HIGH RESOLUTION X-RAY SPECTROSCOPY WITH THE DOUBLE CRYSTAL SPECTROMETER

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HIGH RESOLUTION X-RAY SPECTROSCOPY WITH THE DOUBLE CRYSTAL SPECTROMETER

W. Sauder

To cite this version:

W. Sauder. HIGH RESOLUTION X-RAY SPECTROSCOPY WITH THE DOUBLE CRYS- TAL SPECTROMETER. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-83-C9-86.

�10.1051/jphyscol:1987911�. �jpa-00227217�

Colloque C9, supplement au n012, Tome 48, d6cembre 1987

HIGH RESOLUTION X-RAY SPECTROSCOPY WITH THE DOUBLE CRYSTAL SPECTROMETER

W. C. SAUDER

Department of Physics and Astronomy, Virginia Military Institute, Lexington, VA 24450, U.S.A.

R 6 s d

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Abstract.

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1. Introduction.

The monolithic double crystal spectrometer (MDCS) is an x-ray spectrometer in which two plane surfaces parallel to chosen diffracting planes are carved into a single crystal mono1 i th. Obviously, wavelength scanning cannot be achieved by varying the dihedral angle between the two surfaces as is done in the classical double crystal spectrometer (DCS). Rather, tuning is effected by rotating the monolith about an axis that (1) lies in the plane defined by the normals to the two diffracting planes (the plane of dispersion in the DCS); and (2) is perpendic- ular to the incident beam. Thus the tuning angle for the MDCS i s in the so-called vertical divergence direction of the DCS.

The device is capable of high resolution as was demon- strated by the spectrum of CuKa obtained with a silicon MDCS [I].

The structure visible in Fig. 1 on both the low energy side of Ka, and the high energy side of Ka, has not been well-delineated

,

by observers employing other ;.= 9

types of spectrometers. On the ^.m other hand, the data of Fig. 1

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are compatible with these other .- 3

++ #

++

+ ⁺ + ⁺ + ⁺ + + + ⁺

observations. In an attempt to + +

account for this situation, we have carried out a full deriva- tion of the MDCS apparatus window, so that the theoretical resolutions of the MDCS and DCS can be compared.

Fig. 1

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Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987911

C9-84 JOURNAL DE PHYSIQUE

2. Theory of Two Crystal Spectrometers.

The use of the Ewald Sphere to describe geometrical diffraction theory is well known [2]. In this geometrical representation of Bragg's law, the incident and diffracted wave vectors form the sides of an isosceles triangle, while a reciprocal lattice vector forms the base. This notion is often extended to dynamical theory by using internal wave vectors for the sides of the triangle [3].

But it is also possible to characterize dynamical diffraction by a vector triangle consisting of external wave vectors and a modified vector H' in reciprocal space:

H' = H (1 + ~ 8 / t a n e ) , (1) where H is a reciprocal lattice vector of length l/d, e is the incident grazing angle, and A0 is the deviation of 8 from the Bragg angle.

Two successive diffractions take place in the MDCS. As illustrated by Fig. 2, the incident, intermediate, and outgoing wave vectors define the sides of an equilateral tetrahedron having a base formed by the two modified reciprocal lattice vectors and their s um

H' = H I 1 + Hz'. ₍₂₎

4

The direction of the incident beam can be

specified by a pair of angles, $ and $,. The ^I I

first is the vertical divergence angle between

'4

I(

the wave vector and the base of the H' tetrahedron, while the second is the angle

between the projection of k, on the base plane ^{Fig. 2}

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and the vector H,'

.

The geometry of the tetrahedron provides a relationship between the wave- length (the reciprocal of the magnitude of each of the wave vectors) and the tun- ing angle $:

A = 2 sin a cos

H' ⁹ (3)

where a is the angle between H,' and Hz'. Notice that, for $I sufficiently small, the dispersion of the MDCS can be quite large; the derivative of Eq. (3) yields

d$/dA =

-

The intensity of the radiation emerging from the MDCS is determined by the reflection coefficients C, and C of the two diffracting planes. These coeffi- cients, in turn, dependpupon Aetp;nd Ae,, respectively. Because the planes of incidence for H,' and Hz' are not parallel, polarization mixing takes place, so that polarization states p and p' must be specified for the intensity reflection coefficients. For a perfectly coll imated incident beam with polarization compo- nents of intensity In and Is, the transmitted intensity is a sum of four terms:

'i

where p and p ' each range over two polarization states, and the polarization coef- ficients PDQ' are established by a mixing angle a':

. .

P p p l = cosZ a', p = p'

= sin2 a', p

*

The MDCS is normally operated with small $; under these conditions, a' n, so that little polarization mixing takes place.

