Scanning monochrometer crystal and method of formation

A doubly-curved crystal for use in a scanning monochromator is oriented with respect to a reference plane containing source and image locations of the monochromator. The crystal has concave planes of lattice points and a concave crystal surface which satisfy Johannson geometric conditions within the reference plane for a Rowland circle of radius R. The planes of lattice points are substantially spherically curved to a radius of 2R, and the crystal surface is substantially toroidally curved with a radius of substantially 2R within a plane perpendicular to the reference plane. The crystal may be formed by plastically deforming a cylindrically curved crystal blank over a doubly-curved convex die.

The present invention relates generally to the art of radiation diffraction 
and, more particularly, to a scanning monochromator useful in diffracting 
and selectively monochromatizing radiation emanating from a point source. 
It is often desirable to analyze radiation over a range of possible 
wavelengths to determine spectral content. Devices for doing so are called 
scanning monochromators. One circumstance in which scanning monochromators 
are useful is the analysis of fluorescence X-rays in a local X-ray 
excitation scheme of the type disclosed in the above-identified 
application, Ser. No. 549,366, now U.S. Pat. No. 4,599,741 the disclosure 
of which is hereby incorporated by reference. 
Scanning monochromators analyze source radiation by moving a diffracting 
crystal and a suitable detector relative to the source location, typically 
along the circumference of a Rowland circle having a radius much greater 
than the length of the crystal. This causes the angle of incidence 
(.theta.) on the crystal surface to vary continuously through a range over 
which radiation of different wavelengths is constructively reinforced by 
diffraction from planes of atoms in the crystal lattice. The resulting 
radiation is monochromatized at the wavelength that is constructively 
reinforced at any point in time, permitting the intensity of radiation at 
that wavelength to be detected with ease. Systems of this type are 
disclosed in Browning et al., U.S. Pat. No. 3,546,453 and Hara, U.S. Pat. 
No. 3,914,605, and the principles underlying them are described in "The 
Optical Principles of the Diffraction of X-rays", The Crystalline State, 
Vol II, R. W. James (1958). 
Scanning monochromators of the prior art make use of flat or singly-curved 
crystals to provide a useful output over a range of Bragg angles. 
Doubly-curved diffraction crystals have been proposed in non-scanning 
devices to provide accurate focusing and good performance at certain 
specific Bragg angles, but the range of angles over which such crystals 
are useful has been so limited that they have not been onsidered suitable 
for scanning. A scanning monochromator must provide a reasonably 
consistent output over a wide range of Bragg angles. 
Although prior scanning monochromators have high signal-to-background 
ratios in comparison to other instruments, the signal-to-background ratio 
available with flat or singly-curved crystals limits the sensitivity of 
such monochromators to low intensity signals. Therefore, in many 
applications it is desirable to provide a scanning monochromator with 
improved signal-to-background ratio and enhanced detection limits. 
SUMMARY OF THE INVENTION 
The present invention relates to a crystal arrangement for use in a 
scanning monochromator, which monochromator has a source location and an 
image location and is capable of producing relative movement between the 
source location, the image location and the crystal arrangement to vary 
the angle at which radiation emanating from the source location is 
diffracted. The crystal arrangement includes: a crystal positionable 
relative to a first reference plane containing the source and image 
locations, the crystal having concave planes of lattice points and a 
concave crystal surface which satisfy Johannson geometric conditions 
within the first reference plane for a Rowland circle of radius R; the 
planes of lattice points being substantially spherically curved with a 
radius of 2 R within the first reference plane and a radius of 
substantially 2R within a second reference plane which is perpendicular to 
the first reference plane and bisects the Rowland circle; and the crystal 
surface being substantially toroidally curved with a radius of R within 
the first reference plane and a radius of substantially 2R within the 
second reference plane. In a preferred embodiment, the planes of lattice 
points are curved as a true sphere about a preselected point located at 
the circumference of the Rowland circle and opposite to the crystal 
surface, and the substantially toroidally curved surface is defined by 
rotating points on the Rowland circle about said preselected point in a 
direction perpendicular to the Rowland circle. In a further embodiment, 
the substantially spherical curvature of the planes of lattice points is 
only approximate and the crystal surface is curved as a torid defined by 
rotating an arc of radius R about an axis which is tangential to the 
Rowland circle at the preselected point. 
The invention also relates to a method of fabricating the doubly-curved 
crystal described above. The method involves preparing at least one 
crystal lamella having planes of lattice points and having an adjacent 
crystal surface which is curved cylindrically about a preselected axis; 
placing the lamella onto a convex mold curved to a preselected radius in 
the drection of cylindrical curvature and curved in a perpendicular 
direction to substantially twice that radius; covering the lamella with a 
continuous sheet of material able to withstand elevated temperatures; 
heating the mold, the lamella and the sheet to a temperature at which the 
sheet is flexible and at which the lamella can be deformed; and creating a 
partial vacuum beneath the sheet to draw it downwardly against the lamella 
and plastically deform the lamella to give the lamella a concave surface 
which matches the face of the mold. 
