ПОВЕРХНОСТЬ. РЕНТГЕНОВСКИЕ, СИНХРОТРОННЫЕ И НЕЙТРОННЫЕ ИССЛЕДОВАНИЯ, 2014, № 10, с. 79-87
УДК 539.171.4:537.9
PRESENT STATUS AND PROSPECTS FOR SPIN-ECHO SMALL-ANGLE NEUTRON SCATTERING (SESANS) AT PIK NEUTRON SOURCE
© 2014 W. H. Kraan1, L. A. Akselrod2, Yu. O. Chetverikov2, S. V. Grigoriev23, E. V. Moskwin2, V. V. Piyadov2, Kyaw Thu Set3, A. A. Sumbatyan2, E. V. Velichko2, V. N. Zabenkin2
1Retired from Delft University of Technology, 2629 JB Delft, The Netherlands 2Petersburg Nuclear Physics Institute, 188300 Gatchina, St-Petersburg, Russia Petersburg State University, 198504, St-Petersburg, Russia 1E-mail: W.H.Kraan@tudelft.nl Received January 28.2014
We discuss the present status and the prospects of Spin-Echo Small-Angle Neutron Scattering (SESANS) against the background of the available expertise and the neutron source PIK at PNPI (Gatchina). Two options for SESANS instruments are reviewed: (1) monochromatic with п-flipping permalloy foils and (2) with adia-batic radio frequency spin flippers in a white beam, combined with time-of-flight data collection. A software tool for quantitative prediction of the technical properties of option (2) is developed. For both options, we show that a SESANS instrument which can compete with instruments elsewhere, is realistic. For option (2), we suggest a perspective of spin-echo-length range such that neutron interference experiments become feasible.
DOI: 10.7868/S0207352814100084
INTRODUCTION
SESANS (Spin-Echo Small-Angle-Neutron-Scattering) is a technique based on Neutron Spin Echo (NSE) developed in the period 1995-2005 in Delft [1-4]. After starting this technique it was soon internationally recognized and instruments (among others) at Indiana University [5, 6] and ISIS (Oxford) [7] followed. In view of the long standing expertise in polarized neutrons at PNPI (Gatchina) and in view of the construction of the new neutron source PIK it is the wish to realize a similar instrument at PNPI. Such an instrument aims at probing materials (from daily life all the way to advanced technological environments) at a scale of length in real space from 100 A up to the micrometer range. Contrary to the popular techniques SANS and SAXS which provide answers to materials in reciprocal space, SESANS has the advantage of giving answers in real space, in terms of a quantity called "spin-echo-length", denoted 5. Asking and answering questions in real space should give SESANS an advantage for users from disciplines less mathematically oriented than physics.
Extending the expertise in polarized neutrons, first experiments at the WWR-M reactor to make spinecho in a SESANS-type setup were done in 2009 [8]; real SESANS measurements in SiO2-microspheres with the aim to calibrate a basic SESANS setup were done in 2011 [9].
By the nature of the source PIK, an instrument taking the spectrum of this source for input could have the benefit of a high flux of neutrons in the wavelength range 5-10 A. Since the spin-echo-length 5 goes pro-
portional to the square of the neutron's wavelength, this opens the possibility to push the range of 5 of the instrument with "little" effort beyond the limits (1020 ^m) of similar instruments in the world. This high neutron flux also opens the outlook to real-time SESANS to follow evolution of processes in materials (flow patterns, boiling of fluids, drying of paint, processes in samples from earth oil fields).
In a neutron beam with wavelength 5-10 A and beyond a quite different type of experiments opens up. It can be shown [10] that the length 5 is identical with the separation achieved by the instrument between the "parallel" and "anti-parallel" state of the same neutron. Should these states be pushed apart to 100 ^m (0.1 mm), macroscopic quantum experiments are within sight [11].
PRINCIPLE OF SESANS
Fig. 1 shows that a SESANS setup is a NSE setup with precession regions shaped as parallelograms, with anti-parallel fields. The principle is explained in the caption. This is an alternative for the common way to measure Small-Angle Neutron Scattering (SANS).
The precession phases collected before (i = 1) and after (i = 2) scattering while the neutron traverses regions shaped as parallelograms with homogeneous induction B and length L are
= c X BL[ 1 + cot 0O Qi + O (02,04,...)]= c X BL + + (cX BL cot 0O )0, = cX BL + r0;-
y A
z
Fig. 1. Principle of SESANS: After being polarized in P, the polarization of the neutron beam is rotated over n/2 to the xy-plane. Then the polarization will precess in field regions I and II — in opposite directions because the fields are antiparallel. After another rotation over n/2 the neutrons pass through the analyzer A and are counted by detector D. The regions I and II are shaped as parallelograms with faces, making an angle 9q with the x-axis, rendering the precession angles on transmission strongly dependent on the travel direction of the neutron. When a scattering sample S is absent, the precession angles in regions I and II cancel. When a sample is present, scattering over even a small angle 9s disturbs the echo. This is measured and analyzed.
