- Email: [email protected]
The use of polarized neutrons in determining the magnetic scattering by iron and nickel
J. Phys.
Chem. SoIds
Pergamon
Press 1959. Vol. 10. pp. 138-146.
THE USE OF POLARIZED THE MAGNETIC
NEUTRONS IN DETERMINING
SCATTERING
BY IRON AND
NICKEL*
R. NATHANS Pennsylvania
State University,
University
Park, Pennsylvania, Upton, New York
and Brookhaven
National Laboratory,
C. G. SHULL Massachusetts
Institute of Technology,
Cambridge,
Mass.
G. SHIRANE Brookhaven
National Laboratory,
Upton, New York
and A. ANDRESEN Brookhaven National Laboratory,
Upton, New York and Joint Establishment (Received
29 December
for Nuclear Research, Kjeller, Norway
1958)
Abstract-The polarized neutron-beam technique is discussed and utilized in determining the magnetic scattering by iron and nickel. Procedures for obtaining and using polarized beams of monochromatic neutrons are outlined, and experimental data are presented on the performance of two polarizing crystals, Fe304 and 92 Co-8 Fe alloy. The reflectivity of neutrons of different polarization states by single crystals of iron and nickel has been studied and used for determining the magnetic scattering amplitudes at various scattering angles. Comparison is then made between the observed magnetic form values and theoretical values derived from calculated 3d wave functions. 1. INTRODUCTION
THE advantages in using polarized beams of neutrons in diffraction studies of magnetic materials have been known for some time, and the present report (a) summarizes some of the experiences gained in modifying a typical neutron-diffraction spectrometer to polarized-beam use and (b) presents experimental data obtained with this technique on the absolute magnetic scattering of the ferromagnetic elements, iron and nickel. The ferromagnetic scattering for these elements as measured with unpolarized radiation is unfortunately rather small in comparison with the nuclear scattering, about 8 and 0.6 per cent, respectively, so that it is difficult to measure this quantitatively for comparison with the expected absolute level or to draw comparisons with the theoretical form-factor varia* This research was supported by the U.S. Atomic Energy Commission, the National Security Agency and the Air Force Office of Scientific Research.
tion. Much of this difficulty is surmounted by the use of polarized radiation, whereby it becomes possible to measure accurately very small magnetic scattering cross-sections. In addition to the enhanced sensitivity of the polarized-beam method, two other advantages are worthy of mention. It is possible by these techniques to determine the absolute sign of the magnetic scattering amplitude associated with a particular magnetic lattice reflection, thereby resolving certain ambiguities that arise in the determination of a magnetic crystal structure. This has been utilized in a study of the magnetic structure existing in ordered FesAl and is described in a companion publication.(r) Additionally there exist interesting changes in polarization upon magnetic scattering as predicted by the early theory, and these have been confirmed by experiment and utilized to establish(z) details of the magnetic axis orientation
in
the
CraOs and a-FesOs. 138
antiferromagnetic
materials
POLARIZED
NEUTRONS
IN
DETERMINING
~~NIQ~
MAGNETIC
SCATTERING
139
(3)
predicted from the known scattering amplitudes and to use such polarized radiation in a quantitative study of the magnetic scattering from iron and nickel. It is apparent from the form of equations (3) and (4) that the contribution of a weak magnetic scattering amplitude to the scattering cross-section will be more pronounced for the ~l~ized-earn case. For the production of a monochromatic polarized beam, we have used Bragg scattering by a suitable ferromagnetic crystal which exhibits the favorable balance of nuclear and magnetic scattering indicated above. The (220) reflection of Fe304 has been known to produce high polarization,@) and a study has been made of the polarizing efficiency of this crystal as a function of crystal thickness, lack of magnetic saturation in the crystal and the presence of extinction effects. Additionally when single crystals of a cobalt-iron alloy became available, for which a favorable balance of the scattering amplitudes also occurs, these were studied and their superiority over Fe304 crystals was demonstrated.
With unpolarized incident radiation, the scattering cross-section becomes half of the sum of the two spin-state cross-sections or
A schematic diagram of our experimental arrangement is shown in Fig. 1. It consists of a modified double-crystal neutron spectrometer in
2.
GENERAL
CON-
AND
HALPERN and JOHNSON have shown,@) in a com-
prehensive treatment of the magnetic scattering theory, that the scattering cross-section for a magnetic ion is given by: o = ~(b~+p2q2+2b~
* A),
(11
where 6 and p are the nuclear and magnetic scattering amplitudes, A is the unit vector describing the neutron polarization directed along its spin axis, and q = e(e * x)-w
(2)
with e the unit scattering vector and K the unit vector along the spin direction of the magnetic ion. If K is arranged perpendicular to the scattering vector e, then IqI = q becomes unity, and the crosssections for the two neutron spin states (analyzed such that q * Ih = &l> that make up an unpolarized neutron beam can be written as a+=en(b+p)swithq*h=
+l
and cr- = 4-n(b-p}2 with q * h = -1.
3.
POLARIZBD BEAM SFECI’ROMBTER
which can be considered to be the sum of the nuclear and magnetic scattering cross-sections. It is obvious that a magnetic ion for which b is closely equal to p will scatter neutrons of only one spin state under the above conditions and hence a high polarization P will result in the scattered beam, where p=-
k--12 IlSIZ
and 4 and Ia are the intensities of the two neutron spin states. Unequal scattering amplitudes will of course yield only partial polarization. The detection and utilization of such a polarized neutron beam has been discussed in the litera. ture,@*s) and the present investigation was undertaken as an extension of these studies. It was desired to study the polarizing efficiency of various crystals for quantitative comparison with that
Fro. 1. Schematic diagram of polarized neutron spectrometer.
which the first crystal is replaced with a magnetized polarizing crystal and the second crystal serves as a polarization analyzer or a specimen crystal. It is necessary to magnetize the polarizing crystal to insure that x and hence q above be unidirectional, and for this a permanent magnet giving a field of 2200 Oe over a g-in. gap was used. The reflected
140
R.
NATHANS,
C.
G.
