The asymmetric Johann spectrometer and its application to plasma diagnostic in tokamaks

Nuclear Instruments and Methods in Physics Research 221 (1984) 453-459 North-Holland, Amsterdam

453

THE ASYMMETRIC JOHANN SPECTROMETER AND ITS APPLICATION TO PLASMA D I A G N O S T I C IN T O K A M A K S Rosario B A R T I R O M O , F r a n c e s c a B O M B A R D A * a n d Ruggero G I A N N E L L A Associazione E U R A T O M - E N E A sulla Fusione, Centro Ricerche Energia Frascati, C.P. 65 - 00044 Frascati, Rome, Italy

Received 17 December 1982 and in revised form 4 July 1983

The application of an asymmetric version of the Johann crystal spectrometer to the X-ray spectroscopy of a tokamak plasma is considered. Limits of resolution caused by various mechanisms of instrumental broadening are shown to improve with respect to the symmetrical mounting. The problem of the spectrometer throughput is considered and provisions to increase its value are given.

1. Introduction

High resolution X-ray spectroscopy of tokamak plasmas has recently gained great importance as a versatile diagnostic tool. Determination of ion temperature has been obtained from the Doppler broadening of the emission line of helium-like iron impurities [1]; electron temperature and ionization equilibrium have been evaluated from satellite to resonance line intensity ratios [2]. All these measurements have been obtained by means of Bragg spectrometers in the Johann configuration [3]. In this paper we show that it is possible to increase the quantity of light collected from the source with a modified version of this instrument that utilizes an asymmetrically cut crystal. As a matter of fact this modification of the Johann mounting was first used by Guiner [4] in diffraction experiments in order to shorten the distance of the X-ray source from the monochromator crystal while accommodating the experimental apparatus at a larger distance. In sect. 2 the asymmetrical Johann mounting is illustrated and its geometrical aberration and wavelength broadening are discussed. The spectrometer luminosity is considered in sect. 3 and in sect. 4 a practical example is presented. Finally in sect. 5 we discuss our conclusions.

Rowland circle of radius R. X-ray photons of wavelength h incident on the crystal at an angle O B relative to the diffracting planes given by the Bragg relation are focussed on the Rowland circle [5]. Photons of different wavelengths are focussed at different points and a position sensitive detector allows measurement of the spectrum of the plasma in a range of wavelength simultaneously. In the asymmetric Johann mounting the analysing crystal is cut with its surface at an angle • with the diffracting planes. Bragg diffraction by such a crystal is said to be asymmetric and in this case the angle O 1 = O B + • between the incident beam and the crystal surface is no longer equal to the angle of the reflected beam O2 = O B - 4. Furthermore the distances from the source to the crystal and from the crystal to the detector are no longer the same, their ratio being equal to the asymmetry parameter b = sin O1/sin O 2. A dynamical treatment of X-ray diffraction by crystals leads to an expression for the integrated reflec-

CRYSTAL

2. The asymmetrical Johann mounting

The principle of operation of a Bragg spectrometer in the Johann configuration is illustrated in Fig. 1" the crystal is bent to a radius 2R and has a focussing

* Guest graduate student.

0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Fig. 1. Schematicof a Johann spectrometer for plasma diagnostic.

454

R. Bartiromo et aL /' <4.D'mmetrlc Johann spectrometer

~°1-. . . . . . . . . . . . . . . . 0.8 ~. NI][ '

l i

with q). To a good approximation the integrated rcflectivity is inversely proportional to {h : :~ •

PH 0.6 I P0 0.4 0.2 ~i

0.0 [

0 0B

Fig. 2. Diffraction patterns at different values of the asymmetry angle ~. Reflection from the 223 plane of a-quartz at X = 1.85 ,~. The values of • and 9? relative to each curve are as follows:

I II III IV V

4,(o)

,.~

- 52.57 -26.28 0 26.28 52.57

2.32x 10 5 1.56x 10 s 1.25×10 5 9.93× 10 6 6.00× l0 -6

Only the normal polarization component has been considered.

tivity .~ as a function of b [6]. In fig. 2 we show an example of how the diffraction pattern changes with ~; the curves are relative to the 223 reflection from a thick plate of a-quartz at X = 1.85 A [7]. The variation of the maximum of the reflectivity is moderate even at ~ - _+O a while the width wo and therefore the integrated reflectivitv, decreases sensibly

3.5 F....... I 3.o 2.5 [ i

2.17 il 1.5 ~-

:

-7

t

I

i [.

