Remarks on a Johann spectrometer for exotic-atom research and more

    Remarks on a Johann spectrometer for exotic-atom research and more Detlev E. Gotta, Leopold M. Simons PII: DOI: Reference:

S0584-8547(16)30017-9 doi: 10.1016/j.sab.2016.03.006 SAB 5045

To appear in:

Spectrochimica Acta Part B: Atomic Spectroscopy

Received date: Revised date: Accepted date:

11 November 2015 9 March 2016 11 March 2016

Please cite this article as: Detlev E. Gotta, Leopold M. Simons, Remarks on a Johann spectrometer for exotic-atom research and more, Spectrochimica Acta Part B: Atomic Spectroscopy (2016), doi: 10.1016/j.sab.2016.03.006

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Detlev E. Gotta1

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Remarks on a Johann spectrometer for exotic-atom research and more

Institut f¨ ur Kernphysik, Forschungszentrum J¨ ulich GmbH, D-52425 J¨ ulich, Germany

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Leopold M. Simons

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Laboratory for Particle Physics, Paul Scherrer Institut, CH-5232 Villigen, Switzerland

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Abstract

General properties of a Johann-type spectrometer equipped with spherically bent crystals are described leading to simple rules of thumb for practical use.

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They are verified by comparing with results from Monte-Carlo studies and

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demonstrated by selected measurements in exotic-atom and X-ray fluorescence research.

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Keywords: curved crystal spectrometer, exotic atoms, X-ray fluorescence

1. Introduction

The advent of high-intensity pion and antiproton beams allowed to per-

form ultimate precision measurements of exotic-atom X-rays by means of crystal spectrometers. Vice versa, also fluorescence studies of standard atoms and ions benefit from adapting and developing further established methods of X-ray diffraction to the needs of exotic-atom spectroscopy. Exotic atoms are formed when a heavier negatively charged particle like a muon, pion, or antiproton is bound in the Coulomb field of a nucleus. Because of their large mass the dimensions of such atoms are closer to nuclear than to atomic scales. The study of the elementary systems formed with hydrogen, i. e. muonic, pionic, and antiprotonic hydrogen (µH, πH, and p ¯H), requires 1 Corresponding

author: [email protected]

Preprint submitted to Journal of LATEX Templates

March 11, 2016

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the measurement of X-rays with energies in the few keV range. The X-rays stem from the final steps of a de-excitation cascade of the exotic atom, which is formed in high-lying atomic states. Hence, measurements reveal properties of

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the interplay of the captured particle with remaining electrons in high-lying and

of such systems may be found in ref. [1].

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nuclear properties from the low-lying transitions. A more detailed description

A variety of problems may be tackled by ultimate resolution spectroscopy

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of fluorescence radiation. Examples are the dependence of X-ray line shapes on the chemical environment [2, 3], the relationship of various precise X-ray

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standards [4, 5, 6], or studies of highly-charged ions [7, 8] which in turn allow decisive tests of methods used for QED calculations [9]. In order to minimise absorption losses in the crystal material, measurements

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in the few keV range require the use of reflection-type spectrometers, and only silicon and quartz crystals fulfil the requirements for ultimate-resolution spec-

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troscopy. Their resolving power is of the order of 104 . Consequently, the integrated reflectivity RI of such a crystal material is typically about 100 µrad in

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first order diffraction.

In spite of the high beam intensity provided by modern particle accelerators,

an inherent challenge in exotic-atom research is the low count rate together with demanding background conditions. Typical count rates of exotic-atom X-rays are in the range of 20 − 100 per hour where the flux of pions amounts to a few 108 /s (Paul Scherrer Institut, Switzerland [10]) and up to 106 /s for antiprotons (at LEAR, CERN [11] until shut down by the end of the year 1996). Such high fluxes produce an enormous beam-induced background level. Therefore, an apparatus for ultimate-resolution spectroscopy constitutes a compromise between conserving the intrinsic resolution of ideal single crystals and sufficiently high efficiency to achieve acceptable measurement times together with mechanical long-term stability. The experimental approach consists of a Johann-type Bragg spectrometer equipped with spherically bent silicon and quartz crystals. Such a set-up allows to record simultaneously a finite energy interval according to the width 2

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of the X-ray source when using a correspondingly extended X-ray detector. Large bending radii of about 3 m minimize aberrations and result in a sufficient distance to the X-ray source allowing for an effective shielding of the X-ray

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detector.