The crystal of the present invention monochromatizes and substantially 
focuses a much higher proportion of the radiation impinging upon it from a 
point source than either the flat or the singly-curved crystals of the 
prior art. This is illustrated graphically in the drawings, wherein FIGS. 
5A and 5B depict the diffracting surfaces of a doubly-curved crystal of 
the present invention and a singly-curved ("cylindrical") crystal of the 
prior art, respectively. The outlined areas represent portions of the two 
surfaces which are oriented such tat radiation emanating from a point 
source within a central plane will impinge upon them at angles which 
deviate from a given Bragg angle by no more than a preselected amount. Of 
course, only the radiation which impinges on the crystal at substantially 
the Bragg angle, and which is focused on the image location, is useful in 
a scanning monochromator. Thus, a much larger solid angle of useful 
radiation is subtended by the crystal of the present invention, increasing 
the signal-to-background ratio of the system. This difference is even more 
pronounced when the crystal of the present invention is compared to a flat 
crystal, for which only radiation impinging on a curved line passing 
through the center of the crystal reaches the detector. 
Doubly-curved crystals of the type described herein are fabricated by 
plastically deforming singly-curved crystals having preselected 
cylindrical configurations of lattice planes and crystal surface. 
Deformation is accomplished by bending a crystal against a die or a mold 
in a process wherein bending forces are distributed as uniformly as 
possible over the crystal to avoid damage. Applicant uses a flexible sheet 
drawn downwardly by a partial vacuum to force the crystal over the mold.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Referring to FIG. 1, a scanning monochrometer 10 constructed according to a 
preferred embodiment of the present invention contains a crystal 
arrangement 12 and a detector 14 mounted for movement relative to a source 
16 of electromagnetic radiation to be analyzed. Movement of the crystal 
arrangement 12 and the detector 14 is controlled to vary the angle 
(.theta..sub.i) at which radiation emanating from the source 16 impinges 
upon the crystal, while maintaining the detector in position to detect 
radiation diffracted at an angle (.theta..sub.d) equal to the angle of 
incidence. This "scanning" movement is illustrated in FIG. 2, wherein a 
crystal 18 and the detector 14 are shown in a first (full-line) condition 
in which source radiation is incident on the crystal at an angle 
(.theta..sub.i1), and a second (broken-line) condition in which source 
radiation is incident on the crystal at a smaller angle (.theta..sub.i2). 
The detector counts quanta of diffracted radiation as a function of the 
angle of incidence on the crystal. 
For purposes of the present invention, the radiation emanating from the 
source location 16 can be any form of electromagnetic radiation having a 
wavelength (.lambda.) no more than twice the lattice parameter (d) of the 
crystal. The radiation most often comprises X-rays, gamma rays or a 
neutron beam. 
The crystal 18 may be formed of one or more crystal lamellae positioned 
side-by-side to provide the unique crystal geometry of the present 
invention. In either case, the crystal 18 satisfies Johannson geometric 
conditions within the plane of a Rowland circle of radius R which contains 
the source 16 and the detector 14, and has planes of crystal lattice 
points and a concave crystal surface which are curved appropriately in the 
direction perpendicular to the plane of the Rowland circle. 
The geometry of the crystal 18 in the plane of the Rowland circle is shown 
in FIG. 3A. The crystal 18 has convex planes of lattice points 20 curved 
to a radius of 2R about an origin O located on the circumference of the 
Rowland circle and opposite to the crystal, and has a convex surface 22 
curved to a radius of R about the center of the Rowland circle. In 
accordance with Johannson conditions, radiation emanating from the source 
16 encounters the planes 20 at a uniform angle over the surface 22 and are 
focused at an image location I corresponding to the detector 14 of FIGS. 1 
and 2. All radiation emanating from the source S and subtended by the 
crystal 18 within the plane of the Rowland circle are focused onto the 
detector 14. 
In a plane perpendicular to and bisecting the Rowland circle, the planes of 
lattice points 20 and the crystal surface 22 are curved to radii 
substantially equal to 2R. This profile is illustrated generally in FIG. 
3B. Thus, the planes 20 are curved substantially spherically to a radius 
of substantially 2R and the surface 22 is curved substantially toroidally 
with a radius of R in the plane of the Rowland circle and a radius of 
substantially 2R in a plane perpendicular to the Rowland circle. In a 
preferred embodiment, the toroidal contour of the surface 22 is defined by 
rotating the circular profile of the surface in the plane of the Rowland 
circle about an axis which is tangential to the Rowland circle and 
contains the origin O. 
As used herein, the term "substantially spherically curved" defines a shape 
of crystal planes in which the radii of curvature in two perpendicular 
directions can vary up to 20 percent from each other without defeating the 
utility of the crystal in a scanning monochromator. The planes of lattice 
points are preferably curved as a sphere with radii equal to precisely 2R 
(twice the radius of the Rowland circle) both in the plane of the Rowland 
circle and in a plane perpendicular to and bisecting the Rowland circle. 
