Oxford School on Neutron Scattering (2019)

Practical Neutron Scattering (2 lectures) Ross Stewart (ISIS Neutron and Muon Source)

Recommended Books:

• C G Windsor, Pulsed Neutron Scattering

This is an excellent - and very complete - text on the practicalities of (pulsed) neutron scattering, which can nevertheless also be applied to continuous neutron scattering in many cases. Covers a lot of material that other texts often miss out concerning the practicalities of neutron scattering

• H J C Berendsen, Data and Error Analysis

This is really a must for all experimental scientists! Explains fully the concepts of error presentation, estimation and propagation. Includes a full account of Poisson statistics.

Contents

1. Before the experiment: Optimisation of sample size and shape1.1. scattering units 1.2. number density and scattering length density 1.3. transmission of neutrons 1.4. absorbing materials - optimum sample thickness 1.5. optimising sample shape 1.6. sample holders

2. During the experiment: Counting Neutrons2.1. neutron histograms 2.2. systematic error & random uncertainty 2.3. probability distributions - expectation, variance, covariance 2.4. combining measurements 2.5. the Poisson distribution - error estimation

3. After the experiment: Neutron Data Correction3.1. normalisation to monitor / time / current 3.2. normalisation of the detector 3.3. absolute normalisation 3.4. background subtraction

�1

1. Before the Experiment: Optimising sample size & shape Calculating and optimising the size and shape of your sample is most important in neutron scattering. Neutron beams worldwide are commonly low in flux - much much lower than x-ray beams for example. So most of the time it is important to maximise the number of neutrons scattered by your sample - and hence the size of your sample - and therefore the number of scattering units. This is not always the case however.

• Sometimes your sample may scatter too strongly for the detectors (dead-time)• Perhaps you have absorbing materials in your sample - leading to self-attenuation• Most modelling procedures make the assumption that the scattered neutrons have been

scattered only once by the sample. This is never true - but we still want to try and minimise multiple scattering if we can

Correction routines do exist (e.g. in Mantid) in order to correct for sample self-attenuation and multiple scattering, but these aren’t commonly used. This is because they are tricky and time-consuming calculations. So most of the time we rely on minimising these effects.

1.1 Scattering unitsIn principle we can define the sample scattering unit in any way we like. Often it is convenient to use the sample chemical formula unit for this. We then can find the cross-sections per formula unit.

For example, the table below shows the cross-sections and formulas weights for one formula unit of poly-ethylene.

σcoh is the coherent cross-sectionσinc is the incoherent cross-sectionσabs is the absorption cross-section (*wavelength dependent)Mf is the formula weight

The cross-sections per formula unit are just the sum over all the nuclei and can be found in tables on-line.

1.2 Number density and scattering length densityNumber of scattering units in the sample (mass, m) is given by

Where NA = 6.022 x 1023 mol-1 is Avogadro’s Number. Similarly the number density is given by

where ρ is the density of the sample.

�2

σcoh (b) σinc (b) σabs (b) Mf (amu)

C 5.55 0.001 0.3326 12.011

H 1.7568 80.26 0.0035 1.0079

C2H4 18.1272 321.042 0.6792 28.0536

Ns =m

MfNA (1.1)

ns =⇢

MfNA (1.2)

If we express NA in units of x1024 mol-1 then we get the number density in units of x1024 cm-3 = (barn.cm)-1. The reason for the choice of these units will become clear later on.

with ρ in units of g cm-3 Mf in g mol-1 and ns in (barn.cm)-1.

Example:For poly-ethylene ρ = 0.93 g cm-3 Mf = 28.054 g mol-1 so ns ~ 0.02 (barn.cm)-1 or 2 x 1022 cm-3The scattering length density is defined as the total scattering length per unit volume.

where the subscript i runs over the number of atoms in the volume of interest, Vm. The dimensions of the SLD are inverse area and given commonly in cm-2. Since the volume Vm is arbitrary, it makes sense to recast it in terms of the mass of a formula unit, Mf, of the material divided by the density ρ.

The scattering length density can now be written as

Example:Poly-ethylene: SLD ~ 2 x 1022 x (2 x 6.65x10-13 + 4 x -3.73x10-13) = -3.24x10-9 cm-2

1.3 Transmission of neutronsIt is important to know that the chances of a neutron getting through your sample are high. To calculate the transmission of a thin slab-shaped sample, we refer to the figure below of a thin slab of area A, and thickness dz. The neutron intensity incident on the slab is Iz and the transmitted intensity is Iz + dIz . (the change in intensity dIz is negative)

The slab consists of molecules, with a total cross-section of σT per formula unit. Therefore, from the definition of the number density there are ns A dz formula units in the slab. So the total absorption cross-section is

σT ns A dz.

The fraction of neutrons absorbed is the total absorption cross-section divided by the dimensional cross-section (the area A). Therefore,

�3

ns =⇢

1.661Mf

SLD =⇢NA

Mf

X

i

bi = ns

X

i

bi<latexit sha1_base64="3DYkyW6d6YT3RDgfFSbxIcBo2zg=">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</latexit>

Vm =Mf

⇢NA<latexit sha1_base64="Sy/zPFpoI7RF++fjs9SkJ2cA4YQ=">AAACBHicbVDLSsNAFJ3UV62vqMtuBovQVUmqoBuh4saNUsE+oAlhMp20Q2cmYWYilNCFG3/FjQtF3PoR7vwbp20W2nrgwuGce7n3njBhVGnH+bYKK6tr6xvFzdLW9s7unr1/0FZxKjFp4ZjFshsiRRgVpKWpZqSbSIJ4yEgnHF1N/c4DkYrG4l6PE+JzNBA0ohhpIwV2uR1weAG9SCKc3QTRJPPkMIa3weUEBnbFqTkzwGXi5qQCcjQD+8vrxzjlRGjMkFI910m0nyGpKWZkUvJSRRKER2hAeoYKxInys9kTE3hslD6MYmlKaDhTf09kiCs15qHp5EgP1aI3Ff/zeqmOzv2MiiTVROD5oihlUMdwmgjsU0mwZmNDEJbU3ArxEJk8tMmtZEJwF19eJu16zT2p1e9OK41qHkcRlMERqAIXnIEGuAZN0AIYPIJn8ArerCfrxXq3PuatBSufOQR/YH3+ACdClw8=</latexit>

SLD =

Pi bi

Vm<latexit sha1_base64="eGQdhBgzpqagMIBa3LYHT+SNgbs=">AAACEHicbVDLSsNAFJ3UV62vqEs3g0XsqiRV0I1Q0IULFxXtA5oQJtNJO3QmCTMToYR8ght/xY0LRdy6dOffOGmz0NYDFw7n3Mu99/gxo1JZ1rdRWlpeWV0rr1c2Nre2d8zdvY6MEoFJG0csEj0fScJoSNqKKkZ6sSCI+4x0/fFl7ncfiJA0Cu/VJCYuR8OQBhQjpSXPPHY4UiPB07ubqwxeQCcQCKeOTLhHoe/RDKaw4/HMM6tW3ZoCLhK7IFVQoOWZX84gwgknocIMSdm3rVi5KRKKYkayipNIEiM8RkPS1zREnEg3nT6UwSOtDGAQCV2hglP190SKuJQT7uvO/Hw57+Xif14/UcG5m9IwThQJ8WxRkDCoIpinAwdUEKzYRBOEBdW3QjxCOhKlM6zoEOz5lxdJp1G3T+qN29Nqs1bEUQYH4BDUgA3OQBNcgxZoAwwewTN4BW/Gk/FivBsfs9aSUczsgz8wPn8A35GcZQ==</latexit>

(1.3)

(1.4)

(1.5)

(1.6)

dIzIz

= ��Tnsdz<latexit sha1_base64="sMHAZLjryOyAfCvVA+6hySWRAEU=">AAACDHicbVDNSgMxGMz6W+tf1aOXYBF6sexWQS9CwYveKvQPusuSzWbb0CS7JFmhXfoAXnwVLx4U8eoDePNtTNs9aOtAwjAzH8k3QcKo0rb9ba2srq1vbBa2its7u3v7pYPDtopTiUkLxyyW3QApwqggLU01I91EEsQDRjrB8Gbqdx6IVDQWTT1KiMdRX9CIYqSN5JfKbiQRzsI7fzzJphe8hmeuon2O/CYUvoLh2KTsqj0DXCZOTsogR8MvfblhjFNOhMYMKdVz7ER7GZKaYkYmRTdVJEF4iPqkZ6hAnCgvmy0zgadGCWEUS3OEhjP190SGuFIjHpgkR3qgFr2p+J/XS3V05WVUJKkmAs8filIGdQynzcCQSoI1GxmCsKTmrxAPkGlHm/6KpgRnceVl0q5VnfNq7f6iXK/kdRTAMTgBFeCAS1AHt6ABWgCDR/AMXsGb9WS9WO/Wxzy6YuUzR+APrM8ftiuaqw==</latexit>

(1.7)

Transmission through a slab

Integrating both sides of 1.7 with respect to z,

For a finite slab of total thickness, t, the limits of integration are z=0, t with I0 at z=0 and I1 at z=t. Therefore,

Finally, we take the exponential of both sides to find the expression for the neutron transmission

This is a form of the Beer-Lambert Law for optical attenuation. For neutrons, the productis termed the attenuation coefficient (sometimes denoted by μ) and commonly given in units of cm-1.