SHULL,
monochromatic neutrons pass along a magnetic collimator which maintains a uniform field of 150 Oe, in the same sense as the polarizing field, throughout the trajectory between the polarizing crystal and the second crystal. An electromagnet served for magnetizing the analyzing or test crystal with fields up to 7500 Oe over a Q-in gap. Considerable care had to be taken to control the magnetic field over the entire path from polarizer to analyzer to prevent induced transitions from one neutron spin state to another, thereby affecting the degree of polarization in the beam. For certain types of measurement it was desirable to depolarize the beam completely, and this was accomplished by passing the beam through a thin foil of unmagnetized iron (0.020 in thick). In such an unmagnetized foil, the random direction of the magnetic domains cause many spin transitions and the emerging beam is completely depolarized. Provision was also incorporated into the measurement technique for reversing the neutron polarization relative to the sense of magnetization for the analyzing or test crystal. Merely reversing the analyzing magnetic field direction does not accomplish this, since the neutron polarization would tend to follow the field reversal along its trajectory. If the field reversal region can be made small enough so that the neutron transit time through this region is small compared to its Larmor precession time, the neutron polarization will be maintained and the polarization will be reversed with respect to the analyzing field as desired. This method has been used by BURGYet al.@) In the present experiments it was found difficult to meet the stringent requirements of this procedure, and accordingly an alternative method of invoking the neutron spin transition through resonant spin flipping was utilized.* In this procedure a radio-frequency field, applied at right angles to the neutron polarization, is matched in frequency to the Larmor precessional frequency of the neutron spin in a uniform magnetic field. If the strength of the r.f. field is adjusted properly relative to the neutron transit time, complete transition is possible, and this has been confirmed to a high degree by experiment. Thus, by merely turning on or off the r.f. flipper, the neutron polarization can be kept parallel or antiparallel to * STANFORD et al.(S)have utilized this technique.
G.
SHIRANE
and
A.
ANDRESEN
the common field direction in the polarizing, intermediate, and analyzing regions. It is of convenience in treating the experimental measurements to make use of a quantity S, commonly called the shim ratio and defined as the ratio of the scattering intensity obtained with a polarized beam of polarization Pr to that obtained with a depolarized beam. Thus:
with I+ and I_ being the scattered intensity with incident neutron polarization being parallel and antiparallel to the magnetization of the scattering crystal and 10 the scattered intensity with a depolarized incident beam. This term has been called the shim ratio because in practice it is obtained by inserting a thin shim of unmagnetized ferromagnetic material in the beam thereby depolarizing it. Of usefulness is a related quantity R called the polarization ratio defined as the ratio of I+ to I_ so that (7) For a test crystal whose polarizing efficiency is Pz, defined in the sense of equation (S), it can be easily shown that s = l+PtP,
(8)
where it is to be recognized that the intrinsic polarizing efficiency Pz will depend upon the degree of extinction present in the crystal, since this will affect the spin-state scattered intensities. Pf represents the degree of polarization in the neutron beam incident on the test crystal. For the extinction-free case, Pz will be simply’ related to the cross-sections : Q+-
Pz
= -=-.
6-
o++(I-
%’ bz+Pz
(9)
Equation (8) can be used for assessing the relative polarizing efficiencies of different crystals, but a knowledge of Pi is necessary to determine the absolute efficiency.
It has been found experimentally that sizeable depolarization of the neutron beam accompanies its passage through crystals, and hence that it is necessary to include a depolarization factor in
POLARIZED
equation
NEUTRONS
IN
DETERMINING
(8). It is convenient to define thepolarixaD, such that
&on ~~ff~~~.~,
P =
PO*D T=POexp(-_tcnt)
MAGNETIC
141
SCATTERING
measurements were next made of the polarization transmission D for different crystals, so that the true polarizing efficiencies could be assessed. This was performed by placing the test crystal
with P being the transmitted polarization, .FQ the incident polarization, pP the linear depolarization coefficient, and t the traversed thickness of the crystal. With this definition, equation (8) becomes
~q-4-~p~)l~z
s = 1+&-&U-
(10)
and for small depolarization l6 0
1 CRYSTAL
The polarization transmission wifl be strongly dependent upon the magnetization of the scattering crystal, approaching unity as the magnetization approaches saturation. With less than saturation, the neutron passage from one domain to another will be effective in changing the neutron polarization state as HALPERN and HOLSTEIN(~) have considered. 4,
~~E~~ ~FIC~NC~
ON POLARIZING OF CRYSTALS
Experiments were first performed on a number of Fe304 polarizing and analyzing crystals cut from natural magnetite ingots. These crystals were cut with different shape factors, ranging from blocks to plates to pillars, and with different thicknesses so as to permit study of the internal depolarization effects. With a common polarizing crystal whose polarizing efficiency was later determined, the shim ratio 5’ of a series of analyzing crystals was measured as a function of analyzing field strength and crystal thickness, Fig. 2 shows the variation of S obtained with a series of crystal pillars of common area (19-l x 3.2 mm), but with different thicknesses with each maintained in a magnetic field of 6900 Oe. The variation of S with thickness shows that the effective polarizing efficiencies of these pillars decreases as the thickness is increased, and this is considered to result from the increased internal depolarization within the crystal. The alternative explanation that such a variation could result from increased extinction effects was eliminated by a calculation making use of the known mosaic distribution in these crystals.
Fro,
2.
/ 2
L-l
4
3
THICKNESS,
5
mm
Variation of shim ratio S with Fe804 crystal thickness.
directly in front of, and in the same magnetic field as, an analyzing crystal. The polarized beam from the polarizer then passed through the test crystal, but was not scattered by it, before being Bragg scattered by the analyzer. measurements of the shim ratio with and without the test crystal could then be used to determine directly the polarization
MAGNETIC
FIELD!
oeeteds
FIG. 3. Dependence of polarization transmission D upon magnetic field applied to crystal fur two different Fe304 crystals. The relative magnetization for crystal (B) is also shown.
transmission of the test crystal. This was studied as a function of magnetic field strength, and typical results are shown in Fig. 3 for two crystals. Crystal (A), of dimensions 19.2 x 3.2 mm area and l-24
142
R.
NATHANS,
C.
G.
SHULL,
G.
mm thick was cut so that a [loo] axis was parallel to the long dimension and the magnetic field was applied in this direction. Crystal (B), of dimensions 16.5 x 3.5 mm area and 2.1 mm thick, was cut with a [ill] direction parallel to the long dimension and again the magnetic field was applied in this direction. Also shown in Fig. 3 are measured values for the relative magnetization of crystal (B). It is seen that the polarization transmission is exceedingly sensitive to departure from magnetic saturation, as had been predicted by HALPERN and HOLSTEIN(7)and studied experimentally by HUGHES and colleagues,@,*) Since the easy direction of magnetization in magnetite is along the [ll l] direction, the depolarization effects are expected to be somewhat less for crystals magnetized in this fashion and the data of Fig. 3 demonstrate this. I
I
I
MAGNETIC
I
FIELD,
I
I
I
I
SHIRANE
magnetization.
and
A.
