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Ray tracing of curved-crystal X-ray optics for spectroscopy on fast ion beams
Nuclear Instruments and Methods in Physics Research A272 (1988) 895-905 North-Holland, Amsterdam
895
RAY TRACING OF CURVED-CRYSTAL X-RAY OPTICS FOR SPECTROSCOPY ON FAST ION BEAMS Heinrich F . BEYER and Dieter LIESEN GSI, Postfach 11 05 52, D-6100 Darmstadt, FRG Received 3 March 1988
A method of ray tracing is presented which allows a detailed analysis of the X-ray line profile observed with a curved-crystal spectrometer for both a stationary or a fast X-ray source . In particular, the effect of finite dimensions of crystal and source and their interplay with a finite velocity vector of the source is discussed for Johann and Johansson spectrometers . It is shown how same parameters cause distortions and shifts of the reflection and how far they can be resolved by using a two-dimensional position-sensitive X-ray detector .
1 . Introduction In accelerator-based experiments X-ray sources are provided which are interesting because through their precise spectroscopy quantum-electrodynamic and relativistic effects [1] may be studied in highly charged few-electron systems . Another important issue is the spectroscopy as a diagnostic tool in upcoming heavy-ion storage rings [2] employing electron cooling . Because these sources are very weak as compared to other laboratory X-ray generators, a high detection efficiency in addition to the high wavelength precision is needed . A limited range of ions may be produced as slow recoil ions [3] whereas in general the X-ray emitting ions move with a velocity which may be a large fraction of the speed of light. Curved-crystal spectrometers have experienced an increase of accuracy and efficiency in recent years due to several developments . High-precision laser interferometers are employed [4] for the required angular measurements . Wavelength standards [5] of a few ppm accuracy exist for calibration in crystal diffraction spectrometry. Large specimens of highly perfect crystals can be bent with high accuracy [6] . Position-sensitive X-ray detectors [7] with spatial resolutions below 100 jAm can be built . However, geometrical aberrations due to the finite size of the crystal and of the source may cause a systematic shift and broadening of the spectral line . Most of the aberrations can be made small by a careful design of an instrument or the results of a measurement can be corrected provided the type of aberration and their magnitudes are known . This problem is not new. For transmission spectrometers a detailed analysis was made by Schwitz et al. [8], whereas the Bragg case 0168-9002/88/$03 .50 © Elsevier Science Publishers B .V . (North-Holland Physics Publishing Division)
(Johann [9] and Johansson [10] geometry) was studied by Zschornack et al . [111 . In the present paper we investigate, for the Bragg case, the shape and absolute intensity of a diffraction line . All crystal and source dimensions can be made nonzero and both source and detector can be placed at a position off from the Rowland circle . Characteristic for a fast-beam experiment, the source may furthermore travel in any given direction at any given speed . The many parameters being simultaneously nonzero make the problem sufficiently complex precluding a rigorous analytical calculation . Consequently, ray tracing using the Monte Carlo method represents the most adequate method of designing a spectrometer configuration for accelerator-based X-ray spectroscopy . 2. Definition of the problem and assumptions We consider a focussing Bragg spectrometer with a concave crystal either in the Johann or in the Johansson version . The geometrical arrangement of the problem is illustrated in fig . 1 where most of the relevant parameters are indicated . For consistency, we will keep close to the notation as used in refs. [8] and [111 . Both the source and the crystal, before bending, are assumed to be rectangular parallelepipeds . The depth, width, and height of the source are denoted xo, yo , z o and the corresponding dimensions of the crystal are denoted ro , to , h o , respectively . The radius of curvature of the crystal is denoted by R . The detector is represented by a rectangular screen of width Xo and height Yo and is oriented perpendicular to the "central" X-ray lying in the Rowland-circle plane and being reflected at the crystal's apex . Such an orientation of the detector is a
896
H. F. Beyer, D. Liesen / Ray tracing of curoed-crystal X-ray optics t,
Fig. 1 . The geometry of a curved-crystal spectrometer in the Johann or Johansson version with a position-sensitive X-ray detector and a fast source as provided by accelerator-based experiments .
good choice if one is interested in a small wavelength interval only but in a high wavelength resolution . Detector, source, and crystal are symmetric with respect to the Rowland-circle plane as shown in fig. 1 for a spectrometer setting 0 . For our calculations we make the following assumptions : - The source consists of uniformly distributed emission points Q(x, y, z) with -x o /2 + x off -< x -< x o/2 + xoff, - yo/2 !!~ y :!~ yo/l and - z o/2 < z < z o/2 . They have a constant velocity vion defined by its absolute value I Vi .. I = P co , where c o is the velocity of light, and by its in-plane angle Own and its out-of-plane angle 'Pion as defined in fig . 1 . The center of the source can be shifted by the amount x off inside the Rowland circle. For xoff < 0, it is outside the Rowland circle . The emission characteristic S(a) in the emitter frame of reference can be chosen to be strictly monochromatic, a Gauss function or an exponential as a function of X-ray energy with a width AE. . In the emitter frame, X-ray emission is assumed to be isotropic . - The crystal consists of scattering points B(r, t, h) with 0<_ r<_ ro , -t o/2 <- t < t o/2, and . They are distributed exponentially in the direc:5h<-ho/2 ho/2 tion of the crystal depth r and uniformly in the two other dimensions . The exponential takes care of the X-ray absorption in the crystal leading to an extinction length
X-ray emitted by a source point Q(x, y, z) and reflected at a scattering point B(r, t, h) . B B is the Bragg angle defined by Bragg's law na=2d sin BB,
(2)
where l is the wavelength of the X-ray in the laboratory frame of reference. n and d denote the order of diffraction and the lattice interplanar spacing, respectively. The function F(B; - BB) can be constructed from measured and calculated crystal-structure data . In our calculations we will approximate it by a Gaussian F(g;-9 )=Fo exp~B
(B; - BB) 2
2a 2
~,
(3)
with a peak reflectivity FO and a standard deviation a. The corresponding width of the rocking curve is taken to include all broadening effects due to the treatment of the crystal including bending . - The detector is position-sensitive in two dimensions X and Y. Broadening due to a finite depth of the detector is not considered. It could be evaluated by calculating the reflection profile as a function of detector position, because we provide the possibility to move the detector inside (doff > 0) or outside (doff < 0) the Rowland circle . This option is important for some applications with a fast source where, due to the Doppler effect, a focus can occur away from the Rowland circle.
dr=sin 0, where Ft is the relevant X-ray linear extinction coefficient . It turns out that in most cases ro and dr can be set to zero for soft X-rays because the small penetration depth has a negligible effect compared to other geometrical aberrations . Incident X-rays are specularly reflected with a probability given by the emission pattern (or rocking curve) F(B ; - Ba), where B i = 8; (x, y, z, r, t, h) is the effective glancing angle for an
3. Method of calculation The intensity distribution in the detector is calculated in the following way . For every single X-ray defined by coordinates x, y, z, r, t, h we can calculate the position X, Y in the detector where the X-ray is reflected to . For the determination of the reflectivity according to eq . (3) we need to know the effective
H.F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
897
glancing angle B i (x, y, z, r, t, h) and the Bragg angle B B which, through eq. (2) and a Lorentz transformation, is a function of the wavelength X ion measured in the ion's frame of reference . Therefore, the intensity in the detector is given by the integral
spectively . The distance between Q and B is (see triangle BQH) ] (9) u= [(R'+r-w)2+ o 2 t/2 ,
I (X, Y) = I V
u= [(R'+r-w)2+v2+(h-z)211/2,
f
F(Bi(X) - OB(X,
XS(xion)
dX
Xion))
dXion,
(4 )
where X = (x, y, z, r, t, h) and the integration runs over the source and crystal volume and over the spectral distribution of the emitter. The 7-dimensional integral (4) is solved by the Monte Carlo method.
and the distance between Q and B is
where we used the distance between K and B being R'+ r with R
,_ R - ~R
for the Johann geometry, for the Johansson geometry .
COS T
(11)
The effective glancing angle B i is then given by sin Bi=(R'+r-w)/u,
3.1 . The geometrical problem For calculating the effective glancing angle B i and the locus X, Y where a single X-ray hits the detector we refer to ref. [11] and to fig. 2 showing a projection into the Rowland circle plane and the auxiliary quantities used. All points shown lie in the Rowland circle plane . For our calculations we need the following quantities (fig. 2) : b = R COS 0, and
(12)
and its projection Bi into the Rowland circle plane is
sin Bi = (R'+ r -
w)/u .
(13)
Considering triangle SKW we can write the distance between W and B as p=R'+r-R
COS
B/COS(B-T) .
q can be composed of
(14)
plus p sin(B - T)
q=R(sin 0-sin T/COS(B-T))-d off +p sin( 0-T) . (15) With these quantities the horizontal coordinate X of the image point in the detector reads (X = DE + ED)
It can be shown [111 that
X=p Cos(B-T)+q tan(B-B i -T),
u=(b-y) sin(B+T)-x cos(B+T),
and the corresponding vertical position is
and w=(b-y) COS(B+T)+x sin(B+T) .
