**A Monte Carlo study of the performance of HYSPEC**

**Introduction**

Simulations of HYSPEC (the crystal-time-of-flight hybrid spectrometer)
were performed using the MCSTAS (Monte Carlo Simulation of Triple Axis
Spectrometers, ref 1) code as well as the NISP(Neutron Instrument
Simulation Package) Monte Carlo (MC) code developed at Los Alamos (2)
for incident energies of 5, 15, 30, 60, and 90meV. The neutron flux and
the time and energy distributions of the neutrons at the sample position
were obtained from the simulations.

**Section 1** contains the layout of HYSPEC as well as the MCST (a
multichopper straight-through) spectrometer with the same dimensions as
HYSPEC. The MCST was used as a standard of comparison when studying the
performance of HYSPEC. **Section 2** contains the results of the
simulations done to optimize and evaluate HYSPEC performance. **Section
3** presents a comparison of HYSPEC with the other direct geometry
spectrometers, the Cold Neutron Chopper Spectrometer (CNCS), the Wide
Angle Chopper Spectrometer (ARCS), and the High Resolution chopper
spectrometer (HRCS). **Section 4** contains the results of a comparison
between the MCSTAS and NISP Monte Carlo codes, and **Section 5**
describes the simulations performed to ensure that the PG crystal
reflectivity used in the HYSPEC simulations has the same value as the
reflectivity measured in the Shapiro Chesser experiments(3). An
experiment done by I.Zaliznyak to measure the intensity gain due to a
curved PG crystal was also simulated. The results will be presented
here.

**Section 1**

The layouts of the HYSPEC and MCST spectrometers are shown in Fig 1.
Also provided are descriptions of the various components of HYSPEC and
the MCST model.

**1.1 Description of HYSPEC**

**Moderator**

Both HYSPEC and the MCST are assumed to view a 10cm by 12cm cryogenic,
coupled, super-critical hydrogen moderator. Monte Carlo (MC) simulations
for this moderator as well as the other SNS moderators have been
performed by E. Iverson. The data used in our simulations are contained
in the file 'hl211f_td_05.dat', which can be downloaded from the SNS
website (ref. 4). Iverson simulates the neutron flux (n/pulse/Ev/ster)
emitted by the moderator for different energy ranges. A separate time
distribution of the emitted neutrons for each energy range is also
listed in his moderator tables. The MC source interpolation scheme used
in the NISP simulations was developed by P.A. Seeger and in the MCSTAS
simulations by G. Granroth.

**The guide system**

The HYSPEC guide system consists of four sections of 3̀θ~c,~
supermirror-coated guide, and each section of guide has a width of 4cm.
Section G1 is a 5m expander, the entrance height is 12.8cm, and the exit
height is 15cm. G1 starts 1.5m from the moderator surface. Section G2 is
a curved guide, 11m long with a 4cm by 15cm cross-section. For most of
our simulations we have given G2 a curvature that will provide - at the
PG crystal monochromator - an 8cm offset from the line of sight of the
moderator. Guide section G3A is a 2m long expander, its height increases
from 15 to 18cm. Section G3B is a converger decreasing in height from 25
to 10cm. It starts 15cm downstream of the PG crystal monochromator, and
is 1.1m long. Immediately downstream of G3 is a 15cm long collimator.
The guide system ends 40cm from the sample.

![](media/image1.jpeg){width="5.81875in" height="4.2444444444444445in"}

**Fig. 1. Primary spectrometers for HYSPEC and the MCST**.

**Choppers**

In both models space is provided in the guide immediately after G1 for a
T~0~ chopper but its chopping function wasn't included in the
simulations. Chopper T3, the frame overlap chopper, is placed between
guides G2 and G3A. It is assumed to be a single-blade disc chopper of
radius 30cm rotating at the source frequency of 60Hz. Chopper T2, the
incident energy and burst width defining chopper, is positioned at the
end of guide G3A. It is assumed to be a counter-rotating pair of disc
choppers of radius 30cm rotating at integral multiples of the 60 Hz
source frequency. In most of our simulations it was assigned a
rotational frequency of 300Hz. The Fermi chopper T4 has thus far not
been included in the simulations.

**The vertical-focusing PG crystal monochromator**

The curved, pyrolytic graphite (PG) crystal monochromator was given
dimensions 30cm wide and 30cm high, a horizontal mosaic spread of 60 min
and a vertical mosaic spread of 24 min. Most simulations were performed
using the PG002 reflection but in a few special cases the PG004
reflection was used. For the PG interplanar spacing d was assigned the
value 3.3539Ǻ and the angle θ~B~ required to select a particular
incident wavelength λ was then calculated from the Bragg equation

2dsinθ~B~ = n λ.

