Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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244 Part B Atoms

Since Rydberg atoms are easily perturbed by electric

ﬁelds, it is hardly a surprise that collisions of charged

particles with Rydberg atoms have large cross sections.

In cold Rydberg atom samples, these large cross sections

can lead to the spontaneous evolution to a plasma, since

the macroscopic positive charge of the cold ions can trap

any liberated electrons, leading to impact ionization for

a large part of the Rydberg atom sample [14.38, 39].

14.7 Autoionizing Rydberg States

The bound Rydberg atoms considered thus far are

formed by adding the Rydberg electron to the ground

state of the ionic core. It could equally well be added to

an excited state of the core [14.40]. Figure 14.6 shows

the energy levels of the ground 5s state of Sr

and the

excited 5p state. Adding an n electron to the 5s state

yields the bound Sr 5sn state, and adding it to the ex-

cited 5p state gives the doubly excited 5pn state, which

is coupled by the Coulomb interaction to the degenerate

5s



continuum. The 5pn state autoionizes at the rate

n

given by [14.41]

n

= 2π|5pn|V|5s



|

, (14.35)

where V denotes the Coulomb coupling between the

nominally bound 5pn stateandthe5s



continuum.

A more general description of autoionization can be

found in Chapt. 25.

A simple picture, based on superelastic electron scat-

tering from the Sr

5p state, gives the scaling of the

autoionization rates of (14.35) with n and .Then

Rydberg electron is in an elliptical orbit, and each time

Fig. 14.6 Sr

5s and 5p states (—), the Rydberg states of

Sr converging to these two ionic states are shown by (—),

the continuum above the two ionic levels (///). The 5pn

states are coupled to the 5s



continua and autoionize

it comes near the core it has an n-independent probabil-

ity γ



of scattering superelastically from the Sr

5p ion,

leaving the core in the 5s state and gaining enough en-

ergy to escape from the Coulomb potential of the Sr

core. The autoionization rate of the 5pn state is ob-

tained by multiplying γ



by the orbital frequency of the

n state, 1/n

to obtain

n



. (14.36)

Equation (14.36) displays the n dependence of the

autoionization rate explicitly and the  dependence

through γ



.As increases, the closest approach of the

Rydberg electron to the Sr

is at a larger orbital radius,

so that superelastic scattering becomes progressively

less probable, and γ



decreases rapidly with increas-

ing . The simple picture of autoionization given above

implies a ﬁnite probability of autoionization each time

the n electron passes the ionic core, so the probabil-

ity of an atom’s remaining in the autoionizing state

should resemble stair steps [14.42], which can be di-

rectly observed using mode locked laser excitation and

detection [14.43].

Toa ﬁrst approximation, the Sr 5pn states can be de-

scribed by the independent electron picture used above,

but in states converging to higher lying states of Sr

the independent electron picture fails. Consider the Sr

 ≥ 4 states of n > 5. They are essentially degenerate,

and the ﬁeld due to an outer Rydberg electron converts

the zero ﬁeld  states to superpositions much like Stark

states. The outer electron polarizes the Sr

core, so that

the outer electron is in a potential due to a charge and

a dipole, and the resulting dipole states of the outer elec-

tron display a qualitatively different excitation spectrum

than do states such as the 5pn states, which are well de-

scribed by an independent particle picture [14.44]. When

both electrons are excited to very high-lying states, with

the outer electron in a state of relatively low , the clas-

sical orbits of the two electrons cross. Time domain

measurements, made using wave packets, show that in

this case autoionization is likely to occur in the ﬁrst orbit

of the outer electron [14.45].

Part B 14.7

Rydberg Atoms References 245

References

14.1 H. E. White: Introduction to Atomic Spectra

(McGraw-Hill, New York 1934)

14.2 H. A. Bethe, E. A. Salpeter: Quantum Mechanics of

One and Two Electron Atoms (Academic, New York

1975)

14.3 U. Fano: Phys. Rev. A 2, 353 (1970)

14.4 M. L. Zimmerman, M. G. Littman, M. M. Kash,

D. Kleppner: Phys. Rev. A 20, 2251 (1979)

14.5 S. A. Bhatti, C. L. Cromer, W. E. Cooke: Phys. Rev. A

24,161(1981)

14.6 R. R. Freeman, D. Kleppner: Phys. Rev. A 14,1614

(1976)

14.7 A. Lindgard, S. E. Nielsen: At. Data Nucl. Data Tables

19, 534 (1977)

14.8 E. S. Chang: Phys. Rev. A 31, 495 (1985)

14.9 W. E. Cooke, T. F. Gallagher: Phys. Rev. A 21,588

(1980)

14.10 S. Haroche, J. M. Raimond: Radiative properties

of Rydberg states in resonant cavities. In: Ad-

vances in Atomic and Molecular Physics,Vol.20,

ed. by D. Bates, B. Bederson (Academic, New York

1985)

14.11 F. B. Dunning, R. F. Stebbings: Experimental stud-

ies of thermal-energy collisions of Rydberg atoms

with molecules. In: Rydberg States of Atoms with

Molecules, ed. by R. F. Stebbings, F. B. Dunning

(Cambridge Univ. Press, Cambridge 1983)

14.12 D. S. Bailey, J. R. Hiskes, A. C. Riviere: Nucl. Fusion

5, 41 (1965)

