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80 Gero Vogl and Bogdan Sepiol

Fig. 2.7. Two-line spectrum of

at 1020

◦

C in [113] direction.

of (1.9 ± 0.1) : 1 [48], which agrees with conventional QEMS [21], but the

uncertainty of the NRS measurement is considerably smaller.

As pointed out in Sect. 2.2.1 the spectra of a B2 alloy may seem to

consist of one single Lorentzian even though diﬀusion takes places via NN

jumps if the asymmetry parameter α is large. For highly ordered Fe

50.5

49.5

α should be high due to the relatively low concentration of antistructure

defects and a still large α can be expected for the still well ordered Fe

Therefore, the amount of excess iron atoms was raised further and Fe

was measured, which is still B2 ordered but – due to the composition –

exhibits a considerable amount of disorder [22]. The spectra of this alloy

clearly consist of two Lorentzians (Fig. 2.7) and the complete set of spectra

measured at diﬀerent orientations can only be ﬁtted under the assumption

of NN jumps from the iron to the aluminium sublattice and vice versa.

This leads to a consistent picture of diﬀusion in Fe

1−x

alloys. Start-

ing near the ideal B2 composition FeAl we ﬁnd that iron atoms diﬀuse via

NN jumps to antistructure sites with a remarkably short residence time on

the aluminium sublattice (and hence a large α). With increasing iron con-

tent the residence time of iron atoms on the aluminium sublattice increases

and thus (at comparable temperatures from 1020

◦

C to 1090

◦

C) the asym-

metry parameter α decreases from 30 over 7.0 to 2.45 (Fe

50.5

49.5

,Fe

and Fe

, respectively). In “disordered” Fe

Al iron diﬀuses on a Bravais

lattice, i.e. α =1.

In the interpretation of the diﬀusion mechanism in near-stoichiometric

FeAl another QEMS result is of prime importance. The relative intensity of

the M¨oßbauer spectrum measured as a function of the angle between the

[001] axis and the γ-radiation wave vector shows a clear minimum in the

[113] direction (Fig. 2.8). There is only one possible interpretation of this

result: iron jumps are NN-jumps via short occupation of the Al sublattice.

If the short residence time is much smaller than the M¨oßbauer time window

(τ

= 141 ns), everything that happens within this extremely short time is

invisible for M¨oßbauer spectroscopy and only the start and end positions

2 Diﬀusion Studied by Interference of γ-Rays, X-Rays and Neutrons 81

Fig. 2.8. Angular dependence of

the M¨oßbauer line intensity for (a)

50.5

49.5

and (b)Fe

enter into the M¨oßbauer spectra. The only eﬀect on the spectrum is the

intensity minimum in the direction characteristic for NN jumps.

It should be pointed out that the picture obtained by QEMS and NRS

can today be conﬁrmed by independent experimental results. Even theoretical

simulations, which contradicted this picture in the past [49], today conﬁrm

it. On the basis of [50] and [51], one can exclude the six-jump cycle as a

mechanism of iron diﬀusion due to much too high concentration of vacancies

and antistructure atoms at temperatures where microscopic studies have been

performed. Neither mechanisms like an antistructure-bridge mechanism and

a triple-defect mechanism are operative (e.g. [51, 52]). The energy barrier

for direct Fe jumps on its own sublattice have been calculated by F¨ahnle et

al. [53]. The energy barrier for the next-nearest neighbour jump was 2.16 eV,

which is only slightly larger than the experimental migration energy, but

for the direct third-nearest neighbour jump the calculated energy was 6 eV,

which entirely excludes this possibility. F¨ahnle proposed a modiﬁed six-jump

cycle where two atoms jump in a correlated manner avoiding in this way the

creation of the Al vacancy [48, 53]. Recent defect-structure simulations [54],

which take into account the defect vibrational entropies and the eﬀect of

magnetism, conﬁrm the hybrid character of stoichiometric B2 FeAl

and a

considerable amount of Al antistructure atoms. These results conﬁrm the NN

jump as the most reasonable elementary diﬀusion jump in B2 FeAl.

Thus the conclusion for Fe diﬀusion in Fe

1−x

alloys is: at stoichiom-

etry and also far-oﬀ stoichiometry the elementary Fe jump is a jump into a

This means that stoichiometric FeAl is neither a pure antistructure type system

nor a pure triple defect type system (see [2]).