ANDRESEN
It can be easily shown that PZ ----=P&ax>
M MS
and a comparison is illustrated in Fig. 4 of the intrinsic polarizing efficiency and the relative magnetization in a limited low field region. As seen in Fig. 4, the maximum value for the intrinsic polarizing efficiency of crystal (B) is seen to be about 95 per cent. A number of different Fe304 crystals have been studied, and some variation in this maximum value has been noted. Values as high as 98 per cent have been obtained, this having been measured with the particular crystal used in the early stugy of reference (4). Since all of the crystals studied have been cut from natural mineralogic ingots of unmeasured purity, these relatively small variations are not unexpected. Similar studies have been carried out on polarizing and analyzing crystals of cobalt-iron alloy. For the composition 92 atomic per cent cobalt, this crystallizes from the melt* in the face-centered cubic phase, and by calculation, the intrinsic polarizing efficiency in the (111) reflection should be 99.1 per cent. In this phase, magnetic saturation
oersteds
FIG. 4. Polarizing efficiency of Fe304 crystal function of applied magnetic field.
(B) as a
Once the depolarization factors have been evaluated, a pair of crystals having the same intrinsic polarizing efficiency can be used as a polarizing-analyzing combination to determine the individual polarizing efficiency. Data obtained in this fashion for crystal (B) above are illustrated in Fig. 4. Here the polarizing efficiency of this crystal has been evaluated from shim ratio measurements as a function of applied magnetic field. Also shown in the figure is the variation of the intrinsic polarizing efficiency after correction for internal depolarization through the use of the polarization transmission data for this crystal from Fig. 3. Even after correction for depolarization effects, the intrinsic polarizing efficiency should fall off in the low-field region because of the reduction in crystal
a
1
2 MAGNETIC
3 FIELD,
4
5
6
7
k-oersteds
FIG. 5. Polarizing efficiency of cobalt-iron crystal as a function of applied magnetic field.
is approached with rather low fields. Fig. 5 illustrates the variation of the intrinsic polarization efficiency as a function of field for a crystal of dimensions 19.2~ 3.4 mm in area and 1.1 mm in thickness, with the magnetic field applied along the * We are much indebted to Messrs. R. M. BOZORTH and A. J. WILLIAMS, of the Bell Telephone Laboratories, for supplying such a crystal to us.
POLARIZED
NEUTRONS
IN
DETERMINING
[I101 direction parallel to the long axis. This is seen to approach a value of 99-100 per cent. It has been found in general that the depolarizing effects are smaller for the case of the metallic crystals than has been observed for magnetite and hence smaller magnetic fields can be tolerated. It should be mentioned that the neutronpolarization direction obtained with the metallic crystals is reversed from that obtained with the (220) Peso4 reflection. This follows from the fact that the magnetic ions responsible for the scattered intensity in (220) Fe304 are directed opposite to the applied field on the crystal, since it is a ferrimagnetic substance and these ions are in the minority. In contrast the magnetic ions in the metallic crystal are all parallel to the applied field direction. Thus the polarized neutrons in the (220) Fe304 reflection have their magnetic moments aligned parallel to the field on the crystal, whereas in the (111) cobaltiron reflection, the neutron moments are antiparallel to the field direction. This absolute sense of direction follows from a theoretical assay of the absolute signs of the magnetic and nuclear scattering amplitudes and has been examined experimentally in a neutron STERN-GERLACH type of experiment by SHERWOODet al.(s) A decided advantage to the use of cobalt-iron polarizing crystals lies in their greater neutron reflectivity than is the case for the (220) Fe304 reflection. This arises because of the more favorable crystal-structure factor and the greater density of the magnetic scattering ions. Moreover the higherorder contaminant neutrons are largely absent in the metallic-crystal case. The magnetic structure of Fe304 is such that the (440) reflectivity is approximately 15 times larger than the (220) reflectivity, and hence in scattering neutrons of wavelength ,! from the (220) reflection there is a strong component of h/2 scattered simultaneously. The h/2 component is partially polarized in the opposite direction, and it is necessary to absorb this with a suitable neutron filter. This was done in all of the above Fe304 measurements with a plutonium filter (useful when h = 1.05 A) and this necessarily absorbs some of the desired intensity as well. This problem does not arise in the case of the (111) cobalt-iron polarizing reflection, since the (222) crystal-structure factor is of the same order of size as the (111) and no enhancement of the /\/2 component is thus obtained.
MAGNETIC
SCATTERING
143
5. MAGNETIC SCATTERING RE!XJLTS ON IRON AND NICKEL
The study of the angular dependence of the absolute magnetic scattering in these metals was made by measuring the polarization ratios (R) for the various Bragg reflections present. As a typical example of such a measurement, Fig. 6 shows the
. ANTI-PARALLEL FIELD
COUNTER
TO ON CRYSTAL
ANGLE
FIG. 6. Intensity in the (111) nickel reflection for incident neutron beams of different polarizations.
intensities observed for the nickel (111) reflection with incoming neutrons polarized first parallel and then antiparallel to the magnetic field on the crystal. The sizeable intensity change for the two orientations of scattered neutrons (1.5 : 1) is to be compared with the O-6 per cent obtained in the usual technique with unpolarized neutrons. From the observed values of &kl the intrinsic polarizing efficiencies for various Bragg reflections can be evaluated after the necessary corrections have been made for extinction and depolarization. The final result is a value for the ratio of the magnetic scattering amplitude to the nuclear scattering amplitude, that is ph&l/b&kl. Since the coherent nuclear scattering amplitudes for these two metals are well established (1 a03 x 10-1s cm for nickel and 0.96 x lo-12 cm for iron) and are independent of angle, the magnitude of phkl follows. In the study of both of these metals, the measurements were made, for the most part, with single-crystal pillars placed in the high magnetic field of the analyzing magnet. Intensities were recorded over short-time intervals alternately with r.f. field on and off. Each run was repeated several
144
R.
NATHANS,
C.
G.
SHULL,
times, and the uncertainty in the final results was established from the standard deviations between the different runs. Since the data accumulation extended over a considerable period of time, it was necessary to establish the stability of the incident neutron polarization and the efficiency of the r.f. coil in reversing the neutron polarization. For this purpose the (111) reflection of a cobalt-iron crystal was used as a standard. Measurements of the polarization ratio for this standard were made before and after each run on the test crystal, and the required adjustments, although small, were made to the observed polarization ratio on the test crystal.
G.
SHIRANE
and
A.
ANDRESEN
of the two groups of 3d electrons in nickel. Such a separation can result from the intra-atomic exchange forces which will be more effective for greater number of positive spin electrons than for the lesser number of electrons in the negative spin state.
