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Towards interferometric Fourier spectroscopy
ELSEVIER
Physica B 213&214 (1995) 830--832
Towards interferometric Fourier spectroscopy H. Rauch Atominstitut der Osterreichischen Universithten, A-I020 Wien. Austria
Abstract
Most neutron interferometric experiments carried out up till now have dealt with fundamental physics applications. But it was known from the beginning that the interferometric method can open new horizons for solid state physics research as well. Samples with density or magnetic fluctuations cause inhomogeneous phase shifts which cause a measurable loss of contrast of the interference pattern. Phase topography can be realized without absorbing or scattering neutrons at the object. Very small energy changes (down to 10-1T eV) become measurable by means of the magnetic neutron Josephson effect. By measuring the neutron coherence ~unction, one gets direct access to the correlation function G(r, t) instead of measuring its Fourier transform S(Q, ~o). Various examples of related investigations are discussed together with their future perspectives. The essentials of future neutron Fourier spectroscopy are formulated.
1. Introduction
Since the invention of neutron interferometry [1], this technique has been used primarily for fundamental physics investigations [2-4], whereas applications for condensed-matter research have not been exploited extensively yet. Now neutron interferometry has been introduced to many laboratories and its operation becomes routine and, therefore, strong effort should be put onto its use for the broad field of condensed-matter research. In neutron interferometry the phase of the neutron wave becomes an observable which is influenced by any momentum or/and energy change the beam experiences during its interaction with the sample. Any phase change is measured by a superposition with a coherent reference beam not being affected by the sample. In many cases a much higher sensitivity than in usual spectrometry can be achieved because now the usual constraints do not exist any more; namely that the momentum tenergy) change AQ (AE) has to be larger than the momentum
(energy) width ~k (~E) of the beam. The new method shows many similarities to NMR and optical Fourier spectroscopy (e.g. Ref. [5]). The index of refraction n is given by the particle density N, the coherent scattering length be, the neutron wavelength )~ and the sample thickness D. This defines also the phase shift Z and the spatial shift A of the wave trains l . = ( n - - 1)kD= - N b c 2 D = A . k .
(1)
Any variation of this quantity representing an inhomogeneous phase shifter (powdered samples, magnetic domains, metal hydrites) cause a reduction of the interference contrast due to a spatial variation of the phase shift [6]. This produces results comparable with smallangle scattering investigations and provides the basis for phase topography [7], whose sensitivity is, roughly speaking, higher by a factor of 1000 ( ~ bc2/at; o-t being the total attenuation cross-section) than usual radiographic inspection methods.
0921-4526/95/$09.50 i~+: 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 2 9 4 - 4
H. Rauch / Physica B 213&214 (1995) 830-832 2. Basic relations
the sample (e.g. Ref. [9]). When the transmitted beam is included one gets the following momentum distribution:
The intensity of the beam in the forward direction behind the interferometer stems from wave functions originating from beam paths I and II, which are equal in amplitude and phase because they are reflected reflected transmitted (RRT) and transmitted reflected reflected (TRR), respectively:
Io ~c Iq%(0) + ~"~(,~)12 ~ 1 + IF(A)lcos (A .k)
(2)
where one beam is assumed to be phase shifted by an amount X = A.k compared to the reference beam. The wave function can be represented by a wave packet whose square value gives the momentum distribution function g(k). The autocorrelation function of the wave function defines the spatial coherence function which is directly related to the Fourier transform of the momentum distribution and which reduces the contrast at high interference order
F(A) = (~*(0)~(A)) = fy(k)eik'~dk.
831
(3)
Thus, the coherence function is a measurable quantity when the spatial phase shift is varied. In practice IF(0)I may differ from unity due to various imperfections which are unavoidable in any experimental set-up. Therefore, the normalized coherence function is used IF'(A)I = IF(A)I/IF(O)I. In the case of Gaussian momentum distributions centered around ko and having momentum widths 6k~, the coherence function also has a Gaussian shape whose widths define the coherence lengths A~ which fulfil the Heisenberg uncertainty relations
A~rki = ½.
gin(Q) = e z'oy(Q) + (1 - e-Z'°)(S(Q)* g(Q))
(4)
which gives, after some algebraic calculations, the coherence functions with and without sample: IFm(A)l = e sin + (1 - eS~°)p(A).
