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Luminosity class of neutron reflectometers
Nuclear Instruments and Methods in Physics Research A 834 (2016) 197–204
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Luminosity class of neutron reflectometers N.K. Pleshanov Neutron Research Department, Petersburg Nuclear Physics Institute, NRC “Kurchatov Institute”, Gatchina, 188300 St. Petersburg, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 22 April 2016 Received in revised form 6 August 2016 Accepted 8 August 2016 Available online 9 August 2016
The formulas that relate neutron fluxes at reflectometers with differing q-resolutions are derived. The reference luminosity is defined as a maximum flux for measurements with a standard resolution. The methods of assessing the reference luminosity of neutron reflectometers are presented for monochromatic and white beams, which are collimated with either double diaphragm or small angle Soller systems. The values of the reference luminosity for unified parameters define luminosity class of reflectometers. The luminosity class characterizes (each operation mode of) the instrument by one number and can be used to classify operating reflectometers and optimize designed reflectometers. As an example the luminosity class of the neutron reflectometer NR-4M (reactor WWR-M, Gatchina) is found for four operation modes: 2.1 (monochromatic non-polarized beam), 1.9 (monochromatic polarized beam), 1.5 (white non-polarized beam), 1.1 (white polarized beam); it is shown that optimization of measurements may increase the flux at the sample up to two orders of magnitude with monochromatic beams and up to one order of magnitude with white beams. A fan beam reflectometry scheme with monochromatic neutrons is suggested, and the expected increase in luminosity is evaluated. A tuned-phase chopper with a variable TOF resolution is recommended for reflectometry with white beams. & 2016 Elsevier B.V. All rights reserved.
Keywords: Neutron reflectometers Luminosity and resolution Reference luminosity Luminosity class Fan beam reflectometry Tuned-phase chopper
1. Introduction Neutron reflectometry is used for detailed studies of physical and chemical processes at the interfaces, bio-processes with interaction and transfer of substances through interfaces, membranes, etc. New data unavailable with other research techniques have been obtained. Comparatively low luminance of neutron sources implies an efficient transport of neutrons from the source to the sample and the use of efficient position sensitive detectors (PSD). The relationship between resolution and luminosity for specular reflection measurements is shortly discussed in papers [1,2]. In papers [3,4] the luminosity of the reflectometer was coupled to the momentum transfer resolution Δq. It revealed the possibilities to increase luminosity without loss in resolution and substantiated new optimization criteria for neutron reflectometers. However, the luminosity defined in this manner depends not only on the resolution Δq, but also on the momentum transfer
q = (4π sin θ )/λ ≅ 4πθ /λ ,
(1)
where λ is the neutron wavelength, θ is the glancing angle assumed (usually, it does not exceed several degrees). When angular and wavelengths distributions of neutrons in the incident beam E-mail address: [email protected] http://dx.doi.org/10.1016/j.nima.2016.08.022 0168-9002/& 2016 Elsevier B.V. All rights reserved.
with divergence Δθ and monochromaticity, or time-of-flight resolution, Δλ are Gaussian, one obtains
Δq/q =
(Δθ /θ )2 + (Δλ /λ )2
(2)
(here and further the formulas are written in terms of the standard deviations, which are less than full widths at half maximum in
2 2 ln 2 ≅ 2.355 times). Without loss of generality, we assume that the scattering plane is horizontal, so the slits and the reflecting surface are vertical, and the resolution is defined by the horizontal divergence. The present paper is an elaboration on the papers [3,4]. In Sections 2 and 3 the formulas that relate neutron fluxes at the sample for measurements with differing resolutions are derived, accordingly, for double diaphragm and Soller beam collimation. These formulas are used in Section 4 for introduction of ‘reference luminosity’, independent of q and Δ q, and the methods of its assessment from the intensities measured at the reflectometer. In Section 5 the luminosity class is introduced to compare different reflectometers; the luminosity class of the reflectometer NR-4M (Gatchina) is assessed and the possibility to optimize measurements at this reflectometer is considered. The results obtained are summarized in Conclusion.
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and the resolution
2. Measurements with double diaphragm collimators
Δq can be found from (7) with λ ¼ λe:
2
In measurements with two diaphragms collimation the neutron flux at the sample is [4]:
Φ(λ ) = hs h2
d d L A(λ ) 1 2 Δλ , Ls + L L L
(3)
where hs and h2 are the heights of the source and the diaphragm slit before the sample, d1 and d2 are the widths of the slits of the first and second collimation diaphragms, Ls and L are the distances from the source to the first diaphragm and between diaphragms, respectively. The spectral density A(λ) of the flux incident onto the sample depends on the luminance and the spectral density of the source emission, λ-dependent transmission of neutron guides and focusing optics, width of the monochromator peak, transmission of the collimation system and, generally speaking, on the sample size. It is evident that the flux at the sample for measurement conditions mentioned above may be assessed from the flux experimentally measured with a beam formed with diaphragm slits of known width d1,e and d2,e, with known mean wavelength λe and spectral width Δλe:
Φ(λ ) =
A(λ ) d1 d2 Δλ Φe(λ e). A(λ e) d1,e d2,e Δλ e
(4)
Here and throughout the paper the quantities with a subscript ‘e’ refer to an experiment carried out to calibrate the neutron flux at a given reflectometer. The measurement in the calibration experiment is made in the absence of the sample and with a beam with known parameters. It implies that all quantities with the subscript ‘e’ are known from the experiment or can be found directly from the experiment. What is more important, they can be used to assess the flux at the reflectometer for a beam with differing parameters. The divergence of the respective beams (standard Gaussian deviations) is [2] 2 2 Δθ = (2L )−1 (d12 + d22)/3 , Δθe = (2L )−1 (d1,e + d2,e )/3 ,
Φ=
εg Δλ ⎛ Δθ ⎞ ⎜ ⎟ Φe. εg,e Δλ e ⎝ Δθe ⎠
(9)
Measurements at reflectometers with white beams are carried out with using the TOF technique at a fixed glancing angle θ and with the same divergence Δθ for all wavelengths. The role of monochromaticity of neutrons plays the TOF resolution. Further we consider the time-of-flight technique only at a steady flux reactor to avoid analyzing the details of the realization of the technique at pulsed sources, which are not essential for us. By definition, A(λ) is the spectral flux density in the beam incident onto the sample in the absence of the chopper. Then the time-averaged spectral flux density of the beam chopped into pulses by the chopper is
A chop (λ ) = A(λ )τ (λ )/Tp = A(λ )Δλ(λ )/λ max ,
(10)
where account is taken of the relation of the neutron pulse width τ (λ ) = LbhΔλ(λ ) /mn with the TOF resolution Δλ(λ) and of the relation of the pulse period Tp = Lbhλ max /mn with the maximum wavelength λmax of neutrons registered by the detector before the arrival of the next pulse (Lb is the TOF base, h is the Planсk constant, mn is the neutron mass). In the absence of the chopper the integral flux incident onto the sample (cf. with (3)) is (0) Φint = hs h2
J0 =
∫0
∞
L d1 d2 J , Ls + L L L 0
(11)
A(λ )dλ .
(12)
By analogy with (7), one finds the relationship between the integral fluxes in the absence of the chopper for the beams with the same spectral flux density A(λ): (0) Φint =
(5)
2 εg ⎛ Δθ ⎞ (0) ⎜ ⎟ Φint ,e. εg,e ⎝ Δθe ⎠
(13)
The integral flux with the chopper is related to the integral flux of the same beam without the chopper:
hence 1/2 ⎛ d 2 + d 2 ⎞1/2 d1 ⎡ 1 + (d2/d1)2 ⎤ Δθ 2 ⎟ ⎢ ⎥ = = ⎜⎜ 21 = ⎟ 2 Δθe d1,e ⎢⎣ 1 + (d2,e/d1,e)2 ⎥⎦ ⎝ d1,e + d2,e ⎠
Φint =
1/2 d ⎡ 1 + (d2/d1)−2 ⎤ ⎥ . = 2 ⎢ −2 d2,e ⎢⎣ 1 + (d2,e/d1,e) ⎥⎦
(0) Φint
J0
(6)
Substituting d1/d1,e and d2/d2,e expressed in terms of Δθ/Δθe from (6) into (3), one finds the flux in the incident beam with arbitrary values of λ, Δλ and Δθ:
Φint ,e = =
∫0
(0) Φint ,e
J0
∞
A chop (λ )dλ =
∫0
(0) Φint
1
J0 λ max
∫0
∞
A(λ )Δλ(λ )dλ ,
(14)
∞
A chop, e(λ )dλ
(0) Φint ,e
1
J0
λ max ,e
∫0
∞
A(λ e)Δλ e(λ e)dλ e.
(15)
2
A(λ ) εg Δλ ⎛ Δθ ⎞ Φ(λ ) = ⎜ ⎟ Φe(λ e), A(λ e) εg,e Δλ e ⎝ Δθe ⎠ −1
(
(7) −1
)
εg = 2( d1/d2 + d2/d1) , εg,e = 2 d1,e/d2,e + d2,e/d1,e
.
(8)
The quantities εg and εg,e (geometric optimality of measurements with the respective beams) characterize a decrease in the intensity of the beams in comparison with the maximum flux available for a given divergence of the beams, when d1 ¼d2 and d1, e ¼d2,e (εg ¼ εg,e ¼1). As a rule, at a given reflectometer working with a monochromatic beam, all measurements are carried out with the same mean wavelength. The flux of monochromatic neutrons for the conditions of reflection from the sample with the momentum transfer q
The single disk chopper secures a TOF resolution equal at all wavelengths, Δλ(1) ¼ const. The chopper with two synchronously rotating disks [5] secures measurements with (Δλ/λ) ¼ const at all wavelengths (actually, the range is limited by a maximum wavelength that most of the time can be made sufficiently large and may be ignored in our considerations without loss of generality; see Ref. [2] for further details on application of double disk choppers in neutron reflectometry) and, as a consequence, with one and the same relative resolution Δq/q. The value of Δλ/λ can be changed by varying the rotation frequency and the distance between the disks. The technical complexity did not allow to gradually change these parameters, so several disks located at different distances are used [6,7]. The choice of different pairs of disks provides a discrete change of the ratio (Δλ/λ).