14.13 M. G. Littman, M. M. Kash, D. Kleppner: Phys. Rev.

Lett. 41,103(1978)

14.14 R. R. Jones, T. F. Gallagher: Phys. Rev. A 38,2946

(1988)

14.15 R. R. Freeman, N. P. Economou, G. C. Bjorklund,

K. T. Lu: Phys. Rev. Lett. 41, 1463 (1978)

14.16 W. P. Reinhardt: J. Phys. B 16, 635 (1983)

14.17 J. Gao, J. B. Delos, M. C. Baruch: Phys. Rev. A 46,

1449 (1992)

14.18 T. F. Gallagher: Rydberg Atoms (Cambridge Univ.

Press, Cambridge 1994)

14.19 A. R. P. Rau: J. Phys. B 12, L193 (1979)

14.20 R. R. Jones, L. D. Noordam: Adv. At. Mol. Opt. Phys.

38, 1 (1997)

14.21 G. Alber, P. Zoller: Phys. Rep. 199, 231 (1991)

14.22 J. B. M. Warntjes, C. Wesdorp, F. Robicheaux,

L. D. Noordam: Phys. Rev. Lett. 83, 512 (1999)

14.23 D. Kleppner, M. G. Littman, M. L. Zimmerman: Ryd-

berg atoms in strong ﬁelds. In: Rydberg States

of Atoms and Molecules, ed. by R. F. Stebbings,

F. B. Dunning (Cambridge Univ. Press, Cambridge

1983)

14.24 W. R. S. Garton, F. S. Tomkins: Astrophys. J. 158,839

(1969)

14.25 A. Holle, J. Main, G. Wiebusch, H. Rottke,

K. H. Welge: Phys. Rev. Lett. 61, 161 (1988)

14.26 B. E. Sauer, M. R. W. Bellerman, P. M. Koch: Phys.

Rev. Lett. 68, 1633 (1992)

14.27 R. V. Jensen, S. M. Susskind, M. M. Sanders: Phys.

Rept. 201, 1 (1991)

14.28 P. Pillet, H. B. van Linden van der Heuvell,

W. W. Smith, R. Kachru, N. H. Tran, T. F. Gallagher:

Phys. Rev. A 30, 280 (1984)

14.29 P. Fu, T. J. Scholz, J. M. Hettema, T. F. Gallagher:

Phys. Rev. Lett. 64, 511 (1990)

14.30 M. Nauenberg: Phys. Rev. Lett. 64, 2731 (1990)

14.31 A. Krug, A. Buchleitner: Phys. Rev. A 66,053416

(2002)

14.32 E. Fermi: Nuovo Cimento 11,157(1934)

14.33 T. F. Gallagher, S. A. Edelstein, R. M. Hill: Phys. Rev.

A 15, 1945 (1977)

14.34 F. Gounand, J. Berlande: Experimental studies

of the interaction of Rydberg atoms with atomic

species at thermal energies. In: Rydberg States

of Atoms and Molecules, ed. by R. F. Stebbings,

F. B. Dunning (Cambridge Univ. Press, Cambridge

1983)

14.35 T. F. Gallagher: Phys. Rept. 210, 319 (1992)

14.36 I. Mourachko, D. Comparat, F. de Tomasi,

A. Fioretti, P. Nosbaum, V. M. Akulin, P. Pillet: Phys.

Rev. Lett. 80, 253 (1998)

14.37 W. R. Anderson, J. R. Veale, T. F. Gallagher: Phys.

Rev. Lett. 80, 249 (1998)

14.38 M. P. Robinson, B. Laburthe-Tolra, M. W. Noel,

T. F. Gallagher, P. Pillet: Phys. Rev. Lett. 85, 4466

(2000)

14.39 S. K. Dutta, J. R. Guest, D. Feldbaum, A. Walz-

Flannigan, G. Raithel: Phys. Rev. Lett. 86, 3993

(2001)

14.40 T. F. Gallagher: J. Opt. Soc. Am. B 4, 794 (1987)

14.41 U. Fano: Phys. Rev. 124, 1866 (1961)

14.42 X. Wang, W. E. Cooke: Phys. Rev. Lett. 67, 696 (1991)

14.43 S. N. Pisharody, R. R. Jones: Phys. Rev. A 65, 033418

(2002)

14.44 U. Eichmann, V. Lange, W. Sandner: Phys. Rev. Lett.

68, 21 (1992)

14.45 S. N. Pisharody, R. R. Jones: Science 303, 813 (2004)

Part B 14

247

Rydberg Atom

15. Rydberg Atoms in Strong Static Fields

Confronting classical and quantum mechanics

in systems whose classical motion is chaotic is

one of the fundamental problems of physics, as

evidenced by the enormous outpouring of research

during the last three decades [15.1, 2]. Highly

excited Rydberg atoms in external ﬁelds [15.3]play

a prominent role in this quest because they are

the best known examples of quantum systems

whose classical counterpart is chaotic. For a wide

variety of ﬁeld conﬁgurations and ﬁeld strengths,

their spectra can be measured to high precision.

At the same time, since their Hamiltonians are

known analytically, they are equally amenable

to accurate theoretical investigations using either

classical or quantum mechanics.

