- Email: [email protected]
TCD-extending the limits of neutron diffraction
14 May 2001
Physics Letters A 283 (2001) 243–248 www.elsevier.nl/locate/pla
TCD-extending the limits of neutron diffraction Eduard Bakshi 1 School of Biophysical Sciences and Electrical Engineering, Swinburne University of Technology, P.O. Box 218, Hawthorn 3122, Melbourne, Australia Received 4 October 2000; received in revised form 26 February 2001; accepted 5 April 2001 Communicated by J. Flouquet
Abstract The focus of the Letter is to discuss ways of overcoming hitherto fundamental limitations imposed by nature on our ability to study structures by diffraction. TCD, which is based on diffraction of particles in groups correlated in time and space, could be the answer to some of the problems. The physics of TCD is discussed, which is fundamentally different to all diffraction methods presently used. Neutrons are the most appropriate particles to “benefit” from TCD. 2001 Elsevier Science B.V. All rights reserved. PACS: 61.12.-q; 61.12.Ex; 61.12.Bt Keywords: Neutron diffraction; Neutron scattering; TCD
1. Introduction X-ray and neutron diffraction are the most commonly used experimental techniques for the study of atomic and molecular structures. However, X-rays and neutrons have fundamental limitations, which in turn impose limitations on the type of experiments carried out, the data obtainable and the interpretation of the data. The limitations that we are going to discuss are not commonly perceived as such, but rather the opposite, although on a close inspection the assertion that they are in fact limitations will become apparent. So what are these limitations? In the case of X-ray the major limitation is the fundamental relationship between the energy and the wavelength underpinned by Planck’s constant! ConE-mail address: [email protected] (E. Bakshi). 1 Fax: 3-9819-0856.
sider an X-ray diffraction experiment. The choice for wavelength is normally dictated by interatomic distances, but the energies of these X-rays are too small to perform an experiment with a sample which is anything but small. 2 In the case of neutrons the problem is different. The energies of thermal neutrons are similar to the energies of phonons in a solid, but the accessible range of thermal excitations is limited by the fundamental relationship between wavelength and energy (determined by the mass of the neutron). Experiments which rely on the use of very long wavelength neutrons are limited by the large absorption cross-section for neutrons in many materials. As far as hydrogen is concerned, all diffraction techniques have “failed” it. X-rays cannot “see” hydrogen well because the X-ray scattering cross-section for an atom 2 This problem is further exacerbated for materials with high
electron density.
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 2 3 9 - 0
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E. Bakshi / Physics Letters A 283 (2001) 243–248
is proportional to the square of the electron density of the atom. For neutrons, the thermal neutron scattering cross-section for hydrogen is by far the largest, but only two percent of it contributes to the coherent scattering cross-section. The rest is responsible for incoherent background. 3 The answer to these seemingly unrelated problems can be found in “time correlated diffraction”. The mathematics of TCD has already been described [1]. Here the physics of the technique will be explored as well as its limitations.
2. Discussion on TCD All diffraction techniques are based on the principle that each particle in a diffraction beam is diffracted by a structure entirely independently [2]. The result of diffraction can be broadly divided into two parts: coherent and incoherent. The coherent part carries all the structural information, whereas the incoherent component has no dependence on the reciprocal lattice vector and therefore carries no structural information. 4 On the other hand, if periodic density variation for the particles is introduced into a particle beam, then the incoherent scattering can “come out” with the structural information encoded in it [1]. The experiment can be based on the following principle. The first step is to impose a periodic density variation on a beam of particles, followed by a diffraction experiment, which in turn is followed by analysis of the residual density variation. There are a number of ways to impose periodic density variation on a particle beam; the simplest one (at least from a mathematical point of view) is to modulate it. The analysis of residual density variation can also be done with a second modulator (demodulator).
3 In the case of biological systems the incoherent background is
so large that the only way to perform successful experiments is to substitute hydrogen for deuterium. This in itself is a difficult task and also there is the danger of modifying the bond and therefore changing the structure. 4 It is important to note that incoherent scattering does carry certain types of non-structural information but it is of a limited nature and therefore often disregarded.