The radius of curvature R~c~ was calculated to focus the beam at the
sample axis 1.8m downstream of the crystal using the relation

R~c~ = L~ms~ \* τ~m~ / k~i~ ,

where L~ms~ is the monochromator-sample distance (1.8m), τ~m~ = 2π/d,
and k~i~ = 2π/λ.

**Sample and sample environment**

We defined a right cylinder 2cm high with a radius of 1cm to be our
standard sample. Since the HYSPEC guides are 4cm wide, a very uniform,
monochromatic and order-free neutron flux is produced at the sample
position over a 4cm by 4cm area. The collimators immediately upstream
and downstream of the sample are positioned so as to allow a free space
80cm in diameter around the sample that will be needed for the special
equipment (cryostats, furnaces, magnets, etc) that will define the
sample environment.

**1.2 Description of the MCST**

The moderator-sample distance for the MCST model is the same as that for
HYSPEC; i.e. 21.8m. Also the MCST guides were -- in so far as possible
-- made identical to those for HYSPEC. They had the same supermirror
coating and, to the extent possible, the same dimensions. Curved guide
section G2 was subdivided into G2A and G2B to accommodate a
pulse-shaping chopper T1; its chopping function was not, however,
included in the simulations. Chopper T2 is positioned 2m upstream of the
sample just as in HYSPEC. The major difference between the two models is
that both MCST guides G3A (2m long) and G3B (1.25m long) are convergers.
We chose the exit heights of G3A and G3B to get the most possible flux
on a 2cm high sample for a incident energy of 15meV. As in HYSPEC, a
15cm collimator is placed at the end of G3B and a 40cm space is provided
between the end of the collimator and the sample axis.

**Section 2**

**HYSPEC performance**

HYSPEC is designed for studies that require not only a large flux on a
small sample over the energy range 5-90meV but the lowest attainable
background. To achieve optimal data collection rates it was important to
be sure that

(1) it uses the moderator that emits the largest number of neutrons in
    this energy range,

(2) that these neutrons are optimally transported to the sample,

(3) that the beam focusing mechanism is efficient over a wide energy
    range and can focus the beam on to small samples,

(4) and that we have a detector bank with a large angular acceptance.

The Monte Carlo simulations that were performed

(1) show that the curved crystal monochromator focuses tall beams more
    efficiently than a converging guide compressor (section 2.1),

(2) justify our choice of moderator by comparing the performance of
    HYSPEC with coupled H~2~ and decoupled H~2~O moderators (section
    2.2) and

(3) investigate the effect of guide curvature and guide coating on
    HYSPEC performance (section 2.3).

**Section 2.1**

**Advantages of a curved crystal monochromator**

The design of the instrument and the choice of beam focusing mechanism
was dictated by the relatively large size of the moderator and the small
size of the sample. Since the moderators are 10cm wide and12cm high, the
ideal instrument would extract a beam 10cm wide and 12cm high and focus
all the emitted neutrons on a 2cm by 2cm sample. The uniqueness of
HYSPEC lies in the fact that a vertically focusing monochromator is used
in a time-of-flight instrument. This allows full advantage to be taken
of the moderator height and vertically focuses the greatest number of
neutrons on a small sample. Horizontal focusing by horizontally
converging guide sections is not currently being used in HYSPEC so we
presumably have something more to gain by making optimum use of the full
width of the moderator.

The advantages (and possible disadvantages) of a curved crystal over a
traditional compressor were explored in detail by making a series of
comparisons between HYSPEC and its MCST counterpart which differs from
HYSPEC in that it uses a converging guide for focusing the neutron beam.
The flux on sample for HYSPEC and the MCST spectrometer are compared in
Fig 2. Except at 5meV the curved crystal is seen to do a better job of
focusing the incident beam.

![](media/image2.wmf)

Fig 2. Comparison of the HYSPEC and MCST spectrometers.

The relative efficiencies of the curved crystal and compressing guides
were studied both as a function of energy and of guide height. It was
found that for a fixed guide height (12 cm) a compressor guide was most
efficient at low energies. The use of a compressor did not, however,
result in any significant gain in neutron flux for energies above 60meV.
At 15meV the simulations showed that if the guide height was equal to or
less than 8cm a compressor guide focused the beam on a 2 cm high sample
more efficiently than a curved crystal. If the beam height exceeded 8
cm, however, the curved crystal performance was decidedly better.

The efficiency of a vertically curved crystal depends on its peak
reflectivity and on the intensity gain due to vertical focusing. The
part of the gain due to curvature is well described by a simple,
geometrical algorithm, which makes the value of the peak reflectivity
the most important input in evaluating the performance of a curved
crystal. Fortunately, PG crystal reflectivities have been measured
(Shapiro and Chesser, ref 4) over an energy range of 5-40meV, the lower
part of the energy range in which HYSPEC would operate. Simulations of
these experiments were therefore performed to determine MC crystal
parameters that would match the measured peak reflectivities. These same
parameters were then used in the HYSPEC simulations to evaluate
instrument performance. Since no experimental data were available for
energies over 40meV, the 40meV value of the peak reflectivity was used
for the 60 and 90meV simulations. This may result in an overestimation
of the neutron flux at energies over 40meV. We have submitted a proposal
to LANSCE to make additional measurements of the reflectivities of PG
monochromators.

