novel measuring methods in neutron interferometry

Physica B 151 (1988) 96-102 North-Holland, Amsterdam

NOVEL MEASURING METHODS IN NEUTRON INTERFEROMETRY

D. TUPPINGER, H. RAUCH and J. SUMMHAMMER Atominstitut der Osterreichischen Universitiiten, A-1020 Wien, Austria

Perfect crystal interferometry is becoming a standard tool for precise measurements of coherent scattering lengths. So far accuracy was limited by the resolution achieved with the determination of the wavelength and the wavelength distribution. The newly tested nondispersive sample arrangement avoids this shortcoming because the phase shift becomes independent of the wavelength. Arbitrarily shaped samples can be measured by an adapted Christiansen filter method, where the scattering length density of the powdered sample is matched to the scattering length density of a surrounding liquid. The following values for the bound coherent scattering lengths were determined: bc(Bi ) = 8.5165(62)fm by the standard method, bc(Bi ) = 8.521(4)fm by the nondispersive method and bc(Si ) = 4.157(3)fm by the Christiansen filter method. A phase sensitivity of 2.2 × 10 -~ was achieved, which could easily be increased to a level of 10 6

1. Introduction

The neutron interferometer, operating with widely separated, coherent neutron subbeams [1], allows precise measurements of neutron scattering lengths [2-6].

Designed as a monolithically perfect crystal with three parallel crystal plates - the beam split- ter, the mirror and the analyser on a common base - the neutron interferometer works with matter waves in the following way: an incoming neutron beam with a mean wavelength A 0 and a certain wavelength distribution AA/A 0 is split, recombined and superposed within the neutron interferometer. By changing the optical path difference between the two coherent subbeams, e.g. by inserting a sample in one of the subbeams, interference effects can be obtained. The sample modifies a wave function

~I,ll.___> ff/l,ll eiX , (1)

where the indices I and II refer to the two subbeams and X is the complex phase shift. The real part of X is given by

Xo = - A o N b ' ~ D ~u , (2)

if local field corrections and the term r 2 3

( N b ~ ) AoDCf f are neglected [7]. b" is the real part

of the coherent neutron nucleus scattering length, b c, which we are interested in and which is, generally, a wavelength-dependent quantity [8, 9]. De, is the path difference of the two subbeams through the sample and N is the parti- cle density.

The imaginary part X" is correlated with ab- sorption processes and with all those scattering processes within the sample, which attenuate the neutron intensity in the forward direction. Sum- marizing these effects in an attenuation cross section o',, X" is given by

X"- ~rtNDeff (3) 2

It is possible to calculate X" from known sample parameters or to determine X" by transmission experiments, which can be performed within the interferometer by blocking one coherent subbeam and measuring the intensity in both exit beams, with and without the sample [10].

The neutron intensity I o ( X ) of the forward beam behind the interferometer is given by

I0(X) = lfJ~+ qjI0~ eixla

I0 = ~- e-V'(cosh X" + cos X0). (4)

In this case, the wave functions of both sub-

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D. Tuppinger et al. / Novel measuring methods in neutron interferometry 97

beams (I, II) are twice deflected and once trans- mitted and, therefore, equal in the empty interferometer ( q / = ~0IoI).

Assuming a Gaussian wavelength distribution, an additional attenuation factor is introduced in eq. (4), which results from the wavelength de- pendence of the phase shift in eq. (2) [11],

ea =exp[_[AhA]2X0 2] )t o _1 2 " (5)

The roughness (AD) of the surfaces of the phase shifter causes an additional attenuation term

eAo=exp[--(2,rr AD 2 (6)

/7 posiTiON B (STA

It i

POSITION A ~ POSIT/ON C / (NON DISPERSIVE)

IN TERFEROIVlE TER CRYSTAL

5cm

Fig. 1. Sketch of a skew symmetric perfect crystal interferometer and indication of various phase shift measuring methods (A-standard rotation, B-standard sweep, C - nondispersive).

where D~ = 2~r/Nb'hA o defines the hA-thickness of the sample.

The intensity of the forward beam, as measured with a real interferometer can be written as

( °elf ) t0=A+Bcos2 +4,+x; • (7)

A is the average intensity and B is the amplitude of the interference pattern including the reduction factors of the amplitude, exp(-x") , eao and ea,, and a factor arising from imperfection in the interferometer crystal and the whole set-up. 4, is an internal phase parameter of the interferometer; (2~Deff/D,) and t '; are phase shifts pro- duced by samples put in the interferometer in different ways (fig. 1). This skew symmetrically cut interferometer offers two advantages com- pared with the symmetrically cut one: (1) the distance between the two crystal plates for large samples (up to 5 cm), and (2) the fact that the two subbeams are parallel over a rather long distance.

