alpha_{s}(2-n)+d_{s});$
(31)

  3. c)

for NF–F

$w_{in}\dfrac{D_{opt}-w_{in}}{L_{in}}=V_{in}^{NF}=V_{out}^{F}=V_{s}=2d_{s}\alpha_{s};$
(32)

  4. d)

for F–F

$w_{in}\dfrac{D_{opt}}{L_{in}}=V_{m}=V_{in}^{F}=V_{out}^{F}=V_{s}=2d_{s}\alpha_{s}.$
(33)

From here it is possible to derive functions $D_{opt}(w_{in})$, linking
optimal moderator size and NTS entrance size via instrument parameters. We
call these functions Curves of Optimal and Full Sample Illumination (COFSIs),
where “optimal” refers to minimal background and “full” refers to maximal
sample flux. Table 1 contains expressions for COFSIs obtained for different
types of NTSs and types of collimation before the sample: for the double-slit
collimator (using Eq. (16)) and for Soller or natural collimation ($n=1$).

$D_{opt}=$ |  NTS entrance
---|---
F | NF
NTS exit | F |  $\dfrac{2d_{s}\alpha_{s}L_{in}}{w_{in}}$ | $\dfrac{2d_{s}\alpha_{s}L_{in}}{w_{in}}+w_{in}$
| NF
---
double-slit
$\dfrac{2d_{s}\alpha_{s}L_{in}}{w_{in}}\cdot\dfrac{n}{n-1}$ |  $\dfrac{2d_{s}\alpha_{s}L_{in}}{w_{in}}\cdot\dfrac{n}{n-1}+w_{in}$
| NF
---
Soller or natural
$\dfrac{2d_{s}\alpha_{s}L_{in}}{w_{in}}+\dfrac{4\alpha_{s}^{2}L_{in}L_{out}}{w_{in}}$ | $\dfrac{2d_{s}\alpha_{s}L_{in}}{w_{in}}+\dfrac{4\alpha_{s}^{2}L_{in}L_{out}}{w_{in}}+w_{in}$
Table 1:  COFSIs for four extreme cases of NTS.

As an example, four extreme cases of COFSIs for the instrument with following
parameters: $d_{s}=10$ mm, $\alpha_{s}=0.5$°, $n=1$, $L_{in}=2000$ mm and
$L_{out}=500$ mm are shown in Fig. 7.

Figure 7: Example of COFSIs for the particular neutron instrument (see
parameters in the text) for four extreme cases of NTS.

For any particular type of NTS one can calculate COFSIs basing only upon
instrument parameters and nothing else. COFSIs allow to find optimal moderator
size for any NTS entrance size. Such moderator provides three important
advantages for the instrument:

  1. 1.

full sample illumination (in all cases), allowing to achieve maximal sample
flux for given instrument parameters and moderator brilliance;

  2. 2.

minimal (for PS NF exit) or none (for PS F exit) background at the sample
position, allowing to improve signal-to-noise ratio and detect weak signals;

  3. 3.

minimal (for PS NF entrance) or none (for PS F entrance) background along the
NTS, thus reducing the shielding cost.

For each of NTS types there is only one optimal moderator size for given NTS
entrance size. Any deviation from the optimal moderator size leads to the loss
of above mentioned advantages (this is considered in details in Sec. 4).

As shown in Fig. 7 for the NTSs with PS focusing entrance COFSIs are
hyperbolas (red and blue curves) and for the NTSs with PS non-focusing
entrance they are the sum of hyperbolic and linear functions (yellow and
violet curves). One can see that PS F entrance allows for the optimal use of
very small moderators (in the range of tenths of mm) with very high
brilliance, such as narrow para-H2 moderators.

Consider NTSs with PS non-focusing entrance. Corresponding COFSIs define the
minimal optimal moderator size $D_{opt}^{min}$, which is obtained for NTS
entrance size $w_{in}^{min}$. For any smaller moderator it is impossible to
achieve full and optimal sample illumination. It will be shown in Sec. 4.3
that this size should be as small as possible for an efficient use of para-H2
moderators.

NTS with PS non-focusing entrance may have either PS focusing or non-focusing
exit. In case of NF–F NTS the minimum is reached when (see Table 1)

$w_{in}^{min}=\sqrt{2d_{s}\alpha_{s}L_{in}}.$ (34)

In case of NF–NF NTS with double-slit collimation

$w_{in}^{min}=\sqrt{2d_{s}\alpha_{s}L_{in}\dfrac{n}{n-1}}$ (35)

and for NF–NF NTS with Soller or natural collimation

$w_{in}^{min}=\sqrt{2d_{s}\alpha_{s}L_{in}+4\alpha_{s}^{2}L_{in}L_{out}}.$
(36)

Substituting $w_{in}^{min}$ from Eqs. (34–36) into corresponding expressions
in Table 1 we obtain that in all cases:

$D_{opt}^{min}=2w_{in}^{min}.$ (37)

COFSIs for different values of sample size $d_{s}$ and required resolution
$\alpha_{s}$ are presented in Fig. 8. All COFSIs minima are on one line
corresponding to $D_{opt}=2w_{in}$ (shown as dashed line). Neutron instruments
with smaller $d_{s}$ or $\alpha_{s}$ perform optimally with smaller
moderators.