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The paper discusses general features of such spectrometers using a simple geometrical picture giving at hand rules of thumb to assess a typical set-up (Sec. 2). The validity of approximations is exemplified by means of experi-

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mental results and by Monte-Carlo studies (Sec. 3). The mechanical set-up is sketched in Sec. 4. Examples from exotic-atom spectroscopy are referenced for

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light antiprotonic and pionic atoms as well as applications in the field of X-ray fluorescence (Sec. 5).

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2. Imaging properties

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Crystal spectrometers exploit the fact, that a regular atomic lattice diffracts

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intensity only for wavelengths λ fulfilling the Bragg condition "  2 # 2d δ · sin ΘB , nλ = 2d 1 − nλ

(1)

where n is the order of diffraction, d the lattice spacing perpendicular to the diffracting plane, and ΘB the Bragg angle. The correction term δ = 1−nr takes into account the change of the index of refraction inside the crystal material with nr being the real part of the index of refraction [12]. δ itself depends on λ and is of the order 10−3 in the considered energy range [13, 14]. Energy E and wavelength λ are related by E =

hc λ .

In the so called Johann-type set-up the diffracting planes are bent to a radius Rc . The benefit of the focusing properties of such curved crystal spectrometers was described first for the cylindrical case [15, 16]. The major advantage is the possibility to measure simultaneously a complete energy interval given a (preferentially homogeneous) extended source together with an extended detector (Fig. 1). Different wavelengths are accepted from different locations of the source which correspond to different locations in the focal plane. Consequently, 3

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the impact point of the diffracted X-rays in the detector plane constitutes a

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wavelength or energy spectrum in the direction of dispersion.

Figure 1: Source and image locations of two different wavelengths λ1 < λ2 for a Johann-type set-up (symmetric Bragg case). A spherically bent Bragg crystal allows in addition partial vertical focusing.

Varying ΘB , i. e. λ, the location of the minimum width of a bundle reflected

under ΘB from a curved mirror is a circle with diameter Rc , the Rowland circle. The distance of the crystal centre to the focal point on the Rowland circle is given by Rc · sin ΘB (focal condition). From the angular dispersion,

dλ dΘ

=

λ tan ΘB

(neglecting the index of refraction

shift), one derives the (local) dispersion in a detection plane at the focus F to be

dλ dx

=

dλ dθ

·

1 Rc ·sin ΘB ,

where x is the direction of dispersion in the plane

perpendicular to the direction CF (see Fig. 2). In terms of the X-ray energy E, the relations are

dE dΘ

= − tanEΘB and

dE dx

=

dE dθ

·

1 Rc ·sin ΘB .

In reality, pure specular reflection must be extended to a finite angular range

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around the Bragg angle ΘB determined by a given wavelength λ. In this range, the acceptance of the crystal is weighted according to the shape of the rocking curve. The rocking curve, characterized by its width ωc at full-width-half max-

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imum (FWHM), represents the angular spread of the diffraction pattern for a

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particular crystal plane (Fig. 4). ωc may be regarded as the intrinsic resolution of a plane crystal. The integral of the rocking curve, the integrated reflectivity RI , represents the total reflected intensity. The rocking curve can be calculated

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by means of diffraction theory, e. g., by using the code XOP [17]. A complication when using Johann set-ups is a precise absolute angular cali-

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bration. It requires the exact knowledge of the orientation of the reflecting plane which meets enormous technical difficulties. Therefore, using a calibration line with a Bragg angle as close as possible is mandatory for ultimate energy deter-

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mination. The calibration is then achieved by the angular distance measured

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by means of the position difference in the direction of dispersion x. For cylindrically bent crystals, the image height is approximately twice the crystal height plus the source height. Vertical focusing is achieved by crystal

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bending in the vertical direction z as realized in the so called v. H´amos geometry [18]. The (vertical) bending radius required is Rc · sin2 ΘB , i. e., simultaneous complete focusing is achieved for spheres only in the limit of back reflection ΘB = 90◦ . However, the ideal case of an elliptical surface of highest quality meets very demanding fabrication techniques and, in addition, fixes the geometry to one particular Bragg angle. Therefore, spherical bending is commonly used. For small Bragg angles ΘB , the image height is still comparable with the one of a cylindrically bent crystal, but with increasing ΘB significant vertical focussing can be achieved (see Figs. 3 and 4). A detailed analytical elaboration of imaging properties of spherically bent crystals has been given by Eggs and Ulmer [19]. Such studies have been extended, including also Monte-Carlo based methods, by various authors (see, e. g., [20, 21, 22, 23]). As shown in Fig. 2a, the focusing condition Rc · sin ΘB is distorted by the deviation of the crystal surface from the Rowland circle. The displacement at the Rowland circle is usually expressed as an angular shift, the so called Johann 5

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Figure 2: Johann shift and symmetries in spherical-mirror imaging (see text).