This chapter is restricted to a description of

Rydberg atoms in strong static ﬁelds. Related

15.1 Scaled-Energy Spectroscopy.................. 248

15.2 Closed-Orbit Theory ............................. 248

15.3 Classical and Quantum Chaos ................ 249

15.3.1 Magnetic Field .......................... 249

15.3.2 Parallel Electric

and Magnetic Fields .................. 250

15.3.3 Crossed Electric

and Magnetic Fields .................. 250

15.4 Nuclear-Mass Effects ............................ 251

References .................................................. 251

information on atoms in strong ﬁelds can be found

in Chapt. 13 of this Handbook, on Rydberg atoms

in Chapt. 14, and on the interaction of atoms with

strong laser ﬁelds in Chapt. 74.

Different conﬁgurations of external ﬁelds have been

studied: (i) an electric ﬁeld, which in hydrogen leads

to integrable classical dynamics [15.4, 5] (ii) a mag-

netic ﬁeld, which produces a transition from regular to

chaotic classical dynamics and which sparked the in-

terest in Rydberg atoms as prototype examples for the

study of the quantum-classical correspondence [15.6–

15] and references therein (iii) parallel electric and

magnetic ﬁelds [15.16, 17] (iv) crossed electric and

magnetic ﬁelds which break all continuous symme-

tries of the unperturbed atom and thus allow one to

study the transition from regularity to chaos in three

coupled degrees of freedom [15.18–21] and references

therein.

The hydrogen atom is the prototype example for

states with a single highly excited electron under the

inﬂuence of strong external ﬁelds. For an electron

in the hydrogen ground state, the inﬂuence of exter-

nal electric or magnetic ﬁelds becomes comparable

to that of the nuclear Coulomb ﬁeld when the ﬁeld

strengths are in the order of the atomic units of electric

ﬁeld strength, F

= e/(4πε

) = 5.142 206 42 (44) ×

V/m, or magnetic ﬁeld strength, B

= /





2.350 517 42 (20) ×10

T, which is far beyond exper-

imental reach. However, the relative importance of

the external ﬁelds scales with the principal quan-

tum number n as n

F and n

B, so that for

highly excited atoms, laboratory ﬁelds can easily be

“strong”. Atomic units will be used throughout this

chapter.

In a non-hydrogenic atom, the inﬂuence of the

inner-shell electrons can be summarized by means of

a short-range effective core potential or a set of quan-

tum defects [15.22]. For laboratory ﬁeld strengths, the

core is too small to be appreciably inﬂuenced by the

external ﬁelds. For this reason, the ﬁeld-free quantum

defects can be used to model core effects even in the

presence of external ﬁelds [15.23].

Part B 15

248 Part B Atoms

15.1 Scaled-Energy Spectroscopy

The Hamiltonian for a hydrogen atom in a

z-directed

magnetic ﬁeld and an electric ﬁeld of arbitrary orienta-

tion is

H =

−

+ r · F = E ,

(15.1)

where ρ

= x

and L

is the angular momentum

component along the magnetic ﬁeld axis. The dynamics

depends on three parameters: the ﬁeld strengths F and B

and the energy E. We can reduce the number of indepen-

dent parameters to two if we exploit a scaling property

of the Hamiltonian: In terms of the scaled quantities

r = w

−2

r ,

p = w p

E = w

E ,

F = w

F (15.2)

with the scaling parameter

w = B

−1/3

, (15.3)

the scaled Hamiltonian reads

H =

−

r ·

F =

E . (15.4)

The scaled dynamics thus depends only on two param-

eters, the scaled energy

E and the scaled electric ﬁeld

strength

F. Instead of the above scaling with the mag-

netic ﬁeld strength, which is the most common one,

equivalent scaling prescriptions with the electric ﬁeld

strength or the energy can be used [15.24].

The way of recording an atomic spectrum that is

most suitable for the investigation of quantum-classical

correspondence is scaled-energy spectroscopy. A scaled

spectrum consists of a list of eigenvalues w

of the scal-

ing parameter (15.3) characterizing the quantum states

for a given scaled energy

E and scaled electric ﬁeld

strength

F. It offers the advantage that the underlying

classical dynamics does not change across the spectrum,

which makes the spectrum more easily accessible to

a semiclassical interpretation (Sect. 15.2). For this rea-

son, scaled-energy spectroscopy has been adopted in

numerous experimental [15.4, 5, 8, 18] and theoretical

investigations.

To obtain a theoretical description of a scaled spec-

trum, the Schrödinger equation must be rewritten in

terms of the scaling parameter w. In the case of a single

external ﬁeld, either electric or magnetic, this procedure

leads to a generalized eigenvalue equation for the scal-

ing parameter w [15.25]. In the presence of both electric

and magnetic ﬁelds, the scaled spectrum is described

by a quadratic eigenvalue equation that has become

tractable only recently [15.26].

In a non-hydrogenic atom, the extent of the core

imposes an absolute length scale and thus breaks the

scaling symmetry. However, if the extent of the Ryd-

berg electron’s orbital is large, the size of the core can be

neglected and the scaling behavior is restored. This ren-

ders scaled-energy spectroscopy a useful concept also

for non-hydrogenic atoms [15.4, 5, 18].