(i = 1, 2), as can be seen with elementary geometry. (The constant c equals 4n^nmn/h2 = 4.63 x 1014 T-1 • m-2, with
and mn neutron mass and magnetic moment, respectively.) 9, are the angles between the x-axis and the flying directions in the field regions. The term T9, is the angle labeling term. For rectangular field shape it reduces to zero in first order.
From Eq. (1) we can exactly calculate the offset in the NSE experiment. Suppose a neutron of wavelength X is scattered by 92 — 91 = 9s in the y-direction
(that is wave vector transfer Qy = — sin(0^2) «
~ 2nsin(9s/X), then the precession phase difference between the NSE-arms will be r9s. If we divide Qy out of the angle labeling term r9s (with approximation sin 9s = 9s), we get the quantity
S = rQJQy
(2)
of dimension length, identical with the "spin-echo-length" mentioned in the Introduction. Substituting r = cXBLcot90 gives for precession fields shaped as parallelograms:
8(00, L, B, X) =
— c X2 LBcot0o 2 n
(3)
P(S, Hsc) after the analyzer as a function of the field Hsc in a small coil inside one of the fields. (The field Hsc is not related with 5, as set by the instrument parameters). Scanning the parameter 5 is done by varying one of the parameters 90, L, B, X over the range which the design of the instrument allows.
The polarization P(5) (dropping subscript max) obtained in such a scan, in terms of the sample's properties £ and dZ/dD (total and differential macroscopic cross section), in the limit of vanishing thickness dt (such that Zdt < 1, which means only one scattering event) is [4]:
^ = i - s dt + ^r d-
S dt + d-l\\dS-cos ( Qy S)dQydQv k jJail
kC
(4)
where P0 is the polarization of the empty instrument and k0 = 2n/X. To describe a sample of finite thickness
t, we replace the expression
dP
P(5) - Po
Po
by the differen-
tial — and Eq. (4) takes the shape of a differential
equation. Integrating it over dt implies that multiple scattering is included. Then, Eq. (4) can be rewritten: P(5) = P0exp(-£t[1 - G(5)]), where
which depends on the instrument parameters 90, L, B and X.
Notice that the nominator and denominator in (2) don't depend on the specific direction 9X in region I, nor on the position on the y-axis of the neutron entering into that region. This means that all neutrons making up a parallel sub-beam of given 9; inside the incident beam and, idem, in the scattered beam have the same value for the quantity 6. This has the important consequence that good collimation of the incident and scattered beam is not required to set a specific value for 6.
In the practice of a SESANS measurement, at given setting of 6, one measures the maximum amplitude Pmax(6) of the damped oscillating polarization signal
G (S) = S- JJdl-cos ( QyS) dQydQz. S k0
(5)
G(6) is called "SESANS correlation function", which for systems with inhomogeneous density (for example, particulate systems) is related to the density correlation function p(r) [12, 13].
Because of its statistical character, a high resolution in 6 is often not necessary, especially for 6 larger than the typical length scales in the sample. In our instrument we aim at 5%.
The term T9,- in Eq. (1) arises in fact from the shaded areas in the precession regions in Fig. 1, the remaining parts are superfluous. This is the basis for several other options [4] to realize angle labeling. The op-
0
(a)
-L -
B ®
Г = cXBLcote0
(b)
L
: в = о
Г = 2clBLcote0
B
(c)
L
Г = 2cXBLcote0
Fig. 2. Options for angle labeling and expressions for the coefficient r in Eq. (1): a - the parallelogram separates a region with field from a region with B = 0; b - r is twice as high because the precessions (not B) at both sides of the diagonal n-flipping foil are opposite; c - the factor 2 is due to "zero field precession" (ZFP) [19] in the region between the adiabatic RF flippers denoted "RF" at twice the rate in the magnets around the RF coils. By stray fields the factor 2 is in practice less.
n
n
tions relevant here are (b) and (c) in Fig. 2. We assume that the magnetic fields in the instrument are along z (vertical). Then the scattering plane in option (b) is the vertical xz-plane; in (c) it is the horizontal xy-plane.
To realize precession phases and in the arms of a SESANS instrument with interesting range of spin echo length 5, Maxwell's laws present a problem: the neutron beam should pass through strong magnetic fields, with transitions through gradients rendering the field integrals in-homogeneous over the beam cross section. This will give undesired spread in the phases
and This problem can be overcome by shaping the precession devices exactly as option (a): bounding them by (neutron "transparent") aluminum foils carrying currents up to (say) 100 A and short-circuiting the generated magnetic flux by a soft-magnetic yoke. Then the gradients becom
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