(10)
(8)
Q and B are projections of the source point Q(x, y, z) and of the diffraction point B(r, t, h), re-
Y=h+q h-z /COS(6-B i -T) . u
(16)
(17)
3.2 . Transformation of the wavelength to the laboratory system
In order to evaluate the wavelength X in the laboratory frame of reference we have to determine the angle a between the emitter velocity vion and the direction of the emitted X-ray. According to fig . 1, the direction of the velocity is defined by angles (Pion and Y'ion . We suppose now that the direction of a single X-ray is given in an analogous way by angles ox and Y'x . With the aid of fig . 3 we can write (18)
c 2 =a 2 +b 2 -2ab COS a,
where a, b, and c have only a local meaning defined by the figure . On the other hand, c 2 can be expressed as c2 = a2
COS 2~ion
-2ab Fig. 2 . A projection of the geometry into the Rowland-circle plane for calculation of the image point of a single X-ray originating from a source point Q(x, y, z) and being reflected at a scattering point B(r, t, h) . Q and h are the projections of Q and B, respectively .
+ 62 COS2t~x
COS `,ion COS 4, x COS(OX - 4ion)
+(a sin ip ion - b sin =a2+b2-2ab{COS +Sin 4x Sin
Y'ion ) -
jpx
)2
'Pion COS Y'x COS(ox - 0ion)
(19)
89 8
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
we get Cos a = [ 1 - (( h - z )/u
)21
1/2
X COO - B; + T - 0inn) Cos 41x +(h-z)/u
sin
`Yion .
(23)
This result can be inserted into the Lorentz transformation X =X ion( l-
a2 ) -1/2(1- a COS a),
(24)
where X ion is the wavelength of a single X-ray in the emitter frame of reference. 3.3. Solid angle and efficiency
Fig. 3. Auxiliary plot to calculate the angle a between the direction of a travelling ion and the direction of a single X-ray emitted by the ion.
Comparing eqs. (18) and (19) yields COs
a = COs
Sl =
`rion COs `Yx COS(Ox - (Pion )
+sin
9' x
stn
(20)
4' ion-
Upon substitution (see fig. 2) of 0x=0-6,+T,
(21)
and sin
We are not only interested in the relative distribution of X-rays but also in the absolute intensity in the detector at a given source strength, i.e . we want to determine the spectrometer's absolute efficiency. The geometrical solid angle of the crystal as seen from the center of the source is given by
(22)
¢x=(h-z)/u,
to h o
sin 0
(R sin 0 - xnff )
2 .
Because in practical cases the source-to-crystal separation is large as compared to crystal and source dimensions, eq. (25) is a good approximation for all source points and we do not have to perform an exact integration over solid-angle elements dS2. For a fast source we also have to take into account the relativistic transformation of the solid angles [12] . This is achieved
run -100 switch = n-total = theta-deg x0 = y0 = z0 = x-off = rO = t0 = hO = XO = YO = d-off = rad = sigma = d2-crystal beta = phi-ion-deg = psi-ion-deg = delta-e = lambda-max = n-det-x = n-det-y =
1 5E6 60 2 2 2 0 0 5 5 1 15 0 1000 fE-4 4 0 0 0 0 3 .4641 30 30
. . . . . .
omega = overflow = integral =
2 .2972E-6 1 .27E-f0 0 .0541966
. . . . . . . . . . . . . mean solid angle devided by 4n . . . . . . . . . . . . . . . . . intensity outside detector . . . . . . . . . .integrated intensity in detector
.
. . . . . . . . . . . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
(25)
. . . . . .
. . . . . . . . . 0 = johann, 1 = johansson . . . . . . . . . . . . total number of x rays . . . spectrometer setting in degrees . . . . . . . . . . . . . . . . source depth in mm . . . . . . . . . . . . . . . . source width in mm . . . . . . . . . . . . . . . . source heightin mm . . . . . . . source displacement into R .C . in mm . . . . . . . . . . . . . . . . . . . . . . . crystal depth in mm . . . . . . . . . . . . . . . . . . . . . . . crystal width in mm . . . . . . . . . . . . . . . . . . . . . . crystal height in mm . . . . . . . . . . . . . . . . . . . . . . detector width in mm . . . . . . . . . . . . . . . . . . . . . detector height in mm . . . . . . . . . detector displacement inside R .C . .. radius of curvature in mm . diffraction pattern (angular width) sigma . . . . . . . . . . . . . . . . . . . . . . grating constant 2 d --- . . ion-beam velocity v/c ion-beam : angle in plane of Rowland circle . . . . . . . . . . . . . . out-of-Plane angle (degrees) . . . . . . . C .M . width (eV) of emission pattern . . . . . . . . . . . . . peak wavelength in Angstroems . . number of pixles in horizontal direction . . . . number of pixles in vertical direction
Fig. 4. Chart of input and output parameters for the program IMAGE.
H.F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
by multiplying, for each single X-ray, the reflected intensity by a2 ; on _ aotab
(1
1 _P2 h _ ,8 a) 2 COS
.
20,1
89 9
Influence of the Crystal Rocking Curve
(26)
The overall spectrometer efficiency reads Ndet Not 47T
(27)
with Ndet/Nat the number of X-rays reflected into the detector divided by the total number of X-rays hitting the crystal. 3.4. Numerical procedure
By the numerical procedure one tries to exactly simulate the situation of a real experiment . For this purpose, the computer program IMAGE [131 was developed which evaluates eq . (4). In the program a random grid is defined in the 7-dimensional space of the source and crystal volume and of the source wavelength, i .e . one runs through a loop over a number Not of X-rays in which the independent variables are random numbers . Within the loop all the analytical expressions given above are calculated . For each single X-ray, the effective angle B; is calculated giving the position in the detector according to eqs. (16) and (17) . The detector area itself is digitized in an Nx X NY matrix of small rectangular bins in which the intensity is collected. There is one extra overflow bin collecting the intensity of all those X-rays which are reflected from the crystal but which do not hit the detector. The intensity accumulated in the individual bins originates from single events with reflected intensity given by eq . (3) in which the effective glancing angle Bi is inserted as given by eq . (12) and the Bragg angle B B defined by eqs. (2) and (23), (24). The solid-angle correction of eq. (26) is applied to each single event. In the following, we will present some numerical results calculated with the program IMAGE . Fig . 4 contains a chart of the input and output parameters used for a test run and their explanations . 4. Results of calculations and discussion 4.1 . Stationary source
In this section we consider only those cases in which the source is at rest (,l3 = 0) . Because the results of calculations can easily be scaled with the absolute size of the spectrometer the radius of curvature of the crystal is set constant to R = 1 m throughout this investigation. First, we show the influence of the crystal rocking curve
Distance X (mm) Fig. 5 . X-ray profiles (integrated over the detector height) displaying the influence of the crystal rocking curve with various values of the standard deviation a . for small crystal and source dimensions . The actual input parameters can be found in fig. 4. One-dimensional X-ray profiles, integrated over the detector height, are shown in fig. 5 for various values of a entering the diffraction pattern of eq . (3). The peak reflectivity is Fo = 0.5 in all cases. One can see how the profile gets narrower with decreasing a and that the peak height stays about constant . The effect of the crystal width for the Johann case can be observed in figs . 6 and 7. Fig. 6 displays profiles at a constant spectrometer Bragg angle 0 = 30 ° and varying crystal width t o illustrating clearly the Johann defect of a distorted profile if the crystal becomes too wide . For comparison is shown the profile for "exact" focussing in case of the Johansson geometry with the crystal width being to = 100 mm. Fig. 7 shows the change of the profile with changing Bragg angle. The peak asymmetry and centroid shift is largest at small Bragg angles .
8
2 0
-0 .5
0.0 0 .5 1 .0 1 .5 Distance x (mm) Fig. 6. X-ray profiles displaying the effect of a finite crystal width t o for a Johann spectrometer at 0 = 30 ° . For compari son the profile for a Johansson geometry with t o =100 mm is also shown (dashed line).
900
H. F Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics run 101
10 8
s
6
4 3 2 1
4 2
10" 10-' 10- ' 10- ' 10- ' 0
0
-0 .4-0 .2 0 .0 0.2 0.4 0 .6 0.8 1 .0 Distance x (mm) Fig. 7 . X-ray profiles displaying the effect of a finite crystal width t o = 50 mm at various Bragg angles 0.
The peak centroid position was calculated by Zschornack et al . [111 by an approximate analytical integration of sin 6;, as given by eq. (12), over single nonzero dimensions . Noting that the dispersion can be approximately written as
ax =
a- R sin
2o
cos 6 ' where Ax is a shift in the detector, one obtains
ax = -(24R cos
B) -1 (h 2
+zo - t 2 cos 20) .