(a)

(b)

Figure 8: COFSIs for NF-NF NTS of the instrument with $L_{in}=2000$ mm,
$L_{out}=500$ mm: (a) with $\alpha_{s}=0.5$° and varied $d_{s}$; (b) with
$d_{s}=10$ mm and varied $\alpha_{s}$. Dashed line corresponds to
$D_{opt}=2w_{in}$, on which all COFSI minima are located.

Distance $L_{in}$ between moderator and NTS entrance also can be minimised to
reduce minimal optimal moderator size. At high-flux sources $L_{in}$ is about
1.5–2 m because of the potential radiation or thermal damage to neutron
optics. Much smaller $L_{in}$ are accessible at compact neutron sources.

##  4 Deviations from the optimal size of moderator

###  4.1 Analytic calculations of sample flux

Let us now discuss the case when the point corresponding to any given
combination of moderator and NTS entrance sizes is above or below COFSI.
Optimal and full sample illumination conditions (18) and (19) applied at the
sample position are still hold, but the requirement of the optimal moderator
size (see Table 1) is violated.

We start with consideration of the NTS entrance illumination (see Fig. 9). For
the PS non-focusing entrance the truly accepted PS volume $V_{in}\cap V_{m}$
is the intersection of the PS volume $V_{m}$ (shown in blue) illuminating the
NTS entrance and PS volume $V_{in}$ (inside red dashed line) potentially
acceptable by NTS.

Figure 9: Comparison of phase volume $V_{m}$ provided by the moderator (shown
in blue) and phase volume $V_{in}$ potentially acceptable by the NTS entrance
(inside red dashed line). Only the intersection of those two is accepted by
the NTS.

NTS entrance over- or under-illumination is determined solely by how large
moderator size $D_{m}$ is in comparison to NTS entrance size $w_{in}$, that in
turn defines divergences $\alpha_{m}^{\prime}$ and $\alpha_{m}^{\prime\prime}$
(see Fig. 5a). Four cases can be distinguished:

  1. a)

$\alpha_{m}^{\prime\prime}<\alpha_{m}^{\prime}<\phi_{in}$ (Fig. 9a). This case
corresponds to the moderator being significantly smaller than optimal. NTS
accepts the whole beam (blue area), however is still under-illuminated because
the incoming beam divergence is too low:

$V_{in}\cap
V_{m}=V_{m}=w_{in}(\alpha_{m}^{\prime}+\alpha_{m}^{\prime\prime}).$ (38)

  2. b)

$\alpha_{m}^{\prime}\geq\phi_{in}$ and $\alpha_{m}^{\prime\prime}>\phi_{in}$
(Fig. 9b). This case corresponds to optimal (or larger than optimal) moderator
providing high enough beam divergence. NTS accepts the whole potentially
acceptable beam (red dashed line) and is fully illuminated (or even over-
illuminated):

$V_{in}\cap V_{m}=V_{in}=2w_{in}\phi_{in}.$ (39)

  3. c)

$\alpha_{m}^{\prime}\geq\phi_{in}$ and
$-\phi_{in}\leq\alpha_{m}^{\prime\prime}\leq\phi_{in}$ (Fig. 9c). This is the
intermediate case that corresponds to the moderator being slightly less than
optimal. NTS accepts only a part of the incident beam because of its too high
divergence, however at the same time remains under-illuminated. Given that the
parallelogram inclination is equal to
$\dfrac{w_{in}}{\alpha_{m}^{\prime}-\alpha_{m}^{\prime\prime}}$, PS volume
accepted by the NTS can be defined as

$V_{in}\cap
V_{m}=2w_{in}\phi_{in}-\dfrac{(\phi_{in}-\alpha_{m}^{\prime\prime})^{2}w_{in}}{(\alpha_{m}^{\prime}-\alpha_{m}^{\prime\prime})}.$
(40)

  4. d)

$\alpha_{m}^{\prime}>\phi_{in}$ and $\alpha_{m}^{\prime\prime}<-\phi_{in}$
(Fig. 9d). This is the case of even smaller moderator than in previous case:

$V_{in}\cap
V_{m}=2\phi_{in}(\alpha_{m}^{\prime}+\alpha_{m}^{\prime\prime})\dfrac{w_{in}}{\alpha_{m}^{\prime}-\alpha_{m}^{\prime\prime}}.$
(41)

