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

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Interface with Nuclear Physics 90.3 Muon-Catalyzed Fusion 1363

escaping electron takes offno kinetic energy, and (3) ion-

ization occurs with unit probability if the approach is

closer then R

. The ﬁrst failing is easy to remedy. The

cross section with curved adiabatic trajectories (AI-ct)

is just [90.96]

AI-ct

πR



E +

− 0.5a.u.



(90.30)

as long as the collision energy E in the center-of-mass

system is greater than 0.03 a.u. (0.8 eV). Hence trajec-

tory curvature increases the capture cross section by

a factor 1 +

1.06

(for E in a.u.), which is over a factor

of three even at the highest collision energy (0.5a.u.)

where adiabatic capture can occur. For E < 0.03 a.u.,

the centrifugal barrier in the effective potential,

a(eff)

(R, b) = V

(R) +

E , (90.31)

restricts penetration and reduces the cross section below

the value given by (90.30) [90.93].

Cures for the second and third failings are less

trivial. These two assumptions can be avoided by us-

ing the diabatic-states (DS) model [90.97, 98]. The

adiabatic electronic energy no longer increases once

it reaches the continuum ceiling; however, in view of

the µ

−

acceleration by the Coulomb attraction, the

electron cloud actually does not have enough time

0.0 0.2 0.4 0.6 0.8

capt

(πa

)

Relative energy E

(a.u.)

Al-slt

Al-ct

CTMC

TDHF

CQC

Fig. 90.3 Comparison of different capture cross sections

for µ

−

+H collisions: adiabatic ionization with straight-

line trajectories(AI-slt), adiabatic ionization with curved

trajectories (AI-ct), diabatic states with polarized orbital

(DS), classical-trajectory Monte Carlo (CTMC), time-

dependent Hartree-Fock (TDHF), and classical-quantal

coupling (CQC)

to adjust adiabatically. In recognition of this situation

the diabatic electronic energy crosses into the contin-

uum at a distance larger than R

and continues to rise

smoothly. The concomitant probability of ionization is

given by the ionization width, obtained by a Fermi-

golden-rule-like formula. The ﬁrst three approaches are

somewhat specialized models, while the next three are

general methods. The most economical in terms of

computer time is the classical-trajectory Monte Carlo

(CTMC) method [90.96] discussed in Chapt. 58.This

method treats the dynamics of all particles completely

classically. The time-dependent Hartree–Fock (TDHF)

method discussed in Sect. 50.2 is purely quantum mech-

anical [90.99], but neglects correlation, which turns out

to be important in the present problem. This deﬁciency

is remedied by the classical-quantal coupling (CQC)

method, which makes only the seemingly well-justiﬁed

approximation of treating the muon classically while

retaining the quantum treatment of the electron [90.100].

Real µ

−

capture experiments are done with

molecules (H

, DT, etc.). The naive notion that the

cross section is simply twice that of H is quite

unrealistic for slow (v  1a.u.) collisions. In the past

decade there has been a major advance in the under-

standing of µ

−

capture by hydrogen, which is the ﬁrst

step in µCF. The captures by the H atom and the H

molecule, previously thought to differ by less than a fac-

tor of two, have been shown theoretically to be quite

different [90.102]. For a comprehensive review of all

methods used to threat capture of negative particles see

the review article [90.103]. This difference is primarily

due to the vibrational degree of freedom, which enables

the molecule to capture µ

−

at collision energies up to

≈ 40 eV with n

 9, whereas atomic capture cuts off

above ≈ 14 eVwith n

 14. There is a corresponding iso-

tope effect in the molecule, which is absent in the atom.

Experiments may be conducted in the near future on the

analogous capture of antiprotons by H and H

[90.104].

