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Spectroscopic Techniques: Ultraviolet 44.4 Detectors 649

changes in photographic emulsions. The details of the

design of VUV detectors, and the constraints on their

use, are most often determined by the low levels of

transmission of VUV light through suitable window and

semiconducting materials. Nevertheless, VUV detectors

with single photon counting sensitivities and/or imag-

ing capabilities are widely available. Reviews of VUV

detectors can be found in [44.1, 2, 50, 160–163]. Refer-

ences [44.164–166] review the operating characteristics

and calibration of VUV detector standards. A more gen-

eral summary of photon detector technology is presented

in [44.167, 168]. Below we summarize and compare the

most common VUV detection schemes, including the

use of photographic plates, photomultiplier tubes and

vacuum photodiodes, multichannel plate detectors, sili-

con photodiodes, charge-coupled devices, and ionization

chambers.

Photographic Plates

Photographic plates have long been used at VUV

wavelengths for spectroscopic measurements, and they

are still sometimes the detector of choice for both

spectroscopic surveys and high resolution wavelength

measurements. They have the advantages of being

imaging detectors with very high spatial resolution;

spectral line positions can be determined with an

uncertainty of about 1 µm. Plates also have mul-

tiplexing and very ﬂexible integrating capabilities,

being insensitive to ﬂuctuations in signal intensities

during the period of exposure. However, when com-

pared with photoelectric detectors, photographic plates

have the disadvantages of limited dynamic range and

a nonlinear response, which makes radiometric mea-

surements very problematic when plates are used.

The gelatin base used in standard photographic emul-

sions strongly absorbs VUV radiation, so special

plates with no gelatin base are required for VUV

work [44.2].

Photomultiplier Tubes

Photomultiplier tubes (PMTs) [44.169] are used through

the VUV for single-photon-counting, nonimaging, ap-

plications. Dark count rates are low



about 1 s

−1



for solar-blind tubes with 1 cm

photocathodes; pulse

rise times are about 1 to 10 ns ([44.170]foragen-

eral review of amplifying detectors in the VUV). In

the 105 to 200 nm region, useful PMT window ma-

terials are fused silica, MgF

, and LiF; sapphire is

also used in environments where ionizing radiation,

which causes sapphire to ﬂuoresce, is not present.

In the windowless region of the VUV wavelength

range (2 nm ≤ λ ≤ 105 nm), two options are available:

the PMT may be operated bare, i. e., without a win-

dow, or a ﬂuorescent coating can be deposited on the

window to down-convert the VUV light to longer wave-

lengths.

Peak quantum efﬁciencies for VUV photomultiplier

tubes are about 15 to 20%. Some of the most useful

coatings for photocathodes in the VUV are CsI, KBr,

and other alkali halides. These materials combine high

VUV quantum efﬁciencies with a solar blind response –

their relatively high photoelectric work functions result

in long wavelength cutoffs between 150 and 300 nm.

Solar blind PMTs have low dark count rates and have

minimal response to stray ambient light.

Bare PMTs, although used in some applications,

are constrained by the degradation of many photo-

cathode and dynode surfaces upon exposure to air

and humidity. Metal surfaces such as tungsten and

aluminum/Al

[44.162] must be used rather than

the higher efﬁciency alkali halide coated surfaces.

Sodium salicylate is the most commonly used VUV-

visible conversion phosphor; others include liumogen,

terphenyl, and coronene [44.2, 171, 172]. The con ver-

sion efﬁciency of sodium salicylate (peak ﬂuorescence

about 430 nm) is relatively constant for VUV light

from 30 to 200 nm. It displays some aging effects,

which may be associated with contaminants such as

oil vapor in the associated vacuum system [44.2].

The ﬂuorescent decay time of sodium salicylate

is ≈ 10 ns [44.2].