(5)
Ir(n)l
This equation shows that the normalized scatteringlength density (radial distribution function) can be obtained directly from the measurable coherence functions maintaining the advantage that a wide beam and a broad incident wavelength spectrum can be used. A typical arrangement for such a measurement is shown in Fig. 1. A very interesting application may be a direct measurement of the surface profile in neutron reflectometry, but other small-angle scattering phenomena and structural investigations may profit from this new technique too. Another useful method may come up using a doublehumped coherent incident beam described by stationary Schrrdinger-cat-like states. Such states have been identified recently when the wave trains of the coherent beams inside an interferometer have been shifted more than their coherence lengths [10]. In this case, spatially separated coherent packets exist behind the interferometer which exhibit a marked modulation in momentum space. The van-Hove correlation function G(r, t) has to be obtained as a Fourier transform of the scattering function S(Q, ~o) which has to be extracted from the measured spectroscopic data by deconvolution from the resolution function R(Q)
Ira(Q) = R(Q) * S(Q,(o).
(6)
3. Spatial Fourier spectroscopy Various spectrometric methods are used for the measurement of the momentum distribution function g(k). Eqs. (2) and (3) show that it can also be deduced from a measured coherence function IF(A)I. A related experiment has been performed recently in connection with the determination of the coherence length in the vertical direction of an interference experiment [8]. The equivalence of the spectrum measurement by an angular scan with a contrast measurement using different vertically phase shifting materials has been demonstrated. A sample in front of the interferometer changes the momentum distribution (g(k) -~ g~(k)) which is given by the convolution of the resolution function with the scattering function S(Q) which is given as the Fourier transform of the normalized scattering-length density p(r) in
vertical I ~ phase
sample neutron interferometer
L neutron detector
Fig. 1. Proposed arrangement for spatial Fourier spectroscopy to measure the spatial density distribution function of the sample directly.
H. Rauch / Physica B 213&214 (1995) 830-832
832 incident beam x~"~S
!
.,i; --a41 ph.a:e.l
spatial beam structure
phase shifter
r
/,
II " ~
rotator I s~ew_symnlletric ' ~'~ neutron in,erferomefer . . . .
t " ,5o ~,
~o 'w;
so mple " ~ V _~.-~
X[*)
k~-
detectors
::a:"
neutron
Fig. 2. Proposed arrangement for using Schr6dinger-cat-like states as an incident beam onto a condensed matter target for spatial Fourier spectroscopy.
In case of a Gaussian incident beam, the neutrons "feel" in the sample a spatial region of about A ~ (26k)-L around the origin. O n the other hand, a wellseparated coherent double-peaked incident beam "feels" the physical situation in the sample mainly at the origin and at a distance of its separation A, which is adjustable by the phase shift applied inside the interferometer. In the limit of nearly 6-function like double peaks an intensity measurement as a function of A gives the spatial dependence of the van-Hove correlation function directly:
I(A) oc G(A, O).
(7)
It should be mentioned that the dynamic range is increased when a rather polychromatic incident beam is used which increases the intensity considerably too. No m o m e n t u m scan is necessary anymore. Fig. 2 shows a typical arrangement how this technique can be used.
Temporal Fourier spectroscopy can be established as well. In this case, a purely energy-dependent phase shift has to be applied inside the interferometer. This can be achieved by means of a neutron magnetic resonance energy transfer system as it has been tested in the past [11]. Thus, a measurement of the coherence functions provides direct access to the spatial and time dependence of the correlation functions describing the static and dynamic properties of condensed matter. M o r e details of the new method will be given elsewhere [12].
References [1] H. Rauch, W. Treimer and U. Bonse, Phys. Lett. A 47 (1974) 369. I-2] A.G. Klein and S.A. Werner, Rep. Prog. Phys. 55 (1983) 259. [3] H. Rauch, Contemp. Phys. 27 (1986) 345. [4] G. Badurek, H. Rauch and A. Zeilinger (eds.), Matter Wave Interferometry (North-Holland, Amsterdam, 1988). 1_5] A.G. Marshall and F.R. Verdun, Fourier Transformation in NMR, Optics and Mass Spectroscopy (Elsevier, Amsterdam, 1990). [6] H. Rauch and E. Seidl, Nucl. Instr. and Meth. A 255 (1987) 32. 1-7] M. Schlenker, W. Bauchspie6, W. Graeff, U. Bonse and H. Rauch, J. Magn. Magn. Mater. 15-18 (19801 1507. 1_8] H. Rauch, H. W61witsch, R. Clothier, H. Kaiser and S.A. Werner, Phys. Rev. A, in preparation. 1-9] G.E. Bacon, Neutron Diffraction (Clarendon, Oxford, 1975). [10] D.L. Jacobson, S.A. Werner and H. Rauch, Phys. Rev. A 49 (1994) 3196. [I 1] G. Badurek, H. Rauch and D. Tuppinger, Phys. Rev. A 34 (1986) 2600. [12] H. Rauch, Nucl. Instr. and Meth., in preparation.
