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Surface Physics References 1353

much so that a complete catalog of applicable techniques

would be difﬁcult to compile. Many new facilities have

been constructed, with higher brightness sources allow-

ing the development of a large number of new techniques

using synchrotron radiation from the ir through the soft

and hard X-ray regimes [89.53,54].

References

89.1 G.Ertl,J.Küppers:Low Energy Electrons and Sur-

face Chemistry, 2nd edn. (VCH, Weinheim 1985)

89.2 A. Zangwill: Physics at Surfaces (Cambridge Univ.

Press, Cambridge 1988)

89.3 V. Bortolani, N. H. March, M. P. Tosi (Eds.): Inter-

action of Atoms and Molecules with Solid Surfaces

(Plenum, New York 1990)

89.4 H. Lüth: Surfaces and Interfaces of Solids (Springer,

Berlin, Heidelberg 1993)

89.5 D. P. Woodruff, T. A. Delchar: Modern Techniques of

Surface Science, 2nd edn. (Cambridge Univ. Press,

Cambridge 1994)

89.6 S. D. Kevan (Ed.): Angle-Resolved Photoemission:

Theory and Current Applications (Elsevier, Amster-

dam 1992)

89.7 J. C. Fuggle, J. E. Inglesﬁeld: Unoccupied Electronic

States (Springer, Berlin, Heidelberg 1992)

89.8 R. Z. Bachrach (Ed.): Synchrotron Radiation Re-

search: Advances in Surface and Interface Science,

Vol. 1 (Plenum, New York 1992)

89.9 D. A. King, D. P. Woodruff (Eds.): Chemical Physics

of Solid Surfaces and Heterogeneous Catalysis (El-

sevier, New York 1990)

89.10 Y. J. Chabal, F. M. Hoffmann, G. P. Williams (Eds.):

Vibrations at Surfaces (Elsevier, Amsterdam 1990)

89.11 A. R. Burns, E. R. Stechel, D. R. Jennison: Desorption

Induced by Electronic Transitions, DIET V (Springer,

Berlin, Heidelberg 1993)

89.12 P. H. Fuoss, K. S. Liang, P. Eisenberger: In: Syn-

chrotron Radiation Research: Advances in Surface

and Interface Science, Vol. 1, ed. by R. Z. Bachrach

(Plenum, New York 1992)

89.13 R. Wiesendanger, H.-J. Güntherodt: Scanning Tun-

neling Microscopy (Springer, Berlin, Heidelberg

1993)

89.14 D.Briggs,M.P.Seah:Practical Surface Analysis:

Auger and X-Ray Photoelectron Spectroscopy,Vol.1

(Wiley, Chichester 1990)

89.15 P. J. Rous: Prog. Surface Science 39, 3 (1992)

89.16 M. A. Van Hove: Low-Energy Electron Diffraction

(Springer, Berlin, Heidelberg 1986)

89.17 J. M. MacLaren, J. B. Pendry, P. J. Rous, D. K. Saldin,

G. A. Somorjai, M. A. Van Hove, D. Vveden-

sky: Surface Crystallographic Information Service:

A Handbook of Surface Structures (Reidel, Dor-

drecht 1987)

89.18 H. Ibach, D. L. Mills: Electron Energy Loss Spec-

troscopy and Surface Vibrations (Academic, New

York 1982)

89.19 H. Ibach: In: Interaction of Atoms and Molecules

with Solid Surfaces, ed. by V. Bortolani,

N. H. March, M. P. Tosi (Plenum, New York 1990)

89.20 S. Y. Tong, C. H. Li, D. L. Mills: Phys. Rev. B 24,806

(1981)

89.21 L. Sanche: J. Phys. B 23, 1597 (1990)

89.22 R. E. Palmer, P. J. Rous: Rev. Mod. Phys. 64,383

(1992)

89.23 C. L. Bryant, R. P. Messmer (Eds.): Auger Electron

Spectroscopy (Academic, New York 1988)

89.24 L. E. Davis, N. C. MacDonald, P. W. Palmberg,

G. E. Riach, R. E. Weber: Handbook of Auger Spec-

troscopy (Physical Electronics Div., Perkin Elmer

1976)

89.25 H. J. Freund, M. Neumann: In: Angle-Resolved

Photoemission, ed. by S. D. Kevan (Elsevier, Am-

sterdam 1992) p. 319

89.26 K. L. Kliewer: In: Photoemission and the Elec-

tronic Properties of Surfaces, ed. by B. Feuerbacher,

B. Fitton, R. F. Willis (Wiley, Chichester 1978)

89.27 D. P. Woodruff: In: Angle-Resolved Photoemission,

ed. by S. D. Kevan (Elsevier, Amsterdam 1992)

89.28 D. E. Eastman, F. J. Himpsel, J. F. van der Veen:

J. Vac. Sci. Technol. 20, 609 (1982)

89.29 J. E. Rowe: In: Synchrotron Radiation Research,

Vol. 1, ed. by R. Z. Bachrach (Plenum, New York

1992) p. 117

89.30 P. H. Citrin, P. Eisenberger, R. C. Hewitt: Phys. Rev.

Lett. 45, 1948 (1980)