(28)
(29)
Fig. 8 . X-ray profile displaying the effect of a dominating crystal height. Input parameters for this calculation were set to 8=60 ° , x o = yo =z o =2 mm and ro =0, t o =h o =50 mm .
discuss the case in which the crystal height causes the largest distortion . Fig. 8 shows the two-dimensional intensity profile for a Johansson geometry where we used the parameters 0 = 60 °, x o =yo = z o = 2 mm and ro = 0, t o = h 0 = 50 mm . In order to collect the total reflected intensity the detector is 100 mm high . Because of the curvature observed in the image pattern one would loose resolution by integrating over the detector height Y. This is more clearly demonstrated in fig. 9 showing the corresponding integrated profile together with a cluster plot of the data from fig. 8. The dashed line in fig. 9 indicates the peak centroid calculated by use of eq . (29) . In fig. 9 the solid "white" line represents the result of an analytical calculation which gives only the line of gravity of the image pattern . For this calculation one starts from eqs. (12) and (13) which can be combined to
Eq . (29) takes into account the main contributions to the shifts, namely the crystal and source height and the crystal width (h o , z o , and t o , respectively). In table 1 are compared the centroid shifts derived from data like those displayed in figs . 6 and 7, resulting from the Monte Carlo calculations, to the analytical values obtained from eq . (29) . The errors shown represent the statistical uncertainties of the Monte Carlo sin B; = sin B; ( u/u ) . (30) calculations . The total number of X-rays was n tot = 1 X For dominating vertical coordinates eq . (30) becomes 10 6. An excellent agreement is observed between the 1/2 results of the detailed numerical calculation and those sin B i = Sln BJ1 + tan 24, x ] of the analytical expression . In order to increase the solid angle it is essential to [1 + tan2 p (31) make use of a nonnegligible crystal height . We will now
= 2d
Table 1 Centroid shifts in mm calculated with the Monte Carlo method and analytically using eq . (29) ; the crystal height is set to h o =10 mm . The numbers in brackets show the uncertainty in the last digit displayed. 0
15'
Monte Carlo Analytical
0 .095(2) 0 .0963
Monte Carlo Analytical
30 °
Crystal width t o = 50 mm 0 .085(1) 0 .0854
45 °
60 °
75 °
0 .0676(9) 0 .0678
0.0438(7) 0.0438
0 .0112(7) 0 .0109
0 .289(2) 0 .289
0.199(2) 0.200
0 .092(1) 0 .0917
Crystal width t o =100 mm 0 .394(6) 0 .398
0 .355(4) 0 .356
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
run
run
101
8 7 6 5 4 3 2 1 0
901
101
60 40 20
E
0 -20 -40
-60 1 .0 -1 .0 -0 .5 0.0 0 .5 1 .0 0.0 0 .5 X (mm) X (mm) Fig. 9. Same data as in fig. 8: (a) integrated profile; the dashed line marks the peak centroid calculated from eq . (29) ; (b) a cluster plot of the numerical data compared to the analytical calculation (represented by the "white" line) according to eq. (31) . -1 .0
-0 .5
with tan p, = (h - z )/( R sin 0 - xoff ) .
(32)
In eq . (31) tan %P, and sin Bi determine the vertical and horizontal coordinate in the detector, respectively . The very good correspondence of the two results is evident from fig. 9. The width of the profile stays strictly constant over the whole vertical dimension as can be inspected from horizontal cuts taken at various heights. This means that wavelength resolution is preserved and one does not have to cut the crystal and detector height if a two-dimensional position-sensitive detector is used . The profile width observed in figs . 8 and 9 is dominated by the width of the rocking curve with a set to 1 X 10 -4 corresponding to a wavelength resolution of X/4 X = 6500. This value can still be improved if a crystal with a narrower rocking curve can be selected, without running into problems due to geometrical aberrations . Also the influence of the crystal width, in case of the Johann geometry, is quite moderate at the Bragg angle 0 = 60 °
used here. Crystals with to = h0 = 50 mm can well be used in Johann geometry at this kind of wavelength resolution . The variation of the spectrometer efficiency with the source location is demonstrated in fig. 10 . According to eq . (27) the total efficiency is the product of the crystal efficiency and the geometrical solid angle devided by 4~r. In figs. 10a and b, both contributions are shown for source widths of 1 and 5 mm, respectively. Other geometrical dimensions were to = 20 mm, ho = 10 mm, xo = 0.1 mm, zo = 5 mm . For the small source width of yo = 1 mm a pronounced maximum of the crystal efficiency is observed when the source approaches the Rowland circle xoft = 0. This maximum is washed out when the source width is set to yo = 5 mm, leading to a relative maximum of the total efficiency somewhat inside the Rowland circle . We note, however, that operation of a spectrometer with a small source near the Rowland circle leads to a very narrow useful wavelength interval . Therefore, one has to make a compromise between wavelength range and efficiency.
0 .20 U U
w w
0.15
U t~
0.10
U
w
w
0.05 0.00 -1000
-600
-200 0 200 400 600 Offset (mm)
0 .08 0 .07 0 .06 0 .05 0 .04 0 .03 0.02 0.01 0.00 -1000
-600
-200 0 200 400 600 Offset (mm) Fig. 10. Spectrometer efficiency as given by the crystal efficiency times the geometrical solid angle as a function of the source location . For xoff < 0 the source is outside the Rowland circle . The source width was set to (a) yo =1 mm and (b) yo = 5 mm, respectively.
902
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
4 .2. Fast source
We now consider cases with a > 0 . Let us first derive an analytical formula in analogy to eq. (31) for a finite crystal (or source) height but for a fast source . If we replace, in eq . (31), A by the appropriate laboratory wavelength given by eqs. (24) and (23) we obtain sin 9; -
nXion -#2)1/2
2d(l X [1
- N (COS
+sin
4ion
`Y ionCOS(N - 'Pion ) COS t1 X
sin 4,,)] 11 +tan24/ x1
In eq .(33) we have averaged over Cos( cpx - 0ion) - TO _
~
fTo/2
o/2
T
$x
1/z
(33)
according to
COS(Ox - 4)ion )
I,
cl,- b,
(34)
where To = to/(R - xoff/sin 0) . Carrying out the integration and the multiplication of the square brackets in eq. (33) yields sin Bi =
n ~'ion a2)1/z 2d(1 X [(I
1/z - /3(2 + tan24,, ) sin(TO/2)/To
X COs 'rion COs (P ion
+ sin
4'ion
tan
~x)] .
(35)
The first term in the square brackets is the same as for the stationary source causing the curved image. The velocity-dependent part splits into two terms. The first one is a Doppler shift being independent of 41, whereas the second one is proportional to tan 4r, . In the following we will compare our numerical calculations to eq . (35) .
Let us now discuss the results obtained for different orientations of the velocity via with its absolute value set to 8=0.1 . For clarity, fig. 11 gives the specific values of the input parameters used in the following examples . They are kept constant with the exception of the angles $ion, V'mn, and of the corresponding emitter wavelength a ion . The largest broadening effects are observed when an extended source travels with a velocity component tangential to the Rowland circle, i.e. in the y direction. This is clearly demonstrated in fig. 12 for a source travelling in the Rowland-circle plane with ~b ;on = 90 ° . The wide plateau of fig. 12a indicates that in this case an imaging of the whole source width of yo = 2 mm occurs with an accompanying loss of wavelength resolution . For comparison, fig. 12b reveals the shrinking to a narrow line when a narrow source of yo = 0.1 mm width is used. For such a narrow source, which in practice can be represented by a slit, the crystal efficiency can be further improved by putting source and detector to a refocussing position [14] . This means that the wavelength spread, due to the Doppler effect, of X-rays filling the cone of the solid angle is partly compensated by placing the source inside and the detector outside the Rowland circle . For our particular case the offsets are calculated to amount to xoff = 93 .4 mm and d~ff = -119 .2 mm . A similar refocussing has been applied to grating spectrometers [15] . We will not discuss this point any further because it is important only for small sources and for scanning instruments . We continue with an extended source x o =yo = z o = 2 mm and q)ion = 0. Fig. 13 applies for ipion = 0, i.e . the source travels towards the crystal . For this case we obtain a narrow line again. For figs . 14 and 15 the
run 102 ... . . . . .. . .. . .. . ..
switch = n-total = theta-deg x0 = YO = z0 = x-off = rO = to = hO = XO = Yo = d-off = rad = sigma = d2-crystal beta = phi-lion-deg psi-lion-deg de lta-e = lambda-max = n-det-x = n-det-y =
1 5E6 60 2 2 2 0 0 50 50 3 40 0 1000 fE-4 4 0 .1 90 0 0 3 .44674 30 30
omega = overflow = integral =
2 .2972E-4 . . . . . . . . . . . . . mean solid angle devided by 4rr 8 .15412E-3 . . . . . . . . . . . . . . . . . intensity outside detector 5 .90865E-3 . . . . . . . . . .integrated intensity in detector
0 = johann, 1 = johansson . . . . . . . . . . . . . . . . . . . . . total number of x rays . . . . . . . . . . . . spectrometer setting in degrees . . . . . . . . . . . . . . . . . . . . . . . . . source depth in mm . . . . . . . . . . . . . . . . . . . . . source width in mm . . . . . . . . . . . . . . . . . . . . . . . . . source heightin mm . . . . . . . . source displacement into R .C . in mm . . . . . . . . . . . . . . . . . . . . . . . . crystal depth in mm . . . . . . . . . . . . . . . . . . . . crystal width in mm . . . . . . . . . . . . . . . . . . . . . . . crystal height in mm . . . . . . . . . . . . . . . . . . detector width in mm . . . . . . . . . . . . . . . . . . . . . . detector height in mm . . . . . . . . . . detector displacement inside R .C . . . . radius of curvature in mm . . diffraction pattern (angular width) sigma . . . . . . . . . . . . . . . . . . . . . . . grating constant 2 d . ion-beam velocity v/c . .ion-beam : angle in plane of Rowland circle . . . . . . . . . . . . . . . out-of- lane angle (degrees) . . . . . . . . C .M . width (eV~ of emission pattern . . . . . peak wavelength in Angstroems . . . number of pixles in horizontal direction . . . . . number of pixles in vertical direction
Fig. 11 . Parameter values used for the study of a fast source.