**Section 2.2**

**Choice of moderator**

The total neutron intensity at the entrance to the guide system (a 4cm
by 12cm guide starting 1.5m from the moderator) for the cryogenic
coupled-H~2~ moderator and the decoupled H~2~O moderator used by ARCS
and the HRCS is plotted in Fig 3. This figure highlights two points. One
is that the coupled H~2~ moderator delivers more neutrons in the 5-40meV
range than the H~2~O moderator. The second is that the source flux
interpolation schemes used by the two MC codes are reasonably good
approximations to E. Iverson's calculations of the emitted neutron flux.

![](media/image3.wmf)

Fig 3. Total neutron intensity at the entrance to the guide system for
the coupled-H~2~ and the decoupled H~2~O moderators

![](media/image4.wmf)

Fig 4. Neutron flux at the sample position for HYSPEC viewing either a
coupled-H2 or a decoupled H2O moderator, and with a curved or a straight
guide G2.

The neutron flux at the sample position for HYSPEC is shown as a
function of energy in Fig 4. Comparison of Fig 4 with Fig 3 makes it
evident that the red and blue plots in Fig 4 have shapes almost
identical to those of the red and blue curves in Fig 3. The red and blue
curves in Fig 4 correspond to the simulations of HYSPEC with a straight
G2 guide section (no offset at the monochromator) viewing a coupled H~2~
moderator and a H~2~O moderator, respectively. It is clear from Figs 3
and 4 that the cryogenic coupled H~2~ moderator is clearly the better
choice for HYSPEC for optimum performance in the 5-40meV range. This
correspondence also implies that neutrons are transported with the
essentially the same efficiency over the entire energy range when
straight guides are used.

**Section 2.3**

**Study of guide curvature and guide coating**

The black and green curves in Fig 4 represent HYSPEC with guide section
G2 curved to provide an offset of 8cm at the crystal monochromator
position and viewing, respectively, the H~2~ and H~2~O moderators. It is
also clear from Fig 4 that curving the guide reduces its transmission
for energies greater than 30meV. The rationale for the curved guide was
that it moved the crystal monochromator out of the line of sight of the
moderator which would shield it from the fast neutrons and gamma rays.
At this point, however, it is not entirely clear whether an offset of
8cm is enough or whether it would have to be larger to reduce radiation
levels outside the shielding to acceptable values. Although this issue
is not one that could be resolved using the NISP and MCSTAS programs,
simulations could be done to estimate the effect of increased guide
curvature on the flux on sample. The results are shown in Fig 5. Not
surprisingly, the flux at the sample position decreases dramatically at
the higher energies when the curvature of the guide is increased. From
the flux-on-sample point of view, the straight guide would clearly be
the preferred option. Whether shielding will be adequate in this
configuration is yet to be determined.

![](media/image5.wmf)

Fig 5. The effect of guide curvature on the flux on sample.

As of today the decision is that the HYSPEC guides will have a 3θ~c~(Ni)
coating. This basis for this decision have been simulations that show
that the performance of HYSPEC with m=2 guides is much inferior to that
with m=3 guides (see Fig 6). However, especially at energies greater
than 30meV, large gains in flux on sample could be made by going to a
m=3.5 coating. Simulations with m=3 and 3.5 supermirror coatings for
HYSPEC with a curved guide of 8cm offset are also shown in Fig 6. We
note that the transmission of the guides is somewhat sensitive to the
reflectivity function used. In our simulations the reflectivity is
modeled by a trapezoidal function. For m=3 reflectivity is 1.0 for θ ≤
θc, it decreases to 0.9 as θ increases to 3θc, and it has a
tanhyperbolic tail for θ \> 3θc. For the m=3.5 simulation, the
reflectivity is 1.0 for θ ≤ θc, it decreases to 0.65 as θ increases to
3.5θc, and has a tanhyperbolic tail for θ \> 3.5θc. This results in a
slightly reduced transmission for neutrons with wavelengths larger than
1.5A, and an increased transmission for the neutrons with shorter
wavelengths. If it turns out that curved guides have to be used in
HYSPEC to provide adequate shielding, it is clear that for best guide
performance the supermirror coatings should be the best that are
commercially available. Whether straight guides with m\<3 supermirror
coatings could be used without compromising instrument performance is
yet to be determined.