89.31 J. Stöhr, K. Baberschke, R. Jaeger, R. Treichler,

S. Brennan: Phys. Rev. Lett. 47,381(1981)

89.32 A. Bianconi, A. Marcelli: , Vol. 1, ed. by

R. Z. Bachrach (Plenum, New York 1992) p. 63

89.33 J. Stöhr, F. Sette, A. L. Johnson: Phys. Rev. Lett. 53,

1684 (1984)

89.34 A. Shih, V. A. Parsegian: Phys. Rev. A 12,835(1975)

89.35 C. I. Sukenik, M. G. Boshier, D. Cho, V. Sandoghdar,

E. A. Hinds: Phys. Rev. Lett. 70, 560 (1993)

89.36 J. K. Norskov, N. D. Lang: Phys. Rev. B 21, 2136 (1980)

89.37 D. Menzel: In: Chemistry and Physics of Solid Sur-

faces IV, ed. by R. Vanselow, R. Howe (Springer,

Berlin, Heidelberg 1982) p. 102

89.38 H. J. Kreuzer, S. H. Payne: In: Dynamics of

Gas-Surface Interactions, ed. by C. T. Rettner,

M. N. R. Ashfold (Royal Society of Chemistry, Cam-

bridge 1991) p. 220

89.39 P. A. Redhead: Vacuum 12,203(1962)

89.40 J. T. Stuckless, N. Al-Sarraf, C. Wartnaby, D. A. King:

J. Chem. Phys. 99, 2202 (1993)

Part G 89

1354 Part G Aplications

89.41 K. W. Kolasinski: Surface Science: Foundations of

Catalysis and Nanoscience (John Wiley and Sons,

New York 2002)

89.42 G. A. Somorjai: Introduction to Surface Chem-

istry and Catalysis (Wiley-Interscience, New York

1994)

89.43 C. B. Duke (Ed.): Surface Science: The First Thirty

Years (Elsevier Science, New York 1994)

89.44 D. Bonnell (Ed.): Scanning Probe Microscopy and

Spectroscopy: Theory, Techniques, and Applications

(Wiley-VCH, Berlin 2000)

89.45 L. W. Bruch, M. W. Cole, E. Zaremba: Physical Ad-

sorption: Forces and Phenomena (Oxford Univ.

Press, Oxford 1997)

89.46 D. G. Castner, B. D. Ratner: Surface Science 500,28

(2002)

89.47 M. Tirrell, E. Kokkoli, M. Biesalski: Surface Science

500, 61 (2002)

89.48 W. Eberhardt: Surface Science 500, 242 (2002)

89.49 H. Dai: Surface Science 500, 218 (2002)

89.50 M. Knupfer: Surface Science Reports 42, 1 (2001)

89.51 D. M. Kolb: Surface Science 500, 722 (2002)

89.52 H. L. Dai, W. Ho (Eds.): Laser Spectroscopy and Pho-

tochemistry on Metal Surfaces (World Scientiﬁc,

Singapore 1995)

89.53 C. Lamberti: Surface Science Reports 53, 1 (2004)

89.54 A. Baraldi et al.: Surface Science Reports 49,169

(2003)

Part G 89

1355

Interface with

90. Interface with Nuclear Physics

For an atom with a small to moderately large

atomic number Z, the typical length scale a

/Z of

the innermost core orbitals is so much larger than

typical nuclear length scales that the corrections to

the energy levels and wave functions arising from

the non-zero electric charge radius of the nucleus

can accurately be computed using ﬁrst-order

perturbation theory, as is described in Sect. 90.1.

Nonetheless, these relatively small shifts can

sometimes have a profound effect on processes in

atomic and/or nuclear physics, particularly if two

or more energy levels are very close. For example,

as is discussed in Sect. 90.2, the presence of the

electron cloud makes energetically possible the

β-decay of

187

Re to

187

Os, and signiﬁcantly modiﬁes

the energy distribution of products in the β-decay

of tritium in various chemical environments.

Also, electronic screening can greatly enhance the

cross-sections of low-energy nuclear reactions

relative to what they would be for bare nuclei.

In isotopes of hydrogen, the replacement of an

electron by a muon, with m

≈ 207m

,resultsin

a tiny neutral ‘atom’ which can closely approach

another nucleus, thereby catalyzing nuclear fusion.

For example, the rate of deuterium–tritium fusion

is enhanced by 77 orders of magnitude if a single

electron is replaced by a muon. A rich variety of

bound-state properties and scattering processes

for these exotic atoms and molecules has been ex-

tensively investigated, as is reviewed in Sect. 90.3.