I

0

/ sin O~

1

60svm

An example of full dynamical calculation of these quantities is given in fig. 3. It is worth noting that for an asymmetrical mounting the angular width of acceptance ~0, through which the crystal integrated reflectivity is determined, differs from the angular divergence ~oH of the diffracted beam which affects the resolving power of the crystal. This latter quantity can be deduced from the same curve of ~o. by simply reversing the angle q~; therefore ~oH.= { b ( ( b ) ¢% ~ ( _- ~

=b(q))

sinO, sin 0 2

"

(2)

The possibility of continuously controlling the crystal integrated reflectivity through the asymmetry degree can be exploited in order to improve the throughput of the spectrometer still maintaining its resolving power. A further advantage of the asymmetrical Johann mounting is represented by the possibility of choosing the incidence and reflection angles at the crystal surface quite independently of O B in order to optimize the geometrical aberrations due to the finite length l~. and height h~ of the crystal 15]. The broadening due to l~ can be expressed as [8,9]: (~-)L=(

lc ,2cot O2

-T -c°t o.

(3)

It is interesting to note that when O 2 = ~'/2 then AXE = 0 , but this result can be exploited only when O n > ~r/4, otherwise a reflection configuration would be impossible. In Appendix A a detailed formulation is given to find an analytical expression for the aberration due to the finite height of the crystal h~ and the detector hD:

/ ~

~,

/ ~ , ~ / . /

./

/./

1.17 0.5 '1

sym

¢%

./'

~

•I'l'i''''l

L_( • ~ -80 -60 -40 -20

0

X l 20

x , 40

A 60

L

] 80

(degrees) Fig. 3. Curves of reflecting power ( ), intrinsic angular width (---) and parameter b ( - . - ) as a function of the asymmetry angle. The first two curves are normalized to their values at q) = 0. Reflection from the 223 plane of a-quartz at X = 1.85

This value is lower than the corresponding broadenmg in the symmetrical mounting for ~ ranging from 0 to 2 0 B - or, the m i n i m u m being reached at the upper limit where 0 2 = ~r/2 and AX L = 0 as well. As far as geometrical aberrations are concerned, an asymmetrical mounting with O 2 = ~r/2 has a better resolving power when compared to a symmetrical Johann or Johansson [10] one. In fact, in the latter case the focusing defects due to l¢ are avoided by grinding the crystal surface to a radius 2R, but those due to h~ are the same as in the Johann mounting. Alternatively, as a consequence of the reduced geometrical broadening of the asymmetric mounting, a crystal with larger dimensions can be used while main-

R. Bartiromo

et al. / Asymmetric

455

Johann spectrometer

from the asymmetry degree, we have shown that all the other factors are smaller when @ < 0, see fig. 4. The optimum instrumental resolution is then obtained when 0, = 7r/2.

3. The spectrometer luminosity

0

’ -60

-40

-20

I

I

I

0

20

40

The spectrometer luminosity plays an important role when thinking of its applications to pulsed machines. In order to evaluate the instrument throughput, it is necessary in the first place to define the vertical dimension of the source with respect to the height of the detector and the crystal. If the detector height ho is such as to satisfy the inequality

Cp(degrees) Fig. 4. This figure shows the behaviour of the different contribution to the resolving power of a Johann spectrometer when the angle @Jchanges. Reflection from the 221 plane of a-quartz at h = 1.85A.