![](media/image6.wmf)

Fig 6. Effect of guide coating on flux on sample for HYSPEC

**Section 3**

**Comparison of HYSPEC with the CNCS, ARCS, and the HRCS**

Simulations for ARCS (the wide angle Fermi chopper spectrometer) and the
HRCS (high resolution chopper spectrometer) were performed by G.
Granroth (ref 5) using MCSTAS. The flux on sample and the energy
resolution were calculated from the simulation results. Simulations for
the CNCS were performed at BNL in collaboration with J.V. Pearce.
Simulations of the CNCS have also been performed independently by Pearce
using the VITESS MC code and his results agree with the BNL results to
within 20% (ref 6). The flux on sample for CNCS, ARCS and HRCS were
normalized to the energy resolution of HYSPEC before comparison.

**Details of the CNCS simulations**

The CNCS spectrometer configuration was obtained from "Optimization of
the guide configuration of the Cold Neutron Chopper Spectrometer", J.V.
Pearce, D.G. Narehood, and P.E. Sokol (ref.7). A disc chopper with two
counter-rotating blades (essentially identical to the one being used in
the HYSPEC simulations) was included in the setup to select the incident
energy and to determine time and energy distribution at the sample
position.

According to ref. 7, the CNCS guide configuration consists of 9 guides,
starting 1.5m from and ending at 35.95m from the moderator. Guide 4 is a
15m curved guide and it moves the sample out of the line of sight of the
neutrons emitted from the moderator. The guide width and height are
constant (5cm by 10cm) till after 30m. Three funnels then focus the beam
in both the horizontal and vertical directions to a final width of 1.5cm
and height of 5.0cm. In our simulations all guide coatings are assumed
to be m=3.5 supermirrors, with the same reflectivity function as that
specified in ref. 7.

Ref. 7 also lists the positions of the choppers that will be used in the
CNCS to select the incident energy, to shape the neutron pulse, and to
prevent frame overlap. As a simplification only one chopper, the
counter-rotating disk chopper pair, is included in the simulations,
since it is this chopper that selects the incident energy and determines
the time and energy distribution at the sample position. The width of
the chopper slot (i.e. the width at the center of the illuminated
opening) is taken to be equal to the exit width of guide 7, which is
approximately 2.2cm. MCSTAS assumes that the height of the slot is equal
to that of the guide placed immediately upstream of the chopper. All
simulations were performed assuming a chopper frequency of 300Hz.

According to ref.7, the sample is located 25cm downstream of the last
guide (guide 9). In our simulations a detector is placed at the sample
position, and the neutron intensity (n/s) at this detector is used to
calculate the neutron flux (n/cm^2^/s). Also the time and energy
distribution at the sample position are 'determined' using appropriate
time and energy monitors. The full width at half maximum of the time
distribution is called Δt.

The sample-detector distance is assumed to be 3.5m, and this value is
used to calculate the time (T) required for the most probable neutron to
reach the detector. Because the energy resolution of the instrument is
primarily determined by the sample-to-detector time of flight, the
energy resolution at an incident energy E is reasonably well
approximated by the simple expression ΔE/E = 2 \*Δt/T~.~

**Instrument Parameters**

The wide-angle Fermi chopper spectrometer (ARCS) is described in ref 8
and the High resolution chopper spectrometer (HRCS) in ref 9. The
instrument parameters for the four direct geometry spectrometers are
compared in Table 1.