The measuring positions A and B shown in fig. 1 are routinely used in neutron interferometry. Both require samples with parallel surfaces. In measuring position A the sample is rotated in the two coherent beams [2]. Thereby, the effective thickness of the sample, Def f (eq. (7)), is changed. Measuring position B is also a standard one, where, as in position A, an auxiliary phase

shifter is rotated in both subbeams, thus generat- ing the interference pattern. At each position of the phase shifter, two intensity measurements have to be performed, one with the sample placed into one subbeam and the other without the sample. In this way, phase instabilities are cancelled. X6 is the phase difference between the two interference patterns and is determined modulo 2~r only (X~---~X~+ 2~rm; m is an in- teger). With known parameters )t 0, N, m and Dell, the value b' c can be calculated (eq. (2)). In the case of a gaseous sample, one can obtain additional information about b" by varying the pressure and hence N [3]. The accuracy achiev- able is usually limited to about 5 x 10 -4, mainly due to the errors involved in determining the mean wavelength.

The motivation to overcome this limitation led to a new measuring scheme with a nondispersive sample position (position C in fig. 1). The phase shift X~ becomes in this case independent of wavelength [12]. In addition X~ is nearly independent of the wavelength spread, which re- duces the attenuation factor eaa (eq. (5)) by orders of magnitude. Therefore very thick samples can be inserted into one subbeam without loosing contrast of the interference pattern, which increases the precision considerably. The nondispersive measuring scheme needs samples with extremely smooth and parallel surfaces, which satisfy optical criteria.

98 D. Tuppinger et al. / Novel measuring methods in neutron interferometry

Another tested measuring technique permits the use of irregularly shaped samples and has similarities to the Christiansen filter method used for a scattering length measurement based on small-angle scattering experiments [13, 14]. This technique is also independent of wavelength.

2. Results

2.1. Dispers ive s a m p l e a r r a n g e m e n t

The mean wavelength of neutrons was )t o = 1.9178(10) A, as determined with a dispersive monochromator - in te r fe rometer arrangement. From the transmission ~ = 0.675 = e -x', the first amplitude reduction factor in eq. (4) is obtained. The roughness, AD, of both surfaces of the sample is 15 ixm, which yields eao = 0.79.

The attenuation of the amplitude B 0 of the interference pattern was measured (eq. (7)), as well as the difference (Xe'xp modulo 2~r) of the phase values of the interference pattern with the

sample, ~b + X~, and the interference pattern without the sample, ~b + X'. X'a is the calculable phase shift of the air replaced by the sample, which gives for dry air X" = 3.039(6)rad [2, 4]. One obtains

t t v Xexp + 2"rrm = X 0 - X a ( 8 )

From eq. (2) X0 and b~ can be calculated. For a Bi-sample with a thickness of

3.5006(17) cm, a ratio B / B o = 0.0645(32) was determined experimentally, whereas the calculated ratio is ( B / B o ) c = 0.064 (fig. 2, upper part).

The value of the interference order m = 256 was estimated using the most precise b c values from the literature (bc=8 .5307(20) fm [15]). From the measured phase difference X:xp = 3.079(59) rad one obtains a new value for the coherent scattering length of Bi, b c = 8.5165(62) fm. Errors in thickness ( ~ D / D o = 5 ×

10 -4) and in the mean wavelength (SA/A0= 5.2 × 10 -4) contributed primarily to the error.

a2 11000

~8006

50O(

2OO( i

~- 1500

1ooo

500

'1 I l t I

J" I ? ~. I ' t i i 1 /

: \ / \ / / .~ "/ \ I

'.J \J ~d I : ,

I

\ '? ~ "'! "/ ~ I

',j \,~

0 i i I '2 - 2 - 0 1 O 0 0.1 0

nDAI (turn)

b)

Fig. 2. Sketch of the experimental sample arrangement and typical results for the dispersive (a), and for the nondispersive (b) sample arrangement, and reference interference pattern without sample (dashed lines).