Similarly, we investigate the case of the PS focusing NTS entrance. Note that
if the moderator size is optimal $V_{m}=V_{in}$. If the NTS entrance is under-
illuminated (moderator is smaller than optimal), no parts of $V_{m}$ can be
outside of $V_{in}$ and if the NTS entrance is over-illuminated (moderator is
larger than optimal), there are no parts of $V_{in}$ not filled with $V_{m}$.
This considerably simplifies the answer for the accepted PS volume:

$V_{in}\cap V_{m}=\begin{cases}V_{m},&D<D_{opt}\\\ V_{m,opt},&D\geq
D_{opt},\end{cases}$ (42)

where $V_{m,opt}$ is PS volume provided bu the moderator of optimal size.

If the sample is fully illuminated then according to Eq. (1) sample flux
$\Phi_{s}$ is the product of the PS volume $V_{s}$ required by the instrument
and the brilliance $b_{out}$ of the delivered neutron beam:

$\Phi_{s}=b_{out}V_{s}.$ (43)

Note that here the brilliance $b_{out}$ is averaged over the PS volume
$V_{s}$; this is important if the PS volume is not filled uniformly. In our
model of the ideal lossless NTS the reason for that can only be under-
illumination of its entrance. Any lacuna in potentially accepted PS volume at
the NTS entrance is reproduced in some form at its exit. Inhomogeneities in
$V_{out}$ are in turn reproduced at the sample position. Only in rare cases
these inhomogeneities are not presented in $V_{s}$, e.g. for the straight
neutron guide. Disregarding such cases, one can write:

$\dfrac{b_{out}}{B}=\dfrac{V_{in}\cap V_{m}}{V_{in}},$ (44)

where $B$ is the moderator brilliance.

It is possible now to create colour maps depicting sample flux $\Phi_{s}$
(Fig. 10). Here instrument parameters are $d_{s}=10$ mm, $\alpha_{s}=0.5$ °,
$L_{in}=2000$ mm and $L_{out}=500$ mm. All points lying on the COFSI (red
line) are equivalent in the sense that they provide maximal possible sample
flux and minimal background both at the sample position and along the NTS. For
points above the COFSI sample flux is still maximal, however there is an
increased background along the NTS, that may require an additional shielding.
Note that since OFSI conditions are true no additional “useless” are brought
to the sample position. For points below the COFSI sample flux is reduced due
to the under-illumination of NTS entrance. The COFSI is actually the curve
enveloping the region of maximal flux.

(a)

(b)

Figure 10: Colour maps of sample flux: (a) for NTS with PS focusing entrance;
(b) for NTS with PS non-focusing entrance. COFSI is shown in red thick line.
Sample flux is normalized on its maximal value independently for both panels.

###  4.2 Monte-Carlo simulations of sample flux

As we defined the NTS as all optical elements positioned between moderator and
sample, some of them, like non-ideal neutron guides, crystal monochromators,
etc., may introduce transmission losses. Such realistic NTS with brilliance
transfer less than unity reduces brilliance at the sample $b_{out}$ compared
to the moderator brilliance $B$.

Furthermore, in practice it is not known a priori what type of PS focusing
possesses particular NTS, for example F–F or NF–F. COFSI for such NTS is
somewhere between COFSIs for extreme cases (see Fig. 7).

To find COFSI for a realistic NTS and corresponding sample flux map it is
required to perform Monte-Carlo simulations that allow us to determine the
neutron beam brilliance at the sample position for different combinations
$(w_{in},D_{m})$. This can be done in two steps:

  1. 1.

For each NTS entrance size $w_{in}$ we select the geometric parameters of the
NTS (e.g. elliptic guide foci positions or the mosaicity of crystal
monochromator) in such way that the divergence of the beam leaving the NTS is
equal to required instrument resolution (condition (18)) and beam size at the
sample position is just enough to illuminate the sample (condition (19)).

  2. 2.

For each $w_{in}$ (i.e. the NTS geometry) the series of Monte-Carlo
calculations is performed to obtain sample flux while varying moderator size
$D_{m}$.

We have implemented this algorithm using VITESS simulation package [21] and
calculated sample flux maps for the instrument with following parameters:
$L_{in}=2000$ mm, $L_{out}=500$ mm, $d_{s}=10$ mm and $\alpha_{s}=1$°. 100 m
long neutron guide has elliptic shape and constant $m=3$ wall coating.
Calculations are performed for different neutron wavelengths $\lambda$, which
define critical angles of reflection from the guide walls. Since the
divergence at the exit of the guide is fixed to provide optimal sample
illumination, the guide wall inclination depends on $\lambda$. In other words
the guide geometry is different not only for each guide entrance size
$w_{in}$, but also for each neutron wavelength $\lambda$.

