International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 6.1, pp. 590-593

Section 6.1.2. Magnetic scattering of neutrons

P. J. Browna

6.1.2. Magnetic scattering of neutrons

| top | pdf |

6.1.2.1. Glossary of symbols

| top | pdf |

[m_n] Neutron mass
[m_e] Electron mass
γ Neutron magnetic moment in nuclear magnetons (−1.91)
[\mu_B] Bohr magneton
[\mu_N] Nuclear magneton
[r_e] Classical electron radius [\mu_Be^2/4\pi m_e]
[\specialfonts{\bsf P}_i] Electron momentum operator
[\specialfonts{\bsf S}_e] Electron spin operator
[\specialfonts{\bsf S}_n] Neutron spin operator
[\specialfonts{\bsf M}(r)] Magnetization density operator
k Scattering vector (H/2π)
[{\hat{\bf k}}] A unit vector parallel to k
[{\bf r}_n] A lattice vector
g A reciprocal-lattice vector (h/2π)
[\boldtau] Propagation vector for a magnetic structure
[{\bf \bar s}_n] A unit vector parallel to the neutron spin direction
q, q Initial and final states of the scatterer
σ, σ′ Initial and final states of the neutron
Eq Energy of the state q

6.1.2.2. General formulae for the magnetic cross section

| top | pdf |

The cross section for elastic magnetic scattering of neutrons is given in the Born approximation by [\eqalignno{{{\rm d}\sigma\over{\rm d}\Omega}&=\bigg({m_n\over2\pi\hbar}\bigg)^2\bigg|\bigg\langle q'\sigma'\bigg|\int V({\bf R})\exp(i{\bf k}\cdot{\bf R})\,{\rm d}{\bf R}^3\bigg| q\sigma\bigg\rangle\bigg|^2\cr &\quad\times\delta(E_q-E_{q '}).&(6.1.2.1)}]V(R) is the potential of a neutron at R in the field of the scatterer. If the field is due to N electrons whose positions are given by [{\bf r}_i,i=1,\,N], then [\specialfonts\eqalignno{V({\bf R})&=4\gamma\mu_B\mu_N\bigg\{\sum^N_{i=1}\displaystyle{({\bf R}-{\bf R}_i){\bsf P}_i\over|{\bf R}-{\bf r}_i|^3}{{\bsf S}_i\over|{\bf R}-{\bf r}_i|^3}\cr &\quad+{3{\bsf S}_i\cdot({\bf R}-{\bf r}_i)\over|{\bf R}-{\bf r}_i|^5}+8\pi{\bsf S}_i\delta({\bf R}-{\bf r}_i)\bigg\}\cdot{\bsf S}_n.& (6.1.2.2)}]V(R) is more simply written in terms of a magnetization density operator [\specialfonts{\bsf M}({\bf r})], which gives the magnetic moment per unit volume at r due to both the electron's spin and orbital motions. The potential of (6.1.2.2)[link] can then be written (Trammell, 1953[link]) [\specialfonts\eqalignno{V({\bf R})&={2\gamma\mu_N{\bf S}_n\over\pi^2}\cdot\bigg\{\int\limits^\infty_0\!\!