90.3.3 Muonic Atom Deexcitation

and Transfer

The muon is captured in a highly excited state but nor-

mally must reach the 1s conﬁguration of the heavier

isotope (in case of mixtures like D/T) before the muonic

molecule is formed. In the 1s conﬁguration there are two

hyperﬁne levels – the ground state with the nuclear and

−

spins antiparallel and an excited state with spins

parallel. Resonant molecular formation rates in the two

states can be quite different and also depend strongly

on the atom’s kinetic energy. Thus there are several

Part G 90.3

1364 Part G Applications

types of muonic atom collisions that must be taken into

account: (1) elastic scattering in the ground and excited

states, (2) isotopic transfer in excited states, (3) deex-

citing transitions (which may also occur radiatively),

(4) isotopic transfer in the 1s state, and (5) hyperﬁne

transitions. Cross sections for most of these processes

have been calculated. The bulk of the calculations

have been done by expanding in adiabatic (or modi-

ﬁed adiabatic) eigenfunctions, but there also exist some

calculations using the coupled-rearrangement-channel,

Faddeev, and hyperspherical approaches (see [90.84]for

references).

The cascade of the initially formed muonic atom,

especially in mixtures, is a complicated process not yet

completely characterized. It constitutes a crucial part

of the d − t µCF cycle in that it determines the par-

ameter q

in (90.26). This parameter is essential to

experimental analysis, but it was evident that early cal-

culations yielded values of q

too small to be consistent

with experiments. Recent calculations [90.105] suggest

the explanation is that the excited muonic atoms are

not thermalized. Epithermal atoms have three effects

here: (1) the normal transfer rates are smaller, (2) the

transfer is reversible down to lower principal quantum

numbers n where E still exceeds the threshold for excita-

tion of the next-higher level, and (3) excited-state [from

(tµ)

∗

] resonant formation of ( dtµ)

∗

molecules that can

predissociate back to dµ is enhanced [90.106]. (Note:

the isotopic energy splittings are 134.7/n

for dµ-pµ,

182.8/n

for tµ-pµ, and 48.0/n

for tµ-dµ.) The q

determined by competition between transfer and deex-

citation, which depend on the kinetic energies that result

from further competition between superelastic deexci-

tation and thermalizing elastic collisions. It appears that

the stage of the cascade most crucial to q

is n ≈ 4

for normal muon transfer and n = 2 for the resonant

sidepath.

For muons, the elastic cross sections are more dif-

ﬁcult to calculate than the inelastic ones. The inelastic

transitions occur at short range (a few a

)wherethe

effects of electronic structure are negligible. However,

electronic effects are not negligible for low energy

(< 1 eV) elastic scattering where λ

≈ 1 a

.Theyhave

been taken into account for ground state but not yet

excited state scattering. In doing so, it is not neces-

sary to solve the general problem directly because of

the following simpliﬁcations: (1) this energy is below

the vibrational threshold so the molecular target can

be taken as a rigid rotor and (2) the relative smallness

of the muonic atoms makes the sudden approximation

adequate.

If the 1s state is reached without muon transfer to

the heavier isotope already having occurred, the trans-

fer takes signiﬁcant time and plays an important role in

determining the tritium fraction c

that optimizes the fu-

sion yield. All of the 1s isotopic-exchange cross sections

display the characteristic ≈ 1/v velocity dependence at

thermal energies so that the corresponding rate vσ is

independent of temperature.

In muon-catalyzed d–d and d–t fusions, the reso-

nant molecular formation rates in different hyperﬁne

structure (hfs) states can differ by two or more orders

of magnitude at low T due to their different energy

levels. The hfs also has important effects on thermaliza-

tion and diffusion via the different elastic cross sections.

Under usual µCF experimental conditions, the hyper-

ﬁne quenching (or “spin ﬂip”) is irreversible; the hfs

splittings are 0.1820, 0.0485, and 0.2373 eV for pµ,

dµ,andtµ, respectively. Theoretically, it is expected

that transitions between hfs levels mainly occur in sym-

metric collisions since muon exchange sufﬁces in such

collisions [90.107]; e.g.,

tµ(↑↑) + t(↓) → t(↑) + tµ(↓↑)

(90.32)

(the usual terminology here is “muon exchange” al-

though it might seem more logical to refer to the reaction

as “triton exchange” since it is the identity of the tritons

that enables the reaction). As in the case of the iso-

topic exchange cross sections, the behavior is ≈ 1/v at

thermal energies, so the rates are nearly independent of

temperature.