However, acceptable results can be attained when the planes of lattice 
points are curved to a radius as small as 1.6R (20 percent less than 2R) 
in the plane perpendicular to the Rowland circle. Since the radius within 
the plane of the Rowland circle must equal 2R in order to satisfy 
Johannson conditions there, the resulting geometry is not curved as a 
"true sphere". Rather, it is toroidally curved with radii so nearly equal 
in perpendicular planes that it performs almost as well as the spherical 
case. 
Likewise, the term "substantially toroidally curved" signifies a crystal 
surface which need not define a true torus. The surface can be defined by 
rotating points on the Rowland circle about a single point within planes 
perpendicular to the plane of the Rowland circle, rather than rotating the 
curve in the plane of the Rowland circle about a line to form a true 
torus. The radius within a plane perpendicular to the Rowland circle can 
vary up to 20 percent from the optimum value. In a preferred embodiment, 
the crystal surface is curved to a radius of R in the plane of the Rowland 
circle and to a radius of 2R in a plane perpendicular to and bisecting the 
Rowland circle. In accordance with the guidelines given above, the radius 
in the plane perpendicular to the Rowland circle may be as small as 1.6R 
without defeating the utility of the crystal for scanning. 
Thus, the optimum geometry according to the teachings of the present 
invention is one in which the curvature of the planes of lattice points is 
as close as possible to that of a true sphere of radius 2R and the 
curvature of the crystal surface is as close as possible to that of a 
torid of radius R in the plane of the Rowland circle and radius 2R 
perpendicular to the Rowland circle. Any deviation from this geometry by 
reducing he radii of curvature within planes perpendicular to the Rowland 
circle approaches the toroidal geometry disclosed in the above-referenced 
patent application, Ser. No. 549,366, in which Johannson geometric 
conditions are met in all planes containing source and image locations for 
a specific predefined wavelength of radiation. In such cases, the 
signal-to-background ratio is enhanced for radiation having wavelengths 
very close to the predefined wavelength, at the expense of performance 
over the broader spectrum. Of course, "tuning" of this type must be 
minimized if a crystal is to remain useful in scanning monochromators. The 
highly tuned nature of prior doubly-curved geometries is largely 
responsible for the longstanding belief that this type of doubly curved 
crystal is not useful in scanning monochromators. 
The doubly-curved crystal 18 receives a large proportion of the radiation 
incident on it from the point source S at angles very close to the Bragg 
angle (.theta..sub.B) at which radiation from the point source impinges 
upon the center of the crystal 18, and monochromatizes the radiation 
according to established diffraction principles. At the same time, it 
focuses the diffracted radiation onto the image location I as a short, 
curved line. This is substantiated analytically by calculating the 
deviation of the sine of the incidence angle (.theta.=.theta..sub.B 
+.DELTA..theta.) at any given point on the crystal surface from the sine 
of .theta..sub.B, and comparing it to a similar deviation calculated for 
the case of a cylindrically bent crystal which satisfies Johannson 
conditions within the plane of the Rowland circle. 
Calculation of the Deviation of Sin (.theta.+.DELTA..theta.) from Sin 
.theta. 
The geometry for this calculation is shown in FIG. 4, wherein: R is the 
radius of the Rowland circle (for simplicity, the radius R is taken to be 
unity); S is the position of the radiation source; O is the center of 
curvature of the planes of crystal lattice points; M is the midpoint of 
the crystal; P.sub.1 is an arbitrary point on the crystal surface; .theta. 
is the angle of incidence of the rays from S to M on the crystal planes 
(this is the Bragg angle, .theta..sub.B, for a symmetrical crystal); .phi. 
is the angle between the normal to the crystal planes and the ray from S 
to P.sub.1 ; .alpha. is an angle in the plane of the focal circle; and 
.delta. is an angle perpendicular to the focal circle. The planes of 
crystal lattice points are assumed to be curved as a true sphere of radius 
2R and the convex crystal surface is formed by rotating each point P' on 
the circumference of the Rowland circle individually about the point O 
through an angle .delta. perpendicular to the plane of the Rowland circle. 
When the point P.sub.1 coincides with M, cos .phi.=sin .theta.. We would 
like to find cos .phi. as a function of .theta., .alpha., and .delta.. The 
calculation proceeds as follows: 
In the triangle OSM, 
EQU &lt;OSM=.pi./2 and 
EQU &lt;SOM=.theta.; .thrfore.OS=cos .theta.. (1) 
In the triangle SQO, 
EQU &lt;SQO=.pi./2 by construction, 
EQU OQ=cos .theta. cos (.theta.-.alpha.) (2) 
EQU SQ=cos .theta. sin (.theta.-.alpha.). (3) 
In the triangle OMP', 
EQU &lt;OP'M=.pi./2; .thrfore.OP'=cos .alpha.. 