Example:Often it is desirable to limit the total scattered fraction of neutrons to less than 10%. This ensures that multiple scattering is a small additional component of the neutron counts. To calculate how thick the 10% scattering sample should be we set Eq. 1.10 to

Where σS denotes the total scattering cross-section, and we have assumed no absorption.

So. A 10% scattering poly-ethylene sample with σS = 339.2 barns ns = 0.02 (barn cm)-1 - we see now the usefulness of this strange unit

Has thickness t = 0.1 / 339.2 / 0.02 = 0.015 cm

So only around 0.15 mm thickness of poly-ethylene scatters 10% of the beam. This indicates the strength of the scattering of neutrons from hydrogen

1.4 Absorbing materials - optimum sample thicknessThere are a few isotopes of elements which prove especially challenging for performing neutron scattering measurements, due to their massive absorption cross-sections. They are - however -

often very useful materials use in neutron instrument shielding and detectors.

The table on the left shows examples of particularly highly absorbing materials (in their naturally occurring isotopic combinations). These large absorptions are due to nuclear resonances in these materials, and hence the absorption cross-sections are, in general, dependent on the incident neutron wavelength. The tabulated numbers on the left are the σabs values at a wavelength λ = 1.8 Å. At low energies σabs is taken to be linear in wavelength,

Sometimes it is possible to choose a non-absorbing isotope (e.g. B11 has a very low absorption cross-section σabs = 0.005 b).

�4

ln(Iz) = ��Tnsz + C<latexit sha1_base64="XKx5mFLrJYnoPYB0vT286e9O/ds=">AAACB3icbVDLSsNAFJ3UV62vqEtBBotQEUtSBd0IhW50V6EvaEKYTCd16GQSZiZCG7pz46+4caGIW3/BnX/jtM1CWw9cOJxzL/fe48eMSmVZ30ZuaXlldS2/XtjY3NreMXf3WjJKBCZNHLFIdHwkCaOcNBVVjHRiQVDoM9L2B7WJ334gQtKIN9QwJm6I+pwGFCOlJc88dBgv3XqjE3gNzxxJ+yHyGpB7Eo7gKax5ZtEqW1PARWJnpAgy1D3zy+lFOAkJV5ghKbu2FSs3RUJRzMi44CSSxAgPUJ90NeUoJNJNp3+M4bFWejCIhC6u4FT9PZGiUMph6OvOEKl7Oe9NxP+8bqKCKzelPE4U4Xi2KEgYVBGchAJ7VBCs2FAThAXVt0J8jwTCSkdX0CHY8y8vklalbJ+XK3cXxWopiyMPDsARKAEbXIIquAF10AQYPIJn8ArejCfjxXg3PmatOSOb2Qd/YHz+ACt9ltw=</latexit>

ln(I1)� ln(I0) = (��Tnst+ C)� (��Tns0 + C)

= �ns�T t<latexit sha1_base64="uvIu0MWAQD8K75sq6pJYKW9KW24=">AAACQnicbZBLSwMxFIUzvq2vqks3F4vSIi0zKuhGENzorkKrlU4ZMmnahmYyQ3JHKMXf5sZf4M4f4MaFIm5dmD4QrR4IHL5zL0lOmEhh0HWfnKnpmdm5+YXFzNLyyupadn3jysSpZrzKYhnrWkgNl0LxKgqUvJZoTqNQ8uuwezbIr2+5NiJWFewlvBHRthItwShaFGRvfKnyF4FXKI6MW4DdE8gXfSPaEQ0qoAIDCHtwVoDiJHdH3Pczu3ACxQH6zjHI5tySOxT8Nd7Y5MhY5SD76DdjlkZcIZPUmLrnJtjoU42CSX6X8VPDE8q6tM3r1ioacdPoDyu4gx1LmtCKtT0KYUh/bvRpZEwvCu1kRLFjJrMB/C+rp9g6bvSFSlLkio0uaqUSMIZBn9AUmjOUPWso08K+FViHasrQtp6xJXiTX/5rrvZL3kFp//Iwd5of17FAtsg2yROPHJFTck7KpEoYuSfP5JW8OQ/Oi/PufIxGp5zxzib5JefzC52WqQ0=</latexit>

(1.8)

(1.9)

T =I1I0

= exp(��Tnst)<latexit sha1_base64="Q/gyoNHOyb3asgO0unF+IHB2UIg=">AAACFnicbVDLSsNAFJ3UV62vqEs3g0Woi5akCroRCm50V6EvaEKYTCft0MkkzEzEEvoVbvwVNy4UcSvu/BsnbRbaeuBeDufcy8w9fsyoVJb1bRRWVtfWN4qbpa3tnd09c/+gI6NEYNLGEYtEz0eSMMpJW1HFSC8WBIU+I11/fJ353XsiJI14S01i4oZoyGlAMVJa8sxqC15BJxAIp7eePdXNmmYKeYgrVUfSYYi8FuSehOoUljyzbNWsGeAysXNSBjmanvnlDCKchIQrzJCUfduKlZsioShmZFpyEklihMdoSPqachQS6aazs6bwRCsDGERCF1dwpv7eSFEo5ST09WSI1Eguepn4n9dPVHDpppTHiSIczx8KEgZVBLOM4IAKghWbaIKwoPqvEI+QzkjpJLMQ7MWTl0mnXrPPavW783KjksdRBEfgGFSADS5AA9yAJmgDDB7BM3gFb8aT8WK8Gx/z0YKR7xyCPzA+fwDyzJyj</latexit>

(1.10)

�Tns<latexit sha1_base64="TqAzqFFe2lYpbVnafVTPQ+BOhl4=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRahp7JbBT0WvHis0C9olyWbZtvQJLsm2UIp/R1ePCji1R/jzX9jtt2Dtj4YeLw3w8y8MOFMG9f9dgpb2zu7e8X90sHh0fFJ+fSso+NUEdomMY9VL8SaciZp2zDDaS9RFIuQ0244uc/87pQqzWLZMrOE+gKPJIsYwcZK/kCzkcBBC8lAl4Jyxa25S6BN4uWkAjmaQflrMIxJKqg0hGOt+56bGH+OlWGE00VpkGqaYDLBI9q3VGJBtT9fHr1AV1YZoihWtqRBS/X3xBwLrWcitJ0Cm7Fe9zLxP6+fmujOnzOZpIZKsloUpRyZGGUJoCFTlBg+swQTxeytiIyxwsTYnLIQvPWXN0mnXvOua/XHm0qjmsdRhAu4hCp4cAsNeIAmtIHAEzzDK7w5U+fFeXc+Vq0FJ585hz9wPn8A5IuRcA==</latexit>

exp(��Snst) = 0.9

) �Snst ' 0.1<latexit sha1_base64="ZgEh++sJnVa1jtvNAjPrvzyZoVU=">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</latexit>

element σabs

Gd 49700 b

B 767 b

Cd 2520 b

Li 70 b

Ir 425 b

Dy 994 b

Sm 5922 b3He 5333 b

�abs(�) =���=1.8

abs

1.8 (1.11)

Some of the above absorber materials, especially Cd and B, are used for shielding instruments and bits of your sample holder that you do not want to scatter from.