2.2. Nondispersive sample arrangement

The relation between the effective thickness of the sample, Def f, and the thickness D 0, in the nondispersive sample arrangement is Deff= D0/sin 0 ° (fig. 2). With the Bragg equation, ;t o = 2dhk I sin 0 °, eq. (2) becomes

t _ _ t

go - -2dhktNbcDo , (9)

further value of the coherent neutron nucleus scattering length of Bi: b c =8.521(4)fm. The error in this case is due to the uncertainty in the sample thickness 8D and a possible misalignment of the sample surfaces with respect to the reflect- ing Bragg planes of the interferometer. The precision in the sphase shift determination is ~Xo/ X0 = 2.2 x 10- and could be further increased by using longer measuring times.

which is now independent of the mean wavelength ;t0 [9, 12]. Therefore it is no longer necessary to measure A 0 in order to deduce b~ from the experimental result X0.

The interferometer crystal produces an aniso- tropic wavevector distribution which is very nar- row in the direction parallel to G [16]. This causes a smaller attenuation in the amplitude of the interference pattern, which is determined by the defocusing effect and which is about 100 to 1000 times smaller than eAA [12]. Thus much more sample material can be introduced without significantly loosing contrast of the interference pattern.

The experiments were performed with bismuth samples from the same stock as used in the "dispersive" experiments, which had a purity of 5N9 [17]. The sample thickness was Do= 1.6899(7) cm, which corresponds to an effective length Def f = 3.3798(14) cm. All measurements were performed with an auxiliary aluminium phase shifter to generate the interference pattern.

The attenuation of the amplitude B 0 is, in this case, much less than with the dispersively ar- ranged sample (B/B o =0.351(16); fig. 2, lower part). Transmission experiments yield a value VriTI0 = 0.604 = e x p ( - x " ). The reduction factor due to the thickness variation is eao = 0.6 (AD = 21 Ixm), which again shows agreement between the observed and the calculated values. The mean wavelength was A 0 = 1.9225(18)/~. The sample causes a phase shift of about 248 × 2"rr. The experimentally observed phase difference between the two relevant interference patterns was X~,p = -0.176(37)rad and the calculated phase shift of replaced dry air was X'a = 2.940(7)rad. From eq. (9) one can deduce a

2.3. Christiansen filter measuring method

This method is based on small-angle scattering effects [13] and was frequently used to measure b c values in an adapted small-angle scattering apparatus [14]. The filter consists of a mixture of liquid and bulk material whose index of refraction is different from that of the liquid. There- fore, small-angle scattering in the filter occurs, causing an intensity reduction and a large phase variation of the beam. By changing the index of refraction of the liquid, it can be matched to that of the sample. This results in the disappearance of small-angle scattering and the recovery of the interference contrast. Such a liquid is prepared by mixing two components with significant differ- ences in the index of refraction. By varying the relative content of each component a wide range of values for the index of refraction, or equival- ently, for the scattering length density (Nbc)L, can be obtained. At the matching point, the scattering length density of the liquid, (Nbc)L, equals that of the sample material, (Nbc) s. If one measures (Nbc) L separately or calculates it, one finds the value of b~ of the sample.

In our experiments, we inserted a filter in one coherent subbeam and observed the small-angle scattering effect by measuring the broadening of the rocking curve of the monochromator-interferometer spectrometer. We also observed a reduction of the interference pattern due to the inhomogeneous phase shifts over the cross section of the subbeam, which occurred unless the filter was matched.

Both effects have an influence on the measured pattern, mainly in reducing the amplitude. The reduction due to small-angle scattering is accounted for in the attenuation factor

100 D. Tuppinger et al. I Novel measuring methods in neutron interferometry

e x p ( - x " ) . The inhomogeneous phase shift causes an additional reduction factor e, [18]:

= 1 2

F

Ill + s i n ( E l ( x , y ) ) d x dy , (10) F

where F is the cross section area of the subbeam passing through the filter, f (x , y) the local thickness of the sample material and E = A0[(Nb~)s - (Nb~)L]. E = 0 means that the filter is matched and homogeneous and the reduction vanishes, because e~ = 1.

The signal one measures to detect the homogeneity of such a filter is the maximum height of the amplitude, B , of the interference pattern. It is obvious that one has to normalize B to the amplitude B o of the reference scan without filter (fig. 3).

The ratio B I B o is

B I B 0 : EAA ~r e x p ( - X") ( 11 )

if one neglects a possible surface roughness of the quartz glass cuvette in which the sample- liquid column is kept. Two arrangements of the filter inside the interferometer are possible: first- ly, the column is in one subbeam, and secondly, the column is in one subbeam, but simultaneous- ly the liquid with the same thickness is in the other subbeam (fig. 3). In the second arrangement, the reduction described by ea~ becomes negligible near the matching point, because about the same phase shift occurs in both subbeams. The measured normalized amplitude (eq. (11)) has to be corrected due to the systematical- ly varying values e x p ( - x " ) which depend on the specific liquids inside the filter. Then the data points should be on a Gaussian shaped curve which is symmetric around the interesting matching point.