90.3.4 Muonic Molecule Formation

Until the prediction by Vesman [90.108]ofareso-

nant formation for ddµ, it was thought that all muonic

molecules were formed by an Auger process of the type

dµ+ D

→[(ddµ)de]

+ e

−

. (90.33)

Unlike the resonant process for ddµ and dtµ,the

nonresonant process generally depends weakly on the

temperature of the target, the hyperﬁne state of the

muonic atom, and the “spectator” atom X in the mol-

ecule DX,whereX can be H, D, or T. The nonresonant

rate at low (liquid hydrogen) temperature for ddµforma-

tion is about 3×10

−1

and for dtµ is about 6 × 10

−1

These rates are competitive with the resonant rates at

low T for dµ(↑↓) + D

and tµ(↑↑) + D

,butare2to

3 orders of magnitude smaller than the resonant rates for

dµ(↑↑) + D

and tµ(↑↓) + D

, respectively. The (↑↓)

state is the ground state; thus hfs quenching plays an

important role in low-temperature experiments, espe-

cially for ddµ. At room temperature, resonant formation

Part G 90.3

Interface with Nuclear Physics 90.3 Muon-Catalyzed Fusion 1365

is dominant for both the ground and excited hyperﬁne

states of dµ and tµ.

In the Vesman mechanism the binding energy of the

muonic molecule goes into rovibrational excitation of

the electronic host molecule instead of into ionization

of a molecular electron. The process is resonant since

the collision energy must be tuned to match the energy

of the ﬁnal discrete state. For the compound molecule

formed, two sets of rovibrational quantum numbers are

needed, e.g.,

(tµ)

+[D

]

→



(dtµ)

dee



, (90.34)

where (K

,ν

),and(K

,ν

) are the initial and ﬁnal

quantum numbers of the electronic molecule, (J,v)are

the quantum numbers of the muonic molecule, F is

the spin of the muonic atom, and S is the total spin

of the muonic molecule. The energetics of this process

is shown in Fig. 90.4.

The resonant condition is achieved at the collision

energy



res

(

tµ+ D

)

= 

[dtµ]

+ E

[(dtµ)dee]−E

] ,

(90.35)

where it is explicitly recognized that ( J,v)= (1, 1) is the

only muonic level that can satisfy the resonant energy

condition and that only ν

= 0 is populated at ordinary

temperatures. Accurate calculations require values of



res

to within about 0.1 meV. The rovibrational ener-

gies E

Kν

of the electronic molecule, as well as the

Coulomb contributions to the binding energy of the

muonic molecule, 

, are now known to this high ac-

curacy. However, 

is subject to corrections due to

relativity, vacuum polarization, nuclear charge distribu-

tions and polarizabilities, the hyperﬁne interaction, and

Table 90.1 Resonant (quasiresonant if negative) collision energies 

res

(in meV) calculated using (90.35)

dµ + D

tµ + D

tµ + DT

[(ddµ)

dee]

= 7

[(dtµ)

dee]

= 2

[(dtµ)

dee]

= 3

[(dtµ)

tee]

= 3

F, S, K

, K



res

F, S, K

, K



res

F, S, K

, K



res

F, S, K

, K



res

,0,1 52.7 0, 1, 0, 1 −14.0 0, 1, 0, 1 277.1 0, 1, 0, 1 163.8

,0,1 76.9 0, 1, 0, 2 −4.3 1, 0, 0, 1 223.5 1, 0, 0, 1 110.2

,0,1 4.2 0, 1, 0, 3 10.3 1, 1, 0, 1 226.9 1, 1, 0, 1 113.6

,0,1 28.4 0, 1, 1, 2 −11.7 1, 2, 0, 1 233.3 1, 2, 0, 1 120.0

,1,0 40.9 0, 1, 1, 3 2.9

,1,0 65.1

,1,2 54.1

,1,2 78.3

Note: kT = 1meVforT = 11.6K

tµ+ D

res

J=1,

v=1

= 3

= 2

= 1

= 0

= 0 K

(dtµ) dee

[(dtµ)dee]

dtµ

]