Vacuum Photodiodes

Vacuum photodiodes consist of a photocathode and

an anode, with no amplifying dynode chain. The cur-

rent leaving the photocathode is measured. (A review

of vacuum photodiode designs and performance can

be found in [44.173].) Because of their stability, spa-

tial uniformity, and portability, vacuum photodiodes are

used as radiometric transfer-standard detectors through-

out much of the VUV [44.162, 174, 175]. There is no

gain in vacuum photodiodes, so minimum signal lev-

els that can be accurately measured are on the order

of 10

photons/s. For fast timing applications, vacuum

photodiodes have been designed with risetimes as short

as about 60 ps [44.160].

Microchannel Plates

The microchannel plate (MCP) detector is a photo-

emissive array detector that combines the single-photon

counting sensitivity of a PMT with high resolution imag-

ing capability [44.161–163, 170, 176, 177]. An MCP

Part C 44.4

650 Part C Molecules

consists of an array of semiconducting glass channels

with diameters of about 10 to 25 µm and length-to-

diameter ratios of about 50 : 1. Electrons ejected from

the front surface of an MCP via the photoelectric effect

are accelerated through the channels; repeated colli-

sions with the channel walls result in an ampliﬁcation

of the charge by about 10

. The exiting charge clouds

are detected by a variety of position-sensing anode

structures.

Bare MCPs have quantum efﬁciencies of about

10% for λ ≤ 100 nm and a long wavelength cut-

off about 120 nm [44.178]. Alkali halide coatings

increase the quantum efﬁciency to about 20%

and extend MCP sensitivities toward longer wave-

lengths [44.178–180]. Feedback instabilities pro-

duced by positive ions created in the channels

during the electron cloud ampliﬁcation are mini-

mized through two common channel geometries; the

chevron conﬁguration, where two or more straight-

channel MCPs oriented at different angles are cas-

caded [44.176], or a conﬁguration with one set of curved

channels [44.181,182].

Readout schemes for determining the position of

individual detected photons rely on either direct de-

tection of the resulting electron cloud, or conversion

of the electron cloud into visible photons via a phos-

phor to produce an optical image [44.170, 183]. Direct

detection schemes include centroid-detecting anodes

such as the wedge and strip design [44.178, 184],

cross strip anodes [44.185]. Other designs are described

in [44.186, 187]. Discrete anode arrays that digitally lo-

cate event positions include the MAMA (multianode

microchannel array) [44.188, 189] and the CODACON

systems [44.190]. Spatial resolutions of 15 µmare

possible.

Silicon Photodiodes

Broad use of silicon photodiodes at VUV wave-

lengths has traditionally been limited by the strong

absorption of VUV photons in the outer SiO

pas-

sivation layer that covers the p–n junction of these

devices. Standard silicon detectors are sensitive through-

out the infrared and visible regions and also in the

soft X-ray (λ<2 nm) and X-ray regions. Signiﬁcant

improvements in silicon photodiode sensitivities in

the VUV are realized by thinning the SiO

passi-

vation region to thicknesses of about 5 to 10 nm.

References [44.173, 191]reviewVUV semiconductor

photodiode designs; references [44.192, 193] describe

the development of non-silicon-based photodiodes, e.g.,

wide bandgap materials such as GaN, for this spec-

tral range. References [44.194–197] report devices with

a quantum efﬁciency of 120% (electron–hole pairs per

incident photon) at 100 nm. The development of such

photodiodes with appreciable VUV sensitivity and good

temporal stability has led to their use as radiometric

transfer standards [44.198, 199]. Because silicon pho-

todiodes respond strongly to radiation throughout the

visible, ir, and X-ray regions, a method for rejection

of stray light is essential for their effective use in the

VUV. Reference [44.200] reports on the use of thin-ﬁlm

ﬁlters, deposited on the photodiode surface, to restrict

the bandpass of the radiation impinging on the diode

to selected VUV wavelengths. Wide bandgap semicon-

ductors reject visible light with their natural solar-blind

response [44.199].

Charge-Coupled Devices

CCDs are widely used for low light level imaging ap-

plications throughout the visible and near-ir regions. In

the VUV, charge-coupled devices suffer from strong

absorption both by surface gate structures and by the in-

active passivation layer [44.170]. Two approaches are

employed to overcome these limitations: the CCD front

surface is overcoated with a photon down-converting

phosphor [44.201,202], or a thinned and surface-treated

CCD is back-illuminated with VUV light [44.203, 204].

