Jess H. Brewer

Canadian Institute for Advanced Research

and Dept. of Physics & Astronomy,

Univ. of British Columbia

Vancouver, BC, Canada

At the end of the µ+ radiation track

Since the muonium (μ+e- or Mu) atom was first observed in condensed matter in the 1960s, there has been controversy over the mechanism by which Mu forms when an energetic positive muon (μ+) is stopped in an insulator or semiconductor. Epithermal or "prompt" Mu may form if a μ+ of ~ 102 eV strips an electron from an atom or molecule of the medium; thermal or "delayed" Mu formation (DMF) occurs when an electron liberated by the μ+ earlier in its ionization track is captured by the muon's Coulomb attraction. In the past decade we have found a means of resolving this controversy in many cases: suppression of the characteristic Mu precession signals by applied electric fields is a unique signature of DMF. This simple μSR technique thus provides a "microscope" with which to look at the last 10-100 nm of the μ+ ionization track in various insulators and semiconductors.

OUTLINE

Old Stories

Diamagnetic & Paramagnetic Species

“Themes” in µ+SR

µ+ in Gases, Metals, Liquids & Nonmetals

Spurs vs. Hot Atoms

Delayed Muonium Formation

Cryoliquids & Cryocrystals

A New Method: rapidly switched electric field

Sparse Tracks: s-N2, most Insulators

Dense Tracks: RGL/S, Al2O3, Semiconductors

“Themes” in µ+SR

Muonium as light Hydrogen

(Mu = µ+e-) (H = p+e-)

Mu vs. H atom Chemistry:

- gases, liquids & solids

- Best test of reaction rate theories.

- Study “unobservable” H atom rxns.

- Discover new radical species.

Mu vs. H in Semiconductors:

- Until recently, µ +SR → only data on

metastable H states in semiconductors!

The Muon as a Probe

Quantum Diffusion: µ + in metals (compare H+); Mu in nonmetals (compare H).

Probing Magnetism: unequaled sensitivity

- Local fields: electronic structure; ordering

- Dynamics: electronic, nuclear spins

Probing Superconductivity: (esp. HTcSC)

- Coexistence of SC & Magnetism

- Magnetic Penetration Depth λ

- Coherence Length ξ

Species of µ+ States

“Diamagnetic” µ+ precesses at ωµ/2π [MHz] = 135.5•B [T] e.g. “free” µ+, Mu- = µ+e-e-, molecular ions (N2µ+), compounds (MuHO)

Muonium (Mu = µ+e-): singlet state looks depolarized; triplet state precesses at ωMu ≈ 103•ωµ split by HF interaction. In solids, Mu may be vacuum-like [spherical, with AHF ≈ 4463 MHz], shallow [AHF tiny!] or in between [usually with lattice symmetry imposed on e- wavefunction].

Radical (e.g. C6H6Mu•): complicated frequency spectrum due to “sharing” e- with entire molecule; multiple HF interactions will effectively depolarize the spin system unless a strong external magnetic field is applied.

Typical Timescales

for GAS at STP

µ+ & Mu in Gases

µ+(several MeV)

IONIZ ATION (Bethe-Bloch)

~30 keV

e-

e-

e- CAPTU RE/LOSS

COMPE TITION

~200 eV

Mu excited states

(metastable 2s)

Mu*

g

e- CAPTURE

Charge Exchange

ELASTIC & INELASTIC SCATTERING

Thermal µ+ or molecular ion

Thermal Mu

“HOT”

RXNS

(?)

Diamagnetic Molecules

SPIN EXCHANGE

Depolarized

Mu

CHEMICAL

REACTIONS

µ+ Precession

Mu Precession

Thermal e- CAPTURE is possible in the GAS phase only via donor atoms whose ionization potential is ‹ 13.5 eV.

~ 10 ns

~ 0.1 ns

~ 10 ns

~ µs

Typical

Timescales

µ+ & Mu in Metals

~ 10 ps

~ 1 ps (?)

µ+(several MeV)

IONIZ ATION (Bethe-Bloch)

~30 keV

e-

e-

Plasmons

Excitons

Phonons

Magnons

Interstitial µ+

Substitutional µ+

“ T H E R M A L I Z A T I O N “

“Electronic damage”

is rapidly healed.

