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The Use of Specular, Polarized Neutron Reflectivity To Determine Magnetic Density Profiles† William Barford* Department of Physics and Astronomy, The University of Sheffield, Sheffield, S3 7RH, United Kingdom Received October 23, 2002. In Final Form: February 6, 2003 The Born and distorted wave Born approximations are used to derive expressions for the specular, reflected intensity of polarized neutrons from magnetic surfaces. Both of these approximations are expected to provide qualitative information on the in-plane magnetic structures. In particular, the spin-flipped component is proportional to the Fourier transform of the in-plane transverse magnetization, while the parallel-spin component is proportional to the Fourier transform of the sum of the in-plane longitudinal magnetization and the nonmagnetic scattering density. The distorted wave Born approximation is more valid near to a critical edge than the Born approximation. Within the distorted wave Born approximation the reflected parallel intensity can also be directly inverted to obtain the sum of the nonmagnetic and in-plane longitudinal magnetization. However, this approximation fails beyond a first-order perturbation in the potential, signified by the violation of flux conservation.
I. Introduction Specular neutron reflectivity is a widely used and powerful approach to determine density profiles. It is has been particularly applied to determine the density profiles of nonmagnetic systems at interfaces, such as the liquidvapor interface, polymer melt interfaces, polymer adsorption, etc. Many such examples may be found in this volume of Langmuir. Reviews of neutron scattering may be found in refs 1 and 2. The Born and the distorted wave Born approximations can both be used to interpret the neutron reflectivity to provide direct, but usually qualitative, insight into the density profile. In both cases the reflected intensity is related to the Fourier transform of the density profile. In fact, the distorted wave Born approximation provides a prescription for directly inverting the data to obtain the density profiles and, via the phase relations, to obtain information of the surface coverage. The distorted wave Born approximation is usually the method of choice, as, to first order, this describes the reflected intensity at the critical edge.3,4 However, this method cannot be extended beyond single scattering (i.e., a first-order expansion in the perturbing potential)4 and thus provides qualitative, rather than quantitative, predictions for strong scatterers. On the other hand, while the Born approximation can be formally extended to multiple scattering, it completely fails to describe the critical edge. These deficiencies in the various Born approximations explain why numerical transfer matrix techniques are usually favored to interpret density profiles, especially for strong scatterers. Magnetic systems are strong neutron scatterers and are also studied by neutron reflectivity. Polarized neutron reflectivity is a particularly powerful tool, enabling * E-mail address:
[email protected]. † Part of the Langmuir special issue dedicated to neutron reflectometry. (1) Russell, T. P. Mater. Sci. Rep. 1990, 5, 171. (2) X-ray and Neutron Reflectivity: Principles and Applications; Daillant, J., Gibaud, A., Eds.; Springer: Berlin, 1999. (3) Bouchard, E.; Farnoux, B.; Sun, X.; Daoud, M.; Jannink, G. Europhys. Lett. 1986, 2, 315. (4) Crowley, T. L. Physica A 1993, 195, 354.
Figure 1. The plane, magnetic surface lies in the y-z plane at x ) 0. The applied field, B, is oriented along zˆ . I take the incident beam to be polarized parallel to B.
information to be obtained on the magnetization in the plane of the surface. In this paper, I develop the Born and distorted wave Born approximations to interpret the polarized neutron reflectivity from planar, magnetic systems, such as thin films or magnetic multilayers. As stated above, these methods are expected to provide qualitative information on the magnetic density profiles, for example, the wavelength of magnetic structures normal to the surface. Consider a smooth, plane surface, oriented in the y-z plane. xˆ is normal to the surface. The component of the wave vector normal to the surface is k ) (2π/λ) sin(θ), where θ is the angle of incidence. The magnetic field, B, is oriented along zˆ . This is illustrated in Figures 1 and 2. I suppose that the incident neutron beam is initially polarized parallel to B. In the next section, I develop the Born expansion to first order in the perturbation. This is applied to magnetic systems in section III, where the Born and distorted wave Born approximations are also developed. Finally, section IV concludes the paper.