Physica B 213&214 (1995) 830--832
Towards interferometric Fourier spectroscopy H. Rauch Atominstitut der Osterreichischen Universithten, A-I020 Wien. Austria
Abstract
Most neutron interferometric experiments carried out up till now have dealt with fundamental physics applications. But it was known from the beginning that the interferometric method can open new horizons for solid state physics research as well. Samples with density or magnetic fluctuations cause inhomogeneous phase shifts which cause a measurable loss of contrast of the interference pattern. Phase topography can be realized without absorbing or scattering neutrons at the object. Very small energy changes (down to 10-1T eV) become measurable by means of the magnetic neutron Josephson effect. By measuring the neutron coherence ~unction, one gets direct access to the correlation function G(r, t) instead of measuring its Fourier transform S(Q, ~o). Various examples of related investigations are discussed together with their future perspectives. The essentials of future neutron Fourier spectroscopy are formulated.
1. Introduction
Since the invention of neutron interferometry [1], this technique has been used primarily for fundamental physics investigations [2-4], whereas applications for condensed-matter research have not been exploited extensively yet. Now neutron interferometry has been introduced to many laboratories and its operation becomes routine and, therefore, strong effort should be put onto its use for the broad field of condensed-matter research. In neutron interferometry the phase of the neutron wave becomes an observable which is influenced by any momentum or/and energy change the beam experiences during its interaction with the sample. Any phase change is measured by a superposition with a coherent reference beam not being affected by the sample. In many cases a much higher sensitivity than in usual spectrometry can be achieved because now the usual constraints do not exist any more; namely that the momentum tenergy) change AQ (AE) has to be larger than the momentum
(energy) width ~k (~E) of the beam. The new method shows many similarities to NMR and optical Fourier spectroscopy (e.g. Ref. [5]). The index of refraction n is given by the particle density N, the coherent scattering length be, the neutron wavelength )~ and the sample thickness D. This defines also the phase shift Z and the spatial shift A of the wave trains l . = ( n - - 1)kD= - N b c 2 D = A . k .
(1)
Any variation of this quantity representing an inhomogeneous phase shifter (powdered samples, magnetic domains, metal hydrites) cause a reduction of the interference contrast due to a spatial variation of the phase shift [6]. This produces results comparable with smallangle scattering investigations and provides the basis for phase topography [7], whose sensitivity is, roughly speaking, higher by a factor of 1000 ( ~ bc2/at; o-t being the total attenuation cross-section) than usual radiographic inspection methods.
0921-4526/95/$09.50 i~+: 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 2 9 4 - 4
H. Rauch / Physica B 213&214 (1995) 830-832 2. Basic relations
the sample (e.g. Ref. [9]). When the transmitted beam is included one gets the following momentum distribution:
The intensity of the beam in the forward direction behind the interferometer stems from wave functions originating from beam paths I and II, which are equal in amplitude and phase because they are reflected reflected transmitted (RRT) and transmitted reflected reflected (TRR), respectively:
Io ~c Iq%(0) + ~"~(,~)12 ~ 1 + IF(A)lcos (A .k)
(2)
where one beam is assumed to be phase shifted by an amount X = A.k compared to the reference beam. The wave function can be represented by a wave packet whose square value gives the momentum distribution function g(k). The autocorrelation function of the wave function defines the spatial coherence function which is directly related to the Fourier transform of the momentum distribution and which reduces the contrast at high interference order
F(A) = (~*(0)~(A)) = fy(k)eik'~dk.
831
(3)
Thus, the coherence function is a measurable quantity when the spatial phase shift is varied. In practice IF(0)I may differ from unity due to various imperfections which are unavoidable in any experimental set-up. Therefore, the normalized coherence function is used IF'(A)I = IF(A)I/IF(O)I. In the case of Gaussian momentum distributions centered around ko and having momentum widths 6k~, the coherence function also has a Gaussian shape whose widths define the coherence lengths A~ which fulfil the Heisenberg uncertainty relations
A~rki = ½.
gin(Q) = e z'oy(Q) + (1 - e-Z'°)(S(Q)* g(Q))
(4)
which gives, after some algebraic calculations, the coherence functions with and without sample: IFm(A)l = e sin + (1 - eS~°)p(A).