µ+ Precession

repelled by positive ions

(very rare)

In each case, the muon is screened by conduction electrons.

~ µs

All the interesting physics . . .

OR

Radicals

Typical

Timescales

µ+(several MeV)

IONIZ ATION (Bethe-Bloch)

~ 10 keV

e-

e-

e- CAPTU RE/LOSS

COMPE TITION

~ 20 eV

Thermal µ+ or Molecular Ion

THERMAL e- CAPTURE

Charge Exchange

E L A S T I C & I N E L A S T I C S C A T T E R I N G

Thermal Mu 

Diamagnetic Molecules

SPIN EXCHANGE

Depolarized

Mu

CHEMICAL

REACTIONS

µ+ Precession

Mu Precession

~ 10 ps

~ 1 ps (?)

~ 1 ps (?)

~ µs

e-

e-

PHONONS

EXCITONS

 Thermal Mu sometimes forms in “shallow” states.

These may be long-lived if electron mobility is high.

~ ps to ~ µs

µ+ & Mu in Liquids and Solid Insulators & Semiconductors

OR

Radicals

“HOT”

RXNS

(?)

Mu

Dia (Slow)

Electron

mobility

Orientational

ordering

Dia (Fast)

Shows strong T-dependence mirroring that of

electron mobility,

which in turn reflects orientational ordering.

A separate diamagnetic component “relaxes” too fast for any dipolar mechanism

 “Delayed” Mu Formation?

Muonium Formation
in Solid Nitrogen

Slava Storchak ~ 1990 (TRIUMF)

Muonium in Superfluid Helium:

Solid squares:

pure 4He @ 1.7 K

Solid circles:

pure 4He @ 0.7 K

Open circles: 4He +	0.2% 3He @ 0.5 K

Line: best fit to viscous drag on µ+ “snowball” moving to e- “bubble”:

‹Reµ› ≈ 400 nm; width of distribution is of comparable size.

Dima Eshchenko

thesis ~1990

b phase (high T): free rotors

a phase (low T): orient. ordered

Fast DMF in a phase from free,

delocalized radiolysis electrons.

Slow DMF in b phase from

electronic polarons  dephasing

reduces Mu signal amplitude in TF:

Positive electric field E

(i.e. in the muon beam direction) inhibits DMF; negative E

(in the opposite direction) enhances DMF.

DMF in Solid Nitrogen:

Dia

Mu

Slava Storchak -1990's (TRIUMF)

DMF in solid Neon:
B -dependence

T = 22K

E = 0

Spatial distribution of “initial”

e- positions relative to µ+ leads

to distribution of delay times

for start of Mu precession

 “dephasing” reduces amplitude

and shifts phase of Mu signal.

DMF in Solid Neon

Circles: Diamagnetic fraction

Triangles: Muonium fraction

Filled: directly observed Mu

Open: corrected for dephasing

Squares: Apparent initial phase

of Mu precession shifts due to

finite formation time as electrons

arrive from various distances.

(For large E, only nearby e- survive.)

T = 22 K, TF = 44 G

‹Reµ› ≈ 50 nm

Localization in Liquid Neon:

Sum of Fast & Slow DMF

Delocalized fraction = Fast DMF Mu fraction

External electric field E prevents e- localization in bubbles.

Crosses: Sum of all signals

Circles: Stable Diamagnetic signal

Triangles: Muonium signal (mainly Fast DMF)

Stars: Slow DMF fraction

Squares: Sum of Fast & Slow DMF

lS = Slow DMF rate

(from e- in bubbles)

T =24.9 K, TF=51.6G

DMF in Liquid Neon:

‹Reµ› ≈ 30 nm

Storchak & Eshchenko - 1990's (TRIUMF)

Radiolysis:

“Knockon” electrons cause

secondary ionization.

Ionization

electrons

stop ~ Re

from + ions.

Possible exceptions:

solids with extremely

high electron mobilities.

Z +

µ+ stops

Reμ

(Mu)

e- loss

Distribution of “initial” e- positions

may overlap position where μ+ stops.

e- pickup

(Actual path is probably less straight, but still “forward”.)