In Fig. 1 a possible layout of the experimental set-up is shown. The most appropriate particle for this type of experiment is a neutron. 5 The experiment could work as follows: A monochromatic, homogeneous and collimated beam of neutrons emerges from a source. These neutrons subsequently pass through the high-frequency neutron beam modulator. (The details regarding the actual frequency will be discussed later in this Letter.) Let us assume that the modulator operates in such a way that the transmission of the neutrons can be described by ∼ cos2 (ωm t), 6 where ωm is the modulation frequency. Following the initial modulation, the neutron beam is passing through a sample, where the scattering process takes place. The scattered neutrons in turn pass through the second modulator, which operates at the same frequency as the first one, 7 and they end up on the neutron detector. The time delay (t) between the operation of the modulators is controlled, and is equal to the time-offlight for the neutrons between the modulators (tfl ) (strictly speaking, t = tfl + πn/ωm , where n = 0, ±1, ±2, . . .). The result of the experiment can be described in terms of a time-averaged 8 neutron incoherent 9 scattering cross-section. For a simple case of a structure with one nucleus per lattice point, the calculated cross-
5 Here we are going to take in to account only spin incoherent and isotope incoherent scattering cross-sections for neutrons. The experiment can be also be built for X-rays, but its use would be restricted to mixed systems such as mixed compounds and alloys. 6 Here the choice of the function is dictated simply by mathematical considerations, i.e., the simplicity of the derivation. In fact it can be any other positive periodic function. It is likely that in the case of a real experiment the choice of function will be different, dictated by the optimum experimental set-up and its inherent limitations. 7 The frequency is the same only for elastic scattering experiments. The transmission function does not have to be the same. It has been shown [1] that the most appropriate function for the experimental set-up and conditions described here is a square-wave function. 8 The time variable will be present in a calculated cross-section, due to modulation, therefore the only meaningful way to describe the result of the experiment will be in terms of time-averaged quantity. 9 This Letter deals only with the effect of correlation on elastic incoherent scattering. The effect of correlation on inelastic scattering is in itself a separate subject, outside the scope of this Letter.
E. Bakshi / Physics Letters A 283 (2001) 243–248
245
Fig. 1.
section is [1] 1 2 2 (1) 1 + δ(κ m − τ ) , b − b 4 where b is the scattering length of a nucleus, and the brackets denote the average value of its distribution in the structure; τ is the reciprocal lattice vector of the structure; κ m is the modulated scattering vector, which is defined as follows: κ m = κ m − κ m ,
(2)
κ m
where κ m and are the modulated wave vectors defining the direction of the modulated wavefronts before and after the scattering event respectively, and having a magnitude of 10 |κ m | = 2π/λm ,
(3)
where λm is the modulated wavelength. In the specific case of the modulated wave vector being parallel to the direction of the travelling neutrons, the λm can be expressed as λm = 2πh/(mλωm ),
(4)
where m and λ are the mass and the DeBroglie wavelength of a neutron. It is useful to express the λm in terms of the energy of a neutron (ε): λm = (2π/ωm )(2ε/m)1/2. 10 In the case of elastic scattering, |κ | = |κ |. m m
(5)
Let us investigate expression (1). This cross-section is dependent on the reciprocal lattice vector of the structure (τ ), and therefore carries information on the structure. It is important to remind oneself that this is the incoherent crosssection, which normally does not carry any structural information. The normal coherent scattering cross-section carries information on the distribution of the average scattering lengths b 2 . Cross-section (1) carries information on b2 − b 2 , but since 2 2 b − b 2 = b − b , it carries information on the deviation of the scattering length from its average value. The b 2 and (b − b )2 contain complementary information on the structure. It is of particular interest when the b 2 is small compared to b2 . Then cross-section (1) will carry information on b2 . An example would be an experiment with a structure consisting of hydrogen. The coherent scattering cross-section of hydrogen (4πb 2 ) is less then 2 barn, on the other hand, the total scattering cross-section (4πb2 ) is ∼ =82 barn (far exceeding that of any other nucleus). Therefore for such a system b 2 b2 , and cross-section (1) will carry information primarily on b2 . The other important property of cross-section (1) is its dependence on a modulated scattering vector
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E. Bakshi / Physics Letters A 283 (2001) 243–248
(κ m ). The relationship between κ m , κ m , κ m and λm (as defined by (2) and (3)) is exactly the same as it would be for “normal” κ , κ , κ and λ. On the other hand, the relationship between energy (ε) and λ, is different to the relationship between ε and λm . The DeBroglie expression λ = h/(2εm)1/2
(6)
“imposes” a rigid correlation between the energy and the wavelength of a particle. It can be compared with (5), where the relationship between λm and ε is largely determined by ωm , and can be varied by the experimental set up. It is possible to visualise experiments in which neutrons with energy ε and wavelength λ are modulated to produce λm (λm λ) and then diffracted by a structure, carrying the information on the structure by λm “in to” the incoherent scattering cross-section. It is important to recognise that although the information is carried by λm all other processes, such as absorption, will happen according to the energy (ε).
