X-Ray Data Booklet

Section 4.1 MULTILAYERS AND CRYSTALS

James H. Underwood

(This section is also available as a PDF file.)

A. MULTILAYERS

By means of modern vacuum deposition technology, structures consisting of alternating layers of high- and low-Z materials, with individual layers having thicknesses of the order of nanometers, can be fabricated on suitable substrates. These structures act as multilayer interference reflectors for x-rays, soft x-rays, and extreme ultraviolet (EUV) light. Their high reflectivity and moderate energy bandwidth (10 < E/DE < 100) make them a valuable addition to the range of optical compo­nents useful in instrumentation for EUV radiation and x-rays with photon energies from a few hundred eV to tens of keV. These multilayers are particularly useful as mirrors and disper­sive elements on synchrotron radiation beamlines. They may be used to produce focal spots with micrometer-scale sizes and for applications such as fluorescent microprobing, microdif­fraction, and microcrystallography. Multilayer reflectors also have a wide range of applications in the EUV region, where normal-incidence multilayer reflectors allow the construction of space telescopes and of optics for EUV lithography.

Ordinary mirrors, operating at normal or near-normal angles of incidence, do not work throughout the EUV and x-ray regions. The reason is the value of the complex refractive index, n = 1 – dib, which can be close to unity for all materi­als in this region; d may vary between 10–3 in the EUV region to 10–6 in the x-ray region. The Fresnel equations at normal incidence show that the reflectivity R2 = |(n– 1)/(n + 1)|2 is very small. However, the more general Fresnel equations show that x-rays and EUV radiation can be reflected by mirrors at large angles of incidence. Glancing (or grazing) incidence is the term used when the rays make a small angle (a few degrees or less) with the mirror surface; the complement (90° – i) of the optical angle of incidence i is called the glancing (or grazing) angle of incidence. Glancing-incidence mirrors are discussed in Section 4.2.

Although the normal-incidence intensity reflectivity R2 of a surface might be 10–3 or 10–4, the corresponding amplitude reflectivity R = (n – 1)/(n + 1), the square root of the intensity reflectivity, will be 1/30 to 1/100. This implies that, if the re­flections from 30–100 surfaces could be made to add in phase, a total reflectivity approaching unity could be obtained. This is the multilayer principle. As shown in Fig. 4-1, a multilayer reflector comprises a stack of materials having alternately high and low refractive indices. The thicknesses are adjusted so that the path length difference between reflections from successive layer pairs is equal to one wavelength. Hence, x-ray/EUV mul­tilayers are approximately equivalent to the familiar “quarter-wave stacks” of visible-light coating technology. The equiva­lence is not exact, however, because of the absorption term b, which is usually negligible for visible-light multilayers. This absorption reduces the multilayer reflectivity below unity and requires the design to be optimized for highest reflectivity. Particularly critical is the value of G, which is the ratio of the thickness of the high-Z (high electron density, high |n|) layer to the total thickness of each layer pair. This optimi­zation nor­mally requires modeling or simulation of the multi­layer. Most thin-film calculation programs, even if designed for the visible region, will perform these calculations if given the right com­plex values of the refractive index. Alternatively, there are a number of on-line calculation programs available; links can be found at http://www-cxro.lbl.gov/.

Either elemental materials or compounds can be used to fabricate multilayer reflectors. The performance obtained from a multilayer depends largely on whether there exists a fortuitu­ous combination of materials having the right refractive indices (good contrast, low absorption), whether these materials can be deposited in smooth thin layers, and whether they remain stable (low reactivity, low diffusion) in the deposited state. The roughness of the underlying substrate is also of prime impor­tance; an rms roughness of the same order of magnitude as the layer

 

 

Fig. 4-1.      Schematic of a multilayer reflector of n bilayer pairs. The parameters l, q, and d are chosen to satisfy the familiar Bragg equation, but the rela­tive thicknesses of the high- and low-Z materials are also critical in optimizing reflectivity.  The total reflectivity is the vector sum of the complex reflec­tion coefficients at each interface, with the different path lengths taken into account.

 

thicknesses will spoil the performance of most coatings. “Superpolished” substrates, with roughness s ≈ 0.1 nm are preferred. On such substrates, the peak reflectivity of the coat­ings can approach 80% to 90% of theoretical predictions.

