osns - diffraction from magnetic materials - diffrac… · diffraction from magnetic materials....

$: OSNS - Diffraction from magnetic materials - Diffrac… · Diffraction from magnetic materials. 16th Oxford School on Neutron Scattering | Navid Qureshi THE EUROPEAN NEUTRON SOURCE$
I N S T I T U T L A U E L A N G E V I N

T H E E U R O P E A N N E U T R O N S O U R C E16th Oxford School on Neutron Scattering | Navid Qureshi

Navid Qureshi

Diffraction from magnetic materials


MotivationImpact of magnetic neutron scattering

correlated with

‣ availability of computing tools

‣ relevant topics in new materials

H. Rietveld presents a least-squares program

able to refine crystal and magnetic structures

J. Rodríguez Carvajal presents the FullProf

packagePublications

Year



correlated with





correlated with



Methods and computing programs



correlated with



Multiferroics



correlated with



Superconductors



correlated with



Nanoparticles



correlated with



Order phenomena in manganites


Outline

‣ Reminder: Nuclear scattering scattering amplitude, Fourier transform

‣ Magnetic scattering scattering potential, directional dependence

‣ Magnetic structures Ferro, antiferro, cycloids, helices, …

‣ Scattering from a unit cell Magnetic structure factor, interaction vector

‣ Experimental procedure


Conventions for this lectureWave vector and scattering vector

: initial wavevector

: final wavevector

: momentum transfer, scattering vector

: reciprocal lattice vector

Elastic scattering:

kf<latexit sha1_base64="B7J8YulRjkgvkKb/1iea7956Y20=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgQkqigi4LblxWsA9oQplMb9qhk0mYmQgl9DfcuFDErT/jzr9x2mahrQcGDufcyz1zwlRwbVz32ymtrW9sbpW3Kzu7e/sH1cOjtk4yxbDFEpGobkg1Ci6xZbgR2E0V0jgU2AnHdzO/84RK80Q+mkmKQUyHkkecUWMl34+pGYVRPp72o3615tbdOcgq8QpSgwLNfvXLHyQsi1EaJqjWPc9NTZBTZTgTOK34mcaUsjEdYs9SSWPUQT7PPCVnVhmQKFH2SUPm6u+NnMZaT+LQTs4y6mVvJv7n9TIT3QY5l2lmULLFoSgTxCRkVgAZcIXMiIkllClusxI2oooyY2uq2BK85S+vkvZl3buqew/XtcZFUUcZTuAUzsGDG2jAPTShBQxSeIZXeHMy58V5dz4WoyWn2DmGP3A+fwBhN5HV</latexit>

ki<latexit sha1_base64="/13hgp5wwDC3KNNbVjc6HKIMtB0=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgQkqigi4LblxWsA9oQplMb9qhk0mYmQgl9DfcuFDErT/jzr9x2mahrQcGDufcyz1zwlRwbVz32ymtrW9sbpW3Kzu7e/sH1cOjtk4yxbDFEpGobkg1Ci6xZbgR2E0V0jgU2AnHdzO/84RK80Q+mkmKQUyHkkecUWMl34+pGYVRPp72eb9ac+vuHGSVeAWpQYFmv/rlDxKWxSgNE1TrnuemJsipMpwJnFb8TGNK2ZgOsWeppDHqIJ9nnpIzqwxIlCj7pCFz9fdGTmOtJ3FoJ2cZ9bI3E//zepmJboOcyzQzKNniUJQJYhIyK4AMuEJmxMQSyhS3WQkbUUWZsTVVbAne8pdXSfuy7l3VvYfrWuOiqKMMJ3AK5+DBDTTgHprQAgYpPMMrvDmZ8+K8Ox+L0ZJT7BzDHzifP2XDkdg=</latexit>

k<latexit sha1_base64="PhXbyvGIO65nFp2zHvXvHjgxBz4=">AAAB8XicbVBNS8NAFHypX7V+VT16WSyCBymJCnosePFYwdZiG8pmu2mXbjZh90Uoof/CiwdFvPpvvPlv3LQ5aOvAwjDzHjtvgkQKg6777ZRWVtfWN8qbla3tnd296v5B28SpZrzFYhnrTkANl0LxFgqUvJNoTqNA8odgfJP7D09cGxGre5wk3I/oUIlQMIpWeuxFFEdBmI2n/WrNrbszkGXiFaQGBZr96ldvELM04gqZpMZ0PTdBP6MaBZN8WumlhieUjemQdy1VNOLGz2aJp+TEKgMSxto+hWSm/t7IaGTMJArsZJ7QLHq5+J/XTTG89jOhkhS5YvOPwlQSjEl+PhkIzRnKiSWUaWGzEjaimjK0JVVsCd7iycukfV73Lure3WWtcVbUUYYjOIZT8OAKGnALTWgBAwXP8ApvjnFenHfnYz5acoqdQ/gD5/MH49aQ/A==</latexit>

