Neutron rocking curves in non-absorbing crystals

10 April 1995 PHYSICS

EL.SEVIER

LETTERS

A

Physics Letters A 200 ( 1995) 73-7s

Neutron rocking curves in non-absorbing crystals Toshio Takahashi, Mitsuhiro Hashimoto Institutefor Solid State Physics. University of Tokyo, Roppongi, M&to-ku.

Tokyo 106. Japan

Received 1November 1994; accepted for publication 7 February 1995 Communicated by A. Lagendijk

Abstract

A theoretical and experimental study has been done on the relationship between Darwin and Ewald curves. The Ewald formula is described in the frame of successive reflections at the front and rear surfaces, and a relation of RE= 2Rnl( 1 + R,,)~ has been obtained between the Ewald formula R, and the Darwin formula R, in the case of non-absorbing crystals. Both the Darwin and Ewald curves have been observed by controlling the successive reflections by using a tailless incident beam obtained khrough a channel-cut monochromator in the double-crystal parallel setting of Si crystals.

Diffraction phenomena of neutrons by a perfect crystal are explained by the dynamical theory [l-3] and experiments on neutron rocking curves in the Bragg case have already been studied [l-7]. However, the relationship between the Darwin formula [ 81 and the Ewald formula [ 91 which give the reflectivity of neutrons in the Bragg case is seldom studied except for the work on X-ray diffraction [ IO]. In this Letter, we show that the two formulae are related by a simple relation in the case of non-absorbing crystals and that neutron rocking curves which obey either of the two formulae can be observed by controlling the experimental condition. The Darwin formula is expressed as

=[IyI-(y’-l)“‘]’ in the case parameter beam from tion effect

for(y(>,l,

RE = 1,

(1)

of non-absorbing crystals [ 111. Here y is a which gives the deviation of the incident the Bragg condition corrected for the refrac[ 1, 1 1 ] . Total reflection occurs in the region

0375-9601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDlO375-9601(95)00121-2

for Iv] =G1 ,

=1-(1-y-‘)“‘,

for ]y{ >I.

(2)

The profile of the two rocking curves given by Eqs. ( 1) and (2) differs considerably in their tail arts, and 1 the Ewald formula gives twice the value of the Darwin formula at their far tails. The reflectivity of neutrons in the case of thick crystals is usually described by Eq. (2) [l-3], ~which is conveniently derived from the reflectivity of finite crystals, R=

for ly(
R,, = 1 .

of ]y ) < 1. In contrast, the Ewald formula is described as

sin2[A(y2y*-

1 +sin2[A(y2-

1)“2] l)“‘]

’

(3)

by taking the average over an oscillation on the crystal thickness [ Ill. Here A is related with the extinction distance An and the crystal thickness D by A + ITDIA,. It is noted that this procedure is valid only khen the reflected waves at the front and rear surface/s overlap coherently. However, this condition is broken in the

74

T. Takahashi, M. Hashimoto / Physics Letters A 200 (1995) 73-75

Fig. 1. Illustration

of the diffraction

process

in the case of

non-

absorbing crystals. ordinary experimental situation where the crystal thickness, being larger than 100 pm, is much larger than the coherence length of neutrons. Then the diffraction process may be understood as follows. Fig. 1 shows the illustration of the diffraction process. The reflective intensity R, should correspond to the Darwin formula Fiven by Eq. ( 1) . Then the intensity of the transmittedtbeam through the rear surface T, is given by ( 1 -R,) 2 in the case of non-absorbing crystals [ 121. The beam reflected at the rear surface propagates toward the front surface, and the reflective intensity R, is given by ( 1 - RD)*R,,. Thus the intensity of all the reflected beams is given by R = 2 R,, II=1 =R,+(1-R,)2R,(1+R~+R;+...) =- 2R, l+R,,

’