Example:How much Cd is needed to absorb 99.9% of thermal neutrons?

ns = 0.046 (barn.cm)-1 σabs = 2520 barns

Therefore t = - ln(T) / σabs / ns = - ln(0.001) / 115.9 = 0.6 mm

If a sample has some absorption then it’s not quite so simple to work out the optimal thickness for scattering as it was for poly-ethylene using Eq. 1.10. In general, it will not be possible to get a 10% scatterer, because the sample will become too thick and the neutrons will not escape. If we assume a uniform shape to the sample (e.g. a slab-shaped sample for SANS or a cylinder for diffraction) then we can make a rough approximation that all neutron traverse the sample with the same path length. In that case we can write the fraction of neutrons scattered as

Where Tabs is the absorption dependent sample transmission given by Eq. 1.10 with the absorption cross-section σabs and TS is the scattering dependent transmission given by Eq. 1.10 with the (total) scattering cross-section σT. Expanding 1.12 we get

So - in order to maximise the scattered fraction of neutrons, we differentiate 1.13 and set to zero to find the turning point and therefore the optimal thickness of the absorbing sample

Example:Manganese metal has a moderately large absorption cross-section of 13.3 b @ 1.8 Å. Suppose we are doing an experiment on the IN5 time-of-flight spectrometer at the ILL with a incident neutron wavelength of λ = 5.1 Å. What in this case would be the optimum sample thickness?

σabs = 13.3 * 5.1 / 1.8 = 37.7 barns σT = 2.15 barns ns = 0.08 (barn cm)-1

Therefore

= 3.2 mm

This occurs at a sample transmission of 38%For broad wavelength bands - optimise for maximum desired wavelength.

�5

⌃ = Tabs ⇤ (1� TS)<latexit sha1_base64="gnVauiLwI5xL5Kn3qmhZRg9ZFLA=">AAACCnicbVDLSsNAFJ3UV62vqEs3o0WogiWpgm6EghuXlT6hCWEyndahM0mYmQglZO3GX3HjQhG3foE7/8ZJm4W2HrhwOOde7r3HjxiVyrK+jcLS8srqWnG9tLG5tb1j7u51ZBgLTNo4ZKHo+UgSRgPSVlQx0osEQdxnpOuPbzK/+0CEpGHQUpOIuByNAjqkGCkteeah06QjjuA1bHmJw5G6FzxBvkzT04p91vKaJ55ZtqrWFHCR2DkpgxwNz/xyBiGOOQkUZkjKvm1Fyk2QUBQzkpacWJII4TEakb6mAeJEusn0lRQea2UAh6HQFSg4VX9PJIhLOeG+7syOlfNeJv7n9WM1vHITGkSxIgGeLRrGDKoQZrnAARUEKzbRBGFB9a0Q3yOBsNLplXQI9vzLi6RTq9rn1drdRbleyeMoggNwBCrABpegDm5BA7QBBo/gGbyCN+PJeDHejY9Za8HIZ/bBHxifP0nFmUY=</latexit>

(1.12)

⌃ = T (1� exp[�ns�T t])= exp[�ns�abst]� exp[�ns(�abs + �T )t] (1.13)

d⌃dt

= ns(�abs + �T ) exp[�ns(�abs + �T )t]� ns�abs exp[�ns�abst] = 0

) exp[�ns�T t] =�abs

�abs + �T

) t =ln(�abs + �T )� ln(�abs)

ns�T

(1.14)

t =ln(37.7 + 2.15)� ln(37.7)

0.08 ⇤ 2.15<latexit sha1_base64="YDodGrYWrtRzuube1dULgKLbN8s=">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</latexit>

Scattered neutrons from Mn metal at λ = 5.1 Å

1.5 Optimising the sample shapeSo far we have considered only the transmission of the direct beam at zero scattering angle. However what we really need to know is what is the transmission of the scattered neutrons as a function of scattering angle. In order to calculate that we need to know the shape of the sample accurately, and then average over all the possible scattering paths.

As an example (and the only example which is tractable) we take the case of a slab shaped sample (which we assume is much wider than it is thick).

The incident neutrons make an angle γ with the face of the slab, and the neutrons scatter from a thin layer of thickness dx at a depth x within the slab.

We can write the initial and final path lengths before and after scattering as

The incoming flux φ0 is attenuated along L1

The outgoing beam is attenuated along L2 (and here we assume that the wavelength doesn’t change)

Outgoing number of neutrons is proportional to cross-section into solid angle ΔΩ, flux, ns, thickness of slab Δt and is attenuated along L2

So total neutrons scattered is found by integrating above wrt. x between x=0 and x=t, and then the transmission is given by dividing by the total neutrons scattered in the limit of zero-absorption, N0. Finally we end up with

This is one of the very few solvable cases for transmission as a function of scattering angle. And, as you can see it is a complex expression. It also assumes that the slab is infinitely wide, which is of course not really the case.

Below is plotted the transmission given by expression 1.14 for the case of a monochromatic beam at normal incidence - i.e. γ = 90° with a thickness of 1 cm, and a range of values of the attenuation coefficient of between 0.01 and 0.5 cm-1.

�6

L1 =x

sin �

L2 =t� x

sin(� � 2✓)

� = �0 exp(�ns�t)

= �0 exp

✓�ns�x

sin �

◆L1)

T2 = exp

✓�ns�

t� x

sin(� � 2✓)

◆

= �0ns exp✓�ns�x

sin �

◆dx

sin �

✓d�

d⌦

◆exp

✓� ns�(t� x)

sin(� � 2✓)

◆�⌦

�N = �ns�t

✓d�

d⌦

◆T2�⌦

T =1

ns�t(cosec� � cosec(� � 2✓))

⇢exp

✓� ns�t

sin(� � 2✓)

◆� exp

✓�ns�t

sin �

◆�(1.14)

Scattering from an “infinite” slab

At low angles the 2θ dependence is weak with the transmission being well approximated by the zero-angle transmission. However, as the scattering angle approaches 90° there is a very strong absorption. For this reason, slab shaped samples are popular in SANS experiments, and often in neutron spin-echo measurements. Beyond 2θ = 90° there is another equivalent expression for the transmission.

With the advent of large detector arrays which commonly surround the sample (in the equatorial plane) cylindrical sample geometries have become more common. Sometimes an annular cylinder is used to fill beam, but minimise absorption. In these cases there is no analytical expression for the transmission as a function of angle - it must be calculated numerically (i.e. lookup tables or Monte-Carlo algorithms are commonly used)

In the case of a solid cylinder or sphere with an attenuation coefficient of < 1 cm-1, a good approximation (better than 0.5% accuracy) is given by the expression

Where R is the radius of the cylinder / sphere and the coefficients a1, b1, a2 and b2 have the values,

See, A Hewat, Acta. Cryst. A 35 (1975) p248

This gives a much flatter transmission as a function of angle, and so is a good sample shape for neutron instruments which count neutrons over a wide angular range. Commonly, the sample transmission is assumed to be constant over the whole range of scattering angles in this geometry. Below is the transmission through a 1cm radius cylinder for a range of attenuation coefficients between 0.01 and 0.5 cm-1

�7

T = exp��(a1 + b1 sin2 ✓)ns�R� (a2 + b2 sin2 ✓)(ns�R)2

(1.15)

a1 b1 a2 b2

cylinder 1.7133 -0.0368 -0.0927 -0.3750

sphere 1.5108 -0.0315 -0.0951 -0.2898

Transmission of 1cm thick slab at normal incidence to the beam

1.6 Sample HoldersChoice of material of the sample holders / containers is crucial and depends on the type of neutron measurement being attempted. This also applies to external sample environments (such as cryostats, pressure cells, etc.)

Ideally the sample holders and sample environment should be made as thin as possible to reduce the mass of extraneous materials in the beam. They should also be shielded where possible using absorbing materials such as Cd or B4C (boron carbide).

The table below shows a selection of sample holders for different applications.

�8

Material Comment

Powder diffraction vanadium no Bragg peaks, large σinc some absorption

Inelastic scattering Al / Cu low scattering cross-sections, low incoherent scattering (Cu for low T)Polarized neutrons Al / Cu

Liquids / glasses vanadium / TiZr no Bragg peaks TiZr is a “null-matrix”

SANS Quartz very low small angle scattering

Single crystals Small pin (Al) sample holds own shape

Transmission of 1cm radius solid cylinder

2. During the experiment: Counting Neutrons

2.1 Neutron Histograms.Neutron instruments include detectors which count neutrons as a function of scattering angle and/or neutron time-of-flight. Note that the scattering angle is in general composed of two variables - in the case of spherical polar coordinates these are the in-plane and azimuthal angles θ and φ. In both of these cases the neutron data is stored - in memory and on disk - in the form of a histogram.

A histogram is a representation of data in which the data is sorted in increasing order as a function of some continuous variable, but which as been subdivided into predetermined intervals - often called ranges or bins. There is a mini-tutorial on histograms at: https://www.mathsisfun.com/data/histograms.html

The detector solid angle element, dΩ, naturally defines a histogram bin width. Remember that the solid angle is just defined by an area (in this case, the detector window) divided by the squared distance between the sample and the detector. For time-of-flight the histogram bins are determined by the counting electronics - and can generally be set as broad or as narrow as desired. There is no requirement that the histogram bin ranges be constant, though in practice they often are. If the bin-widths are not the same, then the neutron counts must be divided by the bin widths to create a distribution function.

