Defects and Diffusion in Solids: or The Drunken Sailor’s Walk and The Atomic Slide Puzzle

Phys 188: Freshman SeminarNovember 15, 2007

The Atomic Slide Puzzle:

Prof. Gary S. Collins

10 steps 60 steps

Random walk on a square grid

Chance of getting back to the lamp post in d-dimensions

(no muggers)

Diffusion is random motion and mixing of atoms or molecules

Gases: billiard-ball collisionsperfume aroma spreads in roomcarbon monoxide poisoning

Liquids: atoms “swim”sugar cube dissolves in cup of coffeeink drop in still glass of wateragitated motion of small particles in water (“Brownian motion”)

Solids: atoms normally frozen; diffusion mechanisms complicatedNiMH battery in cell phones; hydrogen diffuses in/out of nickel metaloxidation of metals (rust)phase separation (hardening of quenched steel)doping of semiconductors with electrically active impurities

Early observations of diffusion

Robert Brown, English Botanist (1773-1858)

In 1827, observed under a microscope that pollen grains in water were in a constant state of agitation. Predates idea that matter is composed of atoms.

He later proved motion was inanimate by observing similar effects in water trapped inside quartz inclusions for millions of years old.

“Brownian motion” not studied much in the 19th century; no theory

He also:was first to observe cells in plantshelped map Tasmania

Brownian motion of fat droplets in milk

Film by David Walker, amateur microscopist, England

On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat

“In this paper it will be shown that, according to the molecular-kinetic theory of heat, bodies of a microscopically visible size suspended in liquids must, as a result of thermal molecular motions, perform motions of such magnitudes that they can be easily observed with a microscope. It is possible that the motions to be discussed here are identical with so-called Brownian molecular motion; however, the data available to me on the latter are so imprecise that I could not form a judgment on the question….

Albert Einstein, German physicist (1879-1955) published atheory of “Brownian Motion” in 1905

Einstein at 26

Marian von Smoluchowski, Polish physicist (1872-1917) independently published his theory of diffusion also in 1905.

Jean Perrin, French physicist (1870-1942)

Stimulated by Einstein’s suggestion that agitated motion could be observed easily under a microscope, he studied Brownian motion in detail.

The Atoms, 1909 book

Jean Perrin, French physicist (1870-1942)

Perrin, The Atoms (pp. 109-110)

“Einstein and Smoluchowski have defined the activity of the Brownian movement in the same way. Previously we had been obliged to determine the "mean velocity of agitation" by following as nearly as possible the path of a grain. Values so obtained were always a few microns per second for grains of the order of a micron.

But such evaluations of the activity are absolutely wrong. The trajectories are confused and complicated so often and so rapidly that it is impossible to follow them; ... Similarly, the apparent mean speed of a grain during a given time varies in the wildest way in magnitude and direction, and does not tend to a limit as the time taken for an observation decreases, ...”

“…the Brownian movement [is] completely irregular in all directions”

“It can be proved that the mean displacement of a grain is doubled when the time is increased fourfold; it becomes tenfold when the time is increased a hundredfold and so on. More precisely, it is proved that the mean square x2 of the horizontal displacement during the time t increases in proportion to that time.”

“The same result holds for half this square or the mean square x2 of the projection of the horizontal displacement along an arbitrary horizontal axis. In other words, for a given kind of grain (in a given fluid) the quotient x2/t is constant. ... this quotient characterizes the activity of the Brownian movement for any particular grain.”

txD

2

≅ Dtxrms =

unlike vtxrms =“the diffusivity”

Perrin, The Atoms (pp. 110-111)

Hot blobs of lead

X-ray movie, Paul Preuss, LBL

Leapin' blobs o' lead!

Nanoparticles of molten lead inside solid aluminum move just as described by Albert Einstein in his classic 1905 paper on Brownian motion. The much larger faceted inclusion at upper right is also molten, but is confined by the solid matrix.

