8 Mössbauer Spectroscopy in External Magnetic Fields
425
precise μ e f f values on the Al-rich side. Theoretically calculated hyperfine fields B
th
tot
consist of two contributions: a nearly concentration independent contribution from
the 3d electrons B
th
core and one from the valence electrons (4s and 4p) B
th
val which
increases linearly with decreasing Fe content and has opposite sign (Fig. 8.45). For
YFe 2 B
th
val is small and total hyperfine field is direct proportional to the Fe moment
as usually assumed [151]. Surprisingly, with increasing Al concentration the valence
electrons are strongly positive polarized. By varying the used lattice constant for
the calculations allows to estimate the volume dependence of the theoretical hyperfine field. E.g. calculation of Y 2 Fe 3 Al with the measured lattice constant of YFe 2
(7.36 Å) instead of 7.48 Å corresponds to a concentration change from x = 0.75 to
x = 1.0. This calculated concentration dependence of theoretical hyperfine fields is
shown as short solid lines in Fig. 8.45. Their slopes are nearly independent of x and
cannot be the reason for the concentration dependence of the hyperfine fields. The
negative slope of the volume dependence of B
th
tot explains the decrease of experimentally obtained Curie constant B
M B
ind with x for fixed number of nearest neighbour
Fe atoms. Because the theoretical calculations are done on ordered compounds with
experimentally obtained lattice constants, the results can only describe one of the
different Fe surroundings used in the Mössbauer analysis. Nevertheless B
th
tot shows
the same trend on number of Fe nearest neighbours as the experimentally obtained
Curie constant C
M B
ind . To get now local susceptibilities, the experimentally obtained
induced hyperfine fields have to be decomposed into the core contribution B core
and the valence contribution B val . The local susceptibility is only related to B core ,
which can be compared with the local Fe moment, which originates from the 3d
electrons. Because both B ind /B a and the susceptibility χ
mag determined from the
magnetic measurements follow a Curie-Weiss law, one can assume that B core also
follows such behaviour namely B core /B a = C
M B
core /(T -θ ). As a consequence B val /B a
must also follow a Curie-Weiss law. With help of the band-structure calculations a
separation of B core and B val is possible. Since the theoretical valence contribution
decreases linearly with x to nearly zero for YFe 2 (Fig. 8.45), similar linear behaviour
is assumed for the valence contribution of the experimentally obtained mean Curie
constant ¯
C
M B
val . To determine the slope of the concentration dependence of ¯
C
M B
val ,
one assumes that the core part of the mean Curie constant obtained from Mössbauer
measurements ¯
C
M B
core is directly proportional to the magnetically determined Curie
constant C mag . Relating C mag to one Fe atom shows that both mean ¯
C
M B
core and mean
¯
C mag /Fe are linearly increasing with x (Fig. 8.46), indicating that the direct proportionality between mean ¯
C
M B
core and mean C mag is consistent with the data. There is
only one hyperfine coupling constant relating B ind to μ Fe for all x. The concentration
dependence of the core contribution of the induced hyperfine field, determined from
the Mössbauer investigations, is in perfect agreement with the concentration dependence of the effective moment, determined from the bulk magnetic measurements.
If one now assumes that the valence contribution of the Curie constant for each Fe
surrounding is linear, the values C
M B
val and C
M B
core for the individual Fe environments
can be calculated from the respective measured induced hyperfine fields B ind in the
same way as was just done for the mean ¯
C
M B
val values. The local susceptibilities are
now given by χ loc = B core /B a with B core = C
M B
core B a /(T − θ) (Inset Fig. 8.46). For
425
precise μ e f f values on the Al-rich side. Theoretically calculated hyperfine fields B
th
tot
consist of two contributions: a nearly concentration independent contribution from
the 3d electrons B
th
core and one from the valence electrons (4s and 4p) B
th
val which
increases linearly with decreasing Fe content and has opposite sign (Fig. 8.45). For
YFe 2 B
th
val is small and total hyperfine field is direct proportional to the Fe moment
as usually assumed [151]. Surprisingly, with increasing Al concentration the valence
electrons are strongly positive polarized. By varying the used lattice constant for
the calculations allows to estimate the volume dependence of the theoretical hyperfine field. E.g. calculation of Y 2 Fe 3 Al with the measured lattice constant of YFe 2
(7.36 Å) instead of 7.48 Å corresponds to a concentration change from x = 0.75 to
x = 1.0. This calculated concentration dependence of theoretical hyperfine fields is
shown as short solid lines in Fig. 8.45. Their slopes are nearly independent of x and
cannot be the reason for the concentration dependence of the hyperfine fields. The
negative slope of the volume dependence of B
th
tot explains the decrease of experimentally obtained Curie constant B
M B
ind with x for fixed number of nearest neighbour
Fe atoms. Because the theoretical calculations are done on ordered compounds with
experimentally obtained lattice constants, the results can only describe one of the
different Fe surroundings used in the Mössbauer analysis. Nevertheless B
th
tot shows
the same trend on number of Fe nearest neighbours as the experimentally obtained
Curie constant C
M B
ind . To get now local susceptibilities, the experimentally obtained
induced hyperfine fields have to be decomposed into the core contribution B core
and the valence contribution B val . The local susceptibility is only related to B core ,
which can be compared with the local Fe moment, which originates from the 3d
electrons. Because both B ind /B a and the susceptibility χ
mag determined from the
magnetic measurements follow a Curie-Weiss law, one can assume that B core also
follows such behaviour namely B core /B a = C
M B
core /(T -θ ). As a consequence B val /B a
must also follow a Curie-Weiss law. With help of the band-structure calculations a
separation of B core and B val is possible. Since the theoretical valence contribution
decreases linearly with x to nearly zero for YFe 2 (Fig. 8.45), similar linear behaviour
is assumed for the valence contribution of the experimentally obtained mean Curie
constant ¯
C
M B
val . To determine the slope of the concentration dependence of ¯
C
M B
val ,
one assumes that the core part of the mean Curie constant obtained from Mössbauer
measurements ¯
C
M B
core is directly proportional to the magnetically determined Curie
constant C mag . Relating C mag to one Fe atom shows that both mean ¯
C
M B
core and mean
¯
C mag /Fe are linearly increasing with x (Fig. 8.46), indicating that the direct proportionality between mean ¯
C
M B
core and mean C mag is consistent with the data. There is
only one hyperfine coupling constant relating B ind to μ Fe for all x. The concentration
dependence of the core contribution of the induced hyperfine field, determined from
the Mössbauer investigations, is in perfect agreement with the concentration dependence of the effective moment, determined from the bulk magnetic measurements.
If one now assumes that the valence contribution of the Curie constant for each Fe
surrounding is linear, the values C
M B
val and C
M B
core for the individual Fe environments
can be calculated from the respective measured induced hyperfine fields B ind in the
same way as was just done for the mean ¯
C
M B
val values. The local susceptibilities are
now given by χ loc = B core /B a with B core = C
M B
core B a /(T − θ) (Inset Fig. 8.46). For