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1 2

·



bc 2Rc

2 · cot ΘB being quadratic in

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shift, and reads in leading order ∆ΘJ =

the horizontal extension bc of the crystal, where bc is the full width [16, 19, 20].

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In this order, the shift is equal for the left and right side of the vertical axis z and always occurs on the low (high) wavelength (energy) side. Seen from a

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second point of view, ∆ΘJ also represents the range of the glancing angle ∆ΘG on the crystal sphere seen from the focus point on the Rowland circle. Hence, bc 2

of the crystal determines the range

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in the symmetry plane the half diameter

of source points SS′ which fulfil the reflection condition ΘB . Above and below the symmetry plane, the Johann shift decreases with the decreasing effective

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horizontal width due to the circular crystal boundary (see Fig. 2c). By weighting the shift ∆ΘJ with the available reflecting area as a function of distance from the axis z, the average shift ∆ΘJ is obtained. It results in 1 3

· ∆Θmax for a cylindrically bent plate and J

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∆ΘJ =

1 4

· ∆Θmax for a disk, where J

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∆Θmax is the maximum shift occurring at the left or right boundary ± b2c . J Considering a small (vertical) crystal extension above and below the symmetry plane, the geometry represents the cylindrically bent case with a crystal

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of width bc . The intensity distribution caused by the Johann shift is ∝ 1t , where t is the coordinate along the tangent on the Rowland circle [16] (Fig. 2a - distribution along FF′ ). Coordinates t and x are related by x = t · sin ΘB . The total line shape results from the convolution of this intensity distribution and the rocking curve.

Reflection conditions remain unchanged for a spherical mirror for rotations

of the plane SCF around the axis defined by OC (Fig. 2b). The crystal width bc corresponds to the crystal diameter. Accepted source points move along a circle SS′′ with radius Rc sin ΘB cos ΘB with the reflecting band on the crystal surface rotating correspondingly. Considering only the reflection properties of the crystal, one obtains full cylindrical symmetry for source and image points around the axis OC. However, in any practical application source and detector extensions are limited. Hence, crystal, source and detector centres naturally define a symmetry plane SCF. Obviously, with a limited source height a large

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fraction of the reflecting crystal area is not used. Projecting the circle to the detection plane perpendicular to CF yields an ellipse, the curvature of which when intersecting the symmetry plane SCF corresponds to a circle with radius

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rapex = Rc · cos ΘB (apex radius).

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A second band of accepted source points SS∗ is achieved by rotating around the axis lying in the symmetry plane SCF through the origin O perpendicular to the axis OC (Fig. 2c). The radius of the band is Rc cos2 ΘB and oriented per-

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pendicular to the circle in the previous case. When projected to the detection plane, the apex radius rapex is again Rc · cos ΘB . This is immediately under-

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stood by the fact that all affected points on the crystal surface are in principle accessible also by the previous rotation around OC. In this case, the crystal height determines the effective source height.

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The total image is obtained when applying the rotations displayed in Fig. 2b

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to all cases shown in 2c and vice versa. Therefore, the complete crystal surface is usable even with moderate source heights. In each case, one obtains a set of ellipses with their apex moving along the main circle with radius Rc · cos ΘB .

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bc The vertical axes of the ellipses are tilted up to an angle arctan 2R with respect c

to the symmetry plane SCF. Taking into account the requirement to aim at bc ultimate resolution, any set-up must consider the condition arctan 2R < 1◦ , c

i. e.,

bc Rc

< 10−2 . In this way, the effect of the fan out of the tilted ellipses can

be adapted to the angular spread given by the rocking curve (see Sec. 3). The weighted accepted angular range accepted by any point on the crystal’s

surface is given by the rocking curve. As a rule of thumb, the total resolution ∆Θexp of a Johann set-up maybe estimated by the average Johann broadening and the rocking curve width ωc as long as the fan out of the tilted circles can be ignored. ∆Θexp is expected to be in the range q (ωc )2 + (∆ΘJ )2 ≤ ∆Θexp ≤ ωc + ∆ΘJ ,