10.1021/la026740y CCC: $25.00 © 2003 American Chemical Society Published on Web 03/21/2003
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gn is the nuclear Lande´ factor of the neutron ()1.9132), and µn is the nuclear magneton ()ep/2mp, where mp is the proton mass). In the Born approximation one sets V ˆ (x) ) 0. However, in the distorted wave Born approximation, V ˆ (x) satisfies
V ˆ (x) ) V ˆ 0θ(x)
(7)
where θ(x) is the Heavyside step function. V0 describes the scattering from a nonmagnetic system which acts as a host for the magnetic system. If F and b are the number density and scattering length of the host material, respectively, then Figure 2. The incident (I), reflected (R), and transmitted (T) waves. The incident and transmitted angles are θ1 and θ2, respectively. In the Born approximation θ2 ) θ1, while θ2 * θ1 in the distorted wave Born approximation.
II. Perturbation Theory We seek an approximate solution to the Hamiltonian
H ˆ )H ˆ (0) + U ˆ
|ψ(0) ∑ n 〉 n*m
1 E(0) m
-
E(0) n
〈ψ(0) ˆ (x)|ψ(0) n |U m 〉 (2)
U(x) )
ψm(x) )
+
) ψ(0) m (x) +
(
∫∑ n*m
E(0) m
-
E(0) n
Uz(x) )
)
(0) ψ(0) n (x)ψn (x′)
U(x′)ψ(0) m (x′) dx′
(0) ∫G(x,x′;E(0) m )U(x′)ψm (x′) dx′
(0) ∫G(x,x′;E(0) m )U(x′)ψm (x′) dx′
(11)
ψ(x) )
( ) (
ψ+(x) ψ (0)(x) + ψ+(1)(x) + ‚‚‚ ) +(0) ψ-(x) ψ- (x) + ψ-(1)(x) + ‚‚‚
)
(12)
where, using eq 4, the first-order corrections are
(5)
III. Magnetic Systems
(
ψ+(1)(x) ψ-(1)(x)
pˆ 2x - gnµnµ0σˆ ‚B + V ˆ (x) H ˆ (0) ) 2m
)
)
(
ψ
(0)
(x′)
∫G((x,x′;E(0))U(x′) ψ+(0)(x′) -
)
dx′
(13)
The unperturbed wave function is polarized parallel to B, so ψ-(0)(x) ) 0. Thus
The unperturbed Hamiltonian is
pˆ 2x - gnµnµ0σˆ zB + V ˆ (x) 2m
(10)
The wave function is expressed in its spinor components as
(4)
In the next section, I drop the index m from the wave function and energy.5
)
2πp2 ∆Nbφ(x) - gnµnµ0Mz(x) m
and
(3)
where G(x,x′;E(0) m ) satisfies the differential equation (0) (H(0) - E(0) m )G(x,x′;Em ) ) -δ(x - x′)
(9)
Uy(x) ) -ignµnµ0My(x)
where the term in large parentheses defines the Green function, G(x,x′;E(0) m ). The first-order correction is thus
ψ(1) m (x) )
2πp2 ∆Nbφ(x) - gnµnµ0σ‚M|(x) m
where M| is the magnetization in the plane of the surface. ∆Nb is the difference in the scattering length density between the magnetic and nonmagnetic host materials, while φ(x) is the volume fraction of the magnetic material. Note that in the Born approximation, where there is no host material, ∆Nb ) Nb(x) and φ(x) ) 1. For later convenience, I now define
We require the wave function, ψm(x) ≡ 〈x|ψm〉
ψ(0) m (x)
(8)
where Nb ()Fb) is the scattering length density. In a magnetic system the neutrons are scattered by both the magnetic and nonmagnetic ions. The perturbing, scattering potential is,6
(1)
(0) where H ˆ (0) has exact eigensolutions {|ψ(0) ˆ is m 〉, Em } and U the perturbative scattering potential. Then, using Rayleigh-Schro¨dinger perturbation theory, to first order in the perturbation the perturbed state |ψm〉 is
|ψm〉 ) |ψ(0) m〉 +
2πp2 2πp2 Fb ) Nb m m
V0 )
ψ+(1)(x) )
∫G+(x,x′;E(0))Uz(x′)ψ+(0)(x′) dx′
(14)
ψ-(1)(x) )
∫G-(x,x′;E(0))Uy(x′)ψ+(0)(x′) dx′
(15)
and
(6)
where m is the neutron mass, σˆ is the Pauli spin operator, (5) Higher order terms in the Born expansion may be obtained by using the higher order expansions in the Rayleigh-Schro¨dinger perturbation theory.
where Uz(x) and Uy(x) are defined in eqs 10 and 11. Equations 14 and 15 are the master equations, from which I derive the reflected intensities in the next two subsections. (6) Fermon, C.; Ott, F.; Menelle, A. Chapter 5 of ref 2.