N.K. Pleshanov / Nuclear Instruments and Methods in Physics Research A 834 (2016) 197–204
Taking into consideration the dependence Δλ(λ) in (14) and (15) and relation (13), one obtains the relationship between the integral fluxes for measurements at a given reflectometer, i.e. for the beams with the same spectral flux density A(λ) and the same choppers (the number of disks is shown with the upper index): (1) Φint
2 λ max ,e εg Δλ (1) ⎛ Δθ ⎞ (1) = ⎜ ⎟ Φint ,e 1 ( ) λ max εg,e Δλ e ⎝ Δθe ⎠
λ max ,e εg (Δλ /λ ) ⎛ Δθ ⎞ (2) ⎜ ⎟Φ λ max εg,e (Δλ /λ )e ⎝ Δθe ⎠ int ,e
(2) Φint =
Δλ (1) (0) Φ , λ max int
(17)
Φ=
dc Δλ Δθ Φe. dc,e Δλ e Δθe
(19)
∫0
∞
A(λ )λdλ⋅
⎛ ⎜ ⎝
∫0
∞
⎞−1 A(λ )dλ⎟ , ⎠
(20)
characterizes the white beam spectrum. Say, for the beam of neutrons thermalized at a temperature T with the Maxwellian flux density function A(λ), which is maximum at the wavelength
λ peak =
h2/5mnkT , one has
λ m = λ peak 5π /8 ≅ 1.401λ peak .
(21)
Note that formulas (18) and (19) can be used for experimental determination of Δλ(1) and λm from measurements of the integral intensities without chopper and with chopper. Also from (18) and (19) we obtain the following relationship between the integral fluxes for the choppers with one and two disks for the same neutron pulse frequency, the same initial beam and the same collimation geometry: (1) Φint =
Δλ (1)/λ m (2) Φ . (Δλ /λ ) int
(22)
(9′)
In the case of measurements with the white beams the integral flux incident onto the sample without the chopper is (0) Φint = hc dcΔβ⋅Δθ⋅J0 .
(11′)
Its relation to the integral flux with the chopper is given by Eq. (14). As a consequence, the integral fluxes after double disk and single disk choppers are given by the same formulas (18–22). The integral fluxes for different measurement conditions are related as follows: (1) Φint =
λ max ,e dc Δλ (1) Δθ (1) , Φ λ max dc,e Δλ e(1) Δθe int ,e
(16′)
(2) Φint =
λ max ,e dc (Δλ /λ ) Δθ (2) . Φ λ max dc,e (Δλ /λ )e Δθe int ,e
(17′)
where the mean wavelength, introduced as
λm =
(7′)
where the demanded and measured fluxes are for the beams prepared with collimators of width, accordingly, dc and dc,e. For the measurements with the monochromatic beams (λ ¼ λe)
(18)
(Δλ /λ ) (0) Φ , λ max /λ m int
A(λ ) dc Δλ Δθ Φe(λ e), A(λ e) dc,e Δλ e Δθe
(16)
(cf. with (7)). As in the case with monochromatic beams, one can (1) (2) or Φint measured use the experimental integral intensities Φint ,e ,e at a beam with known divergence Δθe and TOF resolution (Δλe or (Δλ/λ)e) and maximum wavelength λmax,e, to extract the integral flux for a beam with arbitrary parameters. From (14) with Δλ ¼ Δλ(1) ¼ const and Δλ ¼(Δλ/λ)λ ¼ const λ (Δλ(1) and Δλ are standard Gaussian deviations) one obtains, respectively, (1) Φint =
systems and focusing devices, the sample height. Further, Δβ and hc are assumed to be constant. Hence we obtain the formula for the flux incident onto the sample:
Φ(λ ) =
2
(2) Φint =
199
4. Reference luminosity The formulas obtained may be used to assess the flux for different reflectometers in certain measurement conditions, equivalent from the point of view of resolution. The fluxes defined in this manner may serve as a measure of luminosity of reflectometers. This quantity can be named as ‘reference luminosity’. To avoid ambiguities, the quantities Δq, Δθ , Δλ will always be regarded as standard Gaussian deviations. For the reflectometers at beams of monochromatic neutrons with a working wavelength λ define the reference luminosity 3 as the maximum flux achievable for measurements with standard (reference) values of the momentum transfer (q0) and the resolution (Δq0). The quantity q0 agrees with a glancing angle θ0 ¼q0λ/(4π). With a double diaphragm collimation, the neutron flux is maximum for εg ¼1 and when the conditions [4]
Δθ /θ =
2 (Δλ /λ ) =
2/3 (Δq/q)
(23)
are fulfilled, i.e. 3. Measurements with small angle Soller collimators A small angle Soller collimator forms a beam with a cross section of width dc and height hc, with a vertical Δβ and horizontal Δθ divergence. The neutron flux incident onto the sample is
Φ(λ ) = hc dcΔβ⋅A(λ )⋅Δλ⋅Δθ
(3′)
(here and further the primed numbers are used for the formulas concerning measurements with Soller collimators; they correspond to the numbers of similar formulas for measurements with two diaphragms). The vertical divergence Δβ depends on the source height, on the parameters of transport neutron guide
Δθ0 =
2/3 θ0(Δq0/q0) =
2/3 λΔq0, 4π
Δλ 0 =
1/3 λ(Δq0/q0),
(24)
we obtain from (9) that
3=
2 Δq03 Φe 1 Δλ 0 ⎛ Δθ0 ⎞ λ3 . ⎜ ⎟ Φe = 2 εg,e Δλ e ⎝ Δθe ⎠ 24π 3 q0 εg,eΔλ eΔθe2
(25)
Now, knowing 3 , one can find the flux for measurements with arbitrary Δλ and Δθ:
Φ = 24π 2 3 εg
ΔλΔθ 2 q0 λ3
Δq03
3. (26)
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N.K. Pleshanov / Nuclear Instruments and Methods in Physics Research A 834 (2016) 197–204
Using (2) and such a quantity as the optimality of measurements with diaphragms [4]
(
κ=
−3/2
)
εd = (3 3 /2)κ 1 + κ 2
,
(27)
Δλ /λ Δλ q = , Δθ /θ Δθ 4π
(28)
the flux can also be written in terms of q and
Δq:
with (23). So the reference luminosity of the reflectometer will be related not with Δq0, but with a standard relative resolution (Δq/ q)0. Define the reference luminosity as the maximum integral flux available for measurements with (Δq/q)0 ¼ Δq0/q0 and with a standard maximum wavelength λmax,0. Then, defining θ0 ¼q0λm/ (4π), one obtains from (23):
Δθ0 =
2/3 λ mΔq0, 4π
(Δλ /λ )0 =
1/3 (Δq/q)0 =
1/3 (Δq0/q0),
(35)
and from (17):
3
q ⎛ Δq ⎞ ⎟⎟ 3. Φ = εgεd 0 ⎜⎜ q ⎝ Δq0 ⎠
(29)
According to (2), the resolution Δq may be achieved with numerous pairs of Δλ and Δθ. The optimal pair can be found from conditions (23). The choice of Δλ and Δθ is often limited by the technical possibilities designed in the reflectometer. According to (5), the beam divergence Δθ may be achieved with numerous pairs of d1 and d2; the factor εg in (29) gives the decrease of the flux below the maximum flux with d1 ¼ d2 (εg ¼1). The above-introduced definition of reference luminosity is not suited for the reflectometers at white beams. First, the values of the momentum transfer differ at different wavelengths, so the choice of the standard momentum transfer q0 makes no sense. Second, as it has been argued earlier [4], TOF measurements with the same Δq at all wavelengths are not possible because of technical problems connected with the requirement to simultaneously fulfill the conditions Δθ ∝ λ and Δλ ∝ λ2/θ . In the case of measurements with a single disk chopper the conditions of statistically optimal measurements may be fulfilled only at one wavelength, so measurements are always carried out in the regime not optimal as regards the relationship between resolution and intensity. To define the reference luminosity in this case, require that the measurements be optimal (εd ¼ 1) at the mean wavelength λm. Then the optimality of measurements at other wavelengths is [4]: 2⎞−3/2
εd(λ ) =
⎛λ ⎞ λm ⎛ 2 ⎜ + 1 ⎜ m ⎟ ⎟⎟ λ ⎜⎝ 3 3⎝ λ ⎠ ⎠
.
(30)
2
λ ¼ λm
(1) λ m3 1 λ max ,e Δλ 0 ⎛ Δθ0 ⎞ (1) = ⎜ ⎟Φ εg,e λ max ,0 Δλ e(1) ⎝ Δθe ⎠ int ,e 24π 2 3 λ max ,0 q0 (1) Φint ,e
εg,eΔλ e(1)Δθe2
(1) Φint
(2) Φint ,e
εg,e(Δλ /λ )eΔθe2
. (36)
Hence (2) Φint = 24π 2 3 εg
λ max ,0 (Δλ /λ )Δθ 2 q0 λ m2
λ max
Δq03
3. (37)
3
(2) Φint = εgεd
λ max ,0 q0 ⎛ Δqm ⎞ ⎟⎟ 3, ⎜⎜ λ max qm ⎝ Δq0 ⎠
(38)
where Δqm = qm (Δθ /θ )2 + (Δλ /λ )2 . Now with Δλ/λ ¼ const, the quantity εd does not depend on the wavelength. Similar considerations can be made for measurements with Soller collimators, and one can derive the respective formulas by employing the results from Section 3 and the conditions of maximum flux in measurements with the standard parameters, now including a standard Soller collimator width dc,0. With Soller collimators the flux of monochromatic neutrons is maximum, when [4]
1 (Δq0/q0). 2
(23′)
Therefore,
3=
dc,0 Δλ 0 Δθ0 dc,e Δλ e Δθe
Φe =
2 λ2 dc,0 Δq0 Φe , 8π dc,e q0 Δλ eΔθe
(25′)
λ max ,e Δq03 Φ = 8π
.
