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STRUCTURE STUDY OF AMORPHOUS Fe-B AND Ni-Fe-B
RAPIDLY QUENCHED METALS S. Steeb, H. Warlimont (eds.) © Elsevier Science Publishers B.V., 1985
487
STRUCTURE STUDY OF AMORPHOUS Fe-B AND Ni-Fe-B Gy. Faigel, E. Svab Central Research Institute for Physics, H-1525 Budapest 114, P.O.B. 49, Hungary The quasi-crystalline model (QC) is discussed with respect to the correlation function. The dif ference in the chemical short range order measured by high resolution neutron diffraction on Feg,B..g and (^ a c^e 28^77^23 metallic glasses is interpreted on the basis of QC calculations. 1. INTRODUCTION
2. CORRELATION FUNCTION FROM THE
The quasi-crystalline (QC) model
proved to
be a succesful tool for the interpretation of
QUASI-CRYSTALLINE MODEL The basic assumption of the QC model is that
hyperfine-field distribution obtained from
the first neighbour environment of an atom in
Mbssbauer measurements in the case of iron based
an amorphous alloy is similar to that of a
transition metal(TM) - metalloid(M) type amor
crystalline compound existing at the composi
phous alloys. Although the hyperfine field dis
tion investigated.
tribution is one of the most sensitive quan
To find this crystalline phase is not triv
tities related to the local atomic environment,
ial, however, the study of the crystallization
it contains indirect information on the struc
process may give important clue to it.
ture. Therefore an important test of the QC
Our model is based on principles similar to
model is how it reproduces the experimental
those of the one dimensional calculation of
pair correlation function g(r). 2 3 Two papers ' were published so far in which
Prins . In our case however, two further suppo
the g(r) calculated on the basis of the QC model
for the correlation function in three dimensions.
was compared with the results of X-ray diffrac
i/ The probability of the position of the
sitions were necessary to obtain usable formula
tion measurements. However, the problems of the
first neighbours of an atom are given by Gaus
realisation of QC model calculations have not
sian like distributions centered on the crys
been discussed.
talline sites.
In this paper a brief survey of the QC model is given with respect to the description of the
It means, that the deviations from the exact crystalline positions are spherically symmetric.
pair correlation function in particular con
The exact mathematical treatment of the problem
sidering the applied assumptions, their phy
would require the three particle correlations,
sical meaning and the limitations. The reduced 4 pair correlation function (RPCF) of Fe 8 1 B 1 g
but they cannot be taken into account because
Nl
Fe
B
and ( 55 35)77 23 G a l l i c glasses obtained from high resolution time-of-flight (TOF) neutron diffraction measurements are compared with the results of the QC model. The results of small angle neutron scattering (SANS) measurements on Fe-B system are presented supporting the idea of the model calculation.
their exact form is unknown. Our calculation can be carried out consistently only for those lattices in which the first neighbours of an atom are not the first neighbours of each others. From the above assumption an algorithm has been constructed in order to determine the atomic density function p(r). The result is a
488
Gy. Faigel and E. Svdb
sum of Gauss distributions centered on the
where the sum has to be performed for all atomic
crystalline sites. The square of the second
distances r. while the broadening of the Gaus
momentum of these Gaussians is proportional to
sians, o(r) has the following form as a func
the number of the steps needed to get from the
tion of the distance (r) from the atom in the
center atom to a given point of the lattice.
origo
The basic structure of p(r) for a two dimen
a2
sional simple square lattice is illustrated in
a2(r)=^r o
Fig. 1. The crosses represent the lattice and
(2)
where r is the nearest neighbour distance and 3 o is its distribution width. o In the case of more complicated systems one
a
has to be yery careful in the interpretation of the curves obtained from formula (1). The neglection of the three particle correlations can cause a relatively small deviation in the first peak of g(r). But both of the simplifications introduced in i/ and ii/ result increasing de viations at larger distances. Therefore, we may use the QC model as a guide to predict the kind of changes expected in the FIGURE 1 Equi-a surfaces in a two dimensional square lattice: the squares represent the exact sur faces and the spheres show those used in our approximation the squares determine the equi-a surfaces. It means that the Gaussians centered on the lat tice points being on the same square have equal halfwidth. The pair correlation function is de termined by the angular average of p(r). This operation cannot be easily performed for arbitrary equi-a surface, therefore a fur ther approximation was used to simplify our calculation as it is described below. ii/ Let the equi-a surfaces be spheres (see Fig. 1). In this case the angular average can be treated analitically and the final expression for the pair correlation function is the fol lowing 2 2 -,
-.
r. + r
rr.