82 Gero Vogl and Bogdan Sepiol

NN vacancy on an antistructure site. The Fe residence time on this antistruc-

ture site is very short close to stoichiometry and becomes longer until being

comparable to the residence time at a regular site with increasing deviation

from stoichiometry, i.e. the larger is the disorder.

Diﬀusion of Nickel in NiGa and of Cobalt in CoGa

In order to obtain a consistent picture of the elementary diﬀusion jump in

B2 alloys, neutron scattering was added to the repertoire of methods.

QENS was used by Kaisermayr et al. to study diﬀusion in two B2 alloys:

NiGa [55] and CoGa [56]. NiGa and CoGa are very suitable for QENS because

the incoherent neutron scattering cross section [6] of Ga is much smaller than

that of Ni or Co. Thus the scattered intensity can be exclusively attributed

to neutron scattering at nickel and Co nuclei, respectively. The QENS mea-

surements were performed at ILL on single crystals with 51.2 at.% Ni and on

polycrystals with 57 and 62 at.% Ni. Figure 2.9 shows the spectrum measured

at 1400 K. From the Q-dependence of the quasielastic line broadening the Ni

jump mechanism was determined. The very weak concentration dependence

of the Ni diﬀusivity observed by Donaldson and Rawlings [57] was conﬁrmed

over a wide concentration range on the Ni-rich side of the B2 phase. How-

ever, the next-nearest neighbour jumps of Ni [57] could not be found. On

the contrary, the QENS experiment showed that the Ni atoms diﬀuse via

nearest-neighbour jumps. In the near-stoichiometric composition the best ﬁt

of the Q-dependence was achieved assuming eﬀective jumps of Ni atoms with

the preference of [100] over [110] jumps with an admixture of [111] jumps (ra-

tio 4 : 2 : 1, respectively). From the occupation times of both B2 sublattices

Kaisermayr et al. [55] derived the antistructure-atoms concentration and,

assuming triple defects

as a dominant defect structure of NiGa, very high

vacancy concentrations in both alloys. These results are in agreement with va-

cancy concentrations extrapolated from dilatometric measurements (see [55]

for more references).

Very recently the jump model of Ni atoms into antistructure-defect sites

was conﬁrmed on the same NiGa crystal but with higher accuracy using

the neutron spin-echo technique [32]. Figure 2.10 shows ﬁts to the spectra

obtained from the single crystal for two wave vectors. For a wave vector

Q =1.8

−1

which coincides with the reciprocal superlattice lattice point the

next nearest neighbour (NNN) model (dashed line) gives a very unsatisfactory

ﬁt to the experimental data. A perfect agreement with the experiment can

be achieved with NN jumps, i.e. jumps to antistructure sites (solid line).

QENS measurements on CoGa [56] were performed at ILL on single crys-

tals with 54 and 64 at.% Co. For both compositions a maximum of the qua-

sielastic line broadening has been observed near a reciprocal lattice point

corresponding to a B2 superlattice [100] reﬂection. This can be explained

A triple defect consists of two A-sublattice vacancies and one antistructure atom

on B-sublattice.

2 Diﬀusion Studied by Interference of γ-Rays, X-Rays and Neutrons 83

Fig. 2.9. Quasielastic spectrum ob-

tained from a Ni

single crystal

at 1400 K. The thick solid and dashed

lines represent two Lorentzians, dot-

ted line the resolution function.

Fig. 2.10. Time correlation

function measured on the

52.5

47.5

single crystal at

1130

◦

C. Inset: B2-unit cell with

a Ni atom jumping into a NN

vacancy on a Ga site (solid

arrow) and into a NNN vacancy

on the Ni sublattice (dashed

arrow).

84 Gero Vogl and Bogdan Sepiol

only by the presence of a second Lorentzian arising from jumps to antistruc-

ture sites. The elementary jump of Co in CoGa has therefore been identiﬁed

as a nearest-neighbour jump between regular and antistructure sites. From

the residence times on the antistructure sites high vacancy concentrations

were deduced (see e.g. [58]). Moreover, on the basis of the diﬀusion coeﬃ-

cient resulting from the line broadening at high Q values and from tracer

measurements a correlation factor for Co diﬀusion not smaller than approx-

imately 0.6 was found [56,59]. Hence, Co diﬀusion in CoGa seems to be, like

Fe in FeAl alloys, surprisingly eﬀective. This again suggests that it cannot be

described in terms of deﬁned cycles (six-jump, four-jump cycle), percolation-

driven diﬀusion (antistructure-bridge mechanism) or a correlated triple-defect

mechanism as proposed by Stolwijk et al. [60]. Note, however, that van Om-

men and Miranda [61] suggested that NN Co and Ga atoms alone can explain

the experimental results on activation enthalpy in the same manner as the

triple-defect mechanism.