dc ΔλΔθ q0 3, dc,0 λ2 Δq02
(26′)
(31)
Hence the flux for arbitrary measurement conditions from the reference luminosity: (1) Φint = 24π 2 3 εg
2 λ max ,e Δq03 λ m2 1 λ max ,e (Δλ /λ )0 ⎛ Δθ0 ⎞ (2) ⎜ ⎟ Φint ,e = εg,e λ max ,0 (Δλ /λ )e ⎝ Δθe ⎠ 24π 2 3 λ max ,0 q0
Δθ0/θ0 = Δλ 0/λ =
Substituting the parameters obtained from (24) with into (16), we find
3=
3=
λ max ,0 Δλ (1)Δθ 2 q0 λ max
λ m3
3, 3
Δq0
3 λ max ,0 q0 ⎛ Δqm ⎞ ⎟⎟ 3, ⎜⎜ = εgεd,m λ max qm ⎝ Δq0 ⎠
qm = 4πθ /λ m, Δqm = qm (Δθ /θ )2 + (Δλ (1)/λ m)2 ;
(32)
Φ=
2 dc q0 ⎛ Δq ⎞ ⎟⎟ 3, εc ⎜⎜ dc,0 q ⎝ Δq0 ⎠
(29′)
where the optimality of measurements with Soller collimators [4]
εc = 2(κ + κ −1)−1 (33)
is used (the parameter κ is defined above in (30)). For the measurements with a white beam and two types of choppers:
(34)
εd,m is also defined at the wavelength λm by substitution of the above parameters into (12). When Δλ/λ ¼const (double disk chopper), the conditions of statistically optimal measurements may be fulfilled at all wavelengths by the choice of the beam divergence Δθ in accordance
(27′)
3=
dc,0 λ max ,e Δλ 0(1) Δθ0 dc,e λ max ,0 Δλ e(1)
(1) Φint = 8π
Δθe
(1) Φint ,e =
(1) 2 λ m2 dc,0 λ max ,e Δq0 Φint ,e , ( 8π dc,e λ max ,0 q0 Δλ e 1)Δθe
dc λ max ,0 Δλ (1)Δθ q0 3, dc,0 λ max λ m2 Δq02
(31′)
(32′)
N.K. Pleshanov / Nuclear Instruments and Methods in Physics Research A 834 (2016) 197–204
2 q ⎛ Δq ⎞ dc λ max ,0 εc,m 0 ⎜⎜ m ⎟⎟ 3; dc,0 λ max qm ⎝ Δq0 ⎠
(1) Φint =
3=
dc,0 λ max ,e (Δλ /λ )0 Δθ0 dc,e λ max ,0 (Δλ /λ )e Δθe (2) Φint ,e
(Δλ /λ )eΔθe (2) Φint = 8π
(2) Φint
(2) Φint ,e =
(33′) 2 λ m dc,0 λ max ,e Δq0 8π dc,e λ max ,0 q0
,
(36′)
dc λ max ,0 (Δλ /λ )Δθ q0 3, dc,0 λ max λm Δq02
(37′)
(38′)
Note that with a single disk chopper the quantity εc,m is the optimality of measurements at the mean wavelength λm. When the measurement conditions with Soller collimators (23′) are optimal at the wavelength λm, the optimality of measurements at other wavelengths is given by [4]
εc(λ ) =
2 . λ /λ m + λ m /λ
(30′)
There is a correlation between the width and the divergence of the beam only for the double diaphragm collimation. As a consequence, with the Soller and double diaphragm collimation, the flux at the sample is proportional, respectively, to A(λ)λ2 and A(λ) λ3 as stated earlier [4], or to (Δq)2 and (Δq)3 as derived above (Δq is the resolution at a given q for a monochromatic beam or at a given qm for a white beam). In conclusion, define the reference luminosity of reflectometers for measurements with small samples. Obviously, this quantity depends on the sizes of the reflecting surface of the sample, and we define a standard sample with a square surface with a side w0. The corresponding reference luminosity 3(w0) can be assessed from the value of 3 , which is defined now from the conditions of measurements with the standard sample, and the calibration experiment is made with the beam limited by the slit of height w0 at the sample position. Say, formula (29) should be used for a monochromatic beam and collimation with two diaphragms. To assess the reference luminosity, we substitute q ¼q0, Δq ¼ Δq0 and εd ¼ 1. With a small sample the role of d2 actually plays the quantity w0θ0; the beam with a given divergence Δθ0 is formed by setting the width of the first slit d1, (often significantly) exceeding w0θ0 (θ0 and Δθ0 are defined by formulas (24)). The value of w0θ0 defines the cross section of the sub-beam intercepted by the surface of the standard sample. In view of the facts mentioned above, substituting d2 ¼w0θ0 into expression (8) for εg, we find from (29) that −1
3(w0) = εg 3 = 2( d1/w0θ0 + w0θ0/d1) 3
(
−1
)
= 2 4πd1/w0q0λ + w0q0λ /4πd1
It can be shown that formulas (39), (40) and (40′) will be valid for measurements at white beams in the respective slit geometry, provided that λ is replaced with λm. Estimate the possible gain in luminosity, when small angle Soller collimators are used. To this effect, note that each channel of Soller collimator with a length Lch may be regarded as a double diaphragm system with slits of width wch. Proceeding from formula (3), one has the following flux after the Soller collimator designed [4] with traps for neutrons and with the full channel width 3wch:
Φс(λ ) ∝ Ta
2 d λ max ,0 q0 ⎛ Δqm ⎞ ⎟⎟ 3. = c εc ⎜⎜ dc,0 λ max qm ⎝ Δq0 ⎠
3.
(39)
When d1 c w0θ0,
3(w0) ≅ (2w0θ0/d1)3 = (w0q0λ /2πd1)3.
(40)
To assess the reference luminosity with the Soller collimation of the monochromatic beam, we substitute q¼ q0, Δq ¼ Δq0, εc ¼1, dc ¼w0θ0 into formula (29′) and obtain
3(w0) = (w0θ0/dc,0)3 = (w0q0λ /4πdc,0)3.
(40′)
As well as in the case of the double diaphragm collimation, the quantity 3 is assumed to be defined for the beam limited at the sample position with a slit of height w0.
201
2 wch dc , (L s + L ch )L ch 3wch
(41)
where Ta is the beam attenuation in the wafers of silicon or other weakly absorbing material, from which the Soller collimator with a total width dc is assembled. According to the same formula (3), the flux behind the two diaphragms with slits of equal width d is
Φd(λ ) ∝
d2 (L s + L )L
(42)
(we assume that the first slit is at the same distance Ls from the source as the Soller collimator). Hence the flux ratio for the beams with the same divergence Δθ = wch/( 6 Lch ) = d/( 6 L ) is
Φс(λ ) 1 d ⎛ L⎞ = Ta c ⎜ 1 + ⎟. 3 d⎝ Ls ⎠ Φd(λ )
(43)
The flux ratio exceeds unity for sufficiently small distance from the source and sufficiently high resolution, so that the value of d = 6 LΔθ is small in comparison with dc.
5. Luminosity class The neutron flux for different measurement conditions has been shown in the previous Section to be proportional to the reference luminosity defined for standard measurement conditions. Thus, the luminosity of a reflectometer can be characterized by a single number: the reference luminosity 3 . In principle, one and the same reflectometer may be characterized by several values of the reference luminosity, one for each mode of operation: with monochromatic and white beams, with non-polarized and polarized neutrons, with small and large samples. The reference luminosity can certainly be used to compare the operating reflectometers. Moreover, fixing the standard measurement conditions, one can classify neutron reflectometers according to luminosity for each designed mode of operation. To the effect, the following definition of luminosity class of reflectometers is suggested here:
CR = log10 3,
(44)
where 3 is specified by the assigned parameters: q0 ¼1 nm 1, Δq0 ¼0.01 nm 1, λmax,0 ¼1 nm, dc,0 ¼1 mm. When indicating the luminosity class of the reflectometers for the mode of measurements with small samples, it is sufficient to specify the size w0 of the square surface of the standard sample, e.g., CR(10 mm). The choice of the values of the standard quantities is defined by the requirement of the convenience of their usage. Say, the resolution Δq0/q0 ¼ 0.01 is rarely achieved in practice. More often the reflectometers are built for measurements with moderate and low resolution. In this case along with the luminosity class CR, one can point out the reference luminosity of the reflectometer for a typical resolution Δq at the momentum transfer q0. This value is easy to assess multiplying 3 by (Δq/Δq0 )3 in measurements with double diaphragm collimation and by (Δq/Δq0)2 in measurements with Soller collimation of the beam.
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As an example, assess the luminosity class of the neutron reflectometer NR-4M at the thermal beamline 13 of the reactor WWR-M (Gatchina) for each of the 4 modes of its operation: with monochromatic and white beams, with non-polarized and polarized neutrons. All assessments are made on the basis of published experimental data [8]. In all modes of operation the beam is collimated by two diaphragms with the slits of width d1,e ¼d2,e ¼0.2 mm (εg,e ¼1); the distance between the diaphragms is 1155 mm, and according to (5) the beam divergence is Δθe ¼0.07 mrad. For the operation mode with a monochromatic non-polarized beam (λ ¼0.19 nm, Δλe = 0.074λ/(2 2 ln 2 ) = 0.006 nm, Φe ¼ 200 s 1), with formulas (25) and (44) we find the reference luminosity 3 ¼ 113 s 1 and the luminosity class CR ¼2.1. For the operation mode with a monochromatic polarized beam (λ ¼0.20 nm, Δλe = 0.065λ/(2 2 ln 2 ) = 0.0055 nm, Φe ¼110 s 1) the same formulas lead to 3 ¼78 s 1 and CR ¼ 1.9. Surprisingly, the working wavelengths 0.19 nm and 0.20 nm practically coincide with the wavelength 0.189 nm, at which the product A(λ)λ3 is maximum (according to the non-polarized neutron spectrum). Even though the spectral flux density A(λ) is maximum near 0.1 nm, an ‘intuitive’ choice of the wavelengths as a compromise between resolution and intensity turned out to be optimal. The relative resolution Δq/q and the optimality εd for measurements with the monochromatic non-polarized beam are represented in Fig. 1 as functions of q¼4πθ/λ. The ratio Δq/q decreases with the growth of the glancing angle θ. The measurements are optimal (εd ¼1) at the glancing angle 0.08°, εd falls to 0.17 at 0.5° (q¼0.58 nm 1), to 0.042 at 1° (1.15 nm 1) and to 0.011 at 2° (2.3 nm 1). It means that the flux at the sample may be increased, accordingly, in 6, 24 and 90 (! ) times without any loss in resolution. So considerable a loss in intensity is, to a large extent, due to the use of collimation diaphragms with the slits of constant width. To receive evidence, calculate the fluxes with variable slit widths. When Δλ/λ is constant, varying the slit widths proportionally to the glancing angle θ, one obtains the same values of Δq/q and εd for all q. It is a consequence of the proportionality of Δθ and the slit widths: d1 = d2 = 6 3Δθ . In particular, εd ¼ 1 for all q with Δθ = 2 θ Δλ /λ . In this case for measurements with the monochromatic non-polarized beam we obtain Δq/q ¼0.07. A more acceptable resolution Δq/q ¼0.04, close to the reference value Δq/ q ¼0.0315 in Fig. 1 at large q, is achieved with Δθ = 0.557 2 θ Δλ /λ , when εd ¼ 0.78. the slit width is proportional to q and increases up
Fig. 1. The relative resolution Δq/q and the optimality εd as functions of q for measurements at the reflectometer NR-4M (reactor WWR-M, Gatchina) with the monochromatic non-polarized beam collimated by the slits with a fixed width 0.2 mm.