�9

This is a neutron time-of-flight distribution from LET (ISIS). It is a distribution because the neutron counts have been divided by the time-of-flight bin width (giving units of μs-1). The bin width is 200 μs.

This is a neutron scattering angle histogram from D7 (ILL). The Y-axis is just raw neutron counts, and the scattering angle bin size is ~1°

2.2 Systematic error and random uncertaintyThe measured histograms in a neutron experiment are subject to experimental error and uncertainty.

Systematic error can arise due to bad calibration and setup of the detectors (e.g. non-linearity), incorrect calibration of the detector angles or the time-of-flight counting electronics. It could even be said that it could be due to a bad sample…

The best way to check for systematic errors in your measurement is to repeat it. Perhaps on a different batch of sample and/or on a different instrument. Most rigorous is independent corroboration of the measurement. Unfortunately, independent corroboration and experimental repetition is very rare in the neutron scattering - since beam time is scarce and priority is given to experiments not already attempted.

Random uncertainty (often this is incorrectly called random error) is due to the unpredictability of the measurement. In the case of neutron scattering, we cannot predict in advance how many neutrons will be scattered into a detector in a certain time interval. We can only provide a probability that this will occur. The way to reduce random uncertainty in such a measurement is to repeat the measurement many times - and build up an average number of neutron counts per bin, and a spread in that value. The more times we repeat the measurement, the smaller the spread in the measured values. Multiple repetitions of a measurement build up a so-called probability distribution - of which the measured values are random samples.

2.3 Probability distributions - Expectation, Variance, CovarianceNeutrons arriving in a detector in a certain time interval (the counting time) can be considered as random samples of a probability distribution p(n), where n is the count number and is therefore always an integer. The expectation of n - also called the mean value of the probability distribution is given by

The variance of the distribution is defined as the expectation of the squared difference from the mean

The square root of the variance of the distribution is called the standard deviation and it is this that is used as an estimate of the random uncertainty.

Lastly - imagine that we have performed two measurements and we want to add them up.

First measurement: mean: μ1, variance: σ12Second measurement: mean: μ2, variance: σ22

Summed measurements: mean: μT, variance: σT2

The expectation of the summed measurement is:

�10

µ = E[n] =X

n

np(n)<latexit sha1_base64="f6KxsS6TNwbeHDIK+aRQP3MIjWM=">AAACA3icbVDLSgMxFM3UV62vUXe6CRahbspMFXQjFEVwWcE+YGYYMmmmDU0yQ5IRSim48VfcuFDErT/hzr8xbWehrQcu93DOvST3RCmjSjvOt1VYWl5ZXSuulzY2t7Z37N29lkoyiUkTJyyRnQgpwqggTU01I51UEsQjRtrR4Hritx+IVDQR93qYkoCjnqAxxUgbKbQPfJ7BS3jjicA0X2U8FFDAtCJOQrvsVJ0p4CJxc1IGORqh/eV3E5xxIjRmSCnPdVIdjJDUFDMyLvmZIinCA9QjnqECcaKC0fSGMTw2ShfGiTQlNJyqvzdGiCs15JGZ5Ej31bw3Ef/zvEzHF8GIijTTRODZQ3HGoE7gJBDYpZJgzYaGICyp+SvEfSQR1ia2kgnBnT95kbRqVfe0Wrs7K9ev8jiK4BAcgQpwwTmog1vQAE2AwSN4Bq/gzXqyXqx362M2WrDynX3wB9bnD8m8lbE=</latexit>

�2 = E[(n� µ)2] =X

n

(n� µ)2p(n)<latexit sha1_base64="5raYFmApuK1cZWd+r9YiVx9Ez7I=">AAACGnicbZDLSgMxFIYz9VbrrerSTbAI00XLzCjoRiiK4LKCvcDMtGTStA1NMkOSEUrpc7jxVdy4UMSduPFtTC8Lbf0h8POdczg5f5QwqrTjfFuZldW19Y3sZm5re2d3L79/UFdxKjGp4ZjFshkhRRgVpKapZqSZSIJ4xEgjGlxP6o0HIhWNxb0eJiTkqCdol2KkDWrn3UDRHkctD17CG98WsAQDnhZbXmhAoFLeFtAWpRmDiS2K7XzBKTtTwWXjzk0BzFVt5z+DToxTToTGDCnlu06iwxGSmmJGxrkgVSRBeIB6xDdWIE5UOJqeNoYnhnRgN5bmCQ2n9PfECHGlhjwynRzpvlqsTeB/NT/V3YtwREWSaiLwbFE3ZVDHcJIT7FBJsGZDYxCW1PwV4j6SCGuTZs6E4C6evGzqXtk9LXt3Z4XK1TyOLDgCx8AGLjgHFXALqqAGMHgEz+AVvFlP1ov1bn3MWjPWfOYQ/JH19QMgg50u</latexit>

µT = E[n1 + n2] =X

n1,n2

(n1 + n2)p(n1,2)

=X

n1

n1p(n1) +X

n2

n2p(n2)

= µ1 + µ2<latexit sha1_base64="g+Lad/RA/N4JeXuDeoza2ldERZE=">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</latexit>

(2.1)

(2.2)

(2.3)

While the variance of the summed measurement is:

where cov(n1,n2) is the called the covariance of n1 and n2.

We can simplify the expression for the covariance:

If the two measurements, n1 and n2 are independent, then the covariance is zero. This is because two variables are said to be independent if their joint expectation is equal to the product of their expectation. In this case;

Following this, we can say, from the definition of the variance above, that if two variables are independent and are added (or subtracted), then their uncertainties add in quadrature

2.4 Combining measurementsIn order to combine the results from various measurements in different ways, we start with the expression for the total differential.

where f is a function of variables x1, x2, … . In what follows we take f to be a combination of neutron measurements, each of which is independent from the other so that we can take the covariance to be zero. Under this assumption (and also, making the assumption that the uncertainties are small - for a first order expansion) we can take the square of Eq. 2.8 to obtain

2.4.1 Addition / Subtraction This is just a check on what we did in the previous section. Two measurements n1 and n2 added together.

2.4.1 Multiplication (or division) by a constantThis happens when we normalise data (against monitor counts or time for example; see §3.1).

�11

�2T = E[(nT � µT )

2] = E[(n1 � µ1 + n2 � µ2)2]

= E[(n1 � µ1)2 + (n2 � µ2)

2 � 2(n1 � µ1)(n2 � µ2)]

= �21 + �2

2 + 2E[(n1 � µ1)(n2 � µ2)]

= �21 + �2

2 + 2 cov(n1, n2)<latexit sha1_base64="1YzwrULV/rsr9HWwy7w7yLQwj3g=">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</latexit>

cov(n1, n2) = E[(n1 � µ1)(n2 � µ2)]

= E[n1n2 � n1µ2 � n2µ1 + µ1µ2]

= E[n1n2]� E[n1]E[n2]� E[n1]E[n2] + E[n1]E[n2]

= E[n1n2]� E[n1]E[n2]<latexit sha1_base64="BYHjiomiqcGFz6xWCISalge7InY=">AAACqXicfVFdS8MwFE3r15xfUx99CQ7FMTfaKuiLMBTBBx8U3ByupaRZpsEkLUkqjLL/5m/wzX9jGvfgpnghueeecw9JbpKMUaU979NxFxaXllcqq9W19Y3Nrdr2Tk+lucSki1OWyn6CFGFUkK6mmpF+JgniCSOPyetVqT++EaloKh70OCMRR8+CjihG2lBx7T3kSL9IXuD0bXIkYv9YxEEDHl7A60FZtkKex37DwMDCoBHBMKxa3ciGhi2TrQRb0NTWAJvQZsvPOyLTaHFU7mU5UzVnxf+91bhW99qeDfgb+FNQB9O4i2sf4TDFOSdCY4aUGvhepqMCSU0xI5NqmCuSIfyKnsnAQIE4UVFhJz2BB4YZwlEqzRIaWvano0BcqTFPTGc5VzWvleRf2iDXo/OooCLLNRH4+6BRzqBOYfltcEglwZqNDUBYUnNXiF+QRFibzy2H4M8/+TfoBW3/pB3cn9Y7l9NxVMAe2AdHwAdnoANuwB3oAuwcOrdO1+m5Tffe7btP362uM/Xsgplw8RdIucP+</latexit>