(2)

where the average geometrical broadening is approximated by the average Johann shift ∆ΘJ . For setting up an experiment, the condition ωc ≈ ∆ΘG and ωc ≤ 2 · ∆ΘJ 8

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allows the efficient use of the crystal area and preserves the intrinsic resolution of the plane crystal, respectively. Consequently, in order to exploit the resolving  2 bc power of 104 , the order parameter of the defocusing should be 2R ≤ 10−4 . c

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Given an ideal mounting, the crystal’s angular acceptance is about the width

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ωc of the rocking curve. Considered at the focal distance Rc · sin ΘB , it defines a circular ring of radius Rc sin ΘB cos ΘB around the axis OC having a width of ∆xs = ωc · Rc · sin ΘB . In the cases discussed here (30◦ . ΘB . 70◦ ), an X-ray

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source overlaps only with a small part of the circular ring. Typical apertures defining a source are of circular or rectangular shape with diameter ds or height

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hs . Hence, the effective source area matching the crystal acceptance can be estimated to be ∆As = ∆xs · ds or ∆xs · hs .

∆Ω 4π

(3)

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where

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As a rule of thumb, the efficiency of the Johann set-up is given by   ∆Ω ∆As ηc = · foverlap , · P· 4π As

is the solid angle of the crystal with respect to the source, P the peak

reflectivity of the crystal (including absorption), and foverlap the fraction of the

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diffracted image overlapping with the detector area (see Figs. 3 and 4). The ratio

∆As As

may be rewritten as

∆As As

= fS ·

ωc ωs ,

where ωs is the an-

gular width of the source with respect to the crystal center in the direction of dispersion. The weight factor fs is 1 and  target aperture, respectively. Obviously, P ·

reflectivity RI ≈ 34 P ωc [24].

2 for rectangular and spherical π  ωc is related to the integrated ωs

As a typical example for the spectrometer efficiency ηc , we quote an exper-

iment with antiprotonic 3 He (see Sec. 5). Equation (3) yields for the efficiency ηc = 5.5 · 10−7 , which agrees well with the value ηc = (7 ± 3) · 10−7 obtained in a dedicated calibration measurement. A count rate estimate itself implies formation rate of the exotic atom, X-ray line yield, and X-ray absorption in target or detector windows. Such quantities depend strongly on beam conditions and physical properties of the particular exotic atom [1]. Furthermore, the quantum efficiency of the X-ray detector must be taken into account. 9

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3. Monte-Carlo studies A specially tailored Monte-Carlo code has been developed to allow for a

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more detailed understanding of imaging properties. The studies are based on the approach, that the crystal’s angular acceptance essentially determines the

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use of a given target and detector geometry. The crystal selects only impact angles at its surface layers, but not an absolute position in space like X-ray

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origin or impact point on a detection system. A ray representing the incoming photon is diffracted from any point on the surface with an angular distribution weighted according to the shape of the rocking curve.

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For experiment planning and analyses, apertures at arbitrary positions both on the source and the detector side allow for the simulation of a dedicated setup. Vertical displacement and rotation of the crystal both around its vertical

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and horizontal axis allow to take into account misalignments or intended ad-

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justments of the image in the detection plane. Similarly, the detection plane can be shifted along and perpendicular to the direction crystal-detector and, in

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addition, rotated.

Pure geometrical broadening corresponds to a rocking curve of vanishing

width. In order to study this contribution, Monte-Carlo simulations were first performed by using a δ-function like rocking curve. Hit patterns in the detector plane are shown in Fig. 3 when using a source defined by a circular aperture of diameter 100 mm placed at the Rowland circle. The detector distance is chosen according to the focal condition. In Fig. 3 is shown (from left to right): (i) The full height of the image is ≈ 140 mm, where for a cylindrically bent crystal an image height of ≈ 280 mm is expected. FWHM of the curvature-corrected reflection corresponds to 102 meV (in cases (i)-(iii)). This width is about a factor of 3 larger than expected from the Johann shift only. It is understood, that for such large Bragg angles the fan out of the tilted ellipses already dominates in contrast to smaller Bragg angles. The complete image contains 568.000 hits. 10

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Figure 3: Monte-Carlo simulations (using 106 crystal points) of the hit pattern in the detector plane for a δ-function like rocking curve. The crystal’s bending radius is Rc = 2982 mm, its diameter bc = 90 mm, ΘB = 61.5◦ , and the calculated radius in the apex rapex = 1422 mm.