EQU OP.sub.1 =cos .alpha. by construction. (4) 
Using the law of cosines for the triangle OPQ, 
EQU (P.sub.1 Q).sup.2 =(OP.sub.1).sup.2 +(OQ).sup.2 -2(OP.sub.1)(OQ)cos 
.delta.(5) 
From equations 1, 2 and 4, this becomes 
EQU (P.sub.1 Q).sup.2 =cos.sup.2 .alpha.+cos.sup.2 .theta. cos.sup.2 
(.theta.-.alpha.) -2 cos .alpha. cos .theta. cos (.theta.-.alpha.)cos 
.delta.. (6) 
Using the Pythagorean theorem for triangle SPQ, 
EQU (SP.sub.1).sup.2 =(P.sub.1 Q).sup.2 +cos.sup.2 .theta. sin.sup.2 
(.theta.-.alpha.) and 
EQU (SP.sub.1).sup.2 =cos.sup.2 .alpha.+cos.sup.2 .theta.-2 cos .alpha. cos 
.theta. cos (.theta.-.alpha.)cos .delta.. (7) 
Using the law of cosines for triangle SP.sub.1 O, 
EQU cos.sup.2 .theta.=cos.sup.2 .alpha.+(SP.sub.1).sup.2 -2(SP.sub.1) cos 
.alpha. cos .phi.. (8) 
From Equations 7 and 8 we obtain: 
##EQU1## 
If .delta.=0, this reduces to the Johannson case and it is found that cos 
.phi.=sin .theta.. Now, let cos .delta.=(1-.delta..sup.2 /2). Recognizing 
that cos .phi.=sin (.theta.+.DELTA..theta.), Equation 9 becomes: 
##EQU2## 
Where 
EQU A=cos .alpha.-cos.sup.2 .theta. cos .alpha.- cos .theta. sin .theta. sin 
.alpha., 
EQU A.perspectiveto.1-.alpha..sup.2 /2-cos.sup.2 .theta. (1-.alpha..sup.2 /2) 
-.alpha. cos .theta. sin .theta.; 
and 
EQU B.sup.2 =cos.sup.2 .theta.-2 cos.sup.2 .alpha. cos.sup.2 .theta.-2 sin 
.alpha. cos .alpha. sin .theta. cos .theta., 
EQU B.sup.2 .perspectiveto.1-.alpha..sup.2 +cos.sup.2 
.theta.-2(1-.alpha..sup.2)cos.sup.2 .theta.-2 .alpha. (1-.alpha..sup.2 
/2)sin .theta. cos .theta.. 
Since .delta..sup.2 occurs in the second terms of the numerator and the 
denominator of Equation 10, we substitute for A and B, retaining only 
first order terms in the Taylor series expansion. We obtain: 
##EQU3## 
Interpretation of Results 
There are five factors that determine the effective area of a curved 
crystal used in a scanning monochromator. These factors are: (a) the 
accuracy of curvature of the crystal planes; (b) the degree to which the 
crystal surface conforms to the ideal surface; (c) the degree to which 
radiation penetrates below the crystal surface before being diffracted; 
(d) the width of the rocking curve for the crystal; and (f) the deviation 
(.DELTA..theta.) of the angle of incidence from .theta. at the surface for 
various regions of the surface. In this discussion we are concerned only 
with the latter two of these considerations. 
The foregoing derivation gives sin(.theta.+.DELTA..theta.). This may be 
related to the deviation from the Bragg angle .theta. as follows: 
EQU sin(.theta.+.DELTA..theta.)=sin .theta. cos .DELTA..theta.+cos .theta. sin 
.DELTA..theta.. 
For small .DELTA..theta., 
EQU .DELTA..theta.=(cos .theta.).sup.-1 [sin (.theta.+.DELTA..theta.)-sin 
.theta.]. (12) 
Using Equation 11, 
##EQU4## 
For optimum signal-to-background ratio, the maximum value of .DELTA..theta. 
should be less than or approximately equal to the width (w) of the rocking 
curve at half maximum. Typically, w has values ranging from 
3.multidot.10.sup.-5 to 5.multidot.10.sup.-4 radians, depending on the 
type of crystal and its surface treatment. For crystals that are 
plastically deformed, as is often the case when small focal circle radii 
are used, w can be as small as 5.multidot.10.sup.-4 radians if the 
starting material is a nearly perfect crystal and no polygonization occurs 
as a result of heat treatment. 
For monochromators used in X-ray spectrochemical analysis, another factor 
to be considered is the natural width of the characteristic X-ray lines. 
For K lines with photon energies (E) of 1-10 keV, a typical value for the 
full width at half maximum (.DELTA.E) is 3.multidot.10.sup.-4 .multidot.E. 