Thus the conclusion is again: transition metal (Ni, Co) atoms in NiGa and

CoGa jump via antistructure sites. Their residence time on the antistructure

sites is the shorter the closer to stoichiometry the alloy is.

Summary of Diﬀusion of Atoms

of Transition Elements in B2 Ordered Alloys

– Diﬀusion always proceeds via jumps into a vacancy on a nearest-neighbour

site.

– This is true for stoichiometric and non-stoichiometric alloys.

– The residence time on the antistructure site increases with deviation from

stoichiometry.

– The diﬀusivity increases with deviation from stoichiometry.

Alloys (Fe

Si Structure)

Iron Diﬀusion in Fe

The D0

superstructure of Fe

Si is more complicated than the B2 superstruc-

ture (see Fig. 2.11 inset). The cubic supercell consists of eight bcc cells. It

can be divided into four sublattices, i.e. three iron sublattices (α

, α

and γ

sublattice) and one silicon (β) sublattice. The lattice constant of these sub-

lattices of the supercell is twice as large as the lattice constant of the small

bcccells.Eachironatomonaγ site has eight iron neighbours. The two iron

lattices α

and α

are energetically identical (and will be referenced below

as “the α sublattices”). They both have four silicon and four iron nearest

neighbours but the conﬁguration of these neighbours around α

is mirrored

in comparison to the neighbours surrounding an α

site. Silicon atoms on

the β sublattice are surrounded by eight iron neighbours. In Fe

Si the su-

perstructure is stable up to the melting point due to a very high ordering

2 Diﬀusion Studied by Interference of γ-Rays, X-Rays and Neutrons 85

Fig. 2.11. D0

structure of Fe

Si (2/8 of elementary cell) with iron NN (solid

arrow) and NNN (broken arrow) jumps between regular α and γ lattice sites. Ori-

entation dependence of time-integrated intensity of delayed NRS counts with lines

ﬁtted assuming NN (solid line) and NNN (broken line) iron jumps.

energy, while less ordered oﬀ-stoichiometric Fe

1−x

(x<0.25) undergoes

aD0

↔ B2 phase transition before reaching the solidus line (excess iron

atoms in oﬀ-stoichiometric Fe

1−x

are placed on silicon sites).

M¨oßbauer studies were the ﬁrst to detect that despite the high degree of

order in Fe

Si, iron diﬀusion in this compound is very fast [62] (see Arrhenius

plots in Chap. 1) and – most surprisingly – becomes slower with decreasing

order. This was very soon later conﬁrmed by tracer studies [63], and later-

on by studies with NRS [26]. These facts lead to two questions: Is a special

diﬀusion mechanism operative in Fe

Si and what causes the decrease in diﬀu-

sion coeﬃcient with increasing iron content? Experiments using QEMS and

phonon spectroscopy help to solve these problems.

In principle, jumps between all four sublattices are possible, i.e. the spec-

tra of Fe

Si should consist of four Lorentzians. The spectra measured by

QEMS and NRS turn out to be simpler. In contrast to the situation in B2

structures an iron atom in Fe

Si can diﬀuse exclusively between iron sub-

lattices via NN jumps without creating antistructure defects. Thus in highly

ordered Fe

Si no antistructure jumps are observed and the jump matrix re-

duces to a 3 ×3 matrix, i.e. the number of lines reduces to three Lorentzians.

All spectra measured at Fe

Si are well ﬁtted by use of a model with a 3 × 3

jump matrix. No jumps directly between α are found. From Fig. 2.11 it is

clear that iron jumps only between NN sites and, moreover, the lower limit

of NNN-jumps contribution can be ﬁtted. The contribution of NNN-jumps is

below 5% of the NN-jump rate, in other words an upper limit for the direct

NNN-jump diﬀusivity is below 5% of the total iron diﬀusivity.