(a) Nickel The single crystal of nickel was cut so that it could be magnetized along the [llO] direction. The pillar measured 19.1 x 3.2 mm in area and O-55 mm in thickness. A chemical analysis revealed only small traces of impurities. Calculations using the known mosaic distribution and measurements taken at several different neutron wavelengths showed the extinction effects to be negligible. As an additional check, the polarization ratio for the most intense nickel reflection was repeated on a pillar of ~lycrystalline nickel and yielded the same result as the single-crystal sample. Finally, by the method outlined above, it was demonstrated that there was no significant depolarization of the neutron beam within the nickel sample. Table 1 lists the resultant values of p, the magnetic scattering amplitude, and f, the magnetic form factor for the various Bragg reflections on which data were taken. These figures are plotted in Fig. 7 as a function of 4rr sin 8/h. In the forward direction the magnetic scattering amplitude is proportional to the room-temperature saturation moment of nickel, and this value is shown on the curve. For comparison with the neutron data, we have drawn in the Hartree-Fock calculated atomic form factor for the 3d electrons in Cut, which possesses the same number of outer electrons as nickel. A significant variation between the two curves is evident. Recalling that the neutron magnetic scattering comes from the difference between the electrons in the two spin states in the 3d shell in nickel, while the Cu+ data represents the sum of all the 3d electrons, the curves in Fig. 7 suggest that there is a difference in the electronic configurations
46
sin e/h,
A-’
FIG. 7. Observed magnetic scattering amplitudes different nickel reflections and comparison retical form-factor curve.
for with theo-
Table 1. Nickel magnetic scattering amplitudes and form-factor values for observed reflections
kkl
(lO-42cm)
(111) P3 (220)
0~1140&0@015 0~1~4~0~0010 0.0646&0.0016 0+)459$,0~0012 0+418~0~0015 0-0234~0~0021 O-0258 rtO+O30 0~0160:~0~0035
(311) (222) (4001 (3311 (333)
f 0.678f0.006 0~4193t0*010 0*297&0+08 0.271 +O.OlO O-15210.014
(b) Iron In obtaining the iron data, we used a single crystal of an iron-silicon alloy (3.9 weight per cent silicon). This crystal again was cut in the form of a thin pillar (19.2~ 3.9 mm in area and 0.41 mm thick) with its long axis as a [loo] direction. A part of the data was collected on two crystals of widely differing thickness for the purpose of detecting and estimating the effects of extinction. These effects were studied, in this case, by a procedure somewhat different from the standard techniques. Of
POLARIZED
NEUTRONS
IN
DETERMINING
particular use with polarized beams, this method utilizes the fact that the magnitude of the structure factor of the test crystal, and hence the degree of extinction, is changed by reversing the direction of the incident neutron polarization without altering the sample. If now one records the polarization ratio, R, of a particular reflection for a fixed counter angle as the crystal is rotated off the Bragg peak, the presence of extinction becomes quite evident.
>’ CRYSTAL
AMOLE,
W.
145
SCATTERING
mosaic distribution and not a result of the spread in the wavelength of the scattering neutrons.
Table 2. Iron magnetic scattering amplitudes and farm-factor values for observed reJectio?zs . . _I_-._ kkt
(110)
I
MAGNETIC
(200) (220) (310) (400)
(to-~
cm)
0.334OrtO.0027 0~2180&0~0014 0.095 1 rfr0.0024 0.0609 5 0+@20 0.0348~0*01~
f
-
0619&0,006 0~405&0~004 0~176&0*005 0~113~0+04 0~065*0~020
Our results for iron are summarized in Table 2 and illustrated in Fig. 9. Also drawn in are the form-factor curves calculated by WOOD and PRATT(~)) for the free iron atom (3d+5, 3d_, 4~s).
36”
Fro. 8. Polarization ratio as a function of crystal rocking angle for thick and thin iron crystals.
Fig. 8 shows the results of such a series of measurements taken on two iron crystals for the (110) reflection. For the thicker crystal (Fe No. 1) the polarization ratio at the Bragg peak is suppressed. As the crystal is rotated off the peak, where fewer mosaic domains in the crystal are in a position to reflect, the suppression of R caused by the secondary extinction is reduced-in the limit approaching the extinction free value for crystal angles sufficiently removed from the Bragg peak position. For Fe No. 2, a crystal only one-third as thick, we observed a polarization ratio which remains unchanged as the crystal is rotated, and has a value equal to the limiting ratio observed for the much thicker crystal. This test for extinction strongly suggests that crystal Fe No. 2, which was used for the main body of the iron data, is free of extinction. It should be noted that this technique for estimating extinction is useful only in cases where the width of the observed Bragg reflection is due to the
47-5 sin@%
FIG. 9. Observed magnetic scattering amplitudes for different iron reflections and comparison with theoretical form-factor curves.
These theoretical curves have been normalized to agree with the experimental plot in the forward direction. The f+ curve represents the electronic configuration of the five 3d+ electrons, while the curve f_ corresponds to the single 3d_ electron. If, as in nickel, the neutron data were to represent the difference between the spin density in two antiparallel electron groups, one would expect the experimental points in Fig. 9 to faI1 somewhat above both the f, and f_ curves. Instead, we find the experimental points to be in good agreement with the f, curve representing the electrons in the positive
146
R.
NATHANS,
C.
G.
SHULL,
G.
spin group. Such agreement is consistent with a model for iron in which the magnetic spin density comes from electrons in a single spin state only. This result has been strongly indicated in the recent report of WEISS and DEMARCO(~~) based on their measurements of X-ray scattering factors in the first transition group. A further interpretation of the combined neutron and X-ray results for nickel and iron will be found in the companion paper following this report. Acknowledgement-We
wish
to
thank
Dr.
MCREYNOLDS for conceptual and executional during early stages of this research.
A.
W.
assistance
REFERENCES 1. NATHANS R., PICOTT M. T. and SHULL C. G., J. Phys. Chem. Solids 6, 38 (1958).
SHIRANE
and
A.
ANDRESEN
2. NATHANS R., RISTE T.,
SHIFZANE G. and SHULL C. G. In the press. 3. HALPERN 0. and JOHNSONM. H., Phys. Rev. 55,898
(1939). 4. SHULL C. G., Phys. Rev. 81,626 (1951). 5. STANFORD C. P., STEPHENSONT. E., COCHRANL. W. and BERNSTEINS., Phys. Rev. 94, 374 (1954). 6. BURCY M. T., HUGH= D. J., WALLACE J. R., HELLER R. B: and WOOLF W. E., Phys. Rev. 80, 953 (1950). 7. HALPERN 0. and HOLSTEIN T., Phys. Rev. 59, 960 (1941). 8. HUGH= D. J., WALLACEJ. R. and HOLTZMANR. H., Phys. Rev. 73, 1277 (1948). 9. SHERWOODJ. E., STEPHENSONT. E. and BERNSTEIN S., Phys. Rev. %, 1546 (1954). 10. WOOD J. H. and PRATT G. W., Phys. Rev. 107, 995 (1957). 11. WEISS R. J. and DE MARCO J. J., Rev. Mod. Pkys.
30, 59 (1958). 12. WEISS R. J. and FREEMAN A. J., J. Phys. Chem. Soli& 10,147 (1959).