G<latexit sha1_base64="mF8hcLEUKb0JTMp0GFUPovWfIRA=">AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgQsqMCrosuNBlBfvAtpRMeqcNzWSGJCOUoX/hxoUibv0bd/6NmXYW2nogcDjnXnLu8WPBtXHdb6ewsrq2vlHcLG1t7+zulfcPmjpKFMMGi0Sk2j7VKLjEhuFGYDtWSENfYMsf32R+6wmV5pF8MJMYeyEdSh5wRo2VHrshNSM/SG+n/XLFrbozkGXi5aQCOer98ld3ELEkRGmYoFp3PDc2vZQqw5nAaambaIwpG9MhdiyVNETdS2eJp+TEKgMSRMo+achM/b2R0lDrSejbySyhXvQy8T+vk5jgupdyGScGJZt/FCSCmIhk55MBV8iMmFhCmeI2K2EjqigztqSSLcFbPHmZNM+r3kXVu7+s1M7yOopwBMdwCh5cQQ3uoA4NYCDhGV7hzdHOi/PufMxHC06+cwh/4Hz+AK0ikNg=</latexit>

|ki| = |kf | = k<latexit sha1_base64="F4YxrzJhIIQkOJ4n8XV45PICf8Y=">AAACD3icbVDLSsNAFL3xWesr6tLNYFFcSElU0I1QcOOygn1AG8JkOmmHTh7MTISS9g/c+CtuXCji1q07/8ZJmkVtPXDhcM693HuPF3MmlWX9GEvLK6tr66WN8ubW9s6uubfflFEiCG2QiEei7WFJOQtpQzHFaTsWFAcepy1veJv5rUcqJIvCBzWKqRPgfsh8RrDSkmuejLsBVgPPT4cTl6ExutE1I/m5NHTNilW1cqBFYhekAgXqrvnd7UUkCWioCMdSdmwrVk6KhWKE00m5m0gaYzLEfdrRNMQBlU6a/zNBx1rpIT8SukKFcnV2IsWBlKPA053ZpXLey8T/vE6i/GsnZWGcKBqS6SI/4UhFKAsH9ZigRPGRJpgIpm9FZIAFJkpHWNYh2PMvL5LmedW+qNr3l5XaWRFHCQ7hCE7BhiuowR3UoQEEnuAF3uDdeDZejQ/jc9q6ZBQzB/AHxtcvtrSbuA==</latexit>


Nuclear scatteringScattering cross section

dn|{z}ns�1

= �|{z}ncm�2s�1

· d⌦|{z}1

·�(✓,')| {z }cm2

<latexit sha1_base64="lZypFLWk2gl4ol98oaj130phTMQ=">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</latexit>

𝜎 has the unit of a surface

usually in barns = 10-24 cm2

Number of neutrons n detected in solid angle ⌦<latexit sha1_base64="wmuaksdsi8WKe/JAndUeSpkLXXc=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgQcKuCnoMePFmBPOAZAmzk95kzDyWmVkhLPkHLx4U8er/ePNvnCR70MSChqKqm+6uKOHMWN//9gorq2vrG8XN0tb2zu5eef+gaVSqKTSo4kq3I2KAMwkNyyyHdqKBiIhDKxrdTP3WE2jDlHyw4wRCQQaSxYwS66Rm907AgPTKFb/qz4CXSZCTCspR75W/un1FUwHSUk6M6QR+YsOMaMsoh0mpmxpICB2RAXQclUSACbPZtRN84pQ+jpV2JS2eqb8nMiKMGYvIdQpih2bRm4r/eZ3UxtdhxmSSWpB0vihOObYKT1/HfaaBWj52hFDN3K2YDokm1LqASi6EYPHlZdI8rwYX1eD+slI7y+MooiN0jE5RgK5QDd2iOmogih7RM3pFb57yXrx372PeWvDymUP0B97nD1mPjug=</latexit>



The wave function at a spatial position r = sum of transmitted and scattered spherical wave function.

vscatk (r) = eikr + fk(✓,')eikr

r<latexit sha1_base64="AJOQ7wuRf/Fi9beL7BqaG/Wr7vU=">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</latexit>