Such techniques have resulted in CCD VUV quantum

efﬁciencies that rival those of photoemissive devices

such as microchannel plates. A comparison of CCD and

MCP performance in the VUV is given by [44.178];

they conclude that MCPs are most appropriate for low

light-level imaging where photon counting is applicable,

while the larger dynamic range and readout capabili-

ties of CCDs make them suitable for higher light level

applications.

Ionization Chambers

In gas ionization chambers, the impact of VUV pho-

tons with gas atoms or molecules produces electron–ion

pairs which are then collected by the application of an

appropriate voltage [44.205]. The photoionization ef-

ﬁciency and the detection efﬁciency approach 100%

for photon energies above the ionization threshold

of the gas [44.2, 162]. With appropriate gain, ion-

ization chambers can detect individual photons (e.g.,

the Geiger counter); these chambers require relatively

high gas pressures (100 to 2000 Torr) and there-

fore cannot be used in the windowless region of

the VUV. The double ionization chamber described

by [44.206] serves as a primary standard detector from 5

to 100 nm [44.162].

Part C 44.4

Spectroscopic Techniques: Ultraviolet 44.5 Optical Materials 651

44.5 Optical Materials

The design of VUV instrumentation is dictated in large

part by the constraints of VUV optical materials. Trans-

mission through bulk materials is limited to λ  105 nm,

the short wavelength transmission limit of LiF. Normal

incidence reﬂectance from metal surfaces and coatings

decreases dramatically at short wavelengths; polarizers

and narrow-band interference ﬁlters are relatively difﬁ-

cult to produce because of the lack of materials with

suitable optical constants. Below we brieﬂy summa-

rize the properties and uses of VUV optical materials.

Comprehensive discussions can be found in [44.1,2,50].

Windows

Few bulk materials transmit light below 200 nm; none

transmit light from about 2 nm to 105 nm. In the 105

to 200 nm wavelength region, the most common win-

dow materials are synthetic fused quartz or suprasil

(with a short λ transmission cutoff about 160 nm), sap-

phire (145 nm), CaF

(125 nm) MgF

(112 nm), and

LiF (105 nm). Compilations of the transmission char-

acteristics of these and other materials are presented

in [44.2, 50, 131, 207,208]. Single crystal windows with

dimensions up to 10 cm can be obtained for most of

these materials. Sapphire ﬂuoresces when struck by ion-

izing radiation and is therefore not suitable for certain

environments.

Capillaries

In the windowless region, 2 to 105 nm, it is standard

practice to use differentially pumped slits or circular

apertures to isolate different pressure regions, e.g., to iso-

late an absorption cell from a vacuum spectrometer. This

technique has the drawback of requiring large pumping

capacities and limiting optical throughputs. Thin ﬁlms

(see below) can be used below 105 nm as windows, but

they provide limited transmission in narrow bandpasses

and cannot support large pressure differentials. Refer-

ence [44.209] describes a differentially pumped MCP

capillary array as an alternative window in the VUV.

The capillaries are typically ≈ 3 mm in length and have

diameters of 10 to 100 µm. Measured optical transmis-

sions are in the 20 to 50% range throughout the VUV;

tradeoffs must be considered between optical through-

put and gas conductance. The main limitation to an MCP

window is its small angular aperture.

Thin Films

Below 105 nm, thin (about 10 to 200 nm) metallic ﬁlms

are used as transmission ﬁlters with bandpasses of ap-

proximately 10 to 50 nm. Details of ﬁlter properties

are given in [44.208, 210]. A review of thin ﬁlm ﬁlter

preparation techniques is presented in [44.211]. Refer-

ence [44.212] describes composite metallic ﬁlters (e.g.,

Al/Ti/C, Ti/Sb/Al), developed for the Extreme Ultra-

violet Explorer satellite, with transmission bandpasses

of about 10 to 20 nm and high rejection at wavelengths

of strong VUV geocoronal lines.