903
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
run
102
a)
1 .210' 1 .010-, 8.0 10-" 6 .0 10-° 4.0 10-° 2 .0 .10`
Fig. 12 . Intensity profile for a fast source travelling in the Rowland-circle plane with Con = 0 ° and source was (a) yo = 2 mm and (b) yo = 0.1 mm, respectively.
Own =
90 ° . The width of the
run 104
Y6 ) 7-
10
,moo -05
0:
lOcol
Fig. 13 . Intensity profile for a fast source travelling towards the crystal, Con ='Pion = 0 °. velocity vi points out of the Rowland-circle plane . with 'Pion = 30 ° and 4, ;o = 90 °, respectively . As expected from eq . (35) the line gets slanted with the slope being proportional to ,Q sin Con. The slanting has already been observed in beam-foil [161 and beam-gas [171 experiments . A comparison between the numerical data and eq . (35) is shown in figs . 15 and 16 .
Fig. 14. Same as fig. 13 for ¢;o = 0 ° and
~;oa =
30 ° .
For a detailed comparison, we calculated the center of gravity x m of horizontal cuts of the numerical data taken at various heights in the detector . In fig. 16 the difference between the position -'anal determined analytically and the center of gravity xn m is plotted. A perfect agreement within the statistical accuracy of the numerical procedure is observed . In our example run 106
run 106
nn
x 0
X (mm)
.
0 -2
I I
-4 -6
I 4 0 2 x anal . - x num. (I-tm) Fig. 16 . The data of fig. 15 plotted as the difference between the center of gravity determined analytically from eq . (35) and the one derived from the numerical data at various heights in the detector. The error bars indicate the statistical uncertainty. n
Fig. 15 . Cluster plot of the intensity pattern for O . = 0 ° and ; ¢;o = 90 °. The "white" line represents the analytical calculation according to eq . (35) .
4 2
-4
-2
904
H. F. Beyer, D. Liesen / Ray tracing of curued-crystal X-ray optics
fluctuations of about ± 1 p m correspond to a wavelength precision of about 7 x 10 -7 . The same quality of
run 107
correspondence between analytical and numerical data
was obtained for a variety of cases, including stationary
sources, whenever eq . (35) is applicable . This holds true
even if we allow other geometrical dimensions to come
into effect simultaneously . For instance, let us consider
the case of fig. 15 but with a Johann crystal of 50 mm
width and a source height of 10 mm. We obtain the same slanted line which, however, displays some broadening due to the finite crystal width and source height . The
corresponding center-of-gravity shift has been proven numerically to be constant over the whole detector height and to be well described by eq . (29).
In table 2 we summarize the performance of the spectrometer for various source arrangements . The corresponding parameters are /3 = 0.1, $ion = 0 °, 4ion =
0 ° , 30 ° , 90 ° , and xoff = 0 and 100 mm . Other parameters are identical to those given in fig. 11 . The crystal efficiency given in table 2 does not include the enhance-
aQ ion /aQ lab of eq . (26) . For the source travelling towards the crystal, ip ;on = 0, the same crystal efficiency is obtained as for the same source at rest and by the transformation of the solid angle 22% of ment of the solid angle
intensity is gained . If the velocity vector is turned out of the Rowland circle the intensity (which is proportional to the crystal efficiency times the solid-angle enhancement) drops. The relative decrease is much weaker when the source is placed 100 mm inside the Rowland circle .
The wavelength resolution dX/a was derived from the FWHM line width of horizontal cuts of the intensity profile. In table 2 are listed the results obtained for cuts
Fig. 17. Intensity distribution in the detector for an emission pattern being an exponential with a maximum at 3286 eV and a decay constant of 3 eV . Other parameters are identical to those used in fig. 14 .
near the upper or lower edge of the profile. The line width stays practically constant over the whole profile height allowing wavelength resolutions comparable to those obtained for a stationary source . In our example, the resolution degrades by about 20% when the velocity
direction turns from 41 ion = 0 ° to 'P ;an = 90 ° . We finally give an example where, in the emitter frame of reference, the emission pattern is not a delta
function but is given by an exponential as a function of X-ray energy. The decay constant is set to 3 eV with the
maximum intensity taken at an X-ray energy of 3286 eV . The geometrical parameters are identical to those used for generating fig. 14, i.e . /3 = 0.1, 't` ;o = 0 °, Con
taken in the Rowland-circle plane and for those taken
= 30 o . The intensity profile obtained is shown in fig. 17 displaying the decaying intensity towards decreasing wavelength or increasing X.
Table 2 Spectrometer performance with a stationary source and with a source travelling with a velocity /3 = 0.1 . Parameters not specified in the table can be found in fig. 11 .
5. Concluding remarks
Source on R.C. xoff = 0
Crystal efficiency
a'n,on/aszlab
(aX/X)xl0 4 in plane out of plane
Stationary source
Fast source ,Q = 0.1, 'Peon = 0 ° =0' 30' 90-,
0.0545 1 .00
0.0543 1 .22
0.0238 1.19
0.0126 0.99
1.41 1.45
1 .43 1.42
1.52 1.50
1 .71 1 .70
Source inside R.C . xoff =100 mm Crystal efficiency
aSkn/aQlab
(da/a)x10 4 in plane out of plane
0.0193 1 .00
0.0193 1 .22
0.0174 1 .19
0.0112 0.99
1 .36 1 .48
1.36 1.42
1.48 1.58
1 .72 1 .70
In this article we have systematically analyzed the effects of geometrical aberrations combined with Doppler shifts as encountered in curved-crystal spectrome-
try of fast X-ray sources. A detailed ray tracing was performed for Bragg spectrometers operated with a stationary or a fast source . In some limiting cases it is possible to predict accurately the center-of-gravity positions of the X-ray profile in the detector, whereas in the general case and for the study of line shapes it is necessary to perform detailed numerical calculations . The results of this paper can be used to optimize the parameters in the design of a new X-ray experiment . Detailed knowledge of the aberrations we were discuss-
ing can also help to analyze the results of existing and future experiments. We emphasize that both spectrometer efficiency and wavelength resolution can be improved by a careful design of an experiment . Huge crystals can be used,
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
without loss of wavelength resolution, even for a fast source, when a large two-dimensional position-sensitive detector is employed . Of particular importance is the orientation of the source velocity. If one stays away from a velocity having a component tangential to the Rowland circle, i.e . if ¢io = 0, it is possible to work with an extended source and to obtain a narrow intensity pattern in the detector. With the use of a twodimensional detector not only the emitter wavelength is measured but also the ion velocity and the observation angle. This is important for diagnostics of ion beams by means of X-ray spectroscopy . Acknowledgements The authors enjoyed fruitful discussions with Richard Deslattes. We also received excellent assistance from Ingolf Giese in solving many computing problems .
[4]
[6]
[8] [9] [10] [ill [12] [13]
References [1] See for instance : S.J. Brodsky and P.J . Mohr, in : Structure and Collisions of Ions and Atoms, vol. 3, ed . I.A . Sellin (Springer, Berlin, 1978). [2] For a survey see: B. Franzke, IEEE Trans. Nucl. Sci . NS-32 (1985) 3297 . [3] H.F . Beyer and R. Mann, in : Progress in Atomic Spectroscopy, part C, eds . H.J . Beyer and H. Kleinpoppen (Plenum, New York, 1984) p. 397.
[14] [15] [16] [17]
905
R.D . Deslattes, E.G. Kessler, W.C . Sauder and A. Henins, Ann. Phys . (N .Y .) 129 (1980) 378 . E.G . Kessler, R.D . Deslattes and A. Henins, Phys . Rev . 19 (1979) 215 ; J.A . Bearden, Rev. Mod. Phys. 39 (1967) 78 . R.D . Deslattes and E.G. Kessler, in : Atomic Inner-Shell Physics, ed . B. Crasemann (Plenum, New York, 1983) and references cited therein. E. Kellog, P. Henry, S. Murray, L. van Speybroeck and P. Bjorkholm, Rev. Sci. Instr. 47 (1976) 282; R.A. Boie, J. Fischer, Y. Inagaki, F.C . Merritt, V. Radeka, L.C. Rodgers and D.M. Xi, Nucl . Instr. and Meth . 201 (1982) 93 ; B.P . Duval, J. Barth, R.D . Deslattes, A. Henins and G.G . Luther, Nucl . Instr. and Meth . 222 (1984) 274. W. Schwitz, Nucl . Instr. and Meth. 154 (1978) 95 . H.H . Johann, Z. Phys. 69 (1931) 185. T. Johansson, Z. Phys . 82 (1933) 507. G. Zschornack, G. Miiller and G. Musiol, Nucl . Instr. and Meth . 200 (1982) 481. J.D . Jackson, Classical Electrodynamics (Wiley, New York, 1967). The program IMAGE was developed with the Graphics Numerics Operation Method GNOM . Data presentation was facilitated making use of the TOPDRAW library created by U. Post, University of Giessen. H.F. Beyer, unpublished (1982) . J.O . Stoner and J.A . Leavitt, Appl . Phys . Lett. 18 (1971) 368; ibid . 18 (1971) 477. C.J . Hailey, R.E. Stewart, G.A . Chandler, D.D . Dietrich and R.J . Fortner, J. Phys. B18 (1985) 1443. R.D . Deslattes, R. Schuch and E. Justiniano, to be published.