Fig. 2. The ratio of the flux for the diaphragm slits with a variable width (see the text) to the flux for the slits with a fixed width (0.2 mm) as a function of q for measurements with monochromatic neutrons at the reflectometer NR-4M. Due to the variation of the slit widths, Δq/q ¼0.04 and εd ¼0.78 for all q (when the slit widths are fixed, these quantities depend on q as shown in Fig. 1).
to 2.5 mm at q ¼2.3 nm 1. The ratio of the flux for the slits with a variable width to the flux Φe ¼ 110 s 1 for the slits with a fixed width 0.2 mm is represented in Fig. 2. It is to be emphasized that the increase in the flux (by two order of magnitude at large q!) is due to optimization of measurements only: the reference luminosity (luminosity class) of the reflectometer remains the same. At small q the flux is smaller than Φe because of reduction of the slit widths below 0.2 mm to retain the ratio Δq/q ¼0.04. To work with white beams at the reflectometer NR-4M, a single disk chopper is used with TOF resolution Δλ(1) ¼0.006 nm, mean λm ¼ 0.168 nm and maximum wavelength wavelength λmax ¼0.632 nm. The initial flux for the white non-polarized beam (0) ¼15,000 s 1; according to formula (18) the flux after the is Φint ,e (1) ¼ 142 s 1, and according to formula (31) the rechopper is Φint ,e ference luminosity is 3 ¼35 s 1, the respective luminosity class (0) being CR ¼1.5. The initial flux for the white polarized beam is Φint ,e (1) ¼5600 s 1, hence Φint ¼53 s 1, 3 ¼ 13 s 1, CR ¼1.1. ,e Calculated Δq/q and εd (solid curves) for TOF measurements at the glancing angles 0.25° and 0.5° are represented in Fig. 3 as
Fig. 3. The relative resolution Δq/q and the optimality εd of TOF measurements at the reflectometer NR-4M (reactor WWR-M, Gatchina) with non-polarized beam at the glancing angles 0.25° and 0.5° as functions of q. Solid curves: d1 ¼ d2 ¼ 0.2 mm, Δλ(1) ¼0.006 nm (designed parameters for all glancing angles). Dotted curves: for optimized parameters d1 ¼ d2 ¼0.4 mm, Δλ(1) ¼0.0044 nm (θ ¼0.25°) and d1 ¼d2 ¼ 0.6 mm, Δλ(1) ¼0.0033 nm (θ¼ 0.5°).
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203
functions of q¼ 4πθ/λ for the wavelengths from 0.08 nm to λmax. The relative resolution is worse at larger q. The dependences εd(q) at different angles overlap and merge into one (solid) curve. The measurement optimality εd falls to 0.03 at q ¼1.3 nm 1. For the mean wavelength λm and the glancing angles 0.25° and 0.5°, reεd ¼ 0.4, Δq/q ¼0.041 and spectively, qm ¼0.326 nm 1, 1 qm ¼ 0.65 nm , εd ¼ 0.12, Δq/q ¼0.036. To optimize the TOF measurements at the glancing angles 0.25° and 0.5°, require that at the wavelength λm: (a) εd ¼1 and (b) Δq/ q ¼0.04 (0.25°) and Δq/q ¼0.03 (0.5°). The parameters of the beams can be obtained from (23) and (5):
Δλ (1) =
1/3 λ(Δq/q), Δθ =
2/3 θ (Δq/q), d1 = d2 =
6 3Δθ .
(45)
Respectively, Δλ ¼0.0044 nm, d1 ¼d2 ¼0.4 mm (0.25°) and Δλ(1) ¼0.0033 nm, d1 ¼d2 ¼0.6 mm (0.5°). According to (16), the integral fluxes at the sample increase in 3 and 5 times. With the resolution slightly relaxed, the gain in intensity reaches one order of magnitude. Again the increase in the flux is achieved by optimization of measurements; the reference luminosity (luminosity class) of the reflectometer does not change. The results of calculations of Δq/q and εd with the optimized parameters of the beams are represented with dotted curves in Fig. 3. Thus, the luminosity class defines only the potential of the reflectometer; to which extent this potential can be used depends on designed-in technical options. The above example with the reflectometer built in 80's seems to be quite instructive. More technical options are inserted into modern reflectometers, which can be used to optimize measurements. One of such options is the use of choppers with two disks [2]. Relationship (22) can be used to compare the integral fluxes for the choppers with one and two disks. The integral fluxes at the sample will be equal, provided that (Δλ/λ) ¼ Δλ(1)/λm. If Δθ is chosen according to formula (23), we obtain for a chopper with two disks that εd ¼1 at all wavelengths, and Δq/q is the same at all wavelengths and coincides with Δq/q for a single disk chopper at λ ¼ λm. In particular, Δq/q ¼0.03 for the optimized beam parameters obtained above for the TOF measurement at θ ¼ 0.5°. Although the measurements with a single disk chopper are not optimal at all q (see the corresponding dashed curve εd in Fig. 3), the total flux at the sample is the same as that with the double disk chopper, when εd ¼1. The reason is that a decrease in the flux at λ 4 λm (at small q) in comparison with the flux after the double disk chopper is compensated by an increase in the flux at λ o λm (at large q). Such redistribution of the flux is accompanied by improvement of the resolution at the wavelengths λ 4 λm (Δq/ q o0.03) worsening the resolution at the wavelengths λ o λm (Δq/ q 40.03). Since the neutron reflectivity, as a rule, falls down as q 4, or even faster because of interfacial roughness, the redistribution of the flux with an increase in the number of neutrons reflected with larger q may give priority to the single disk chopper with a variable slit width, i.e. when the TOF resolution can be tuned as required for a certain resolution Δq. The task of finding an appropriate constructional solution in this case seems to be much simpler than a formidable task of building double disk choppers with a TOF resolution tuned by changing the distance between the disks. In particular, a chopper made of two closely spaced rotating disks with a tunable phase of rotation (‘tuned-phase chopper’) is equivalent to a single disk chopper with a variable slit width. Indeed, a constant phase shift in rotation of one of the disks will decrease the efficient slit width of the chopper by reducing the overlap of the opposite slits on the rotating disks and will improve the TOF resolution. In the same manner the drum chopper [4] suggested for fan beam TOF measurements can also be made with a tuned-phase variable resolution. (1)
Fig. 4. A scheme of the fan beam reflectometry with monochromatic neutrons. Solid and dashed lines show extreme trajectories of neutrons traversing two slits with widths d1 and d2 {d1 and registered within the spatial resolution δdet of PSD. Trajectories of neutrons without the sample and trajectories of neutrons reflected from the sample are shown. Neutrons with different glancing angles are registered in different parts of PSD.
It is impossible in one paper to encompass all measurement schemes and describe all specific details in classification of the reflectometers according to the luminosity. However, one may suppose that, in most cases, the generalization will make no difficulties. As an example, consider a scheme of measurements of specular reflection with a fan beam of monochromatic neutrons (a similar scheme was proposed earlier [4] for TOF measurements). In this scheme (Fig. 4) the slit with a width d1 is a virtual source, the slit near the sample with a width d2 { d1 and the horizontal coordinate resolution δdet of PSD work as two collimation slits. Then it is easy to evaluate the angular resolution 2 Δθ = (2Ldet )−1 (d22 + δdet ) /3 (Ldet is the distance from PSD to the slit near the sample) and optimize measurements. The luminosity gain
factor is Ω/Δθ, where Ω = (2L )−1 (d12 + d22) /3 is the fan angle, and instead of (25) one obtains
3=
Δq03 λ3 Ω Φe 2 24π 3 q0 εg,dΔλ eΔθe3
(46)
with the geometric optimality factor −1
εg,d = 2( d2/δdet + δdet/d2) .
(47)
For a given resolution Δq the flux in the fan beam will be maximum with δdet = d2 = 6 LdetΔθ , where Δθ = 2/3 θ Δq /q. Note that the spatial resolution of PSD must be sufficiently high; it is impossible to diminish a large δdet, whereas one can multiply increase a small δdet by summarizing the intensities in neighboring channels of PSD.
6. Conclusion In order to increase the flux at the sample without worsening the resolution, the possibilities to increase the vertical divergence during transportation of neutrons from the source are usually considered, incl. the beam condensation by a neutron guide tapering in height and the vertical focusing of the beam onto the sample [9,10]. At the same time, the additional possibilities to increase the flux without worsening the resolution by tailoring the beam optimal in the horizontal plane are largely ignored. In the papers [3,4] the luminosity of the reflectometer was coupled with the momentum transfer resolution Δq. Some possibilities to increase the luminosity without loss of resolution were revealed and new criteria of optimization of neutron
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reflectometers substantiated. The reference luminosity introduced in the present paper as a maximum flux for standard resolution parameters essentially differs from the luminosity defined as a flux for any given resolution Δq. The reference luminosity 3 characterizes the luminosity of the instrument by one number and can be used to classify operating reflectometers [11] and optimize designed reflectometers. To meet this end, the luminosity class CR ¼ log10 3 has been defined. It is to be emphasized that the luminosity class determines only the measurement potential of the reflectometer. The maximum flux available for a given resolution depends on the technical options designed in the reflectometer. The luminosity class CR of the neutron reflectometer NR-4M (reactor WWR-M, Gatchina) has been determined for 4 modes of operation: 2.1 (monochromatic non-polarized beam), 1.9 (monochromatic polarized beam), 1.5 (white non-polarized beam), 1.1 (white polarized beam). It has been shown that optimization of measurements will increase the neutron flux at the sample up to two orders of magnitude with monochromatic beams and up to one order of magnitude with white beams. Small angle Soller collimators have been shown (Section 4) to increase the luminosity of the instrument providing they are close to the (virtual) source. Even more promising may be their use in small angle neutron scattering stations, where the beams with large cross sections are used. Though, small angle Soller collimators are still to be worked out. Of practical interest may also be the reasoning (Section 5) in favor of using single disk choppers with a variable width of the slits as a trade-off increasing the luminosity for the white beam modes of the reflectometer instead of using
technically more complicated double disk choppers with a variable distance between the disks.
Acknowledgments The work was supported by the Federal target program (FTP) of Ministry of Education and Science of Russian Federation (project No. RFMEFI61614X0004).