Fig. 90.4 Diagram of energy levels for the resonant reac-

tion tµ+[D

]

=0,ν

→[( dtµ)

F=0,S=1

J=1,v=1

dee]

. The rovibra-

tional quantum numbers are designated by (J,v) for the muonic

molecule and (K,ν)for the electronic molecules

the ﬁnite size and shape of the muonic molecule in the

complex.The present overall accuracyis ≈ 1meV.Some

of the resulting values of 

res

are given in Table 90.1.The

calculated cross section for reaction (90.34)issharply

peaked at E

res

, but must be averaged over a kinetic

energy distribution (e.g. Maxwellian) to obtain the ob-

servable rate. Still, the rate will display a characteristic

resonant dependence on T.

Because the E

res

are different for each target

molecule (D

,DT,T

, ···), the effective molecular

formation rate in a mixture depends on the molecu-

lar composition in addition to the isotopic fractions

, c

, ···) if the target is not in chemical equilibrium.

Part G 90.3

1366 Part G Applications

The rate of ddµ and dtµ resonant molecular forma-

tion is calculated from

(T ) = N







d 2π |i|H



| f|

× f(, T ) I( − 

, T)



, (90.36)

where N is the target density, i|H



| f is a transition

matrix element, f(, T) is the collisional energy distri-

bution, 

is the energy of the unperturbed resonance,

and I(∆, T ) is the intensity at energy ∆ relative to the

unperturbed energy. H



has usually been taken to be the

dipole interaction (“post” form of the rearrangement-

collision Hamiltonian using the dtµ bound state as the

zeroth-order Hamiltonian [90.109]),



= e

d· E , (90.37)

where d is the dipole operator of the dtµ (or ddµ)

system and E is the electric ﬁeld at the dtµ (or ddµ)

center of mass due to the “spectator” nucleus and elec-

trons [90.110]. Conservation of angular momentum thus

requires [90.111]

L+ K

= J+ K

(90.38)

where L is the orbital angular momentum of relative

motion for tµ+ D

in reaction (90.34). At low T, L = 0

is predominant so that K

= K

± 1. This is simply the

case for dµ+ D

→ (ddµ)dee where the most proba-

ble transition is (K,ν)= (0, 0) → (1, 7). For dtµ the

vibrational state of the electronic molecule changes by

only ∆ν

= 2 or 3 instead of 7, so the matrix element

of (90.37) and the resulting rate are considerably larger

than for ddµ. However, it can be seen in Fig. 90.4 that

if D

is in its ground state (K

= 0), the ﬁrst level en-

ergetically accessible for (dtµ)dee has K

= 3. If, as

proves to be adequate in the case of ddµ, the intensity

distribution I is taken to be a δ function, the lower lev-

els are eliminated from (90.36). There are two possible

solutions to this problem, whose importances have not

been fully resolved: (1) the less likely L > 0 collisions

contribute or (2) the levels with smaller K

play a role

even though they lie “below threshold”.

The latter case is termed “quasiresonant”. The-

oretically the levels below threshold can contribute

(1) directly if they are broadened so that they extend

to positive energy [90.112–114] or (2) indirectly if

conﬁgurations with different K

are mixed [90.115].

Broadening can occur either inhomogeneously due to

the ﬁnite lifetime (mainly with respect to Auger emission

of an electron in the complex) or homogeneously due

to collisions with neighboring molecules. Interactions

with neighboring molecules also can mix the different

states, so the K

= 3 state may “borrow” some in-

tensity from the lower K

states. Three-body molecular

formation facilitated by neighboring molecules leads to

a density dependence of the formation rate (normalized

to LHD) that has been observed in experiments.