taining the resolving power of the instrument, but appreciably increasing its luminosity (see sect. 3). Another important contribution to the image broadening on the focal circle is given by nonuniformities of the curvature radius across the crystal surface. Cauchois [l I] gives the following expression for the spectral broadening due to curvature defects: -Ah

i x

1 co

= I,I*r (2R)2

- 2Rl cot Q,,

i -X

1 i

WH

tan 0, z-z=

sin 0,

%ym

tan 8,

/--

sin 0,

’

(5)

As the linear dispersion changes with the degree of asymmetry the spectral resolving power of the instrument changes too:

8xcot 0, 2 R sin 0, ’

h’=h,

2R sin 0, + d 2Rsin8,

’

it can be considered

as unlimited in this direction. The instrument throughput can be evaluated with the results of Appendix B from which the following expression is found: Lm = aGh,ho sin Qi 2R sin 0,

where 12r - 2RJ represents the maximum deviation of the curvature radius. Two more factors affect the spectral resolution of the Johann spectrometer, namely the crystal intrinsic resolving power and the space resolution Sx of the detector. The first is due to the angular broadening of the reflected beam wn and is equal to AX

where d is the distance of the source from the focal point of the Rowland circle (positive if the source is outside the circle), then the crystal height determines the acceptance angle in the vertical direction. If the source has a vertical extension greater than

(6)

This shows that the detector contribution to the spectral broadening is greater than in the symmetrical geometry, as long as QI > 0, whereas for @ ( 0 it decreases continuously down to a minimum value, again corresponding to 0, = n/2. Apart from curvature defects that are independent

i 2.5 2.0 1.5 l:o 0.5 -

- -60

-60

-40

-20

0

20

40

60

80

@ (degrees) Fig. 5. A plot of the luminosity versus the angle Cpfor a limited source () and an extended source (- - -). The reflection from the 221 plane of a-quartz at X = 1.85 A has been considered. The curves are normalized to their value at @ = 0.

456

R. Barliromo tel al. /' A,~Tmrnelrtc Johann spectrometer

T h e ratio with the c o r r e s p o n d i n g value for the symmetrical m o u n t i n g is given b y L~

_

L~m

~

sinO~

_/sinO~

~sym sin 02

-

V sin O 2 '

a n d is s h o w n as a d o t t e d line in fig. 5; the curve g r o w s s m o o t h l y with q~. A d i f f e r e n t result is o b t a i n e d w h e n the source d i m e n sion is limited so that h~ < h'. In this situation the t h r o u g h p u t is f o u n d to be

L=

2~ hc ,,2 x + T

l ;2_;;

-

l,:h~h~ sin O, 2 R sin Oa + d '

×

and

~

L

Lsym

~sym

sin

~}2

sin O~ '

w h e n d - 0. In this case it can b e seen in fig. 5 that there is a gain w h e n q~ < 0.

4. A practical e x a m p l e

Fig. 6. The dashed area is accessible to photons of wavelength ~, diffracted by a thin slab of crystal. The plane of the figure is perpendicular to the normal to the diffracting planes.

A s an e x a m p l e , let us evaluate the resolving p o w e r a n d the t h r o u g h p u t o f an a s y m m e t r i c version o f the existing J o h a n n s p e c t r o m e t e r for the p l a s m a d i a g n o s t i c o n the F r a s c a t i t o k a m a k . T h e d i f f r a c t i n g e l e m e n t is a p l a t e o f a - q u a r t z (/c = 10 cm, h c = 4 cm), cut parallel to the 223 p l a n e ( d = 1.0148

•~) a n d b e n t to a radius 2 R = 385 cm. T h e X - r a y source used for testing e m i t t e d a t ~ = 1.79 •~ ( C o K,,) resulting in a Bragg angle O B = 61.88 °. T h e e x p e r i m e n t a l value for the d e t e c t o r c o n f u s i o n w i d t h was 8 x = 0.4 m m .