Paul Preuss, LBL

Simulation of diffusion in gases and liquids

(1) Mixing of Ne and Ar atoms

(Atomic Microscope software)

Diffusion smoothes out concentration differences

How atoms move in solids Vacancy mechanism

(Metals, most elements) (Small atoms in solids, hydrogen, lithium, oxygen…)

Interstitial mechanism

More diffusion mechanisms

Vacancy Interstitial Interstitialcy Kick-out

Frank-Turnbull Direct exchange

Condensation of gases into liquids and solids

(1) Kr and Ar mixture (demixing)(2) Na and Cl mixture (ordering)

Diffusion in solid Ar with a few impurities

(Atomic Microscope software)

Macroscopic study of diffusion: concentration profiles

−=

Dtx

DtA

txc4

exp2

),(2

π

Radioactive tracer atoms—thin layer or sandwich

“Drunken sailor’s walk”

Concentration of tracer atoms c(x) has Gaussian profile :

Diffusivity D is property ofmedium, diffusing species,temperature, pressure, …

root-mean-square diffusion length:

DtR 2=

Example: Diffusion of radioactive 59Fe atoms in Fe3Al

H. Mehrer, Materials Transactions, JIM, 37, 1259 (1996).

)/exp(0 TkQDD B−=

−=

Dtx

DtA

c4

exp2

2

πD depends on temperature T and

activation energy Q

Dtxxc /~)](log[ 2−

Migration barrier for atom movement in solids

Vacancy diffusion dominant mechanism in most solids

Temperature Dependences of Diffusivity give Q

Inverse

Slopes equal activation energies Q

Simulation of diffusion in solids

(1) solidification of two elements2d: separation into two phases (Ar and Kr)3d: ordering into compound (Na and Cl)

(2) diffusion 2d: Ar with a few atoms of Kr and Ne

(maybe see vacancies)

(Atomic Microscope software)

Vacancy motion mixes up atoms

Ordering of light and dark blue “atoms”

15-Puzzle, software by Jerry Brons and Harry Broeders

The meaning of D

wfD 261 l=

Microscopic model for diffusivity:

= jump distance

w= jump frequency (inverse of mean residence time)

f= correlation coefficient

l

Case Study 1: the surface atomic slide puzzle

Science News Online Week of Feb. 24, 2001; Vol. 159, No. 8

Seeming sedate, some solid surfaces seethe (Peter Weiss)

Scientists have long thought that the surface atoms of a solid fit together snugly and stably, like floor tiles. Now they'refinding that surface atoms of copper--and maybe other materials --roam randomly and widely within their orderly grid. A hole's zigzags (blue line) shuffle indium (yellow) and copper (red) atoms.

The findings, from independent teams in the Netherlands and New Mexico, may shed light on how materials form layers on other materials. An important application of such layering is the fabrication of integrated circuits and micromachines, such as tiny pressure and motion sensors.

The results also warn that future nanometer-scale components may suffer from the newfound atomic rearrangements, says Raoul van Gastel of Leiden University in the Netherlands. He and his colleagues stumbled upon the surface commotion while studying how indium impurities affect copper-crystal growth. In the Feb. 19 Physical Review Letters, the Dutch team reports using a scanning tunneling microscope to observe the positions of indium atoms within a copper surface. The scientists scanned the same 400-atom-by-400-atom region as often as every 2 seconds.

They noted that a patch of surface would remain unchanged for many scans. Then, suddenly, many indium atoms would move within the next few images. The movements suggested that atoms were hopping up and moving across the surface before dropping back in. Or, because the surfaces were warm enough, agitated indium atoms were moving into holes left in the surface layer when copper atoms occasionally hopped off the surface's edge, van Gastel says.

"One of things that surprised us was the indium atoms made so-called long jumps" of up to five grid spaces between images, he recalls. The jumps' timing and lengths, together with other evidence that indium atoms can't easily enter the surface layer, pointed to leftover holes.