Experimentally, the resonant dtµ formation has now

been observed directly [90.116]. Previously the experi-

mental evidence for this mechanism derived from the

magnitude and the temperature dependence of the µCF

cycling rate. The new experiment, at TRIUMF, ob-

tained the energy-dependent molecular-formation rate

by measuring the time of ﬂight between a cryogenic

layer where the tµ atom was formed and a second cryo-

genic layer where the dtµ molecule was formed and

fusion occurred.

90.3.5 Fusion

Usually the nuclear fusion rate in muonic molecules

is calculated by a separable method; i. e., the united-

atom limit of the molecular wave function is determined

ignoring nuclear forces and then simply multiplied

by a single number extracted from nuclear scattering

experiments [90.117]. Fusion of d − t is strongly dom-

inated by the I

resonance of

He. For dtµ we

have

dtµ

= A lim

→0



|ψ

dtµ

, (90.39)

where A is simply related to the low-energy limit of the

astrophysical S factor by

A =

πe

lim

E→0

S(E), (90.40)

where M

is the reduced mass of the nuclei. S(E) is

usually obtained by ﬁtting the d+ t fusion cross section

observed in beam experiments to the form of (90.18).

The above formulation has yielded fusion rates in

good agreement with another formulation – more ac-

curate in principle – where a complex molecular wave

function and energy are obtained directly incorporating

the nuclear forces. In the latter approach,

dtµ

=−2Im(E

dtµ

)/ , (90.41)

where E

dtµ

is the complex eigenvalue (the imaginary

part is −Γ/2, where Γ is the width). The nuclear

effect in this formulation is taken into account by

two different techniques: (1) using a complex opti-

cal potential [90.118, 119] (Sect. 48.2), and (2) using

the nuclear R-matrix as an interior boundary condi-

tion [90.120–122] (Sect. 47.1.5). In the optical potential

Part G 90.3

Interface with Nuclear Physics 90.3 Muon-Catalyzed Fusion 1367

method, a short-range complex potential, determined

by ﬁtting experimental nuclear scattering data, is added

to the three-body Coulomb potential; the real part de-

scribes elastic scattering and the imaginary (absorptive)

part describes the fusion reaction. Then the eigenvalue

problem is solved over all space with a regular bound-

ary condition at the origin. In the R-matrix method,

the same nuclear scattering data are used to determine

a complex boundary condition at a distance characteris-

tic of the nuclear forces, say r

= a

where a

≈ 5fm.

The muonic eigenvalue problem is then formulated with

the boundary condition at r

= a

and solved over

the space excluding r

< a

. The two methods can be

used with similar basis sets to expand the wave func-

tion, which can also be used to calculate the sticking

probability (Sect. 90.3.6).

The relation of other µCF cross sections to normal

beam experiments is somewhat more complicated. In

the cases of ddµ and ttµ, the fusion may occur in J=1

states, since the J=1 to J=0 transition is forbidden in

molecules with identical nuclei. In this case the rele-

vant information from beam experiments resides in the

p-wave anisotropy, which is relatively small at low en-

ergies where σ is dominated by the s wave. It is then

necessary to carry out the analysis in terms of the partial-

wave transition amplitudes rather than the ﬁt of the

integrated σ via the S factor [90.123].

Fusions in pdµ and ptµ present a different compli-

cation [90.117]. There is a signiﬁcant E0 contribution

from muon conversion in addition to the M1 γ -ray

contribution seen in p+ dandp+ t beam experiments

(see Sect. 12.1 for discussion of multipole moments).

The E0 contribution cannot be expressed through cross

sections observed in the beam experiments, but has been

determined using the bound-state nuclear wave func-

tions of

He (or

He) and scattering wave functions of

p + d(orp+ t). For pdµ there is the additional com-

plication that two different p − d spin states contribute

signiﬁcantly. For ptµ, theory [90.117] predicts that the

probability of the fusion energy going into a e

−

pair

is competitive with that of muon conversion, though the

former has not yet been observed.