A reader interested in the interface between

atomic and nuclear physics should also consult

90.1 Nuclear Size Effects in Atoms ................1356

90.1.1 Nuclear Size Effects

on Nonrelativistic Energies .........1356

90.1.2 Nuclear Size Effects

on Relativistic Energies ..............1357

90.1.3 Nuclear Size Effects

on QED Corrections ....................1358

90.2 Electronic Structure Effects

in Nuclear Physics ................................1358

90.2.1 Electronic Effects on Closely

Spaced Nuclear Energy Levels......1358

90.2.2 Electronic Effects

on Tritium Beta Decay................1358

90.2.3 Electronic Screening

of Low Energy Nuclear Reactions . 1359

90.2.4 Atomic and Molecular Effects

in Relativistic Ion–Atom

Collisions .................................1359

90.3 Muon-Catalyzed Fusion........................1359

90.3.1 The Catalysis Cycle .....................1361

90.3.2 Muon Atomic Capture ................1362

90.3.3 Muonic Atom Deexcitation

and Transfer .............................1363

90.3.4 Muonic Molecule Formation .......1364

90.3.5 Fusion .....................................1366

90.3.6 Sticking and Stripping................1367

90.3.7 Prospectus................................1369

References ..................................................1369

Chapt. 27 (Quantum Electrodynamics), Chapt. 28

(Tests of Fundamental Physics), and Chapt. 29

(Parity Nonconserving Effects in Atoms) in this

Handbook.

That nuclei are not inﬁnitesimally small, structure-

less particles causes small but perceptible shifts in

the electronic structure of atoms and molecules. Even

for nuclei with small nuclear charge Z, the effects

are readily detectable through modern high precision

spectroscopy, and their magnitude grows as Z

14/3

Conversely, the presence of electrons tightly bound

to atomic nuclei can alter the ordering of nuclear

energy levels or make them unstable to β decay.

Atomic effects can also inﬂuence nuclear branch-

ing ratios into the product channels. Nominally small

atomic effects have been shown to affect the compli-

cated chain of nuclear reactions responsible for the

generation of energy in the sun. Setting bounds to

the rest mass of the neutrino from the endpoint of

the β-decay spectrum of tritium requires a precise

understanding of atomic and molecular structure and

scattering processes.

Part G 90

1356 Part G Applications

The great disparity between nuclear scales of energy

(several MeV) and distance (10

−5

to 10

−4

Å) and the

corresponding atomic scales (several eV and 1 Å, respec-

tively) usually allows the separate treatment of nuclear

and atomic effects. However, since not absolute ener-

gies but energy differences determine the magnitudes

of perturbative effects, near coincidences in energy dif-

ferences can greatly enhance the interplay between the

two regimes. Such comparable differences of energies

account for the important role of nuclear structure in

the Lamb shift splitting between the 2s

1/2

and 2p

1/2

states of hydrogen (see Chapt. 27), and the inﬂuence of

atomic structure on nuclear processes (see Sects. 90.1

and 90.2).

For the case of muonic atoms and molecules, the

interplay is enhanced by the much larger mass of a muon

relative to an electron. This decreases the distance scale

by a factor of m

and increases the energy scale by

a factor of m

. Small corrections such as the vacuum

polarization part of the Lamb shift are ampliﬁed even

more (≈ (m

)

for low Z).

Besides the areas where atomic physics effects play

an important role in nuclear physics, or vice versa, it

is worth remembering that atomic and molecular physi-

cists and nuclear physicists can beneﬁt from knowing

the theoretical techniques which have been developed in

each others’ ﬁelds. For example, it is well-known that

group theoretical methods are widely employed in for-

mulating and solving many-body problems in nuclear,

atomic, and molecular physics. To take another case,

the coupled-cluster method, which was ﬁrst proposed in

the late 1950s by Coester and Kummel in the context of

nuclear theory [90.1–3], was applied a decade later to

electronic structure problems in atomic and molecular

physics and quantum chemistry by Cizek, Paldus,and

Shavitt [90.4–6], and in the 1970s and 1980s was widely

developed by Bartlett and coworkers at the University of

Florida. Quite recently, ‘quantum halos’, which are very

loosely bound states for which most of the probability

density is spread diffusely over the classically forbidden

region, have been treated in a uniﬁed manner for both

nuclear and molecular systems [90.7].

90.1 Nuclear Size Effects in Atoms

90.1.1 Nuclear Size Effects on

Nonrelativistic Energies

Interest in the inﬂuence of a ﬁnite nuclear charge distri-

bution on the energy levels of the hydrogen atom goes

back to the measurement of the Lamb shift [90.8–11],

and even earlier indications that the ﬁne structure of

hydrogen did not quite agree with the predictions of the

Dirac equation for a point nucleus [90.12–15]. The ﬁnite

proton size does in fact raise the energy of the 2s

1/2

state

relative to 2p

1/2

, but the shift is only ≈ 0.012% of the

dominant electron self-energy contribution (Chapt. 27).

It must nevertheless be taken into account in high preci-

sion tests of QED.

Early derivations were given by several au-

thors [90.16–19] and generalized by Zemach [90.20]

(see also [90.21]) to a form involving integrals over the

nuclear electric and magnetic form factors. The basic re-

sult is illuminated by the following argument. Let ρ(r)

be the electron density, which may have no spatial sym-

metry properties in the particular case of a polyatomic

molecule, and ρ

) be the charge density of a nucleus,

which obeys



) = Z . (90.1)

Assume that ρ

) has no permanent electric dipole

moment, so that



) = 0 . (90.2)

By writing the Coulomb potential for a pointlike nucleus

−

=−



Zδ

(3)

)

|r− r

(90.3)

the ﬁrst-order shift of the electronic energy due to the

replacement of the pointlike nucleus by an extended

nucleus is

∆E

nuc



Zδ

(3)

) − ρ

)

|r− r

ρ(r).