Only depends on the scattering

potential .

fk(✓,')<latexit sha1_base64="BIZkHoNkvOfq+Grffrs01iITa3o=">AAAB/HicbVBNS8NAEN3Ur1q/oj16CRahQimJCnosePFYwX5AE8Jmu2mWbjZhd1IIpf4VLx4U8eoP8ea/cdvmoK0PBh7vzTAzL0g5U2Db30ZpY3Nre6e8W9nbPzg8Mo9PuirJJKEdkvBE9gOsKGeCdoABp/1UUhwHnPaC8d3c702oVCwRj5Cn1IvxSLCQEQxa8s1q6I/rLkQUcMOdYJlG7MI3a3bTXsBaJ05BaqhA2ze/3GFCspgKIBwrNXDsFLwplsAIp7OKmymaYjLGIzrQVOCYKm+6OH5mnWtlaIWJ1CXAWqi/J6Y4ViqPA90ZY4jUqjcX//MGGYS33pSJNAMqyHJRmHELEmuehDVkkhLguSaYSKZvtUiEJSag86roEJzVl9dJ97LpXDWdh+taq1HEUUan6AzVkYNuUAvdozbqIIJy9Ixe0ZvxZLwY78bHsrVkFDNV9AfG5w8f/ZRb</latexit>

V (r)<latexit sha1_base64="2oFztpCCnfp5VoK6zrEuOWGXBYc=">AAAB9HicbVDLSgMxFL1TX7W+qi7dBItQQcqMCrosuHFZwT6gHUomzbShmWRMMoUy9DvcuFDErR/jzr8x085CWw8EDufcyz05QcyZNq777RTW1jc2t4rbpZ3dvf2D8uFRS8tEEdokkkvVCbCmnAnaNMxw2okVxVHAaTsY32V+e0KVZlI8mmlM/QgPBQsZwcZKfqvai7AZBWGqZuf9csWtuXOgVeLlpAI5Gv3yV28gSRJRYQjHWnc9NzZ+ipVhhNNZqZdoGmMyxkPatVTgiGo/nYeeoTOrDFAolX3CoLn6eyPFkdbTKLCTWUS97GXif143MeGtnzIRJ4YKsjgUJhwZibIG0IApSgyfWoKJYjYrIiOsMDG2p5ItwVv+8ippXda8q5r3cF2pX+R1FOEETqEKHtxAHe6hAU0g8ATP8ApvzsR5cd6dj8Vowcl3juEPnM8fZJORyA==</latexit>



‣ mediated by strong force, short ranged (fm = 10-15 m)

‣ neutron wavelength much larger (10-10 m) cannot probe internal structure scattering is isotropic

‣ the interaction between the neutron and the atomic nucleus is

represented by the Fermi pseudo-potential, a scalar field that is 0

except very close to the nucleus V (r) =2⇡~2mn

b�3(r)<latexit sha1_base64="71yGz+YYUKXXmQawjH+4Un/QqZU=">AAACJXicbVDLSgMxFM34rPU16tJNsAgVpMy0gi4UCm5cVrAP6LQlk2ba0CQzJBmhDPMzbvwVNy4sIrjyV8y0XdTWA4HDOeeSe48fMaq043xba+sbm1vbuZ387t7+waF9dNxQYSwxqeOQhbLlI0UYFaSuqWakFUmCuM9I0x/dZ37zmUhFQ/GkxxHpcDQQNKAYaSP17NtG0eNID/0gkekFvINeIBFOyl5EvaGPZLecJrwnUt/rE6ZRt7IY79kFp+RMAVeJOycFMEetZ0+8fohjToTGDCnVdp1IdxIkNcWMpHkvViRCeIQGpG2oQJyoTjK9MoXnRunDIJTmCQ2n6uJEgrhSY+6bZLaiWvYy8T+vHevgppNQEcWaCDz7KIgZ1CHMKoN9KgnWbGwIwpKaXSEeIlOTNsXmTQnu8smrpFEuuZWS+3hVqF7O68iBU3AGisAF16AKHkAN1AEGL+ANfICJ9Wq9W5/W1yy6Zs1nTsAfWD+/jy2lQA==</latexit>

advantage: neutron senses atomic position and not the electron cloud (bonds)



The scattering amplitude is related to the Fourier transform of the potential function.