Table 1

(AX~ { 1¢ )2 cot O2 cot OB - ~ - }, = \ 4--R 2

~-

h

~

4R

d

2RsinO2

O F = 61.88 ° 02 = 61.88 °

01 = 33.76 ° O 2 = 90 °

2.4X 10 5

0.08 X 10-5

' } 2 sin2 02

10,

6.0×10

CD=~I2r--2RIcotO

n

s

5.3 ×10 5

2.4×10 ' 5

2.4 ×10 -5

4.1×10 -6

5.5

I/2

lchch s ~

L = "~sYm-'~ -

sin O 2 sinO]

)<10 - 6

R. Bartiromo et al. / Asymmetric Johann spectrometer

CRYSTAL ROWLAND~ B ~

'V SOURCE

FiB. 7. The angle~ h correspondsto the acceptanceangleof the crystal in the diffracting planes. The rays OA' an(] OR' correspond to ~0 = ~o. The curvature defects as measured in visible light give a contribution ( A X / X ) c o = 2.4 × 10-5; the computed crystal angular width is too = 1.3 × 10-5 and the integrated reflectivity is ~ = 9.9 × 10 -6. In table 1 are listed all the contributions to the instrument resolving power and the throughput for the symmetric mounting and for the case of ~b = O B - ~r/2 (that is O 2 = qr/2); the throughput has been evaluated assuming that the source vertical dimension h S is equal to the crystal height h c. For the asymmetric mounting, (AX/X)L has been evaluated assuming a spectral range A X / X of 10 -2 for the instrument and the maximum value has been considered. In a situation where the detector resolving power represents an important limiting factor, an increase in resolution of about 40% is achievable; had the detector played a negligible role, this improvement would have been of about 70%. The throughput increase obtained is entirely due to the increased crystal integrated reflectivity determined by the asymmetric cut, as discussed in sect. 1. However, when the maximum of luminosity is required, it is better to increase the crystal length I c so as to maintain the resolving power, with an increase of throughput that can be of the order of a factor ten.

5. Conclusions The results obtained so far show that an asymmetrical mounting should sensibly improve the characteristics of Johann spectrometers used for tokamak plasma diagnostics. The kind of asymmetry to be chosen will essentially depend upon the source dimensions and the range of variation allowed for the crystal to source distance. With the tokamaks of the present generation, a reasonable lay-out of a high resolution instrument involves typical distances and curvature radii of a few meters, as suggested by the need of high linear dispersion and the space limitations around the torus. In this case, geometrical aberrations and resolving power of the detector are still important, so that an

457

optimal mounting could be realized with O 2 = ~r/2 ( ~ < 0). This would imply a reduction of the length of the input arm of the spectrometer with an improvement of the throughput, while radiation damage can still be overlooked. The inverse situation is met with the tokamak of the next generation like JET, where radiation screening requires much longer input arms; on the other hand, a relatively compact instrument is desirable, together with a luminosity as high as possible. In this case an asymmetric mounting with • > 0 can be envisaged in order to shorten the detector arm, the geometrical aberrations being usually negligible, owing to the large curvature radii necessary in this situation.

Appendix A We consider the rays diffracted by a narrow vertical band at the crystal center; each point of this band can reflect over a surface of a cone of angular half width O B, the cone axis being directed along the normal n to the diffracting planes. The intersection of each cone with a plane perpendicular to n and passing through the focal point is a circle, fig. 6, whose radius is p = 2 R sin O 2 cos O a. The light collected at a point ( x , y ) in this plane is proportional to the total length of the circles intersected by the elemental surface d x d y : dxdy

3 ( x , y ) cc

(A.1)

However, 3 ( x , y ) is zero when x > p or when y > he~2 + ¢p2 _ x 2 or y < p ~ - - ' x 2 - he~2, see fig. 6. The light collected by a detector o f height hD at the location x is obtained by integrating eq. (A.1) over y. We obtain 0 for x > p, 3h~