Chem. SoIds
Pergamon
Press 1959. Vol. 10. pp. 138-146.
THE USE OF POLARIZED THE MAGNETIC
NEUTRONS IN DETERMINING
SCATTERING
BY IRON AND
NICKEL*
R. NATHANS Pennsylvania
State University,
University
Park, Pennsylvania, Upton, New York
and Brookhaven
National Laboratory,
C. G. SHULL Massachusetts
Institute of Technology,
Cambridge,
Mass.
G. SHIRANE Brookhaven
National Laboratory,
Upton, New York
and A. ANDRESEN Brookhaven National Laboratory,
Upton, New York and Joint Establishment (Received
29 December
for Nuclear Research, Kjeller, Norway
1958)
Abstract-The polarized neutron-beam technique is discussed and utilized in determining the magnetic scattering by iron and nickel. Procedures for obtaining and using polarized beams of monochromatic neutrons are outlined, and experimental data are presented on the performance of two polarizing crystals, Fe304 and 92 Co-8 Fe alloy. The reflectivity of neutrons of different polarization states by single crystals of iron and nickel has been studied and used for determining the magnetic scattering amplitudes at various scattering angles. Comparison is then made between the observed magnetic form values and theoretical values derived from calculated 3d wave functions. 1. INTRODUCTION
THE advantages in using polarized beams of neutrons in diffraction studies of magnetic materials have been known for some time, and the present report (a) summarizes some of the experiences gained in modifying a typical neutron-diffraction spectrometer to polarized-beam use and (b) presents experimental data obtained with this technique on the absolute magnetic scattering of the ferromagnetic elements, iron and nickel. The ferromagnetic scattering for these elements as measured with unpolarized radiation is unfortunately rather small in comparison with the nuclear scattering, about 8 and 0.6 per cent, respectively, so that it is difficult to measure this quantitatively for comparison with the expected absolute level or to draw comparisons with the theoretical form-factor varia* This research was supported by the U.S. Atomic Energy Commission, the National Security Agency and the Air Force Office of Scientific Research.
tion. Much of this difficulty is surmounted by the use of polarized radiation, whereby it becomes possible to measure accurately very small magnetic scattering cross-sections. In addition to the enhanced sensitivity of the polarized-beam method, two other advantages are worthy of mention. It is possible by these techniques to determine the absolute sign of the magnetic scattering amplitude associated with a particular magnetic lattice reflection, thereby resolving certain ambiguities that arise in the determination of a magnetic crystal structure. This has been utilized in a study of the magnetic structure existing in ordered FesAl and is described in a companion publication.(r) Additionally there exist interesting changes in polarization upon magnetic scattering as predicted by the early theory, and these have been confirmed by experiment and utilized to establish(z) details of the magnetic axis orientation
in
the
CraOs and a-FesOs. 138
antiferromagnetic
materials
POLARIZED
NEUTRONS
IN
DETERMINING
~~NIQ~
MAGNETIC
SCATTERING
139
(3)
predicted from the known scattering amplitudes and to use such polarized radiation in a quantitative study of the magnetic scattering from iron and nickel. It is apparent from the form of equations (3) and (4) that the contribution of a weak magnetic scattering amplitude to the scattering cross-section will be more pronounced for the ~l~ized-earn case. For the production of a monochromatic polarized beam, we have used Bragg scattering by a suitable ferromagnetic crystal which exhibits the favorable balance of nuclear and magnetic scattering indicated above. The (220) reflection of Fe304 has been known to produce high polarization,@) and a study has been made of the polarizing efficiency of this crystal as a function of crystal thickness, lack of magnetic saturation in the crystal and the presence of extinction effects. Additionally when single crystals of a cobalt-iron alloy became available, for which a favorable balance of the scattering amplitudes also occurs, these were studied and their superiority over Fe304 crystals was demonstrated.
With unpolarized incident radiation, the scattering cross-section becomes half of the sum of the two spin-state cross-sections or
A schematic diagram of our experimental arrangement is shown in Fig. 1. It consists of a modified double-crystal neutron spectrometer in
2.
GENERAL
CON-
AND
HALPERN and JOHNSON have shown,@) in a com-
prehensive treatment of the magnetic scattering theory, that the scattering cross-section for a magnetic ion is given by: o = ~(b~+p2q2+2b~
* A),
(11
where 6 and p are the nuclear and magnetic scattering amplitudes, A is the unit vector describing the neutron polarization directed along its spin axis, and q = e(e * x)-w
(2)
with e the unit scattering vector and K the unit vector along the spin direction of the magnetic ion. If K is arranged perpendicular to the scattering vector e, then IqI = q becomes unity, and the crosssections for the two neutron spin states (analyzed such that q * Ih = &l> that make up an unpolarized neutron beam can be written as a+=en(b+p)swithq*h=
+l
and cr- = 4-n(b-p}2 with q * h = -1.
3.
POLARIZBD BEAM SFECI’ROMBTER
which can be considered to be the sum of the nuclear and magnetic scattering cross-sections. It is obvious that a magnetic ion for which b is closely equal to p will scatter neutrons of only one spin state under the above conditions and hence a high polarization P will result in the scattered beam, where p=-
k--12 IlSIZ
and 4 and Ia are the intensities of the two neutron spin states. Unequal scattering amplitudes will of course yield only partial polarization. The detection and utilization of such a polarized neutron beam has been discussed in the litera. ture,@*s) and the present investigation was undertaken as an extension of these studies. It was desired to study the polarizing efficiency of various crystals for quantitative comparison with that
Fro. 1. Schematic diagram of polarized neutron spectrometer.
which the first crystal is replaced with a magnetized polarizing crystal and the second crystal serves as a polarization analyzer or a specimen crystal. It is necessary to magnetize the polarizing crystal to insure that x and hence q above be unidirectional, and for this a permanent magnet giving a field of 2200 Oe over a g-in. gap was used. The reflected
140
R.
NATHANS,
C.
G.