A hole's zigzags (blue line) shuffle indium (yellow) and copper (red) atoms.van Gastel et al./PRL

STM movie of indium atoms on Cu(001) surface

Each yellow protrusion in the movie is the STM-image of a single, embedded indium atom. The vertical scale is grossly exaggerated; the indiums "stick out" by only 40 pm. The fine corrugation on the red terrace is the atomic structure of the Cu(001) substrate. This movie was recorded at a low speed of 20 s/image. However, even in our fastest movies (up to 18 images/s) the long jumps of the embedded indium atoms appeared completely instantaneous to the STM. Typical jump lengths are a few lattice spacings.

Science News Online Week of Feb. 24, 2001; Vol. 159, No. 8

(continued)

When an atom escapes the surface's edge, the researchers propose, a neighboring atom fills the vacancy, causing the hole to shift by one grid space. That process repeats some 100 million times per second, shoving the vacancy around.

"You know the [toy] slide puzzles with 15 numbers and one missing? It's the perfect analogy," says Brian S. Swartzentruber of Sandia National Laboratories in Albuquerque. "To get numbers in the right places, you move the vacancy around."

According to the Dutch model, a vacancy pops up every half minute or so in any particular surface patch. Each vacancy's rapid zigzagging affects many atoms, some of them repeatedly. These multiply disturbed atoms appear to take long leaps. Nearly all copper atoms in the surface layer also shuffle around, but they're indistinguishable from each other, so their motions can't be tracked.

Just a few fast-moving vacancies could account for all the motion. At room temperature, "one in 6 billion atoms in the surface is missing--that's like one person in the entire population of the Earth," van Gastel says. Overall, the surface motion dislodges any given atom about once every 30 or 40 seconds, he says.

Using a different method, Swartzentruber says he and his coworkers at Sandia came to "basically . . . the same conclusion" as the Dutch group did. In an unpublished study, the Sandia team used a scanning tunneling microscope tip to track one palladium atom at a time within a copper surface.

Surface scientists "are very excited" about the findings, comments Karina Morgenstern of the Free University of Berlin. "We should go back . . . and see if some unexplained things might be due to this phenomenon," she says. For instance, chemists make wide use of metal surfaces as catalysts. Morgenstern speculates that some catalytic behavior may be influenced by the newfound surface motion.

References:van Gastel, R., et al. 2001. Nothing moves a surface: Vacancy mediated surface diffusion. Physical Review Letters86(Feb. 19):1562.

Simulation of atom movement on Cu surface

From accurate measurements of the statistics of the observed jump lengths and of the waiting times between successive jumps, Raoul van Gastel deduced that the mobility of the indium atoms is caused by the rapid, two-

dimensional diffusion of a very low density of monatomic vacancies (missing copper atoms), through the first copper layer. Due to their ultrahigh diffusion rate, these vacancies remain "invisible" for the STM at room temperature.

Case Study 2: atom jump frequencies inside solids

Crystal structure of In3La

X

Y

Z

L12 (Cu3Au) structure

Motion on sublattice of “yellow” sites In3La

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

One unit cell Four unit cells

+1/2 h

-5/2 h

-3/2 h

+5/2 h

-1/2 h

+3/2 h

2/5±

2/3±

2/1±

2/5

1ω

2ω 3ω

Spin 5/2 nucleus, US football shape, in electric field gradient

Positive ion

Energy of orientation of quadrupole moment in electric field gradient

“EFG”

Nuclear quadrupole interaction

+1/2 h

-5/2 h

-3/2 h

+5/2 h

-1/2 h

+3/2 h

2/5±

2/3±

2/1±

2/5

1ω

2ω 3ω

t0 100 ns

Spin precessions in time domain observed by PAC

Spin 5/2 nucleus, US football shape, in electric field gradient

Nuclear quadrupole interaction

Perturbed angular correlation of gamma rays (PAC)

1 1 1

Cd

ec

247 keV

173 keV

120 ns3/2

1 1 1

In (4.0 d)