(1)
pair correlation function, mainly in the struc ture of the first coordination shell. 3. LOCAL ORDER IN AMORPHOUS Fe-B AND Ni-Fe-B: MODEL AND EXPERIMENT The high resolution TOF neutron diffraction measurements on FegiB^Q
and
(^a^e2E>h7^23
metallic glasses have shown differences in the short range structure. To clarify the origin of the experimentally observed differences in the first peak of the correlation function, relating to different chemical short range order, the experimental curves of the RPCF are interpreted on the basis of the quasi-crystalline model. Fe-B system First the case of Fe7t-B?[- composition has been studied, where the first step of crystal lization takes place without composition changes by a transformation into metastable tetragonal Fe 3 B (Ni^P structure) . The atomic position parameters of this structure have not been de-
Structure study of amorphous Fe-B and Ni-Fe-B termined yet, thus we started from those of the o
489
symmetric inhomogeneities were supposed and the G u i n i e r - f i t of the form
crystalline Ni-P. The model calculations were extended for the
A exp(-Q2R|j/3)
[I-I
(3)
description of the off-stoichiometric Feft,B.Q composition. It was supposed that the RPCF is
was used. The results clearly indicate the ex
the sum of two spectra. One of them corresponds
istence of small clusters of about 6 8 (±i 8)
to the stoichiometric Fe7C-B?{- glass with the
giration radius in the samples of off-stoichio
structure described above and the other one to
metric compositions. The number of these clus
the pure iron glass based on fee structure. It
ters is increasing with the increase of the
may be expected that a dense random packed type
iron concentration (see Fig. 3/b), in agreement
structure could give a better approximation of
with the model calculation which gives linear
the pure iron surrounding. Remaining however,
A(x).
within the frame of the QC model, it is natural to start from the close packed fee structure of the iron phase, stable at high temperature. The experimental and the calculated RPCF are shown in Fig. 2. The tetragonal Fe 3 B and the fee iron has been characterized by lattice parameters and distribution width: a=8.62 8, c=4.28 8, a =0.08 8 and a=3.64 8, a =0.1 8, respectively.
^3
F
i(Q)
«100-X
a.
Bx
■ 14 *19 • 25
(0
b > ' rV)
zLU
HSZ
(/> z <4. w
X >K
X ^
^+C "*xj—^
measurement model calculation
•
^ ^ ^ f c# ^ f c ^ ^ f . • • t* * r H
1
1
1
1—.—i
.—
In
I
0l3 0.5 0.7 09 SCATTERING VECTOR , Q [ A - 1 ]
'
FIGURE 3/a SANS intensity of the Fe-B system. The solid lines represents the Guiner fit
1
2
3 4 5 ATOMIC DISTANCE , r[A]
6
FIGURE 2 RPCF of the Feg^Big glass as measured by neu tron diffraction and from QC model calculation There exists an additional experimental evidence supporting the idea that the Fe atoms have two types of surrounding in the off-stoic hiometric Fe-B glasses: Fig. 3/a shows the in tensity curves of the SANS measurements per formed on Fe, nn _ B glasses at x=25, 19 and 14 at%. In the evaluation procedure spherically
1 1
2 3 4 5 6 7 8 9 10 CONCENTRATION, (25-X)[ at %1
11
FIGURE 3/b Concentration dependence of A (see text). The horizontal and the vertical bars indi cate the errors of the chemical analysis and the SANS measurements, respectively
Gy. Faigel and E. Svab
490
Ni" B intermetallic compound were used to model
Ni-Fe-B system In the case of the (NigcFe-j-UyEL-
meta
^lc
the nearest neighbour environment on the above
glass we started from a supposed local order
glasses, respectively. The main differences of
similar to that of the orthorhombic Ni' B inter-
the G(r) spectra are reflected in the relative
metallic compound. As far as there are no data
positions, the height ratio and the widths of
available for the lattice- and position parame
the TM-B and TM-TM distributions. Probably the
ters for the case when nickel atoms are replaced
agreement between the experiment and the cal
by iron atoms, we have used the corresponding 9 data of the pure Ni'B intermetallic compound . Furthermore it is assumed that the small devia
culation could be improved for even more dis tant coordination shells, if more accurate parameters of the corresponding crystalline
tion of the glass composition from the stoichio-
phases were available. On the other hand the
metric T M 7 5 B 2 5 doesn't cause any significant
approximations applied in the model calculation
change in the short range order. The calculated
(see i/ and ii/ in part 2) may also lead to
and measured RPCF are shown in Fig. 4. Lattice
deviations at larger distances between the cal
parameters, a=4.45 8, b=5.43 8, c=6.66 8 and
culated and measured curves.
distribution width a =0.095 8, were used in the o calculation.