SHULL,
monochromatic neutrons pass along a magnetic collimator which maintains a uniform field of 150 Oe, in the same sense as the polarizing field, throughout the trajectory between the polarizing crystal and the second crystal. An electromagnet served for magnetizing the analyzing or test crystal with fields up to 7500 Oe over a Q-in gap. Considerable care had to be taken to control the magnetic field over the entire path from polarizer to analyzer to prevent induced transitions from one neutron spin state to another, thereby affecting the degree of polarization in the beam. For certain types of measurement it was desirable to depolarize the beam completely, and this was accomplished by passing the beam through a thin foil of unmagnetized iron (0.020 in thick). In such an unmagnetized foil, the random direction of the magnetic domains cause many spin transitions and the emerging beam is completely depolarized. Provision was also incorporated into the measurement technique for reversing the neutron polarization relative to the sense of magnetization for the analyzing or test crystal. Merely reversing the analyzing magnetic field direction does not accomplish this, since the neutron polarization would tend to follow the field reversal along its trajectory. If the field reversal region can be made small enough so that the neutron transit time through this region is small compared to its Larmor precession time, the neutron polarization will be maintained and the polarization will be reversed with respect to the analyzing field as desired. This method has been used by BURGYet al.@) In the present experiments it was found difficult to meet the stringent requirements of this procedure, and accordingly an alternative method of invoking the neutron spin transition through resonant spin flipping was utilized.* In this procedure a radio-frequency field, applied at right angles to the neutron polarization, is matched in frequency to the Larmor precessional frequency of the neutron spin in a uniform magnetic field. If the strength of the r.f. field is adjusted properly relative to the neutron transit time, complete transition is possible, and this has been confirmed to a high degree by experiment. Thus, by merely turning on or off the r.f. flipper, the neutron polarization can be kept parallel or antiparallel to * STANFORD et al.(S)have utilized this technique.
G.
SHIRANE
and
A.
ANDRESEN
the common field direction in the polarizing, intermediate, and analyzing regions. It is of convenience in treating the experimental measurements to make use of a quantity S, commonly called the shim ratio and defined as the ratio of the scattering intensity obtained with a polarized beam of polarization Pr to that obtained with a depolarized beam. Thus:
with I+ and I_ being the scattered intensity with incident neutron polarization being parallel and antiparallel to the magnetization of the scattering crystal and 10 the scattered intensity with a depolarized incident beam. This term has been called the shim ratio because in practice it is obtained by inserting a thin shim of unmagnetized ferromagnetic material in the beam thereby depolarizing it. Of usefulness is a related quantity R called the polarization ratio defined as the ratio of I+ to I_ so that (7) For a test crystal whose polarizing efficiency is Pz, defined in the sense of equation (S), it can be easily shown that s = l+PtP,
(8)
where it is to be recognized that the intrinsic polarizing efficiency Pz will depend upon the degree of extinction present in the crystal, since this will affect the spin-state scattered intensities. Pf represents the degree of polarization in the neutron beam incident on the test crystal. For the extinction-free case, Pz will be simply’ related to the cross-sections : Q+-
Pz
= -=-.
6-
o++(I-
%’ bz+Pz
(9)
Equation (8) can be used for assessing the relative polarizing efficiencies of different crystals, but a knowledge of Pi is necessary to determine the absolute efficiency.
It has been found experimentally that sizeable depolarization of the neutron beam accompanies its passage through crystals, and hence that it is necessary to include a depolarization factor in
POLARIZED
equation
NEUTRONS
IN
DETERMINING
(8). It is convenient to define thepolarixaD, such that
&on ~~ff~~~.~,
P =
PO*D T=POexp(-_tcnt)
MAGNETIC
141
SCATTERING
measurements were next made of the polarization transmission D for different crystals, so that the true polarizing efficiencies could be assessed. This was performed by placing the test crystal
with P being the transmitted polarization, .FQ the incident polarization, pP the linear depolarization coefficient, and t the traversed thickness of the crystal. With this definition, equation (8) becomes
~q-4-~p~)l~z
s = 1+&-&U-
(10)
and for small depolarization l6 0
1 CRYSTAL
The polarization transmission wifl be strongly dependent upon the magnetization of the scattering crystal, approaching unity as the magnetization approaches saturation. With less than saturation, the neutron passage from one domain to another will be effective in changing the neutron polarization state as HALPERN and HOLSTEIN(~) have considered. 4,
~~E~~ ~FIC~NC~
ON POLARIZING OF CRYSTALS
Experiments were first performed on a number of Fe304 polarizing and analyzing crystals cut from natural magnetite ingots. These crystals were cut with different shape factors, ranging from blocks to plates to pillars, and with different thicknesses so as to permit study of the internal depolarization effects. With a common polarizing crystal whose polarizing efficiency was later determined, the shim ratio 5’ of a series of analyzing crystals was measured as a function of analyzing field strength and crystal thickness, Fig. 2 shows the variation of S obtained with a series of crystal pillars of common area (19-l x 3.2 mm), but with different thicknesses with each maintained in a magnetic field of 6900 Oe. The variation of S with thickness shows that the effective polarizing efficiencies of these pillars decreases as the thickness is increased, and this is considered to result from the increased internal depolarization within the crystal. The alternative explanation that such a variation could result from increased extinction effects was eliminated by a calculation making use of the known mosaic distribution in these crystals.
Fro,
2.
/ 2
L-l
4
3
THICKNESS,
5
mm
Variation of shim ratio S with Fe804 crystal thickness.
directly in front of, and in the same magnetic field as, an analyzing crystal. The polarized beam from the polarizer then passed through the test crystal, but was not scattered by it, before being Bragg scattered by the analyzer. measurements of the shim ratio with and without the test crystal could then be used to determine directly the polarization
MAGNETIC
FIELD!
oeeteds
FIG. 3. Dependence of polarization transmission D upon magnetic field applied to crystal fur two different Fe304 crystals. The relative magnetization for crystal (B) is also shown.
transmission of the test crystal. This was studied as a function of magnetic field strength, and typical results are shown in Fig. 3 for two crystals. Crystal (A), of dimensions 19.2 x 3.2 mm area and l-24
142
R.
NATHANS,
C.
G.
SHULL,
G.
mm thick was cut so that a [loo] axis was parallel to the long dimension and the magnetic field was applied in this direction. Crystal (B), of dimensions 16.5 x 3.5 mm area and 2.1 mm thick, was cut with a [ill] direction parallel to the long dimension and again the magnetic field was applied in this direction. Also shown in Fig. 3 are measured values for the relative magnetization of crystal (B). It is seen that the polarization transmission is exceedingly sensitive to departure from magnetic saturation, as had been predicted by HALPERN and HOLSTEIN(7)and studied experimentally by HUGHES and colleagues,@,*) Since the easy direction of magnetization in magnetite is along the [ll l] direction, the depolarization effects are expected to be somewhat less for crystals magnetized in this fashion and the data of Fig. 3 demonstrate this. I
I
I
MAGNETIC
I
FIELD,
I
I
I
I
SHIRANE
magnetization.
and
A.