+-------------+-------------+-------------+------------+-------------+
|             | **HYSPEC**  | **CNCS**    | **ARCS**   | **HRCS**    |
+-------------+-------------+-------------+------------+-------------+
| Moderator   | Coupled 20K | Coupled 20K | Decoupled  | Decoupled   |
|             | su          | su          | poisoned   | poisoned    |
|             | percritical | percritical | H~2~O      | H~2~O       |
|             | H~2~        | H~2~        |            |             |
+-------------+-------------+-------------+------------+-------------+
| Beamline \# | 15          | 5           | 18         | 17          |
+-------------+-------------+-------------+------------+-------------+
| Moder       | 21.8m       | 36.2m       | 13.6m      | 17.5m       |
| ator-sample |             |             |            |             |
| distance    |             |             |            |             |
+-------------+-------------+-------------+------------+-------------+
| Guide       | Supermirror | m=3.5       | M=3-3.6    |             |
| coating     | m=3         |             |            |             |
+-------------+-------------+-------------+------------+-------------+
| Guide       | entrance    | entrance 5  |            |             |
| apertures   | 4x12.8      | x10         |            |             |
|             |             |             |            |             |
|             | main 4 x 15 | main        |            |             |
|             | cm^2^       | 5x10cm^2^   |            |             |
|             |             |             |            |             |
|             | exit 4 x 10 | exit 1.5x5  |            |             |
|             | cm^2^       | cm^2^       |            |             |
+-------------+-------------+-------------+------------+-------------+
| Intended    | 4 (w) x 2   | 1.5 (w) x   | 5.0(w) x   | 5.0(w) x    |
| beam size   | (h) cm2     | 5(h) cm2    | 5.0(h) cm2 | 5.0(h) cm2  |
| at sample   |             |             |            |             |
+-------------+-------------+-------------+------------+-------------+
| Samp        | 4.5m        | 3.5m        | 2.5m       | 5.5m        |
| le-detector |             |             |            |             |
| distance    |             |             |            |             |
+-------------+-------------+-------------+------------+-------------+
| Angular     | 0.27 SR     |             | 3.1 SR     | 1.61 SR     |
| acceptance  |             |             |            |             |
|             | 60°         |             | -40° \< 2θ | -30° \< 2θ  |
|             | horizontal  |             | \< 140°    | \< 60°      |
|             |             |             | horizontal | horizontal  |
|             | ±7.5°       |             |            |             |
|             | vertical    |             | ±30°       | ±30°        |
|             |             |             | vertical   | vertical    |
+-------------+-------------+-------------+------------+-------------+
| Incident    | 5-90meV     | 2-50meV     | 10-1000meV | 15-1000meV  |
| energy      |             |             |            |             |
| range       |             |             |            |             |
+-------------+-------------+-------------+------------+-------------+
| Energy      | 0.03        |             | 0.02\      | ΔE/E= 1.5%  |
| resolution  | \<ΔE/E\<0.1 |             | <ΔE/E\<.05 |             |
+-------------+-------------+-------------+------------+-------------+
| Q           | ΔQ/         |             |            |             |
| resolution  | Q\~0.01-.03 |             |            |             |
+-------------+-------------+-------------+------------+-------------+
| Maximum     | 1.1 x 10^7^ | 5.6 x 10^6^ | 7.8 x      | 9.6 x 10^5^ |
| flux at     |             |             | 10^5^      |             |
| sample      | at 15meV    | at 5meV     |            | at 100meV   |
|             |             |             | at 100meV  |             |
+-------------+-------------+-------------+------------+-------------+

Table 1.Comparison of the instrument parameters for the four direct
geometry spectrometers- CNCS, ARCS, HRCS and HYSPEC.

**Results**

![](media/image7.wmf)

**Fig. 7.** The flux on sample as a function of incident energy for
HYSPEC (red and black), CNCS (green), ARCS(dark blue) and HRCS(light
blue).CNCS, ARCS and HRCS results were normalized to the energy
resolution of HYSPEC for the comparison.

The flux on sample for HYSPEC, CNCS, ARCS and HRCS is plotted in Fig. 7.
It is clear from this figure that HYSPEC has a larger flux on sample
than the other instruments from 5 to 60meV, even if we were to stay with
a 8cm offset configuration of the instrument. However, if we can provide
adequate shielding for HYSPEC in the straight guide (no offset)
configuration we can get a better flux on sample even in the 60-90meV
range.

**Section 4**

**Comparison of NISP with MCSTAS**

Initially, all design simulations for HYSPEC were done with NISP (2).
When the 15meV simulations were repeated with the MCSTAS (1) code,
however, the flux on sample turned out to be a factor of four smaller
than that obtained from NISP. Comparing the output element by element,
most of the difference appeared to be coming from the guide transmission
and PG crystal reflectivity algorithms, but there were also smaller
differences arising from the chopper and collimator algorithms and, in
addition, the neutron flux at the guide entrance closest to the
moderator was not the same. A detailed comparison of each algorithm used
in the HYSPEC simulation helped us get a better understanding of why the
results differed. Part of the reason for the lack of acceptable
agreement turned out to be differences in the algorithms. The other part
was that errors were found in the two codes. Those in NISP have since
been corrected by P.A. Seeger; latest version incorporates these
changes. E. Farhi (at ILL) and K. Leffman (at RISOE) have been informed
of the errors we found in MCSTAS. Below we briefly describe these NISP,
MCSTAS comparisons.

**Moderators**

Monte Carlo (MC) simulations for all moderators have been made by E.
Iverson. The data for the coupled H~2~ moderator used in our HYSPEC
simulations are contained in the file 'hl211f_td_05.dat', which can be
downloaded from the SNS website(ref. 4). Iverson simulates the neutron
flux (n/pulse/Ev/sr) emitted by the moderator for different energy
ranges. His moderator tables also list the time distribution of the
emitted neutrons for each energy range. The MC source interpolation
scheme used in the NISP simulations was developed by P.A. Seeger and in
MCSTAS by G. Granroth. To compare the two interpolation schemes the
neutron intensity at the entrance of a 4cm wide, 12cm high guide
starting 1.5m from the moderator was generated by both NISP and MCSTAS.
The resulting intensities were compared to each other as well as to an
approximate value obtained from E. Iverson\'s moderator calculations
assuming a linear interpolation over energy bins. It was found that the
NISP interpolation scheme, which used a wavelength-dependent sampling
method, did not originally give good agreement with the Iverson data.
The sampling algorithm was then modified by P.A. Seeger; the new NISP
values were in better agreement both with MCSTAS and with Iverson\'s
data. The results are shown in Fig 3.