ACKNOWLEDGEMENT We are \/ery grateful to Dr. L. Granasy for
1 G(r)
*
the valuable discussions and the critical read
(Ni65Fe35)77B23
Jl| ill
measurement model calculation
!i
i li
T
1
1
REFERENCES 1. I. Vincze, D.S. Boudreaux, M. Tegze, Phys. Rev. B19 (1979) 4896.
1
1 rs IlAs 11 1 / ^-« fi\ r x I' \ / ' \ / ' '^ IV?/ Vv T
*
v
I
,
1
,
ing of the manuscript.
1 —
2 3 4 5 ATOMIC DISTANCE , r [A]
FIGURE 4 RPCF of the (Ni'65Fe35)77B23 glass as measured by neutron diffraction and from the QC model calculation 4. CONCLUSIONS The main features of the first peak splitt ing of the RPCF seen by high resolution TOF neutron diffraction on Fe-B and Ni-Fe-B metal lic glasses could be reproduced by the model calculation. The crystal structure of the tetragonal Fe3B and that of the orthorhombic
2. Gy. Faigel, W.H. de Vries, H.J.F. Jansen, M. Tegze, I. Vincze, Proc. Conf. on Metallic Glasses: Science and Technology, (Budapest, 1980) Vol. 1 (1981) p. 275. 3. S. Aur, T. Egami, I. Vincze, Proc. 4th Int. Conf.Rapidly Quenched Metals (ed. T. Masumoto and K. Suzuki, The Japan Institute of Metals, Sendai, Japan 1982) Vol. 1, p. 309. 4. E. Svab, N. Kro6, S.N. Ishmaev, I.P. Sadikov, A.A. Chernyshov, Solid State Comm. 44 (1981) 1151. 5. E. Svab, N. Kro6, S.N. Ishmaev, I.P. Sadikov, A.A. Chernyshov, Solid State Comm. 46 (1983) 351. 6. J.A. Prins, Naturwissenschaften 19 (1931) 435. 7. T. Kemeny, I. Vincze, B. Fogarassy, S. Arajs, Phys. Rev. B20 (1979) 476. 8. U. Herold, U. Kbster, Z. Metallkde. 69 (1978) 326. 9. Aronson, Acta Cryst. 15 (1962) 878.
487
STRUCTURE STUDY OF AMORPHOUS Fe-B AND Ni-Fe-B Gy. Faigel, E. Svab Central Research Institute for Physics, H-1525 Budapest 114, P.O.B. 49, Hungary The quasi-crystalline model (QC) is discussed with respect to the correlation function. The dif ference in the chemical short range order measured by high resolution neutron diffraction on Feg,B..g and (^ a c^e 28^77^23 metallic glasses is interpreted on the basis of QC calculations. 1. INTRODUCTION
2. CORRELATION FUNCTION FROM THE
The quasi-crystalline (QC) model
proved to
be a succesful tool for the interpretation of
QUASI-CRYSTALLINE MODEL The basic assumption of the QC model is that
hyperfine-field distribution obtained from
the first neighbour environment of an atom in
Mbssbauer measurements in the case of iron based
an amorphous alloy is similar to that of a
transition metal(TM) - metalloid(M) type amor
crystalline compound existing at the composi
phous alloys. Although the hyperfine field dis
tion investigated.
tribution is one of the most sensitive quan
To find this crystalline phase is not triv
tities related to the local atomic environment,
ial, however, the study of the crystallization
it contains indirect information on the struc
process may give important clue to it.
ture. Therefore an important test of the QC
Our model is based on principles similar to
model is how it reproduces the experimental
those of the one dimensional calculation of
pair correlation function g(r). 2 3 Two papers ' were published so far in which
Prins . In our case however, two further suppo
the g(r) calculated on the basis of the QC model
for the correlation function in three dimensions.
was compared with the results of X-ray diffrac
i/ The probability of the position of the
sitions were necessary to obtain usable formula
tion measurements. However, the problems of the
first neighbours of an atom are given by Gaus
realisation of QC model calculations have not
sian like distributions centered on the crys
been discussed.
talline sites.