ANDRESEN
It can be easily shown that PZ ----=P&ax>
M MS
and a comparison is illustrated in Fig. 4 of the intrinsic polarizing efficiency and the relative magnetization in a limited low field region. As seen in Fig. 4, the maximum value for the intrinsic polarizing efficiency of crystal (B) is seen to be about 95 per cent. A number of different Fe304 crystals have been studied, and some variation in this maximum value has been noted. Values as high as 98 per cent have been obtained, this having been measured with the particular crystal used in the early stugy of reference (4). Since all of the crystals studied have been cut from natural mineralogic ingots of unmeasured purity, these relatively small variations are not unexpected. Similar studies have been carried out on polarizing and analyzing crystals of cobalt-iron alloy. For the composition 92 atomic per cent cobalt, this crystallizes from the melt* in the face-centered cubic phase, and by calculation, the intrinsic polarizing efficiency in the (111) reflection should be 99.1 per cent. In this phase, magnetic saturation
oersteds
FIG. 4. Polarizing efficiency of Fe304 crystal function of applied magnetic field.
(B) as a
Once the depolarization factors have been evaluated, a pair of crystals having the same intrinsic polarizing efficiency can be used as a polarizing-analyzing combination to determine the individual polarizing efficiency. Data obtained in this fashion for crystal (B) above are illustrated in Fig. 4. Here the polarizing efficiency of this crystal has been evaluated from shim ratio measurements as a function of applied magnetic field. Also shown in the figure is the variation of the intrinsic polarizing efficiency after correction for internal depolarization through the use of the polarization transmission data for this crystal from Fig. 3. Even after correction for depolarization effects, the intrinsic polarizing efficiency should fall off in the low-field region because of the reduction in crystal
a
1
2 MAGNETIC
3 FIELD,
4
5
6
7
k-oersteds
FIG. 5. Polarizing efficiency of cobalt-iron crystal as a function of applied magnetic field.
is approached with rather low fields. Fig. 5 illustrates the variation of the intrinsic polarization efficiency as a function of field for a crystal of dimensions 19.2~ 3.4 mm in area and 1.1 mm in thickness, with the magnetic field applied along the * We are much indebted to Messrs. R. M. BOZORTH and A. J. WILLIAMS, of the Bell Telephone Laboratories, for supplying such a crystal to us.
POLARIZED
NEUTRONS
IN
DETERMINING
[I101 direction parallel to the long axis. This is seen to approach a value of 99-100 per cent. It has been found in general that the depolarizing effects are smaller for the case of the metallic crystals than has been observed for magnetite and hence smaller magnetic fields can be tolerated. It should be mentioned that the neutronpolarization direction obtained with the metallic crystals is reversed from that obtained with the (220) Peso4 reflection. This follows from the fact that the magnetic ions responsible for the scattered intensity in (220) Fe304 are directed opposite to the applied field on the crystal, since it is a ferrimagnetic substance and these ions are in the minority. In contrast the magnetic ions in the metallic crystal are all parallel to the applied field direction. Thus the polarized neutrons in the (220) Fe304 reflection have their magnetic moments aligned parallel to the field on the crystal, whereas in the (111) cobaltiron reflection, the neutron moments are antiparallel to the field direction. This absolute sense of direction follows from a theoretical assay of the absolute signs of the magnetic and nuclear scattering amplitudes and has been examined experimentally in a neutron STERN-GERLACH type of experiment by SHERWOODet al.(s) A decided advantage to the use of cobalt-iron polarizing crystals lies in their greater neutron reflectivity than is the case for the (220) Fe304 reflection. This arises because of the more favorable crystal-structure factor and the greater density of the magnetic scattering ions. Moreover the higherorder contaminant neutrons are largely absent in the metallic-crystal case. The magnetic structure of Fe304 is such that the (440) reflectivity is approximately 15 times larger than the (220) reflectivity, and hence in scattering neutrons of wavelength ,! from the (220) reflection there is a strong component of h/2 scattered simultaneously. The h/2 component is partially polarized in the opposite direction, and it is necessary to absorb this with a suitable neutron filter. This was done in all of the above Fe304 measurements with a plutonium filter (useful when h = 1.05 A) and this necessarily absorbs some of the desired intensity as well. This problem does not arise in the case of the (111) cobalt-iron polarizing reflection, since the (222) crystal-structure factor is of the same order of size as the (111) and no enhancement of the /\/2 component is thus obtained.
MAGNETIC
SCATTERING
143
5. MAGNETIC SCATTERING RE!XJLTS ON IRON AND NICKEL
The study of the angular dependence of the absolute magnetic scattering in these metals was made by measuring the polarization ratios (R) for the various Bragg reflections present. As a typical example of such a measurement, Fig. 6 shows the
. ANTI-PARALLEL FIELD
COUNTER
TO ON CRYSTAL
ANGLE
FIG. 6. Intensity in the (111) nickel reflection for incident neutron beams of different polarizations.
intensities observed for the nickel (111) reflection with incoming neutrons polarized first parallel and then antiparallel to the magnetic field on the crystal. The sizeable intensity change for the two orientations of scattered neutrons (1.5 : 1) is to be compared with the O-6 per cent obtained in the usual technique with unpolarized neutrons. From the observed values of &kl the intrinsic polarizing efficiencies for various Bragg reflections can be evaluated after the necessary corrections have been made for extinction and depolarization. The final result is a value for the ratio of the magnetic scattering amplitude to the nuclear scattering amplitude, that is ph&l/b&kl. Since the coherent nuclear scattering amplitudes for these two metals are well established (1 a03 x 10-1s cm for nickel and 0.96 x lo-12 cm for iron) and are independent of angle, the magnitude of phkl follows. In the study of both of these metals, the measurements were made, for the most part, with single-crystal pillars placed in the high magnetic field of the analyzing magnet. Intensities were recorded over short-time intervals alternately with r.f. field on and off. Each run was repeated several
144
R.
NATHANS,
C.
G.
SHULL,
times, and the uncertainty in the final results was established from the standard deviations between the different runs. Since the data accumulation extended over a considerable period of time, it was necessary to establish the stability of the incident neutron polarization and the efficiency of the r.f. coil in reversing the neutron polarization. For this purpose the (111) reflection of a cobalt-iron crystal was used as a standard. Measurements of the polarization ratio for this standard were made before and after each run on the test crystal, and the required adjustments, although small, were made to the observed polarization ratio on the test crystal.
G.
SHIRANE
and
A.
ANDRESEN
of the two groups of 3d electrons in nickel. Such a separation can result from the intra-atomic exchange forces which will be more effective for greater number of positive spin electrons than for the lesser number of electrons in the negative spin state.