(90.4)

Since the Fourier transform, deﬁned by

(k) =



−ik·r

), (90.5)

preserves inner products within a factor of (2π)

and

maps convolutions to simple products, the integral

in (90.4) reduces to

(2π)



k [Z −

(k)]

4π

ρ(k),

(90.6)

Part G 90.1

Interface with Nuclear Physics 90.1 Nuclear Size Effects in Atoms 1357

where the hats denote the Fourier transforms of the dens-

ities and 4π/k

is the Fourier transform of the Coulomb

potential 1/r.SinceZ =



), the energy shift

can be reexpressed as

(2π)



(1− e

−ik·r

)ρ

)

4π

ρ(k),

(90.7)

which is still an exactﬁrst-order perturbation expression.

Since typical nuclear length scales are much smaller

than typical nonrelativistic atomic length scales, it is

legitimate to expand the exponential in a Taylor series.

The zeroth-order term, −1, is canceled by the +1. The

linear term, ik· r

, contributes nothing by the hypothesis

that the nuclear charge distribution has no permanent

electric dipole moment. The ﬁrst nonvanishing term is

(2π)



(k· r

)

4π

ρ(k).

(90.8)

If ρ(r) is nonzero at the nucleus, then for large k the lead-

ing behavior of

ρ(k) is that of a spherically symmetric

s-wave with a radial dependence proportional to k

−4

The angular integration in the variable k leads to the re-

placement of (k· r

)

by its average value

,sothat

expression (90.8) reduces to

2π

(2π)



ρ(k)



), (90.9)

which can be further simpliﬁed by observing that

(2π)



ρ(k) = (2π)



k e

i0·k

ρ(k) = ρ(0),

(90.10)

and by deﬁnition



) = Z





(90.11)

thus yielding the ﬁnal expression

∆E

nuc

2π

ρ(0)





(90.12)

with

ρ(0) =





πn

−3

(90.13)

for a hydrogenic ion with reduced mass µ. This deriva-

tion is independent of the speciﬁc nuclear model or

the assumption of spherical symmetry of the electron

density. Since





scales as Z

2/3

, ∆E

nuc

then scales

as Z

14/3

. For a molecule with several nuclei, the contri-

butions (90.12) from each nucleus should be summed.

For the helium atom, ρ(0) =δ(r

) + δ(r

)

can be accurately calculated from high preci-

sion variational wave functions (Chapt. 11). For the

ground state, ρ(0)  (µ/m

)

[3.620 8586 −

0.182 37(µ/M)]a

−3

where M is the nuclear mass. Re-

sults for other states up to n = 10 are tabulated in [90.22].

Combined with high precision isotope shift measure-

ments, the results can be used to extract differences in

nuclear radii for pairs such as



He,



Li, and



D [90.23–26]. The method has recently been ap-

plied to the short-lived, neutron-rich nuclei

He,

Li,

and

Li [90.27,28].

Expression (90.12) works well for atoms with

small Z, since relativistic corrections to the electron

density are small. However, it breaks down for heavier

nuclei, for which relativistic wave functions are needed.

90.1.2 Nuclear Size Effects

on Relativistic Energies

The preceding analysis breaks down for relativistic wave

functions because they are singular at a point nucleus,

making ρ(0) inﬁnite. In this case, the Dirac equations

with Hamiltonians H

and H for the point nucleus and

distributed nucleus cases, respectively, can be combined

to obtain

(E − E

)Ψ

†

= Ψ

†

HΨ

− Ψ

†

. (90.14)

If a ﬁnite radius r

is now chosen such that H = H

outside the sphere r = r

, then this equation can be

integrated from r

outward to yield [90.29]

∆E

nuc

c(gf

− fg

)

r=r



∞

(gg

+ ff

)dr

(90.15)

where f and gare the large and small radial components

of Ψ (Chapt. 9), and the numerator is the surface term

that remains after integrating by parts the cα · p term

in H. The units are

c/a

= αm

. The solutions can

be further expanded in terms of Bessel functions, or the

Dirac equation can simply be integrated numerically.

For hydrogenic ions up to moderately large Z,the

results are reasonably well represented by [90.30,31]

∆E

nuc

(Zα)



,0

+ C

(Zα)









(90.16)

Part G 90.1

1358 Part G Applications

with γ =



1− (Zα)



1/2

,andC

 0.50, 1.38, and

0.1875 for the 1S

1/2

,2S

1/2

, and 2P

1/2

states, respec-

tively. Extensions to higher-order terms are discussed

in [90.32]. The above formula was used in the tabula-

tions of Mohr [90.31] for 10 ≤ Z ≤ 40, while Johnson

and Soff [90.33] used the numerical integration method

for Z up to 110. The nuclear electric and magnetization

density distributions are tabulated in [90.34], nuclear

moments in [90.35], and nuclear masses in [90.36]. In

the absence of better data, the rms nuclear radius can be

estimated from





1/2

≈ 0.777A

1/3

+ 0.778± 0.06 fm,

where A is the atomic mass number.

