Introduction to Wavelength Frame Multiplication#

This notebook aims to explain the concept of wavelength frame multiplication (WFM), why is it used, how it works, and what results can be expected from using WFM at a neutron beamline.

Much of the material presented here was inspired by / copied from the paper by Schmakat et al. (2020), which we highly recommend to the reader, for more details on how a WFM chopper system is designed.

[1]:

import numpy as np
import matplotlib.pyplot as plt
import scipp as sc
from scipp import constants
import scippneutron as scn
import ess.wfm as wfm
import ess.choppers as ch

Note

The source code used to create the figures below can be viewed by downloading this notebook (this is hidden from the documentation pages for the sake of clarity).

The long ESS pulse#

Instruments at a pulsed neutron source, assuming an idealized rectangular source pulse in time of length \(t_{P}\), have a resolution given by

\[\frac{\Delta \lambda}{\lambda} = \frac{t_{P}}{t} = \frac{t_{P}}{\alpha \lambda z_{\rm det}} ~, ~~~~~(1)\]

where \(\lambda\) is the neutron wavelength, \(t\) is time, \(\alpha = m_{\rm n}/h = 2.5278 \times 10^{-4}~{\rm s}\unicode{x212B}^{-1}{\rm m}^{-1}\) is the ratio of the neutron mass and the Planck constant, and \(z_{\rm det}\) is the distance from the source to the detector.

Here, we have assumed that the neutron velocity \(v\) is related to its wavelength via the de Broglie equation

\[\lambda = \frac{h}{m_{\rm n}v} = \frac{1}{\alpha v} ~.\]

A natural consequence of this is that the wavelength resolution \(\Delta \lambda / \lambda\) becomes finer with increasing wavelength.

This also means that the resolution is poor for a long-pulsed source such as the ESS, compared to that of a short-pulse facility, such as ISIS. A good way to visualize this is using a time-distance diagram, which can represent the paths taken by the neutrons from the source to the detector.

[3]:

figure1()

../../_images/techniques_wfm_introduction-to-wfm_5_0.png

As illustrated in Fig. 1, with a long pulse, two neutrons with very different wavelengths (\(\lambda_{1} < \lambda_{2}\)) can reach the detector at the exact same time, if they originated from a different part of the pulse.

The problem cannot be resolved at the detector. According to the difference between time recorded at the detector and the measured start of the source pulse, both neutrons have the same wavelength. The detector recording system has no way of knowing that this is not the reality or adjusting for this. We instead look at the pulse generation to find ways to better measure the wavelength of our neutrons.

So what is WFM anyway?#

Within the concept of wavelength frame multiplication (WFM), each source pulse is chopped into a number of sub pulses referred to as wavelength frames, where each wavelength frame \(N\) contains a subsequent part of the spectrum of the source pulse.

The main reason for using the WFM concept is to redefine the burst time \(t_{P} = \Delta t\) as implied by Eq. (1), in order to match the required wavelength resolution of the experiment. A secondary objective, or constraint of the first, is to utilize as much of the source pulse as possible.

[4]:

figure2()

../../_images/techniques_wfm_introduction-to-wfm_8_0.png

A closer look#

The ESS pulse shape#

At a real beamline, the pulse shape is not rectangular, but has rising and falling edges, as shown in Fig. 3.

[5]:

figure3()

../../_images/techniques_wfm_introduction-to-wfm_10_0.png

Here we define several important quantities:

The pulse \(t_0\) is defined as the point in time at which the pulse is bright enough for the purposes of the experiment.
\(t_{\rm A}\) is the time when the flux has fallen down to a level below the required brightness.
\(t_{\rm P} = t_{\rm A} - t_{0}\) is the portion of the pulse that is used for the measurements, the analog of the pulse length for the ideal rectangular pulse above.
\(t_{\rm B}\) marks the end of the pulse; i.e. the time when the flux is considered to be effectively zero.

Using a single WFM chopper#

The effective burst time \(\Delta t\) is defined by a WFM chopper (WFMC), as illustrated in Fig. 4, for the two limiting wavelengths \(\lambda_{\rm min}\) and \(\lambda_{\rm max}\) of a single wavelength frame \(N\).

The wavelength frame is re-limited in a predefined time window by at least one frame overlap chopper (FOC) that inhibits the overlap of neutrons from various frames, as indicated by the dashed lines in Fig. 4. Their wavelength is labeled with \(\lambda_{\rm min}^{'}\) and \(\lambda_{\rm max}^{'}\). The FOC is also removes undesired neutrons with the wrong wavelength that arise from the rising and falling edges at the beginning and at the end of the source pulse. Although their intensity is small, neutrons with an undesired wavelength would lead to an increased background.