15.2 Closed-Orbit Theory

Among the most remarkable effects strong external

ﬁelds produce in Rydberg atoms are the Quasi-Landau

oscillations: Close to the ionization limit, the photoab-

sorption spectrum of Ba I in a magnetic ﬁeld shows

regular oscillations [15.6]. This phenomenon was given

a convincing interpretation by Starace [15.27]and

embedded by Du and Delos [15.28, 29]andBogo-

molny[15.30] into the general framework of closed-orbit

theory, which has since become the central interpreta-

tive tool for a description of Rydberg spectra in external

ﬁelds [15.4,5,8,18]). Recently, it has also been used for

the computation of Rydberg spectra [15.31, 32].

Closed-orbit theory represents an atomic photo-

absorption spectrum as a superposition of regular

oscillations, each of which is related to a closed orbit

of the underlying classical dynamics, i. e., to an orbit

that starts and ends at the position of the nucleus. The

period of an oscillation is given by the return time of

the associated closed orbit (divided by

), the amplitude

is determined on the one hand by the initial state and

the polarization of the exciting photon, and on the other

hand by the stability of the closed orbit. Once the ini-

tial state is speciﬁed, both can therefore be calculated

within classical mechanics.

A spectrum that contains contributions from many

closed orbits can look enormously complicated. Its

Fourier transform, on the other hand, consists of a series

of isolated peaks that can be identiﬁed with the contribu-

tions of individual closed orbits. The Fourier transform

thus provides the means to identify the crucial dynam-

Part B 15.2

Rydberg Atoms in Strong Static Fields 15.3 Classical and Quantum Chaos 249

ics underlying a complicated spectrum. If a spectrum

is recorded at constant external ﬁeld strength, however,

this analysis is inhibited by the fact that the oscilla-

tions are not strictly harmonic because both the return

time of a closed orbit and the recurrence amplitude as-

sociated with it vary across the spectrum. This is the

principal reason why scaled-energy spectroscopy (see

Sect. 15.1) is customarily used. In a scaled spectrum,

the period of an oscillation is given by the scaled ac-

tion of the corresponding orbit and is ﬁxed across the

spectrum.

Although initially devised for atoms in magnetic

ﬁelds [15.28–30], closed-orbit theory is equally applica-

ble to atoms in electric [15.33] as well as parallel [15.34]

or crossed [15.34, 35] electric and magnetic ﬁelds. In

the case of non-hydrogenic atoms, the inﬂuence of the

ionic core can be modelled either by means of an ef-

fective classical potential [15.35, 36]orintermsof

quantum defects [15.37]. Recently, closed-orbit theory

has even been applied to the spectra of simple molecules

in external ﬁelds [15.38].

Since a non-hydrogenic core is much smaller than

the extent of a closed orbit (which is comparable to the

size of the atomic Rydberg state), it does not apprecia-

bly modify the shape of the orbit. The peaks observed

in a hydrogen spectrum are therefore also observed in

the corresponding spectrum of a non-hydrogenic atom,

although their strengths may be altered considerably

(core shadowing) [15.37]. The principal effect of a core

is to scatter the electron returning along one closed

orbit into the initial direction of another, so that con-

catenations of hydrogenic closed orbits appear in the

spectrum [15.37]. For this reason, the closed orbits of

the hydrogen atom in external ﬁelds are the crucial in-

gredient for the interpretation of any Rydberg spectrum.

They have been systematically studied for hydrogen

in magnetic [15.39, 40] as well as electric [15.41]and

crossed [15.42, 43]ﬁelds.

15.3 Classical and Quantum Chaos

In the absence of external electric and magnetic

ﬁelds, the classical dynamics described by the atomic

Hamiltonian (15.1) is integrable and completely de-

generate [15.44]. When external ﬁelds are present,

a transition to classical chaos can be observed whose

details depend on the precise ﬁeld conﬁguration (see be-

low). It is characterized by the break-up of invariant tori

and the appearance of irregular regions in the classical

phase space.

Chaos, as understood in classical mechanics, does

not exist in closed quantum systems [15.45]. Never-

theless, in the dynamics of a quantum system clear

indications of regularity or chaos in the underlying clas-

sical system can be found [15.2]. Most prominent among

them is the statistical distribution of nearest-neighbor

energy level spacings (NNS). In a classically chaotic

system, energy levels show avoided crossings. Level

repulsion is statistically reﬂected by NNS following

a Wigner distribution

P(S) =

−πS/4

(15.5)

that restricts small spacings. On the other hand, in-

tegrable systems possess a complete set of quantum

numbers, so that levels are allowed to cross. This gives

rise to a Poissonian NNS distribution

P(S) = e

−S

(15.6)

that favors small spacings. For mixed regular-chaotic

systems, a transition from a Poisson to a Wigner NNS

distribution is found [15.14].

15.3.1 Magnetic Field

An atom exposed to a magnetic ﬁeld possesses rota-

tional symmetry around the magnetic ﬁeld axis, which

leads in classical mechanics to the conservation of the

angular-momentum component L

along the ﬁeld axis

and in quantum mechanics to a good magnetic quantum

number m = L

. The dynamics of the rotation coordi-

nate can thus be separated, leaving two coupled degrees

of freedom.