90.3.6 Sticking and Stripping

The fundamental mechanism of muon loss from the

catalysis cycle, other than by particle decay, is via stick-

ing to a helium nucleus,

He(≡ α) or

He, produced in

the fusion reaction. Especially in the case of dtµ,where

the charged particle is fast and the sticking probabil-

ity is already small, subsequent collisions may strip the

muon.Thus the sticking probability is determined by two

steps

dtµ

α + µ+ n

αµ(3.5MeV)+n

1− ω

α + µ

αµ(thermal) .

1− R

(90.42)

The initial sticking probability ω

depends only on

intramolecular dynamics, but the stripping conditional

probability depends on collisions. (Note that ω

is not

the sticking in the zero-density limit since R is still ﬁnite

in this limit.) The net sticking is then

= ω

(1− R). (90.43)

Since the nuclear reaction is very rapid compared

with the atomic and molecular dynamics, the probability

of sticking in a given state ν is given adequately by the

sudden approximation,





( f )

|ψ

(i)





, (90.44)

where the initial wave function ψ

(i)

is the normalized

molecular wave function in the limit r

→ 0andthe

ﬁnal wave function ψ

( f )

is given by

( f )

= φ

nm

(r) e

iq·r

(90.45)

in which φ

nm

is an atomic wave function of



Heµ



and the plane wave with momentum q represents its

motion with respect to the initial molecule (recoil deter-

mined by conservation of energy and momentum). The

total sticking is then



. (90.46)

The



Heµ



wave function is known analytically

since it is hydrogenic. Most of the labor goes into

determination of the muonic molecule wave function.

In the Born–Oppenheimer approximation this is sim-

ply “



Heµ



” and results in





= 1.20% for

dtµ [90.124]. More accurate nonadiabatic calculations

show that the muonic motion lags behind that of the

nuclei and reduces ω

to 0.886% [90.125, 126]. After

inclusion of nuclear effects, the best current theoretical

value of ω

is 0.912% [90.122,127].

Since the ground state (Heµ)

ion is bound by

11 keV, it takes a quite energetic collision to strip off

Part G 90.3

1368 Part G Applications

the muon. The reactivation fraction R is determined

basically by competition between collisional processes

that slow down the muonic ion and those that lead

to stripping. Calculation of R requires a full kinetic

treatment of the fast (Heµ)

ion, starting with its dis-

tribution among various states (1s, 2s, 2p, ···, n ≈ 10).

The most important processes are stopping power (due

mainly to ionization of the medium) and muon ioniza-

tion or transfer in collisions of the (Heµ)

ion with

an isotope of H, but inelastic (excitation and deexci-

tation), Auger deexcitation, and -changing collisions

as well as radiative deexcitation are also involved. The

initial sticking occurs mostly in the 1s state (77% of

the αµ’s from dtµ fusion), but the excited states have

larger ionization cross sections. Most of the muons

stripped from αµ originally stuck in the 1s state, but

a signiﬁcant number are promoted to excited states be-

fore being ionized (so-called “ladder ionization”). The

metastable 2s state is signiﬁcant for its role in pro-

longing the excited state populations. The resulting

values of R and ω

are shown as a function of density

in Fig. 90.5.

Most experiments on d-t µCF have been done with

neutron detection [90.137–139], where λ

and the muon

loss probability per cycle W can be deduced from the

time structure of the neutron emissions. The analysis

is indirect and requires a theoretical model. What is

actually measured is the product Wφλ

; the extraction

of ω

requires corrections for other loss mechanisms

and separate determination of λ

. Thus it is desirable

to have other experimental diagnostics. Two types of

Table 90.2 Comparison of sticking values

Source φ ω

(%) ω

(%)