The incident beam must be collimated in the $-direction, but need not be so in the $,-direction. This collimation can be described by a slit function S($-$,) that specifies the transmission of the incident beam about an angle $,, the tuning

source spectrum g(h) i s then computed b y i n t e g r a t i n g over a.11 d i r e c t i o n s i n t h e d i v e r g i n g i n c i d e n t beam and over a l l wavelengths:

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The angular s e t t i n g $, can be converted t o an e q u i v a l e n t wavelength s e t t i n g A, by means o f Eq. ( 3 ) . I f t h i s i s done, then considerable m a n i p u l a t i o n o f Eq.(7) leads t o a c o r r e l a t i o n i n t e g r a l i n v o l v i n g t h e apparatus window R:

The f u n c t i o n R i s a sum o f f o u r p o l a r i z a t i o n terms, each o f which i s a convo- l u t i o n o f t h e form

Rppl ( z ) =

I

-

where

F i n a l l y , t h e double c r y s t a l f u n c t i o n C i s a c o n v o l u t i o n o r a c o r r e l a t i o n i n t e - g r a l o f the two c r y s t a l r e f l e c t i o n ~ ~ ' c o e f f i c i e n t s :

The upper ( p l u s ) s i g n i s used i f b o t h $, and $, a r e p o s i t i v e , w h i l e t h e lower s i g n i s taken i f e i t h e r $, o r

?,

The f u n c t i o n E(E) i n Eq. (9) i s r e l a t e d t o t h e s l i t f u n c t i o n ; f o r a symmetric s l i t i t i s j u s t

E(E) = S/$

,

= 0 otherwise. (11)

Here, q~ i s a f u n c t i o n of 5: $ = + (9,'

-

L e t a, be t h e r e l a t i v e w i d t h (e.g. 6h/h) o f C,

.

PP' w 1 l c o t $11

W(l) = I C O ~ $11 + [ c o t $,I (12)

A corresponding r e l a t i o n s h i p holds between t h e second c r y s t a l c o n t r . i b u t i o n w(2) and t h e r e l a t i v e w i d t h w,. As a r e s u l t t h e r e s o l u t i o n o f t h e apparatus window depends upon b o t h wl and u,; each a r e weighted by cotangent f a c t o r s .

T h i s a n a l y s i s a p p l i e s t o t h e DCS as w e l l as t o t h e MDCS, except t h a t , f o r t h e DCS, a becomes t h e t u n i n g v a r i a b l e and go i s constant. For a p e r f e c t l y a l i g n e d system, go i s equal t o zero; t h a t i s , $, i s a measure o f s l i t misalignment.

Furthermore, t h e r e i s no p o l a r i z a t i o n m i x i n g i n t h e DCS, because t h e planes o f incidence o f t h e two c r y s t a l s are p a r a l l e l . F i n a l l y , 0, and 8, (as w e l l as I$, and 8,) are complimentary angles, so t h a t , f o r example, [ c o t $,I = t a n 8,.

3. Comparison o f Resolutions.

One now has a b a s i s f o r comparing t h e r e s o l u t i o n s o f t h e MDCS and DCS. There a r e t h r e e f a c t o r s t h a t can y i e l d h i g h e r r e s o l v i n g powers f o r t h e MDCS than f o r t h e DCS :

(a) The MDCS i s mechanically more s t a b l e than t h e DCS a g a i n s t v i b r a t i o n and thermal f l u c t u a t i o n s , both o f which tend t o smear o u t f e a t u r e s i n t h e observed spectrum.

C9-86 JOURNAL DE PHYSIQUE

(b) In either an MDCS or a DCS employing a lower order and a higher order diffraction, there is an effect due to the fact that the higher order diffraction width is less than that for the lower order. The weighting illustrated by Eq.

(12) means that the higher order diffraction contributes more heavily to the window width. This effect increases with $,, so that an MDCS has a higher resolv- ing power than a DCS employing the same diffraction planes.

(c) The function E appears in the convolution integral of Eq. (9). A mea- sure of the contribution by the slit system to the window width is provided by the second moment of E [4].

(13) (For convenience, we have taken E to be normalized.) For a symmetrical slit system, Eq. (13) can be reduced to

$2) = ~ ( 4 )

-

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In a perfectly aligned DCS system, the third term drops out (because $, = 0), so that better collimation of the input beam is needed for the MDCS in order to match the slit contribution in an equivalent DCS.

However, the most difficult alignment to make for the DCS is that of minimiz-

ing [ 5 ] : In fact, in most precision DCS spectroscopy, the stated slit mis-

alignment is of the order of the angular width of the slit system [6]. For a collimation system consisting of two slits of equal height and a slit misalignment equal to the angular width of the slit system, Eq. (14) leads to a four-fold increase in the contribution of the slit width to the overall resolution.

4. Work i n Progress.

Now that we better understand why the resolution of the MDCS can exceed that of the DCS, the fact that we have found structure in CuKa not seen by others seems plausible. Given the refined data of Fig. 1 and the theoretical results of sec- tion 2, we have begun to reanalyze our previous results. Fourier techniques are being applied to strip the apparatus window from the observed data in order to improve the fitting of a theoretical model for the structure to the spectrum.

Furthermore, a repetition of the CuKa experiment has begun using an optimum MDCS. This, along with an improved collimation system, will lead to a factor of six enhancement of resolution.

References

[l] Sauder, W. C., Huddle, J. R., Wilson, J. D., and LaVilla, R . E., Phys. Lett.

63A (1977) 313.

[2] G e s , R. W. in Solid State Physics, ed. by F. Seitz and D. Turnbull (Academ- ic Press, New York, 1955), Vol. 15, p. 55.

[3] Batterman, B. W. and Cole, H. Rev. Mod. Phys.

36

[4] Sauder, W. C., J. Appl. Phys.

37

[5] Bearden, J. A., and Thomsen, J. S., J. Appl. Crystallogr.

4

[6] Bearden, J. A., Henins, A., Marzolf, J. G., Sauder, W. C. and Thomsen, J.

S., Acta Crystallogr.

A24