The quantum numbers l and n that characterize

the pure hydrogen states both break down in a mag-

netic ﬁeld. However, the ﬁeld affects them differently:

Whereas l breaks down extensively even for small ﬁelds

(l-mixing region), the breakdown of n is only achieved

through considerably stronger ﬁelds or higher ener-

gies (n-mixing region). Since chaotic dynamics can

only exist in at least two degrees of freedom, a sin-

gle n-manifold of electronic energy levels does not

have enough degrees of freedom to support chaos.

Chaos can develop only when different n-manifolds

mix (intermanifold chaos). The regular dynamics that

prevails as long as n is approximately conserved is

reﬂected in the existence of a second adiabatic con-

Part B 15.3

250 Part B Atoms

stant of motion (apart from the energy), which is given

by [15.46, 47]

Λ = 4A

−5A

, (15.7)

in terms of the Runge–Lenz vector

√

−2E



( p× L−L× p) −



(15.8)

and is conserved to second order in the magnetic ﬁeld

strength.

As the magnetic ﬁeld strength increases, corres-

ponding to an increase in the scaled energy from

−∞ toward zero, the classical dynamics changes from

regular to almost entirely chaotic [15.13, 14]. For

positive scaled energies above

=0.328 782 ...,com-

pletely hyperbolic dynamics is reached [15.48]. In step

with the onset of classical chaos, the quantum NNS

distribution changes from a Poisson to a Wigner distri-

bution [15.13, 14].

15.3.2 Parallel Electric and Magnetic Fields

In parallel ﬁelds, as in a pure magnetic ﬁeld, an atom

retains rotational symmetry around the ﬁeld axis. It

therefore shows a similar transition from regular dynam-

ics to intermanifold chaos at scaled energies

E ≈ 0. At

small ﬁeld strengths, a second-order adiabatic invariant

akin to (15.7) is given by [15.49, 50]

= 4A

−5(A

−β)

+5β

, (15.9)

where the parameter

β =

(15.10)

measures the relative strengths of the electric and mag-

netic ﬁelds.

15.3.3 Crossed Electric and Magnetic Fields

In nonaligned electric and magnetic ﬁelds, the rotational

symmetry of the ﬁeld-free atom is broken completely,

so that all three degrees of freedom are coupled. The an-

gular momentum quantum numbers l and m break down

extensively even at small ﬁeld strengths; the principal

quantum number n follows only gradually. Even when

n is approximately conserved, however, in the crossed-

ﬁelds atom two coupled degrees of freedom remain.

They allow the occurrence of chaotic dynamics within

a single n-manifold (intramanifold chaos) [15.51, 52].

The intramanifold dynamics can conveniently be

described in terms of the vectors [15.53]

(L+ A),

(L− A),

(15.11)

that obey independent angular momentum Poisson

bracket (or, in quantum mechanics, commutator) re-

lations. For ﬁxed n, I

and I

are restricted to the

spheres

= I

(15.12)

They span a four-dimensional space that is a convenient

representation of the intramanifold phase space.

Within a given n manifold, the position vector r

can be replaced with −

nA [15.54]. Using this re-

placement, we can re write the contributions to the

Hamiltonian (15.1) that are linear in the ﬁeld strengths

lin

= ω

· I

+ω

· I

(15.13)

with the constant vectors

(B−3nF),

(B+3nF).

(15.14)

The ﬁrst-order Hamiltonian H

lin

describes a precession

of the vectors I

and I

around ω

and ω

, respectively,

and preserves the integrability of the dynamics. Intra-

manifold chaos arises only if the quadratic contribution

to the Hamiltonian (15.1) is taken into account. It can be

detected either in classical mechanics [15.51, 52]orin

quantum mechanics via its imprint on the intramanifold

NNS distribution [15.52].

The properties of the crossed-ﬁelds hydrogen atom

above the ionization threshold provide an example of

a chaotic scattering system. Classically, chaotic ion-

ization manifests itself in a fractal dependence of the

electron escape time on the initial conditions [15.55].

Experimentally, a distinction has been made between

“prompt” electrons that ionize fast, and “delayed” elec-

trons that ionize only after more than 100 ns [15.20]. The

latter can be interpreted as electrons that undergo chaotic

scattering and circle the nucleus many times before they

escape. A detailed classical model of chaotic scatter-

ing was presented in [15.56]. In quantum mechanics,

chaotic scattering can be identiﬁed through the occur-

rence of Ericson ﬂuctuations in the above-threshold

spectrum [15.55].

Part B 15.3

Rydberg Atoms in Strong Static Fields References 251

15.4 Nuclear-Mass Effects

So far, only the relative motion of the electron with

respect to the ionic core has been described. This is

appropriate if the nucleus can be assumed to be inﬁnitely

heavy and thus not to take part in the motion. To include

the effects of a ﬁnite nuclear mass, the description must

start from the coupled two-body Hamiltonian and then

work toward a separation of the internal dynamics from

the center-of-mass (CM) motion.

It turns out that in the presence of a magnetic

ﬁeld, unlike the ﬁeld-free two-body problem, a com-

plete separation of the relative and CM motions is

impossible. Instead, only a pseudo-separation can be

achieved, where the relative and CM motions remain

coupled through a new constant of motion called the

pseudomomentum K [15.57]. This coupling introduces

a number of novel effects into the dynamics (see [15.58]

for a detailed discussion).

