Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 59

Gibbs free energy of binding for vacancy-solute complex

Gibbs free energy of vacancy formation

Gibbs free energy of vacancy migration

Gibbs free energy of formation of an interstitial solute from

a substitutional one

binding enthalpy of vacancy-solute complex

enthalpy of vacancy formation

enthalpy of vacancy migration

enthalpy of vacancy-solute exchange

∆H activation enthalpy of self-diﬀusion

∆H

activation enthalpy of substitutional impurity diﬀusion

∆H

activation enthalpy of hydrogen diﬀusion

∆H

activation enthalpy of deuterium diﬀusion

∆H

activation enthalpy of tritium diﬀusion

j diﬀusion ﬂux (number of particles per second and unit

area)

components of diﬀusion ﬂux

l jump length of an atomic jump

∆Q ∆H

− ∆H

R gas constant

jump vector of an atom

R total displacement vector of an atom





mean square displacement of atoms





mean square displacement of atoms for a true random walk

entropy of vacancy formation

entropy of vacancy migration

∆S diﬀusion entropy

t time

T temperature

x position

position of Matano plane

mole fractions of A and B in a binary A-B alloy

Z coordination number

direction cosine of diﬀusion ﬂux with axis i =1, 2, 3

coeﬃcient of activity of species i

Γ jump rate of an atom in a certain direction

Θ angle between diﬀusion ﬂux and crystal axis

chemical potential of species i

attempt frequency (≈ Debye frequency)

τ =1/ΓZ mean residence time of an atom in a coordination lattice

with coordination number Z

−1

mean jump rate of an atom in any direction on a lattice

Φ thermodynamic factor

ω jump rate of vacancy in the matrix

60 Helmut Mehrer

jump rate for rotation of vacancy-solute complex in a dilute

(fcc) alloy

jump rate of vacancy-solute exchange

dissociation jump rate of vacancy-solute complex

association jump rate of vacancy-solute complex

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2 The Elementary Diﬀusion Step

in Metals Studied by the Interference

of Gamma-Rays, X-Rays and Neutrons

Gero Vogl and Bogdan Sepiol

2.1 Introduction

The elementary diﬀusion jump in solids can be traced down with a bundle

of methods from solid state physics. In this chapter ﬁrst the possibilities for

studying diﬀusion with the quasielastic version of M¨oßbauer spectroscopy and

neutron scattering are discussed. Since the ﬁrst edition of this book two new

methods were adopted for studying the elementary diﬀusion jump. Whereas

as quasielastic methods M¨oßbauer spectroscopy and neutron scattering both

work in the energy domain, the new methods work in the time domain.

The new methods are nuclear resonance scattering (NRS) of synchrotron

radiation, i. e. a daughter of M¨oßbauer spectroscopy using synchrotron ra-

diation, and neutron spin echo spectroscopy. Nuclear resonance scattering

was adopted to diﬀusion studies in 1996, i.e. shortly before the appearance

of the ﬁrst edition and was shortly mentioned there. Neutron spin echo had

not been applied to diﬀusion in crystalline systems before 2001. The spec-

trum of systems studied has considerably widened, ﬁrstly by the addition of

these methods, secondly due to more experiments performed with quasielas-

tic neutron scattering at improved spectrometers. This chapter is therefore

considerably renewed compared to the ﬁrst edition.

Our contribution will report on vacancy-induced diﬀusion in metals and

alloys. The sensitivity of the methods cited in the preceding paragraph to the

elementary step of diﬀusion on a microscopic scale is their outstanding virtue.

Together with nuclear magnetic resonance measurements (see Chaps. 9 and

10) they are the representatives of diﬀusion methods provided by nuclear

solid state physics. These methods are “microscopic” in the sense that they

directly study jump vectors and jump rates (cf. Fig. 2.1). In addition we can

calculate the “macroscopic” diﬀusion coeﬃcients from these quantities and

compare them with literature data obtained by tracer diﬀusion measurements

(Chap. 1 and [1, 2]), but this is not the main virtue of microscopic methods.

Of course there are other mechanisms proposed for diﬀusion than va-

cancy mechanisms (see Chap. 1). Some of them will be shortly discussed for

the cases of self-diﬀusion in titanium but – just as it turns out in these exam-

ples – the most important process of self-diﬀusion in metals is the vacancy

mechanism [3]. We will only discuss the basic results, more sophisticated ar-

66 Gero Vogl and Bogdan Sepiol

Fig. 2.1. Microscopic methods study

the frequency and the displace-

ment vectors of elementary diﬀusion

jumps.

gumentation is found in the cited literature. For a general review on diﬀusion

in solids see, e.g. [4, 5] and Chap. 1 of this book.