**Guides**

In long guides the contribution of unreflected
(i.e.\"straight-through\") neutrons can be neglected and the
transmission and angular acceptance of the guides will be effectively
determined by the reflectivity function used in the simulations. The
NISP algorithm for guides has 100% reflectivity for θ=θ~c~(Ni), the
reflection probability decreases to a value R~c~ at mθ~c~(Ni) -
decreasing proportional to \[θ-mθ~c~(Ni)\] -- and, finally, there is a
so-called "Fresnel tail" for θ\>mθ~c~(Ni). Our simulations assumed m=3
guides with R~c~ assigned a value of 0.9 i.e.

R(NISP)= 1.0 for θ≤θ~c~(Ni),

= 1.0 -- α\[θ-m\*θ~c~(Ni )\] θ~c~\<θ≤mθ~c~(Ni),

= {\[1-sqrt(1-eta)\]/\[1+sqrt(1-eta)\]}2 for θ\>m\*θ~c~(Ni)

with α=(1.0-Rc)/2θ~c~(Ni), and eta=\[m\*θ~c~(Ni)/θ\]2 .

The MCSTAS reflectivity function, on the other hand, is defined as a
function of the scattering wave vector Q= \|k~i~-k~f~\| ≈ (4π/λ)θ.
According to the MCSTAS manual, "R=R~0~ for Q\<Q~c~, where Q~c~
=(4π/λ)θ~c~(Ni) is the critical scattering wave vector for Ni*.* At
larger values of Q, the reflectivity starts falling linearly with a
slope α until a cutoff at Q=m\*Q~c~. The width of the cutoff is denoted
W". The reflectivity is given by the following formula

R(MCSTAS) = R~0~ if Q=Qc,

= 0.5 R~0~{1-tanh\[(Q-m\*Qc)/W\]}{1.0- α(Q-Qc)} if Q\>Qc.

Unfortunately, this reflectivity function doesn't yield the trapezoidal
reflectivity curve shown in the MCSTAS manual nor does it result in a
linear decrease in reflectivity (with a slope α) as the value of Q
increases from Q~c~ to m\*Q~c~. At Q=m\*Qc the value of
tanh\[(Q-m\*Q~c~)/W\]=0.0. Therefore, at Q=m\*Q~c~ the value of the
reflectivity R=0.5 R~c~, not R=R~c~ as is shown in the MCSTAS manual
figure. The reflectivity decrease is also faster than a linear
dependence on (Q-m\*Q~c~). Furthermore, any attempt to increase the
reflectivity tail by increasing the value of W, results in suppressing
the reflectivity for values of Q between Q~c~ and m\*Q~c~. The
discrepancy between the figure and the equations leads the user to
assume that a trapezoidal reflectivity function is being used, when, in
fact that is not the case.

The more significant problem in MCSTAS is that the variable R~0~ is
missing in the code for the guide components "Guide.comp" and
"Channeled-guide.comp" for the Q\>Q~c~ line. This will cause a
discontinuity at θ=θ~c~ (Q=Q~c~) if R~0~ is not equal to 1. It is
essential that the variable R~0~ be inserted in the MCSTAS code for the
all guide components. The detailed form of the "tail function", however,
does not appear to have a significant impact on the transmission of the
guides that were used in our simulations. However, if the guides were
longer, or the dimensions were such that the neutrons made many bounces,
the cumulative effect of the tail function might turn out to be
significant.

When the transmission of neutrons through the guides as simulated by
NISP and MCSTAS was compared, it was found that the NISP guide algorithm
indicated that substantially more neutrons were being transmitted. The
discrepancy could not be explained away by the differences in the
reflectivity function. Moreover, the neutrons exiting the NISP guides
were seen to have an unrealistically large vertical divergence. The only
reasonable explanation for both these observations appeared to be that
the NISP software was calculating an incorrect value (too large by a
factor of √2) for the critical angle of the supermirror guides. This
was, in fact, the case and after the error was corrected the NISP and
MCSTAS calculated guide transmissions were in better agreement. However,
the discrepancy did not disappear altogether, and the MCSTAS calculated
transmissions are consistently slightly smaller.

**Choppers**

There is a difference between the disc chopper algorithms of NISP and
MCSTAS. The MCSTAS disc choppers have a wedge-shaped opening, and the
user is asked to specify the slot width as well as the radius of the
disc with respect to the midpoint of the guide immediately upstream of
the chopper. Since the disc blade opening is wedge-shaped, even when the
chopper is in the fully open configuration some of the neutrons exiting
the guide will not go through the opening. NISP, on the other hand,
assumes a rectangular disc chopper opening of the same height as the
guide hence all neutrons exiting the guide can also pass through the
chopper opening. Since disc chopper openings tend to be wedge-shaped
rather than rectangular, the NISP chopper algorithm overestimates the
chopper transmission. Thus we believe the MCSTAS algorithm to be the
more realistic of the two.

