Evolution of spin excitations into the superconducting state in FeTe1-xSex
ABSTRACT The origin of the superconducting state in the recently discovered Fe-based materials is the subject of intense scrutiny. Neutron scattering and NMR measurements have already demonstrated a strong correlation between magnetism and superconductivity. A central unanswered question concerns the nature of the normal-state spin fluctuations that may be responsible for the pairing. Here we present inelastic neutron scattering measurements from large single crystals of superconducting and non-superconducting Fe1+yTe1-xSex. These measurements indicate a spin fluctuation spectrum dominated by two-dimensional incommensurate excitations extending to energies greater than 250;meV. Most importantly, the spin excitations in Fe1+yTe1-xSex have four-fold symmetry about the (1, 0) wavevector (square-lattice (pi,pi) point). Moreover, the excitations are described by the identical wavevector and can be characterized by the same model as the normal-state spin excitations in the high-TC cuprates. These results demonstrate commonality between the magnetism in these classes of materials, which perhaps extends to a common origin for superconductivity.
- [show abstract] [hide abstract]
ABSTRACT: We report a systematic investigation of Ba[Fe(1-x)Co(x)]2As2 based on transport and 75-As NMR measurements, and establish the electronic phase diagram. We demonstrate that doping progressively suppresses the uniform spin susceptibility and low frequency spin fluctuations. The optimum superconducting phase emerges at x_c~0.08 when the tendency toward spin ordering completely diminishes. Our findings point toward the presence of a quantum critical point near x_c between the SDW (spin density wave) and superconducting phases.Journal of the Physical Society of Japan 01/2009; 78(1). · 2.09 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: The excitations responsible for producing high-temperature superconductivity in the copper oxides have yet to be identified. Two promising candidates are collective spin excitations and phonons(1). A recent argument against spin excitations is based on their inability to explain structures observed in electronic spectroscopies such as photoemission(2-5) and optical conductivity(6,7). Here, we use inelastic neutron scattering to demonstrate that collective spin excitations in optimally doped La2-xSrxCuO4 are more structured than previously thought. The excitations have a two-component structure with a low-frequency component strongest around 18 meV and a broader component peaking near 40 - 70 meV. The second component carries most of the spectral weight and its energy matches structures observed in photoemission(2-5) in the range 50 - 90 meV. Our results demonstrate that collective spin excitations can explain features of electronic spectroscopies and are therefore likely to be strongly coupled to the electron quasiparticles.Nature Physics 02/2007; · 19.35 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Since the discovery of high-transition-temperature (high-T(c)) superconductivity in layered copper oxides, extensive effort has been devoted to exploring the origins of this phenomenon. A T(c) higher than 40 K (about the theoretical maximum predicted from Bardeen-Cooper-Schrieffer theory), however, has been obtained only in the copper oxide superconductors. The highest reported value for non-copper-oxide bulk superconductivity is T(c) = 39 K in MgB(2) (ref. 2). The layered rare-earth metal oxypnictides LnOFeAs (where Ln is La-Nd, Sm and Gd) are now attracting attention following the discovery of superconductivity at 26 K in the iron-based LaO(1-x)F(x)FeAs (ref. 3). Here we report the discovery of bulk superconductivity in the related compound SmFeAsO(1-x)F(x), which has a ZrCuSiAs-type structure. Resistivity and magnetization measurements reveal a transition temperature as high as 43 K. This provides a new material base for studying the origin of high-temperature superconductivity.Nature 07/2008; 453(7196):761-2. · 38.60 Impact Factor
Evolution of spin excitations into the superconducting
state in FeTe1-xSex
M. D. Lumsden1, A. D. Christianson1, E. A. Goremychkin2,3, S. E. Nagler1, H. A.
Mook1, M. B. Stone1, D. L. Abernathy1, T. Guidi3, G. J. MacDougall1, C. de la Cruz4,
A. S. Sefat1, M. A. McGuire1, B. C. Sales1, & D. Mandrus1
1Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA. 2Argonne
National Laboratory, Argonne, Illinois 60439, USA. 3ISIS Facility, Rutherford
Appleton Laboratory, Chilton, Didcot OX11 0QX, UK. 4The University of Tennessee,
Knoxville, Tennessee 37996, USA.