Sect. 2.2 introduces to the formalism, which relates jump diﬀusion of

atoms to the experimental methods, starting with diﬀusion on Bravais lat-

tices and ending with diﬀusion on the non-Bravais lattices of ordered phases.

Sect. 2.3 discusses experimental results, Sect. 2.4 sums up and concludes.

2.2 Self-Correlation Function and Quasielastic Methods

In this chapter we use the “scattering terminology” for all methods. Thus for

M¨oßbauer spectroscopy we use Q instead of k for the wave vector and the

scattering function S(Q,ω) instead of the absorption cross section σ

(k,ω).

The advantage of directly comparable results for all methods outweighs the

inconvenience of an unusual terminology for M¨oßbauer spectroscopy. As the

formalism is discussed in detail in Chaps. 3, 18 and, e. g., in [6] we will only

give a short introduction to the most important equations.

The simple-minded picture of atoms jumping on the lattice can be pre-

cisely formulated applying the Van Hove correlation function [7]. Displace-

ments of atoms in space and time are described by the self-correlation function

(r,t). This is the probability density to ﬁnd an atom displaced by a vec-

tor r within a time interval t, averaged over all possible starting positions

Van Hove [7] introduced the formalism of the self-correlation function for the

scattering of thermal neutrons, Chudley and Elliott [8] used it for neutron

Note that here r denotes a displacement (in our case the jump vector) and not

a position.

2 Diﬀusion Studied by Interference of γ-Rays, X-Rays and Neutrons 67

scattering in liquids and Singwi and Sj¨olander [9] for neutron scattering as

well as for the resonant absorption of γ-radiation (M¨oßbauer spectroscopy).

We are interested in the self-correlation function because the incoherent scat-

tering function S(Q,ω) is the double Fourier transform of G

(r,t)inspace

and time

S(Q,ω)∝Re





∞

dt exp



−



2

+iω







dr exp(iQ · r)G

(r,t)



. (2.1)

All prefactors have been omitted since they are diﬀerent for neutron scatter-

ing (see Chap. 3) and M¨oßbauer spectroscopy (see [10]). For neutron scat-

tering, Q is the scattering vector and ω the energy transfer. As ω is very

small the scattering process is nearly elastic or “quasielastic”, that is why

the method is called quasielastic neutron scattering (QENS). In analogy we

speak of quasielastic M¨oßbauer spectroscopy (QEMS) because the method is

basedonthesametheoryasQENS,ω again being very small. The term

t/2 has no meaning for QENS, it is speciﬁc for QEMS: Γ

is the natural

width of the nuclear M¨oßbauer transition. Here S(Q,ω) is the cross section

for resonant nuclear absorption of γ radiation. Remember, that in any case

the calculated S(Q,ω) has to be convoluted with the resolution of the given

experimental set-up.

(r,t) contains both, diﬀusional motion and motion induced by lat-

tice vibrations. Under certain conditions usually fulﬁlled in metals these two

contributions can be separated. The vibrational part gives the Debye-Waller

(Lamb-M¨oßbauer) factor f

which rules the total intensity of scattered (res-

onantly absorbed) radiation while the diﬀusional part determines the shape

of the spectra.

Another important consequence of (2.1) is that due to the product Q · r

the scattering function S(Q,ω) depends on the relative orientation between

the scattering vector Q and the jump vector r and hence on the orientation

of the crystal lattice. Thus full information is obtained only by series of

measurements on single crystals at various orientations

At the ﬁrst glance it seems possible to measure spectra of interesting

materials, then to perform an inverse double Fourier transform and arrive

directly at the self-correlation function G

(r,t). Unfortunately, this is not

possible because of the ﬁnite experimental resolution. The loss of information

due to the ﬁnite resolution can not be reversed by an analytical deconvolution

of the measured spectra. Thus one has to choose a diﬀerent way. Starting

from a diﬀusion model we calculate S(Q,ω), perform a convolution with

the resolution function and compare the obtained “theoretical” spectra with

the measured spectra. At the end of a selection process the most probable

diﬀusion mechanism remains.

In the case of non-crystalline samples (glasses, ﬂuids etc.) there is no such

anisotropy.