A closer look at the final charge exchange:

µ-e-E geometry:

Trajectories for E = 1 kV/cm:

e- capture boundaries

for E = 1/16 to 16 kV/cm:

e- constantly loses momentum

to the medium, and so follows

“lines of E” at constant speed.

Viscous flow model:

“Looks like a nail to me.”

Weaker E-dependence implies

mean distance of e- from µ+

is shorter than in Ne.

Mu phase shift is smaller

than in solid Ne, indicating

faster DMF, as expected

for closer electrons.

Very small fraction of

“X” (slow DMF).

DMF in Liquid Argon

‹Reµ› ≈ 30 nm

DMF in Solid & Liquid Argon

Filled circles: 78 K (solid) Open squares: 84 K (liquid)

Distribution of initial e- locations is similar in both phases.

Mean distance from µ+ is shorter than in Ne.

Track electrons more plentiful than in Ne; all delocalized & fast!

No

Slow

DMF.

Diamagnetic

‹Reµ› ≈ 30 nm

Storchak &

Eshchenko

(1990's)

DMF in “Ordinary” Liquids:

So far (to my knowledge) NO DATA on liquid water.

Theoretical simulations by Siebbeles et al. predict DMF with

‹Reµ› ~ 60 -150 nm for n-hexane.

Data from ISIS on n-hexane show no effect of E ≤ 15 kV/cm.

David Walker steadfastly maintains that DMF does not occur

in “real” liquids.

Note different horizontal & vertical scales!

‹Reµ› ≈ 15 nm

‹Reµ› ≈ 40 nm

‹Reµ› ≈ 20 nm

‹Reµ› ≈ 60 nm

“Ordinary” Solids: Insulators & Semiconductors

xtal

fused

Sodium Chloride:

Quartz:

294 K

141 K

300 K

Early tests with fused quartz did not use high enough electric fields to see effect.

‹Reµ› ~ 15 nm

‹Reµ› ~ 20 nm

“Ordinary” Solids: INSULATORS

At a temperature where Mu* (BC) state starts to ionize thermally, apply an electric field E. This inhibits DMF and preserves a larger diamagnetic fraction; but the effect is small.

‹Reµ› ~ 40 nm

Elemental Semiconductors: Silicon

3-frequency precession at B ≈100 G

Zero electric field

Positive electric field applied

AHF ≈ 140 kHz

BHF ≈ 10 G

Formation of shallow state

inhibited by electric field.

‹Reµ› ~ 40 nm

II-VI Semiconductors: CdS & Shallow States

‹Reµ› ~ 40 nm

Extreme sensitivity of DMF (Mu* BC state) to electric field

 ‹Reµ› ≥ 60 nm, suggesting E-ionization of extremely shallow precursor.

Enhanced effect at low temperature & low magnetic field.

III-V Semiconductors: GaAs & Weakly Bound States

DMF (Mu0BC state) is enhanced by strong magnetic field as

radiolysis electrons “freeze out” into localized Landau levels.

Magnetic Freezeout of Electrons into Muonium in GaAs

Expanded view

of Dia fraction

Mu0BC

Dia

Slava Storchak

2003

High temperature:

DMF inhibited by E

‹Reµ› ~ 10 nm

Low temperature:

DMF much slower and

fraction subject to DMF decreased by modest E

suggesting conversion of

Mu- to Mu by E field.

DMF (?) in Sapphire (Al2O3):

This could explain why no Mu is seen in many oxide insulators (e.g. Calcite); perhaps Mu- is often the preferred state in such materials.

CONCLUSIONS

In gases, Mu is always formed in a collision between the µ+ and an atom or molecule from which it “steals” an electron. This may take place “promptly” at high energy (~keV) or “delayed” (thermally, with impurity species). Ionization products from the muon's track do not play a role. Radicals may be formed.

In metals, the µ+ is screened by conduction electrons; there is no Mu in the usual sense of a freely evolving 2-spin system. Ionization etc. “heals” before it can affect the µSR signal.

In semiconductors & insulators (including most liquids) Mu is most likely to be formed by capture of radiolysis electrons by the µ+ that created them in its ionization track (DMF). Some “prompt” Mu formation may also occur.