3. Experimental considerations The resolution of the experiment is a function of the modulated wavelength (λm ). The λm , according to (5), is a function of modulation frequency and the energy of the neutrons. The range for λm is dictated by the techniques available for neutron beam modulation [3–9]. For thermal neutrons, within the limitations imposed by present technology, the limit for λm is between 1,000 and 10,000 Å. This seems to suggest that even within the present technology, the technique can be used to study microstructures, to overlap with light scattering and optical microscopy. However, for practical realization of the experiment the other important factor has to be taken into account and that is the degree of monochromaticity of the initial beam. It is clear that the initial spread in wavelength λ will eventually lead to the nullifying of the neutron density variation (produced by modulation). Therefore the dimensions of the experimental setup (in this case, the distance traveled by the neutrons between the modulators, d) are limited by this factor.
This distance is approximately equal to d ≈ (λm /2)(λ/λ).
(7)
This would seem to impose a severe limitation on the size of experimental setup, and restrict the technique to long λm , with a high degree of monochromaticity for the initial beam. But this is not necessarily the case. Until this point, it has been assumed that the direction of the modulated wavefront and the direction of the travelling neutrons is the same. But it does not have to be. It is important to remember that we are dealing here with incoherent scattering and therefore, strictly speaking, the direction of the travelling neutrons is not important. What is important is the direction of the modulated wavefront as per (1) and (2). By making the direction of the modulated wavefront different to the direction of the neutrons themselves, the problem associated with (7) can be largely overcome. This can be illustrated by the following example. The example itself is only to illustrate the geometrical concept and should not be considered as a real experiment (although, in principle, it can be built and tested, which could be of interest). Let us modulate a neutron beam by means of reflection from a surface, as shown in Fig. 2. A polarised monochromatic neutron beam (described by wavevector κ ) is scattered from the surface (A) undergoing total external reflection at a critical angle αc . The surface A is a thin film, the magnetic state of which can be switched between two different states at a high rate. (By state, we refer to a spin-state, 11 or to the type of magnetic order/disorder, or simply to the direction of magnetisation in space.) As a consequence, the refractive index of the surface varies in time, and so does the critical angle αc . The time-changes to the critical angle will result in the reflected neutron beam being modulated in time. 12
11 As an example, the family of compounds exhibiting LISCO (light-induced spin-crossover) effect, normally would have a metal ion/s, which would change its spin-state as a result of the shot pulse of light of a certain energy. The spin-state of this metal ion/s can be changed from high to low as well as from low to high, at a high rate. In the high spin-state, some of the compounds exhibit the ordered, antiferromagnetic phase [12–14]. 12 It should be set up for total external reflection for the larger critical angle of the states.
E. Bakshi / Physics Letters A 283 (2001) 243–248
247
Fig. 2.
The direction of individual neutrons reflected from the surface is described by the wave-vector κ . On the other hand, κ m can be described in terms of κ m⊥ and κ m|| . It is easy to see from the geometry that λm⊥ = λm sin(αc ),
(8a)
λm||
(8b)
= λm cos(αc ).