The inﬂuence of the CM motion on the internal dy-

namics is twofold: on the one hand, the motion of the

atom in the magnetic ﬁeld causes an induced electric

ﬁeld (motional Stark effect). On the other hand, the ki-

netic energy of the CM motion gives rise to an additional

conﬁning potential for the internal motion that could, in

principle, locate the electron at a large distance from the

nucleus, and produce atomic states with a huge dipole

moment.

Conversely, the motion of the CM is driven

by the internal motion, most strongly so in the

case of vanishing pseudomomentum. It thus re-

ﬂects the transition from regular to chaotic internal

dynamics: A regular internal motion leads to a reg-

ular CM motion, whereas chaotic internal dynamics,

for K = 0, give rise to a classical diffusion of the

CM.

References

15.1 M. C. Gutzwiller: Chaos in Classical and Quantum

Mechanics (Springer, Berlin, Heidelberg 1990)

15.2 F. Haake: Quantum Signatures of Chaos, 2nd edn.

(Springer, Berlin, Heidelberg 2000)

15.3 T. F. Gallagher: Rydberg Atoms (Cambridge Univ.

Press, Cambridge 1994)

15.4 A. Kips, W. Vassen, W. Hogervorst: Phys. Rev. A 59,

2948 (1999)

15.5 R. V. Jensen, H. Flores-Rueda, J. D. Wright,

M. L. Keeler, T. J. Morgan: Phys. Rev. A 62,053410

(2000)

15.6 W. R. S. Garton, F. S. Tomkins: Astroph. J. 185,839

(1969)

15.7 K. T. Lu, F. S. Tomkins, W. R. S. Garton: Proc. Roy.

Soc. London Ser. A 362, 421 (1978)

15.8 J. Main, G. Wiebusch, K. Welge, J. Shaw, J. B. Delos:

Phys. Rev. A 49, 847 (1994)

15.9 R. J. Elliott, G. Droungas, J.-P. Connerade: J. Phys. B

28, L537 (1995)

15.10 R. J. Elliott, G. Droungas, J.-P. Connerade, X.-H. He,

K. T. Taylor: J. Phys. B 29, 3341 (1996)

15.11 J. C. Gay: The structure of Rydberg atoms in strong

static ﬁelds. In: NATO Advanced Study Institute Se-

ries B: Physics, Vol. 143, ed. by J. P. Briand (Plenum

Press, New York 1986) pp. 107–152

15.12 J. C. Gay: Hydrogenic systems in electric and mag-

netic ﬁelds. In: The spectrum of atomic hydrogen:

Advances, ed. by G. W. Series (World Scientiﬁc, Sin-

gapore 1988) pp. 367–446

15.13 H. Friedrich, D. Wintgen: Phys. Rep. 183,37

(1989)

15.14 H. Hasegawa, M. Robnik, G. Wunner: Prog. Theor.

Phys. Suppl. 98, 198 (1989)

15.15 D. Delande: Chaos in atomic and molecular physics.

In: Chaos and Quantum Physics, Session LII of Les

Houches, ed. by M. J. Giannoni, A. Voros, J. Zinn-

Justin (North-Holland, Amsterdam 1991) pp. 665–

726

15.16 M. Courtney, H. Jiao, N. Spellmeyer, D. Kleppner,

D. Gao, J. B. Delos: Phys. Rev. Lett. 74, 1538 (1995)

15.17 I. Seipp, K. T. Taylor, W. Schweizer: J. Phys. B 29,1

(1996)

15.18 G. Raithel, M. Fauth, H. Walther: Phys. Rev. A 44,

1898 (1991)

15.19 J.-P. Connerade, M.-S. Zhan, J. Rao, K. T. Taylor:

J. Phys. B 32, 2351 (1999)

15.20 S. Freund, R. Ubert, E. Flöthmann, K. Welge,

D. M Wang, J. B Delos: Phys. Rev. A 65,053408

(2002)

15.21 T. Uzer: Phys. Scr. 90, 176 (2001)

15.22 H. Friedrich: Theoretical Atomic Physics (Springer,

Berlin, Heidelberg 1998)

15.23 J. Rao, K. T. Taylor: J. Phys. B 30, 3627 (1997)

15.24 H. Friedrich: Scaling properties for atoms in ex-

ternal ﬁelds. In: Atoms and Molecules in Strong

External Fields, ed. by P. Schmelcher, W. Schweizer

(Plenum Press, New York 1998) pp. 153–167

15.25 T. S. Monteiro, G. Wunner: Phys. Rev. Lett. 65, 1100

(1990)

15.26 J. Rao, K. T. Taylor: J. Phys. B 35, 2627 (2002)

15.27 A. F. Starace: J. Phys. B 6, 585 (1973)

15.28 M. L. Du, J. B. Delos: Phys. Rev. A 38, 1896 (1988)

Part B 15

252 Part B Atoms

15.29 M. L. Du, J. B. Delos: Phys. Rev. A 38,1913

(1988)

15.30 E. B. Bogomolny: Sov. Phys. JETP 69,275

(1989)

15.31 J. Main, G. Wunner: Phys. Rev. A 59, R2548 (1999)

15.32 T. Bartsch, J. Main, G. Wunner: J. Phys. B 36,1231

(2003)