One can show that Eq. (4) is equivalent formula. Then a simple relation,

(4) to the Ewald

mated by 0 0 2 reflection of PC, were incident on the first crystal Cl. A channel-cut Si crystal using five successive reflections was adopted as the first crystal to reduce extremely the tail part of the incident beam on the second crystal C2; otherwise the distinction between the Darwin and Ewald curves is not easy due to the convolution effect. The beam size in front of the second crystal was 5 mmw X 10 mmH. The experiments were performed using two sample crystals with thicknesses of 10 mm and 1.4 mm, much larger than the coherence length. The first experiment was done using the crystal of 10 mm thickness. A slit S2 was placed behind the sample crystal so that only the reflected beam R, is detected by a ‘He detector. In the second experiment, the crystal of 1.4 mm thickness was used and the slit S2 was removed so as to detect significant parts of all the reflected beams R,. In order to get the reflectivity, the diffracted intensity was normalized by the intensity of the incident beam. Both the intensities of the incident and diffracted beams were measured by the same detector. The experimental result is shown in Figs. 3a and 3b. Solid and broken curves were calculated based on Eq. ( 1) and Eq. (5)) respectively, with the convolution of the reflectivity curve of the first crystal, Ri. The experimental results agree well with the calculated ones. Recent studies on crystal truncation rod scattering [ 131 reveal that the far tails of the rocking curve have information on the structure of crystal surfaces. Thus the present work indicates that the experimental con-

~RD RE= l+R,,’ is obtained between the Darwin and Ewald formulae. Similarly, the intensity of all the transmitted beams is given by

C2

Sl

0 D )

Cl

--

I

s2

The experiment was done at the beam-line Cl-3 (ULS) in JRR-3M, which has been constructed for experiments on ultra-small angle neutron scattering and neutron optics. The double-crystal parallel setting of 1 1 1 reflection of Si crystals is used as illustrated in Fig. 2. Neutrons with the wavelength of 4.73 A, monochro-

1

ClD Fig. 2. Experimental arrangement of the double-crystal parallel setting; first crystal Cl, second crystal C2, Cd slits SI and S2, and 3He detector D.

T. Tukahashi. M. Hashimoto / Physics Letters A 200 (1995) 73-75

r

7.5

1.000

(W

0.100 > k > 6 !f! L a: 0.010

-20

-15

-10

-5

0

5

10

15

! 20

ANGLE(SEC)

I

0.001

i -20

-15

-10

-5

0

5

10

15

20

ANGLE(SEC)

Fig. 3. Rocking curves of neutrons in linear scale (a) and logarithmic scale (b). Solid circles mean the reflectivity of a IO mm qhick crystal with slit S2 installed. Open circles represent the reflectivity of a 1.4 mm thick crystal while the slit S2 was removed. Solid and broken curves were calculated based on Eqs. ( I ) and (5 ) , respectively.

dition should be strictly controlled in experiments on neutron truncation rod scattering. In conclusion, a simple relation of R,= 2R,l ( I+ R,,) was derived between the Ewald formula R, and the Darwin formula RD in the case of non-absorbing crystals and the rocking curves confirming the relation were observed using neutron diffraction by Si crystals.

References 1II H. Rauch and D. Petrascheck,

in: Topics in current physics, Vol. 6. Neutron Diffraction, ed. H. Dachs (Springer, Berlin, 1978) p. 303. 121 V.F. SearsCan. J. Phys.56 (1978) 1261.

[3] V.F. Sears. Neutron optics (Oxford Univ. Press. Oxford. 1989). [4] S.G. Shull, J. Appl. Cryst. 6 (1973) 257. [5] S. Kikuta, 1. Ishikawa, K. Kohra and S. Hoshino. .I. Phys. Sot. Japan 39 (1975) 471. [ 61 S. Kikuta. T. Takahashi. K. Nakayama. Y. Fujii and S. Hoshino, J. Phys. Sot. Japan 45 (1978) 1065. 171 W. Treimer, W. Fielder and B. Biickert, Phys. Lett. A I10 (1985) 173. [8\ C.G. Darwin, Philos. Mag. 27 (1914) 675. [9] P.P. EwaId, Ann. Phys. (Leipzig) 54 (1917) 557 IlO1N. Kato, in: X-ray diffraction, ed. L.V. A&off (McGraw-Hill, New York, 1974) p. 222. 1111W.H. Zachariasen, Theory of X-ray diffraction in crystals (Dover, New York, 1945) lI21 U. Bonse and M. Hart, Z. Phys. 194 ( 1966) 1. (13 K. Al Usta, H. Dosch and J. Peisl. Z. Phys. B 79 ( 1990) 409. and references therein.