References [1] V.O. de Haan, J. de Blois, P. van der Ende, H. Fredrikze, A. van der Graaf, M. N. Schipper, A.A. van Well, J. van der Zanden, Nucl. Instrum. Methods A 362 (1995) 434. [2] A.A. van Well, H. Fredrikze, Physica B 357 (2005) 204. [3] N.K. Pleshanov, J. Surf. Investig.: X-Ray Synchrotron Neutron Tech. 10 (2016) 790. [4] N.K. Pleshanov, Nucl. Instrum. Methods A 820 (2016) 146. [5] A.A. van Well, Physica B 180–181 (1992) 959. [6] R.A. Campbell, H.P. Wacklin, I. Sutton, R. Cubitt, G. Fragneto, Eur. Phys. J. 126 (2011) 107. [7] M. James, A. Nelson, S.A. Holt, T. Saerbeck, W.A. Hamilton, F. Klose, Nucl. Instrum. Methods A 632 (2011) 112. [8] V.G. Syromyatnikov, N.K. Pleshanov, V.M. Pusenkov, A.F. Schebetov, V.A. Ul’yanov, Ya.A. Kasman, S.I. Khakhalin, M.R. Kolkhidashvili, V.N. Slyusar, A.A. Sumbatyan, Preprint PNPI-2619, Gatchina, 2005, 47 pp. [9] T. Hils, P. Boeni, J. Stahn, Physica B 350 (2004) 166. [10] J. Stahn, U. Filges, T. Panzner, Eur. Phys. J. Appl. Phys. 58 (2012) 11001. [11] Adrian R. Rennie, 2016, 〈http://www.reflectometry.net/reflect.htm〉.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Luminosity class of neutron reflectometers N.K. Pleshanov Neutron Research Department, Petersburg Nuclear Physics Institute, NRC “Kurchatov Institute”, Gatchina, 188300 St. Petersburg, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 22 April 2016 Received in revised form 6 August 2016 Accepted 8 August 2016 Available online 9 August 2016
The formulas that relate neutron fluxes at reflectometers with differing q-resolutions are derived. The reference luminosity is defined as a maximum flux for measurements with a standard resolution. The methods of assessing the reference luminosity of neutron reflectometers are presented for monochromatic and white beams, which are collimated with either double diaphragm or small angle Soller systems. The values of the reference luminosity for unified parameters define luminosity class of reflectometers. The luminosity class characterizes (each operation mode of) the instrument by one number and can be used to classify operating reflectometers and optimize designed reflectometers. As an example the luminosity class of the neutron reflectometer NR-4M (reactor WWR-M, Gatchina) is found for four operation modes: 2.1 (monochromatic non-polarized beam), 1.9 (monochromatic polarized beam), 1.5 (white non-polarized beam), 1.1 (white polarized beam); it is shown that optimization of measurements may increase the flux at the sample up to two orders of magnitude with monochromatic beams and up to one order of magnitude with white beams. A fan beam reflectometry scheme with monochromatic neutrons is suggested, and the expected increase in luminosity is evaluated. A tuned-phase chopper with a variable TOF resolution is recommended for reflectometry with white beams. & 2016 Elsevier B.V. All rights reserved.
Keywords: Neutron reflectometers Luminosity and resolution Reference luminosity Luminosity class Fan beam reflectometry Tuned-phase chopper
1. Introduction Neutron reflectometry is used for detailed studies of physical and chemical processes at the interfaces, bio-processes with interaction and transfer of substances through interfaces, membranes, etc. New data unavailable with other research techniques have been obtained. Comparatively low luminance of neutron sources implies an efficient transport of neutrons from the source to the sample and the use of efficient position sensitive detectors (PSD). The relationship between resolution and luminosity for specular reflection measurements is shortly discussed in papers [1,2]. In papers [3,4] the luminosity of the reflectometer was coupled to the momentum transfer resolution Δq. It revealed the possibilities to increase luminosity without loss in resolution and substantiated new optimization criteria for neutron reflectometers. However, the luminosity defined in this manner depends not only on the resolution Δq, but also on the momentum transfer
q = (4π sin θ )/λ ≅ 4πθ /λ ,
(1)
where λ is the neutron wavelength, θ is the glancing angle assumed (usually, it does not exceed several degrees). When angular and wavelengths distributions of neutrons in the incident beam E-mail address: [email protected] http://dx.doi.org/10.1016/j.nima.2016.08.022 0168-9002/& 2016 Elsevier B.V. All rights reserved.
with divergence Δθ and monochromaticity, or time-of-flight resolution, Δλ are Gaussian, one obtains
Δq/q =
(Δθ /θ )2 + (Δλ /λ )2
(2)
(here and further the formulas are written in terms of the standard deviations, which are less than full widths at half maximum in
2 2 ln 2 ≅ 2.355 times). Without loss of generality, we assume that the scattering plane is horizontal, so the slits and the reflecting surface are vertical, and the resolution is defined by the horizontal divergence. The present paper is an elaboration on the papers [3,4]. In Sections 2 and 3 the formulas that relate neutron fluxes at the sample for measurements with differing resolutions are derived, accordingly, for double diaphragm and Soller beam collimation. These formulas are used in Section 4 for introduction of ‘reference luminosity’, independent of q and Δ q, and the methods of its assessment from the intensities measured at the reflectometer. In Section 5 the luminosity class is introduced to compare different reflectometers; the luminosity class of the reflectometer NR-4M (Gatchina) is assessed and the possibility to optimize measurements at this reflectometer is considered. The results obtained are summarized in Conclusion.
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and the resolution
2. Measurements with double diaphragm collimators
Δq can be found from (7) with λ ¼ λe:
2
In measurements with two diaphragms collimation the neutron flux at the sample is [4]:
Φ(λ ) = hs h2
d d L A(λ ) 1 2 Δλ , Ls + L L L
(3)
where hs and h2 are the heights of the source and the diaphragm slit before the sample, d1 and d2 are the widths of the slits of the first and second collimation diaphragms, Ls and L are the distances from the source to the first diaphragm and between diaphragms, respectively. The spectral density A(λ) of the flux incident onto the sample depends on the luminance and the spectral density of the source emission, λ-dependent transmission of neutron guides and focusing optics, width of the monochromator peak, transmission of the collimation system and, generally speaking, on the sample size. It is evident that the flux at the sample for measurement conditions mentioned above may be assessed from the flux experimentally measured with a beam formed with diaphragm slits of known width d1,e and d2,e, with known mean wavelength λe and spectral width Δλe:
Φ(λ ) =
A(λ ) d1 d2 Δλ Φe(λ e). A(λ e) d1,e d2,e Δλ e
(4)
Here and throughout the paper the quantities with a subscript ‘e’ refer to an experiment carried out to calibrate the neutron flux at a given reflectometer. The measurement in the calibration experiment is made in the absence of the sample and with a beam with known parameters. It implies that all quantities with the subscript ‘e’ are known from the experiment or can be found directly from the experiment. What is more important, they can be used to assess the flux at the reflectometer for a beam with differing parameters. The divergence of the respective beams (standard Gaussian deviations) is [2] 2 2 Δθ = (2L )−1 (d12 + d22)/3 , Δθe = (2L )−1 (d1,e + d2,e )/3 ,
Φ=
εg Δλ ⎛ Δθ ⎞ ⎜ ⎟ Φe. εg,e Δλ e ⎝ Δθe ⎠
(9)
Measurements at reflectometers with white beams are carried out with using the TOF technique at a fixed glancing angle θ and with the same divergence Δθ for all wavelengths. The role of monochromaticity of neutrons plays the TOF resolution. Further we consider the time-of-flight technique only at a steady flux reactor to avoid analyzing the details of the realization of the technique at pulsed sources, which are not essential for us. By definition, A(λ) is the spectral flux density in the beam incident onto the sample in the absence of the chopper. Then the time-averaged spectral flux density of the beam chopped into pulses by the chopper is
A chop (λ ) = A(λ )τ (λ )/Tp = A(λ )Δλ(λ )/λ max ,
(10)
where account is taken of the relation of the neutron pulse width τ (λ ) = LbhΔλ(λ ) /mn with the TOF resolution Δλ(λ) and of the relation of the pulse period Tp = Lbhλ max /mn with the maximum wavelength λmax of neutrons registered by the detector before the arrival of the next pulse (Lb is the TOF base, h is the Planсk constant, mn is the neutron mass). In the absence of the chopper the integral flux incident onto the sample (cf. with (3)) is (0) Φint = hs h2
J0 =
∫0
∞
L d1 d2 J , Ls + L L L 0
(11)
A(λ )dλ .
(12)
By analogy with (7), one finds the relationship between the integral fluxes in the absence of the chopper for the beams with the same spectral flux density A(λ): (0) Φint =
(5)
2 εg ⎛ Δθ ⎞ (0) ⎜ ⎟ Φint ,e. εg,e ⎝ Δθe ⎠
(13)
The integral flux with the chopper is related to the integral flux of the same beam without the chopper:
hence 1/2 ⎛ d 2 + d 2 ⎞1/2 d1 ⎡ 1 + (d2/d1)2 ⎤ Δθ 2 ⎟ ⎢ ⎥ = = ⎜⎜ 21 = ⎟ 2 Δθe d1,e ⎢⎣ 1 + (d2,e/d1,e)2 ⎥⎦ ⎝ d1,e + d2,e ⎠
Φint =
1/2 d ⎡ 1 + (d2/d1)−2 ⎤ ⎥ . = 2 ⎢ −2 d2,e ⎢⎣ 1 + (d2,e/d1,e) ⎥⎦
(0) Φint
J0
(6)
Substituting d1/d1,e and d2/d2,e expressed in terms of Δθ/Δθe from (6) into (3), one finds the flux in the incident beam with arbitrary values of λ, Δλ and Δθ:
Φint ,e = =
∫0
(0) Φint ,e
J0
∞
A chop (λ )dλ =
∫0
(0) Φint
1
J0 λ max
∫0
∞
A(λ )Δλ(λ )dλ ,
(14)
∞
A chop, e(λ )dλ
(0) Φint ,e
1
J0
λ max ,e
∫0
∞
A(λ e)Δλ e(λ e)dλ e.
(15)
2
A(λ ) εg Δλ ⎛ Δθ ⎞ Φ(λ ) = ⎜ ⎟ Φe(λ e), A(λ e) εg,e Δλ e ⎝ Δθe ⎠ −1
(
(7) −1
)
εg = 2( d1/d2 + d2/d1) , εg,e = 2 d1,e/d2,e + d2,e/d1,e
.
(8)
The quantities εg and εg,e (geometric optimality of measurements with the respective beams) characterize a decrease in the intensity of the beams in comparison with the maximum flux available for a given divergence of the beams, when d1 ¼d2 and d1, e ¼d2,e (εg ¼ εg,e ¼1). As a rule, at a given reflectometer working with a monochromatic beam, all measurements are carried out with the same mean wavelength. The flux of monochromatic neutrons for the conditions of reflection from the sample with the momentum transfer q
The single disk chopper secures a TOF resolution equal at all wavelengths, Δλ(1) ¼ const. The chopper with two synchronously rotating disks [5] secures measurements with (Δλ/λ) ¼ const at all wavelengths (actually, the range is limited by a maximum wavelength that most of the time can be made sufficiently large and may be ignored in our considerations without loss of generality; see Ref. [2] for further details on application of double disk choppers in neutron reflectometry) and, as a consequence, with one and the same relative resolution Δq/q. The value of Δλ/λ can be changed by varying the rotation frequency and the distance between the disks. The technical complexity did not allow to gradually change these parameters, so several disks located at different distances are used [6,7]. The choice of different pairs of disks provides a discrete change of the ratio (Δλ/λ).