E[n1n2] = E[n1]E[n2]<latexit sha1_base64="lRlHkEgUX/aQ4IECJv9iQXrVDPM=">AAACBHicbZBNS8MwGMfT+TbrW9XjLsEh7DTaKuhFGIjgcYJ7ga6UNEu3sDQtSSqMsoMXv4oXD4p49UN489uYbj3o5gMJP/7/5yF5/mHKqFS2/W1U1tY3Nreq2+bO7t7+gXV41JVJJjDp4IQloh8iSRjlpKOoYqSfCoLikJFeOLku/N4DEZIm/F5NU+LHaMRpRDFSWgqsmglvPB44PHB9eLVgv7hd3wysut205wVXwSmhDspqB9bXYJjgLCZcYYak9Bw7VX6OhKKYkZk5yCRJEZ6gEfE0chQT6efzJWbwVCtDGCVCH67gXP09kaNYymkc6s4YqbFc9grxP8/LVHTp55SnmSIcLx6KMgZVAotE4JAKghWbakBYUP1XiMdIIKx0bkUIzvLKq9B1m85Z0707r7caZRxVUAMnoAEccAFa4Ba0QQdg8AiewSt4M56MF+Pd+Fi0Voxy5hj8KePzB1eqlUU=</latexit>

�2T = �2

1 + �22

<latexit sha1_base64="Hwh/pGVTwF+2IExfMKV11rrNRY4=">AAACD3icbVDLSsNAFJ3UV62vqEs3g0UpCCWJgm6EghuXFfqCNg2T6aQdOpOEmYlQQv/Ajb/ixoUibt2682+ctBG09cLAedzLnXv8mFGpLOvLKKysrq1vFDdLW9s7u3vm/kFLRonApIkjFomOjyRhNCRNRRUjnVgQxH1G2v74JvPb90RIGoUNNYmJy9EwpAHFSGnJM09LPUmHHHmNvgOvYU5sTc5+iNN3Sp5ZtqrWrOAysHNQBnnVPfOzN4hwwkmoMENSdm0rVm6KhKKYkWmpl0gSIzxGQ9LVMEScSDed3TOFJ1oZwCAS+oUKztTfEyniUk64rzs5UiO56GXif143UcGVm9IwThQJ8XxRkDCoIpiFAwdUEKzYRAOEBdV/hXiEBMJKR5iFYC+evAxaTtU+rzp3F+VaJY+jCI7AMagAG1yCGrgFddAEGDyAJ/ACXo1H49l4M97nrQUjnzkEf8r4+Aa6I5nY</latexit>

df =X

i

@f

@xidxi

<latexit sha1_base64="e+up3RSzocHLiTD+vAWWZAj0lRk=">AAACGnicbVBNS8MwGE7n16xfVY9egkPYabRT0Isw8OJxgvuAtZQ0TbewNC1JKo6y3+HFv+LFgyLexIv/xnQrqJsPJDw8z/u+yfsEKaNS2faXUVlZXVvfqG6aW9s7u3vW/kFXJpnApIMTloh+gCRhlJOOooqRfioIigNGesH4qvB7d0RImvBbNUmJF6MhpxHFSGnJtxwzjOAldGUW+xS6kUA4d1MkFEUMRtMffu/Taagv07dqdsOeAS4TpyQ1UKLtWx9umOAsJlxhhqQcOHaqvLyYixmZmm4mSYrwGA3JQFOOYiK9fLbaFJ5oJYRRIvThCs7U3x05iqWcxIGujJEayUWvEP/zBpmKLryc8jRThOP5Q1HGoEpgkRMMqSBYsYkmCAuq/wrxCOl4lE6zCMFZXHmZdJsN57TRvDmrteplHFVwBI5BHTjgHLTANWiDDsDgATyBF/BqPBrPxpvxPi+tGGXPIfgD4/Mbdaqgdw==</latexit>

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)(df)2 = �2f =

X

i

✓@f

@xi

◆2

�2xi

<latexit sha1_base64="YX+dEHC9TQrrCXwvj4OzWUrLrzM=">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</latexit>

(2.11)

(2.10)

f = n1 ± n2

@f/dn1 = 1

@f/dn2 = 1

) �2f = �2

n1+ �2

n1<latexit sha1_base64="UjtEWzRsY3D0qnGfqUVmtvoj8cw=">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</latexit>

f = k ⇤ n1

df/dn1 = k

) �f = k ⇤ �n1<latexit sha1_base64="K36qb/sYntHAW2EilMn1MKm4ZlM=">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</latexit>

2.4.2 Multiplication or division of measurementsMultiplication of neutron measurements is not a common operation… but division is very common. Neutron data is very commonly divided by a vanadium measurement. It is used for correcting the data for detector efficiency, solid angle and flux profile (see §3.2)

Let’s deal with multiplication first - for this we need to remember the product rule for differentiation (https://www.mathsisfun.com/calculus/derivatives-rules.html)

For two measurements multiplied together;

We can divide both sides by f2 to obtain

So, in the case of multiplied measurements the combined fractional uncertainty is the addition of the individual fractional uncertainties in quadrature.

For the case of division of data sets we might want to use the quotient rule (see above link). However, I can never remember that one - so I just keep using the product rule…

This looks a lot more complicated, until we again divide both sides by f2 to give

which is Eq. 2.14 again. So in the case of both multiplication and division of measurements, the fractional uncertainties are added in quadrature.

2.5 The Poisson distribution and error estimationIf we repeat a neutron measurement a few times how do we calculate the mean and the variance. Both of these depend on the shape of the probability distribution p(n) which (remember) is the probability of counting n neutrons as a function of n.

�12

d(u.v)

dx= u.

dv

dx+ v.

du

dx<latexit sha1_base64="SMMFoFtw7mnp4RhK7nEJyvmKSms=">AAACIXicbZDLSsNAFIYn9VbrLerSzWARKkJIqmA3QsGNywr2Am0pk8mkHTq5MJdiCX0VN76KGxeKdCe+jJMmgrb+MPDznXM4c343ZlRI2/40CmvrG5tbxe3Szu7e/oF5eNQSkeKYNHHEIt5xkSCMhqQpqWSkE3OCApeRtju+TevtCeGCRuGDnMakH6BhSH2KkdRoYNZ6Pkc48WBFWZPzWeI9zuANVFaOJxm5gJMfojJSGphl27IXgqvGyU0Z5GoMzHnPi7AKSCgxQ0J0HTuW/QRxSTEjs1JPCRIjPEZD0tU2RAER/WRx4QyeaeJBP+L6hRIu6O+JBAVCTANXdwZIjsRyLYX/1bpK+rV+QsNYSRLibJGvGJQRTOOCHuUESzbVBmFO9V8hHiGdhNShpiE4yyevmlbVci6t6v1VuV7J4yiCE3AKKsAB16AO7kADNAEGT+AFvIF349l4NT6MedZaMPKZY/BHxtc3hkijCw==</latexit>

(2.12)

f = n1.n2

@f/dn1 = n2

@f/dn2 = n1

) �2f = n2

2�2n1

+ n21�

2n2

<latexit sha1_base64="YDRFihccHQ+v3F4kX2qJ7NJB+kg=">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</latexit>

(2.13)

✓�f

f

◆2

=

✓�n1

n1

◆2

+

✓�n2

n2

◆2

<latexit sha1_base64="3nljiJpF5XSBBAQ+kkqcYfTvxlU=">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</latexit>

(2.14)

f =n1

n2

@f/dn1 = 1/n2

@f/dn2 = �n1/n22

) �2f =

�2n1

n22

+n21�

2n2

n42

<latexit sha1_base64="QblgqlfeCk82Vv3+xrRM7NXAp0U=">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</latexit>

(2.15)

✓�f

f

◆2

=

✓�n1

n1

◆2

+

✓�n2

n2

◆2

<latexit sha1_base64="3nljiJpF5XSBBAQ+kkqcYfTvxlU=">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</latexit>

pP (n) =µn exp(�µ)

n!<latexit sha1_base64="w6tSGHprPRSYUsXbNE18bV+KEC8=">AAACDnicbZDLSsNAFIYnXmu9RV26GS2FdGFJqqAboeDGZQV7gSaGyXTSDp1MwsxELCFP4MZXceNCEbeu3fk2Ti8Lbf1h4OM/53Dm/EHCqFS2/W0sLa+srq0XNoqbW9s7u+befkvGqcCkiWMWi06AJGGUk6aiipFOIgiKAkbawfBqXG/fEyFpzG/VKCFehPqchhQjpS3fLCd+w+IVeAndUCCcuVF6x6FLHhLrRHMlz/hRDn2zZFftieAiODMogZkavvnl9mKcRoQrzJCUXcdOlJchoShmJC+6qSQJwkPUJ12NHEVEetnknByWtdODYSz04wpO3N8TGYqkHEWB7oyQGsj52tj8r9ZNVXjhZZQnqSIcTxeFKYMqhuNsYI8KghUbaUBYUP1XiAdIx6J0gkUdgjN/8iK0alXntFq7OSvVrVkcBXAIjoEFHHAO6uAaNEATYPAInsEreDOejBfj3fiYti4Zs5kD8EfG5w+q6ZqI</latexit>

(2.16)

Whenever the measurement involves counting objects (like the number of flowers in a field per unit area, or the number of neutrons in a detector per unit time), then p(n) takes the form of the Poisson distribution. Then integrating this according to Eqs. 2.1 and 2.2 we find the mean and variance,

This means that the standard deviation estimate of the uncertainty in neutron counts is just given by the square-root of the number of counts.