(5)
Ir(n)l
This equation shows that the normalized scatteringlength density (radial distribution function) can be obtained directly from the measurable coherence functions maintaining the advantage that a wide beam and a broad incident wavelength spectrum can be used. A typical arrangement for such a measurement is shown in Fig. 1. A very interesting application may be a direct measurement of the surface profile in neutron reflectometry, but other small-angle scattering phenomena and structural investigations may profit from this new technique too. Another useful method may come up using a doublehumped coherent incident beam described by stationary Schrrdinger-cat-like states. Such states have been identified recently when the wave trains of the coherent beams inside an interferometer have been shifted more than their coherence lengths [10]. In this case, spatially separated coherent packets exist behind the interferometer which exhibit a marked modulation in momentum space. The van-Hove correlation function G(r, t) has to be obtained as a Fourier transform of the scattering function S(Q, ~o) which has to be extracted from the measured spectroscopic data by deconvolution from the resolution function R(Q)
Ira(Q) = R(Q) * S(Q,(o).
(6)
3. Spatial Fourier spectroscopy Various spectrometric methods are used for the measurement of the momentum distribution function g(k). Eqs. (2) and (3) show that it can also be deduced from a measured coherence function IF(A)I. A related experiment has been performed recently in connection with the determination of the coherence length in the vertical direction of an interference experiment [8]. The equivalence of the spectrum measurement by an angular scan with a contrast measurement using different vertically phase shifting materials has been demonstrated. A sample in front of the interferometer changes the momentum distribution (g(k) -~ g~(k)) which is given by the convolution of the resolution function with the scattering function S(Q) which is given as the Fourier transform of the normalized scattering-length density p(r) in
vertical I ~ phase
sample neutron interferometer
L neutron detector
Fig. 1. Proposed arrangement for spatial Fourier spectroscopy to measure the spatial density distribution function of the sample directly.
H. Rauch / Physica B 213&214 (1995) 830-832
832 incident beam x~"~S
!
.,i; --a41 ph.a:e.l
spatial beam structure
phase shifter
r
/,
II " ~
rotator I s~ew_symnlletric ' ~'~ neutron in,erferomefer . . . .
t " ,5o ~,
~o 'w;
so mple " ~ V _~.-~
X[*)
k~-
detectors
::a:"
neutron
Fig. 2. Proposed arrangement for using Schr6dinger-cat-like states as an incident beam onto a condensed matter target for spatial Fourier spectroscopy.
In case of a Gaussian incident beam, the neutrons "feel" in the sample a spatial region of about A ~ (26k)-L around the origin. O n the other hand, a wellseparated coherent double-peaked incident beam "feels" the physical situation in the sample mainly at the origin and at a distance of its separation A, which is adjustable by the phase shift applied inside the interferometer. In the limit of nearly 6-function like double peaks an intensity measurement as a function of A gives the spatial dependence of the van-Hove correlation function directly:
I(A) oc G(A, O).
(7)
It should be mentioned that the dynamic range is increased when a rather polychromatic incident beam is used which increases the intensity considerably too. No m o m e n t u m scan is necessary anymore. Fig. 2 shows a typical arrangement how this technique can be used.
Temporal Fourier spectroscopy can be established as well. In this case, a purely energy-dependent phase shift has to be applied inside the interferometer. This can be achieved by means of a neutron magnetic resonance energy transfer system as it has been tested in the past [11]. Thus, a measurement of the coherence functions provides direct access to the spatial and time dependence of the correlation functions describing the static and dynamic properties of condensed matter. M o r e details of the new method will be given elsewhere [12].
References [1] H. Rauch, W. Treimer and U. Bonse, Phys. Lett. A 47 (1974) 369. I-2] A.G. Klein and S.A. Werner, Rep. Prog. Phys. 55 (1983) 259. [3] H. Rauch, Contemp. Phys. 27 (1986) 345. [4] G. Badurek, H. Rauch and A. Zeilinger (eds.), Matter Wave Interferometry (North-Holland, Amsterdam, 1988). 1_5] A.G. Marshall and F.R. Verdun, Fourier Transformation in NMR, Optics and Mass Spectroscopy (Elsevier, Amsterdam, 1990). [6] H. Rauch and E. Seidl, Nucl. Instr. and Meth. A 255 (1987) 32. 1-7] M. Schlenker, W. Bauchspie6, W. Graeff, U. Bonse and H. Rauch, J. Magn. Magn. Mater. 15-18 (19801 1507. 1_8] H. Rauch, H. W61witsch, R. Clothier, H. Kaiser and S.A. Werner, Phys. Rev. A, in preparation. 1-9] G.E. Bacon, Neutron Diffraction (Clarendon, Oxford, 1975). [10] D.L. Jacobson, S.A. Werner and H. Rauch, Phys. Rev. A 49 (1994) 3196. [I 1] G. Badurek, H. Rauch and D. Tuppinger, Phys. Rev. A 34 (1986) 2600. [12] H. Rauch, Nucl. Instr. and Meth., in preparation.