(a)

(b)

(c)

(d)

Figure 11: Sample flux maps (resolution 5 mm, relative error $\leq 5\%$)
obtained using MC calculations. Sample flux is normalized on the calculated
value when $b_{out}=B$. Colour lines depict COFSIs for extreme cases of NTS.

Consider simulated sample flux maps shown in Fig. 11. They are in a good
agreement with our analytical calculations presented earlier in Fig. 10b. As
mentioned above in Sec. 4.1 the COFSI envelops the region of maximal flux and
such an envelope can now be compared to analytically calculated COFSIs for
four extreme cases of neutron optics (shown in solid lines in Fig. 11). We can
conclude that elliptic guides considered for the simulation are very close to
the NTSs of NF–F type.

Thus, the Monte-Carlo analysis allows rather easy classification of realistic
NTSs with respect to their phase space focusing properties.

In the region of large $w_{in}$ the inclination of the guide exit wall can be
so large that it is impossible to satisfy OFSI conditions (18) and (19). Since
beam divergence at the guide exit depends both on the wall inclination and
wavelength, this problem should be more pronounced for larger wavelengths.
Indeed, for $\lambda=$ 1–2 Å OFSI-compliant guides exist and provide some
sample flux for each pair ($w_{in}$, $D_{m}$) for all $w_{in}\leq 150$ mm as
shown in Figs. 11a,b.

For increased wavelength of $\lambda=3$ Å the region where OFSI-compliant
guides exist shrinks (limited by the purple stripe in Fig. 11c). With increase
of wavelength up to $\lambda=3.2$ Å this region shrinks further (Fig. 11d),
and for $\lambda\geq 3.5$ Å disappears, i.e. there are no OFSI-compliant guide
solutions at all.

Another deviation from analytical predictions lies in the region of small
$w_{in}$. Here the guide must capture relatively large beam divergence and
that is the reason why all analytically calculated COFSIs rise up fast here.
However, accepted beam divergence for short wavelengths can be too small due
to limited critical angle. Comparing sample flux maps for $\lambda=3$ Å and
$\lambda=1$ Å (Figs. 11c and a), one can see that for large wavelength sample
flux quickly rises with increase of the guide entrance size and reaches
maximal value $\Phi_{s}^{max}=1$, while for shorter wavelength sample flux
rises much slower and its maximal value is about $\Phi_{s}^{max}=0.6$. Fig.
11b depicts the intermediate case for $\lambda=2$ Å.

These Monte-Carlo simulations highlight the fact that COFSIs solutions are
chromatic. Though the expressions in Table 1 are derived in a purely geometric
way, in practice $\phi_{out}$ usually depends on neutron wavelength. It means
that both optimal moderator size and optimal guide geometry are actually
different for different wavelengths.

To sum up, Monte-Carlo simulations allow for the generalization of the
developed COFSI method by taking into account realistic reflectivity losses
(non-ideal neutron transport) and practical guide geometries with various PS
focusing properties. Well-designed NTS with minimal transmission losses and
high enough reflection angle at its entrance, provides sample flux map very
close to analytical predictions from Sec. 4.1.

###  4.3 Sample flux in case of low-dimensional para-hydrogen moderator

Sample flux maps shown in Figs. 10,11 are obtained under the assumption of
size-independent moderator brilliance $B$. In practice this corresponds well
to large deuterium moderators at research reactors.

However, rather different situation occurs for para-H2 moderators, which
exhibit the strong size dependence of brilliance: smaller moderators provide
significantly higher brilliance than larger ones [1]. For our further
considerations we use data from [22], Fig. 18, normalized for 100 mm large
moderator. We fitted the brilliance with the function

$B_{fit}(D_{m})=\dfrac{p_{1}D_{m}+p_{2}}{D_{m}^{2}+q_{1}D_{m}+q_{2}},$ (45)

where $D_{m}$ is measured in cm, $p_{1}=10.4271$, $p_{2}=38.3069$,
$q_{1}=3.7588$ and $q_{2}=4.9990$. This function is shown in Fig. 12.

Figure 12: Dependence of the brilliance for para-H2 moderator on its size.

Now let us consider the same instrument as described in Sec. 4.1 but employing
the above-mentioned para-H2 moderator. To account for its brilliance we
multiply the sample flux maps obtained above (Fig. 10) by function in Eq.
(45). Obtained sample flux maps are shown in Fig. 13.

(a)

(b)

Figure 13: Sample flux maps in case of employing para-H2 moderator for F–F and
NF-NF neutron transport systems. COFSIs are shown in red. Flux scales for (a)
and (b) are different to keep a high colour contrast. Sample flux is
normalized on the value when $B=1$.