(a) Nickel The single crystal of nickel was cut so that it could be magnetized along the [llO] direction. The pillar measured 19.1 x 3.2 mm in area and O-55 mm in thickness. A chemical analysis revealed only small traces of impurities. Calculations using the known mosaic distribution and measurements taken at several different neutron wavelengths showed the extinction effects to be negligible. As an additional check, the polarization ratio for the most intense nickel reflection was repeated on a pillar of ~lycrystalline nickel and yielded the same result as the single-crystal sample. Finally, by the method outlined above, it was demonstrated that there was no significant depolarization of the neutron beam within the nickel sample. Table 1 lists the resultant values of p, the magnetic scattering amplitude, and f, the magnetic form factor for the various Bragg reflections on which data were taken. These figures are plotted in Fig. 7 as a function of 4rr sin 8/h. In the forward direction the magnetic scattering amplitude is proportional to the room-temperature saturation moment of nickel, and this value is shown on the curve. For comparison with the neutron data, we have drawn in the Hartree-Fock calculated atomic form factor for the 3d electrons in Cut, which possesses the same number of outer electrons as nickel. A significant variation between the two curves is evident. Recalling that the neutron magnetic scattering comes from the difference between the electrons in the two spin states in the 3d shell in nickel, while the Cu+ data represents the sum of all the 3d electrons, the curves in Fig. 7 suggest that there is a difference in the electronic configurations
46
sin e/h,
A-’
FIG. 7. Observed magnetic scattering amplitudes different nickel reflections and comparison retical form-factor curve.
for with theo-
Table 1. Nickel magnetic scattering amplitudes and form-factor values for observed reflections
kkl
(lO-42cm)
(111) P3 (220)
0~1140&0@015 0~1~4~0~0010 0.0646&0.0016 0+)459$,0~0012 0+418~0~0015 0-0234~0~0021 O-0258 rtO+O30 0~0160:~0~0035
(311) (222) (4001 (3311 (333)
f 0.678f0.006 0~4193t0*010 0*297&0+08 0.271 +O.OlO O-15210.014
(b) Iron In obtaining the iron data, we used a single crystal of an iron-silicon alloy (3.9 weight per cent silicon). This crystal again was cut in the form of a thin pillar (19.2~ 3.9 mm in area and 0.41 mm thick) with its long axis as a [loo] direction. A part of the data was collected on two crystals of widely differing thickness for the purpose of detecting and estimating the effects of extinction. These effects were studied, in this case, by a procedure somewhat different from the standard techniques. Of
POLARIZED
NEUTRONS
IN
DETERMINING
particular use with polarized beams, this method utilizes the fact that the magnitude of the structure factor of the test crystal, and hence the degree of extinction, is changed by reversing the direction of the incident neutron polarization without altering the sample. If now one records the polarization ratio, R, of a particular reflection for a fixed counter angle as the crystal is rotated off the Bragg peak, the presence of extinction becomes quite evident.
>’ CRYSTAL
AMOLE,
W.
145
SCATTERING
mosaic distribution and not a result of the spread in the wavelength of the scattering neutrons.
Table 2. Iron magnetic scattering amplitudes and farm-factor values for observed reJectio?zs . . _I_-._ kkt
(110)
I
MAGNETIC
(200) (220) (310) (400)
(to-~
cm)
0.334OrtO.0027 0~2180&0~0014 0.095 1 rfr0.0024 0.0609 5 0+@20 0.0348~0*01~
f
-
0619&0,006 0~405&0~004 0~176&0*005 0~113~0+04 0~065*0~020
Our results for iron are summarized in Table 2 and illustrated in Fig. 9. Also drawn in are the form-factor curves calculated by WOOD and PRATT(~)) for the free iron atom (3d+5, 3d_, 4~s).
36”
Fro. 8. Polarization ratio as a function of crystal rocking angle for thick and thin iron crystals.
Fig. 8 shows the results of such a series of measurements taken on two iron crystals for the (110) reflection. For the thicker crystal (Fe No. 1) the polarization ratio at the Bragg peak is suppressed. As the crystal is rotated off the peak, where fewer mosaic domains in the crystal are in a position to reflect, the suppression of R caused by the secondary extinction is reduced-in the limit approaching the extinction free value for crystal angles sufficiently removed from the Bragg peak position. For Fe No. 2, a crystal only one-third as thick, we observed a polarization ratio which remains unchanged as the crystal is rotated, and has a value equal to the limiting ratio observed for the much thicker crystal. This test for extinction strongly suggests that crystal Fe No. 2, which was used for the main body of the iron data, is free of extinction. It should be noted that this technique for estimating extinction is useful only in cases where the width of the observed Bragg reflection is due to the
47-5 sin@%
FIG. 9. Observed magnetic scattering amplitudes for different iron reflections and comparison with theoretical form-factor curves.
These theoretical curves have been normalized to agree with the experimental plot in the forward direction. The f+ curve represents the electronic configuration of the five 3d+ electrons, while the curve f_ corresponds to the single 3d_ electron. If, as in nickel, the neutron data were to represent the difference between the spin density in two antiparallel electron groups, one would expect the experimental points in Fig. 9 to faI1 somewhat above both the f, and f_ curves. Instead, we find the experimental points to be in good agreement with the f, curve representing the electrons in the positive
146
R.
NATHANS,
C.
G.
SHULL,
G.
spin group. Such agreement is consistent with a model for iron in which the magnetic spin density comes from electrons in a single spin state only. This result has been strongly indicated in the recent report of WEISS and DEMARCO(~~) based on their measurements of X-ray scattering factors in the first transition group. A further interpretation of the combined neutron and X-ray results for nickel and iron will be found in the companion paper following this report. Acknowledgement-We
wish
to
thank
Dr.
MCREYNOLDS for conceptual and executional during early stages of this research.
A.
W.
assistance
REFERENCES 1. NATHANS R., PICOTT M. T. and SHULL C. G., J. Phys. Chem. Solids 6, 38 (1958).
SHIRANE
and
A.
ANDRESEN
2. NATHANS R., RISTE T.,
SHIFZANE G. and SHULL C. G. In the press. 3. HALPERN 0. and JOHNSONM. H., Phys. Rev. 55,898
(1939). 4. SHULL C. G., Phys. Rev. 81,626 (1951). 5. STANFORD C. P., STEPHENSONT. E., COCHRANL. W. and BERNSTEINS., Phys. Rev. 94, 374 (1954). 6. BURCY M. T., HUGH= D. J., WALLACE J. R., HELLER R. B: and WOOLF W. E., Phys. Rev. 80, 953 (1950). 7. HALPERN 0. and HOLSTEIN T., Phys. Rev. 59, 960 (1941). 8. HUGH= D. J., WALLACEJ. R. and HOLTZMANR. H., Phys. Rev. 73, 1277 (1948). 9. SHERWOODJ. E., STEPHENSONT. E. and BERNSTEIN S., Phys. Rev. %, 1546 (1954). 10. WOOD J. H. and PRATT G. W., Phys. Rev. 107, 995 (1957). 11. WEISS R. J. and DE MARCO J. J., Rev. Mod. Pkys.
30, 59 (1958). 12. WEISS R. J. and FREEMAN A. J., J. Phys. Chem. Soli& 10,147 (1959).