The electronic structure of Mu in matter is quite varied. In gases, liquids and insulators it resembles the free Mu atom in vacuum, but in semiconductors there are Mu states ranging from isotropic, deeply bound Mu through intermediate bond-centred Mu* to true “shallow” states in which the electron is entirely “out in the lattice”.

Mu- may form in many oxides like calcite; this would explain the absence of Mu.

Invited talk at Seaborg Symposium in honor of Don Fleming – ACS meeting in Anaheim, CA, USA, 30 March 2004.

I will actually say both more and less than what you see here.

This is the story we tell Philistines who ask, “Why on earth do you want to study muons? What are they good for?” As it happens, they are excellent magnetic probes in solids and unprecedented as a light isotope of hydrogen; but the real reason we (or I, at least) study muons in matter is an obsessive fascination with the idea of being able to “see” what they do!

I believe this obsession has helped me look past the standard themes and notice the things muons do (as they slow down and stop in matter) that grown-in hydrogen or other magnetic resonance probes do not.

In metals, the ubiquitous diamagnetic state consists of a positive muon screened by conduction electrons.

In insulators & semiconductors there is a rich variety of muonium states, most of which have the symmetry of the lattice imposed on the electronic wavefunction; in truly “shallow” states the electron is almost entirely “out in the lattice” and will have the same effective mass etc. as free electrons in the conduction band.

In gases and liquids, both diamagnetic and muonium states are observed but there are also paramagnetic molecules incorporating the muon and its “extra” (unpaired) electron; such radicals have many interesting properties of their own.

The best understood cases are gases (where the muon stopping processes may be very complicated but are sufficiently spread out in time and isolated in space that they can usually be distinguished from each other) and metals (where the actual disturances of the medium are probably even more complex but “heal” so fast that none of them have any measurable effect).

All condensed nonmetals have these common features; they differ only in the mobility of free carriers and muons, the sensitivity to impurities and/or disorder, and the density of states for assorted elementary excitations – which can vary by orders of magnitude!

In the early days of Muonium Chemistry (1970's) there was a protracted debate over whether the “residual” diamagnetic fraction in liquids and solids was the result of thermal or epithermal (“hot atom”) reactions. The answer was obvious from the outset: “Yes ! ”

Delayed Muonium Formation (DMF) was first observed in solid nitrogen after it was noticed that the muonium formation probability varied dramatically with temperature and was strongly correlated with the mobility of free electrons, suggesting an interaction with the only free electrons around: those liberated in the muon's ionization track!

In liquid helium the electron forms a huge bubble and the muon a “snowball” of He atoms; the latter is the smaller object, so in this case (only!) the muon moves to the electron rather than the other way around. Moreover, the snowball moves fast enough to effect DMF only in the superfluid state where there is no viscosity until the snowball approaches the speed of sound in l-He. The muon stops an extraordinarily large distance (nearly half a micron!) from its last radiolysis electron, mainly due to the very high ionization potential of He.

The muonium signal amplitude and apparent initial phase both vary strongly with magnetic field in the high temperature b phase, indicating DMF on a time scale comparable to the muonium precession period in a few gauss. In the low temperature a phase (where electron mobility is much higher) the DMF is apparently too fast for such dephasing to be effective; but the anisotropic effect of an electric field in the alpha phase shows that the

muon stops on average well “downstream” of its last radiolysis electron. A more refined model is needed.

Solid neon shows effects similar to those seen in solid β-nitrogen.

The finite time of DMF in s-Ne leads to a large phase shift of the Mu precession signal; as DMF is suppressed by an electric field, only the electrons closest to the muon participate; they arrive much earlier and thus the phase shift is also suppressed by the electric field.

The fraction of electrons localized in bubbles is also larger in stronger electric fields, suggesting a role of E in the localization process.

In liquid neon, free electrons tend to form bubbles, after which their motion is so slow that DMF from such localized electrons is manifest only in an otherwise inexplicable relaxation rate of the diamagnetic precession signal; this is referred to as “slow DMF”. Electrons which remain delocalized long enough to reach the muon in time to contribute to the Mu precession signal constitute “fast DMF”. Both types of electrons are pulled away from the muon by an applied electric field E, incleasing the diamagnetic fraction at the expense of the DMF fraction. Again only the nearby electrons contribute for high E fields, and these reach the muon sooner, giving a higher λS.