With the Fermi pseudo potential for neutron scattering from a nucleus

Neutron scattering from a nucleus is isotropic!

b is in the order of 10-12 cm

fk(✓,') = � 1

4⇡

2µ

~2

ZV (r)e�ikrd3r

<latexit sha1_base64="6K5Lg3ALed7S5pVoGgyuvm7y8tM=">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</latexit>

V (r) =2⇡~2mn

b�3(r)<latexit sha1_base64="71yGz+YYUKXXmQawjH+4Un/QqZU=">AAACJXicbVDLSgMxFM34rPU16tJNsAgVpMy0gi4UCm5cVrAP6LQlk2ba0CQzJBmhDPMzbvwVNy4sIrjyV8y0XdTWA4HDOeeSe48fMaq043xba+sbm1vbuZ387t7+waF9dNxQYSwxqeOQhbLlI0UYFaSuqWakFUmCuM9I0x/dZ37zmUhFQ/GkxxHpcDQQNKAYaSP17NtG0eNID/0gkekFvINeIBFOyl5EvaGPZLecJrwnUt/rE6ZRt7IY79kFp+RMAVeJOycFMEetZ0+8fohjToTGDCnVdp1IdxIkNcWMpHkvViRCeIQGpG2oQJyoTjK9MoXnRunDIJTmCQ2n6uJEgrhSY+6bZLaiWvYy8T+vHevgppNQEcWaCDz7KIgZ1CHMKoN9KgnWbGwIwpKaXSEeIlOTNsXmTQnu8smrpFEuuZWS+3hVqF7O68iBU3AGisAF16AKHkAN1AEGL+ANfICJ9Wq9W5/W1yy6Zs1nTsAfWD+/jy2lQA==</latexit>

|fk(✓,')| = b<latexit sha1_base64="1sWirUCD0ePZJv0N//7D21ddJTM=">AAACAHicbVBNS8NAEN34WetX1IMHL8EiVCglUUEvQsGLxwr2A5oQNttNu3SzCbuTQkl78a948aCIV3+GN/+N2zYHbX0w8Hhvhpl5QcKZAtv+NlZW19Y3Ngtbxe2d3b198+CwqeJUEtogMY9lO8CKciZoAxhw2k4kxVHAaSsY3E391pBKxWLxCKOEehHuCRYygkFLvnk8Dv1B2YU+BVxxh1gmfXY+vg18s2RX7RmsZeLkpIRy1H3zy+3GJI2oAMKxUh3HTsDLsARGOJ0U3VTRBJMB7tGOpgJHVHnZ7IGJdaaVrhXGUpcAa6b+nshwpNQoCnRnhKGvFr2p+J/XSSG88TImkhSoIPNFYcotiK1pGlaXSUqAjzTBRDJ9q0X6WGICOrOiDsFZfHmZNC+qzmXVebgq1Sp5HAV0gk5RGTnoGtXQPaqjBiJogp7RK3oznowX4934mLeuGPnMEfoD4/MHPpqWGg==</latexit>


Nuclear scatteringComparison to X-rays

The amplitude of the scattered wave (the Fourier transform of the potential function) is called the atomic form factor f (X-rays) or scattering length b (neutrons).

Nucleus ~10-15 m

Electron shell ~10-10 m

advantage with neutrons: scattered intensity does not drop with increasing scattering angle



Scattering lengths (analog to X-ray form factor)

superposition of resonance scattering with slowly increasing potential scattering due to atomic weight

advantages:

contrast between neighbouring elements light elements can be measured easily

isotope effect (bH=-3.7, bD=6.8)



Scattering lengths (analog to X-ray form factor)

superposition of resonance scattering with slowly increasing potential scattering due to atomic weight

advantages:

contrast between neighbouring elements light elements can be measured easily

isotope effect (bH=-3.7, bD=6.8)

K+

Cl-Example KCl:

scattering lengths of K and Cl are very different strong contrast

X-rays would see a primitive cell with half the lattice constant


Magnetic scatteringElectron magnetic moment

Magnetic dipole moment in classical electrodynamics

µ = I ·A<latexit sha1_base64="SGDC0ONJB3FjiX4lIuLs+FurSZI=">AAAB+HicbVDLSsNAFJ34rPXRqEs3g0VwISVRQTdCxY3uKtgHNKFMJpN26DzCzESooV/ixoUibv0Ud/6N0zYLbT1w4XDOvdx7T5Qyqo3nfTtLyyura+uljfLm1vZOxd3da2mZKUyaWDKpOhHShFFBmoYaRjqpIohHjLSj4c3Ebz8SpakUD2aUkpCjvqAJxchYqedWAp7BK3gHAxxLA697btWreVPAReIXpAoKNHruVxBLnHEiDGZI667vpSbMkTIUMzIuB5kmKcJD1CddSwXiRIf59PAxPLJKDBOpbAkDp+rviRxxrUc8sp0cmYGe9ybif143M8llmFORZoYIPFuUZAwaCScpwJgqgg0bWYKwovZWiAdIIWxsVmUbgj//8iJpndb8s5p/f16tnxRxlMABOATHwAcXoA5uQQM0AQYZeAav4M15cl6cd+dj1rrkFDP74A+czx+q4JG5</latexit>