Figure 14: Cross-section of sample flux map shown in Fig. 13b.

While all pairs $(w_{in},D_{m})$ belonging to the COFSI (shown in red) are
still equivalent from the illumination point of view as they provide full
sample illumination and minimal over-illumination, the sample flux is however
different for each pair. While 100 mm moderator provides sample flux
$\Phi_{s}=1$, smaller moderator allows to achieve even higher flux at the
sample. In case of F–F type NTS it is possible to achieve very large sample
flux of $\Phi_{s}=6.5$ for this particular instrument. This result highlights
the importance of potential future search for NTSs with PS focusing entrances.

In case of NF–NF type NTS COFSI has a minimum at $w_{in}^{min}=25.5$ mm,
$D_{m}^{min}=51$ mm, where sample flux reaches a value of $\Phi_{s}=1.8$.
Competing trends of sample under-illumination and increasing moderator
brilliance provide maximal sample flux somewhere directly below the COFSI
minimum as can be seen in Fig. 14. For this particular instrument maximal
sample flux is $\Phi_{s}=2.3$, a gain of 1.28 over one reached with full
sample illumination.

Since maximal sample flux is reached below the COFSI, it means the sample is
under-illuminated. Then PS volume $V_{s}$ is not filled homogeneously, that
usually results in an irregular beam divergence profile with multiple dips and
peaks (see details in Sec. 4.1 before Eq. (44), also see [23], Fig. 1). For
low-resolution instruments, which integrate flux over large beam divergence,
this “gothic”-like structure is not problematic. For high-resolution
instruments, when $\alpha_{s}$ is comparable to peaks (dips) width and the
angular precision of optical elements positioning, it could prove to be
disastrous. The choice whether to work with the under-illuminated sample
should be carefully considered in each particular case. Alternative “safe”
option is to choose moderator size equal to COFSI minimum, which provides full
sample illumination with slightly reduced sample flux.

##  5 Practical applications of the developed method

###  5.1 Case of an instrument with variable parameters

The optimization method described in this paper allows to choose the moderator
size basing on predefined instrument parameters, however quite often
instruments operate in several modes with varied parameters.

Consider, for example, small-angle scattering instrument (SANS) performing an
experiment over a wide range of the momentum transfer. Variable collimation
base must be used to cover the whole required range, meaning that angular
resolution is changed during the experiment. Other parameters like $d_{s}$ and
$n=1.5$ (first collimation slit is twice as large as the sample) are constant
and NF–NF type NTS is used.

The instrument optics and moderator size can be optimized only for one
specific value $\alpha_{s}^{*}$. We define all other possible resolutions
$\alpha_{s}=k\alpha_{s}^{*}$, which corresponds to the required PS volume
$V_{s}=2d_{s}\alpha_{s}$.

Fig. 15 shows PS representation of neutron beam at the sample position. PS
volume $V_{out}$ (shown in blue) is optimized for optimal (i.e. minimal flux
of undesirable neutron, see Sec. 3.2 for details) and full sample illumination
($k=1$). This optimization graphically corresponds to equal slopes of
$V_{out}$ and $V_{s}$ (shown with red dashed line) sides. Since different
collimation bases are used for $k\neq 1$, the shape of $V_{out}$ is also
different: its skewness directly depends on the distance between the optics
exit and sample. Note that slopes of both volumes sides are equal in all
cases, since they both depend linearly on $k$.

Figure 15: PS representation of the neutron beam at the sample (shown in blue)
and instrument requirements (shown with red dashed line) for different values
$k$.

If $k<1$, then $V_{s}$ is fully filled (Fig. 15a):

$V_{s}\cap V_{out}=2kd_{s}\alpha_{s}^{*}.$ (46)

Indeed, the sample is fully illuminated, however not optimally since too much
of excessive PS volume $V_{out}$ is presented at the sample position, meaning
relatively high flux of undesirable neutrons. For optimal sample illumination
divergence $\phi_{out}$ should have been smaller by a factor $k$.

If $k>1$, then $V_{s}$ is not fully filled (Fig. 15c). Using that $n=1.5$ and
$\phi_{out}=n\alpha_{s}$ as in Eq. (18), we obtain:

$V_{s}\cap
V_{out}=\begin{cases}3d_{s}\alpha_{s}^{*}-d_{s}\alpha_{s}^{*}\dfrac{(3-k)^{2}}{4},&1<k\leq
3\\\ 3d_{s}\alpha_{s}^{*},&k>3.\par\end{cases}$ (47)

Thus, in this case sample is under-illuminated and effectively the instrument
operates with better than required resolution.