In this paper a brief survey of the QC model is given with respect to the description of the
It means, that the deviations from the exact crystalline positions are spherically symmetric.
pair correlation function in particular con
The exact mathematical treatment of the problem
sidering the applied assumptions, their phy
would require the three particle correlations,
sical meaning and the limitations. The reduced 4 pair correlation function (RPCF) of Fe 8 1 B 1 g
but they cannot be taken into account because
Nl
Fe
B
and ( 55 35)77 23 G a l l i c glasses obtained from high resolution time-of-flight (TOF) neutron diffraction measurements are compared with the results of the QC model. The results of small angle neutron scattering (SANS) measurements on Fe-B system are presented supporting the idea of the model calculation.
their exact form is unknown. Our calculation can be carried out consistently only for those lattices in which the first neighbours of an atom are not the first neighbours of each others. From the above assumption an algorithm has been constructed in order to determine the atomic density function p(r). The result is a
488
Gy. Faigel and E. Svdb
sum of Gauss distributions centered on the
where the sum has to be performed for all atomic
crystalline sites. The square of the second
distances r. while the broadening of the Gaus
momentum of these Gaussians is proportional to
sians, o(r) has the following form as a func
the number of the steps needed to get from the
tion of the distance (r) from the atom in the
center atom to a given point of the lattice.
origo
The basic structure of p(r) for a two dimen
a2
sional simple square lattice is illustrated in
a2(r)=^r o
Fig. 1. The crosses represent the lattice and
(2)
where r is the nearest neighbour distance and 3 o is its distribution width. o In the case of more complicated systems one
a
has to be yery careful in the interpretation of the curves obtained from formula (1). The neglection of the three particle correlations can cause a relatively small deviation in the first peak of g(r). But both of the simplifications introduced in i/ and ii/ result increasing de viations at larger distances. Therefore, we may use the QC model as a guide to predict the kind of changes expected in the FIGURE 1 Equi-a surfaces in a two dimensional square lattice: the squares represent the exact sur faces and the spheres show those used in our approximation the squares determine the equi-a surfaces. It means that the Gaussians centered on the lat tice points being on the same square have equal halfwidth. The pair correlation function is de termined by the angular average of p(r). This operation cannot be easily performed for arbitrary equi-a surface, therefore a fur ther approximation was used to simplify our calculation as it is described below. ii/ Let the equi-a surfaces be spheres (see Fig. 1). In this case the angular average can be treated analitically and the final expression for the pair correlation function is the fol lowing 2 2 -,
-.
r. + r
rr.
(1)
pair correlation function, mainly in the struc ture of the first coordination shell. 3. LOCAL ORDER IN AMORPHOUS Fe-B AND Ni-Fe-B: MODEL AND EXPERIMENT The high resolution TOF neutron diffraction measurements on FegiB^Q
and
(^a^e2E>h7^23
metallic glasses have shown differences in the short range structure. To clarify the origin of the experimentally observed differences in the first peak of the correlation function, relating to different chemical short range order, the experimental curves of the RPCF are interpreted on the basis of the quasi-crystalline model. Fe-B system First the case of Fe7t-B?[- composition has been studied, where the first step of crystal lization takes place without composition changes by a transformation into metastable tetragonal Fe 3 B (Ni^P structure) . The atomic position parameters of this structure have not been de-
Structure study of amorphous Fe-B and Ni-Fe-B termined yet, thus we started from those of the o
489
symmetric inhomogeneities were supposed and the G u i n i e r - f i t of the form
crystalline Ni-P. The model calculations were extended for the
A exp(-Q2R|j/3)
[I-I
(3)
description of the off-stoichiometric Feft,B.Q composition. It was supposed that the RPCF is
was used. The results clearly indicate the ex
the sum of two spectra. One of them corresponds
istence of small clusters of about 6 8 (±i 8)
to the stoichiometric Fe7C-B?{- glass with the
giration radius in the samples of off-stoichio
structure described above and the other one to
metric compositions. The number of these clus
the pure iron glass based on fee structure. It
ters is increasing with the increase of the
may be expected that a dense random packed type
iron concentration (see Fig. 3/b), in agreement
structure could give a better approximation of
with the model calculation which gives linear
the pure iron surrounding. Remaining however,
A(x).
within the frame of the QC model, it is natural to start from the close packed fee structure of the iron phase, stable at high temperature. The experimental and the calculated RPCF are shown in Fig. 2. The tetragonal Fe 3 B and the fee iron has been characterized by lattice parameters and distribution width: a=8.62 8, c=4.28 8, a =0.08 8 and a=3.64 8, a =0.1 8, respectively.
^3
F
i(Q)
«100-X
a.