For all materials, the critical angle for thermal neutrons αc is only of the order of 10 of arc [15,16], and therefore the geometrical factor (sin(αc )) is of the order of 10−3 . Therefore λm⊥ ≈ λm /103 ,
(9a)
λm||
(9b)
≈ λm .
The reflected modulated wavefront is parallel to the surface A, and is described by the modulated wavevector κ m⊥ . The wavelength of the reflected modulated wavefront (λm⊥ ) is three orders of magnitude shorter compared to λm . κ m⊥ and λm⊥ are the only parameters relevant to TCD (see (2) and (3)), and expressions (4) and (5) should be modified accordingly. It is important to realise that κ m⊥ and λm⊥ describe the “correlated behavior” of neutrons in “groups”, the propagation of the groups in relation to each other. There are no neutrons travelling in the κ m⊥ direction, neutrons
can only travel in the κ direction. The only entity travelling in the κ m⊥ direction is the “correlation” between groups of neutrons. It is important to visualise the structure of the modulated beam. The neutrons will be travelling along the beam in the κ direction, the modulated wavefront (“correlation”) will be travelling across the beam in κ m⊥ , almost perpendicular to κ . The speed of propagation of the modulated wavefront is only a fraction of the speed of the neutrons, reduced by the same geometrical factor sin(αc ). Coming back to the question of dimensions of the experimental setup (7), it is easy to see that since κ defines the direction of travelling neutrons and κ ||κ m , the only relevant parameter for d is λm . Eq. (7) can be rewritten as follows: d ≈ (λm /2)(λ/λ).
(10)
The difference of three orders of magnitude between λm⊥ and λm would greatly reduce the constraints on the experiment. The other factor to consider is the effect of modulation on monochromaticity. This effect which relates to Heisenberg’s uncertainty principle, has been extensively discussed elsewhere [10,11]. The problem becomes apparent at high ωm at which the λm is approaching λ. This latest problem can also be largely overcome by the method discussed above, since TCD
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E. Bakshi / Physics Letters A 283 (2001) 243–248
is concerned with the size of λm⊥ , and at the same time λm can be kept sufficiently large compared to λ. 4. Possible applications The technique could find applications in many areas of neutron diffraction. For example, it could be used in the areas where long wavelength neutrons are required (at present, the domain of small and ultra-small angle scattering instruments). It could extend the use of neutrons all the way to the optical region, and fill the gap which traditional methods cannot access. It could also be used to study materials in which incoherent scattering dominates the scattering process, such as in biological macromolecules. The other possible application is in the field of magnetically disordered or “strangely ordered” systems in which the time-averaged or space-averaged magnetic moment per site is small or equal to zero. A good example of this is a paramagnetic system, in which magnetic scattering is entirely incoherent. This is because in any chosen direction in a lattice, the averaged magnetic scattering length bm is equal to zero. But the average of the square of the magnetic scattering length 2 will not be equal to zero. bm It may also find applications in the field of inelastic scattering. This is because the standard inelastic scattering experiment is limited by the fundamental relationship between the energy and the wavelength of neutron (6). TCD has no such limitations.
Acknowledgements The present work was supported by AINSE for a number of years. My sincere thanks to the Neutron Diffraction Group at ILL for inviting me for a longterm stay and the many interesting discussions which contributed to this work. I would like to thank Prof. Otto Scharpf for his enthusiasm and encouragement. I would also like to thank Dr. Peter Cadusch (SUT) for many discussions and being a good soundingboard, Prof. Tony Klein (Melbourne University) for his interest and encouragement.