Sample flux can be calculated according to Eq. (1) as

$\Phi_{s}=b_{out}\times(V_{s}\cap V_{out}),$ (48)

where $b_{out}$ is the brilliance of the delivered neutron beam. This equation
can be rewritten as

$\Phi_{s}=\begin{cases}2Bkd_{s}\alpha_{s}^{*},&k\leq 1\\\
3Bd_{s}\alpha_{s}^{*}-Bd_{s}\alpha_{s}^{*}\dfrac{(3-k)^{2}}{4},&1<k\leq 3\\\
3Bd_{s}\alpha_{s}^{*},&k>3,\end{cases}$ (49)

where $B=b_{out}$ is the moderator brilliance.

Consider an instrument with parameters $d_{s}=10$ mm, $L_{in}=2000$ mm,
$n=1.5$ and varied angular resolution $0.001^{o}\leq\alpha_{s}\leq 1^{o}$. The
NTS and moderator can be optimized only for one chosen resolution
$\alpha_{s}^{*}$. Results of sample flux calculations for different angular
resolutions $\alpha_{s}^{*}$ are presented in Fig. 16.

Let us again consider two types of moderators employed for the illumination of
NTS. In case of large liquid deuterium moderator with size-independent
brilliance (Fig. 16a) it is always beneficial in terms of sample flux to
optimize the NTS for the largest $\alpha_{s}^{*}$, i.e. for the coarse
resolution, thus achieving maximal possible flux at the sample. Then for finer
resolution, that corresponds to $k<1$, the sample flux is reduced
proportionally to $k$, as expected for a tightened resolution.

In case of para-H2 moderator brilliance $B$ depends on its size. If the
instrument was optimized for a specific $\alpha_{s}^{*}$, then the optimal
moderator size is determined using Eqs. (35,37). Corresponding moderator
brilliance can be calculated using Eq. (45).

If the instrument employs such para-H2 moderator, then it may be beneficial to
optimize the NTS and the moderator size for small $\alpha_{s}^{*}$ (see Fig.
16b). In the region of small $\alpha_{s}$ the optimization for small
$\alpha_{s}^{*}$ (blue line) provides the sample flux gain of about 4 compared
to the optimization for large $\alpha_{s}^{*}$ (yellow line). However, in the
region of large $\alpha_{s}$ we get a loss of about 25. Hence, optimal
solution for large $\alpha_{s}^{*}$ leads to losses for small $\alpha_{s}$
mode and vice versa. Optimization of the NTS and moderator for intermediate
$\alpha_{s}^{*}$ (red line) can be seen as an option for compromise.

In a similar way it is possible to analyse other instruments performing
experiments with variable $\alpha_{s}$ or $d_{s}$, e.g. reflectometers
performing $\theta$–$2\theta$ scans.

(a)

(b)

Figure 16: Sample flux for different angular resolutions of the SANS
instrument in the case of using (a) deuterium moderator and (b) para-hydrogen
moderator. Inclined lines in panel (a) are slightly offset vertically for
better presentation.

###  5.2 Case of time-of-flight instrument

Many neutron instruments operate in time-of-flight mode, measuring scattering
of neutrons with different wavelengths in the neutron pulse almost
simultaneously. However, the geometry of NTS remains the same during the pulse
and since the critical angle of reflection of the guide walls’ coating is
proportional to wavelength, the beam divergence $\phi_{out}$ at the NTS exit
changes during the experiment.

According to Eq. (18) optimal and full sample illumination can be achieved
only for unique value of $\phi_{out}^{*}$ and, indeed, only for unique neutron
wavelength $\lambda^{*}$, that in turn determines the optimal moderator size
$D_{opt}$. For other wavelengths $\lambda=p\lambda^{*}$,
$\phi_{out}=p\phi_{out}^{*}$ and the chosen moderator size is not optimal.

For simplicity, we consider the case of NF–NF type NTS and instrument with
Soller collimator ($n=1$), which corresponds to the rectangular phase space
volume $V_{s}$. Expressions for other types of NTSs or for $n>1$ can be
derived in a similar way, however are more cumbersome.

The PS volume $V_{out}$ of the beam at the NTS exit is different from the
optimal one by factor $p$:

$V_{out}=pV_{out}^{*}.$ (50)

The shape of $V_{out}$ at the sample position is shown in Fig. 17.

Figure 17: PS volume $V_{out}$ at the sample position. Different shades of
blue correspond to different values of $p$. Shown with red dashed line is
$V_{s}$.

If $p>1$, then PS volume $V_{s}$ required by the instrument is fully inscribed
in $V_{out}$. Otherwise it is filled only partially, meaning that PS volume
$V_{s}\cap V_{out}$ actually available to the instrument is reduced:

$V_{s}\cap V_{out}=\begin{cases}V_{s},&p\geq 1\\\ pV_{s},&p<1.\end{cases}$
(51)

Excessive PS volume $V_{out}-V_{s}\cap V_{out}$ at the sample position may
provide undesirable background. The ratio of “useful” to “useless” neutrons is
independent from $p$ for $p<1$ and decreases inversely proportional to $p$ for
$p\geq 1$.