\int\limits^\infty_0[{\hat{\bf k}}\times{\bsf M}({\bf r})\times {\hat{\bf k}}]\cr &\quad\times\exp[i{\bf k}\cdot({\bf R}-{\bf r})]\,{\rm d}{\bf k}^3\,{\rm d}{\bf r}^3\bigg\},& (6.1.2.3)}]giving for the cross section, from (6.1.2.1)[link], [\specialfonts\eqalignno{{{\rm d}\sigma\over{\rm d}\Omega}&=(\gamma r_e)^2\bigg|\bigg\langle q\sigma'\bigg|{\bf S}_n\cdot\int [{\hat{\bf k}}\times{\bsf M}({\bf r})\times {\hat{\bf k}}]\cr &\quad\times\exp(i{\bf k}\cdot{\bf r})\,{\rm d}{\bf r}^3\bigg| q\sigma\bigg\rangle\bigg|^2. & (6.1.2.4)}]The unit-cell magnetic structure factor M(k) is defined as [\specialfonts M({\bf k})=\bigg\langle q\textstyle\int\limits_{\rm unit\; cell}{\bsf M}(r)\exp(i{\bf k}\cdot{\bf r})\,{\rm d}{\bf r}^3\bigg| q\bigg\rangle.\eqno (6.1.2.5)]For periodic magnetic structures, [\specialfonts{\bsf M}({\bf r})=\textstyle\sum\limits_{\rm lattice\; vectors}{\bf P}({\bf r}_n\cdot{\boldtau})\cdot {\bsf M}_u({\bf r}-{\bf r}_n),]where P is a periodic function with a period of unity, which describes how the magnitude and direction of the magnetization density, defined within one chemical unit cell by [\specialfonts{\bsf M}_u({\bf r})], propagates through the lattice. The magnetic structure factor m(k) is then given by [{\bf m}({\bf k})=(g-j{\boldtau}){\bf A}(j)\cdot{\bf M}({\bf k}),\eqno (6.1.2.6)] where [{\bf A}(j)] is the jth term in the Fourier expansion of P defined by [{\bf P}({\bf r}\cdot{\boldtau})=\textstyle\sum\limits^\infty_{j=-\infty}{\bf A}(j)\exp\{i(j{\boldtau}\cdot{\bf r})\}\eqno (6.1.2.7)]and the scattering cross section given in terms of the magnetic interaction vector [Q({\bf k})], [Q({\bf k})=\hat {\bf k}\times{\bf m}({\bf k})\times\hat {\bf k},\eqno (6.1.2.8)]is [\specialfonts{\rm d\sigma\over{\rm d}\Omega}=(\gamma r_e){}^2|\langle\sigma'|{\bsf S}_n\cdot{\bf Q}({\bf k})|\sigma\rangle|{}^2.\eqno (6.1.2.9)]Equation (6.1.2.9)[link] leads to two independent scattering cross sections: one for scattering of the neutron with no change in spin state (σ′ = σ) proportional to [\specialfonts|{\bsf S}_n\cdot{\bf Q}({\bf k})|^2], and the other to scattering with a change of neutron spin (`spin flip scattering') proportional to [\specialfonts|{\bsf S}_n\times{\bf Q}({\bf k})|^2]. The sum over all final spin states gives [{\rm d\sigma\over{\rm d}\Omega}=(\gamma r_e){}^2|{\bf Q}({\bf k})|{}^2.\eqno (6.1.2.10)]