15.33 J. Gao, J. B. Delos, M. Baruch: Phys. Rev. A 46, 1449

(1992)

15.34 J.-M. Mao, K. A. Rapelje, S. J. Blodgett-Ford,

J. B. Delos: Phys. Rev. A 48, 2117 (1996)

15.35 K. Weibert, J. Main, G. Wunner: Ann. Phys. (NY)

268, 172 (1998)

15.36 B. Hüpper, J. Main, G. Wunner: Phys. Rev. A 53,744

(1996)

15.37 P. A. Dando, T. S. Monteiro, D. Delande, K. T. Taylor:

Phys. Rev. A 54, 127 (1996)

15.38 A. Matzkin, T. S. Monteiro: Phys. Rev. Lett. 87,

143002 (2001)

15.39 M. A. Al-Laithy, P. F. O’Mahony, K. T. Taylor:

J. Phys. B 19, L773 (1986)

15.40 J. Main, A. Holle, G. Wiebusch, K. H. Welge: Z. Phys.

D 6, 295 (1987)

15.41 J. Gao, J. B. Delos: Phys. Rev. A 49, 869 (1994)

15.42 D. M. Wang, J. B. Delos: Phys. Rev. A 63, 043409

(2001)

15.43 T. Bartsch, J. Main, G. Wunner: Phys. Rev. A 67,

063410 (2003)

15.44 H. Goldstein: Classical Mechanics (Addison-Wesley,

Reading 1965)

15.45 M. V. Berry: Phys. Scr. 40, 335 (1989)

15.46 E. A. Solov’ev: Sov. Phys. JETP 55, 1017 (1982)

15.47 D. R. Herrick: Phys. Rev. A 26, 232 (1982)

15.48 K. T. Hansen: Phys. Rev. E 51, 1838 (1995)

15.49 P. Cacciani, E. Luc-Koenig, J. Pinard, C. Thomas,

S. Liberman: J. Phys. B 21, 3499 (1988) and refer-

ences therein

15.50 P. A. Braun: Rev. Mod. Phys. 65, 115 (1993) and

references therein

15.51 J. von Milczewski, T. Uzer: Phys. Rev. E 55, 6540

(1997)

15.52 J. Main, M. Schwacke, G. Wunner: Phys. Rev. A 57,

1149 (1998)

15.53 M. J. Engleﬁeld: Group Theory and the Coulomb

Problem (Wiley-Interscience, New York 1972)

15.54 W. Pauli: Z. Phys. 36, 339 (1926)

15.55 J. Main, G. Wunner: J. Phys. B 27, 2835 (1994)

15.56 C. Jaffé, D. Farrelly, T. Uzer: Phys. Rev. Lett. 84,610

(2000)

15.57 J. E. Avron, I. W. Herbst, B. Simon: Ann. Phys. (NY)

114,431(1978)

15.58 P. Schmelcher, L. S. Cederbaum: Two interacting

charged particles in strong static ﬁelds. A variety of

two-body phenomenon. In: Structure and Bond-

ing, Vol. 86, ed. by L. S. Cederbaum, K. C. Kulander,

N. H. March (Springer, Berlin 1997) pp. 27–62

Part B 15

253

Hyperﬁne Stru

16. Hyperﬁne Structure

Hyperﬁne structure in atomic and molecular

spectra is a result of the interaction between

electronic degrees of freedom and nuclear

properties other than the dominant one, the

nuclear Coulomb ﬁeld. It includes splittings

of energy levels (and thus of spectral lines)

from magnetic dipole and electric quadrupole

interactions (and higher multipoles, on occasion).

Isotope shifts are experimentally entangled

with hyperﬁne structure, and the so-called

ﬁeld effect in the isotope shift can be naturally

included as part of hyperﬁne structure. Studies

of hyperﬁne structure can be used to probe

nuclear properties, but they are an equally

important probe of the structure of atomic

systems, providing especially good tests of

atomic wave functions near the nucleus. There

are also isotope shifts owing to the mass

differences between different nuclear species,

and the study of these shifts provides useful

atomic information, especially about correlations

between electrons.

Hyperﬁne effects are usually small and often,

but not always, it is sufﬁcient to consider only

16.1 Splittings and Intensities ..................... 254

16.1.1 Angular Momentum Coupling ..... 254

16.1.2 Energy Splittings ....................... 254

16.1.3 Intensities ................................ 255

16.2 Isotope Shifts ...................................... 256

16.2.1 Normal Mass Shift ..................... 256

16.2.2 Speciﬁc Mass Shift ..................... 256

16.2.3 Field Shift ................................ 256

16.2.4 Separation of Mass Shift

and Field Shift .......................... 257

16.3 Hyperﬁne Structure.............................. 258

16.3.1 Electric Multipoles ..................... 258

16.3.2 Magnetic Multipoles .................. 258

16.3.3 Hyperﬁne Anomalies ................. 259

References .................................................. 259

diagonal matrix elements for the atomic or

molecular system and for the nuclear system.

In some cases, however, matrix elements off-

diagonal in the atomic space, even though small,

canbeofimportance;onepossibleresultisto

cause admixtures sufﬁcient to make normally

forbidden transitions possible.