The nature of the superconducting state in the recently discovered Fe-based
superconductors1-3 is the subject of intense scrutiny. Neutron scattering
investigations have already elucidated a strong correlation between magnetism and
superconductivity in the form of a spin resonance in the magnetic excitation
spectrum4-7. A central unanswered question concerns the nature of the normal
state spin fluctuations which may be responsible for the pairing mechanism. Here
we show inelastic neutron scattering measurements of Fe1.04Te0.73Se0.27, not
superconducting in bulk, and FeTe0.51Se0.49, a bulk superconductor. These
measurements demonstrate that the spin fluctuation spectrum is dominated by
two-dimensional incommensurate excitations near the (1/2,1/2) (square lattice
(π,0)) wavevector , the wavevector of interest in other Fe-based superconductors,
that extend to energies at least as high as 300 meV. Most importantly, the spin
excitations in Fe1+yTe1-xSex exhibit four-fold symmetry about the (1,0) (square
lattice (π,π)) wavevector and are described by the identical wavevector as the
normal state spin excitations in the high-TC cuprates8-12 demonstrating a
commonality between the magnetism in these classes of materials which perhaps
extends to a common origin for superconductivity.
The discovery of superconductivity in LaFeAsO1-xFx with TC = 28 K1 sparked a
flurry of scientific activity and TC rapidly increased to ~55 K on replacing La with other
rare earth elements2,12-14. In addition to the RFeAsO family of compounds,
superconductivity was also discovered in the AFe2As23, LiFeAs15, and in the alpha
phase of Fe1+yTe1-xSex16,17. These materials share common structural square layers with
Fe coordinated with either a pnictogen or a chalcogen. The unit cell contains two Fe
atoms generating a reciprocal space rotated by 45 degrees from the conventional square
lattice (see Fig. 1a). In RFeAsO and AFe2As2, the parent compounds exhibit long-range
spin density wave order characterized by the wavevector Q = (1/2,1/2,L) 18-20 Doping
suppresses magnetic order allowing superconductivity to emerge with the concomitant
appearance of a resonance in the spin fluctuation spectrum4-7. However, the resonance
likely contains only a small fraction of the total magnetic spectral weight and,
consequently, understanding the role magnetism plays in the superconductivity of these
materials requires detailed understanding of the higher energy spectrum of magnetic
The Fe1+yTe1-xSex materials are ideal candidates for such a study of the magnetic
excitations as large single crystals, necessary for detailed inelastic neutron scattering
studies, may be grown. However, these materials differ somewhat from other Fe-based
superconductors in that the Fe1+yTe endpoint member orders magnetically with a
structure described by the wavevector (1/2,0,1/2)21 as opposed to (1/2,1/2, L). Despite
this ordering wavevector, superconducting samples of Fe1+yTe1-xSex with higher Se
content exhibit a magnetic resonance at the same (1/2,1/2) wavevector as other Fe-based
superconductors7,22 suggesting commonality in the magnetic response. To explore the
magnetic excitations, inelastic neutron scattering measurements were performed using
the MERLIN spectrometer at the ISIS neutron scattering facility (x=0.27) and the
ARCS spectrometer at the Spallation Neutron Source (x=0.49). Measurements of lower
energy excitations were performed using the HB1 (x=0.27) and HB3 (x=0.49) triple-
axis spectrometers at the High Flux Isotope Reactor. The single crystal samples of
Fe1.04Te0.75Se0.25 and FeTe0.51Se0.49 studied here were prepared as in ref. 23. Bulk
measurements indicate weak, likely filamentary, superconductivity in Fe1.04Te0.73Se0.27
and bulk superconductivity in FeTe0.51Se0.49.
Figure 2a-h summarizes the measured magnetic excitation spectrum for several
energy transfers for both the x=0.27 and x=0.49 samples. Before proceeding to describe
the data in detail we first note that the observed spectrum of magnetic excitations is very
two-dimensional (2d) in nature. For these measurements, wavevector transfer
perpendicular to the (HK0) plane varies with incident energy; consistency of the data
for differing incident energies is evidence for this two-dimensionality. Furthermore, fits
to a 2d model (equation 1 below) can also reproduce the measured spectrum at different
sample rotations (see supplementary information) providing quantitative evidence for
two-dimensionality. This is consistent with recent measurements of the magnetic
resonance in a single crystal of FeSe0.4Te0.6 indicating two-dimensional excitations22.