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Barford
A. The Born Approximation. This is the “contrast matched” situation, where V0 ) 0. The unperturbed wave function satisfies the Schro¨dinger equation
(
-
)
p2 d2 - gnµnB ψ+(0)(x) ) E(0)ψ+(0)(x) 2m dx2
R- is the Fourier transform of the in-plane transverse magnetization. The intensities of the reflected spin components, I( ) | R(|2. Thus
(16)
I+ )
|
|
U (x) exp(i(k+ + k-)x) dx 2 ∫-∞∞ (2m 2 ) y p
with plane wave solutions
(17) I- )
where
2m (0) (E ( gnµnB) p2
(18)
Similarly, the Green function satisfies the differential equation
(
-
)
p2 d2 - E(0) - gµB G((x, x′; E(0)) ) -δ(x - x′) 2m dx2 (19)
and thus
G((x,x′;E(0)) )
)
( )
2m exp(ik((x - x′)) , 2ik( p2
( )
2m exp(-ik((x - x′)) , 2ik( p2
(
∫∞
-∞
for x > x′ (20) for x < x′ (21)
( )
2m Uz(x′) × p2
)
where 2k+ is the change of momentum of the scattered, parallel neutron. The coefficient of exp(-ik+x) is the reflected amplitude for the parallel spins, R+
Uz(x) exp(2ik+x) dx ∫-∞∞ (2m p2 )
i 2k+
(
(
i 2k-
Uy(x′) × ∫-∞∞ (2m p2 )
)
where (k+ + k-) is the change of momentum of the spinflipped scattered neutron. The coefficient of exp(-ik-x) is the reflected amplitude for the antiparallel spins, R-
i 2k-
|
(27)
)
where V(x) is defined in eqs 7 and 8 and Figure 3. The plane wave solutions are
Uy(x) exp(i(k+ + k-)x) dx ∫-∞∞ (2m p2 )
ψ+(0)(x) ) ψ1+(0)(x)θ(-x) + ψ2+(0)(x)θ(x)
ψ1+(0)(x) ) exp(ik1+x) +
(
(29)
(30)
and
(
)
2k1+ exp(ik2+x) k1+ + k2+
ψ2+(0)(x) )
(31)
The wave vectors are defined by the equations
2m (0) (E ( gnµnB) p2
k1(2 )
(32)
and
2m V0 p2
(33)
When k2+ e 0 there is total external reflection and the reflected intensity is unity. The Green function satisfies the differential equation
(
-
(25)
)
k1+ - k2+ exp(-ik1+x) k1+ + k2+
k2(2 ) k1(2 -
exp(i(k+ + k-)x′) dx′ (24)
R- ) -
1 2k-
p2 d2 + V(x) - gnµnB ψ+(0)(x) ) E(0)ψ+(0)(x) (28) 2m dx2
-
(23)
We see that R+ is the Fourier transform of the sum of the nonmagnetic density and the in-plane longitudinal magnetization. This expression rederives the result for nonmagnetic systems when M ) 0. A new result is that in magnetic systems the transverse, in-plane magnetization spin-flips the polarized neutrons. The reflected wave function for the antiparallel spins is
ψ-(1)(x) ) exp(-ik-x) -
(26)
where
exp(2ik+x′) dx′ (22)
R+ ) -
|
B. The Distorted Wave Born Approximation. The distorted wave Born approximation is more appropriate near to a critical edge than the Born approximation. The assumption underlying this approximation is that there is a host material that is responsible for the critical edge and that the system of interest is a perturbation of the host system. A magnetic multilayer might be such an example, where the nonmagnetic system is the host material and the magnetic layers are treated as the perturbation. Figure 3 illustrates the unperturbed scattering potential. The unperturbed wave function satisfies the Schro¨dinger equation
We require the reflected wave in the asymptotic limit, i.e., x f -∞. Using eqs 14, 17, and 21, we have
i ψ+(1)(x) ) exp(-ik+x) 2k+
Uz(x) exp(2ik+x) dx 2 ∫-∞∞ (2m p2 )
and
ψ+(0)(x) ) exp(ik+x)
k(2 )
1 2k+
)
p2 d2 + V(x) - E(0) - gnµnB G((x,x′;E(0)) ) 2m dx2 -δ(x - x′) (34)
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Figure 3. The unperturbed scattering potential. The scattering caused by a perturbation of this potential is described by the distorted wave Born approximation. There is total external reflection when V0 g p2k1+2/2m.