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The uncertainty for rapex and zapex from a parabolic curvature fit (indicated in the corresponding drawing) is of the order ± 3 mm and ± 0.05 mm, respectively. The angular set-up corresponds to the Si 111 reflection of the µH(3p − 1s) X-ray transition (2.25 keV). Note the different scales for horizontal and vertical detector coordinates xd and zd .

(ii) When using a detector of 72 mm height and centred in the symmetry plane 71% of the diffracted intensity is covered. Curvature of the reflection and position of the apex are still well reproduced for such detector limits. (iii) A vertical ”misalignment” of the crystal by + 20 mm results in an upward shift of the reflection. Only 43% of the intensity are recorded. (iv) The intensity for set-up (iii) maybe optimised by tilting the crystal around the horizontal axis through its centre. Now, 63% of the intensity are recorded, the line width, however, increases to 153 meV because fanning out due to the additional tilt of the ellipses becomes significant (see also Fig. 5 - left). 11

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Fig. 4 displays the same set-up but now using the rocking curve as calculated by the code XOP [17] for a plane silicon crystal and the (111) reflection of 2.25 keV X-rays. The polarisation-averaged rocking curve width is found to be ωc = 45

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seconds of arc (FWHM), which corresponds to 218 µrad or 266 meV. As for the

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case of the δ-function like response, the full height of the image is ≈ 140 mm and the detector/image overlap amounts to 71% (ii), 43% (iii), and 63% (iv). FWHM of the reflection is found to about 285 meV for cases (i)-(iii), which

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increases to 345 meV in the last case. The additional broadening for case (iv)

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is now only about 20% because of the finite width ωc (Fig. 5 - right).

Figure 4: Monte-Carlo simulations (using 106 crystal points) of the hit pattern in the detector plane as described in Fig. 3 but using the rocking curve for the Si 111 reflection at 2.25 keV as calculated by the code XOP [17] (insert bottom - left). Uncertainties for rapex and zapex from the parabolic curvature fit are now of the order ± 15 mm and ± 0.15 mm, respectively.

In order to determine the curvature of the image in the detector plane, a fit to a parabola is better suited than an arc of a circle because at the detector limits the influence of higher orders becomes noticeable. On the other hand, the parabola fit xd (y) = A · zd2 + B · zd + C turns out to be sufficient, where 12

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the detector coordinate xd denotes the direction of dispersion (horizontal axis) and zd the height (vertical axis). An ideal adjustment given, the apex of the parabola is located at the detector coordinate origin (xd , zd ) = (0, 0) with an

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apex curvature radius

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rapex ≈ Rc · cos ΘB

(4)

and with a value for the parabola parameter B ≡ 0 (see Sec. 2). In any real

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experiment, the reflection will be shifted and rotated relative to (xd , zd ) = (0, 0), where the parameter C of the parabola fit accounts for a shift along the axis of

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dispersion x. Apex displacement in the vertical direction and apex radius are given by

=

rapex

=

B 2A 1 − . 2A

−

and

(5) (6)

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zapex

A parabola fit yielding values for both B and C significantly different from zero indicates a misalignment of the crystal height relative to the detector, in

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particular when rapex deviates from the theoretical value Rc · cos ΘB . This results usually in an apex shift in the vertical direction. A possible miscut (finite angle between the diffracting planes and the crys-

tal surface) may cause similar shifts than a crystal displacement. On the other hand, a miscut when oriented along the direction of dispersion, actually constitutes the asymmetric Bragg case. Therefore, the parameters cut angle and orientation are included in the Monte-Carlo routine. Thus, simulations together with additional experimental studies allow for the precise characterisation of Bragg crystals [25]. Deviations from the focal condition Rc · sin ΘB maybe mandatory in some cases. E . g., (i) the crystal’s bending radius Rc or a possible miscut is unknown or (ii) the simultaneous measurement of two (or more) lines of different Bragg angle needs to accept a compromise for the detector distance. In case (i), the curvature may be extracted from comparing the reflection convoluted with the calculated response at a given distance with the measured line shape, where the 13

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Figure 5: Distortion of the crystal response by misalignment. Displayed are the most left (i)

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and most right case (iv) of the δ-function like (dots) and Si (111) response (solid line) shown in Figs. 3 and 4. The height misalignment together with the readjustment of the reflection by tilting the crystal induces a broadening and a small asymmetry on the left side, i. e., towards