Below is a plot of the Poisson distribution giving the probability that exactly n neutrons are counted in a given time interval, assuming that the average count-rate is 10 neutrons per time interval.

In this case the estimated standard error would be ±√10 = 3.16 neutrons per time interval.

Obviously the longer we count (in other words, adding up consecutive time intervals) the larger the average number neutrons and uncertainty. However the fractional uncertainty, which for the Poisson distribution is given by

where N is the total (integrated) number of neutrons counted (i.e. the un-normalised mean over many small time intervals), decreases (slowly) as N increases.

Example:Some useful numbers to remember about fractional uncertainty in neutron scattering….If you want to count until the fractional uncertainty is 3% (0.03) then the number of counts required is

Say now that we have decided we actually want to achieve a smaller uncertainty by a factor of 2, i.e. 1.5% (0.015). This means we need

counts. So we have to count 4 times longer to halve the experimental uncertainty.

�13

µP = µ

�2P = µ

<latexit sha1_base64="u05jQ89alMcBqOuM9JMjivLlCos=">AAACCXicbVDLSgMxFM3UV62vUZdugkXoqsxUQTdCwY3LCvYBnXHIpJk2NMkMSUYoQ7du/BU3LhRx6x+482/MtCNo64HLPZxzL8k9YcKo0o7zZZVWVtfWN8qbla3tnd09e/+go+JUYtLGMYtlL0SKMCpIW1PNSC+RBPGQkW44vsr97j2RisbiVk8S4nM0FDSiGGkjBTb0eBq04GXeoedVPEWHHN01frRKYFedujMDXCZuQaqgQCuwP71BjFNOhMYMKdV3nUT7GZKaYkamFS9VJEF4jIakb6hAnCg/m10yhSdGGcAolqaEhjP190aGuFITHppJjvRILXq5+J/XT3V04WdUJKkmAs8filIGdQzzWOCASoI1mxiCsKTmrxCPkERYm/DyENzFk5dJp1F3T+uNm7Nqs1bEUQZH4BjUgAvOQRNcgxZoAwwewBN4Aa/Wo/VsvVnv89GSVewcgj+wPr4B4HuX0A==</latexit>

(2.17)(by definition)

pN

N=

1pN

<latexit sha1_base64="fQhcJd12+7S4A5Md0ta9bR1a0Wc=">AAACEHicbZDLSsNAFIYn9VbrLerSzWARuypJFXQjFNy4kgr2Ak0ok+mkHTq5OHMilJBHcOOruHGhiFuX7nwbp21Abf3hwM93zmHm/F4suALL+jIKS8srq2vF9dLG5tb2jrm711JRIilr0khEsuMRxQQPWRM4CNaJJSOBJ1jbG11O+u17JhWPwlsYx8wNyCDkPqcENOqZx44vCU0ddSchvc4yXfgCz6Cd/fCeWbaq1lR40di5KaNcjZ756fQjmgQsBCqIUl3bisFNiQROBctKTqJYTOiIDFhX25AETLnp9KAMH2nSx34kdYWAp/T3RkoCpcaBpycDAkM135vA/3rdBPxzN+VhnAAL6ewhPxEYIjxJB/e5ZBTEWBtCJdd/xXRIdBigMyzpEOz5kxdNq1a1T6q1m9NyvZLHUUQH6BBVkI3OUB1doQZqIooe0BN6Qa/Go/FsvBnvs9GCke/soz8yPr4BAQ2dvw==</latexit>

1pN

= 0.03

) N ' 1100<latexit sha1_base64="MEebx5/ZEJxbg0WEHTyC3+tOz6c=">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</latexit>

1pN

= 0.015

) N ' 4400<latexit sha1_base64="SuFyYrb6SX3YILI9JG5fOUy2nmY=">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</latexit>

(2.18)

3. After the experiment: Data Correction The correction of neutron data is primarily intended to deal with sources of systematic error. The aim is to turn the measured neutron intensities into measured neutron cross-sections - sometimes on an absolute scale. Do do this we need to know things like -• how long did we count for? • How big are our detectors• what is our sample mass / thickness / volume?• are there any sources of background counts?• what is the sample transmission?The expression for the total number of counts is,

N [neutrons] φ incident neutrons [neutrons] E(λ) detector efficiency [dimensionless] ΔΩ detector solid angle [dimensionless] ns number density [e.g. (barn.cm)-1] t sample thickness [cm] dσ/dΩcross-section [barns] T transmission [dimensionless]

In this part of the lecture we will go through the very basics of correcting neutron data - both at pulsed and continuous sources. Much data correction is done using pre-made scripts in whatever analysis package is available (e.g. Mantid, LAMP, DAVE…).

3.1 Normalisation to monitor / time / currentThe total number of neutrons counted in a measurement will depend on uninteresting factors. The least interesting of these is φ, the number of incident neutrons. Obviously the longer we count for, or the higher the incident flux, the more neutrons we will count. So we need to divide the measured neutron counts by a number, which indicates how many neutrons were available to be scattered.

Option 1: Counting Time If the flux of the neutrons can be taken to be constant in time over the period of the measurement, then it is acceptable to divide the neutron counts by the counting time. However this is only really an option on continuous sources such as the ILL. On spallation sources, the neutron beam fluxes can be quite variable over the experiment.

Option 2: Integrated proton beam current At a spallation source such as ISIS, a common method to normalise data is to use the integrated proton beam current to normalise the data. The beam current is generally recorded in units of μΑ or mΑ, and the time integral of this (which is the total proton charge) is recorded (units of μΑ.Hrs). Assuming that the proton monitors accurately reflect the current and integrated charge, then this can be used to normalise the data. This method is very commonly used at ISIS.

Option 3: Neutron Incident Beam Monitor In most neutron spectrometers, there is a low efficiency detector placed upstream of everything else. This is called the incident beam monitor and it typically has a counting efficiency of around 1 neutron in every 10 000. On continuous sources this monitor will accurately reflect the integrated number of neutrons incident on the sample during the measurement. On pulsed sources, this monitor will - in addition - give the time-of-flight (wavelength) profile of the incident beam. Normalisation to monitor is arguably the most reliable method of the 3 choices, depending on the stability in time of the monitor.

�14

N = �E(�)�⌦nst

✓d�

d⌦

◆T

<latexit sha1_base64="FnFBCg1Zo+9uCfS1yLrBqQxnQK0=">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</latexit>

(3.1)

3.2 Normalisation of the detectorThe detector solid angle, ΔΩ, the wavelength dependent efficiency, E(λ), and the wavelength dependent (and now normalised) intensity, φ0(λ), can be removed from Eq. 3.1 by dividing by a flat scatterer; that is, a sample whose angle (and wavelength) dependent differential cross-section, dσ/dΩ, is a constant.

For most applications, the flat scatterer is chosen to be vanadium. As shown in the table below, the scattering from vanadium is dominated by incoherent scattering (due to the nuclear spin), and is therefore flat.

Unfortunately, vanadium cannot be used to normalise SANS or reflectivity data since there is a lot of small-angle scattering from grains and oxide layers in metallic vanadium. For SANS instruments the scattering from water is sometimes used (flat at low Q), while for reflectivity, glassy carbon can be used.

Examples:Vanadium normalisation on D7 at the ILL. Amorphous silica before and after division by a vanadium measurement. D7 is a monochromatic instrument, so the correction is mostly to do with differences in solid angle. Each individual detector has a different effective solid angle and efficiency - so the data before correction looks pretty ugly.

Vanadium normalisation on POLARIS at ISIS. Below we see the sample (silicon powder) and vanadium spectra as a function of time-of-flight. We are seeing only one detector spectrum (grouped). So this time the correction is mainly to do with energy efficiency and wavelength dependent neutron intensity. The corrected spectrum has been converted into d-spacing.