6.1.2.3. Calculation of magnetic structure factors and cross sections

| top | pdf |

If the magnetization within the unit cell can be assigned to independent atoms so that each has a total moment [\mu_i] aligned in the direction of the axial unit vector [\hat{\bf m}_i], then the unit-cell structure factor can be written [\specialfonts{\bf M}({\bf k})=\textstyle\sum\limits_j\sum\limits_i{\bsf T}_j{\bsf R} _j\cdot\hat{\bf m} _i\mu_i f_i(k)\exp[i{\bf k}\cdot({\bsf R} _j{\bf r}_i+{\bf t} _j)].\eqno (6.1.2.11)][\specialfonts{\bsf R} _j] and [{\bf t} _j] are the rotations and translations associated with the jth element of the space group and [\specialfonts{\bsf T}_j] is an operator that reverses all the components of moment whenever the element j includes time reversal in the magnetic space group. [f_i(k)] is the magnetic form factor of the ith atom (see Subsection 6.1.2.3[link]).

The vector part of the magnetic structure factor can be factored out so that [{\bf m}({\bf k})\quad{\rm becomes}\quad\hat{\bf m}[m({\bf k})],]where [m({\bf k})] is now a scalar. For collinear structures, all the atomic moments are either parallel or antiparallel to [\hat{\bf m}], which in this case is independent of k. The intensity of a magnetic Bragg reflection is proportional to [|{\bf Q}({\bf k})|^2] and [\eqalignno{|{\bf Q}({\bf k})|{}^2&=1-(\hat{\bf m}\cdot\hat{\bf k}){}^2|m({\bf k})|{}^2\cr &=\sin^2\alpha|m({\bf k})|{}^2\cr &=q^2|m({\bf k})|{}^2,& (6.1.2.12)}]where α is the angle between the moment direction [\hat{\bf m}] and the scattering vector k. The factor [1-(\hat{\bf m}\cdot\hat{\bf k}){}^2], often referred to as [q^2], is the means by which the moment direction in a magnetic structure can be determined from intensity measurements. If the intensities are obtained from measurements on polycrystalline samples then the average of [q^2] over all the different k contributing to the powder line must be taken. [\specialfonts\overline {q^2}=1-{1\over n_g}\sum_j({\bsf R}_j\hat{\bf k}\cdot\hat{\bf m})^2,\eqno (6.1.2.13)]the sum being over all [n_g] rotations [\specialfonts{\bsf R}_j] of the point group. [\overline{q^2}] is given for different crystal symmetries by Shirane (1959[link]). For uniaxial groups, the result is [\overline{q^2}=1-\textstyle{1\over2}(\sin{^2}\Psi\sin{^2}\varphi-\cos{^2}\Psi\cos{^2}\varphi),\eqno (6.1.2.14)]where ψ and [\varphi] are the angles between the unique axis and the scattering vector and moment direction, respectively. For cubic groups [{\overline{q^2}}=2/3] independent of the moment direction and of the direction of k.

6.1.2.4. The magnetic form factor

| top | pdf |

The magnetic form factor introduced in (6.1.2.11)[link] is determined by the distribution of magnetization within a single atom. It can be defined by [\specialfonts f({\bf k})={\big\langle q\big|\int{\bsf M}({\bf r})\exp(i{\bf k}\cdot{\bf r})\,{\rm d}r^3\big|q\big\rangle\over\big\langle q\big|\int{\bsf M}({\bf r})\,{\rm d}r^3\big|q\big\rangle},\eqno (6.1.2.15)]where q now represents a state of an individual atom.

In the majority of cases, the magnetization of an atom or ion is due to a single open atomic shell: the d shell for transition metals, the 4f shell for rare earths, and the 5f shell for actinides. Magnetic form factors are calculated from the radial wavefunctions of the electrons in the open shells. The integrals from which the form factors are obtained are [\langle \, j_l(k)\rangle=\textstyle\int\limits^\infty_0U^2(r)\,j_l(kr)4\pi r^2\,{\rm d}r,\eqno (6.1.2.16)]where U(r) is the radial wavefunction for the atom and [j_l(kr)] is the lth-order spherical Bessel function. Within the dipole approximation (spherical symmetry), the magnetic form factor is given by [f(k)=\langle \, j_0(k)\rangle+(1 - 2/g)\langle\, j_2(k)\rangle,\eqno (6.1.2.17)]where g is the Landé splitting factor (Lovesey, 1984[link]). Higher approximations are needed if the orbital contribution is large and to describe departures from spherical symmetry. They involve terms in [\langle \, j_4\rangle\langle\, j_6\rangle] etc. Fig. 6.1.2.1[link] shows the integrals [\langle\, j_0\rangle,\langle\, j_2\rangle], and [\langle\, j_4\rangle] for Fe2+ and in Fig. 6.1.2.2[link] the spherical spin-only form factors [\langle\, j_0\rangle] for 3d, 4d, 4f, and 5f electrons are compared. Tables of magnetic form factors are given in Section 4.4.5[link] .

[Figure 6.1.2.1]

Figure 6.1.2.1 | top | pdf |

The integrals <j0>, <j2>, and <j4> for the Fe2+ ion plotted against [(\sin\theta)/\lambda]. The integrals have been calculated from wavefunctions given by Clementi & Roetti (1974[link]).

[Figure 6.1.2.2]

Figure 6.1.2.2 | top | pdf |

Comparison of 3d, 4d, 4f, and 5f form factors. The 3d form factor is for Co, and the 4d for Rh, both calculated from wavefunctions given by Clementi & Roetti (1974[link]). The 4f form factor is for Gd3+ calculated by Freeman & Desclaux (1972[link]) and the 5f is that for U3+ given by Desclaux & Freeman (1978[link]).