90.1.3 Nuclear Size Effects

on QED Corrections

Recent progress in the experimental study of tran-

sition energies in heavy ions stripped of most of

their electrons [90.37–39] has inspired theoretical

work on modiﬁcations of QED corrections due to

an extended nuclear charge distribution. Calculations

based on propagators expanded in terms of basis

splines [90.40–42] (Sect. 8.1.1) have led to rela-

tively rapid convergence with the number of angular

functions.

90.2 Electronic Structure Effects in Nuclear Physics

90.2.1 Electronic Effects on Closely Spaced

Nuclear Energy Levels

The presence of a nearby cloud of electrons can signiﬁ-

cantly affect nuclear processes involving closely spaced

nuclear energy levels. One of the most dramatic cases

involves the β-decay process

187

Re →

187

Os + e

−

, which is energetically forbidden by about 12 keV

for bare nuclei, but becomes allowed for the neutral

atoms when the difference in electronic binding ener-

gies is included. The nuclear charges are Z = 75 for

187

Re and Z = 76 for

187

Os. There is also the pos-

sibility of the electron being captured into a bound

state of

187

Os, as opposed to the continuum β-decay

process.

The electronic binding energies of heavy atoms

can be estimated from the Thomas–Fermi result

−20.93 Z

7/3

eV for a neutral atom of charge Z.

The difference between the energies of two atoms with

nuclear charges Z + 1andZ, respectively, is then

(Z + 1) − E

(Z) −48.83Z

4/3

eV, (90.17)

which amounts to − 15.4keV at Z = 75. This is more

than sufﬁcient to overcome the 12 keV energy deﬁcit in

the otherwise energetically forbidden β-decay of

187

Re.

The general theory of bound state β-decay is dis-

cussed by Bahcall [90.43], who also calculated the

ratio ρ of bound state β-decay to continuum β-decay for

bare nuclei. In the case of

187

Re →

187

Os, ρ is of impor-

tance in estimating changes in the half-life for β-decay

187

Re under variousconditions of ionization, since the

measured isotope ratios

187



188

Re and

187



188

from terrestrial rocks and meteorites can be used to de-

termine not only the age of the solar system, but also

the age of our galaxy [90.44, 45]. Estimates based on

a modiﬁed TF model [90.46] indicate that ρ  0.01,

and further multiconﬁguration Dirac–Fock calculations

give ρ = 0.005 to 0.007 [90.47–49]. See [90.47–49]for

further details and references.

90.2.2 Electronic Effects

on Tritium Beta Decay

The mass of the neutrino, normally taken to be zero in

the Standard Model, can be determined in principle from

analysis of the β-decay process

T →

+ e

−

An early measurement based on this method [90.50–52]

yielded a neutrino mass of ≈ 25 eV. Several independent

tests of this result were initiated soon thereafter. Since

the experiments are performed not on bare tritons but

on tritium gases and solids under various conditions, it

is essential to understand quantitatively the atomic and

molecular processes that affect the distribution of the

highest-energy electrons produced from various initial

states [90.53,54].

Martin and Cohen [90.55] used a Stieltjes imaging

technique to calculate shake-up and shake-off probabil-

ities for the β-decay of T

into

HeT

. Simultaneously,

extensive calculations were carried out [90.56]us-

ing potential energy curves for the reactant T

and

TH molecules and the product

HeT

and

HeH

molecules and accounting for the production of elec-

tronically and rovibrationally ground and excited ﬁnal

states, as well as resonant states. Nuclear motion was

found to have a small but detectable effect on the re-

sults, and solid-state effects for frozen T

were also

investigated and found to be small. These calculations

played a crucial role in the interpretation of the ex-

periments [90.57–61], which indicated that the neutrino

Part G 90.2

Interface with Nuclear Physics 90.3 Muon-Catalyzed Fusion 1359

mass is less than ≈ 10 eV. In 1998 there was pub-

lished evidence from the super-Kamiokande experiment

that the three ﬂavors of neutrinos oscillate, as further

conﬁrmed by the Sudbury Neutrino Observatory. This

implies that neutrinos have a nonzero rest mass [90.62].

Subsequently, upper bounds of the order of a few eV to

the neutrino mass have been derived from measurements

of tritium beta decay [90.63,64] and from cosmological

considerations [90.65].

90.2.3 Electronic Screening

of Low Energy Nuclear Reactions

The cross section σ(E) for a nuclear reaction involving

charged reactants drops very rapidly for collision ener-

gies E below the Coulomb barrier. A WKB treatment

shows that for low collision energies, the dependence of

σ(E) can be conveniently expressed as

σ(E) = S(E) E

−1

−2πη

, (90.18)

where S(E) is the astrophysical factor, and

η = Z

αc(µ/2E)

1/2

(90.19)

is the Sommerfeld parameter, which depends on the

charge numbers Z

and Z

of the projectile and target

nuclides, their reduced mass µ, and the cm energy E.

For nuclear reactions involving light nuclei, it is found

that S(E) typically varies slowly with E except close

to resonances. Thus the accurate determination of S(E)

at moderately low E can be used to extrapolate σ(E)

to much lower energies, which are beyond the reach of

laboratory experiments but are of great relevance to the

nuclear reactions that occur in stars.