The low energy magnetic response (figure 2a-b) for the x=0.27 sample, is
characterized by two peaks at incommensurate wavevectors near (1/2,1/2). Interestingly
the data does not show four-fold symmetry around this wavevector as expected for a
tetragonal system, but rather form a quartet around the (1,0) wavevector. With
increasing energy, the peaks disperse away from (1/2,1/2) towards (1,0) as shown
schematically in Fig. 1b. At higher energies, the excitations continue to disperse
towards (1,0) but also evolve from spots into rings (Fig. 2c) centred on this wavevector.
Eventually, as shown for an energy transfer of 120 meV in Fig. 2d, the excitations
evolve into broad spots centred at (1,0). For the superconducting x=0.49 sample, the
low energy spectrum appears as a series of asymmetric spots (Fig. 2e). However, data
measured at a higher energy transfer of 22 meV (Fig. 2f) shows incommensurate peaks
similar to those in the x=0.27 sample. This can easily be understood by considering that
the size of the incommensuration away from (1,0) is larger in the x=0.49 sample such
that the pair of peaks around (1/2,1/2) have moved closer together and overlap
significantly. At an energy of 45 meV (Fig. 2g) the magnitude of the incommensuration
appears similar in the two samples but the x=0.49 scattering appears to have not fully
evolved into the rings of scattering present in the x=0.27 sample (Fig. 2c). At high
energies, the scattering is similar in the two samples as can be seen by comparing Fig.
2d and 2h. For both compositions, the excitations persist for energy transfers as large as
300 meV with Q-dependence similar to that shown at 120 meV for all higher energies.
Examination of the wavevector describing these excitations reveals similarities
with the high-TC cuprates. The quartet of peaks are characterized by wavevectors (1±ξ,
±ξ) and (1±ξ, ?ξ) which, in square lattice notation, corresponds to (π±ξ,π) and (π, π±ξ)
as shown in Fig. 1b. This is precisely the same wavevector as the low energy
excitations observed in La2-xSrxCuO48,9 and YBa2Cu3O6+x10,11 indicating remarkable
commonality in the excitation spectrum of these two classes of high-TC
superconductors. Furthermore, the evolution of the scattering from well defined peaks
at low energies to broadened rings at higher energies is a characteristic property of
magnetic excitations in the cuprates24,25. The magnitude of ξ, however, is much larger
in Fe1+yTe1-xSex resulting in low energy excitations displaced away from (1,0) and much
closer to (1/2,1/2).
At low energies, the largest difference between the two concentrations becomes
evident as shown in Fig. 3a-b (E=6 meV). In addition to the incommensurate
excitations, an additional component centered near (1/2,0) is present in the x=0.27
sample (also visible in Fig. 2a). With decreasing energy, the intensity of this
component increases and eventually forms the short range order observed previously for
samples with a similar concentration21,26. It has been suggested27 that excess Fe in Te
rich samples results in local moments that may provide a pair breaking mechanism
which destroys superconductivity. The component of scattering observed near (1/2,0)
is absent in the x=0.49 sample with no excess Fe (Fig. 3b). Furthermore, the scattering
near (1/2,0) exists inelastically for all energies below ~10 meV and, as such, exists well
below the superconducting gap potentially providing a pair breaking mechanism. These
observations are consistent with the additional component near (1/2,0) existing as a
result of the influence of additional Fe in the x=0.27 sample.
To quantify the dispersion of the incommensurate excitations, the data was fit
using the phenomenological Sato-Maki (SM) function28 previously used for the
The specific form for R(Q) in equation 2 describes the incommensurate excitations
where (HC,KC) represents the wavevector about which the excitations are four-fold
symmetric, in this case (HC,KC)=(1,0). As discussed below, δ parameterizes the
dispersion, λ defines the evolution of the spectrum from peaks (large λ) to rings (small
λ), and κ is a broadening parameter.
The result of fits to the data presented in Fig. 2a-h using the SM function
convolved with instrumental resolution are shown in Fig. 2i-l(2m-p) for x=0.27(0.49).
The fits agree well with the measurements over the full range of measured energies.