Coatings

Above 120 nm, the principal broadband reﬂector for

VUV wavelengths is Al with a thin protective over-

coat of MgF

, having a normal incidence reﬂectance of

≈ 80 to 85% [44.2, 213]. The normal incidence reﬂec-

tivities of all materials drop dramatically below about

100 nm. A compilation of coating reﬂectivities at nor-

mal and grazing incidence is presented in [44.214].

Materials with the highest reﬂectivities include Os,

Pt, Au, and Ir, with reﬂectivities of 15 to 30% from

30 to 110 nm [44.2, 215]. [44.216] reviews the prepa-

ration of VUV reﬂectance coatings for diffraction

gratings. Reference [44.217] reports grazing incidence

reﬂection coefﬁcients for Rh, Os, Pt, and Au from

5 to 30 nm; Rh has the highest reﬂectivity in this

region.

Interference Filters and Multilayer Coatings

The development of interference ﬁlters for VUV wave-

lengths has been limited by the availability of coating

materials with both high transmission and apprecia-

ble range of refractive differentials [44.218]. Multilayer

dielectric reﬂectors thus require many layers to compen-

sate for the small reﬂectivities at the material interfaces.

The theory of multilayer reﬂecting optics and optic

designs are reviewed in [44.219]. Reference [44.215]

evaluates the commercially available VUV reﬂectance

ﬁlters, antireﬂection coatings, and neutral density ﬁlters

as of 1983. Developments as of the earlys 1990s are

described in [44.220–223]. More recent advances are

presented in [44.224–227].

Normal incidence optics with multilayer reﬂec-

tion coatings can be used in the nominally grazing

incidence spectral region below 30 nm [44.228–

231]. Reference [44.232] summarizes synthesis pro-

cedures and models of multilayer structures. Ref-

erences [44.233–238] describe normal incidence

gratings coated with Mo/Si multilayers. The perfor-

mances of other multilayer coatings are described

in [44.239–242].

Part C 44.5

652 Part C Molecules

Polarizers

The production and detection of linearly polarized light

in the VUV is discussed in [44.2, 131, 243–247]. Above

about 112 nm, transmission polarizers, based on the

birefringence of MgF

, are employed [44.248, 249].

Reﬂection polarizers must be used below 112 nm. An

analysis of double-reﬂection circular polarizers is given

in [44.250].