For reflected waves we need the solution when x′ > 0 and x < 0, namely
G((x,x′;E(0)) )
( )
2m exp(-i(k1(x - k2(x′)) (35) i(k1( + k2() p2
Substituting eqs 29 and 33 into eq 14 we have
ψ+(1)(x) )
(
exp(-ik1+x) -
2ik1+ (k1+ + k2+)2
∫0 ( ∞
)
where I+F is the Fresnel reflectivity
I+F ≡ |R+(0)|2 )
2m Uz(x′) × p2
)
( ) p2 2m
∫0
kc2
∞ 2
ψ- (x) )
(
2ik1+
∞ 2m ∫ ( p2 ) × 0 )
(k1+ + k2+)(k1- + k2-
)
Uy(x′) exp(i(k2+ + k2-)x′) dx′ (37) where again the term in the large parentheses is the reflected amplitude; now for the antiparallel spins, R-. The reflected intensity for the antiparallel spins is I- ) |R-(1)|2, as R-(0) ) 0. Thus
|
(41)
)
-1 × I+F sin(2k2+x) d(2k2+) (42)
where kc2 ) 4πNb. C. Validity of the Born Approximations. The firstorder Born approximations are valid provided that
ψ(1) , ψ(0)
(43)
A sufficient condition for this is that the reflected amplitude, R(1), satisfies
exp(-ik1-x) -
I- )
(
I+(2k2+)
2 1/2
4(k2 + kc )
where the term in the large parentheses is the reflected amplitude for the parallel spins, R+. Similarly, the spinflipped reflected wave function is (1)
)
2
Equation 40 can be inverted to directly obtain the z dependence of the potential
Uz(x) )
exp(2ik2+x′) dx′ (36)
(
k1+ - k2+ k1+ + k2+
2k1+
(k1+ + k2+)(k1- + k2-
∫∞ 2m U (x) × ) 0 ( p2 ) y
exp(i(k2+ + k2-)x) dx
|
2
|2πNb -
k+ m g µ µ M|, 2 n n 0 z a p
(45)
and
| (38)
(44)
Let us assume that the layer has a thickness of ∼a. Then, for the Born approximation, eq 44 implies that
|
km g µ µ M , n n 0 y a p2
However, the reflected intensity for the parallel spins is
I+ ) | R+(0)|2 + R+(0)(R+(1)* + R+(1))
|R(1)| , 1
(46)
must be satisfied. Similarly, for the distorted wave Born approximation, near to the critical edge (where k2 ≈ 0)
|8π∆Nbφ -
(39)
k1+ 4m gnµnµ0Mz| , 2 a p
(47)
and
Thus
[ (
I+ ) I+F 1 +
4k1+
)
k1+2 - k2+2
Uz(x) × ∫0∞ (2m p2 )
]
sin(2k2+x) dx (40)
|
|
k14m gnµnµ0My , 2 a p
(48)
must be satisfied. These approximations therefore break down as the thickness of the layer increases.
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IV. Discussions and Conclusions In this paper I developed the Born and distorted wave Born approximations for polarized neutron reflectivity from magnetic systems. Both of these approximations are expected to provide qualitative information on the in-plane magnetic structures. In particular, the spin-flipped component is proportional to the Fourier transform of the in-plane transverse magnetization, while the parallel-spin component is proportional to the Fourier transform of the sum of the in-plane longitudinal magnetization and the nonmagnetic scattering density. The distorted wave Born approximation is more valid near to a critical edge than the Born approximation. Within
Barford
the distorted wave Born approximation the reflected parallel intensity can also be directly inverted to obtain the sum of the nonmagnetic and in-plane longitudinal magnetization. However, this approximation fails beyond a first-order perturbation in the potential, signified by the violation of flux conservation. Acknowledgment. I thank J. M. F. Gunn for discussions on the Born approximation and neutron reflectivity. LA026740Y