Using Bragg's law (n.lambda.=2d sin .theta.) and .lambda.=hc/E, it is 
found that 
##EQU5## 
Because the intensity as a function of spectrometer position is the 
convolution of the intensity distribution of the emission line and the 
spectrometer transmission function, it can be seen that a value of 
.DELTA..theta. of approximately 4.multidot.10.sup.-4 radians should be 
close to the value necessary to optimize the signal-to-background ratio. 
Comparison of Results with Cylindrical Johannson Geometry 
Plots of the locus of points on the surface of the crystal of the present 
invention for which .DELTA..theta. is a constant are shown in FIG. 5A for 
a crystal which is 2 cm wide, 4 cm long and has a focal circle 10 cm in 
radius. The area included within these curves is the area for which 
.DELTA..theta. is less than the given value. For comparison, the 
cylindrically curved Johannson case is shown in FIG. 5B. It can be seen 
that the area for a given .DELTA..theta. is greater for the doubly-curved 
crystal than for the cylindrically curved Johannson crystal. It can also 
be seen that the geometric efficiency of the doubly-curved crystal could 
be further improved by truncating the corners of the rectangular crystal. 
The ratio of the crystal area utilized in diffracting useful radiation in 
the geometry of the present invention to the area utilized in the 
cylindrically curved Johannson geometry is shown in FIG. 6 for 
.DELTA..theta.=4.multidot.10.sup.-4 radians. The greatest improvement is 
obtained at large Bragg angles. This is expected since exact 
point-to-point focusing is obtained for .theta.=90 degrees (a physically 
unrealizable case). 
The improvement in the performance of the disclosed doubly-curved crystal 
over the cylindrically curved Johannson crystal is even greater when the 
effects of penetration are taken into account. This is true because the 
planes of lattice points at the surface of the doubly-curved crystal come 
closer to satisfying the Bragg angle for points away from the center of 
the crystal than do the planes in the cylindrically curved Johannson case. 
Therefore, less penetration is required to reach a plane for which the 
Bragg condition is satisfied. For this reason, crystals having elements of 
high atomic number and large scattering factor may be preferred in the 
structure of the present invention over those having low atomic number. 
Such materials yield higher intensities of diffracted radiation, 
permitting even higher signal-to-background ratios. This is opposite to 
the consideration for monochromators using cylindrically curved Johannson 
crystals, in which the materials of low atomic number and low scattering 
factor are often required to achieve a suitable intensity. The 
contribution of diverging rays diffracted by planes below the surface is 
then quite important. 
One material of high atomic number and large scattering factor which can be 
used in the crystal structure of the present invention is crystalline 
germanium. It is advantageous over the more commonly used alkali halide 
materials for the reasons described above. Silicon has an adequate 
scattering factor for these purposes and can be used because it has 
similar properties to germanium with respect to plasiic deformation but is 
much less expensive. 
Toroidal Approximation to the Spherical Case 
The calculations above were made for the case in which the line OP' is 
rotated about point 0 through an angle .delta.. Alternatively, a truly 
toroidal crystal surface can be obtained by rotating the line O'P' about 
the axis 0'0 (FIG. 7) through a comparable angle .delta.'. In the latter 
case, the position of final point (P.sub.2) is not the same as the final 
position obtained above (P.sub.1). This can be seen in FIG. 8, which is a 
plan view of the Rowland circle of FIG. 7. In FIG. 8, the point P' becomes 
P.sub.1 in the optimum spherical case Case A , and the point P' becomes 
P.sub.2 in the more approximate toroidal case (Case B). As before, we take 
R, the radius of the Rowland circle, to be equal to 1. The rectilinear 
coordinates of P.sub.1 and P.sub.2 can then be expressed as follows: 
______________________________________ 
Coordinates of P.sub.1 
Coordinates of P.sub.2 
______________________________________ 
P.sub.1X = cos.sup.2 .alpha. cos .delta. 
P.sub.2X = cos.sup.2 .alpha. cos .delta.' 
P.sub.1Y = cos .alpha. sin .alpha. cos .delta. 
P.sub.2Y = cos .alpha. sin .alpha. 
P.sub.1Z = cos .alpha. sin .delta. 
P.sub.2Z = cos .alpha. cos .alpha. sin 
______________________________________ 
.delta.'. 
noting that: 
P.sub.2 is the same for both cases if sin .delta.=cos .alpha. sin .delta.'; 
P.sub.X does not change to terms of order .alpha..delta..sup.2 ; 
P.sub.Y is smaller for case A by terms of order .alpha..delta..sup.2, as 
can be seen by using a power series expansion; 
##EQU6## 
Thus, for the toroidal surface also, only terms of order 
.alpha..delta..sup.2 will be important in the expression: 
EQU sin .theta.'=sin .theta.[1-K(.alpha..delta..sup.2)]. 
This expression relates the angle (.theta.') at which radiation emanating 
from the source S impinges upon the crystal at the point P.sub.2 of the 
truly toroidal surface to the comparable angle (.theta..sub.B) at the 
center of the crystal. It is similar to an expression obtainable for the 
slightly different surface of the point P.sub.1, except that the value of 
the coefficient K is slightly lower in the truly toroidal case. 