[6]:

figure4()

../../_images/techniques_wfm_introduction-to-wfm_12_0.png

The relative spectral resolution \(\Delta \lambda / \lambda\) at the detector position \(z_{\rm det}\) is defined by the burst time \(\Delta t\) and the time-of-flight \(t(\lambda)\) of the neutrons (see eq. (1)). Because \(\Delta t\) is independent of the wavelength for the case of a single WFM chopper disc, \(\Delta \lambda / \lambda\) depends on \(\lambda\), on the distance \(z_{\rm WFM}\) of the WFM chopper from the source, and on the detector position \(z_{\rm det}\) as depicted in Fig. 4.

The WFM chopper acts as a virtual source, reducing the effective burst time \(\Delta t\), while at the same time also reducing the effective time-of-flight to the detector from \(t(\lambda) = \alpha \lambda z_{\rm det}\) to \(t(\lambda) = \alpha \lambda (z_{\rm det} - z_{\rm WFM})\). Note that this assumes a straight line from WFM choppers to the detector. In the case of scattering from a sample, the first branch of the flight path (\(L_{1}\)) would be modified, while the secondary path (\(L_{2}\)) will remain unchanged.

A resolution that depends on \(\lambda\) is not suited for some applications (such as imaging, reflectometry, …) where a constant \(\Delta \lambda / \lambda\) is much more desirable.

A short example#

We now proceed to illustrate data processing at a WFM beamline in the form of a short example.

Create a beamline#

We first create a beamline with two WFM choppers and 6 wavelength frames, using the wfm.make_fake_beamline helper utility. The detector is placed 60 m from the source.

[9]:

coords = wfm.make_fake_beamline(nframes=6)
ds = sc.Dataset(coords=coords)
f = wfm.plot.time_distance_diagram(ds)

../../_images/techniques_wfm_introduction-to-wfm_18_0.png

The properties of the frames (boundaries in time and wavelength) are computed using the wfm.get_frames function

[10]:

frames = wfm.get_frames(ds)
frames

[10]:

scipp.DataGroup

(frame: 6)

time_correction
scipp
Variable
(frame: 6)
float64
µs
4702.574, 7277.777, ..., 1.408e+04, 1.607e+04
time_min
scipp
Variable
(frame: 6)
float64
µs
1.804e+04, 4.068e+04, ..., 1.005e+05, 1.180e+05
delta_time_min
scipp
Variable
(frame: 6)
float64
µs
113.750, 284.797, ..., 736.629, 868.881
time_max
scipp
Variable
(frame: 6)
float64
µs
3.839e+04, 5.993e+04, ..., 1.168e+05, 1.335e+05
delta_time_max
scipp
Variable
(frame: 6)
float64
µs
284.797, 445.190, ..., 868.881, 992.895
wavelength_min
scipp
Variable
(frame: 6)
float64
Å
1.000, 2.504, ..., 6.476, 7.638
wavelength_max
scipp
Variable
(frame: 6)
float64
Å
2.504, 3.914, ..., 7.638, 8.729
delta_wavelength_min
scipp
Variable
(frame: 6)
float64
Å
0.008, 0.021, ..., 0.055, 0.065
delta_wavelength_max
scipp
Variable
(frame: 6)
float64
Å
0.021, 0.033, ..., 0.065, 0.074
wfm_chopper_mid_point
scipp
Variable
()
vector3
m
[0. 0. 7.]

Create some neutrons#

We create 6 neutrons with known wavelengths, one for each frame. We choose the values for the wavelengths based on the limits given in the frames information above.

[11]:

wavelengths = sc.array(
    dims=['wavelength'], values=[1.5, 3.0, 4.5, 6.0, 7.0, 8.25], unit='angstrom'
)

We assume that all 6 neutrons originated half-way through the source pulse, and we can thus calculate the times at which they hit the detector.

[12]:

# Neutron mass to Planck constant ratio
alpha = sc.to_unit(constants.m_n / constants.h, 's/m/angstrom')
# Distance between the detector pixel and the source
dz = sc.norm(coords['position'] - coords['source_position'])
# Compute arrival times, in microseconds
arrival_times = (
    sc.to_unit(alpha * dz * wavelengths, 'us')
    + coords['source_pulse_t_0']
    + (0.5 * coords['source_pulse_length'])
)
arrival_times

[12]:

scipp.Variable (304 Bytes)

(wavelength: 6)

float64

µs

2.431e+04, 4.706e+04, ..., 1.077e+05, 1.267e+05

Values:
array([ 24310.05718489,  47060.11436978,  69810.17155467,  92560.22873956,
       107726.93352949, 126685.3145169 ])

References#

Schmakat P., Seifert M., Schulz M., Tartaglione A., Lerche M., Morgano M., Böni P., Strobl M., 2020, Wavelength frame multiplication chopper system for the multi-purpose neutron-imaging instrument ODIN at the European Spallation Source, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 979, 164467

Introduction to Wavelength Frame Multiplication

Contents

Introduction to Wavelength Frame Multiplication#

The long ESS pulse#

So what is WFM anyway?#

A closer look#

The ESS pulse shape#

Using a single WFM chopper#

Using a pair of optically blind choppers#

A short example#

Create a beamline#

Create some neutrons#

Wrap the neutron counts and the beamline into a DataArray#

Stitch the frames#

References#