Consider now the situation at NTS entrance. The beam divergence
$\phi_{in}=p\phi_{in}^{*}$, that can be accepted by NTS, also changes
proportionally to neutron wavelength, meaning that potentially acceptable PS
volume is $V_{in}=pV_{in}^{*}$, where $V_{in}^{*}$ — PS volume accepted by the
NTS when $\lambda=\lambda^{*}$ (see Fig. 18 and compare to Fig. 9).

Figure 18: PS volume $V_{in}$ (shown with dashed line) at the NTS entrance.
Different colors of dashed lines correspond to different values of $p$. Shown
in blue is $V_{m}$.

If $p<1$ the NTS entrance is over-illuminated and brilliance $b_{out}$ at the
sample position is equal to moderator brilliance $B$.

If $p>1$ the NTS entrance is under-illuminated and according to Eq. (44)
brilliance $b_{out}$ at the sample position is reduced compared to $B$:

$b_{out}=B\dfrac{V_{in}\cap
V_{m}}{V_{in}}=\begin{cases}\dfrac{V_{m}}{V_{in}}B,&p>\dfrac{\alpha_{m}^{\prime}}{\phi_{in}^{*}},\\\
B-B\dfrac{w_{in}(\phi_{in}^{*})^{2}}{V_{in}}\dfrac{(p-1)^{2}}{\alpha_{m}^{\prime}-\phi_{in}^{*}},&1<p\leq\dfrac{\alpha_{m}^{\prime}}{\phi_{in}^{*}},\\\
B,&p\leq 1.\end{cases}$ (52)

Taking into account Eqs. (22)–(24) and (26) we can rewrite Eq. (52) as:

$b_{out}=\begin{cases}B\dfrac{D_{opt}}{p(D_{opt}-w_{in})},&p\geq\dfrac{D_{opt}+w_{in}}{D_{opt}-w_{in}},\\\
B\left(1-\dfrac{(p-1)^{2}(D_{opt}-w_{in})}{4pw_{in}}\right),&1<p\leq\dfrac{D_{opt}+w_{in}}{D_{opt}-w_{in}},\\\
B,&p\leq 1.\end{cases}$ (53)

Sample flux for any given $\lambda$ can be calculated according to Eq. (1) as

$\Phi_{s}=b_{out}\times(V_{s}\cap V_{out}).$ (54)

Substituting expressions (51) and (53) in Eq. (54) we obtain

$\Phi_{s}=\begin{cases}BV_{s}\dfrac{D_{opt}}{p(D_{opt}-w_{in})},&p\geq\dfrac{D_{opt}+w_{in}}{D_{opt}-w_{in}},\\\
BV_{s}\left(1-\dfrac{(p-1)^{2}(D_{opt}-w_{in})}{4pw_{in}}\right),&1\leq
p<\dfrac{D_{opt}+w_{in}}{D_{opt}-w_{in}},\\\ BV_{s}p,&p<1.\end{cases}$ (55)

Consider an instrument working in the range $\lambda=2-20$ Å. The NTS and the
moderator size are optimized for one specific $\lambda^{*}$. Fig. 19
demonstrates the $\lambda$-dependence of sample flux. Sample flux rises
linearly up to this wavelength and then falls afterwards. For $\lambda^{*}$ on
the border of the $\lambda$ range only half of this curve is presented. There
are two ways to proceed to choose $\lambda^{*}$ for which to optimize the NTS
and moderator size.

  1. 1.

If some particular neutron wavelength is of most interest, then it should be
chosen as $\lambda^{*}$. In this case sample flux reaches maximum exactly when
$\lambda=\lambda^{*}$.

  2. 2.

Alternatively, one can aim for maximal integrated sample flux in single
neutron pulse. From three options shown in Fig. 19 the best one would be
$\lambda^{*}=11$ Å.

Figure 19: Sample flux as a function of neutron wavelength for the NTS and
moderator size optimized for particular neutron wavelengths.

In practice, additional factors should be additionally taken into account in
Eq. (55). First, neutron beam brilliance depends on wavelength because of
wavelength dependence of moderator spectrum and NTS transmission. Exact
expression for this dependence is different for different instruments. Second,
if the constant relative resolution $\dfrac{\Delta\lambda}{\lambda}$ is
employed, then flux at longer wavelengths is relatively higher than at shorter
wavelengths. Finally, the sample scattering power may depend on $\lambda$ as
well. Taken all together, these factors will change the shape of curves shown
in Fig. 19, however the general conclusion about optimization options will not
be affected.