Because it is simpler to manufacture a toroidal mold using conventional 
machine tools and because the results achieved with a toroidal surface are 
very close to those obtained by rotating points on the Rowland circle 
about a point, the truly toroidal surface geometry is preferred in some 
cases. 
Three methods of fabricating diffracting crystals for use in the practice 
of the present invention are outlined in FIG. 10. The first step, 
designated S100, is to obtain at least one crystal lamella 50 suitable for 
use in forming the diffracting crystal 18 of FIG. 3. The lamella is 
preferably cut from a single crystal of bulk material with a string or 
wire saw. The atomic planes of the bulk crystal are preferably flat and 
the lamella is cut to a radius equal to twice that of the Rowland circle 
for which it is intended. Alternatively, a flat crystal segment can be 
ground to the same radius, although this process is time consuming and 
wasteful of material. 
The configuration of the lamella 50 after cutting is shown in FIG. 11A. The 
dimensions of the lamella must be small enough that the desired toroidal 
curvature can be achieved without exceeding the fracture limit of the 
material. Thus, it is necessary in some cases to form the crystal 18 of a 
plurality of distinct lamellae combined together to form a structure 
having the geometry disclosed herein. This can be done in the manner 
disclosed in the above-referenced patent application, Ser. No. 549,366. 
However, the crystals of the present invention are curved to significantly 
greater radii than those of the referenced application, reducing the 
strain involved in forming them. This enables the crystal to be made from 
fewer lamellae than are required in the device of the prior application, 
and often a single crystal lamella. 
In an exemplary embodiment, the lamella 50 used to form the crystal is 
approximately 2 cm wide by 4 cm long, and is thin enough to be bent to the 
required radius. Thus, the maximum permissible thickness is lower for 
smaller radii so that the crystal can be more easily deformed. Taken 
generally, the maximum thickness is preferably (1.times.10.sup.-3)R, where 
R is the radius of the Rowland circle. When the Rowland circle is 10 
inches in radius, the thickness of the lamella may be as great as 10 mils. 
The next step, designated S102, is to polish and treat the surfaces of the 
lamella so that they are damage- and contamination-free. This is 
particularly important for the surface to be placed in tension when the 
lamella is deformed. Cracks and surface imperfections are eliminated by 
polishing, and the surfaces may be treated by any of a variety of 
conventional techniques to aid in deformation. Alkali-halide crystals can 
be treated by soaking in a solvent or suitable etchant to trigger the 
Joffe effect. The crystals are then bathed in a suitable drying liquid to 
remove the solvent or etchant and prevent its evaporation on the crystal 
surface. In the case of NaCl, suitable solvent and drying liquids are 
water and alcohol, respectively. During the entire process, it is 
desirable to avoid exposing the crystal to ozone. 
After the crystal surface has been treated and dried, the crystal is 
plastically deformed according to one of three processes, designated S104, 
S106 and S108, respectively, in FIG. 10. In step S104, illustrated 
schematically in FIG. 11B, the lamella 50 is pressed between upper and 
lower forming dies 52 and 54. This is preferably accomplished under a 
lubricant such as silicone oil. The lower die 54 has a convex upper 
surface (not shown) which conforms to the desired configuration of the 
crystal surface 22 (FIG. 30). Thus, the lower die has a radius of R in the 
plane of the eventual Rowland circle and a radius of substantially 2R in a 
plane which is perpendicular to and bisects the Rowland circle. The lower 
die may be made of silicone rubber to avoid damaging the crystal's 
surface. 
In the alternative step S106, illustrated schematically in FIG. 11C, the 
lamella 50 is bent by pressing it downwardly over a lower convex die 56 
using a forming roller 58. The roller has a rolling element 60 of suitable 
elastomeric material to yieldingly urge the fragile crystal lamella 
against the die. The process is preferably carried out at elevated 
temperatures, as with the other forming processes, to activate appropriate 
slip systems of the crystal. During the process, the roller 58 is passed 
back and forth over the crystal segment in a direction indicated at 62. A 
process of this type for fabricating cylindrically curved crystals is 
discussed in greater detail in Birks, X-Ray Spectrochemical Analysis, 
Appendix 2, pp. 127-131, Interscience Publishers, Inc., New York (1959), 
which is hereby incorporated by reference. 
A further alternative step, S108, is illustrated in FIG. 11D. At present, 
it is the preferred method of crystal fabrication. In step S108, the 
crystal lamella 50 is bent by vacuum-forming it over a convex lower die 64 
at elevated temperatures. The die 69 is positioned on a base plate 66 
which has a plurality of holes 68 for evacuation of air around the die. 