The quality of the fits using the SM function is also demonstrated in Fig. 4d-g for cuts
along the line (1/2±ξ, 1/2?ξ) indicating excellent agreement over a wide range of
wavevector and energy transfer. The same SM function can be used to describe the
additional component in the x=0.27 sample near (1/2,0) with the R(Q) factor (equation
2) rotated by 45 degrees (see supplementary information for the equation describing
both components) and the fits displayed in Fig. 3c as well as Fig. 2i contain both
components again yielding excellent agreement with the data.
The best fit value of δ parameterizes the dispersion of the incommensurate
excitation. If we define the wavevector of the incommensurate excitation as qinc = (1±ξ,
?ξ), the incommensuration, ξ=δ/ √2. The resulting dispersion is shown in Fig. 4a-b and
demonstrates excitations dispersing from a wavevector near (1/2,1/2) (ξ=0.5) towards
(1,0) (ξ=0). The shape of the dispersion is reminiscent of the “hour-glass” dispersion
observed in the cuprates25,29,30. However, unlike the cuprates, the high energy
excitations of Fe1+yTe1-xSex remain centered near (1,0) with no evidence for dispersion
away from (1,0). The difference between the two concentrations is emphasized in Fig.
4c. For energies greater than about 50 meV, the dispersions are consistent for the two
concentrations. However, for lower energies, |ξ| is larger for x=0.49 indicating
excitations displaced closer to the (1/2,1/2) wavevector (ξ=0.5) with larger Se content.
This can also be clearly seen in comparing the cuts in Figs. 4d with 4f (25 meV) and 4e
with 4g (70 meV). The peaks in the 25 meV data are at clearly different wavevectors,
but the distribution of scattering intensity is nearly identical at 70 meV for the two
concentrations. It is interesting to note that the sample (x=0.49) with normal state
excitations closer to the resonance wavevector exhibits bulk superconductivity while
only weak superconductivity is observed in the sample with normal state excitations
displaced further from this wavevector. However, attributing such a wavevector shift to
enhanced superconductivity is complicated by the presence of excess Fe in the x=0.27
Finally, we comment on the nature of the magnetic excitations in these materials.
The itinerant or local moment nature of the magnetism in Fe-based superconductors is
an important, unresolved question. Local moment antiferromagnetic correlations, i.e.
spin waves, would result in rings of scattering in constant energy slices whose diameter
increases with increasing energy transfer. The observed spectrum (Figure 2) consists of
incommensurate peaks at low energy that disperse toward the (1,0) wavevector
inconsistent with expectation for local moment spin waves. Another explanation for the
quartet of peaks around (π,π) in the cuprates was overlapping excitations from a stripe
model30. In this scenario, streaks of intensity produce enhancement at intersecting
wavevectors. The large value of incommensuration in Fe1+yTe1-xSex makes such an
explanation unlikely. To generate the quartet of peaks around (1,0) requires streaks
propagating along H and K. However, such streaks would also generate considerable
overlap between neighboring zones as the incommensuration is large. This is best
shown in Fig. 2b where overlapping stripes along H and K would result in considerable
intensity near (-0.7,-0.7), clearly not present in the data. Ruling out local moment
antiferromagnetism and stripes, it is reasonable to conclude that the observed excitations
for Fe1+yTe1-xSex are predominately itinerant in nature.
1. Kamihara, Y., et. al. Iron-based layered superconductor La[O1-xFx]FeAs (x = 0.05-
0.12) with TC = 26 K. J. Am. Chem. Soc. 130, 3296-3297 (2008).
2. Chen, X. C., et al. Superconductivity at 43 K in SmFeAsO1-xFx. Nature 453, 761-762
3. Rotter, M., Tegel & M., Johrendt, D. Superconductivity at 38 K in the iron arsenide
(Ba1-xKx)Fe2As2. Phys. Rev. Lett. 101, 107006 (2008).
4. Christianson, A. D., et al. Unconventional superconductivity in Ba0.6K0.4Fe2As2 from
inelastic neutron scattering. Nature 456, 930-932 (2008).
5. Lumsden, M. D., et al. Two-dimensional resonant magnetic excitation in
BaFe1.84Co0.16As2. Phys. Rev. Lett 102, 107005 (2009).
6. Chi, S., et al. Inelastic neutron-scattering measurements of a three-dimensional spin
resonance in the FeAs-based BaFe1.9Ni0.1As2 superconductor. Phys. Rev. Lett. 102,
7. Mook, H. A., et al. Neutron scattering patterns show superconductivity in
FeTe0.5Se0.5 likely results from itinerant electron fluctuations. arXiv:0904.2178 (2009).