A large variety of multilayers have been made and their re­flectivities measured over the years. A database of reported re­flectivities has been assembled from surveys taken at the bien­nial Physics of X-ray Multilayer Structures conferences and can be found at the website listed above. For the EUV region around 100 eV, two remarkably successful combinations are molybdenum-silicon and molybdenum-beryllium. With Mo-Si, a normal-incidence reflectivity of 68% has been achieved at a wavelength of 13.4 nm; Mo-Be multilayers have achieved a reflectivity close to 70% at 11.4 nm (see Fig. 4-2). These rela­tively high reflectivities are the basis for current efforts in the field of EUV lithography.

 

 

 

Fig. 4-2.     The reflectivity of two multilayer reflectors at extreme ultraviolet wavelengths.

 


Fig. 4-3.     The reflectivity of a tungsten–boron carbide multilayer at 8048 eV. The parameters d and G are discussed in the text.

 

At hard x-ray wavelengths near 10 keV, a commonly used multilayer is made with tungsten as the high-Z material and boron carbide (B4C) as the low-Z material. A reflectivity of 84% has been achieved with this combination at 8048 eV, the energy of the Cu Ka line (see Fig. 4-3).

B. CRYSTALS

Multilayers are examples of a periodic structure that can be used to analyze short-wavelength electromagnetic radiation. Such structures split the incident beam into a large number N of separate beams; between any beam i and the beam + 1, the optical path difference is constant. After leaving the periodic structure, the beams are recombined and caused to interfere, whereupon the spectrum of the incident radiation is produced.

Dispersion of radiation by a periodic structure is thus formally equivalent to multiple-beam interferometry. Struc­tures that are periodic across their surface and that produce the N interfering beams by division of the incident wave front are called gratings and are treated in Section 4.3. Crystals and multilayer struc­tures produce the N interfering beams by divi­sion of the inci­dent amplitude. The spectrum of the incident radiation is dis­persed in angle according to the Bragg equation nl  = 2d sin q, where n is an integer representing the order of the reflec­tion,l  is the wavelength of the incident radiation, d is the period of the multilayer or crystal structure, and q is the angle of glancing incidence. For a crystal, d is the lattice spac­ing, the perpendicular distance between the successive planes of atoms contributing to the reflection. These planes are desig­nated by their Miller indices [(hkl) or, in the case of crystals belonging to the hexagonal group, (hkil)]. (The value of 2d also represents the longest wavelength that the structure can diffract.)

For 2d values greater than about 25 Å, the choice of natural crystals is very limited, and those available (such as pro­chlorite) are likely to be small and of poor quality. Sputtered or evaporated multilayers can be used as dispersing elements at longer wavelengths. (Langmuir-Blodgett films have fallen into disfavor since the development of vacuum-deposited multilay­ers.)

Table 4-1 is a revision of one compiled by E. P. Bertin [1]. The crystals are arranged in order of increasing 2d spacing.

REFERENCE

1.            E. P. Bertin, “Crystals and Multilayer Langmuir-Blodgett Films Used as Analyzers in Wavelength-Dispersive X-Ray Spectrometers,” in J. W. Robinson, Ed., Handbook of Spectroscopy (CRC Press, Cleveland, 1974), vol. 1, p. 238.

 


Table 4-1.    Data for selected crystals used as dispersive elements in x-ray spectrometers and monochromators. The Miller indices [(hkl), or (hkil) for hexagonal crystals] are given for the diffracting planes parallel to the surface of the dispersive element. A question mark (?) indicates that the crystal is developmental and that the indices have not been ascertained. An asterisk following the indices indicates that literature references to this crystal without specification of (hkl) or 2d are likely to be references to this “cut.” The indicated useful wavelength region lies in the 2q interval between 10° and 140°. The analyzer should be used outside this region in special cases only.



No.



Crystal


Miller indices



2d (Å)



Chemical formula

Useful wavelength region (Å)



Applications, remarks

1

a-Quartz, silicon dioxide

1.624

SiO2

0.142–1.55

Shortest 2d of any practical crystal. Good for high-Z K-lines excited by 100-kV generators.

2

Lithium fluoride

(422)

1.652

LiF

0.144–1.58

Better than quartz  for the same applications.

3

Corundum, aluminum oxide

(146)

1.660

Al2O3

0.145–1.58

Same applications as quartz

4

Lithium fluoride

(420)

1.801

LiF

0.157–1.72

Similar to LiF (422).