References [1] E.N. Bakshi, Meas. Sci. Technol. 1 (1990) 1123. [2] G.A. Biberman et al., Dokl. Akad. Nauk 26 (1949) 185. [3] M. Hart, D.P. Siddons, in: Neutron Interferometry, Proc. Int. Workshop ILL, 5–7 June, 1978. [4] G. Badurek, G.P. Westphal, Nucl. Instr. Meth. 128 (1975) 315. [5] A.D. LeGrand et al., Nucl. Instr. Meth. A 275 (1989) 442. [6] P. Mikula et al., Phys. Stat. Sol. 26 (1974) 691. [7] R. Hock, J. Kulda, Nucl. Instr. Meth. A 338 (1994) 38. [8] P. Mikula et al., Nucl. Instr. Meth. 137 (1976) 23. [9] J. Kulda et al., Nucl. Instr. Meth. 180 (1981) 89. [10] R. Gahler, R. Golub, Z. Phys. B 56 (1984) 5. [11] J. Felber et al., Phys. Rev. A 53 (1) (1996) 319. [12] P. Gutlich, A. Hauser, Coord. Chem. Rev. 97 (1990) 1. [13] P. Gutlich et al., Angew. Chem. Int. Ed. Engl. 33 (1994) 2024. [14] H. Toftlund, Coord. Chem. Rev. 94 (1989) 67. [15] H. Dosch et al., Rev. Sci. Instr. 63 (12) (1992) 5533. [16] H. Dosch, Physica B 192 (1993) 163.
Physics Letters A 283 (2001) 243–248 www.elsevier.nl/locate/pla
TCD-extending the limits of neutron diffraction Eduard Bakshi 1 School of Biophysical Sciences and Electrical Engineering, Swinburne University of Technology, P.O. Box 218, Hawthorn 3122, Melbourne, Australia Received 4 October 2000; received in revised form 26 February 2001; accepted 5 April 2001 Communicated by J. Flouquet
Abstract The focus of the Letter is to discuss ways of overcoming hitherto fundamental limitations imposed by nature on our ability to study structures by diffraction. TCD, which is based on diffraction of particles in groups correlated in time and space, could be the answer to some of the problems. The physics of TCD is discussed, which is fundamentally different to all diffraction methods presently used. Neutrons are the most appropriate particles to “benefit” from TCD. 2001 Elsevier Science B.V. All rights reserved. PACS: 61.12.-q; 61.12.Ex; 61.12.Bt Keywords: Neutron diffraction; Neutron scattering; TCD
1. Introduction X-ray and neutron diffraction are the most commonly used experimental techniques for the study of atomic and molecular structures. However, X-rays and neutrons have fundamental limitations, which in turn impose limitations on the type of experiments carried out, the data obtainable and the interpretation of the data. The limitations that we are going to discuss are not commonly perceived as such, but rather the opposite, although on a close inspection the assertion that they are in fact limitations will become apparent. So what are these limitations? In the case of X-ray the major limitation is the fundamental relationship between the energy and the wavelength underpinned by Planck’s constant! ConE-mail address: [email protected] (E. Bakshi). 1 Fax: 3-9819-0856.
sider an X-ray diffraction experiment. The choice for wavelength is normally dictated by interatomic distances, but the energies of these X-rays are too small to perform an experiment with a sample which is anything but small. 2 In the case of neutrons the problem is different. The energies of thermal neutrons are similar to the energies of phonons in a solid, but the accessible range of thermal excitations is limited by the fundamental relationship between wavelength and energy (determined by the mass of the neutron). Experiments which rely on the use of very long wavelength neutrons are limited by the large absorption cross-section for neutrons in many materials. As far as hydrogen is concerned, all diffraction techniques have “failed” it. X-rays cannot “see” hydrogen well because the X-ray scattering cross-section for an atom 2 This problem is further exacerbated for materials with high
electron density.
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 2 3 9 - 0
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E. Bakshi / Physics Letters A 283 (2001) 243–248
is proportional to the square of the electron density of the atom. For neutrons, the thermal neutron scattering cross-section for hydrogen is by far the largest, but only two percent of it contributes to the coherent scattering cross-section. The rest is responsible for incoherent background. 3 The answer to these seemingly unrelated problems can be found in “time correlated diffraction”. The mathematics of TCD has already been described [1]. Here the physics of the technique will be explored as well as its limitations.
2. Discussion on TCD All diffraction techniques are based on the principle that each particle in a diffraction beam is diffracted by a structure entirely independently [2]. The result of diffraction can be broadly divided into two parts: coherent and incoherent. The coherent part carries all the structural information, whereas the incoherent component has no dependence on the reciprocal lattice vector and therefore carries no structural information. 4 On the other hand, if periodic density variation for the particles is introduced into a particle beam, then the incoherent scattering can “come out” with the structural information encoded in it [1]. The experiment can be based on the following principle. The first step is to impose a periodic density variation on a beam of particles, followed by a diffraction experiment, which in turn is followed by analysis of the residual density variation. There are a number of ways to impose periodic density variation on a particle beam; the simplest one (at least from a mathematical point of view) is to modulate it. The analysis of residual density variation can also be done with a second modulator (demodulator).