N.K. Pleshanov / Nuclear Instruments and Methods in Physics Research A 834 (2016) 197–204
Taking into consideration the dependence Δλ(λ) in (14) and (15) and relation (13), one obtains the relationship between the integral fluxes for measurements at a given reflectometer, i.e. for the beams with the same spectral flux density A(λ) and the same choppers (the number of disks is shown with the upper index): (1) Φint
2 λ max ,e εg Δλ (1) ⎛ Δθ ⎞ (1) = ⎜ ⎟ Φint ,e 1 ( ) λ max εg,e Δλ e ⎝ Δθe ⎠
λ max ,e εg (Δλ /λ ) ⎛ Δθ ⎞ (2) ⎜ ⎟Φ λ max εg,e (Δλ /λ )e ⎝ Δθe ⎠ int ,e
(2) Φint =
Δλ (1) (0) Φ , λ max int
(17)
Φ=
dc Δλ Δθ Φe. dc,e Δλ e Δθe
(19)
∫0
∞
A(λ )λdλ⋅
⎛ ⎜ ⎝
∫0
∞
⎞−1 A(λ )dλ⎟ , ⎠
(20)
characterizes the white beam spectrum. Say, for the beam of neutrons thermalized at a temperature T with the Maxwellian flux density function A(λ), which is maximum at the wavelength
λ peak =
h2/5mnkT , one has
λ m = λ peak 5π /8 ≅ 1.401λ peak .
(21)
Note that formulas (18) and (19) can be used for experimental determination of Δλ(1) and λm from measurements of the integral intensities without chopper and with chopper. Also from (18) and (19) we obtain the following relationship between the integral fluxes for the choppers with one and two disks for the same neutron pulse frequency, the same initial beam and the same collimation geometry: (1) Φint =
Δλ (1)/λ m (2) Φ . (Δλ /λ ) int
(22)
(9′)
In the case of measurements with the white beams the integral flux incident onto the sample without the chopper is (0) Φint = hc dcΔβ⋅Δθ⋅J0 .
(11′)
Its relation to the integral flux with the chopper is given by Eq. (14). As a consequence, the integral fluxes after double disk and single disk choppers are given by the same formulas (18–22). The integral fluxes for different measurement conditions are related as follows: (1) Φint =
λ max ,e dc Δλ (1) Δθ (1) , Φ λ max dc,e Δλ e(1) Δθe int ,e
(16′)
(2) Φint =
λ max ,e dc (Δλ /λ ) Δθ (2) . Φ λ max dc,e (Δλ /λ )e Δθe int ,e
(17′)
where the mean wavelength, introduced as
λm =
(7′)
where the demanded and measured fluxes are for the beams prepared with collimators of width, accordingly, dc and dc,e. For the measurements with the monochromatic beams (λ ¼ λe)
(18)
(Δλ /λ ) (0) Φ , λ max /λ m int
A(λ ) dc Δλ Δθ Φe(λ e), A(λ e) dc,e Δλ e Δθe
(16)
(cf. with (7)). As in the case with monochromatic beams, one can (1) (2) or Φint measured use the experimental integral intensities Φint ,e ,e at a beam with known divergence Δθe and TOF resolution (Δλe or (Δλ/λ)e) and maximum wavelength λmax,e, to extract the integral flux for a beam with arbitrary parameters. From (14) with Δλ ¼ Δλ(1) ¼ const and Δλ ¼(Δλ/λ)λ ¼ const λ (Δλ(1) and Δλ are standard Gaussian deviations) one obtains, respectively, (1) Φint =
systems and focusing devices, the sample height. Further, Δβ and hc are assumed to be constant. Hence we obtain the formula for the flux incident onto the sample:
Φ(λ ) =
2
(2) Φint =
199
4. Reference luminosity The formulas obtained may be used to assess the flux for different reflectometers in certain measurement conditions, equivalent from the point of view of resolution. The fluxes defined in this manner may serve as a measure of luminosity of reflectometers. This quantity can be named as ‘reference luminosity’. To avoid ambiguities, the quantities Δq, Δθ , Δλ will always be regarded as standard Gaussian deviations. For the reflectometers at beams of monochromatic neutrons with a working wavelength λ define the reference luminosity 3 as the maximum flux achievable for measurements with standard (reference) values of the momentum transfer (q0) and the resolution (Δq0). The quantity q0 agrees with a glancing angle θ0 ¼q0λ/(4π). With a double diaphragm collimation, the neutron flux is maximum for εg ¼1 and when the conditions [4]
Δθ /θ =
2 (Δλ /λ ) =
2/3 (Δq/q)
(23)
are fulfilled, i.e. 3. Measurements with small angle Soller collimators A small angle Soller collimator forms a beam with a cross section of width dc and height hc, with a vertical Δβ and horizontal Δθ divergence. The neutron flux incident onto the sample is
Φ(λ ) = hc dcΔβ⋅A(λ )⋅Δλ⋅Δθ
(3′)
(here and further the primed numbers are used for the formulas concerning measurements with Soller collimators; they correspond to the numbers of similar formulas for measurements with two diaphragms). The vertical divergence Δβ depends on the source height, on the parameters of transport neutron guide
Δθ0 =
2/3 θ0(Δq0/q0) =
2/3 λΔq0, 4π
Δλ 0 =
1/3 λ(Δq0/q0),
(24)
we obtain from (9) that
3=
2 Δq03 Φe 1 Δλ 0 ⎛ Δθ0 ⎞ λ3 . ⎜ ⎟ Φe = 2 εg,e Δλ e ⎝ Δθe ⎠ 24π 3 q0 εg,eΔλ eΔθe2
(25)
Now, knowing 3 , one can find the flux for measurements with arbitrary Δλ and Δθ:
Φ = 24π 2 3 εg
ΔλΔθ 2 q0 λ3
Δq03
3. (26)
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Using (2) and such a quantity as the optimality of measurements with diaphragms [4]
(
κ=
−3/2
)
εd = (3 3 /2)κ 1 + κ 2
,
(27)
Δλ /λ Δλ q = , Δθ /θ Δθ 4π
(28)
the flux can also be written in terms of q and
Δq:
with (23). So the reference luminosity of the reflectometer will be related not with Δq0, but with a standard relative resolution (Δq/ q)0. Define the reference luminosity as the maximum integral flux available for measurements with (Δq/q)0 ¼ Δq0/q0 and with a standard maximum wavelength λmax,0. Then, defining θ0 ¼q0λm/ (4π), one obtains from (23):
Δθ0 =
2/3 λ mΔq0, 4π
(Δλ /λ )0 =
1/3 (Δq/q)0 =
1/3 (Δq0/q0),
(35)
and from (17):
3
q ⎛ Δq ⎞ ⎟⎟ 3. Φ = εgεd 0 ⎜⎜ q ⎝ Δq0 ⎠
(29)
According to (2), the resolution Δq may be achieved with numerous pairs of Δλ and Δθ. The optimal pair can be found from conditions (23). The choice of Δλ and Δθ is often limited by the technical possibilities designed in the reflectometer. According to (5), the beam divergence Δθ may be achieved with numerous pairs of d1 and d2; the factor εg in (29) gives the decrease of the flux below the maximum flux with d1 ¼ d2 (εg ¼1). The above-introduced definition of reference luminosity is not suited for the reflectometers at white beams. First, the values of the momentum transfer differ at different wavelengths, so the choice of the standard momentum transfer q0 makes no sense. Second, as it has been argued earlier [4], TOF measurements with the same Δq at all wavelengths are not possible because of technical problems connected with the requirement to simultaneously fulfill the conditions Δθ ∝ λ and Δλ ∝ λ2/θ . In the case of measurements with a single disk chopper the conditions of statistically optimal measurements may be fulfilled only at one wavelength, so measurements are always carried out in the regime not optimal as regards the relationship between resolution and intensity. To define the reference luminosity in this case, require that the measurements be optimal (εd ¼ 1) at the mean wavelength λm. Then the optimality of measurements at other wavelengths is [4]: 2⎞−3/2
εd(λ ) =
⎛λ ⎞ λm ⎛ 2 ⎜ + 1 ⎜ m ⎟ ⎟⎟ λ ⎜⎝ 3 3⎝ λ ⎠ ⎠
.
(30)
2
λ ¼ λm
(1) λ m3 1 λ max ,e Δλ 0 ⎛ Δθ0 ⎞ (1) = ⎜ ⎟Φ εg,e λ max ,0 Δλ e(1) ⎝ Δθe ⎠ int ,e 24π 2 3 λ max ,0 q0 (1) Φint ,e
εg,eΔλ e(1)Δθe2
(1) Φint
(2) Φint ,e
εg,e(Δλ /λ )eΔθe2
. (36)
Hence (2) Φint = 24π 2 3 εg
λ max ,0 (Δλ /λ )Δθ 2 q0 λ m2
λ max
Δq03
3. (37)
3
(2) Φint = εgεd
λ max ,0 q0 ⎛ Δqm ⎞ ⎟⎟ 3, ⎜⎜ λ max qm ⎝ Δq0 ⎠
(38)
where Δqm = qm (Δθ /θ )2 + (Δλ /λ )2 . Now with Δλ/λ ¼ const, the quantity εd does not depend on the wavelength. Similar considerations can be made for measurements with Soller collimators, and one can derive the respective formulas by employing the results from Section 3 and the conditions of maximum flux in measurements with the standard parameters, now including a standard Soller collimator width dc,0. With Soller collimators the flux of monochromatic neutrons is maximum, when [4]
1 (Δq0/q0). 2
(23′)
Therefore,
3=
dc,0 Δλ 0 Δθ0 dc,e Δλ e Δθe
Φe =
2 λ2 dc,0 Δq0 Φe , 8π dc,e q0 Δλ eΔθe
(25′)
λ max ,e Δq03 Φ = 8π
.