Bx
■ 14 *19 • 25
(0
b > ' rV)
zLU
HSZ
(/> z <4. w
X >K
X ^
^+C "*xj—^
measurement model calculation
•
^ ^ ^ f c# ^ f c ^ ^ f . • • t* * r H
1
1
1
1—.—i
.—
In
I
0l3 0.5 0.7 09 SCATTERING VECTOR , Q [ A - 1 ]
'
FIGURE 3/a SANS intensity of the Fe-B system. The solid lines represents the Guiner fit
1
2
3 4 5 ATOMIC DISTANCE , r[A]
6
FIGURE 2 RPCF of the Feg^Big glass as measured by neu tron diffraction and from QC model calculation There exists an additional experimental evidence supporting the idea that the Fe atoms have two types of surrounding in the off-stoic hiometric Fe-B glasses: Fig. 3/a shows the in tensity curves of the SANS measurements per formed on Fe, nn _ B glasses at x=25, 19 and 14 at%. In the evaluation procedure spherically
1 1
2 3 4 5 6 7 8 9 10 CONCENTRATION, (25-X)[ at %1
11
FIGURE 3/b Concentration dependence of A (see text). The horizontal and the vertical bars indi cate the errors of the chemical analysis and the SANS measurements, respectively
Gy. Faigel and E. Svab
490
Ni" B intermetallic compound were used to model
Ni-Fe-B system In the case of the (NigcFe-j-UyEL-
meta
^lc
the nearest neighbour environment on the above
glass we started from a supposed local order
glasses, respectively. The main differences of
similar to that of the orthorhombic Ni' B inter-
the G(r) spectra are reflected in the relative
metallic compound. As far as there are no data
positions, the height ratio and the widths of
available for the lattice- and position parame
the TM-B and TM-TM distributions. Probably the
ters for the case when nickel atoms are replaced
agreement between the experiment and the cal
by iron atoms, we have used the corresponding 9 data of the pure Ni'B intermetallic compound . Furthermore it is assumed that the small devia
culation could be improved for even more dis tant coordination shells, if more accurate parameters of the corresponding crystalline
tion of the glass composition from the stoichio-
phases were available. On the other hand the
metric T M 7 5 B 2 5 doesn't cause any significant
approximations applied in the model calculation
change in the short range order. The calculated
(see i/ and ii/ in part 2) may also lead to
and measured RPCF are shown in Fig. 4. Lattice
deviations at larger distances between the cal
parameters, a=4.45 8, b=5.43 8, c=6.66 8 and
culated and measured curves.
distribution width a =0.095 8, were used in the o calculation.
ACKNOWLEDGEMENT We are \/ery grateful to Dr. L. Granasy for
1 G(r)
*
the valuable discussions and the critical read
(Ni65Fe35)77B23
Jl| ill
measurement model calculation
!i
i li
T
1
1
REFERENCES 1. I. Vincze, D.S. Boudreaux, M. Tegze, Phys. Rev. B19 (1979) 4896.
1
1 rs IlAs 11 1 / ^-« fi\ r x I' \ / ' \ / ' '^ IV?/ Vv T
*
v
I
,
1
,
ing of the manuscript.
1 —
2 3 4 5 ATOMIC DISTANCE , r [A]
FIGURE 4 RPCF of the (Ni'65Fe35)77B23 glass as measured by neutron diffraction and from the QC model calculation 4. CONCLUSIONS The main features of the first peak splitt ing of the RPCF seen by high resolution TOF neutron diffraction on Fe-B and Ni-Fe-B metal lic glasses could be reproduced by the model calculation. The crystal structure of the tetragonal Fe3B and that of the orthorhombic
2. Gy. Faigel, W.H. de Vries, H.J.F. Jansen, M. Tegze, I. Vincze, Proc. Conf. on Metallic Glasses: Science and Technology, (Budapest, 1980) Vol. 1 (1981) p. 275. 3. S. Aur, T. Egami, I. Vincze, Proc. 4th Int. Conf.Rapidly Quenched Metals (ed. T. Masumoto and K. Suzuki, The Japan Institute of Metals, Sendai, Japan 1982) Vol. 1, p. 309. 4. E. Svab, N. Kro6, S.N. Ishmaev, I.P. Sadikov, A.A. Chernyshov, Solid State Comm. 44 (1981) 1151. 5. E. Svab, N. Kro6, S.N. Ishmaev, I.P. Sadikov, A.A. Chernyshov, Solid State Comm. 46 (1983) 351. 6. J.A. Prins, Naturwissenschaften 19 (1931) 435. 7. T. Kemeny, I. Vincze, B. Fogarassy, S. Arajs, Phys. Rev. B20 (1979) 476. 8. U. Herold, U. Kbster, Z. Metallkde. 69 (1978) 326. 9. Aronson, Acta Cryst. 15 (1962) 878.