Theory

[90.122,127] 1.2 0.59 0.91

Neutron Experiments

LANL [90.128] ≈ 1 0.43±0.05±0.06 0.66

PSI [90.129] ≈ 1 0.48±0.02±0.04 0.74

KEK [90.130] 1.2 0.51±0.004 0.78

KEK [90.131] solid 0.421±0.008±0.029 0.65

X-ray Experiments

PSI [90.132] 1.2 0.39±0.10 0.60

KEK [90.130] 1.2 0.34±0.13 0.52

α/αµ Experiments

LANL [90.133] 0.001 – 0.80±0.15±0.12

PSI [90.134] 0.17 0.56±0.04 0.80

Experimental values of ω

without error bars were obtained assuming the theoretical stripping [90.135, 136]

In cases of two error estimates, the ﬁrst is statistical and the second is systematic

The extraction of ω

from the X-ray experiments requires theoretical scaling

Fig. 90.5 Theoretical sticking fraction (solid curve) and

reactivation probability (dashed curve) for d-t µCF

corroborating experiments detect either X-rays from the

αµ formed by sticking or detect the species (α)

and

(αµ)

by the different effects of their double and single

electrical charges.

The theoretical sticking is compared in Table 90.2

with that from all three types of experiments. For a more

meaningful comparison of measurements at different

densities φ, the theoretical R has been used to convert

all values to ω

. The theoretical values are slightly, but

signiﬁcantly, higher than the observations. This discrep-

ancy has not yet been resolved.

One intriguing explanation for the lower-than-

predicted value of ω

may be that a signiﬁcant fraction

of the fusions might occur in muonically excited bound

Part G 90.3

Interface with Nuclear Physics References 1369

or resonant states for which the initial sticking is lower

than in the ground state [90.140].

A recent experiment has systematically studied

muon-catalyzed fusion in solid deuterium and tritium

mixtures as a function of temperature and tritium con-

centration [90.141]. An unexpected decrease in the

muon cycling rate (λ

) and an increase in the muon

loss (W) were observed. The former is likely due to the

freezing out of phonons contributing to the resonance

energy. The latter is especially intriguing. It is incon-

ceivable that ω

for fusion in a given state of dtµ could

depend on temperature, but this observation could im-

ply either an unexpected effect of temperature on the

muonic state in which fusion occurs or an unpredicted

temperature dependence of the thermalization kinetics

(e.g., due to ion channeling). It should be noted that

the experimental analysis does not reject the possibil-

ity of some correlation between the extracted values

of λ

and W [90.140].

90.3.7 Prospectus

There is still a great deal to be learned from the

less-studied µCF cycles, like those of p-d, p-t, and

t-t, but more quantitative work is also needed on some

key processes in the d-t µCF cycle: in particular, three-

body effects on molecular formation at high densities,

the excited-state cross sections and kinetics that go into

the determination of the cascade factor q

,andthere-

maining discrepancy in the sticking factor ω

which

might have a theoretical or experimental resolution.

Experimentally it is of interest to push on to higher

temperatures and densities to see if more surprises lurk

there. There have been a few schemes proposed to en-

hance stripping of stuck muons artiﬁcially, but none has

been subjected to experiments yet.

The currently observed yield of about 150 d-t fu-

sions (releasing 17.6 MeV each) per muon produces

an energy return 25 times the rest-mass energy of the

muon, but is only about one-third of that required for

breakeven in a pure-fusion reactor. This conclusion is

based on the estimated energy cost of producing a muon,

≈ 8 GeV [90.142,143]. Other possible practical uses in-

clude a hybrid (fusion-ﬁssion) reactor [90.142, 143]or

an intense 14 MeV neutron source [90.144].

Apart from such technological applications, the

study of µCF is fruitful for a number of reasons including

(1) bridging the gap between atomic and nuclear physics,

(2) enabling nuclear reactions (including p-waves) at

room temperature, (3) allowing precise studies under

unusual physical conditions, (4) observing a compound

electronic-muonic molecular environment, and (5) ex-

hibiting phenomena spanning nine orders of magnitude

in distance and energy. The experimental possibilities

are far from exhausted even though the holy grail of

pure fusion energy now appears just beyond reach.

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