dc ΔλΔθ q0 3, dc,0 λ2 Δq02
(26′)
(31)
Hence the flux for arbitrary measurement conditions from the reference luminosity: (1) Φint = 24π 2 3 εg
2 λ max ,e Δq03 λ m2 1 λ max ,e (Δλ /λ )0 ⎛ Δθ0 ⎞ (2) ⎜ ⎟ Φint ,e = εg,e λ max ,0 (Δλ /λ )e ⎝ Δθe ⎠ 24π 2 3 λ max ,0 q0
Δθ0/θ0 = Δλ 0/λ =
Substituting the parameters obtained from (24) with into (16), we find
3=
3=
λ max ,0 Δλ (1)Δθ 2 q0 λ max
λ m3
3, 3
Δq0
3 λ max ,0 q0 ⎛ Δqm ⎞ ⎟⎟ 3, ⎜⎜ = εgεd,m λ max qm ⎝ Δq0 ⎠
qm = 4πθ /λ m, Δqm = qm (Δθ /θ )2 + (Δλ (1)/λ m)2 ;
(32)
Φ=
2 dc q0 ⎛ Δq ⎞ ⎟⎟ 3, εc ⎜⎜ dc,0 q ⎝ Δq0 ⎠
(29′)
where the optimality of measurements with Soller collimators [4]
εc = 2(κ + κ −1)−1 (33)
is used (the parameter κ is defined above in (30)). For the measurements with a white beam and two types of choppers:
(34)
εd,m is also defined at the wavelength λm by substitution of the above parameters into (12). When Δλ/λ ¼const (double disk chopper), the conditions of statistically optimal measurements may be fulfilled at all wavelengths by the choice of the beam divergence Δθ in accordance
(27′)
3=
dc,0 λ max ,e Δλ 0(1) Δθ0 dc,e λ max ,0 Δλ e(1)
(1) Φint = 8π
Δθe
(1) Φint ,e =
(1) 2 λ m2 dc,0 λ max ,e Δq0 Φint ,e , ( 8π dc,e λ max ,0 q0 Δλ e 1)Δθe
dc λ max ,0 Δλ (1)Δθ q0 3, dc,0 λ max λ m2 Δq02
(31′)
(32′)
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2 q ⎛ Δq ⎞ dc λ max ,0 εc,m 0 ⎜⎜ m ⎟⎟ 3; dc,0 λ max qm ⎝ Δq0 ⎠
(1) Φint =
3=
dc,0 λ max ,e (Δλ /λ )0 Δθ0 dc,e λ max ,0 (Δλ /λ )e Δθe (2) Φint ,e
(Δλ /λ )eΔθe (2) Φint = 8π
(2) Φint
(2) Φint ,e =
(33′) 2 λ m dc,0 λ max ,e Δq0 8π dc,e λ max ,0 q0
,
(36′)
dc λ max ,0 (Δλ /λ )Δθ q0 3, dc,0 λ max λm Δq02
(37′)
(38′)
Note that with a single disk chopper the quantity εc,m is the optimality of measurements at the mean wavelength λm. When the measurement conditions with Soller collimators (23′) are optimal at the wavelength λm, the optimality of measurements at other wavelengths is given by [4]
εc(λ ) =
2 . λ /λ m + λ m /λ
(30′)
There is a correlation between the width and the divergence of the beam only for the double diaphragm collimation. As a consequence, with the Soller and double diaphragm collimation, the flux at the sample is proportional, respectively, to A(λ)λ2 and A(λ) λ3 as stated earlier [4], or to (Δq)2 and (Δq)3 as derived above (Δq is the resolution at a given q for a monochromatic beam or at a given qm for a white beam). In conclusion, define the reference luminosity of reflectometers for measurements with small samples. Obviously, this quantity depends on the sizes of the reflecting surface of the sample, and we define a standard sample with a square surface with a side w0. The corresponding reference luminosity 3(w0) can be assessed from the value of 3 , which is defined now from the conditions of measurements with the standard sample, and the calibration experiment is made with the beam limited by the slit of height w0 at the sample position. Say, formula (29) should be used for a monochromatic beam and collimation with two diaphragms. To assess the reference luminosity, we substitute q ¼q0, Δq ¼ Δq0 and εd ¼ 1. With a small sample the role of d2 actually plays the quantity w0θ0; the beam with a given divergence Δθ0 is formed by setting the width of the first slit d1, (often significantly) exceeding w0θ0 (θ0 and Δθ0 are defined by formulas (24)). The value of w0θ0 defines the cross section of the sub-beam intercepted by the surface of the standard sample. In view of the facts mentioned above, substituting d2 ¼w0θ0 into expression (8) for εg, we find from (29) that −1
3(w0) = εg 3 = 2( d1/w0θ0 + w0θ0/d1) 3
(
−1
)
= 2 4πd1/w0q0λ + w0q0λ /4πd1
It can be shown that formulas (39), (40) and (40′) will be valid for measurements at white beams in the respective slit geometry, provided that λ is replaced with λm. Estimate the possible gain in luminosity, when small angle Soller collimators are used. To this effect, note that each channel of Soller collimator with a length Lch may be regarded as a double diaphragm system with slits of width wch. Proceeding from formula (3), one has the following flux after the Soller collimator designed [4] with traps for neutrons and with the full channel width 3wch:
Φс(λ ) ∝ Ta
2 d λ max ,0 q0 ⎛ Δqm ⎞ ⎟⎟ 3. = c εc ⎜⎜ dc,0 λ max qm ⎝ Δq0 ⎠
3.
(39)
When d1 c w0θ0,
3(w0) ≅ (2w0θ0/d1)3 = (w0q0λ /2πd1)3.
(40)
To assess the reference luminosity with the Soller collimation of the monochromatic beam, we substitute q¼ q0, Δq ¼ Δq0, εc ¼1, dc ¼w0θ0 into formula (29′) and obtain
3(w0) = (w0θ0/dc,0)3 = (w0q0λ /4πdc,0)3.
(40′)
As well as in the case of the double diaphragm collimation, the quantity 3 is assumed to be defined for the beam limited at the sample position with a slit of height w0.
201
2 wch dc , (L s + L ch )L ch 3wch
(41)
where Ta is the beam attenuation in the wafers of silicon or other weakly absorbing material, from which the Soller collimator with a total width dc is assembled. According to the same formula (3), the flux behind the two diaphragms with slits of equal width d is
Φd(λ ) ∝
d2 (L s + L )L
(42)
(we assume that the first slit is at the same distance Ls from the source as the Soller collimator). Hence the flux ratio for the beams with the same divergence Δθ = wch/( 6 Lch ) = d/( 6 L ) is
Φс(λ ) 1 d ⎛ L⎞ = Ta c ⎜ 1 + ⎟. 3 d⎝ Ls ⎠ Φd(λ )
(43)
The flux ratio exceeds unity for sufficiently small distance from the source and sufficiently high resolution, so that the value of d = 6 LΔθ is small in comparison with dc.
5. Luminosity class The neutron flux for different measurement conditions has been shown in the previous Section to be proportional to the reference luminosity defined for standard measurement conditions. Thus, the luminosity of a reflectometer can be characterized by a single number: the reference luminosity 3 . In principle, one and the same reflectometer may be characterized by several values of the reference luminosity, one for each mode of operation: with monochromatic and white beams, with non-polarized and polarized neutrons, with small and large samples. The reference luminosity can certainly be used to compare the operating reflectometers. Moreover, fixing the standard measurement conditions, one can classify neutron reflectometers according to luminosity for each designed mode of operation. To the effect, the following definition of luminosity class of reflectometers is suggested here:
CR = log10 3,
(44)
where 3 is specified by the assigned parameters: q0 ¼1 nm 1, Δq0 ¼0.01 nm 1, λmax,0 ¼1 nm, dc,0 ¼1 mm. When indicating the luminosity class of the reflectometers for the mode of measurements with small samples, it is sufficient to specify the size w0 of the square surface of the standard sample, e.g., CR(10 mm). The choice of the values of the standard quantities is defined by the requirement of the convenience of their usage. Say, the resolution Δq0/q0 ¼ 0.01 is rarely achieved in practice. More often the reflectometers are built for measurements with moderate and low resolution. In this case along with the luminosity class CR, one can point out the reference luminosity of the reflectometer for a typical resolution Δq at the momentum transfer q0. This value is easy to assess multiplying 3 by (Δq/Δq0 )3 in measurements with double diaphragm collimation and by (Δq/Δq0)2 in measurements with Soller collimation of the beam.
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As an example, assess the luminosity class of the neutron reflectometer NR-4M at the thermal beamline 13 of the reactor WWR-M (Gatchina) for each of the 4 modes of its operation: with monochromatic and white beams, with non-polarized and polarized neutrons. All assessments are made on the basis of published experimental data [8]. In all modes of operation the beam is collimated by two diaphragms with the slits of width d1,e ¼d2,e ¼0.2 mm (εg,e ¼1); the distance between the diaphragms is 1155 mm, and according to (5) the beam divergence is Δθe ¼0.07 mrad. For the operation mode with a monochromatic non-polarized beam (λ ¼0.19 nm, Δλe = 0.074λ/(2 2 ln 2 ) = 0.006 nm, Φe ¼ 200 s 1), with formulas (25) and (44) we find the reference luminosity 3 ¼ 113 s 1 and the luminosity class CR ¼2.1. For the operation mode with a monochromatic polarized beam (λ ¼0.20 nm, Δλe = 0.065λ/(2 2 ln 2 ) = 0.0055 nm, Φe ¼110 s 1) the same formulas lead to 3 ¼78 s 1 and CR ¼ 1.9. Surprisingly, the working wavelengths 0.19 nm and 0.20 nm practically coincide with the wavelength 0.189 nm, at which the product A(λ)λ3 is maximum (according to the non-polarized neutron spectrum). Even though the spectral flux density A(λ) is maximum near 0.1 nm, an ‘intuitive’ choice of the wavelengths as a compromise between resolution and intensity turned out to be optimal. The relative resolution Δq/q and the optimality εd for measurements with the monochromatic non-polarized beam are represented in Fig. 1 as functions of q¼4πθ/λ. The ratio Δq/q decreases with the growth of the glancing angle θ. The measurements are optimal (εd ¼1) at the glancing angle 0.08°, εd falls to 0.17 at 0.5° (q¼0.58 nm 1), to 0.042 at 1° (1.15 nm 1) and to 0.011 at 2° (2.3 nm 1). It means that the flux at the sample may be increased, accordingly, in 6, 24 and 90 (! ) times without any loss in resolution. So considerable a loss in intensity is, to a large extent, due to the use of collimation diaphragms with the slits of constant width. To receive evidence, calculate the fluxes with variable slit widths. When Δλ/λ is constant, varying the slit widths proportionally to the glancing angle θ, one obtains the same values of Δq/q and εd for all q. It is a consequence of the proportionality of Δθ and the slit widths: d1 = d2 = 6 3Δθ . In particular, εd ¼ 1 for all q with Δθ = 2 θ Δλ /λ . In this case for measurements with the monochromatic non-polarized beam we obtain Δq/q ¼0.07. A more acceptable resolution Δq/q ¼0.04, close to the reference value Δq/ q ¼0.0315 in Fig. 1 at large q, is achieved with Δθ = 0.557 2 θ Δλ /λ , when εd ¼ 0.78. the slit width is proportional to q and increases up
Fig. 1. The relative resolution Δq/q and the optimality εd as functions of q for measurements at the reflectometer NR-4M (reactor WWR-M, Gatchina) with the monochromatic non-polarized beam collimated by the slits with a fixed width 0.2 mm.