LIQUID

o - BEAM

I [ INTERFEROMETER CRYSTAL

-- (220) AUXILIARY AI- PHASE SHIFTER

CHRISTIANSEN FIL TER

H - BEAM 5cm t I

3000

2000

tO00

S i ( 5 6 0 - 600,um) w i t hou t $omple

/ ~ ' ~ ~ ' ~

/ ' a / . \ 7 /

/ ' \ " " i " ~ " ' " \ V

C?Ht6 =0.371; = 0 . 3 8 0 = 0 3 9 0 C2CI 4

i I I I J -0.2 -01 0 OI 02

4DAI (mini

t.O

,~ 08 i

0.6 c3

0 2

5 i - p o w d e r { 560 - 600/Jrn )

(Nbc)FWHM :3. 666(62). 109cm" 2

/ ; ' 5 20 22. 5 25 SCATTERING L E N G T H OENSITY (Nbc )L . lO9cm "2

Fig. 3. Typical results for Christiansen filter measurements with different liquids and reference interference pattern without sample (above). In the lower part, the normalized ratio of amplitudes is plotted versus the scattering length density of liquids. Sketch of the arrangement of the filter and an auxiliary phase shifter within the interferometer (left).


Several measurements with different samples were performed: Tungsten in the shape of a cylinder of about 2 to 3mm radius [19]; Te- powder with grain sizes from 100 to 3000 ~m [9]; Si, Bi and glass spheres with grain sizes from 100 to 600 I~m. Here only the Si-measurements are reported.

In fig. 3 the filter arrangement within the interferometer is schematically represented. Very high purity Si-powder was prepared with grain sizes from 560 to 600p, m and from 180 to 200 ixm. The two components of the liquid were C2C14 (purity 99.9%) and C7H16 (purity 99.75%), which, at 20°C, cover a range of scattering length densities from about - 5 . 4 7 x 109 cm -2 to about 30.41 × 109 cm -2. The density of the liquid was determined to a precision of 1 x 1 0 - a g / c m 3 and, therefore, its scattering length density could be calculated from tabulated b~ values [20] with a relative accuracy a(Nbc)L/ (Nb~) L of about 5 x 10 -4. As a test of the scale the absolute values of (Nb~) L were measured independently for some liquids, by the standard measuring method discussed before. The powder column is specified by a package density between 55 and 65%. It requires special installations to introduce the liquids into such a column without forming of gas bubbles and also to dry it after each measurement.

Each Christiansen filter with different mixtures of liquids gives a different measuring signal B/B o (fig. 3, above). Such a scan takes about 3 to 4 hours, with the cross section F of the powder column measuring about 7 x 9 mm 2. The counting rate in the outcoming beam in forward direction was about 35 counts/s. The achieved precision in ~(B/Bo)/(B/Bo) is about 0.02 at the matching point and can be increased further by using longer measuring times. In fig. 3 the normalized contrast B/B o is plotted versus the scattering length density of the liquids (Nb~) L. A proper fit gives the matching point [(Nb~) L = (Nbc)s] with an accuracy ~(Nbc)L/(Nb~) L of about 7 x 10 -4. Therefore, a value for the effective scattering length for Si, b~ = 4.158(3)fm, is obtained. This value is in some disagreement with the most precise be value derived for Si from a dynamical diffraction experiment (b¢ =

4.149(1) fm [21]), but in rather good agreement with a b c value derived from Christiansen filter measurements in a small-angle scattering apparatus [22]. With Si grains of sizes 180 to 200 Ixm, a second value for b~ was observed: (b~ =4.151(8)fm), in agreement with the first result. The characteristic difference between both Si-results is illustrated by the full width at half maximum of the two normalized fits. Smal- ler grain sizes cause broader widths and, therefore, the sensitivity of the method increases with the grain size of the powder, as postulated by eq. (10).

3. Discussion

Both new methods in neutron interferometry are characterized by results independent of the wavelength. The precision of the b c values is no longer limited by the precise knowledge of A 0.