The considerations in this section are related to the choice of the optimal
wavelength for a time-of-flight instrument. We have provided a way to choose
the wavelength to optimize the instrument for in case of time-of-flight
technique is employed. Similar arguments may be applied to monochromatic
instruments, using several selected wavelengths for different tasks.

###  5.3 Case of two instruments sharing the same moderator

Let us consider two instruments both using NF–NF type NTSs and sharing the
same para-H2 moderator. Moderator size should be chosen to provide the best
performance for both instruments. For simplicity, both instruments have the
same parameters: $d_{s}=10$ mm, $L_{in}=2000$ mm and $L_{out}=500$ mm. The
only difference are requirements for angular resolution: one instrument is a
high-resolution (HR) one with $\alpha_{s}=0.1$°, while another is a low-
resolution (LR) one with $\alpha_{s}=0.5$°.

(a)

(b)

Figure 20: Sample flux maps in case of using para-H2 moderator for two
instruments. COFSIs are shown in red. Flux scales for (a) and (b) are
different to keep high colour contrast.

Sample flux maps corresponding to both these instruments are shown in Figs.
20a,b, respectively. Different optimal sizes of moderator correspond to these
instruments (Table 2). For each of them there are two optimal solutions are
provided: one for the case of optimal and full sample illumination (COFSI
minimum in Fig. 20) and one for the case of under-illuminated sample with
highest possible sample flux (yellow spot below COFSI in Fig. 20). Detailed
comparison of these two options is given in Sec. 4.3. Note that despite a
5-times better resolution, the sample flux for high-resolution instrument (if
optimised) is only 2.4–2.8 times less than sample flux for a low-resolution
instrument, thanks to the increased brilliance of para-$H_{2}$ moderator.

| HR instrument | LR instrument
---|---|---
| Optimally and fully
---
illuminated sample
|  $D_{opt}=18$ mm
---
$\Phi_{s}=3.8$
|  $D_{opt}=51$ mm
---
$\Phi_{s}=9$
| Under-illuminated sample;
---
maximal sample flux
|  $D_{opt}=13$ mm
---
$\Phi_{s}=4.1$
|  $D_{opt}=25$ mm
---
$\Phi_{s}=11.5$
Table 2: Optimal moderator sizes and corresponding sample fluxes for HR and LR
instruments.

There are two possibilities for the choice of the moderator size, suitable for
both instruments:

  1. 1.

To optimize the moderator size for the best performance of low-resolution
instrument. Moderator size $D_{m}=51$ mm is optimal if homogeneous sample
illumination is required. For high-resolution instrument this choice means the
sample flux of $\Phi_{s}^{HR}=1.8$. However, this value is significantly lower
than the maximally achievable flux of 4.1.

  2. 2.

To optimize the moderator size for the best performance of high-resolution
instrument. The moderator size $D_{m}=18$ mm is optimal and allows for
homogeneous sample illumination. With such moderator the sample flux at low-
resolution instrument is $\Phi_{s}^{LR}=11$, which is slightly less than the
maximally achievable flux of 11.5. Note, that for such choice of moderator
size, the low-resolution instrument operates with under-illuminated sample.

Particularly, for this pair of instruments we would suggest to optimize the
moderator size for the best performance of high-resolution instrument allowing
to obtain high sample flux for both instruments, because low-resolution
instrument probably can perform reasonably well also with under-illuminated
sample.

Similar considerations can be applied to instrument suits including many
versatile instruments.

##  6 Conclusion

We have developed a simple analytic method to find out optimal combinations of
sizes of moderator and entrance of the neutron transport system (NTS), that
provides the full illumination of the sample with minimum to none over-
illumination (as well as minimum to none background along the NTS) for any
neutron scattering instrument. Only the knowledge of basic instrument
parameters — sample size, angular resolution, distances from the NTS to
moderator and sample, is required for calculations. When employing the low
dimensional para-hydrogen moderators this method allows to find out the unique
optimal solution, which provides the maximum sample flux.

One of the important advantages of this method is that extensive time-
consuming Monte-Carlo simulations, usually employed to tackle such problems,
are not required. The optimizations of moderator and NTS are effectively
decoupled and the number of free parameters for the neutron optics
optimization is reduced. Monte-Carlo analysis can be used as a complimentary
technique taking into account the non-ideal neutron transport and phase space
focusing properties of a particular NTS.

This method can be used at initial steps of neutron sources/instrument design,
for upgrades of NTS in the case of fixed moderator size or during the
exchange/upgrade of neutron moderators to adapt them to parameters of existing
neutron guides delivering neutrons from moderator to sample.

We have also shown that by means of the phase space focusing (F–F) type NTS it
is principally possible to make use of very small para-hydrogen moderators
with significantly enhanced brilliance even for neutron scattering instruments
with a large sample and coarse angular resolution.

## Acknowledgement

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No. 871072.

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