=�e · ve2⇡r

· ⇡r2 =�e · ve

2· r =

�e

2m·mver = � e

2mL

<latexit sha1_base64="f/FdZivhpEIm4Ivndxw5kZ+OIFA=">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</latexit>

Gyromagnetic ratio : ratio between magnetic dipole moment and total angular momentum�<latexit sha1_base64="rb2AirB/XSr2+pnI4GtwbIWRgLA=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8SNhVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/eyura+sbm4Wt4vbO7t5+6eCwYVSmKatTJZRuRWiY4AmrW24Fa6WaoYwEa0bD26nffGLacJU82FHKQon9hMeconVSo9NHKbFbKvsVfwayTIKclCFHrVv66vQUzSRLLBVoTDvwUxuOUVtOBZsUO5lhKdIh9lnb0QQlM+F4du2EnDqlR2KlXSWWzNTfE2OUxoxk5Dol2oFZ9Kbif147s/FNOOZJmlmW0PmiOBPEKjJ9nfS4ZtSKkSNINXe3EjpAjdS6gIouhGDx5WXSuKgEl5Xg/qpcPc/jKMAxnMAZBHANVbiDGtSBwiM8wyu8ecp78d69j3nripfPHMEfeJ8/gUmPAg==</latexit>

� = � e

2m= �µB

~<latexit sha1_base64="UVYUikRPVQWHJWCEKHn6zKnrdSg=">AAACFHicbVDLSgMxFM3UV62vqks3wSIIapmpgm4KRTcuK9gHdEq5k2ba0GRmSDJCGeYj3Pgrblwo4taFO//GtJ2Fth64cHLOveTe40WcKW3b31ZuaXlldS2/XtjY3NreKe7uNVUYS0IbJOShbHugKGcBbWimOW1HkoLwOG15o5uJ33qgUrEwuNfjiHYFDALmMwLaSL3iiTsAIQBX8ZnrSyAJTZOKSKvZyxVx7zpN3KEHMu0VS3bZngIvEicjJZSh3it+uf2QxIIGmnBQquPYke4mIDUjnKYFN1Y0AjKCAe0YGoCgqptMj0rxkVH62A+lqUDjqfp7IgGh1Fh4plOAHqp5byL+53Vi7V91ExZEsaYBmX3kxxzrEE8Swn0mKdF8bAgQycyumAzBpKFNjgUTgjN/8iJpVsrOedm5uyjVTrM48ugAHaJj5KBLVEO3qI4aiKBH9Ixe0Zv1ZL1Y79bHrDVnZTP76A+szx9uN55l</latexit>

µB =e~2me

<latexit sha1_base64="DaiN8n8rPRlKINZGrfy3MSD+tbI=">AAACAnicbVBNS8NAEN3Ur1q/op7ES7AIHqQkVdCLUPTisYL9gCaEzXbSLt1Nwu5GKCF48a948aCIV3+FN/+N2zYHbX0w8Hhvhpl5QcKoVLb9bZSWlldW18rrlY3Nre0dc3evLeNUEGiRmMWiG2AJjEbQUlQx6CYCMA8YdILRzcTvPICQNI7u1TgBj+NBRENKsNKSbx64PPWvr9xQYJKBOwywyLM69yH3zapds6ewFolTkCoq0PTNL7cfk5RDpAjDUvYcO1FehoWihEFecVMJCSYjPICephHmIL1s+kJuHWulb4Wx0BUpa6r+nsgwl3LMA93JsRrKeW8i/uf1UhVeehmNklRBRGaLwpRZKrYmeVh9KoAoNtYEE0H1rRYZYp2G0qlVdAjO/MuLpF2vOWc15+682jgt4iijQ3SETpCDLlAD3aImaiGCHtEzekVvxpPxYrwbH7PWklHM7KM/MD5/AF1jl1c=</latexit>

with

This works well for the electron’s orbital momentum, but its intrinsic spin momentum cannot be explained in the classical approach correction by g-factor