3 In the case of biological systems the incoherent background is
so large that the only way to perform successful experiments is to substitute hydrogen for deuterium. This in itself is a difficult task and also there is the danger of modifying the bond and therefore changing the structure. 4 It is important to note that incoherent scattering does carry certain types of non-structural information but it is of a limited nature and therefore often disregarded.
In Fig. 1 a possible layout of the experimental set-up is shown. The most appropriate particle for this type of experiment is a neutron. 5 The experiment could work as follows: A monochromatic, homogeneous and collimated beam of neutrons emerges from a source. These neutrons subsequently pass through the high-frequency neutron beam modulator. (The details regarding the actual frequency will be discussed later in this Letter.) Let us assume that the modulator operates in such a way that the transmission of the neutrons can be described by ∼ cos2 (ωm t), 6 where ωm is the modulation frequency. Following the initial modulation, the neutron beam is passing through a sample, where the scattering process takes place. The scattered neutrons in turn pass through the second modulator, which operates at the same frequency as the first one, 7 and they end up on the neutron detector. The time delay (t) between the operation of the modulators is controlled, and is equal to the time-offlight for the neutrons between the modulators (tfl ) (strictly speaking, t = tfl + πn/ωm , where n = 0, ±1, ±2, . . .). The result of the experiment can be described in terms of a time-averaged 8 neutron incoherent 9 scattering cross-section. For a simple case of a structure with one nucleus per lattice point, the calculated cross-
5 Here we are going to take in to account only spin incoherent and isotope incoherent scattering cross-sections for neutrons. The experiment can be also be built for X-rays, but its use would be restricted to mixed systems such as mixed compounds and alloys. 6 Here the choice of the function is dictated simply by mathematical considerations, i.e., the simplicity of the derivation. In fact it can be any other positive periodic function. It is likely that in the case of a real experiment the choice of function will be different, dictated by the optimum experimental set-up and its inherent limitations. 7 The frequency is the same only for elastic scattering experiments. The transmission function does not have to be the same. It has been shown [1] that the most appropriate function for the experimental set-up and conditions described here is a square-wave function. 8 The time variable will be present in a calculated cross-section, due to modulation, therefore the only meaningful way to describe the result of the experiment will be in terms of time-averaged quantity. 9 This Letter deals only with the effect of correlation on elastic incoherent scattering. The effect of correlation on inelastic scattering is in itself a separate subject, outside the scope of this Letter.
E. Bakshi / Physics Letters A 283 (2001) 243–248
245
Fig. 1.