5

Calcite, calcium carbonate

(633)

2.02

CaCO3

0.176–1.95

 

6

a-Quartz, silicon dioxide

2.024

SiO2

0.177–1.96

 

7

a-Quartz, silicon dixoide

2.3604

SiO2

0.205–2.25

Transmission-crystal optics.

8

a-Quartz, silicon dioxide

2.451

SiO2

0.213–2.37

 

9

Topaz, hydrated aluminum fluorosilicate

(303)*

2.712

Al2(F,OH)2SiO4

0.236–2.59

Improves dispersion for V–Ni K-lines and rare earth L-lines.

10

Corundum, aluminum oxide, sapphire, alumina

(030)

2.748

Al2O3

0.240–2.62

Diffracted intensity ~2–4X topaz (303) and quartz (203) with the same or better resolution.

11

a-Quartz, silicon dioxide

2.749

SiO2

0.240–2.62

Same applications as topaz (303) and LiF (220).


 



No.



Crystal


Miller indices



2d (Å)



Chemical formula

Useful wavelength region (Å)



Applications, remarks

12

Topaz

(006)

2.795

Al2(F,OH)2SiO4

0.244–2.67

 

13

Lithium fluoride

(220)

2.848

LiF

0.248–2.72

Same applications as topaz (303) and quartz , with 2–4X their diffracted intensity. Diffracted intensity
~0.4–0.8X LiF (200).

14

Mica, muscovite

(331)

3.00

K2O·3Al2O3·6SiO2·2H2O

0.262–2.86

Transmission-crystal optics (Cauchois, DuMond types).

15

Calcite, calcium carbonate

(422)

3.034

CaCO3

0.264–2.93

 

16

a-Quartz, silicon dioxide

3.082

SiO2

0.269–2.94

 

17

a-Quartz, silicon dioxide

3.636

SiO2

0.317–3.47

 

18

Silicon

(220)

3.8403117

Si

0.335–3.66

Lattice period known to high accuracy.

19

Fluorite, calcium fluoride

(220)

3.862

CaF2

0.337–3.68

 

20

Germanium

(220)

4.00

Ge

0.349–3.82

 


21

Lithium fluoride

(200)*

4.027

LiF

0.351–384

Best general crystal for K K- to Lr L-lines. Highest intensity for largest number of elements of any crystal. Combines high intensity and high dispersion.

22

Aluminum

(200)

4.048

Al

0.353–3.86

Curved, especially doubly curved, optics.

23

a-Quartz, silicon dioxide

4.246

SiO2

0.370–4.11

“Prism” cut.

24

a-Quartz, silicon dioxide

4.564

SiO2

0.398–4.35

Used in prototype Laue multichannel spectrometer.

25

Topaz

(200)

4.638

Al2(F,OH)2SiO4

0.405–4.43

 


 



No.



Crystal


Miller indices



2d (Å)



Chemical formula

Useful wavelength region (Å)



Applications, remarks

26

Aluminum

(111)

4.676

Al

0.408–4.46

Curved, especially doubly curved, optics.

27

a-Quartz, silicon dioxide

4.912

SiO2

0.428–4.75

 

28

Gypsum, calcium sulfate dihydrate

(002)

4.990

CaSO4·2H2O

0.435–4.76

Efflorescent: loses water in vacuum to become Plaster of Paris.

29

Rock salt, sodium chloride

(200)

5.641

NaCl

0.492–5.38

S Ka and Cl Ka in light matrixes. Like LiF (200), good general crystal for S K to Lr L.

30

Calcite, calcium carbonate

(200)

6.071

CaCO3

0.529–5.79

Very precise wavelength measurements. Extremely high degree of crystal perfection with resultant sharp lines.

31

Ammonium dihydrogen phosphate (ADP)

(112)

6.14

NH4H2PO4

0.535–5.86

 

32

Silicon

(111)*

6.2712

Si

0.547–5.98

Very rugged and stable general-purpose crystal. High degree of perfection obtainable.

33

Sylvite, potassium chloride

(200)

6.292

KCl

0.549–6.00

 

34

Fluorite, calcium fluoride

(111)

6.306

CaF2

0.550-6.02

Very weak second order, strong third order.

35

Germanium

(111)*

6.532

Ge

0.570–6.23

Eliminates second order. Useful for intermediate- and low-Z elements where Ge Kaemission is eliminated by pulse-height selection.