�15

(3.2)

abund. % σcoh σinc σabs

50V 0.25 7.3 0.5 6051V 99.75 0.0203 5.07 4.9

V - 0.038 5.06 5.04

Nnorm =�0(�)E(�)�⌦nst

�d�d⌦

�T

�0(�)E(�)�⌦ ⇤ constant

=nst

�d�d⌦

�T

constant<latexit sha1_base64="Jw48Phh8HNTBQSIJvfUlnpSpJCo=">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</latexit>

3.3 Absolute normalisationThe cross-section and number density of vanadium is well known,

(dσ/dΩ)V = 5.06 / 4π (barns st-1 f.u-1) ns = 0.0685 (barn cm)-1

therefore we should be able to work out the constant term in Eq. 3.1 provided we know the vanadium thickness (and therefore the transmission).

where TV is the vanadium transmission and tV is the vanadium thickness. Re-arranging we have,

The ratio of sample and vanadium thicknesses and number-densities can be replaced by masses and formula weights if required.

3.4 Background SubtractionThere are many possible sources of background scattering in a typical neutron instrument. These can include; cosmic ray background scattering; neutrons escaping from the source; vacuum windows (often made of aluminium) in the beam path; sample environment such as cryostats, magnets, pressure-cells; sample holders; air in the beam path; …

These are depicted in the diagram below of a typical continuous source powder diffractometer.

�16

40

Time of flight / micro-seconds0 5000 10000 15000 20000

0

5000

10000

15000

20000

25000

a) VanadiumLAD 2T=60°

Time of flight / micro-seconds0 5000 10000 15000 20000

Countspermicro-second

0

2000

4000

6000

8000

10000

12000

14000

16000

b) SiliconLAD 2T=60°

Time of flight / micro-seconds0 5000 10000 15000 20000

0

10000

20000

30000

40000

c) Vitreous GermaniaLAD 2T=60°

d-spacing / Å0 1 2 3 4D

ifferentialcross-section/arb.units

0.00

0.02

0.04

0.06

0.08

0.10

0.12 d) SiliconLAD 2T=60°

Q / Å-10 5 10 15 20 25 30

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

e) Vitreous GermaniaLAD 2T=60°

Time-of-flight Spectra

epithermal

thermal

Countspermicro-second

Countspermicro-second

Differentialcross-section/arb.units

¾Fig. 2.27 Time-of-flight spectra

for a) vanadium,

b) polycrystalline silicon and

c) vitreous germania. Also shown

are the normalised spectra for

d) polycrystalline silicon and

e) vitreous germania.

the data by some kind of smoothing process).

In practice the differential cross-section for

the sample is determined from performing

the following operation with the measured

time-of-flight spectra;

¸̧¹

·¨̈©

§��

backgroundvanadiumcontainersampleresultcorrected

(2.33)

(A fully corrected dataset will also have been

corrected for such effects as attenuation,

multiple scattering and detector dead-time.

A discussion of these corrections is beyond

the scope of this article.) Figure 2.27

illustrates the effect of flux normalisation for

pulsed diffraction data. The vanadium time-

of-flight spectrum is closely related to the

flux distribution � �) O arising from the

moderator. The upturn in � �) O at low

times is due to high energy epithermal

neutrons whilst the broad peak at

intermediate times is the peak of a

Maxwellian distribution whose position

depends on the moderator temperature.

The spectra measured on a reactor

diffractometer do not need to be normalised

to a flux distribution, but a vanadium

standard may still be used in an experiment

in order to achieve an absolute normalisation

of the differential cross-section.

One of the most important concerns for a

diffraction experiment on a disordered

material is the Q-range. Theoretically the

Fourier transformation of Equation 2.28

extends from Q = 0 to infinity, but this is not

possible in reality. By the use of low scattering

angles, 2θ, the data should be measured

down to as low a Q-value as possible, so that

they may be extrapolated reasonably down to

40

Time of flight / micro-seconds0 5000 10000 15000 20000

0

5000

10000

15000

20000

25000

a) VanadiumLAD 2T=60°

Time of flight / micro-seconds0 5000 10000 15000 20000

Countspermicro-second

0

2000

4000

6000

8000

10000

12000

14000

16000

b) SiliconLAD 2T=60°

Time of flight / micro-seconds0 5000 10000 15000 20000

0

10000

20000

30000

40000

c) Vitreous GermaniaLAD 2T=60°

d-spacing / Å0 1 2 3 4D

ifferentialcross-section/arb.units

0.00

0.02

0.04

0.06

0.08

0.10

0.12 d) SiliconLAD 2T=60°

Q / Å-10 5 10 15 20 25 30

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

e) Vitreous GermaniaLAD 2T=60°

Time-of-flight Spectra

epithermal

thermal

Countspermicro-second

Countspermicro-second

Differentialcross-section/arb.units

¾Fig. 2.27 Time-of-flight spectra

for a) vanadium,

b) polycrystalline silicon and

c) vitreous germania. Also shown

are the normalised spectra for

d) polycrystalline silicon and

e) vitreous germania.

the data by some kind of smoothing process).

In practice the differential cross-section for

the sample is determined from performing

the following operation with the measured

time-of-flight spectra;

¸̧¹

·¨̈©

§��

backgroundvanadiumcontainersampleresultcorrected

(2.33)

(A fully corrected dataset will also have been

corrected for such effects as attenuation,

multiple scattering and detector dead-time.

A discussion of these corrections is beyond

the scope of this article.) Figure 2.27

illustrates the effect of flux normalisation for

pulsed diffraction data. The vanadium time-

of-flight spectrum is closely related to the

flux distribution � �) O arising from the

moderator. The upturn in � �) O at low

times is due to high energy epithermal

neutrons whilst the broad peak at

intermediate times is the peak of a

Maxwellian distribution whose position

depends on the moderator temperature.

The spectra measured on a reactor

diffractometer do not need to be normalised

to a flux distribution, but a vanadium

standard may still be used in an experiment

in order to achieve an absolute normalisation

of the differential cross-section.

One of the most important concerns for a

diffraction experiment on a disordered

material is the Q-range. Theoretically the

Fourier transformation of Equation 2.28

extends from Q = 0 to infinity, but this is not

possible in reality. By the use of low scattering

angles, 2θ, the data should be measured

down to as low a Q-value as possible, so that

they may be extrapolated reasonably down to

÷ =

40

Time of flight / micro-seconds0 5000 10000 15000 20000

0

5000

10000

15000

20000

25000

a) VanadiumLAD 2T=60°

Time of flight / micro-seconds0 5000 10000 15000 20000

Countspermicro-second

0

2000

4000

6000

8000

10000

12000

14000

16000

b) SiliconLAD 2T=60°

Time of flight / micro-seconds0 5000 10000 15000 20000

0

10000

20000

30000

40000

c) Vitreous GermaniaLAD 2T=60°

d-spacing / Å0 1 2 3 4D

ifferentialcross-section/arb.units

0.00

0.02

0.04

0.06

0.08

0.10

0.12 d) SiliconLAD 2T=60°

Q / Å-10 5 10 15 20 25 30

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

e) Vitreous GermaniaLAD 2T=60°

Time-of-flight Spectra

epithermal

thermal

Countspermicro-second

Countspermicro-second

Differentialcross-section/arb.units

¾Fig. 2.27 Time-of-flight spectra

for a) vanadium,

b) polycrystalline silicon and

c) vitreous germania. Also shown

are the normalised spectra for

d) polycrystalline silicon and

e) vitreous germania.

the data by some kind of smoothing process).

In practice the differential cross-section for

the sample is determined from performing

the following operation with the measured

time-of-flight spectra;

¸̧¹

·¨̈©

§��

backgroundvanadiumcontainersampleresultcorrected

(2.33)

(A fully corrected dataset will also have been

corrected for such effects as attenuation,

multiple scattering and detector dead-time.

A discussion of these corrections is beyond

the scope of this article.) Figure 2.27

illustrates the effect of flux normalisation for

pulsed diffraction data. The vanadium time-

of-flight spectrum is closely related to the

flux distribution � �) O arising from the

moderator. The upturn in � �) O at low

times is due to high energy epithermal

neutrons whilst the broad peak at

intermediate times is the peak of a

Maxwellian distribution whose position

depends on the moderator temperature.

The spectra measured on a reactor

diffractometer do not need to be normalised

to a flux distribution, but a vanadium

standard may still be used in an experiment

in order to achieve an absolute normalisation

of the differential cross-section.