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smaller energies.

best agreement yields a measure for Rc [26]. In case (ii), in order to extract a line-shape distortion caused by Doppler broadening or the hadronic interaction

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needs the precise and individual knowledge of the response for each line. Fig. 6 displays the development of the line shape for the pionic neon line

πNe(6h − 5g) at various distances. A Monte-Carlo simulation (Fig. 6 - left) is compared to data taken at the meson factory of the Paul Scherrer Institut (Fig. 6 - right). The natural line width of the πNe(6h−5g) transition is negligibly small (< 1 second of arc) compared to the rocking curve width (≈ 13 seconds of arc). The line width is dominated by the geometrical broadening (≈ 26 seconds of arc at focus). Noteworthy, that in a real experiment, as shown in Fig. 6 - right, counting statistics is typically two orders of magnitude smaller than in the Monte-Carlo examples presented here. Consequently, uncertainties of rapex and zapex as obtained from the curvature fit to the data are a factor of about ten larger. For the data set ”focus+1.5 mm”, the parabola fit yields zapex = (1 ± 2) mm and rapex = (1900 ± 70) mm, where the calculated value is rapex = 2082 mm.

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Figure 6: Comparison of a ray-tracing Monte-Carlo simulation and data demonstrating the defocussing effect for the πNe(6h − 5g) line (4.51 keV) using the silicon 220 reflection of a

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crystal with a diameter of 90 mm and a bending radius of Rc = 2982 mm. Note, that the integral of each measurement is adjusted to the one with the maximum count rate (focus

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+1.5 mm).

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4. Set-up 4.1. Mechanical set-up

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A Johann set-up assumes a sufficiently homogeneous extended source in

order to cover a particular wavelength interval. Hence, for long-term precision measurements, the stability of the crystal orientation towards the detector is decisive, whereas small angular changes towards the source are irrelevant. Therefore, the concept for the mechanical set-up is to provide a common rigid

support bar bearing both crystal mounting and detector (Figs. 7 and 8). A highprecision angular encoder having a resolution of 0.13 seconds of arc, integrated in the support bar, monitores the orientation crystal-to-detector. The Braggangle setting is stabilised to ≈ 0.1 seconds of arc by means of a piezoelectric device (see also ref. [3]). The use of identical material with respect to the central axis of the support bar minimizes any tension or torsion, in particular, when temperature can be kept stable to about 2◦ C or better. A second angular encoder (resolution 1.5 seconds of arc) controls the position of the common support towards the X-ray source. In this way, setting (by stepper motors) and monitoring of Bragg angle (orientation crystal-to-detector)

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and movement of the support bar (orientation source-to-crystal) are completely decoupled.

A further requirement was to cover all Bragg angles in the range between

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30◦ − 70◦ . This is achieved by using a ball-bearing support for the complete

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crystal-detector support and the exchange of only one connector flange at the crystal chamber. A set of 7 flanges is sufficient to cover the full range. A vacuum system for the path source-crystal-detector is mandatory because

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of the low X-ray energies. Nevertheless, for a given mechanical set-up fine adjustments, e.g., the focal length must be performed also under measuring

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conditions. A scan of the source’s intensity distribution in the direction of dispersion is achieved, meanwhile keeping the orientation crystal-to-detector, by a rotation of the support bar. Such movements are made possible by one bellow

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before the detector and a hinge consisting of two short bellows at the crystal

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chamber towards the source. Within such a set-up, a Bragg angle variation of ± 4◦ is possible without breaking the vacuum. The vacuum induced forces must be kept minimal in order to stay approxi-

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mately in a mechanical equilibrium during such operations. Therefore, the force on each of the above-mentioned bellows is counteracted by a compensation bellow of identical cross section. In this way, the mechanical set-up is robust under smaller changes of the external conditions and stable enough to allow measurement periods lasting for weeks. 4.2. Bent crystal

The Bragg crystals are manufactured by attaching thin polished silicon or quartz slabs to a concave glass surface of optical quality (about λ/12), which defines the bending radius Rc (Fig. 9 - left). The adhesive force is provided exclusively by molecular forces between the crystal and glass surfaces, which eliminates the distortion of the bent surface by volume effects of any glue. Different glass material is selected for quartz and silicon to match as good as possible the thermal expansion coefficients. The crystal slabs are disks of up to 100 mm in diameter. In the experiment, an aluminium aperture of 95 mm diameter is 16