86 Gero Vogl and Bogdan Sepiol

In oﬀ-stoichiometric samples the excess iron atoms already occupy silicon

sites and thus diﬀusion via antistructure sites is less unfavoured than in stoi-

chiometric Fe

Si. Spectra taken at diﬀerent orientations on oﬀ-stoichiometric

samples could only be ﬁtted using a 4 × 4 jump matrix [62].

In the Fe

Si system relatively soft phonon modes [64] which give low mi-

gration energies are found, but they do not vary with composition. Hence the

composition dependency of D cannot be attributed to phonon characteristics.

The diﬀusion mechanism turns out to be a rather conventional one: Diﬀusion

in Fe

Si is fast due to a very large concentration of vacancies and occurs

via NN jumps from γ to α sites and vice versa. Oﬀ-stoichiometric Fe

1−x

alloys contain much less vacancies [65] and thus diﬀusion is slow. In contrast

to stoichiometric Fe

Si iron jumps between α and β sites (silicon sublattice)

are not only possible in oﬀ-stoichiometric alloys but even more frequent than

jumps between the regular iron sites α and β [62] which was also conﬁrmed

by Monte Carlo simulations [66].

The high vacancy concentration in Fe

Si may be due to the high order-

ing energy: introducing a large number of vacancies is the sole possibility

to increase entropy upon heating without creating antistructure defects. In

oﬀ-stoichiometric Fe

1−x

where already built-in antistructure defects exist

creation of additional defects of this type is less expensive in energy than in

Si and thus less vacancies are formed and diﬀusion is slow.

Nickel Diﬀusion in Ni

Again neutron scattering can serve to widen the ﬁeld of accessible alloys be-

cause neutron measurements are not limited to the diﬀusion of Fe atoms.

Another compound with D0

structure is Ni

Sb (at the high tempera-

tures mandatory for quasielastic measurements). Nickel diﬀusion in Ni

Sb,

which contains many vacancies, together with Cu diﬀusion in Cu

Sn, is the

fastest ever found in an intermetallic alloy

. Typical values at 800

◦

Care

D ≈ 6 × 10

−11

−1

. In contrast to the case of Fe

Si, however, they are

asymmetrically distributed among the sublattices: the vacancies are concen-

trated on the α sublattices [67]. This may lead to a high rate of “far” jumps

between α sites. Ni

Sb is very suitable for QENS because again the inco-

herent neutron scattering cross section of the metalloid (here Sb) is much

smaller for antimony than for the transition element (Ni). Thus the scattered

intensity can be exclusively attributed to neutron scattering at nickel nuclei.

At two temperatures the quasielastic line broadening was measured on a

72.5

27.5

crystal at diﬀerent orientations. Figure 2.12 shows the result for

diﬀerent orientations of the single crystal. The bulk lines are ﬁts according

to the model of NN jumps between γ and α sites. The dotted, dashed-dotted

and dashed lines are calculated under the assumption of jumps between α

sites (see arrows in Fig. 2.11). The corresponding jump lengths in terms of

To the best of our knowledge, it is even the fastest self-diﬀusion in a pure metallic

system on the reduced temperature scale T/T

melting

2 Diﬀusion Studied by Interference of γ-Rays, X-Rays and Neutrons 87

the bcc lattice constant a for these jumps are a (dotted), a/

√

2(dashed-

dotted, along the face diagonal of the cube formed by the α sites, e.g. from

to α

)anda/2 (dashed, along the edges of the α-site cube, e.g. from α

to α

). Under the assumption that nickel atoms jump between γ and α sites

(nearest neighbours) the measured linewidth is well ﬁtted, while other jump

models give markedly worse results. Even in this case the elementary steps

of diﬀusion are NN jumps, proﬁting from the high vacancy concentration on

two of the three nickel sublattices. Details can be found, e.g., in [68].

Fig. 2.12. Linewidths mea-

suredbyQENSonNi

Sb at

690

◦

C (dots) and 780

◦

C (tri-

angles) and comparison with

diﬀerent models.

Summary of the Diﬀusion of Atoms of Transition Elements in D0

Alloys

Here the picture is not at all as consistent as for B2 alloys, it rather demon-

strates considerable diﬀerences and contrasts: fastest diﬀusion of transition

element atoms for stoichiometry in Fe

Si, slowest in Ni

Sb. This appears to

be still a not completely solved problem. First principle density functional

calculations for Ni

Sb [69] completely conﬁrm the experimental results [68],

for Fe

Si work is still underway.