The “charge exchange cycle” is well documented in the gas phase; its importance in the condensed phases (liquids and solids) is more controversial.

One thing we do know is that in every condensed medium where DMF has bee observed (by virtue of its electric field dependence) the effect of E is anisotropic: more muons stop “downstream” of their last radiolysis electron(s) than “upstream” -- though not always by very much!

In a “viscous flow” model the radiolysis electron moves at approximately the speed of sound and follows electric field “lines” in the direction of the vector sum of the muon's Coulomb field and the constant external field. This makes it easy to separate all space into a region where the electron will escape capture and another region from which it will inevitably be catured in DMF. Of course, this model corresponds to zero temperature; at finite T the capture region will be reduced further to the region where the binding energy to the muon is greater than kT. (Unlike for zero applied field, this region will not be spherical.)

DMF in liquid argon resembles that in solid neon but with a shorter mean muon-electron separation. There is little if any formation of localized electron states.

There is not much difference between the liquid and solid phases of condensed argon – at last an example that agrees with naïve expectations!

For some reason the only liquids studied with an applied E field (until very recently) were “cryoliquids”. Water in particular is a surprising omission.

Early results on “ordinary” insulators and semiconductors showed remarkable similarities, differing mainly y the mean muon-electron separation and the extent of the electron spatial distribution (which apparently overlaps the muon position in the case of GaAs).

The mean electron-muon separation is very short in these insulators, but an effect can still be seen because such high electric fields can be applied without breakdown. There is a dramatic difference between crystalline and fused quartz.

Only a small E field could be applied to this Si sample without breakdown; nevertheless it was possible to detect the usual effect on DMF. It is not yet certain whether a shallow precursor is involved in this case, but there is some unusual temperature dependence in the diamagnetic signal at low magnetic field.

Even a low magnetic field produces the “high field limit” for this extremely weak Mu hyperfine coupling. A modest electric field completely suppresses formation of the shallow Mu state.

The shallow Mu state in CdS is unusually sensitive to aplied electric field. This is partly because DMF involves unusually distant radiolysis electrons and partly due to “electrostatic ionization” of the weakly bound Mu state. The latter effect is of course isotropic at high E.

As in CdS, there are two effects of applied E on DMF in GaAs: an isotropic electrostatic ionization of a very shallow precursor state at high E, and a highly anisotropic suppression of the initial capture of radiolysis electrons at low E. Unlike in CdS, the shallow state is not stable, but quickly de-excites into the bond-centred Mu* state, whose amplitude mirrors that of the diamagnetic component as a function of E.

Application of very strong magnetic fields localizes radiolysis electrons in Landau orbitals from which they are more easily captured into the unstable weakly bound states that eventually populate the bond-centered Mu0BC state.

Sapphire is mysterious. At low T, DMF is slow enough to be observed directly as a “relaxation” of the diamagnetic signal. The fraction of muons participating in this process is reduced by a small positive electric field, without a concomitant increase in the stable diamagnetic component, suggesting that a positive E enhances the initial Mu fraction. This makes no sense unless the muon stops preferentially behind its own radiolysis electrons (patently absurd) or the effect of E is to suppress the formation of diamagnetic Mu- in favour of neutral Mu. Diamagnetic RF resonance experiments on n-type Si have revealed the dominance of a Mu- state at high T, so this is not unprecedented; but it is unknown why Mu- should be the preferred state in pure sapphire at low T.

So far (to my knowledge) no Mu signals have been detected in the complicated oxide insulators where µSR is frequently applied to studies of exotic magnetism. This is peculiar, in view of the clear evidence for DMF in almost all simple oxide insulators like quartz and sapphire, and in the ionic alkali halides.

Muon Stopping & Muonium States

Tutorial lecture at Student Workshop preceding muSR2002

Jess Brewer

2002-05-23T08:51:14

Jess H. Brewer

Jess H. Brewer