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Figure 7: Top view of the spectrometer set-up. Here, as an example for an X-ray source, the set-up is shown of a fluorescence target mounted inside a vacuum chamber and excited by means of an X-ray tube. A vacuum system prevents X-ray absorption. For exotic-atom measurements, the source is replaced by a dedicated stop arrangement for particle beams used to optimize the formation of muonic, pionic, or antiprotonic atoms [1, 27, 28, 29, 30].

applied to avoid boundary effects. For smaller Bragg angles, in addition, the horizontal dimension can be reduced to limit the Johann shift (Fig. 9 - right). Reflection adjustment in the detector plane is achieved by rotating the crystal around a vertical and a horizontal axis. 4.3. X-ray detector Charge-coupled devices (CCDs) provide ideal two-dimensional position-sensitive detectors because of the inbuilt granularity. In the experiments mentioned here, CCDs with pixel sizes of 22.5 [31, 32] and 40 µm [33] have been used. For the 17

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Figure 8: Side view of the spectrometer set-up. A common support arm for crystal and X-ray

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detector defines a robust and stable Bragg angle configuration. The hinge towards the X-ray source is made from a double bellow and is drawn rotated for display only (see also Fig. 7).

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geometry chosen in the examples shown in Figs. 3 and 4, one pixel of 40 µm

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corresponds to 18.6 meV in the direction of dispersion. CCDs are operated at a temperature of −100◦ C cooled by means of liquid nitrogen.

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In an accelerator environment, a significant beam-induced background level is present because of nuclear reactions and decay products of muons, pions or antiprotons. This results in a high γ-background which, however, is recognised by the hit pattern created in the sensitive layer of the CCD. Charge created via photo effect by few keV X-rays is deposited in a single or two adjacent pixels only, whereas Compton- or particle-induced events produce larger clusters. Hence, by selecting small clusters background suppression by orders of magnitude becomes feasible [28]. 4.4. Verification of the response function In the few keV range, the intrinsic energy resolution of silicon and quartz is of the order of a few 100 meV. Hence, the experimental verification of the (calculated) spectrometer response requires narrow X-ray lines. Kα fluorescence radiation in the range of 1.7 − 10 keV (Z = 14 − 32), however, has natural line widths of 0.52 − 2.8 eV and narrow nuclear γ-rays of sufficient intensity do not exist. But two other methods can be used for comparison of calculated and 18

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Figure 9: Left - glass lens with an attached (transparent) quartz (10¯ 1) crystal. The thickness of the crystal slabs is 0.2 − 0.3 mm. The white material at the boundary serves as sealing. Right - adjustable crystal support. Vertical and horizontal rotation axes meet in the centre of

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the crystal’s surface. The horizontal limitation by an aperture is applied to limit the Johann

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measured response. The first method exploits information obtained from the measurement itself. For the second one a dedicated set-up is needed which allows crystal characterisation to a much wider extent: (i) Exotic-atom X-rays in the intermediate range of the de-excitation cas-

cade have natural line widths of a few meV or less. Transitions (n − n′ ) with n ≈ 4 − 8 from medium Z elements are well suited, because they

occur at a distance to the nucleus, where finite-size and, if applicable, strong-interaction effects do not influence the line shape, and line yields are already sufficient to achieve the necessary statistics [34, 35]. For example, in the study of the Lα line in antiprotonic hydrogen (1.74 keV), the

p ¯3 He(5g − 4f ) transition (1.69 keV) was used to determine the response of a quartz 100 reflection [28] (Fig. 10). For pionic hydrogen, the (3p − 1s) transitions (2.88 keV) is close in energy to the πC(5g − 4f ) line (2.97 keV), where a dilute target (to prevent self absorption in the gas) is realised by using CH4 [36]. This method suffers from the fact, that measuring periods

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of days are needed to achieve the necessary statistics for a single spectrum. (ii) In few-electron systems, narrow lines occur because the Auger pro-

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cesses dominating the line width of fluorescence X-rays are absent. M1 transitions, e. g., in helium-like sulphur, chlorine or, argon have life times

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as small as 10−8 s [37]. Their energies match almost perfect with the energies of some Lyman transitions in pionic hydrogen and deuterium [29,

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36]. By using an electron-cyclotron resonance source to produce the fewelectron atoms, high count rates are achieved allowing a detailed study of crystal properties [38, 26].