68 Gero Vogl and Bogdan Sepiol

The next subsection deals with the way from a proposed diﬀusion mech-

anism to the spectrum.

2.2.1 Quasielastic Methods:

M¨oßbauer Spectroscopy and Neutron Scattering

Quasielastic neutron scattering is the method promoted in the late ﬁfties

by Brockhouse [11] and for diﬀusion jumps in metals was ﬁrst applied in

1976 [12,13], whereas quasielastic M¨oßbauer spectroscopy started in the late

sixties [14].

If diﬀusion in a solid is fast enough (D in the range from 10

−14

−10

−1

for quasielastic M¨oßbauer spectroscopy and from 10

−13

−7

−1

for quasielastic neutron scattering) this leads to the so-called

diﬀusional broadening of measured M¨oßbauer or neutron scattering spectra.

The M¨oßbauer eﬀect is the following phenomenon: A γ quantum emitted by

a nucleus decaying from an excited to the ground level is resonantly absorbed

at the ground level of a second nucleus of the same isotope. Because of the ex-

treme sharpness of nuclear levels, any energy change of the γ quantum beyond

tenth of µeV (for the nucleus

Fe) – i.e. any recoil or even phonon excitation

– must be absent or be compensated, otherwise the energy of the γ-quantum

would not match and resonance absorption would not occur. For

Fe, by

far the most frequently used isotope in M¨oßbauer spectroscopy, the energy

diﬀerence between excited and ground level, i.e. the energy of the γ quantum,

is 14.4 keV corresponding to a length of its wave vector Q =7.3

−1

.This

can be understood in a roughly simpliﬁed picture: If at low temperatures

the emitting M¨oßbauer nuclei stay on their sites during the emission process,

the energy width of the emitted radiation is ruled by the mean lifetime τ

the excited M¨oßbauer level. If at suﬃciently high temperatures, however, the

mean residence time τ of diﬀusing atoms between two successive jumps is of

the same order of magnitude as or even smaller than the mean lifetime τ

the M¨oßbauer level, on the average each emitting M¨oßbauer atom changes

its position during the emission process. Thus the wave train emitted by a

diﬀusing atom is “cut” to several shorter wave trains which leads to a larger

width (Fig. 2.2). As these wave trains are emitted by one and the same nu-

cleus they are coherent. The interference between wave trains emitted by the

same nucleus depends on the relative orientation between the jump vector

and the direction of the wave vectors. Therefore, in certain crystal directions

the line width is small while in other directions it is very large. In this picture

the broadening will be the larger the stronger the fragmentation of the wave

train is.

An analogous argumentation is possible for QENS where the wave train of

neutrons scattered by a diﬀusing atom is cut into several wave trains. While

QEMS has proved to be an appropriate tool for diﬀusion studies in iron

aluminides and silicides, it can not be applied to other B2 alloys. QENS, on

the other hand, can be applied for diﬀusion investigations in a broader range

2 Diﬀusion Studied by Interference of γ-Rays, X-Rays and Neutrons 69

Fig. 2.2. Simpliﬁed, semi-classical explanation of the diﬀusional line broadening

for QEMS (top) and QENS (bottom). Q: wave vector (QEMS) or scattering vector

(QENS), k

, k

: initial and ﬁnal wave vector.

of alloys, provided the diﬀusivity is fast enough because of about a factor

of hundred lower energy resolution of backscattering spectrometers IN10 or

IN16 at the high ﬂux reactor ILL Grenoble.

Below we will treat this phenomenon in a more formalistic way.

Bravais Lattices

For a continuous medium, neglecting the discrete structure of real material,

the diﬀusion of particles must obey Fick’s equation (see Chap. 1). The solu-

tion of this equation neglects the discontinuous nature of real materials. In

a real material and at not too much elevated temperatures, diﬀusion can be

considered as jump diﬀusion as in the model of Chudley and Elliott [8]. This

model, although originally developed for liquids, found more applications in

solid state diﬀusion with the following assumptions [6]:

1. All sites on which diﬀusion takes place are equivalent, i.e. we have a

Bravais lattice.

2. The diﬀusion is independent of other kinds of motion like vibrations.

3. Only jumps to nearest-neighbour sites are allowed, r

is an ith vector

connecting these sites.

4. The jump time is negligible compared with the residence time τ on the

lattice site.

5. Successive jumps are uncorrelated, i.e. the jump direction of the follow-

ing jump is independent from the preceding one. With other words: the

present state of the system is determined only by the past state at a

particular time (Markov process).