36

Potassium bromide

(200)

6.584

KBr

0.574–6.28

 

37

a-Quartz, silicon dioxide

6.687

SiO2

0.583–6.38

P Ka in low-Z matrixes, especially in calcium. Intensity for P–K K-lines greater than EDDT, but less than PET.


 



No.



Crystal


Miller indices



2d (Å)



Chemical formula

Useful wavelength region (Å)



Applications, remarks

38

Graphite

(002)

6.708

C

0.585–6.40

P, S, Cl K-lines, P Kaintensity > 5X EDDT. Relatively poor resolution but high integrated reflectivity.

39

Indium antimonide

(111)

7.4806

InSb

0.652–7.23

Important for K-edge of Si.


40

Ammonium dihydrogen phosphate (ADP)

(200)

7.5

NH4H2PO4

0.654–7.16

Higher intensity than EDDT.

41

Topaz

(002)

8.374

Al2(F,OH)2SiO4

0.730–7.99

 

42

a-Quartz, silicon dioxide

*

8.512

SiO2

0.742–8.12

Same applications as EDDT and PET; higher resolution, but lower intensity.

43

Pentaerythritol (PET)

(002)

8.742

C(CH2OH)4

0.762–8.34

Al, Si, P. S, Cl Ka. Intensities ~1.5-2X EDDT, ~2.5X KHP. Good general crystal for Al–Sc Ka. Soft; deteriorates with age and exposure to x-rays.

44

Ammonium tartrate

(?)

8.80

(CHOH)2(COONH4)2

0.767–8.4

 

45

Ethylenediamine-d-tartrate (EDDT, EDdT, EDT)

(020)

8.808

 

0.768–8.40

Same applications as PET, but lower intensity, substantially lower thermal expansion coefficient. Rugged and stable.

46

Ammonium dihydrogen phosphate (ADP)

(101)*

10.640

NH4H2PO4

0.928–10.15

Mg Ka. Same applications as PET, EDDT, but lower intensity.

47

Na b-alumina

(0004)

11.24

NaAl11O17

0.980–10.87

 

48

Oxalic acid dihydrate

(001)

11.92

(COOH)2·2H2O

1.04–11.37

 

49

Sorbitol hexaacetate (SHA)

(110)

13.98

1.22–13.34

Applications similar to ADP (101) and gypsum (020). High resolution; stable in vacuum. Available in small pieces only.


 



No.



Crystal


Miller indices



2d (Å)



Chemical formula

Useful wavelength region (Å)



Applications, remarks

50

Rock sugar, sucrose

(001)

15.12

C12H22O11

1.32–14.42

 

51

Gypsum, calcium sulfate dihydrate

(020)*

15.185

CaSO4·2H2O

1.32–14.49

Na Ka. Inferior to KHP, RHP, and beryl. Poor in vacuum (efflorescent).

52

Beryl

15.954

3BeO·Al2O3·6SiO2

1.39–15.22

Difficult to obtain. Good specimens have l/dl ~ 2500–3000 at 12 Å. 2d may vary among specimens.

53

Bismuth titanate

(040)

16.40

Bi2(TiO3)3

1.43–15.65

 

54

Mica, muscovite

(002)*

19.84

K2O·3Al2O3·6SiO2·2H2O

1.73–18.93

Easy to obtain. Easily bent: good for curved-crystal spectrometers, spectrographs.

55

Silver acetate

(001)

20.0

CH3COOAg

1.74–19.08

 

56

Rock sugar, sucrose

(100)

20.12

C11H22O11

1.75–19.19

 

57

Na b-alumina

(0002)

22.49

NaAl11O17

1.96–21.74

 

58

Thallium hydrogen phthalate (THP, TlHP, TAP, TIAP)

(100)

25.9

TlHC8H4O4

2.26–24.7

Same applications as KHP, RHP.

59

Rubidium hydrogen phthalate (RHP, RbHP, RAP, RbAP)

(100)

26.121

RbHC8H4O4

2.28–24.92

Diffracted intensity ~3X KHP for Na, Mg, Al Ka and Cu La; ~4X KHP for F Ka; ~8X KHP for O Ka


60

Potassium hydrogen phthalate (KHP, KAP)

(100)

26.632

KHC8H4O4

2.32–25.41

Good general crystal for all low-Z elements down to O.

61

Octadecyl hydrogen maleate (OHM)

(?)

63.5

CH3(CH2)17OOC(CH)2COOH

5.54–60.6

Ultralong-wavelength region down to C Ka