8. Cheong, S-W., et al., Incommensurate magnetic fluctuations in La2-xSrxCuO4. Phys.
Rev. Lett. 67, 1791-1794 (1991).
9. Mason, T.E., et al., Magnetic dynamics of superconducting La1.86Sr0.14CuO4. Phys.
Rev. Lett. 68, 1414-1417 (1992).
10. Dai, P., Mook, H. A., and Dogan, F., Incommensurate magnetic fluctuations in
YBa2Cu3O6.6, Phys. Rev. Lett. 80, 1738-1741 (1998).
11. Mook, H. A. et al., Spin fluctuations in YBa2Cu3O6.6, Nature 395, 580-582
12. Chen, G. F., et al., Superconductivity at 41 K and Its Competition with Spin-
Density-Wave Instability in Layered CeO1-xFxFeAs. Phys. Rev. Lett. 100 247002
13. Ren, Z. A., et al. , Superconductivity in the iron-based F-doped layered quaternary
compound NdO1 - xFx] FeAs. Europhys. Lett. 82 57002 (2008).
14. Ren Z, et al., Superconductivity at 55 K in Iron-Based F-Doped Layered Quaternary
Compound Sm[O1-xFx] FeAs, Chin. Phys. Lett. 25, 2215.
15. Wang, X. et al., The superconductivity at 18 K in LiFeAs system. Solid State
Comm. 148, 538 (2008).
16. Hsu, F.-C. et al., Superconductivity in the PbO-type structure α-FeSe. PNAS 105,
17. Yeh, K.-W. et al., Tellurium substitution effect on superconductivity of the α-phase
iron selenide. Europhys Lett. 84 (2008) 37002.
18. de la Cruz, C. et al., Magnetic order close to superconductivity in the iron-based
layered LaO1-xFxFeAs systems. Nature 453, 899 (2008).
19. McGuire, M. A. et al., Phase transitions in LaFeAsO: Structural, magnetic, elastic,
and transport properties, heat capacity and Mössbauer spectra. Phys. Rev. B 78, 094517
20. Huang, Q. et al., Neutron-Diffraction Measurements of Magnetic Order and a
Structural Transition in the Parent BaFe2As2 Compound of FeAs-Based High-
Temperature Superconductors. Phys. Rev. Lett. 101, 257003 (2008).
21. Bao, Wei, et al., Tunable (δπ, δπ)-Type Antiferromagnetic Order in α-Fe(Te,Se)
Superconductors. Phys. Rev. Lett. 102, 247001 (2009).
22. Qiu, Y., et al., Spin Gap and Resonance at the Nesting Wavevector in
Superconducting FeSe0.4Te0.6. arXiv:0905.3599 (2009).
23. Sales, B. C., et al., Bulk superconductivity at 14 K in single crystals of Fe1+yTexSe1-
x. Phys. Rev. B 79, 094521 (2009).
24. Stock, C., et al., From incommensurate to dispersive spin fluctuations: The high
energy inelastic spectrum in superconducting YBa2Cu3O6.5. Phys. Rev. B 71, 024522
25. Vignolle, B., et al., Two energy scales in the spin excitations of the high-
temperature superconductor La2-xSrxCuO4. Nature Physics 3, 163 (2007).
26. Wen, J., et al., Coexistence and competition of short-range magnetic order and
superconductivity in Fe1+δTe1-xSex, arXiv: 0906.3774 (2009).
27. Zhang, L., Singh, D. J. & Du, M. H. Density functional study of excess Fe in
Fe1+xTe: Magnetism and doping. Phys. Rev. B 79, 012506 (2009).
28. Sato, H. and Maki, K., Theory of inelastic neutron scattering from Cr and its alloys
near the Néel temperature. Int. J. Magnetism 6, 183 (1974).
29. Hayden, S. M. et al., The structure of the high-energy spin excitations in a high-
transition-temperature superconductor. Nature 429, 531 (2004).
30. Tranquada, J.M. et al., Quantum magnetic excitations from stripes in copper oxide
superconductors. Nature 429, 534 (2004).
Author Contributions Statement: All authors made critical comments on the manuscript. M. L, A. C., E.