895
RAY TRACING OF CURVED-CRYSTAL X-RAY OPTICS FOR SPECTROSCOPY ON FAST ION BEAMS Heinrich F . BEYER and Dieter LIESEN GSI, Postfach 11 05 52, D-6100 Darmstadt, FRG Received 3 March 1988
A method of ray tracing is presented which allows a detailed analysis of the X-ray line profile observed with a curved-crystal spectrometer for both a stationary or a fast X-ray source . In particular, the effect of finite dimensions of crystal and source and their interplay with a finite velocity vector of the source is discussed for Johann and Johansson spectrometers . It is shown how same parameters cause distortions and shifts of the reflection and how far they can be resolved by using a two-dimensional position-sensitive X-ray detector .
1 . Introduction In accelerator-based experiments X-ray sources are provided which are interesting because through their precise spectroscopy quantum-electrodynamic and relativistic effects [1] may be studied in highly charged few-electron systems . Another important issue is the spectroscopy as a diagnostic tool in upcoming heavy-ion storage rings [2] employing electron cooling . Because these sources are very weak as compared to other laboratory X-ray generators, a high detection efficiency in addition to the high wavelength precision is needed . A limited range of ions may be produced as slow recoil ions [3] whereas in general the X-ray emitting ions move with a velocity which may be a large fraction of the speed of light. Curved-crystal spectrometers have experienced an increase of accuracy and efficiency in recent years due to several developments . High-precision laser interferometers are employed [4] for the required angular measurements . Wavelength standards [5] of a few ppm accuracy exist for calibration in crystal diffraction spectrometry. Large specimens of highly perfect crystals can be bent with high accuracy [6] . Position-sensitive X-ray detectors [7] with spatial resolutions below 100 jAm can be built . However, geometrical aberrations due to the finite size of the crystal and of the source may cause a systematic shift and broadening of the spectral line . Most of the aberrations can be made small by a careful design of an instrument or the results of a measurement can be corrected provided the type of aberration and their magnitudes are known . This problem is not new. For transmission spectrometers a detailed analysis was made by Schwitz et al. [8], whereas the Bragg case 0168-9002/88/$03 .50 © Elsevier Science Publishers B .V . (North-Holland Physics Publishing Division)
(Johann [9] and Johansson [10] geometry) was studied by Zschornack et al . [111 . In the present paper we investigate, for the Bragg case, the shape and absolute intensity of a diffraction line . All crystal and source dimensions can be made nonzero and both source and detector can be placed at a position off from the Rowland circle . Characteristic for a fast-beam experiment, the source may furthermore travel in any given direction at any given speed . The many parameters being simultaneously nonzero make the problem sufficiently complex precluding a rigorous analytical calculation . Consequently, ray tracing using the Monte Carlo method represents the most adequate method of designing a spectrometer configuration for accelerator-based X-ray spectroscopy . 2. Definition of the problem and assumptions We consider a focussing Bragg spectrometer with a concave crystal either in the Johann or in the Johansson version . The geometrical arrangement of the problem is illustrated in fig . 1 where most of the relevant parameters are indicated . For consistency, we will keep close to the notation as used in refs. [8] and [111 . Both the source and the crystal, before bending, are assumed to be rectangular parallelepipeds . The depth, width, and height of the source are denoted xo, yo , z o and the corresponding dimensions of the crystal are denoted ro , to , h o , respectively . The radius of curvature of the crystal is denoted by R . The detector is represented by a rectangular screen of width Xo and height Yo and is oriented perpendicular to the "central" X-ray lying in the Rowland-circle plane and being reflected at the crystal's apex . Such an orientation of the detector is a
896
H. F. Beyer, D. Liesen / Ray tracing of curoed-crystal X-ray optics t,
Fig. 1 . The geometry of a curved-crystal spectrometer in the Johann or Johansson version with a position-sensitive X-ray detector and a fast source as provided by accelerator-based experiments .
good choice if one is interested in a small wavelength interval only but in a high wavelength resolution . Detector, source, and crystal are symmetric with respect to the Rowland-circle plane as shown in fig. 1 for a spectrometer setting 0 . For our calculations we make the following assumptions : - The source consists of uniformly distributed emission points Q(x, y, z) with -x o /2 + x off -< x -< x o/2 + xoff, - yo/2 !!~ y :!~ yo/l and - z o/2 < z < z o/2 . They have a constant velocity vion defined by its absolute value I Vi .. I = P co , where c o is the velocity of light, and by its in-plane angle Own and its out-of-plane angle 'Pion as defined in fig . 1 . The center of the source can be shifted by the amount x off inside the Rowland circle. For xoff < 0, it is outside the Rowland circle . The emission characteristic S(a) in the emitter frame of reference can be chosen to be strictly monochromatic, a Gauss function or an exponential as a function of X-ray energy with a width AE. . In the emitter frame, X-ray emission is assumed to be isotropic . - The crystal consists of scattering points B(r, t, h) with 0<_ r<_ ro , -t o/2 <- t < t o/2, and . They are distributed exponentially in the direc:5h<-ho/2 ho/2 tion of the crystal depth r and uniformly in the two other dimensions . The exponential takes care of the X-ray absorption in the crystal leading to an extinction length
X-ray emitted by a source point Q(x, y, z) and reflected at a scattering point B(r, t, h) . B B is the Bragg angle defined by Bragg's law na=2d sin BB,
(2)
where l is the wavelength of the X-ray in the laboratory frame of reference. n and d denote the order of diffraction and the lattice interplanar spacing, respectively. The function F(B; - BB) can be constructed from measured and calculated crystal-structure data . In our calculations we will approximate it by a Gaussian F(g;-9 )=Fo exp~B
(B; - BB) 2
2a 2
~,
(3)
with a peak reflectivity FO and a standard deviation a. The corresponding width of the rocking curve is taken to include all broadening effects due to the treatment of the crystal including bending . - The detector is position-sensitive in two dimensions X and Y. Broadening due to a finite depth of the detector is not considered. It could be evaluated by calculating the reflection profile as a function of detector position, because we provide the possibility to move the detector inside (doff > 0) or outside (doff < 0) the Rowland circle . This option is important for some applications with a fast source where, due to the Doppler effect, a focus can occur away from the Rowland circle.
dr=sin 0, where Ft is the relevant X-ray linear extinction coefficient . It turns out that in most cases ro and dr can be set to zero for soft X-rays because the small penetration depth has a negligible effect compared to other geometrical aberrations . Incident X-rays are specularly reflected with a probability given by the emission pattern (or rocking curve) F(B ; - Ba), where B i = 8; (x, y, z, r, t, h) is the effective glancing angle for an
3. Method of calculation The intensity distribution in the detector is calculated in the following way . For every single X-ray defined by coordinates x, y, z, r, t, h we can calculate the position X, Y in the detector where the X-ray is reflected to . For the determination of the reflectivity according to eq . (3) we need to know the effective
H.F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
897
glancing angle B i (x, y, z, r, t, h) and the Bragg angle B B which, through eq. (2) and a Lorentz transformation, is a function of the wavelength X ion measured in the ion's frame of reference . Therefore, the intensity in the detector is given by the integral
spectively . The distance between Q and B is (see triangle BQH) ] (9) u= [(R'+r-w)2+ o 2 t/2 ,
I (X, Y) = I V
u= [(R'+r-w)2+v2+(h-z)211/2,
f
F(Bi(X) - OB(X,
XS(xion)
dX
Xion))
dXion,
(4 )
where X = (x, y, z, r, t, h) and the integration runs over the source and crystal volume and over the spectral distribution of the emitter. The 7-dimensional integral (4) is solved by the Monte Carlo method.
and the distance between Q and B is
where we used the distance between K and B being R'+ r with R
,_ R - ~R
for the Johann geometry, for the Johansson geometry .
COS T
(11)
The effective glancing angle B i is then given by sin Bi=(R'+r-w)/u,
3.1 . The geometrical problem For calculating the effective glancing angle B i and the locus X, Y where a single X-ray hits the detector we refer to ref. [11] and to fig. 2 showing a projection into the Rowland circle plane and the auxiliary quantities used. All points shown lie in the Rowland circle plane . For our calculations we need the following quantities (fig. 2) : b = R COS 0, and
(12)
and its projection Bi into the Rowland circle plane is
sin Bi = (R'+ r -
w)/u .
(13)
Considering triangle SKW we can write the distance between W and B as p=R'+r-R
COS
B/COS(B-T) .
q can be composed of
(14)
plus p sin(B - T)
q=R(sin 0-sin T/COS(B-T))-d off +p sin( 0-T) . (15) With these quantities the horizontal coordinate X of the image point in the detector reads (X = DE + ED)
It can be shown [111 that
X=p Cos(B-T)+q tan(B-B i -T),
u=(b-y) sin(B+T)-x cos(B+T),
and the corresponding vertical position is
and w=(b-y) COS(B+T)+x sin(B+T) .