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Part C 44

657

Scattering T

Part D

Part D Scattering Theory

45 Elastic Scattering: Classical, Quantal,

and Semiclassical

M. Raymond Flannery, Atlanta, USA

46 Orientation and Alignment in Atomic

and Molecular Collisions

Nils Andersen, Copenhagen, Denmark

47 Electron–Atom, Electron–Ion,

and Electron–Molecule Collisions

Philip Burke, Belfast, UK

48 Positron Collisions

Robert P. McEachran, Canberra, Australia

Allan Stauffer, Toronto, Canada

49 Adiabatic and Diabatic Collision Processes

at Low Energies

Evgueni E. Nikitin, Haifa, Israel

50 Ion–Atom and Atom–Atom Collisions

A. Lewis Ford, College Station, USA

John F. Reading, College Station, USA

51 Ion–Atom Charge Transfer Reactions

at Low Energies

Muriel Gargaud, Floirac, France

Ronald McCarroll, Paris Cedex 05, France

52 Continuum Distorted Wave and Wannier

Methods

Derrick Crothers, Belfast, UK

Fiona McCausland, Belfast, UK

John Glass, Belfast, UK

Jim F. McCann, Belfast, UK

Francesca O’Rourke, Belfast, UK

Ruth T. Pedlow, Belfast, UK

53 Ionization in High Energy Ion–Atom Collisions

Joseph H. Macek, Knoxville, USA

Steven T. Manson, Atlanta, USA

54 Electron–Ion and Ion–Ion Recombination

M. Raymond Flannery, Atlanta, USA

55 Dielectronic Recombination

Michael S. Pindzola, Auburn, USA

Donald C. Grifﬁn, Winter Park, USA

Nigel R. Badnell, Glasgow, United Kingdom

56 Rydberg Collisions: Binary Encounter, Born

and Impulse Approximations

Edmund J. Mansky, Oak Ridge, USA

57 Mass Transfer at High Energies: Thomas Peak

James H. McGuire, New Orleans, USA

Jack C. Straton, Portland, USA

Takeshi Ishihara, Tsukuba, Japan

58 Classical Trajectory and Monte Carlo Techniques

Ronald E. Olson, Rolla, USA

59 Collisional Broadening of Spectral Lines

Gillian Peach, London, UK

659

Elastic Scatter

45. Elastic Scattering: Classical, Quantal,

and Semiclassical

Scattering cross sections determine the rates

at which gas phase processes and chemical

reactions happen, whether in the atmosphere,

or in an industrial reactor. This chapter provides

a handy compendium of equations, formulae,

and expressions for the classical, quantal, and

semiclassical approaches to elastic scattering.

Reactive systems and model potentials are also

considered.

45.1 Classical Scattering Formulae................ 659

45.1.1 Deﬂection Functions .................. 660

45.1.2 Elastic Scattering Cross Section .... 661

45.1.3 Center-of-Mass to Laboratory

Coordinate Conversion ............... 662

45.1.4 Glory and Rainbow Scattering..... 662

45.1.5 Orbiting and Spiraling Collisions.. 662

45.1.6 Quantities Derived

from Classical Scattering............. 663

45.1.7 Collision Action ......................... 663

45.2 Quantal Scattering Formulae ................ 664

45.2.1 Basic Formulae ......................... 664

45.2.2 Identical Particles:

Symmetry Oscillations ................ 666

45.2.3 Partial Wave Expansion.............. 667

45.2.4 Scattering Length

and Effective Range................... 668

45.2.5 Logarithmic Derivatives.............. 670

45.2.6 Coulomb Scattering ................... 671

45.2.7 Resonance Scattering................. 671

45.2.8 Integral Equation for Phase Shift. 673

45.2.9 Variable Phase Method .............. 673

45.2.10 General Amplitudes ................... 674

45.3 Semiclassical Scattering Formulae ......... 675

45.3.1 Scattering Amplitude: Exact

Poisson Sum Representation....... 675

45.3.2 Semiclassical Procedure ............. 675

45.3.3 Semiclassical Amplitudes:

Integral Representation ............. 676

45.3.4 Semiclassical Amplitudes

and Cross Sections ..................... 677

45.3.5 Diffraction and Glory Amplitudes 679

45.3.6 Small-Angle (Diffraction)

Scattering................................. 680

45.3.7 Small-Angle (Glory) Scattering .... 681

45.3.8 Oscillations in Elastic Scattering .. 683

45.4 Elastic Scattering in Reactive Systems .... 683

45.4.1 Quantal Elastic, Absorption

and Total Cross Sections ............. 683

45.5 Results for Model Potentials ................. 684

45.5.1 Born Amplitudes

and Cross Sections

for Model Potentials .................. 689

References .................................................. 689

45.1 Classical Scattering Formulae

Central Field. The total energy E > 0 and orbital angular

momentum L of relative planar motion are conserved.

For a particle of mass M with coordinates (R,ψ), a sym-

metric potential V(R), and asymptotic speed v,

E =

(R)

+V(R)+

2MR

= constant ,

(45.1)

= (2ME)b

= M



(t)

ψ(t)



= constant . (45.2)

Equation (45.2) implies constant areal velocity.

Radial momentum p(R):

p(R; E, b) = (2ME)

1/2



1 −

V(R)

−



1/2

(45.3)

Effective potential:

eff

(R) = V(R) +

2MR

= V(R) +

E . (45.4)

The turning points R

(E, b) are the roots of

E = V

eff

(R). The smallest R

is the distance of clos-

est approach R

(E, b) at the pericenter. The maximum

impact parameter b

and angular momentum L

for

Part D 45