However, electron screening effects can greatly en-

hance cross sections for nuclear reactions as measured in

the laboratory at low energy [90.66], because at least the

target nucleus is almost always surrounded by a cloud of

electrons which screen the Coulomb repulsion between

nuclei. The effect has been observed in various low-

energy reactions such as

He( d, p)

He,

Li(p, α)

He,

Li( d, α)

He, and

Li(p, α)

He [90.67–70]. Since re-

actions in stars involve bare nuclei, the laboratory data

must be carefully corrected for screening effects.

Analysis of the data for the

He(d,p)

He reaction in-

dicates that the effect of screening is always greater than

that predicted in the adiabatic limit [90.71–73]. A more

general theoretical treatment of the d+

Handd+

reactions [90.74], using a time-dependent Hartree–Fock

method for the electrons screening and classical mo-

tion for the nuclei found less enhancement than that

observed. An improved treatment taking account of elec-

tron correlation and quantum-mechanical effects on the

nuclear motion will likely be needed. This remains an

important area of development for the future.

For some recent work on the subject of elec-

tronic screening of low-energy nuclear reactions,

see [90.75–80].

90.2.4 Atomic and Molecular Effects

in Relativistic Ion–Atom Collisions

High-energy accelerators can now produce beams of

atomic ions partly or completely stripped of their elec-

trons, even for Z as high as 92. The collisions of such

beams of highly charged ions with ﬁxed targets involve

a broad array of atomic and molecular processes, such

as excitation, ionization, charge transfer, and, in the

extreme relativistic case, pair production. A similarly

broad array of theoretical techniques is required to study

these topics. A thorough review of them, including com-

parisons with experimental data where available,is given

in [90.81,82].

A topic of particular recent interest is the ﬁrst ex-

perimental observation of the capture of electrons from

electron-positron pair production in the extreme rel-

ativistic collision of a 0.96 GeV / nucleon U

92+

beam

with gold, silver, copper, and Mylar targets [90.83]. The

energy and angular distributions of the positrons were

also measured. For the gold target, the cross section for

capture was nearly as large as that for pair production

without capture, and it was found to vary with the nuclear

charge Z

of the target nucleus roughly as Z

2.8(±0.25)

Neither the dependence on Z

nor the relatively great

probability for capture is in agreement with perturbation

theory, which highlights the need for further exploration

of this exotic system.

90.3 Muon-Catalyzed Fusion

Exotic muonic atoms and molecules are more suit-

able subjects than electronic atoms and molecules for

probing some physical effects. The muon µ

−

is a lep-

tonic elementary particle like the electron except that

it is 206.768 times more massive and has a ﬁnite

lifetime (τ

= 1/λ

,whereλ

is the rate of decay)

of 2.197 µs. This lifetime is amply long for most ex-

periments. In normal atoms, the ﬁne-structure splitting

Part G 90.3

1360 Part G Applications

(due to L· S coupling) is much larger than the hyper-

ﬁne splitting (due to s

nuc

·s

coupling); this relation is

reversed in muonic atoms. Likewise, vacuum polariza-

tion, relativistic, ﬁnite-nuclear-size, and nonadiabatic

effects are enhanced. (Note: the muonic Bohr radius

≈ 1/207 a

is similar in size to the Compton

wavelength

c ≈ 1/137 a

.) Remarkably, muonic

molecules make nuclear fusion possible at room tem-

perature. In the phenomenon of muon-catalyzed fusion

(µCF), there are both indirect and direct interactions be-

tween the atomic and molecular physics and the nuclear

physics. Indirectly, the atomic and molecular densities

and transition rates control the nuclear fusion rates, and,

in turn, the kinetic energies of the fusion products af-

fect the atomic and molecular kinetics. Directly, the

nuclear structure affects some molecular energy levels

that determine important resonant rates and the bound-

ary condition on the muonic wave functions used to

calculate the muon “sticking” loss.

In comparison with µCF, hot fusion schemes are

made difﬁcult by the electrostatic (Coulomb) repulsion

between nuclei. In the two conventional approaches to

controlled fusion, magnetic and inertial conﬁnement,

this barrier is partially surmounted by energetic colli-

sions. (Note: the particle densities N and conﬁnement

times τ in the hot plasmas (T

 10

K) are typically

more than ten orders of magnitude different for these

two schemes, but the product of the two required for

d-t fusion is Nτ

 10

s/cm

in either case. For muon-

catalyzed fusion, effectively Nτ ≈ 10

s/cm

, but this

criterion doesn’t tell the real story.) On the other hand,

in µCF the objective is to tunnel through the barrier

without the beneﬁt of kinetic energy. This feat is en-

abled by binding two hydrogenic nuclei (p, d, or t) in an

exotic molecule like H

with the electron replaced by

a negative muon.