~

2 -- X2

for

02 -

< x < P,

1] 0

R. Bartiromo et al. / Asvmmetrw Johann s;oectrometer

458

where 3 is a constant proportional to the total reflected light. The function f(x) represents the photon distribution along the x axis; however, it is impossible to define its width in the usual way because of the divergence at

tained by means of the linear dispersion

2~

4R

] sin 20~

V(hl~

vh~)2

(A.2)

X=,O,

For this reason we proceed in a different way: we calculate the integrated distribution function g ( x ) that gives the total number of photons collected in the length from p to x: t0

When compared with the value given by eq. (4), eq. (A.2) shows that the resolution value obtained with this procedure is lower by a factor dependent upon a. If a = 0.15 is assumed, in analogy with a Gaussian line shape, and h c = h D, this factor takes a value of about 0.5.

for x > O

Appendix B

-hc

(hD-hc

The light emitted by the elemental surface d l . dh of the source is reflected by the crystal with a reflection coefficient f only in the solid angle AS2. AS2 can be expressed as AS2 = Ag,v • A@h where

~ v = h,:/(2 R sin 01 + d ) , g(x) =

provided that the detector height tion

he, satisfies the rela-

( 2 R sin 82 ) hD>h~ l + 2 R s i n 8 1 + d , and assuming that the source depth is much smaller than 2R sin 81 + d. On the other hand, Aqsh is the angle that corresponds in the diffraction plane to the width of the crystal rocking curve ~00; referring to fig. 7 we obtain AB

=

dARIt h =

2R sin 81w o,

and where A x = O - x << O and h c, h r) << O have been considered and the total flux has been normalized to the unit. If we define for the width of the function f(x) the interval zax where 1 - a of the photons are collected, we obtain in the hypothesis that hc/h o > a

'~q'h = wo( 2R sin

The throughput of the spectrometer is given by r

1

-

-

(hD + hc)2

The width along the direction normal to the ray path is obtained by projecting Ax

(a~),

sin 8 a

(hi:)

4 R sin 8 2 cos 8 B ~

+ h,: t 2 ! 2

(hDq-hc) 2 The corresponding wavelength broadening is ob-

2 R sin 01

fl

,dl

fm:~hc d(22R sin 8 ~

R sin O 1 + d ) d h ,

where I' = t ~ a / 2 n ,

provided the source extension in the diffracting plane is greater than l' (this is obviously true for a tokamak plasma) and h'

X 1-

r

,.~-- dl fwoh ~ - . . . . . . . J,' L' d ( 2 R s i n O 1 +d) dh =

2

81)/d.

2R sin 81 + d 2R sin 8 2

hD,

if the source is higher than h'; otherwise h' = h~.

R. Bartiromo et a L / Asymmetric Johann spectrometer In t h e first case

= h c l c h D sin O 1 2R sin @ 2 "

[3] [4] [5] [6]

In t h e s e c o n d case

hcl~h s sin O 1

[7]

~= ~-~ G ~, ~-d"

References [1] M. Bitter, S. von Goeler et al., Phys. Rev. Lett. 42 (1979) 304. [2] M. Bitter, S. von Goeler et al., Phys. Rev. Lett. 47 (1981) 921.

[8] [9] [10] [11]

459

P. Platz, J. Ramette et al., J. Phys. E 14 (1981) 448. A. Guiner, C.R. Acad. Sci. (Paris) 223 (1946) 31. H. Johann, Z. Phys. 69 (1931) 185. W. Zachariasen, Theory of X-ray diffraction in crystals, (Wiley, New York, 1945) F. Bombarda and R. Giannella, Report 82.2, Ass. E U R A T O M - C N E N sulla Fusione, Centro di Frascati, Italy (1982). J. Witz, Acta Cryst. A 25 (1969) 30. K. Feser and A. Faessler, Z. Physik 209 (1968) 1. V.T. Johansson, Z. Phys. 82 (1933) 507. J. Cauchois and C. Bonnelle in Atomic inner-shell processes, ed., B. Crasemann (Academic Press, New York, 1975).