(10)
(8)
Q and B are projections of the source point Q(x, y, z) and of the diffraction point B(r, t, h), re-
Y=h+q h-z /COS(6-B i -T) . u
(16)
(17)
3.2 . Transformation of the wavelength to the laboratory system
In order to evaluate the wavelength X in the laboratory frame of reference we have to determine the angle a between the emitter velocity vion and the direction of the emitted X-ray. According to fig . 1, the direction of the velocity is defined by angles (Pion and Y'ion . We suppose now that the direction of a single X-ray is given in an analogous way by angles ox and Y'x . With the aid of fig . 3 we can write (18)
c 2 =a 2 +b 2 -2ab COS a,
where a, b, and c have only a local meaning defined by the figure . On the other hand, c 2 can be expressed as c2 = a2
COS 2~ion
-2ab Fig. 2 . A projection of the geometry into the Rowland-circle plane for calculation of the image point of a single X-ray originating from a source point Q(x, y, z) and being reflected at a scattering point B(r, t, h) . Q and h are the projections of Q and B, respectively .
+ 62 COS2t~x
COS `,ion COS 4, x COS(OX - 4ion)
+(a sin ip ion - b sin =a2+b2-2ab{COS +Sin 4x Sin
Y'ion ) -
jpx
)2
'Pion COS Y'x COS(ox - 0ion)
(19)
89 8
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
we get Cos a = [ 1 - (( h - z )/u
)21
1/2
X COO - B; + T - 0inn) Cos 41x +(h-z)/u
sin
`Yion .
(23)
This result can be inserted into the Lorentz transformation X =X ion( l-
a2 ) -1/2(1- a COS a),
(24)
where X ion is the wavelength of a single X-ray in the emitter frame of reference. 3.3. Solid angle and efficiency
Fig. 3. Auxiliary plot to calculate the angle a between the direction of a travelling ion and the direction of a single X-ray emitted by the ion.
Comparing eqs. (18) and (19) yields COs
a = COs
Sl =
`rion COs `Yx COS(Ox - (Pion )
+sin
9' x
stn
(20)
4' ion-
Upon substitution (see fig. 2) of 0x=0-6,+T,
(21)
and sin
We are not only interested in the relative distribution of X-rays but also in the absolute intensity in the detector at a given source strength, i.e . we want to determine the spectrometer's absolute efficiency. The geometrical solid angle of the crystal as seen from the center of the source is given by
(22)
¢x=(h-z)/u,
to h o
sin 0
(R sin 0 - xnff )
2 .
Because in practical cases the source-to-crystal separation is large as compared to crystal and source dimensions, eq. (25) is a good approximation for all source points and we do not have to perform an exact integration over solid-angle elements dS2. For a fast source we also have to take into account the relativistic transformation of the solid angles [12] . This is achieved
run -100 switch = n-total = theta-deg x0 = y0 = z0 = x-off = rO = t0 = hO = XO = YO = d-off = rad = sigma = d2-crystal beta = phi-ion-deg = psi-ion-deg = delta-e = lambda-max = n-det-x = n-det-y =
1 5E6 60 2 2 2 0 0 5 5 1 15 0 1000 fE-4 4 0 0 0 0 3 .4641 30 30
. . . . . .
omega = overflow = integral =
2 .2972E-6 1 .27E-f0 0 .0541966
. . . . . . . . . . . . . mean solid angle devided by 4n . . . . . . . . . . . . . . . . . intensity outside detector . . . . . . . . . .integrated intensity in detector
.
. . . . . . . . . . . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
(25)
. . . . . .
. . . . . . . . . 0 = johann, 1 = johansson . . . . . . . . . . . . total number of x rays . . . spectrometer setting in degrees . . . . . . . . . . . . . . . . source depth in mm . . . . . . . . . . . . . . . . source width in mm . . . . . . . . . . . . . . . . source heightin mm . . . . . . . source displacement into R .C . in mm . . . . . . . . . . . . . . . . . . . . . . . crystal depth in mm . . . . . . . . . . . . . . . . . . . . . . . crystal width in mm . . . . . . . . . . . . . . . . . . . . . . crystal height in mm . . . . . . . . . . . . . . . . . . . . . . detector width in mm . . . . . . . . . . . . . . . . . . . . . detector height in mm . . . . . . . . . detector displacement inside R .C . .. radius of curvature in mm . diffraction pattern (angular width) sigma . . . . . . . . . . . . . . . . . . . . . . grating constant 2 d --- . . ion-beam velocity v/c ion-beam : angle in plane of Rowland circle . . . . . . . . . . . . . . out-of-Plane angle (degrees) . . . . . . . C .M . width (eV) of emission pattern . . . . . . . . . . . . . peak wavelength in Angstroems . . number of pixles in horizontal direction . . . . number of pixles in vertical direction
Fig. 4. Chart of input and output parameters for the program IMAGE.
H.F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
by multiplying, for each single X-ray, the reflected intensity by a2 ; on _ aotab
(1
1 _P2 h _ ,8 a) 2 COS
.
20,1
89 9
Influence of the Crystal Rocking Curve
(26)
The overall spectrometer efficiency reads Ndet Not 47T
(27)
with Ndet/Nat the number of X-rays reflected into the detector divided by the total number of X-rays hitting the crystal. 3.4. Numerical procedure
By the numerical procedure one tries to exactly simulate the situation of a real experiment . For this purpose, the computer program IMAGE [131 was developed which evaluates eq . (4). In the program a random grid is defined in the 7-dimensional space of the source and crystal volume and of the source wavelength, i .e . one runs through a loop over a number Not of X-rays in which the independent variables are random numbers . Within the loop all the analytical expressions given above are calculated . For each single X-ray, the effective angle B; is calculated giving the position in the detector according to eqs. (16) and (17) . The detector area itself is digitized in an Nx X NY matrix of small rectangular bins in which the intensity is collected. There is one extra overflow bin collecting the intensity of all those X-rays which are reflected from the crystal but which do not hit the detector. The intensity accumulated in the individual bins originates from single events with reflected intensity given by eq . (3) in which the effective glancing angle Bi is inserted as given by eq . (12) and the Bragg angle B B defined by eqs. (2) and (23), (24). The solid-angle correction of eq. (26) is applied to each single event. In the following, we will present some numerical results calculated with the program IMAGE . Fig . 4 contains a chart of the input and output parameters used for a test run and their explanations . 4. Results of calculations and discussion 4.1 . Stationary source
In this section we consider only those cases in which the source is at rest (,l3 = 0) . Because the results of calculations can easily be scaled with the absolute size of the spectrometer the radius of curvature of the crystal is set constant to R = 1 m throughout this investigation. First, we show the influence of the crystal rocking curve
Distance X (mm) Fig. 5 . X-ray profiles (integrated over the detector height) displaying the influence of the crystal rocking curve with various values of the standard deviation a . for small crystal and source dimensions . The actual input parameters can be found in fig. 4. One-dimensional X-ray profiles, integrated over the detector height, are shown in fig. 5 for various values of a entering the diffraction pattern of eq . (3). The peak reflectivity is Fo = 0.5 in all cases. One can see how the profile gets narrower with decreasing a and that the peak height stays about constant . The effect of the crystal width for the Johann case can be observed in figs . 6 and 7. Fig. 6 displays profiles at a constant spectrometer Bragg angle 0 = 30 ° and varying crystal width t o illustrating clearly the Johann defect of a distorted profile if the crystal becomes too wide . For comparison is shown the profile for "exact" focussing in case of the Johansson geometry with the crystal width being to = 100 mm. Fig. 7 shows the change of the profile with changing Bragg angle. The peak asymmetry and centroid shift is largest at small Bragg angles .
8
2 0
-0 .5
0.0 0 .5 1 .0 1 .5 Distance x (mm) Fig. 6. X-ray profiles displaying the effect of a finite crystal width t o for a Johann spectrometer at 0 = 30 ° . For compari son the profile for a Johansson geometry with t o =100 mm is also shown (dashed line).
900
H. F Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics run 101
10 8
s
6
4 3 2 1
4 2
10" 10-' 10- ' 10- ' 10- ' 0
0
-0 .4-0 .2 0 .0 0.2 0.4 0 .6 0.8 1 .0 Distance x (mm) Fig. 7 . X-ray profiles displaying the effect of a finite crystal width t o = 50 mm at various Bragg angles 0.
The peak centroid position was calculated by Zschornack et al . [111 by an approximate analytical integration of sin 6;, as given by eq. (12), over single nonzero dimensions . Noting that the dispersion can be approximately written as
ax =
a- R sin
2o
cos 6 ' where Ax is a shift in the detector, one obtains
ax = -(24R cos
B) -1 (h 2
+zo - t 2 cos 20) .