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When comparing calculated and measured response, it turns out that the folding of the intrinsic response of the plane crystal with the geometry by using the ray-tracing Monte-Carlo code requires to include a third contribution. This

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contribution is sufficiently modelled by a Gaussian and is attributed to imper-

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fections in crystal mounting and surface. It was found to amount maximum to about 30% of the contributions rocking curve and geometrical aberration and

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decreases with X-ray energy. Hence, after convolution the total effect of the Gaussian on the line width is marginal. The method is described in refs. [38, 26]. 5. Applications

In the following, some examples from recent exotic-atom and fluorescence

studies are mentioned to illustrate the performance of the spectrometer. One example spectrum is shown to demonstrate the quality of the data: • Resolving the fine-structure splitting of X-ray transitions from the intermediate part of the atomic cascade give access to details of the deexcitation mechanisms of exotic atoms. The example spectrum displays parallel transitions in antiprotonic 3 He (Fig. 10). The energy splitting of the transitions p ¯3 He(5g − 4f ) and p ¯3 He(5f − 4d) was measured to (907 ± 12) meV confirming a QED calculation yielding (913 ± 1) meV [39]. The p ¯3 He(5g − 4f ) line serves also as response function (FWHM = 272 meV) for the measurement of the p ¯H Lα transition [1, 28] (Sec. 4.4). 20

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Figure 10: X-ray transitions from the intermediate part of the de-excitation cascade in antiprotonic 3 He measured at LEAR (from [28]). The intensity ratio of the p ¯ 3 He(5f − 4g) to the p ¯3 He(5g − 4f ) line determined to be (7.7 ± 0.6)% yields immediately (21 ± 2)% for the relative population of the 5f and the 5g state. Bending radius, effective diameter of crystal,

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measuring time was 24 hours.

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and distance X-ray source-to-crystal were Rc = 2979 mm, and 95 mm, and 2.25 m. Total

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• Comparing the line widths of the (3p − 1s) transitions in muonic and pionic hydrogen reveals in both cases a significant line broadening which, however, is of completely different origin. In the case of µH, the increase in

line width is due to Doppler broadening originating from a non-radiative de-excitation step (n − n′ ) between higher lying atomic levels [40]. In πH,

in addition to Doppler broadening one observes an increase of the natural line width stemming from the nuclear force exchanged between the proton and the pion [1, 36].

• New X-ray standards in the energy range 2 − 3 keV became accessible because of the simultaneous measurement of X-rays from several charge states of few-electron systems. The energy determination reaches accuracies of 3 − 25 meV [38, 26, 8]. • Exotic-atom X-rays from the intermediate part of the de-excitation cascade maybe be used as X-ray standards itself as their transition energies

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can be calculated to a precision of 1 meV by means of QED [35, 8]. E. g., the transition π 14 N(5g − 4f ) was used to determine precisely the energy of titanium Kα fluorescence radiation [6], and the line π 16 O(6h − 5g) serves

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as energy calibration in the determination of strong-interaction effects in

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pionic hydrogen [30].

• 3d transition metals show significant chemical effects even for Kα fluores-

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cence radiation both for the line energy and the line shape. A study for the case of manganese is introduced in ref. [3].

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• (4p − 2s) transitions in the alkaline earth elements like barium show a complicated and broad structure because of the interplay of outer shell electrons [41]. Using the Johann-type set-up, the count rate is superior to

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6. Summary

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a scanning device by one order of magnitude [42].

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A Johann-type Bragg spectrometer is described which is set up in robust mechanics in order to achieve sufficient stability for long-term low-rate measurements in exotic-atom research. Furthermore, such a device is perfectly suited for high-statistics applications in X-ray fluorescence studies. Symmetry considerations provide simple rules of thumb for imaging properties and rate estimates to allow a quick assessment of measuring conditions. They are verified by Monte-Carlo ray-tracing calculations based on intrinsic crystal properties and experiment set-up, which include in principle geometrical aberrations to all orders. Applications in various fields of X-ray research are quoted.

Acknowledgements We are grateful to N. Dolfus, H. Labus, B. Leoni, U. Rindfleisch, and K.P. Wieder for the skilful realisation of the spectrometer concept. In the fabrication of the Bragg crystals the inspiring collaboration with Carl Zeiss AG, Oberkochen, Germany, is acknowledged. 22

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[42] Th. Strauch, diploma thesis, Universit¨ at zu K¨ oln, Germany (2005).

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3. Performance demonstrated with novel results in exotic-atom and fluorescence X-ray experiments.