**Vertical Focusing, Curved Crystal Monochromators**

The monochromator algorithms in NISP and MCSTAS are very different.
MCSTAS simulates an infinitely thin single crystal with all Bragg
reflection taking place at a single surface which is imagined to be
composed of a large number of infinitely thin micro-crystals with
orientations that deviate from the nominal crystal orientation with a
Gaussian distribution. The full width at half maximum of the Gaussian
distribution defines the mosaic spread which can be either isotropic or
anisotropic. The variance of the lattice spacing is, however, assumed to
be zero. The probability of finding a particular orientation is thus
proportional to a Gaussian in the angle between the MC selected and the
most probable micro-crystal orientation hence the probability of
reflection is

P~reflect~ = R~0~exp (-d^2^/2σ^2^),

where R~0~ is defined to be the peak reflectivity, σ is the mosaic
spread, and d is the angle that the nominal scattering vector makes with
the most probable scattering vector. Values for R~0~ and for the
horizontal and vertical mosaic distributions are input by the user.

NISP takes a different approach. It assumes an ideally-imperfect crystal
of finite thickness and uses the V. F. Sears\' solution of the Darwin
equations (10). In this algorithm neutrons can Bragg scatter at planes
within the crystal hence a crystal thickness, a layer thickness, and a
reflectivity per layer all have to input by the user. Since each layer
is assumed to consist of micro-crystalline mosaic blocks both horizontal
and vertical mosaic parameters must also be put in by the user. In the
so-called "thick" crystal limit the NISP algorithm produces almost
perfect reflectivity.

HYSPEC simulations using NISP indicated almost perfect PG crystal
reflectivity with what appeared to be reasonable input parameters, a
result in disagreement with the measurements of Shapiro and Chesser (3),
who report values of the PG 002 reflectivity for incident energies
ranging from 4.5 to 40meV. Their measured reflectivities for the
experimental layout they used were approximately 0.8 at 5meV, 0.7 at
15meV, and 0.6 at 25-40meV.

Simulations of the Shapiro Chesser experiment were done with both NISP
and MCSTAS for neutron energies of 5, 15, and 30 meV and the various
parameters adjusted until both matched the experimental results. These
same parameters were then used to do HYSPEC simulations at 5, 15 and
30meV. The 30meV values were also used for the 60 and 90meV simulations
since no experimental results were available to bench mark the
algorithms at these energies. Thus it is not unlikely that the neutron
flux at these energies was overestimated.

In the process of doing these simulations we discovered a problem with
MCSTAS monochromator algorithm. When the sign of the Bragg angle is
changed from positive to negative the reflected intensities differ. The
input files and results were sent to E. Farhi at ILL who indicated he
obtained the same results and would look into the problem. Until the
problem is solved, he advised using only positive values for the Bragg
angle.

**Section 5**

The efficiency of a curved, vertical-focusing crystal depends on two
factors: the reflectivity of the crystal and the intensity gain obtained
by vertically focusing the beam. The gain produced by vertical focusing
of a multi-plate, pseudo-curved PG crystal was measured by I. Zaliznyak
at NIST. Both MCSTAS and NISP simulations of this measurement were in
good agreement with the experiment which gave us confidence that both
algorithms are correct in their treatments of vertical focusing. For
equivalent bench marking of PG crystal reflectivities we used the
Shapiro Chesser measurements.

**Simulation of the Shapiro Chesser measurements**

Their measurements were made on triple axis spectrometer (TAS). The
monochromator was a curved PG (002) crystal, a chromium crystal was
mounted on the sample axis, and the PG crystal whose reflectivity was to
be measured was placed at the analyzer position. The horizontal
collimation for both the in-pile and exit collimators was 20 minutes but
there was no horizontal collimation after the chromium crystal. Also,
there was no vertical collimation before the chromium but a 20 minute
vertical collimator was positioned after the chromium crystal. The
intensity at a detector of fixed area (and at a fixed distance from the
analyzer) was measured when the PG crystal in the analyzer position was
in the beam (and adjusted to give the maximum Bragg-reflected intensity,
then the detector was repositioned in the incident beam, the PG crystal
removed and the intensity again measured. The ratio of the two
intensities then gave the peak reflectivity. Shapiro and Chesser note
that their experimental results were only 10% lower than the values
calculated using equations for an ideally imperfect crystal given by
Bacon (11).

Our simulations reproduced the layout of the BNL H8 TAS. The horizontal
and vertical mosaic of the simulated crystals were assigned the same
values as those for the actual crystals used in the experiment. For the
MCSTAS simulations the value of R~0~ was adjusted until the simulated
peak reflectivity had the same value as the measured reflectivity at 5,
15, and 30 meV. These values of R~0~ were used for the HYSPEC
simulations.