In the nondispersive sample arrangement one observes interference contrast up .to very high orders, because the interferometric measurement becomes decoupled from the wavelength distribution. An accuracy of the phase shift of the sample ~X'/X' of at least 10 -6 can be achieved by using longer measuring times and a broader wavelength distribution to increase the intensity. High quality demands on the sample geometry, surfaces and structure have to be fulfilled and a proper adjustment of the sample has to be car- ried out to take full advantage of the nondisper- sivity of the experiment. High precision data of phase shifts and of effective scattering lengths allow to investigate local field effects, neutron- electron and neutron-gravity interactions, etc.

The Christiansen filter method makes scattering length measurements possible with arbitrarily shaped specimens and small volumes of powdered samples. In a single scan one can measure the normalized amplitude B/Bo, which fixes the matching point (Nbc) L = (Nbc) s as well as the absolute scattering length density of the liquid (Nbc) L. The precision ~bc/b ~ for the bc values of Si reached in the present experiments is about 7 x 10 -4, which can be increased by about one order of magnitude by measuring with better

102 D. Tuppinger et al. / Novel measuring methods in neutron interferometry

counting statistics. A predicted relation between the loss of contrast of the interference pattern and the size of precipitations inside the filter could be verified. Smaller precipitations attenuate the amplitude of the interference pattern less than the larger ones, assuming the same difference in scattering length densities between precipitations and liquid. In this sense the interferometric Christiansen filter method is com- plementary to the Christiansen small-angle scattering method.

The newly developed and tested techniques make the interferometric method more univer- sal, with respect to high precision measurements of the scattering lengths of samples in all three states of aggregation and with all shapes and structures.

Acknowledgements

The hospitality of the Institute Laue-Langevin is gratefully acknowledged. The work was performed within the Dortmund-Grenoble-Wien interferometer cooperation and was financially supported by Fonds zur Frrderung der Wis- senschaftlichen Forschung (project $42/01). The authors wish to thank A. Rumpf (Grenoble/ Dortmund) and H. Obbing (Grenoble/ Dortmund) for their valuable support and G. Schmid (Grenoble) for his technical assistance.

References

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[2] W. Bauspiess, U. Bonse and H. Rauch, Nucl. Instr. Meth. 157 (1978) 495.

[3] H. Kaiser, H. Rauch, G. Badurek, W. Bauspiess and U. Bonse, Z. Phys. A: Atoms and Nuclei 291 (1979) 231.

[4] U. Bonse and U. Kischko, Z. Phys. A: Atoms and Nuclei 305 (1982) 171.

[5] U. Bonse and T. Wroblewski, Nucl. Instr. Meth. A 235 (1985) 557.

[6] H. Rauch, D. Tuppinger, H. W61witsch and T. Wrob- lewski, Phys. Lett. B 165 (1985) 39.

[7] V.F. Sears, Z. Phys. A: Atoms and Nuclei 321 (1985) 443.

[8] R.E. Word and S.A. Werner, Phys. Rev. B 26 (1982) 4190.

[9] H. Rauch and D. Tuppinger, Z. Phys. A: Atoms and Nuclei 322 (1985) 427.

[10] H. Rauch and J. Summhammer, Phys. Lett. A 104 (1984) 44.

[11] H. Rauch and M. Suda, Phys. Stat. Sol. 25a (1974) 495. [12] H. Rauch, E. Seidl, D. Tuppinger, D. Petraschek and

R. Scherm, Z. Phys. B: Condensed Matter 69 (1987) 313.

[13] R.J. Weiss, Phys. Rev. 83 (1951) 379. [14] L. Koester and H. Ungerer, Z. Phys. 219 (1968) 300. [15] L. Koester, W. Waschkowski and A. Klfiver, Physica B

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DISCUSSION

(Q) U. Bonse: How large was the correction due to the air replaced by the sample?

(A) D. Tuppinger: The phase shift caused by the replaced air was calculated to be X'a = 2.940(7) rad for the nondispersive, and X'a = 3.039(6) rad for the dispersive sample arrangement, assuming dry air.

(Q) U. Bonse: How large is A (area enclosed by beams) with the skew symmetric interferometer? Can gravitational effects be neglected during measurements, as they are caused by an

inclination of the experiment with respect to the horizontal during measurements?

(A) D. Tuppinger: The area A was about 16 cm 2 assuming a Bragg angle 0~ = 30 °. Gravitational effects can be neglected because only phase shift measurements relative to the empty intefferometer have been done. At each position of the auxiliary phase shifter two interferograms have been measured, one with the sample and one without the sample. Therefore phase shift instabilities drop out.

novel measuring methods in neutron interferometry

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