section is [1] 1 2 2 (1) 1 + δ(κ m − τ ) , b − b 4 where b is the scattering length of a nucleus, and the brackets denote the average value of its distribution in the structure; τ is the reciprocal lattice vector of the structure; κ m is the modulated scattering vector, which is defined as follows: κ m = κ m − κ m ,
(2)
κ m
where κ m and are the modulated wave vectors defining the direction of the modulated wavefronts before and after the scattering event respectively, and having a magnitude of 10 |κ m | = 2π/λm ,
(3)
where λm is the modulated wavelength. In the specific case of the modulated wave vector being parallel to the direction of the travelling neutrons, the λm can be expressed as λm = 2πh/(mλωm ),
(4)
where m and λ are the mass and the DeBroglie wavelength of a neutron. It is useful to express the λm in terms of the energy of a neutron (ε): λm = (2π/ωm )(2ε/m)1/2. 10 In the case of elastic scattering, |κ | = |κ |. m m
(5)
Let us investigate expression (1). This cross-section is dependent on the reciprocal lattice vector of the structure (τ ), and therefore carries information on the structure. It is important to remind oneself that this is the incoherent crosssection, which normally does not carry any structural information. The normal coherent scattering cross-section carries information on the distribution of the average scattering lengths b 2 . Cross-section (1) carries information on b2 − b 2 , but since 2 2 b − b 2 = b − b , it carries information on the deviation of the scattering length from its average value. The b 2 and (b − b )2 contain complementary information on the structure. It is of particular interest when the b 2 is small compared to b2 . Then cross-section (1) will carry information on b2 . An example would be an experiment with a structure consisting of hydrogen. The coherent scattering cross-section of hydrogen (4πb 2 ) is less then 2 barn, on the other hand, the total scattering cross-section (4πb2 ) is ∼ =82 barn (far exceeding that of any other nucleus). Therefore for such a system b 2 b2 , and cross-section (1) will carry information primarily on b2 . The other important property of cross-section (1) is its dependence on a modulated scattering vector
246
E. Bakshi / Physics Letters A 283 (2001) 243–248
(κ m ). The relationship between κ m , κ m , κ m and λm (as defined by (2) and (3)) is exactly the same as it would be for “normal” κ , κ , κ and λ. On the other hand, the relationship between energy (ε) and λ, is different to the relationship between ε and λm . The DeBroglie expression λ = h/(2εm)1/2
(6)
“imposes” a rigid correlation between the energy and the wavelength of a particle. It can be compared with (5), where the relationship between λm and ε is largely determined by ωm , and can be varied by the experimental set up. It is possible to visualise experiments in which neutrons with energy ε and wavelength λ are modulated to produce λm (λm λ) and then diffracted by a structure, carrying the information on the structure by λm “in to” the incoherent scattering cross-section. It is important to recognise that although the information is carried by λm all other processes, such as absorption, will happen according to the energy (ε).
3. Experimental considerations The resolution of the experiment is a function of the modulated wavelength (λm ). The λm , according to (5), is a function of modulation frequency and the energy of the neutrons. The range for λm is dictated by the techniques available for neutron beam modulation [3–9]. For thermal neutrons, within the limitations imposed by present technology, the limit for λm is between 1,000 and 10,000 Å. This seems to suggest that even within the present technology, the technique can be used to study microstructures, to overlap with light scattering and optical microscopy. However, for practical realization of the experiment the other important factor has to be taken into account and that is the degree of monochromaticity of the initial beam. It is clear that the initial spread in wavelength λ will eventually lead to the nullifying of the neutron density variation (produced by modulation). Therefore the dimensions of the experimental setup (in this case, the distance traveled by the neutrons between the modulators, d) are limited by this factor.
This distance is approximately equal to d ≈ (λm /2)(λ/λ).
(7)
This would seem to impose a severe limitation on the size of experimental setup, and restrict the technique to long λm , with a high degree of monochromaticity for the initial beam. But this is not necessarily the case. Until this point, it has been assumed that the direction of the modulated wavefront and the direction of the travelling neutrons is the same. But it does not have to be. It is important to remember that we are dealing here with incoherent scattering and therefore, strictly speaking, the direction of the travelling neutrons is not important. What is important is the direction of the modulated wavefront as per (1) and (2). By making the direction of the modulated wavefront different to the direction of the neutrons themselves, the problem associated with (7) can be largely overcome. This can be illustrated by the following example. The example itself is only to illustrate the geometrical concept and should not be considered as a real experiment (although, in principle, it can be built and tested, which could be of interest). Let us modulate a neutron beam by means of reflection from a surface, as shown in Fig. 2. A polarised monochromatic neutron beam (described by wavevector κ ) is scattered from the surface (A) undergoing total external reflection at a critical angle αc . The surface A is a thin film, the magnetic state of which can be switched between two different states at a high rate. (By state, we refer to a spin-state, 11 or to the type of magnetic order/disorder, or simply to the direction of magnetisation in space.) As a consequence, the refractive index of the surface varies in time, and so does the critical angle αc . The time-changes to the critical angle will result in the reflected neutron beam being modulated in time. 12
11 As an example, the family of compounds exhibiting LISCO (light-induced spin-crossover) effect, normally would have a metal ion/s, which would change its spin-state as a result of the shot pulse of light of a certain energy. The spin-state of this metal ion/s can be changed from high to low as well as from low to high, at a high rate. In the high spin-state, some of the compounds exhibit the ordered, antiferromagnetic phase [12–14]. 12 It should be set up for total external reflection for the larger critical angle of the states.