(28)
(29)
Fig. 8 . X-ray profile displaying the effect of a dominating crystal height. Input parameters for this calculation were set to 8=60 ° , x o = yo =z o =2 mm and ro =0, t o =h o =50 mm .
discuss the case in which the crystal height causes the largest distortion . Fig. 8 shows the two-dimensional intensity profile for a Johansson geometry where we used the parameters 0 = 60 °, x o =yo = z o = 2 mm and ro = 0, t o = h 0 = 50 mm . In order to collect the total reflected intensity the detector is 100 mm high . Because of the curvature observed in the image pattern one would loose resolution by integrating over the detector height Y. This is more clearly demonstrated in fig. 9 showing the corresponding integrated profile together with a cluster plot of the data from fig. 8. The dashed line in fig. 9 indicates the peak centroid calculated by use of eq . (29) . In fig. 9 the solid "white" line represents the result of an analytical calculation which gives only the line of gravity of the image pattern . For this calculation one starts from eqs. (12) and (13) which can be combined to
Eq . (29) takes into account the main contributions to the shifts, namely the crystal and source height and the crystal width (h o , z o , and t o , respectively). In table 1 are compared the centroid shifts derived from data like those displayed in figs . 6 and 7, resulting from the Monte Carlo calculations, to the analytical values obtained from eq . (29) . The errors shown represent the statistical uncertainties of the Monte Carlo sin B; = sin B; ( u/u ) . (30) calculations . The total number of X-rays was n tot = 1 X For dominating vertical coordinates eq . (30) becomes 10 6. An excellent agreement is observed between the 1/2 results of the detailed numerical calculation and those sin B i = Sln BJ1 + tan 24, x ] of the analytical expression . In order to increase the solid angle it is essential to [1 + tan2 p (31) make use of a nonnegligible crystal height . We will now
= 2d
Table 1 Centroid shifts in mm calculated with the Monte Carlo method and analytically using eq . (29) ; the crystal height is set to h o =10 mm . The numbers in brackets show the uncertainty in the last digit displayed. 0
15'
Monte Carlo Analytical
0 .095(2) 0 .0963
Monte Carlo Analytical
30 °
Crystal width t o = 50 mm 0 .085(1) 0 .0854
45 °
60 °
75 °
0 .0676(9) 0 .0678
0.0438(7) 0.0438
0 .0112(7) 0 .0109
0 .289(2) 0 .289
0.199(2) 0.200
0 .092(1) 0 .0917
Crystal width t o =100 mm 0 .394(6) 0 .398
0 .355(4) 0 .356
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
run
run
101
8 7 6 5 4 3 2 1 0
901
101
60 40 20
E
0 -20 -40
-60 1 .0 -1 .0 -0 .5 0.0 0 .5 1 .0 0.0 0 .5 X (mm) X (mm) Fig. 9. Same data as in fig. 8: (a) integrated profile; the dashed line marks the peak centroid calculated from eq . (29) ; (b) a cluster plot of the numerical data compared to the analytical calculation (represented by the "white" line) according to eq. (31) . -1 .0
-0 .5
with tan p, = (h - z )/( R sin 0 - xoff ) .
(32)
In eq . (31) tan %P, and sin Bi determine the vertical and horizontal coordinate in the detector, respectively . The very good correspondence of the two results is evident from fig. 9. The width of the profile stays strictly constant over the whole vertical dimension as can be inspected from horizontal cuts taken at various heights. This means that wavelength resolution is preserved and one does not have to cut the crystal and detector height if a two-dimensional position-sensitive detector is used . The profile width observed in figs . 8 and 9 is dominated by the width of the rocking curve with a set to 1 X 10 -4 corresponding to a wavelength resolution of X/4 X = 6500. This value can still be improved if a crystal with a narrower rocking curve can be selected, without running into problems due to geometrical aberrations . Also the influence of the crystal width, in case of the Johann geometry, is quite moderate at the Bragg angle 0 = 60 °
used here. Crystals with to = h0 = 50 mm can well be used in Johann geometry at this kind of wavelength resolution . The variation of the spectrometer efficiency with the source location is demonstrated in fig. 10 . According to eq . (27) the total efficiency is the product of the crystal efficiency and the geometrical solid angle devided by 4~r. In figs. 10a and b, both contributions are shown for source widths of 1 and 5 mm, respectively. Other geometrical dimensions were to = 20 mm, ho = 10 mm, xo = 0.1 mm, zo = 5 mm . For the small source width of yo = 1 mm a pronounced maximum of the crystal efficiency is observed when the source approaches the Rowland circle xoft = 0. This maximum is washed out when the source width is set to yo = 5 mm, leading to a relative maximum of the total efficiency somewhat inside the Rowland circle . We note, however, that operation of a spectrometer with a small source near the Rowland circle leads to a very narrow useful wavelength interval . Therefore, one has to make a compromise between wavelength range and efficiency.
0 .20 U U
w w
0.15
U t~
0.10
U
w
w
0.05 0.00 -1000
-600
-200 0 200 400 600 Offset (mm)
0 .08 0 .07 0 .06 0 .05 0 .04 0 .03 0.02 0.01 0.00 -1000
-600
-200 0 200 400 600 Offset (mm) Fig. 10. Spectrometer efficiency as given by the crystal efficiency times the geometrical solid angle as a function of the source location . For xoff < 0 the source is outside the Rowland circle . The source width was set to (a) yo =1 mm and (b) yo = 5 mm, respectively.
902
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
4 .2. Fast source
We now consider cases with a > 0 . Let us first derive an analytical formula in analogy to eq. (31) for a finite crystal (or source) height but for a fast source . If we replace, in eq . (31), A by the appropriate laboratory wavelength given by eqs. (24) and (23) we obtain sin 9; -
nXion -#2)1/2
2d(l X [1
- N (COS
+sin
4ion
`Y ionCOS(N - 'Pion ) COS t1 X
sin 4,,)] 11 +tan24/ x1
In eq .(33) we have averaged over Cos( cpx - 0ion) - TO _
~
fTo/2
o/2
T
$x
1/z
(33)
according to
COS(Ox - 4)ion )
I,
cl,- b,
(34)
where To = to/(R - xoff/sin 0) . Carrying out the integration and the multiplication of the square brackets in eq. (33) yields sin Bi =
n ~'ion a2)1/z 2d(1 X [(I
1/z - /3(2 + tan24,, ) sin(TO/2)/To
X COs 'rion COs (P ion
+ sin
4'ion
tan
~x)] .
(35)
The first term in the square brackets is the same as for the stationary source causing the curved image. The velocity-dependent part splits into two terms. The first one is a Doppler shift being independent of 41, whereas the second one is proportional to tan 4r, . In the following we will compare our numerical calculations to eq . (35) .
Let us now discuss the results obtained for different orientations of the velocity via with its absolute value set to 8=0.1 . For clarity, fig. 11 gives the specific values of the input parameters used in the following examples . They are kept constant with the exception of the angles $ion, V'mn, and of the corresponding emitter wavelength a ion . The largest broadening effects are observed when an extended source travels with a velocity component tangential to the Rowland circle, i.e. in the y direction. This is clearly demonstrated in fig. 12 for a source travelling in the Rowland-circle plane with ~b ;on = 90 ° . The wide plateau of fig. 12a indicates that in this case an imaging of the whole source width of yo = 2 mm occurs with an accompanying loss of wavelength resolution . For comparison, fig. 12b reveals the shrinking to a narrow line when a narrow source of yo = 0.1 mm width is used. For such a narrow source, which in practice can be represented by a slit, the crystal efficiency can be further improved by putting source and detector to a refocussing position [14] . This means that the wavelength spread, due to the Doppler effect, of X-rays filling the cone of the solid angle is partly compensated by placing the source inside and the detector outside the Rowland circle . For our particular case the offsets are calculated to amount to xoff = 93 .4 mm and d~ff = -119 .2 mm . A similar refocussing has been applied to grating spectrometers [15] . We will not discuss this point any further because it is important only for small sources and for scanning instruments . We continue with an extended source x o =yo = z o = 2 mm and q)ion = 0. Fig. 13 applies for ipion = 0, i.e . the source travels towards the crystal . For this case we obtain a narrow line again. For figs . 14 and 15 the
run 102 ... . . . . .. . .. . .. . ..
switch = n-total = theta-deg x0 = YO = z0 = x-off = rO = to = hO = XO = Yo = d-off = rad = sigma = d2-crystal beta = phi-lion-deg psi-lion-deg de lta-e = lambda-max = n-det-x = n-det-y =
1 5E6 60 2 2 2 0 0 50 50 3 40 0 1000 fE-4 4 0 .1 90 0 0 3 .44674 30 30
omega = overflow = integral =
2 .2972E-4 . . . . . . . . . . . . . mean solid angle devided by 4rr 8 .15412E-3 . . . . . . . . . . . . . . . . . intensity outside detector 5 .90865E-3 . . . . . . . . . .integrated intensity in detector
0 = johann, 1 = johansson . . . . . . . . . . . . . . . . . . . . . total number of x rays . . . . . . . . . . . . spectrometer setting in degrees . . . . . . . . . . . . . . . . . . . . . . . . . source depth in mm . . . . . . . . . . . . . . . . . . . . . source width in mm . . . . . . . . . . . . . . . . . . . . . . . . . source heightin mm . . . . . . . . source displacement into R .C . in mm . . . . . . . . . . . . . . . . . . . . . . . . crystal depth in mm . . . . . . . . . . . . . . . . . . . . crystal width in mm . . . . . . . . . . . . . . . . . . . . . . . crystal height in mm . . . . . . . . . . . . . . . . . . detector width in mm . . . . . . . . . . . . . . . . . . . . . . detector height in mm . . . . . . . . . . detector displacement inside R .C . . . . radius of curvature in mm . . diffraction pattern (angular width) sigma . . . . . . . . . . . . . . . . . . . . . . . grating constant 2 d . ion-beam velocity v/c . .ion-beam : angle in plane of Rowland circle . . . . . . . . . . . . . . . out-of- lane angle (degrees) . . . . . . . . C .M . width (eV~ of emission pattern . . . . . peak wavelength in Angstroems . . . number of pixles in horizontal direction . . . . . number of pixles in vertical direction
Fig. 11 . Parameter values used for the study of a fast source.