Determining the corresponding values for NISP was much more difficult
and could not be done in a completely consistent way. In these
simulations the PG crystal was assigned the same thickness (1/16 inch)
and the same horizontal and vertical mosaic spread (24 minutes) as the
crystal used in the experiment. Two microscopic variables - the layer
thickness and reflectivity/layer - then had to be adjusted to obtain a
particular value for the macroscopic peak reflectivity. In fact,
whatever the number of layers the reflectivity/layer could always be
adjusted to generate the same macroscopic peak reflectivity for the
crystal. But these many combinations of variables yielded different
reflectivities when the values of the horizontal or vertical mosaic of
the crystal were changed from 24 minutes to some other value! If the
horizontal and vertical mosaic of the HYSPEC monochromator were to be
fixed at 24 minutes, then we could use the parameters determined from
this simulation, but if the mosaic spread was assigned some other value
than 24 minutes it was unclear which of the many combinations of values
for the number of layers and the reflectivity/layer should be used.
Depending on the choice of parameters, the simulated peak
reflectivitites varied from values much lower to much higher than the
Shapiro Chesser measured value.

**Simulations to compare the intensity gain due to a curved crystal with
experiment**

Both simulations modeled the essential features of the Zaliznyak
experiment at NIST (12). A 12cm by 12cm steady state source was
introduced to produce neutrons with wavelength λ= 4.045A (5meV) and a
Δλ/λ of 0.1, followed (at a distance of 1m) by a ^58^Ni guide 5cm wide,
12cm high and 18.85 m long.. The PG monochromator (24cm wide and 20cm
high and with an isotropic mosaic of 24 minutes) was positioned 20m from
the source. As in the experiment, an 80 minute collimator of length 30cm
was placed 70cm downstream of the PG crystal and a detector was
positioned at the sample position 1.67m downstream of the crystal. Again
following the experiment, the neutron intensity on a 2cm by 2cm detector
at the sample position was simulated first for a flat crystal and then
successively for a crystal with curvatures of 1.9, 2.0, and 2.1m. The
maximum intensity gain of 4 was found for a curvature of 2.0m using both
the NISP and MCSTAS codes, the result is in excellent agreement with the
measured value.

**References**

1.  MCSTAS home page: <http://www.neutron.risoe.dk/mcstas/> , or
    <http://www.ill.fr/tas/mcstas/> .

> McStas, a general software package for neutron ray-tracing
> simulations, K. Lefmann and K. Nielsen, Neutron News 10/3, 20-24,
> 1999.
>
> User and Programmers Guide to the Neutron Ray-Tracing package McStas,
> Version 1.4, July 2000, Kristian Nielsen and Kim Lefmann,
> Riso-R-1175(EN).

2.  The MCLIB Library: Monte Carlo simulation of neutron scattering
    instruments, P.A. Seeger, 13th International Meeting of the
    International Collaboration on Advanced Neutron Sources (ICANS
    XIII), Oct.110-14, 1995, Paul Scherrer Institute, Villigen. PSI
    conference proceedings 95-02, pg 195.

> A user-friendly geometry interface for the Monte Carlo neutron optics
> code MCLIB, T.G.Thelliez, L.L.Daemen, P.A.Seeger, and R.P.Hjelm, ibid,
> pg 307.
>
> The MCLIB Library: new features, P.A.Seeger,Workshop on Methods for
> Neutron Scattering Instrument Design, Lawrence Berkeley Laboratory,
> Berkeley, CA, Sept. 23-27,1996.

3.  Characteristics of pyrolytic graphite as an analyzer and higher
    order filter in neutron scattering experiments, S.M. Shapiro and
    N.J. Chesser, Nuclear Instruments and Methods 101, 183-186 (1972).

4.  The site <http://www.sns.anl.gov/components/moderator/index.shtml>
    can be used to access the HPTS design as of 2/21/01 (HL2 series)
    which was used in our simulations, or the HPTS design as of 7/31/02
    (SCT series).

5.  G. Granroth (private communication).

6.  J.V. Pearce (private communication)

7.  Optimization of the guide configuration of the Cold Neutron Chopper
    Spectrometer, J.V. Pearce, D.G. Narehood, and P.E. Sokol, 24 April,
    2002 (unpublished).

8.  <http://www.sns.gov/users/instrument_data_sheets.pdf>

9.  High Resolution Chopper Spectrometer Summary for IDT, Garrett E.
    Granroth, July 8, 2002.

10. Bragg reflection in mosaic crystals. I. General solution of the
    Darwin equations, V.F. Sears, Acta Cryst. A53, 35-45 (1997). II
    Neutron monochromator properties, Acta Cryst. A53, 46-54 (1997)

11. Neutron diffraction, G.E. Bacon (Oxford University Press,
    London, 1962) pg 65.

12. I. Zaliznyak