E. Bakshi / Physics Letters A 283 (2001) 243–248
247
Fig. 2.
The direction of individual neutrons reflected from the surface is described by the wave-vector κ . On the other hand, κ m can be described in terms of κ m⊥ and κ m|| . It is easy to see from the geometry that λm⊥ = λm sin(αc ),
(8a)
λm||
(8b)
= λm cos(αc ).
For all materials, the critical angle for thermal neutrons αc is only of the order of 10 of arc [15,16], and therefore the geometrical factor (sin(αc )) is of the order of 10−3 . Therefore λm⊥ ≈ λm /103 ,
(9a)
λm||
(9b)
≈ λm .
The reflected modulated wavefront is parallel to the surface A, and is described by the modulated wavevector κ m⊥ . The wavelength of the reflected modulated wavefront (λm⊥ ) is three orders of magnitude shorter compared to λm . κ m⊥ and λm⊥ are the only parameters relevant to TCD (see (2) and (3)), and expressions (4) and (5) should be modified accordingly. It is important to realise that κ m⊥ and λm⊥ describe the “correlated behavior” of neutrons in “groups”, the propagation of the groups in relation to each other. There are no neutrons travelling in the κ m⊥ direction, neutrons
can only travel in the κ direction. The only entity travelling in the κ m⊥ direction is the “correlation” between groups of neutrons. It is important to visualise the structure of the modulated beam. The neutrons will be travelling along the beam in the κ direction, the modulated wavefront (“correlation”) will be travelling across the beam in κ m⊥ , almost perpendicular to κ . The speed of propagation of the modulated wavefront is only a fraction of the speed of the neutrons, reduced by the same geometrical factor sin(αc ). Coming back to the question of dimensions of the experimental setup (7), it is easy to see that since κ defines the direction of travelling neutrons and κ ||κ m , the only relevant parameter for d is λm . Eq. (7) can be rewritten as follows: d ≈ (λm /2)(λ/λ).
(10)
The difference of three orders of magnitude between λm⊥ and λm would greatly reduce the constraints on the experiment. The other factor to consider is the effect of modulation on monochromaticity. This effect which relates to Heisenberg’s uncertainty principle, has been extensively discussed elsewhere [10,11]. The problem becomes apparent at high ωm at which the λm is approaching λ. This latest problem can also be largely overcome by the method discussed above, since TCD
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is concerned with the size of λm⊥ , and at the same time λm can be kept sufficiently large compared to λ. 4. Possible applications The technique could find applications in many areas of neutron diffraction. For example, it could be used in the areas where long wavelength neutrons are required (at present, the domain of small and ultra-small angle scattering instruments). It could extend the use of neutrons all the way to the optical region, and fill the gap which traditional methods cannot access. It could also be used to study materials in which incoherent scattering dominates the scattering process, such as in biological macromolecules. The other possible application is in the field of magnetically disordered or “strangely ordered” systems in which the time-averaged or space-averaged magnetic moment per site is small or equal to zero. A good example of this is a paramagnetic system, in which magnetic scattering is entirely incoherent. This is because in any chosen direction in a lattice, the averaged magnetic scattering length bm is equal to zero. But the average of the square of the magnetic scattering length 2 will not be equal to zero. bm It may also find applications in the field of inelastic scattering. This is because the standard inelastic scattering experiment is limited by the fundamental relationship between the energy and the wavelength of neutron (6). TCD has no such limitations.
Acknowledgements The present work was supported by AINSE for a number of years. My sincere thanks to the Neutron Diffraction Group at ILL for inviting me for a longterm stay and the many interesting discussions which contributed to this work. I would like to thank Prof. Otto Scharpf for his enthusiasm and encouragement. I would also like to thank Dr. Peter Cadusch (SUT) for many discussions and being a good soundingboard, Prof. Tony Klein (Melbourne University) for his interest and encouragement.
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