6.1.2.5. The scattering cross section for polarized neutrons

| top | pdf |

The cross section for scattering of neutrons with an arbitrary spin direction is obtained from (6.1.2.9)[link] but adding also nuclear scattering given by the nuclear structure factor [F({\bf k})], which is assumed to be spin independent. In this case, [\specialfonts{\rm d\sigma\over{\rm d}\Omega}=\langle\sigma|(\gamma r_e){\bsf S}_n\cdot{\bf Q}({\bf k})+F({\bf k})|\sigma\rangle^2,\eqno (6.1.2.18)]the scattering without change of spin direction is [\eqalignno{I^{++}&\propto|F'({\boldkappa})|^2+|\hat {\bf s}_n\cdot{\bf Q}({\bf k})|{}^2\cr &\quad+\hat{\bf s}_n\cdot[{\bf Q}^*({\bf k})F'({\bf k})+{\bf Q}({\bf k})F'^*({\bf k})],& (6.1.2.19)}]and, for the spin flip scattering, [I^{+-}\propto[\hat{\bf s}_n\times{\bf Q}({\bf k})]\cdot[\hat{\bf s}_n\cdot{\bf Q}^*({\bf k})]+\hat {\bf s}_n\cdot[{\bf Q}({\bf k})\times{\bf Q}^*({\bf k})]\eqno (6.1.2.20)]with [F'({\bf k})=F({\bf k})/(\gamma r_e)].

The cross section I++ implies interference between the nuclear and the magnetic scattering when both occur for the same k. This interference is exploited for the production of polarized neutrons, and for the determination of magnetic structure factors using polarized neutrons.

In the classical method for determining magnetic structure factors with polarized neutrons (Nathans, Shull, Shirane & Andresen, 1959[link]), the `flipping ratio' R, which is the ratio between the cross sections for oppositely polarized neutrons, is measured: [R={|F'({\bf k})|{}^2+2P\hat{\bf s}_n\cdot[{\bf Q}({\bf k})F'^*({\bf k})+{\bf Q}^*({\bf k})F'({\bf k})]+|{\bf Q}({\bf k})|{}^2\over|F'({\bf k})|{}^2-2Pe\hat{\bf s}_n\cdot[{\bf Q}({\bf k})F'^*({\bf k})+{\bf Q}^*({\bf k})F'({\bf k})]+|{\bf Q}({\bf k})|{}^2}.\eqno (6.1.2.21)]In this equation, [\hat{\bf s}_n] is a unit vector parallel to the polarization direction. P is the neutron polarization defined as [P=(\langle S^+\rangle-\langle S^-\rangle)/(\langle S^+\rangle+\langle S^-\rangle),]where [\langle S^+\rangle] and [\langle S^-\rangle] are the expectation values of the neutron spin parallel and antiparallel to [\hat{\bf s}_n] averaged over all the neutrons in the beam. e is the `flipping efficiency' defined as e = (2f − 1), where f is the fraction of the neutron spins that are reversed by the flipping process. Equation (6.1.2.21)[link] is considerably simplified when both [F({\bf k})] and [{\bf Q}({\bf k})] are real and the polarization direction is parallel to the magnetization direction, as in a sample magnetized by an external field. The `flipping ratio' then becomes [R={1+2Py\sin^2\rho+y^2\sin^2\rho\over1-2Pey\sin^2\rho+y^2\sin^2\rho},\eqno (6.1.2.22)]with [y=(\gamma r_e)M({\bf k})/F({\bf k})], ρ being the angle between the magnetization direction and the scattering vector. The solution to this equation is [\eqalignno{y&=\{P\sin\rho(Re+1)\pm[P^2\sin^2\rho(Re+1){}^2-(R-1){}^2]{}^{1/2}\}\cr &\quad\times[(R-1)\sin\rho]{}^{-1};& (6.1.2.23)}]the relative signs of [F({\bf k})] and [M({\bf k})] are determined by whether R is greater or less than unity. The uncertainty in the sign of the square root in (6.1.2.23)[link] corresponds to not knowing whether [F({\bf k}) > M({\bf k})] or vice versa.