G., S. N., M. S., D. A., T. G., G. M., C. C., and H. M. all contributed to data collection. A. S. M. M., B.
S., and D. M. contributed to sample synthesis and characterization.
We acknowledge discussions with David Singh. This work was supported by the Scientific User Facilities
Division and the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US
Department of Energy. We acknowledge discussions with David Singh.
Correspondence and requests for materials should be addressed to M. L. (firstname.lastname@example.org).
Figure 1: (a) Reciprocal space for Fe1+yTe1-xSex compounds. Black (red) labels
correspond to wavevectors in the tetragonal (square) reciprocal lattice. Red
circles represent the square lattice (π,0) points where the resonance is
observed while green squares represent the square lattice (π,π) points. (b) The
same reciprocal space diagram as (a) but centred at (1, 0). The blue ovals
show the location of the observed incommensurate excitations and the arrows
represent the direction of dispersion with increasing energy transfer.
Figure 2: Constant energy plots of the magnetic excitations in Fe1.04Te0.73Se0.27,
a-d, and FeTe0.51Se0.49, e-h. All measurements were performed with the c-axis
parallel to the incident beam and the sample temperature was 5 (3.5) K for the
x=0.27 (0.49) measurements. Panels a-d show data from the x=0.27 sample
measured using MERLIN with incident energies of 25, 60, 120, and 250 meV,
respectively. Panels e-h show data from the x=0.49 sample measured using
ARCS with incident energies of 40, 60, 120, and 250 meV, respectively. The
x=0.27(0.49) samples were single crystals with a mass of 16.91(15.45) g. The
data is fit with a model function (equation 1) convolved with instrumental
resolution and the resulting fits are shown in i-l (m-p) for the x=0.27(0.49)
sample. The fit (i) to the 10 meV data (a) for the x=0.27 sample includes two
components rotated by 45 degrees in the H-K plane. All other fits include only a
single component. All plots show a projection of the data onto the H-K plane.
Figure 3: Constant energy plots through the magnetic excitation spectrum at an
energy transfer of 6±1 meV for both the x=0.27 and x=0.49 samples.
Measurements of the x=0.27 (0.49) sample a, (b) were performed with an
incident energy of 25 (40) meV. The temperature was 5 (3.5) K for the x=0.27
(0.49) measurements. This shows an additional component in the x=0.27 data
centred near (1/2, 0) which is absent in the x=0.49 data. The fits to a model
function (equation 1) convolved with instrumental resolution (c-d). Fits to the
x=0.27 data (c) included two components rotated by 45 degrees with respect to
one another in the H-K plane while fits to the x=0.49 data (d) included only the
single, incommensurate component.
Figure 4: Dispersion of the magnetic excitation spectrum in Fe1+yTexSe1-x for
x=0.27 (a) and x=0.49 (b). The dispersion, ξ, is related to δ (see equation 1) as
ξ=δ/ √2. Below 15 meV, the dispersion was extracted from triple-axis
measurements in the (HK0) scattering plane via a convolution of equation 1 with
instrumental resolution. The dispersion above 15 meV was extracted from the
time-of-flight data. For the x=0.27 sample, all data analyzed was measured at
3.5 K (triple-axis) or 5 K (time-of-flight). For the x=0.49 sample, time-of-flight
data analyzed was measured at 3.5 K while the triple-axis data analyzed was
measured at 20 K to avoid scattering from the magnetic resonance. Fits to the
time-of-flight data were performed both with and without inclusion of
instrumental resolution. As resolution was found to have very little effect on the
extracted value of δ, the dispersion was generated without the inclusion of
resolution for simplicity. The fits generate a single value of δ for each energy
transfer. The plotted dispersion is symmetrized to illustrate the symmetry about
the (1, 0) wavevector. For energies above 100 meV, the peak position cannot
be distinguished from ξ=0. The inset, (c), shows a direct comparison of the
dispersion for the two concentrations demonstrating low energy excitations
closer to (1/2,1/2) for the x=0.49 sample. Cuts at 25±3 meV were measured
with incident energy of 120 meV for the x=0.27 (d) and x=0.49 (f) samples.
Cuts at 70±5 meV were measured with incident energy of 250 meV for the
x=0.27 (e) and x=0.49 (g) samples. Solid lines represent fits using equation 1
showing excellent agreement with the data.