When a continuous sheet 70 of elastomeric materal is positioned above the 
lamella and the die, evacuation through the holes 68 causes the sheet to 
be forced downwardly by ambient air pressure against the lamella. The 
lamella is forced smoothly against the die until it conforms to the 
desired configuration. The sheet 70 can be made of any flexible material, 
such as buterate or silicone rubber, which is impermeable to air and 
possesses the tensile strength required to draw the crystal segment 
against the die. The attractiveness of this method lies primarily in the 
fact that the crystal is never contacted on its critical tensile side by a 
die or other rigid element applying a localized force. Rather, the force 
is distributed over the area of the crystal segment, minimizing the chance 
of breakage during the forming process. 
After the lamella is formed by one of the steps S104, S106 or S108, it is 
preferably mounted for use on a concave mounting die 72 to form the 
crystal arrangement 12 of FIG. 1. This process is illustrated in FIG. 11E, 
wherein a crystal segment is engaged at the concave surface thereof by a 
convex die 79 which is preferably made of an elastomeric material such as 
silicon rubber or coated with such a material to avoid scratching the 
crystal segment. The segment is retained in position against the mounting 
die by a suitable adhesive 76. 
FIG. 12 illustrates a specific apparatus 80 for carrying out the 
vacuum-forming process of step S108 in a high temperature environment. The 
apparatus 80 includes a vessel 82 with side walls 84 extending from a 
bottom 86 to an open upper end 88. The walls 84 define a chamber 90 which 
contains a perforated table 92. An upwardly directed convex mold 94 is 
disposed on the table 92 to support the crystal lamella 50, and the open 
end 88 of the vessel is closed by a continuous sheet 96 and a weight 98. 
The vessel 82 and the elements associated with it are made of materials 
able to withstand high temperatures, enabling them to be placed into an 
oven with the lamella 50 in place over the mold 94. The temperature of the 
oven is chosen so that the crystal lamella 50 and the continuous sheet 96 
become flexible but are not damaged. After heat treatment, the apparatus 
80 is removed from the oven and placed on a base 110 so that an opening 
112 in the bottom of the vessel communicates with an upwardly directed 
fluid conduit 114. The chamber 90 is then at least partially evacuated 
through the conduit 114 and the opening 112. Air within the chamber passes 
freely through perforations of the table 92 to the vacuum hose 112, 
drawing the sheet 96 downwardly so that the lamella is effectively 
"pulled" over the mold 94. During this process, the lamella and the sheet 
96 are maintained in a soft, flexible state by the heat of the vessel 82, 
the mold 94 and the weight 98. However, the mass of these elements should 
be kept as small as possible so that the lamella does not remain at 
elevated temperatures any longer than necessary after removal from the 
oven. 
The temperature to which the apparatus 80 is heated depends upon the 
material chosen for the crystal lamella 50. Preferred temperature ranges 
are 600-700 degrees Celsius for germanium (Ge), 800-900 degrees Celsius 
for silicon (Si) and 300-400 degrees Celsius for certain ionic crystal 
materials. Suitable ionic crystal materials include alkali halides such as 
lithium fluoride (LiF), as well as ADP, PET and EDDT. The optimum 
temperature for treatment of lamellae made of a specific material is the 
range within which the material becomes soft enough to be deformed 
plastically without significant damage to its diffracting characteristcs. 
Because the temperature of heat treatment varies widely depending upon the 
crystal material used, the same material cannot be used for the continuous 
sheet 96 in all cases. The continuous sheet 96 must withstand the 
temperatures of heat treatment and be pliable enough at those temperaure 
to be drawn downwardly against the lamella. "Soft" glass is suitable for 
the range encountered with germanium, and heat-resistant glass of the type 
sold under the name "Pyrex" is preferred for the range encountered with 
silicon. In each case, the sheets are rigid at room temperature but become 
flexible at the temperatures of heat treatment. Silicone rubber can be 
used in the range of 300-400 degrees Celsius, as encountered in processing 
ionic crystal materials. The continuous sheet 96 is preferably between a 
fraction of a millimeter and approximately 2 millimeters thick. 
When germanium and silicon are used as the diffracting crystal material, it 
may be necessary in some circumstances to make use of a suitable "parting 
medium" to prevent bonding of the sheet 96 to the lamella 50. Of course, 
the medium must be able to withstand the temperatures of heat treatment. 
The fabrication process described above is designed to produce the 
necessary double curvature of the crystal material without disrupting the 
ability of the material to diffract radiation. In this process, it is 
desirable to produce a network of edge dislocations of the crystal 
lattice, rather than deforming the lattice in a series of "slip bands". 
After fabrication, the surface and atomic planes of the plastically 
deformed crystals can be investigated by X-ray topography, using a micro 
focus X-ray source. 