903
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
run
102
a)
1 .210' 1 .010-, 8.0 10-" 6 .0 10-° 4.0 10-° 2 .0 .10`
Fig. 12 . Intensity profile for a fast source travelling in the Rowland-circle plane with Con = 0 ° and source was (a) yo = 2 mm and (b) yo = 0.1 mm, respectively.
Own =
90 ° . The width of the
run 104
Y6 ) 7-
10
,moo -05
0:
lOcol
Fig. 13 . Intensity profile for a fast source travelling towards the crystal, Con ='Pion = 0 °. velocity vi points out of the Rowland-circle plane . with 'Pion = 30 ° and 4, ;o = 90 °, respectively . As expected from eq . (35) the line gets slanted with the slope being proportional to ,Q sin Con. The slanting has already been observed in beam-foil [161 and beam-gas [171 experiments . A comparison between the numerical data and eq . (35) is shown in figs . 15 and 16 .
Fig. 14. Same as fig. 13 for ¢;o = 0 ° and
~;oa =
30 ° .
For a detailed comparison, we calculated the center of gravity x m of horizontal cuts of the numerical data taken at various heights in the detector . In fig. 16 the difference between the position -'anal determined analytically and the center of gravity xn m is plotted. A perfect agreement within the statistical accuracy of the numerical procedure is observed . In our example run 106
run 106
nn
x 0
X (mm)
.
0 -2
I I
-4 -6
I 4 0 2 x anal . - x num. (I-tm) Fig. 16 . The data of fig. 15 plotted as the difference between the center of gravity determined analytically from eq . (35) and the one derived from the numerical data at various heights in the detector. The error bars indicate the statistical uncertainty. n
Fig. 15 . Cluster plot of the intensity pattern for O . = 0 ° and ; ¢;o = 90 °. The "white" line represents the analytical calculation according to eq . (35) .
4 2
-4
-2
904
H. F. Beyer, D. Liesen / Ray tracing of curued-crystal X-ray optics
fluctuations of about ± 1 p m correspond to a wavelength precision of about 7 x 10 -7 . The same quality of
run 107
correspondence between analytical and numerical data
was obtained for a variety of cases, including stationary
sources, whenever eq . (35) is applicable . This holds true
even if we allow other geometrical dimensions to come
into effect simultaneously . For instance, let us consider
the case of fig. 15 but with a Johann crystal of 50 mm
width and a source height of 10 mm. We obtain the same slanted line which, however, displays some broadening due to the finite crystal width and source height . The
corresponding center-of-gravity shift has been proven numerically to be constant over the whole detector height and to be well described by eq . (29).
In table 2 we summarize the performance of the spectrometer for various source arrangements . The corresponding parameters are /3 = 0.1, $ion = 0 °, 4ion =
0 ° , 30 ° , 90 ° , and xoff = 0 and 100 mm . Other parameters are identical to those given in fig. 11 . The crystal efficiency given in table 2 does not include the enhance-
aQ ion /aQ lab of eq . (26) . For the source travelling towards the crystal, ip ;on = 0, the same crystal efficiency is obtained as for the same source at rest and by the transformation of the solid angle 22% of ment of the solid angle
intensity is gained . If the velocity vector is turned out of the Rowland circle the intensity (which is proportional to the crystal efficiency times the solid-angle enhancement) drops. The relative decrease is much weaker when the source is placed 100 mm inside the Rowland circle .
The wavelength resolution dX/a was derived from the FWHM line width of horizontal cuts of the intensity profile. In table 2 are listed the results obtained for cuts
Fig. 17. Intensity distribution in the detector for an emission pattern being an exponential with a maximum at 3286 eV and a decay constant of 3 eV . Other parameters are identical to those used in fig. 14 .
near the upper or lower edge of the profile. The line width stays practically constant over the whole profile height allowing wavelength resolutions comparable to those obtained for a stationary source . In our example, the resolution degrades by about 20% when the velocity
direction turns from 41 ion = 0 ° to 'P ;an = 90 ° . We finally give an example where, in the emitter frame of reference, the emission pattern is not a delta
function but is given by an exponential as a function of X-ray energy. The decay constant is set to 3 eV with the
maximum intensity taken at an X-ray energy of 3286 eV . The geometrical parameters are identical to those used for generating fig. 14, i.e . /3 = 0.1, 't` ;o = 0 °, Con
taken in the Rowland-circle plane and for those taken
= 30 o . The intensity profile obtained is shown in fig. 17 displaying the decaying intensity towards decreasing wavelength or increasing X.
Table 2 Spectrometer performance with a stationary source and with a source travelling with a velocity /3 = 0.1 . Parameters not specified in the table can be found in fig. 11 .
5. Concluding remarks
Source on R.C. xoff = 0
Crystal efficiency
a'n,on/aszlab
(aX/X)xl0 4 in plane out of plane
Stationary source
Fast source ,Q = 0.1, 'Peon = 0 ° =0' 30' 90-,
0.0545 1 .00
0.0543 1 .22
0.0238 1.19
0.0126 0.99
1.41 1.45
1 .43 1.42
1.52 1.50
1 .71 1 .70
Source inside R.C . xoff =100 mm Crystal efficiency
aSkn/aQlab
(da/a)x10 4 in plane out of plane
0.0193 1 .00
0.0193 1 .22
0.0174 1 .19
0.0112 0.99
1 .36 1 .48
1.36 1.42
1.48 1.58
1 .72 1 .70
In this article we have systematically analyzed the effects of geometrical aberrations combined with Doppler shifts as encountered in curved-crystal spectrome-
try of fast X-ray sources. A detailed ray tracing was performed for Bragg spectrometers operated with a stationary or a fast source . In some limiting cases it is possible to predict accurately the center-of-gravity positions of the X-ray profile in the detector, whereas in the general case and for the study of line shapes it is necessary to perform detailed numerical calculations . The results of this paper can be used to optimize the parameters in the design of a new X-ray experiment . Detailed knowledge of the aberrations we were discuss-
ing can also help to analyze the results of existing and future experiments. We emphasize that both spectrometer efficiency and wavelength resolution can be improved by a careful design of an experiment . Huge crystals can be used,
H. F. Beyer, D. Liesen / Ray tracing of curved-crystal X-ray optics
without loss of wavelength resolution, even for a fast source, when a large two-dimensional position-sensitive detector is employed . Of particular importance is the orientation of the source velocity. If one stays away from a velocity having a component tangential to the Rowland circle, i.e . if ¢io = 0, it is possible to work with an extended source and to obtain a narrow intensity pattern in the detector. With the use of a twodimensional detector not only the emitter wavelength is measured but also the ion velocity and the observation angle. This is important for diagnostics of ion beams by means of X-ray spectroscopy . Acknowledgements The authors enjoyed fruitful discussions with Richard Deslattes. We also received excellent assistance from Ingolf Giese in solving many computing problems .
[4]
[6]
[8] [9] [10] [ill [12] [13]
References [1] See for instance : S.J. Brodsky and P.J . Mohr, in : Structure and Collisions of Ions and Atoms, vol. 3, ed . I.A . Sellin (Springer, Berlin, 1978). [2] For a survey see: B. Franzke, IEEE Trans. Nucl. Sci . NS-32 (1985) 3297 . [3] H.F . Beyer and R. Mann, in : Progress in Atomic Spectroscopy, part C, eds . H.J . Beyer and H. Kleinpoppen (Plenum, New York, 1984) p. 397.
[14] [15] [16] [17]
905
R.D . Deslattes, E.G. Kessler, W.C . Sauder and A. Henins, Ann. Phys . (N .Y .) 129 (1980) 378 . E.G . Kessler, R.D . Deslattes and A. Henins, Phys . Rev . 19 (1979) 215 ; J.A . Bearden, Rev. Mod. Phys. 39 (1967) 78 . R.D . Deslattes and E.G. Kessler, in : Atomic Inner-Shell Physics, ed . B. Crasemann (Plenum, New York, 1983) and references cited therein. E. Kellog, P. Henry, S. Murray, L. van Speybroeck and P. Bjorkholm, Rev. Sci. Instr. 47 (1976) 282; R.A. Boie, J. Fischer, Y. Inagaki, F.C . Merritt, V. Radeka, L.C. Rodgers and D.M. Xi, Nucl . Instr. and Meth . 201 (1982) 93 ; B.P . Duval, J. Barth, R.D . Deslattes, A. Henins and G.G . Luther, Nucl . Instr. and Meth . 222 (1984) 274. W. Schwitz, Nucl . Instr. and Meth. 154 (1978) 95 . H.H . Johann, Z. Phys. 69 (1931) 185. T. Johansson, Z. Phys . 82 (1933) 507. G. Zschornack, G. Miiller and G. Musiol, Nucl . Instr. and Meth . 200 (1982) 481. J.D . Jackson, Classical Electrodynamics (Wiley, New York, 1967). The program IMAGE was developed with the Graphics Numerics Operation Method GNOM . Data presentation was facilitated making use of the TOPDRAW library created by U. Post, University of Giessen. H.F. Beyer, unpublished (1982) . J.O . Stoner and J.A . Leavitt, Appl . Phys . Lett. 18 (1971) 368; ibid . 18 (1971) 477. C.J . Hailey, R.E. Stewart, G.A . Chandler, D.D . Dietrich and R.J . Fortner, J. Phys. B18 (1985) 1443. R.D . Deslattes, R. Schuch and E. Justiniano, to be published.