One of the most important concerns for a

diffraction experiment on a disordered

material is the Q-range. Theoretically the

Fourier transformation of Equation 2.28

extends from Q = 0 to infinity, but this is not

possible in reality. By the use of low scattering

angles, 2θ, the data should be measured

down to as low a Q-value as possible, so that

they may be extrapolated reasonably down to

(3.2)Nsample

Nvanadium=

nst�d�d⌦

�T

0.0685 tV5.064⇡ TV

<latexit sha1_base64="SGwu0nm3wxxiWtxBH5OFf7zPbxY=">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</latexit>

d�

d⌦=

Nsample

Nvanadium

0.0685

ns

TV

T

tVt

5.06

4⇡<latexit sha1_base64="ZZnn59MXfIzY2KNROoug6Ne5e/8=">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</latexit>

(3.3)

One might expect that in order to subtract the background scattering, all you need to do is measure an empty sample holder in exactly the same conditions as the sample measurement (i.e. filled sample holder). This is in fact true for samples which have a very good neutron transmission. In general though the measured neutron counts with a sample in the beam is a sum of three terms,

where, Nm is the measured count, Ns is the count due to sample scattering, NbgA is the count due to background sources which have passed through the sample and is hence attenuated, and NbgB is the count due to background sources which bypass the sample (unattenuated). Since Ns and NbgA are attenuated by the sample they have to be multiplied by the sample transmission Ts. This is depicted in Fig. (a) below.

Fig. (b) depicts the situation for an empty sample holder measurement. In this case we are measuring both the NbgA and NbgB contributions - but the NbgA term is no longer attenuated by the sample. Therefore the empty sample holder counts, NE are;

Fig. (c) shows the situation when we measure a totally absorbing sample (such as Cd) in the sample holder. In this case, the NbgA background term is blocked by the sample, giving;

�17

(a) (b) (c)

Nm = TsNs + TsNbgA +NbgB<latexit sha1_base64="02H1GbYhE9rWCxInGc2OgonwCOc=">AAACDnicbZBNS8MwGMdTX+d8q3r0EhyDgTDaKehFmHrxNCbsDbZS0izdwpK2JKkwSj+BF7+KFw+KePXszW9jtvagm38I/PJ/nofk+XsRo1JZ1rexsrq2vrFZ2Cpu7+zu7ZsHhx0ZxgKTNg5ZKHoekoTRgLQVVYz0IkEQ9xjpepPbWb37QISkYdBS04g4HI0C6lOMlLZcs9xwObyCLVc2XAlPM0i8kXud6luGN6lrlqyqNRdcBjuHEsjVdM2vwTDEMSeBwgxJ2betSDkJEopiRtLiIJYkQniCRqSvMUCcSCeZr5PCsnaG0A+FPoGCc/f3RIK4lFPu6U6O1Fgu1mbmf7V+rPxLJ6FBFCsS4OwhP2ZQhXCWDRxSQbBiUw0IC6r/CvEYCYSVTrCoQ7AXV16GTq1qn1Vr9+eleiWPowCOwQmoABtcgDq4A03QBhg8gmfwCt6MJ+PFeDc+stYVI585An9kfP4AypuZ/Q==</latexit>

NE = NbgA +NbgB<latexit sha1_base64="0fTLVFQuqI+wuLH57u+bdWyes8o=">AAACAnicbZDLSsNAFIZPvNZ6i7oSN4NFKAglqYJuhKoIrkoFe4E2hMl02g6dXJiZCCUEN76KGxeKuPUp3Pk2TtsstPWHgY//nMOZ83sRZ1JZ1rexsLi0vLKaW8uvb2xubZs7uw0ZxoLQOgl5KFoelpSzgNYVU5y2IkGx73Ha9IbX43rzgQrJwuBejSLq+LgfsB4jWGnLNfer7g26QFU38fruZYqOM7xKXbNglayJ0DzYGRQgU801vzrdkMQ+DRThWMq2bUXKSbBQjHCa5juxpBEmQ9ynbY0B9ql0kskJKTrSThf1QqFfoNDE/T2RYF/Kke/pTh+rgZytjc3/au1Y9c6dhAVRrGhApot6MUcqROM8UJcJShQfacBEMP1XRAZYYKJ0ankdgj178jw0yiX7pFS+Oy1UilkcOTiAQyiCDWdQgVuoQR0IPMIzvMKb8WS8GO/Gx7R1wchm9uCPjM8fh3uVhg==</latexit>

(3.4)

(3.5)

NCd = NbgB<latexit sha1_base64="WIaFCnIDZXjngn5lBiaCPvbcT8A=">AAAB+3icbZDLSsNAFIZP6q3WW6xLN4NF6KokVdCNUOzGlVSwtdCGMJlM2qGTCzMTsYS8ihsXirj1Rdz5Nk7bLLT1h4GP/5zDOfN7CWdSWda3UVpb39jcKm9Xdnb39g/Mw2pPxqkgtEtiHou+hyXlLKJdxRSn/URQHHqcPniT9qz+8EiFZHF0r6YJdUI8iljACFbacs3qrZu1/RxdIaTJG7nXuWvWrIY1F1oFu4AaFOq45tfQj0ka0kgRjqUc2FainAwLxQineWWYSppgMsEjOtAY4ZBKJ5vfnqNT7fgoiIV+kUJz9/dEhkMpp6GnO0OsxnK5NjP/qw1SFVw6GYuSVNGILBYFKUcqRrMgkM8EJYpPNWAimL4VkTEWmCgdV0WHYC9/eRV6zYZ91mjendda9SKOMhzDCdTBhgtowQ10oAsEnuAZXuHNyI0X4934WLSWjGLmCP7I+PwBU/6TQQ==</latexit>

(3.6)

Depiction of a typical continuous source powder diffractometer

Combining Eqs. 3.5 and 3.6 we can write,

And therefore re-arranging 3.4 and substituting 3.7 and 3.6 we get,

Note that when the sample transmission is close to 1, then the NCd terms subtract, and all that is required is the empty sample holder measurement. Sometimes, NCd is quite close to zero (e.g. in SANS experiments) but it is still checked. In wide angle diffraction experiments NCd can be very large (almost as large as NE).

�18

NbgA = NE �NCd<latexit sha1_base64="GBfyfA4yc6+7HIVzfEpV7Mh0Lrw=">AAACAnicbZBNS8MwGMfT+TbnW9WTeAkOYRdHOwW9CJMheJIJ7gW2UtI028LStCSpMErx4lfx4kERr34Kb34b060H3Xwg5Mf//zwkz9+LGJXKsr6NwtLyyupacb20sbm1vWPu7rVlGAtMWjhkoeh6SBJGOWkpqhjpRoKgwGOk440bmd95IELSkN+rSUScAA05HVCMlJZc8+DWTbzhVQovIdR4ncKT7G74qWuWrao1LbgIdg5lkFfTNb/6fojjgHCFGZKyZ1uRchIkFMWMpKV+LEmE8BgNSU8jRwGRTjJdIYXHWvHhIBT6cAWn6u+JBAVSTgJPdwZIjeS8l4n/eb1YDS6chPIoVoTj2UODmEEVwiwP6FNBsGITDQgLqv8K8QgJhJVOraRDsOdXXoR2rWqfVmt3Z+V6JY+jCA7BEagAG5yDOrgBTdACGDyCZ/AK3own48V4Nz5mrQUjn9kHf8r4/AGARJV+</latexit>

(3.7)

Nm =1

Ts(Ns �Ncd)� (NE �NCd)

<latexit sha1_base64="2EAUTnuDgj5wOE8Njvr+pRH0i3k=">AAACHnicbZDLSgMxFIYz9VbrbdSlm2AR6qJlpiq6EQpFcCUVeoN2GDJppg3NZIYkI5RhnsSNr+LGhSKCK30b03YW2nogOT/fOYfk/F7EqFSW9W3kVlbX1jfym4Wt7Z3dPXP/oC3DWGDSwiELRddDkjDKSUtRxUg3EgQFHiMdb1yf1jsPREga8qaaRMQJ0JBTn2KkNHLNizs3CVJ4DWHfFwgndpo0XZmWNJZpWd94kJ7CMtTgRicN6hq4ZtGqWLOAy8LORBFk0XDNz/4gxHFAuMIMSdmzrUg5CRKKYkbSQj+WJEJ4jIakpyVHAZFOMlsvhSeaDKAfCn24gjP6eyJBgZSTwNOdAVIjuVibwv9qvVj5V05CeRQrwvH8IT9mUIVw6hUcUEGwYhMtEBZU/xXiEdI2Ke1oQZtgL668LNrVin1Wqd6fF2ulzI48OALHoARscAlq4BY0QAtg8AiewSt4M56MF+Pd+Ji35oxs5hD8CePrB1MKoAA=</latexit>

(3.8)