Since the molecular size is inversely proportional to

the mass of the binding particle, the average distance be-

tween nuclei in ppµ is ≈ 1/200 Å (500 fm) instead of

1 Å as in ppe



i. e., H



. This distance, which would be

reached in a d + d collision at ≈ 3keV (≈ 3×10

K),

is still large compared with the separation of a few

fm where the nuclear strong forces cause fusion, but

fusion occurs rapidly because of the increased vibra-

tional frequency and, more important, the increased

probability of tunneling per vibration. The vibrational

frequency is (m

)

3/2

≈ 3×10

times faster than for

the corresponding electronic molecule. (Note:forcom-

parison, the muonic/electronic energy scales as m

and the rotational energy scales as (m

)

.These

relations [90.84] are based on the Born–Oppenheimer

approximation, which is not very accurate for muonic

molecules.) The effect on the tunneling probability de-

pends on the nuclear masses; for dtµ, which has the

largest nuclear matrix element (astrophysical S fac-

tor, Sect. 90.2.3), the increase is by a factor of ≈ 10

compared with DT, and the consequential fusion rate

is λ

dtµ

≈ 10

−1

Just on the basis of the fusion rate, one would expect

a yield of λ

dtµ

/λ

≈ 10

muon-catalyzed d-t fusions for

the average muon. While this number indeed provides

an upper limit, the actual average number of fusions,

≈ 150 for dtµ, is much smaller and is determined by

the atomic and molecular physics of the catalysis cycle

(though the energy released in the nuclear fusion does

play an important role here). Some of the atomic and

molecular processes in the µCF cycle are quite ordinary,

but others, like atomic capture and resonant molecular

formation, have no counterpart with “normal” atoms.

Muon-catalyzed fusions of all pairs of hydrogen iso-

topes, except two protons, have been observed. Based

on the experiments and theory, the reaction products are:

pdµ →







He(0.005) + µ+ γ(5.49)(92% − 89%)

He(0.20) + µ(5.29)(8% − 11%)

(90.20)

ddµ →







t(1.01) + p(3.02) + µ (41%− 53%)

He(0.82) + µ+ n(2.45)(59% − 47%)

(90.21)

ptµ →











He(0.05) + µ+ γ(19.76)(60% − 76%)

He(0.59) + µ(19.22)(23%− 14%)

He + µ+ e

+ e

−

(19.81total)

(17%− 10%)

(90.22)

(Note: the last reaction of (90.22) has been theoretically

predicted [90.85], but has not yet been observed.)

dtµ →

He(3.54) + µ+ n(14.05) (90.23)

ttµ →

He + µ+ 2n(11.33total). (90.24)

Here the product particle kinetic energies (in MeV) are

given in parentheses. A µ without an energy designated

is a spectator, i. e. serves to bring the nuclei together but

plays no signiﬁcant role in the kinematics of the reac-

tion – such a µ may actually be bound (stuck) to one of

the product nuclei. As indicated, the branching fractions

Part G 90.3

Interface with Nuclear Physics 90.3 Muon-Catalyzed Fusion 1361

Fig. 90.1 Rovibrational energy levels for D

and ddµ.The

J = 0levelsareshownassolid lines and the J > 0levels

are shown dashed.ForD

, all 28 vibrational levels are

displayed, but the associated rotational levels are displayed

only up to the next higher vibrational level.All levelsof ddµ

are displayed; the (J = 1, v = 1) level is barely discernible

below the V = 0axis

depend somewhat on the target parameters (isotopic

composition, density, and temperature) [90.86–88].

Each reaction is of special interest in its own right:

pdµ and ptµ for the contribution of µ conversion, and

ttµ for the correlation of the two ﬁnal state neutrons.

Only the ddµ and dtµ molecular formations are res-

onant; i. e., their formation can occur in a one-body

state because they, and only they, possess a loosely

bound state such that the muonic binding energy can

go into rovibrational energy of the electronic molecule.

That the existence of such a state really is fortuitous

can be seen in Fig. 90.1 where the bound rovibra-

tional states of ddµ(1m.a.u. = 5626.5eV) are compared

with the rovibrational states of D

(1 a.u. = 27.2eV).

Though both ddµ and dtµ can be formed resonantly,

dtµ is unique in having a rapid (as compared with

muon decay) formation rate and also in having a small

sticking loss. The sticking loss is due to the pos-

sibility that the negatively charged muon may form

a bound state with the positively charged fusion prod-

uct. The relatively low branching fraction (< 1%) for

dtµ →

Heµ+ n is due simply to the high speed of the

outgoing

He (α particle).

90.3.1 The Catalysis Cycle

A diagram of the µCF cycle for a d-t mixture is shown

in Fig. 90.2. The basic steps in the cycle are

Fig. 90.2 The simpliﬁed d-t muon-catalyzed fusion cycle.

The times are for density φ = 1 (liquid hydrogen density)

and tritium fraction c

= 0.4. τ

is the cycle time, and τ

the muon-decay time

1. Atomic capture to form dµ or tµ (initially in a highly

excited state, n

 14).

2. Transfer of the µ from d to t, if necessary.

3. Resonant molecular formation, shown schematic-

ally. Here the dtµis so small (in reality) that is can be

considered to be a pseudo-nucleus in the electronic

molecule.