Fig. 2. The ratio of the flux for the diaphragm slits with a variable width (see the text) to the flux for the slits with a fixed width (0.2 mm) as a function of q for measurements with monochromatic neutrons at the reflectometer NR-4M. Due to the variation of the slit widths, Δq/q ¼0.04 and εd ¼0.78 for all q (when the slit widths are fixed, these quantities depend on q as shown in Fig. 1).
to 2.5 mm at q ¼2.3 nm 1. The ratio of the flux for the slits with a variable width to the flux Φe ¼ 110 s 1 for the slits with a fixed width 0.2 mm is represented in Fig. 2. It is to be emphasized that the increase in the flux (by two order of magnitude at large q!) is due to optimization of measurements only: the reference luminosity (luminosity class) of the reflectometer remains the same. At small q the flux is smaller than Φe because of reduction of the slit widths below 0.2 mm to retain the ratio Δq/q ¼0.04. To work with white beams at the reflectometer NR-4M, a single disk chopper is used with TOF resolution Δλ(1) ¼0.006 nm, mean λm ¼ 0.168 nm and maximum wavelength wavelength λmax ¼0.632 nm. The initial flux for the white non-polarized beam (0) ¼15,000 s 1; according to formula (18) the flux after the is Φint ,e (1) ¼ 142 s 1, and according to formula (31) the rechopper is Φint ,e ference luminosity is 3 ¼35 s 1, the respective luminosity class (0) being CR ¼1.5. The initial flux for the white polarized beam is Φint ,e (1) ¼5600 s 1, hence Φint ¼53 s 1, 3 ¼ 13 s 1, CR ¼1.1. ,e Calculated Δq/q and εd (solid curves) for TOF measurements at the glancing angles 0.25° and 0.5° are represented in Fig. 3 as
Fig. 3. The relative resolution Δq/q and the optimality εd of TOF measurements at the reflectometer NR-4M (reactor WWR-M, Gatchina) with non-polarized beam at the glancing angles 0.25° and 0.5° as functions of q. Solid curves: d1 ¼ d2 ¼ 0.2 mm, Δλ(1) ¼0.006 nm (designed parameters for all glancing angles). Dotted curves: for optimized parameters d1 ¼ d2 ¼0.4 mm, Δλ(1) ¼0.0044 nm (θ ¼0.25°) and d1 ¼d2 ¼ 0.6 mm, Δλ(1) ¼0.0033 nm (θ¼ 0.5°).
N.K. Pleshanov / Nuclear Instruments and Methods in Physics Research A 834 (2016) 197–204
203
functions of q¼ 4πθ/λ for the wavelengths from 0.08 nm to λmax. The relative resolution is worse at larger q. The dependences εd(q) at different angles overlap and merge into one (solid) curve. The measurement optimality εd falls to 0.03 at q ¼1.3 nm 1. For the mean wavelength λm and the glancing angles 0.25° and 0.5°, reεd ¼ 0.4, Δq/q ¼0.041 and spectively, qm ¼0.326 nm 1, 1 qm ¼ 0.65 nm , εd ¼ 0.12, Δq/q ¼0.036. To optimize the TOF measurements at the glancing angles 0.25° and 0.5°, require that at the wavelength λm: (a) εd ¼1 and (b) Δq/ q ¼0.04 (0.25°) and Δq/q ¼0.03 (0.5°). The parameters of the beams can be obtained from (23) and (5):
Δλ (1) =
1/3 λ(Δq/q), Δθ =
2/3 θ (Δq/q), d1 = d2 =
6 3Δθ .
(45)
Respectively, Δλ ¼0.0044 nm, d1 ¼d2 ¼0.4 mm (0.25°) and Δλ(1) ¼0.0033 nm, d1 ¼d2 ¼0.6 mm (0.5°). According to (16), the integral fluxes at the sample increase in 3 and 5 times. With the resolution slightly relaxed, the gain in intensity reaches one order of magnitude. Again the increase in the flux is achieved by optimization of measurements; the reference luminosity (luminosity class) of the reflectometer does not change. The results of calculations of Δq/q and εd with the optimized parameters of the beams are represented with dotted curves in Fig. 3. Thus, the luminosity class defines only the potential of the reflectometer; to which extent this potential can be used depends on designed-in technical options. The above example with the reflectometer built in 80's seems to be quite instructive. More technical options are inserted into modern reflectometers, which can be used to optimize measurements. One of such options is the use of choppers with two disks [2]. Relationship (22) can be used to compare the integral fluxes for the choppers with one and two disks. The integral fluxes at the sample will be equal, provided that (Δλ/λ) ¼ Δλ(1)/λm. If Δθ is chosen according to formula (23), we obtain for a chopper with two disks that εd ¼1 at all wavelengths, and Δq/q is the same at all wavelengths and coincides with Δq/q for a single disk chopper at λ ¼ λm. In particular, Δq/q ¼0.03 for the optimized beam parameters obtained above for the TOF measurement at θ ¼ 0.5°. Although the measurements with a single disk chopper are not optimal at all q (see the corresponding dashed curve εd in Fig. 3), the total flux at the sample is the same as that with the double disk chopper, when εd ¼1. The reason is that a decrease in the flux at λ 4 λm (at small q) in comparison with the flux after the double disk chopper is compensated by an increase in the flux at λ o λm (at large q). Such redistribution of the flux is accompanied by improvement of the resolution at the wavelengths λ 4 λm (Δq/ q o0.03) worsening the resolution at the wavelengths λ o λm (Δq/ q 40.03). Since the neutron reflectivity, as a rule, falls down as q 4, or even faster because of interfacial roughness, the redistribution of the flux with an increase in the number of neutrons reflected with larger q may give priority to the single disk chopper with a variable slit width, i.e. when the TOF resolution can be tuned as required for a certain resolution Δq. The task of finding an appropriate constructional solution in this case seems to be much simpler than a formidable task of building double disk choppers with a TOF resolution tuned by changing the distance between the disks. In particular, a chopper made of two closely spaced rotating disks with a tunable phase of rotation (‘tuned-phase chopper’) is equivalent to a single disk chopper with a variable slit width. Indeed, a constant phase shift in rotation of one of the disks will decrease the efficient slit width of the chopper by reducing the overlap of the opposite slits on the rotating disks and will improve the TOF resolution. In the same manner the drum chopper [4] suggested for fan beam TOF measurements can also be made with a tuned-phase variable resolution. (1)
Fig. 4. A scheme of the fan beam reflectometry with monochromatic neutrons. Solid and dashed lines show extreme trajectories of neutrons traversing two slits with widths d1 and d2 {d1 and registered within the spatial resolution δdet of PSD. Trajectories of neutrons without the sample and trajectories of neutrons reflected from the sample are shown. Neutrons with different glancing angles are registered in different parts of PSD.
It is impossible in one paper to encompass all measurement schemes and describe all specific details in classification of the reflectometers according to the luminosity. However, one may suppose that, in most cases, the generalization will make no difficulties. As an example, consider a scheme of measurements of specular reflection with a fan beam of monochromatic neutrons (a similar scheme was proposed earlier [4] for TOF measurements). In this scheme (Fig. 4) the slit with a width d1 is a virtual source, the slit near the sample with a width d2 { d1 and the horizontal coordinate resolution δdet of PSD work as two collimation slits. Then it is easy to evaluate the angular resolution 2 Δθ = (2Ldet )−1 (d22 + δdet ) /3 (Ldet is the distance from PSD to the slit near the sample) and optimize measurements. The luminosity gain
factor is Ω/Δθ, where Ω = (2L )−1 (d12 + d22) /3 is the fan angle, and instead of (25) one obtains
3=
Δq03 λ3 Ω Φe 2 24π 3 q0 εg,dΔλ eΔθe3
(46)
with the geometric optimality factor −1
εg,d = 2( d2/δdet + δdet/d2) .
(47)
For a given resolution Δq the flux in the fan beam will be maximum with δdet = d2 = 6 LdetΔθ , where Δθ = 2/3 θ Δq /q. Note that the spatial resolution of PSD must be sufficiently high; it is impossible to diminish a large δdet, whereas one can multiply increase a small δdet by summarizing the intensities in neighboring channels of PSD.
6. Conclusion In order to increase the flux at the sample without worsening the resolution, the possibilities to increase the vertical divergence during transportation of neutrons from the source are usually considered, incl. the beam condensation by a neutron guide tapering in height and the vertical focusing of the beam onto the sample [9,10]. At the same time, the additional possibilities to increase the flux without worsening the resolution by tailoring the beam optimal in the horizontal plane are largely ignored. In the papers [3,4] the luminosity of the reflectometer was coupled with the momentum transfer resolution Δq. Some possibilities to increase the luminosity without loss of resolution were revealed and new criteria of optimization of neutron
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reflectometers substantiated. The reference luminosity introduced in the present paper as a maximum flux for standard resolution parameters essentially differs from the luminosity defined as a flux for any given resolution Δq. The reference luminosity 3 characterizes the luminosity of the instrument by one number and can be used to classify operating reflectometers [11] and optimize designed reflectometers. To meet this end, the luminosity class CR ¼ log10 3 has been defined. It is to be emphasized that the luminosity class determines only the measurement potential of the reflectometer. The maximum flux available for a given resolution depends on the technical options designed in the reflectometer. The luminosity class CR of the neutron reflectometer NR-4M (reactor WWR-M, Gatchina) has been determined for 4 modes of operation: 2.1 (monochromatic non-polarized beam), 1.9 (monochromatic polarized beam), 1.5 (white non-polarized beam), 1.1 (white polarized beam). It has been shown that optimization of measurements will increase the neutron flux at the sample up to two orders of magnitude with monochromatic beams and up to one order of magnitude with white beams. Small angle Soller collimators have been shown (Section 4) to increase the luminosity of the instrument providing they are close to the (virtual) source. Even more promising may be their use in small angle neutron scattering stations, where the beams with large cross sections are used. Though, small angle Soller collimators are still to be worked out. Of practical interest may also be the reasoning (Section 5) in favor of using single disk choppers with a variable width of the slits as a trade-off increasing the luminosity for the white beam modes of the reflectometer instead of using
technically more complicated double disk choppers with a variable distance between the disks.
Acknowledgments The work was supported by the Federal target program (FTP) of Ministry of Education and Science of Russian Federation (project No. RFMEFI61614X0004).
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