The spatial resolution of a focused image obtained with the plastically 
deformed crystals of the invention depends upon the half-width of the 
rocking curve for the crystal and upon its primary and secondary 
extinction coefficients. Therefore, it may be desirable to resort to 
polygonization of the crystal to fine-tune its efficiency, either up or 
down. Polygonization, an increase in the size of mosaic blocks within a 
crystal lattice, is caused by final heat treatment. It may increase the 
primary extinction coefficient in one dimension by causing a coarsening of 
the structure near the surface. The effect is a trade-off, enhancing 
efficiency in some regions and reducing it in others. It is desirable to 
obtain an optimum size of mosaic blocks to maximize the integrated 
reflection coefficient and the primary extinction coefficient of the 
crystal, as described in R. W. James, "The Optical Principles of the 
Diffraction of X-Rays", The Crystalline State, Volume II, pp. 267-305, G. 
Bell & Sons, Ltd., 1958, which is hereby incorporated by reference. Heat 
treatment for polygonization would take the form of annealing at an 
elevated temperature. Execution of this step is well within the 
capabilities of one skilled in the art of crystal fabrication. 
The rocking curve "half-width", which is actually the full width at half 
maximum, is the angular range over which the intensity of a diffracted 
beam drops to one half its maximum value as the angle of incidence varies 
from the Bragg angle. It is increased by imperfections in the surface and 
atomic planes of a crystal. An increase in the rocking curve half-width 
can have a beneficial effect on the efficiency of the crystal but causes a 
poorer signal-to-background ratio and poorer focus. The rocking curve 
half-width can be increased by surface treatment, if desired. 
Specifically, such treatment can take the form of controlled grinding or 
abrading of the surface. 
In use, the crystal 18 is mounted to the backing member 72 (FIG. 11E) and 
installed as the crystal arrangement 12 in a scanning monochromator such 
as that shown in FIG. 1. Referring to FIG. 1 in detail, the spectrometer 
has a first arm 120 and a second arm 122 mounted for rotation about an 
axis 124 as the spectrometer operates. The crystal arrangement 12 is 
carried near an outer end of the arm 120 and the detector 14 is carried at 
an outer end of the arm 122, both for movement with the arms relative to 
the stationary source location 16. Movement of the arms relative to each 
other and to the source location is controlled by sector gears 126 and 128 
carried by the respective arms. The sector gears are interconnected by a 
pair of spur gears 130 which are mounted for common rotation about a 
second axis 132. The diameters of the sector gears 126 and 128 and the 
spur gears 130 are selected so that the detector 14 moves in the same 
direction as the crystal arrangement 12 and through twice the angle that 
the crystal arrangement moves. When the distance between the diffracting 
crystal surface and the axis 124 is equal to the radius of the Rowland 
circle of the crystal, the mechanism maintains the source location 16, the 
crystal arrangement 12 and the detector 14 at the surface of the Rowland 
circle in the classic diffraction geometry. 
The arms 120 and 122 are biased by a spring 134 acting through a cable 136 
to eliminate backlash of the gear train composed of the sector gears 126 
and 128 and the spur gears 130. The cable 136 passes around a pair of 
pulleys 138 to engage arcuate guide members 140 associated with each of 
the arms. 
Adjustment of the crystal arrangement 12 in a radial direction is permitted 
by an adjusting mechanism 142 on the arm 120. Similarly, alignment of the 
detector 14 relative to the crystal arrangement 12 is maintained by a 
guide rod 144 which is pivotally mounted to the arm 120 at a location near 
the crystal arrangement. The rod passes through a sleeve (not shown) 
carried by the detector 14 to keep the detector pointing toward the 
crystal. 
The spectrometer 10 is operable to maintain the desired relationship 
between the source location 16, the crystal arrangement 12 and the 
detector 14 as relative movement occurs, varying the angle at which 
radiation emanating from the source location impinges upon the surface of 
the crystal 18. The detector 14 may be any form of proportional counter 
capable of measuring the intensity of incoming radiation at each angular 
orientation. The output of the counter thus provides information as to the 
spectral content of source radiation. 
From the foregoing it can be seen that the unique geometry of the crystal 
arrangement 12, in the context of a scanning monochromator, provides a 
signal of higher intensity and better resolution over a broad spectrum of 
wavelengths than is obtainable using flat or singly-curved crystals. It 
thereby permits in-depth analysis of radiation sources which have 
heretofore been too weak for accurate measurement. 
While certain specific embodiments of the present invention have been 
disclosed as typical, the invention is, of course, not limited to these 
particular forms, but rather is applicable broadly to all such variations 
as fall within the scope of the appended claims. For example, dimensional 
and geometric relationships described herein can be varied to some extent 
without interfering with the operation of the spectrometer. The bounds of 
such variation are set out generally herein. Features such as the number 
of crystal lamellae used to form the crystal structure can also be varied 
as needed, to produce the required geometry from a particular material. 
The number of lamellae depends on the charcteristics of the material, the 
size of the crystal and the radius of the Rowland circle associated with 
the crystal. In addition, the scanning monochromator 10 may be of any 
suitable design and need not move the crystal in a circular manner. For 
instance, the crystal might be moved along a linear path to scan a range 
of wavelengths as encountered in commercially available electron probe 
microanalyzers.