4. Nuclear fusion.

5. Sticking (αµ formation) or recycling.

The reaction times shown are at liquid hydrogen

density (φ = 1 in the conventional LHD units) and a tri-

tium fraction c

= 0.4, which is close to the value that

maximizes the number of cycles. The times for muonic

atom formation and deexcitation, ≈ ps, are short com-

pared with the times for muon transfer and molecular

formation, ≈ ns, which in turn are short compared with

the muon decay time ≈ µs.

Thus the time for a cycle is mainly given by the

average time the µ spends as dµ waiting to transfer to t

plus the average time it then spends as tµ waiting to form

dtµ,

≈ τ

dµ

+ τ

tµ

, (90.25)

or, in terms of rates,

≈

dtµ

, (90.26)

Part G 90.3

1362 Part G Applications

where c

and c

are the fractions of deuterium and tritium

+ c

= 1), λ

is the d-to-t transfer rate in the 1s state,

is the fraction of dµ atoms reaching the 1s state

(before transfer), and λ

dtµ

is the molecular formation

rate. The factor q

takes into account the fact that any

transfer in excited states is rapid (of necessity, since it

must compete with the rapid deexcitation).

The cycle rate λ

along with the sticking fraction

constitute the two basic parameters of the catalysis

cycle. The average number of fusions per muon Y is

given by

≈

+ ω

(90.27)

where λ

(≡ 1/τ

) is the muon-decay rate. (Note:more

precisely, W should appear in (90.27) in place of ω

. W

may include other losses, e.g., muon capture by impu-

rities, but we will restrict the present discussion to ω

which is fundamental and normally dominant.) Coinci-

dentally, the limits imposed on the yield by the cycling

rate and by sticking are similar; for φ ≈ 1andT ≈ 300 K,

/λ

≈ 300 and 1/ω

≈ 200 for dtµ. More than 100

muon-catalyzed d-t fusions per muon have been ob-

served. Similar considerations apply to the ddµ cycle,

and, by further coincidence, the two limits are similar

there as well, λ

ddµ

/λ

≈ 7and1/ω

eff(ddµ)

≈ 14. (Note:

the effective sticking probability (per cycle) in the ddµ

cycle takes into account that in only 58% of the fusion

reactions is a

He produced that can remove the µ by

sticking. Sticking to t or p is possible but would facilitate

rather than terminate the cycling.) The four experimen-

tal “knobs” are the temperature (T), density (φ), isotopic

fractions (c

, c

,andc

), as well as the molecular frac-

tions (c

, c

, ···) in the case of a target not in

chemical equilibrium.

Each stage of the cycle is discussed in the following

sections. The reader is referred to reviews [90.84,89–92]

for details of the theoretical and experimental methods

and extensive values of the relevant parameters.

90.3.2 Muon Atomic Capture

The µCF process starts with a free muon, injected

into a mixture of hydrogen isotopes, being stopped to

form a muonic atom (stopping power is discussed in

Sect. 91.1.1). The slowing and capture occur primarily

by ionization, e.g.,

µ+ D →



µ+ d + e

dµ(n) + e .

(90.28)

The muon is captured into an orbital with n 



≈ 14, which has about the same size and

energy as that of the displaced electron.

Methods for hydrogen and helium atoms have

been reviewed in [90.93]; the brief discussion here

emphasizes the correct intuitive understanding. Until

1977 most calculations were done using the Born or

Coulomb–Born approximation [90.94]. These methods

are not very accurate for µ

−

at velocities below 1 a.u.,

but, more importantly, their implementation treated

slowing down and capture inconsistently. The upshot

was prediction of capture of muons typically with ki-

netic energies of hundreds of eV, whereas it turns out that

most captures actually occur at energies below 100 eV.

The perturbative methods fail because of the great

electron charge redistribution that occurs during the cap-

ture process. Six other approaches have led to accurate

treatment:

1. Adiabatic ionization with straight-line trajectories

(AI-slt) [90.95],

2. Adiabatic ionization with curved trajectories (AI-

ct) [90.93,96],

3. Diabatic states (DS) [90.97, 98],

4. Classical-trajectory Monte Carlo (CTMC) [90.96],

5. Time-dependent Hartree–Fock (TDHF) [90.99],

6. Classical-quantal coupling (CQC) [90.100].

The ﬁrst three are models tailored for the muon cap-

ture problem (the ﬁrst two of these specialized for the

hydrogen-atom target). The results for all six methods

are shown in Fig. 90.3.

The early study by Wightman [90.95] shed a great

deal of light on the capture process. His method, known

as adiabatic ionization (AI), followed on the observation

of Fermi and Teller [90.101] that there exists a critical

strength of the dipole eR

, formed by the negative muon

and positive proton at distance R

= 0.639 a

, for bind-

ing the electron. In collisions where the µ

−

approaches

closer than this distance, the electron is assumed to es-

cape adiabatically, and, if the electron carries off more

energy than the muon’s initial kinetic energy, the pµ

atom is formed. This cross section is thus

AI-slt

= πR

, (90.29)

and µ

−

capture results if and only if E < 0.5 a.u., the

target ionization energy.

The AI-slt model has three major shortcomings: (1) it

does not take into account trajectory curvature, which

is caused by the Coulomb attraction of µ

−

toward the

nucleus and can be large at the low trajectory veloci-

ties where capture usually occurs, (2) the adiabatically

Part G 90.3