2.4 Conclusion

The central aim of this chapter was to demonstrate the potential of methods

working with scattering of X-rays, including M¨oßbauer gamma rays, and neu-

trons for determining the elementary diﬀusion jump. These methods either

work in the energy domain or in the time domain.

We have discussed the results of quasielastic M¨oßbauer spectroscopy

(QEMS) and nuclear resonance scattering of synchrotron X-rays (NRS) on

the one hand; quasielastic neutron scattering (QENS) and neutron spin echo

(NSE) on the other hand, for studying the elementary diﬀusion jump in

88 Gero Vogl and Bogdan Sepiol

metallic systems. By way of examples we reported on the determination of

the elementary diﬀusion jump in pure metals, of impurities in dilute alloys

and in ordered intermetallic alloys. We chose mostly examples from our own

work, not aiming at completeness. For the sake of simplicity, which we re-

garded necessary for an overview, we dwelled exclusively on cubic systems.

Thus we did not report on non-cubic lattices as, e.g., the QENS experiments

on Ni diﬀusion in NiSb [70] or QEMS experiments on Fe diﬀusion in FeSb [71],

both hexagonal intermetallic alloys. For more complicated diﬀusion processes

studied by QEMS and QENS we point to our earlier reviews about diﬀusion

studies by M¨oßbauer spectroscopy [10, 72, 73], neutron scattering [74] and

NRS [28]. Reference [10] includes the ﬁrst atomistic results on diﬀusion in

restricted regions, i.e. cages [75], which recently gained great interest in bio-

logical systems; this is the large regime of diﬀusion in conﬁned regions.

As a conclusion for this report limited to long-range diﬀusion in cubic

metals we could choose: the basic step for these processes is the jump of

an atom into a NN vacancy. World appears to be so simple in these simple

cases, but this had ﬁrst to be demonstrated. What is not so simple, is the

time scale for occupation of antistructure sites in the particularly interesting

class of ordered alloys: the whole zoo of possibilities exists, connected to the

degree of order of an alloy and to the way the partners match.

The future appears bright: the highly collimated synchrotron beam will

enable studies of surface diﬀusion, the high coherence of present and more

so of future sources of synchrotron radiation will ﬁnally enable photon cor-

relation spectroscopy to study much slower diﬀusion than possible now. In

neutron spectroscopy considerable progress will be accomplished with the

new spallation sources. Since – and that should not be tacitly omitted be-

cause we want to remain modest – until now diﬀusion has to be rather fast

to be unveiled by the scattering methods.

Notation

a lattice constant (

A2 two component phase A

1−x

; solid solution with b.c.c structure

B2 two component intermetallic phase AB; superstructure basing on a

b.c.c cell with edges occupied by atoms of one species while the body

centred position belongs to atoms of the other species

two component intermetallic phase A

B ; superstructure basing on

eight b.c.c cells; can also be described as simple cubic lattice with a

base of four atoms

D diﬀusion coeﬃcient or diﬀusivity (m

−1

)

f correlation factor

Debye-Waller (Lamb-M¨oßbauer) factor; relative intensity of scat-

tered (resonantly absorbed) radiation

Fe most important M¨oßbauer isotope; interesting M¨oßbauer transition

with energy 14.4 keV and wave vector |Q| =7.3

−1

2 Diﬀusion Studied by Interference of γ-Rays, X-Rays and Neutrons 89

(r,t) probability density to ﬁnd an atom displaced by a vector r from its

starting position within a time interval t averaged over all possible

starting positions

P (r,t) probability density to ﬁnd an atom on the site r at time t

Q scattering vector (QENS) or M¨oßbauer radiation wave vector (QEMS)

S(Q,ω) incoherent scattering function (QENS) or cross section for resonant

nuclear absorption (QEMS)

Γ diﬀusional line broadening

natural line width of the M¨oßbauer transition (Γ

≈ /τ

)

ϑ angle between Q and the [001] crystal axis

mean life time of the excited M¨oßbauer level

mean residence time of an atom between jumps from sublattice i

(τ

−1

–jumprate)

mean residence time between jumps from sublattice i to any nearest-

neighbour site of symmetry j

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