In the diagonal case, one can picture each electron

undergoing elastic scattering from the nucleus and re-

turning to its initial bound state. As pointed out by

Casimir [16.1, 2], however, the internal conversion of

nuclear gamma-ray transitions involves the inelastic

down-scattering from an excited nuclear state to a lower

one as an electron goes from an initial bound state

to the continuum. By further conversion of bound to

continuum states, one sees the connection with elec-

tron scattering from the nucleus – elastic, inelastic, and

break-up. Hyperﬁne structure of outer-shell electronic

states is at the low momentum-transfer end of this chain

of related processes.

Some of the standard textbooks which discuss hyper-

ﬁne structure are [16.3–10] and a few newer texts

[16.11–14]. Especially relevant are [16.15–19]andthe

conference proceedings [16.20–22].

The study of hyperﬁne structure in free atoms, ions,

and molecules is part of the more extensive research

area of hyperﬁne interactions, which includes the study

of atoms and molecules in matter, both at rest, for ex-

ample as part of the structure of a solid, and moving,

such as ions moving through condensed or gaseous mat-

ter. This more general subject also includes the ways

in which atomic electrons shield the nucleus, or anti-

shield it, from external or collective ﬁelds. Thus nuclear

magnetic resonance, nuclear quadrupole resonance,

electron-nuclear double resonance, recoilless nuclear

absorption and emission, nuclear orientation, production

of polarized beams, and many other widely used tech-

niques, are intimately connected with hyperﬁne effects.

Though hyperﬁne effects are ordinarily small in elec-

tronic systems, they can become much larger in “exotic”

atoms: those with a heavier lepton or hadron as the

Part B 16

254 Part B Atoms

“light” particle. Hyperﬁne effects are typically related

to light particle density at the nucleus, or to expectation

values of r

−3

, and thus scale as the cube of the light par-

ticle mass. The study of muonic atoms has contributed

importantly to knowledge of the nuclear charge distri-

bution [16.23–26]. There has been considerable interest

in pionic atoms, where the strong interaction also con-

tributes to hyperﬁne structure (e.g., [16.27, 28]), and

also in kaonic, antiprotonic, and other hadronic “atoms”

[16.29, 30]. See especially [16.31] for recent work on

antiprotonic helium.

Some other examples of interaction between atomic

and nuclear degrees of freedom are discussed in

Chapt. 90.

16.1 Splittings and Intensities

16.1.1 Angular Momentum Coupling

When the nuclear system is in an isotropic environment,

each nuclear state β has a deﬁnite value of nuclear an-

gular momentum I

, where the possible values of I

are related to the number of nucleons (protons plus neu-

trons) in the same way as those for J

are related to

the number of electrons in electronic state |α. The nu-

clear operators, eigenstates, and eigenvalues are related

to each other in the same way as for atomic angular

momentum by

|β=I

+ 1)|β ,

|β=M

|β , (16.1)

in units with = 1. Shift operators move the system

from one M-value to another, as for the atomic system

(see Chapt. 2), and the operator I is the generator of

rotations. When the combined atomic-nuclear system is

considered, in an isotropic environment, it is the total

angular momentum of the combined system deﬁned by

F = J+ I ,

(16.2)

that has deﬁnite values. The state of the combined system

can be labeled by γ ,sothat

|γ =F

+ 1)|γ  ,

|γ =M

|γ  . (16.3)

The shift operators are deﬁned as before, and it is now

F that is the generator of rotations of the (combined)

system, or of the coordinate frame to which the system

is referred.

By the rules of combining angular momenta, the

possible values of the quantum number F are separated

by integer steps and run from an upper limit of J

+ I

to a lower limit of |J

− I

|. The number of possible

eigenvalues F is the smaller of 2J

+ 1and2I

+ 1.

Experimental values of the nuclear quantum number I

may be found in a number of compilations [16.32–34].

16.1.2 Energy Splittings

Electromagnetic interactions between atomic electrons

and the nucleus can be expanded in a multipole series



(N) · T

(e),

≡



k,q

(−1)

(N)T

−q

(e) (16.4)

where T

(N) is an irreducible tensor operator of rank k

operating in the nuclear space, and similarly T

(e) op-

erates in the space of the electrons. Since one is taking

diagonal matrix elements (in the nuclear space, at least)

in states that are to a very good approximation eigen-

states of the parity operator, only even electric multipoles

(E0, E2, etc.) and odd magnetic multipoles (M1, M3,

etc.) contribute to the series. The effects of the par-

ity nonconserving weak interaction are considered in

Chapt. 29.

The term with k = 0 contributes directly to the struc-

ture (and ﬁne structure) of atomic systems, and its

dominant contributions come from the external r

−1

elec-

trostatic ﬁeld of the nucleus. The hyperﬁne Hamiltonian

is deﬁned by subtracting that term to obtain

hfs





(N)T

(i) −



− Ze







k=1

(N) · T

(e), (16.5)

where Z is the nuclear charge number. The difference

between the Ze

/r term(s) and the full monopole term

is called the ﬁeld effect or ﬁnite nuclear size effect in the

isotope shift, and the remaining terms contribute dipole

(k = 1), quadrupole (k = 2), and higher multipoles in

hyperﬁne structure.

Since the hyperﬁne Hamiltonian can be expressed

as a multipole expansion, its contributions to the pattern

of energy levels for the various F values in a given

Part B 16.1