6.1.2.6. Rotation of the polarization of the scattered neutrons

| top | pdf |

Whenever the neutron spin direction is not parallel to the magnetic interaction vector Q(k), the direction of polarization is changed in the scattering process. The general formulae for the scattered polarization are given by Blume (1963[link]). The result for most cases of interest can be inferred by calculating the components of the scattered neutron's spin in the x, y, and z directions for a neutron whose spin is initially parallel to z. For simplicity, y is taken parallel to k; x and z define a plane that contains Q(k). From (6.1.2.18)[link], [\eqalign{S_x&=\textstyle{1\over2}\{[Q_z({\bf k})+F'({\bf k})]Q^*_x({\bf k})\cr &\quad+[Q^*_z({\bf k})+F'^*({\bf k})]Q_x({\bf k})\}/N\cr S_y&={1\over2i}\{[Q_z({\bf k})+F'({\bf k})]Q^*_x({\bf k})\cr &\quad-[Q^*_z({\bf k})+F'^*({\bf k})]Q_x({\bf k})\}/N\cr S_z&=\textstyle{1\over2}\{[Q_z({\bf k})+F'({\bf k})][Q^*_z({\bf k})+F'^*({\bf k})]\}/N\cr N&=|Q_z({\bf k})+F'({\bf k})|{}^2+|Q_x({\bf k})|{}^2.} (6.1.2.24)]

It is clear from this set of equations that [S_x] and [S_y] are zero if [Q_x({\bf k})=0]. Three simple cases may be taken as examples of the use of (6.1.2.24)[link]:

  • (a) A magnetic reflection from a simple antiferromagnet for which Q(k) is real, F(k) = 0; under these conditions, [\eqalign{S_x&=Q_x({\bf k})[Q_z({\bf k})]/|{\bf Q}({\bf k})|{}^2\cr S_y&=0\cr S_z&=\textstyle{1\over2}[Q_z({\bf k}){}^2-Q_x({\bf k}){}^2]/|{\bf Q}({\bf k})|{}^2,}]showing that the direction of polarization is turned through an angle [2\varphi] in the xy plane where [\varphi] is the angle between Q(k) and the initial polarization direction.

  • (b) A satellite reflection from a magnetic structure described by a circular helix for which [Q_x({\bf k})] = [iQ_z({\bf k}),F'({\bf k})] = 0; in this case, [\eqalign{S_x&=0\cr S_y&=Q^2_z({\bf k})/|{\bf Q}({\bf k})|{}^2=\textstyle{1\over2}\cr S_z&=0}]and the scattered polarization is parallel to the scattering vector independent of its initial direction.

  • (c) A mixed magnetic and nuclear reflection from a Cr2O3-type antiferromagnet for which Q(k) is imaginary, [{\bf Q}({\bf k})=] [-{\bf Q}^*({\bf k})], [F({\bf k})] is real. Then, [\eqalign{S_x&=Q_x({\bf k})Q_z({\bf k})/[F'({\bf k}){}^2+|{\bf Q}({\bf k})|{}^2]\cr S_y&=iF({\bf k})Q_x({\bf k})/[F'({\bf k}){}^2+|{\bf Q}({\bf k})|{}^2]\cr S_z&=\textstyle{1\over2}[|Q_z({\bf k})+F'({\bf k})|{}^2-|Q_x({\bf k})|{}^2]\cr &\quad\times[F'({\bf k}){}^2+|{\bf Q}({\bf k})|{}^2]{}^{-1}}]so that in this case the final polarization has components along all three directions.

References

Blume, M. (1963). Polarization effects in the magnetic elastic scattering of slow neutrons. Phys. Rev. 130, 1670–1676.
Clementi, E. & Roetti, C. (1974). Roothaan–Hartree–Fock atomic wavefunctions. Basis functions and their coefficients for ground and certain excited states of neutral and ionized atoms. At. Data Nucl. Data Tables, 14, 177–478.
Desclaux, J. P. & Freeman, A. J. (1978). Dirac–Fock studies of some electronic properties of actinide ions. J. Magn. Magn. Mater. 8, 119–129.
Freeman, A. J. & Desclaux, J. P. (1972). Neutron magnetic form factor of gadolinium. Int. J. Magn. 3, 311–317.
Lovesey, S. W. (1984). Theory of neutron scattering from condensed matter. Vol. 2. Polarization effects and magnetic scattering. The International Series of Monographs on Physics No. 72. Oxford University Press.
Nathans, R., Shull, C. G., Shirane, G. & Andresen, A. (1959). The use of polarised neutrons in determining the magnetic scattering by iron and nickel. J. Phys. Chem. Solids, 10, 138–146.
Shirane, G. (1959). A note on the magnetic intensities of powder neutron diffraction. Acta Cryst. 12, 282–285.
Trammell, G. T. (1953). Magnetic scattering of neutrons from rare earth ions. Phys. Rev. 92, 1387–1393.








































to end of page
to top of page