1

5/2

/2

2 3

1

5/2γ1

γ2

t∆ωP , =0)(γ2

t

γ1

Anisotropy in emission of 2nd γ-ray

Long-lived intermediate state

+1/2 h

-5/2 h

-3/2 h

+5/2 h

-1/2 h

+3/2 h

2/5±

2/3±

2/1±

2/5

1ω

2ω 3ω

t0 100 ns

Spin precessions detected in time domain

Spin 5/2 nucleus, US football shape, in electric field gradient

Quadrupole interaction (spin 5/2)

ttttG 0355

03510

03513

51

2 3cos2coscos)( ωωω +++=

Static PAC perturbation function for axial symmetry

PAC spectrometer

RoutingLogic

Latch

4

valid

MCA

startstop

dela

y

TAC

fast slow

AMP

1-γ 2-γSCA SCA

(x4)

1 2 3CFD

1234

CFD

delay

strobereset

ADC

1

3

4 2

fast slow

AMP

1-γ 2-γSCA SCA

(x4)

1 2 3CFD

1234

CFD

detectors

4 BaF2 scintillation detectors at 90o anglesTurbo-pumped oven to 1200 oC

Nij

channels10240

0 delay time 300 ns

counts

(cadmium metal)

WS

U

PAC hot spots (condensed matter physics)

Sample preparation

Melt high-purity metals with 111In carrier-free activity in arc furnace: ~100 mg spheres. High probe dilution: 1011 atoms of 111In (mole fraction ~10 ppb).

+1/2 h

-5/2 h

-3/2 h

+5/2 h

-1/2 h

+3/2 h

2/5±

2/3±

2/1±

2/5

1ω

2ω 3ω

t0 100 ns

Jumping to sites with different orientations of the EFG leads to decoherence of precessions

time

Spin 5/2 nucleus, US football shape in electric field gradient

Measuring tracer jump frequencies by relaxation of nuclear quadrupole interaction

Nuclear quadrupole interaction

+1/2 h

-5/2 h

-3/2 h

+5/2 h

-1/2 h

+3/2 h

2/5±

2/3±

2/1±

2/5

1ω

2ω 3ω

t0 100 ns

Spin 5/2 nucleus “football” shaped, in electric field gradient

2/5±

2/3±

2/1±

2/5

1ω

2ω 3ωw

Decoherence of spin precessions shows up as

“damping”. Fit jump frequency w.

t0 100 ns

1/w

relaxation

Cd tracer atoms jumping on indium sublattice in LaIn3

0 100 200 3000

1156oC

G2(t)

t (ns)

0

1261oC

G2(t)

0

1340oC

G2(t)

0

1419oC

G2(t)

0

1

629oCG

2(t)

Three relaxation regimes:

419, 629 C: fast fluctuations, w>ωQ

340 C: crossover, w~ωQ

156, 261 C: slow fluctuations, w<ωQ

(More In-rich phase boundary composition)

Zacate, Favrot and Collins, Phys. Rev. Lett. 92, 225901 (2004)

)()exp()( 22 tGwttG static−≅

)3

200exp()(

2

2 tw

tG Qω−≅

• Two samples with slightly different compositions (slightly two-phase fields):

x1: more In-rich (A)x2: less In-rich (B)

• w is ~10-100 times greater at more In-rich boundary composition

• Q1= 0.53(1) eVQ2= 0.81(1) eV10 15 20 25

105

106

107

108

109

4005006008001200

2x

1x

T [K]

w [

Hz]

1/kBT [eV-1]

Zacate, Favrot and Collins, Phys. Rev. Lett. 92, 225901 (2004)

Cd tracer motion in LaIn3: fitted jump frequencies

Summary

Drunken sailor’s walk

A little history of atom movement or diffusionBrown, Einstein, Smoluchowski, Perrin…why it matters

How atoms move in solids (…via vacancies)

SimulationsIntermixing of gasesSolidification (phase separation or ordering)Impurity and vacancy motion in solidsThe 15-puzzle and atomic slide puzzle

Looked at macroscopic and microscopic methods for studying diffusionMeasure concentration profilesSee atoms jump using STM (surface diffusion; atomic slide puzzle)See atoms jump via nuclear relaxation (volume diffusion; In3La)

End

Professor Gary S. Collins

Webster 554

335-1354

mailto:collins@wsu.edu

If this work interests you, contact me! For more info, go to http://defects.physics.wsu.edu/