� = �geµB

~<latexit sha1_base64="6KC019gZtpyC9z3zqRNFAL1rZ/o=">AAACCXicbVDLSsNAFJ34rPUVdelmsAgutCQq6EYounFZwT6gCeFmOkmHziRhZiKU0K0bf8WNC0Xc+gfu/Bunj4W2HrhwOOde7r0nzDhT2nG+rYXFpeWV1dJaeX1jc2vb3tltqjSXhDZIylPZDkFRzhLa0Exz2s4kBRFy2gr7NyO/9UClYmlyrwcZ9QXECYsYAW2kwMZeDEIAvsIncUC9SAIpPJEH18PC64Ugh4FdcarOGHieuFNSQVPUA/vL66YkFzTRhINSHdfJtF+A1IxwOix7uaIZkD7EtGNoAoIqvxh/MsSHRuniKJWmEo3H6u+JAoRSAxGaTgG6p2a9kfif18l1dOkXLMlyTRMyWRTlHOsUj2LBXSYp0XxgCBDJzK2Y9MCkoU14ZROCO/vyPGmeVt2zqnt3XqkdT+MooX10gI6Qiy5QDd2iOmoggh7RM3pFb9aT9WK9Wx+T1gVrOrOH/sD6/AHuD5nM</latexit>

(gL = 1, gS = 2)


Magnetic scatteringElectron magnetic moment

Angular momenta are quantised in units of

spin moment (1 per electron spin)

orbital moment (1 per )

Total moment due to LS coupling :

The magnetic moment of en electron is quantised in

Spin momenta are quantised in units of 1/2

~<latexit sha1_base64="10Uu+MJGiNZ5co/h0xjPZwIDYH0=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBT0WvHisYNpCG8pku2mXbjZhdyOU0N/gxYMiXv1B3vw3btsctPXBwOO9GWbmhang2rjut1NaW9/Y3CpvV3Z29/YPqodHLZ1kijKfJiJRnRA1E1wy33AjWCdVDONQsHY4vpv57SemNE/ko5mkLIhxKHnEKRor+b1RiKpfrbl1dw6ySryC1KBAs1/96g0SmsVMGipQ667npibIURlOBZtWeplmKdIxDlnXUokx00E+P3ZKzqwyIFGibElD5urviRxjrSdxaDtjNCO97M3E/7xuZqLbIOcyzQyTdLEoygQxCZl9TgZcMWrExBKkittbCR2hQmpsPhUbgrf88ippXda9q7r3cF1rXBRxlOEETuEcPLiBBtxDE3ygwOEZXuHNkc6L8+58LFpLTjFzDH/gfP4Av7GOkg==</latexit>

(L = ...,�2~,�~, 0, ~, 2~, ...)<latexit sha1_base64="5PXW7D7bVDXrj5iKTT8OQYVg5iU=">AAACGnicbZDLSgMxFIbP1Futt1GXboJFqFCHmSroRii4ceGigr1AO5RMmmlDMxeSjFCGPocbX8WNC0XciRvfxrQdQVsPhHz8/zkk5/dizqSy7S8jt7S8srqWXy9sbG5t75i7ew0ZJYLQOol4JFoelpSzkNYVU5y2YkFx4HHa9IZXE795T4VkUXinRjF1A9wPmc8IVlrqmk7pBl0iy7LK6KTSGXhYaMhuu4wy+nF033HXLNqWPS20CE4GRciq1jU/Or2IJAENFeFYyrZjx8pNsVCMcDoudBJJY0yGuE/bGkMcUOmm09XG6EgrPeRHQp9Qoan6eyLFgZSjwNOdAVYDOe9NxP+8dqL8CzdlYZwoGpLZQ37CkYrQJCfUY4ISxUcaMBFM/xWRARaYKJ1mQYfgzK+8CI2K5Zxazu1ZsVrO4sjDARxCCRw4hypcQw3qQOABnuAFXo1H49l4M95nrTkjm9mHP2V8fgMiqptK</latexit>

� = �geµB

~<latexit sha1_base64="6KC019gZtpyC9z3zqRNFAL1rZ/o=">AAACCXicbVDLSsNAFJ34rPUVdelmsAgutCQq6EYounFZwT6gCeFmOkmHziRhZiKU0K0bf8WNC0Xc+gfu/Bunj4W2HrhwOOde7r0nzDhT2nG+rYXFpeWV1dJaeX1jc2vb3tltqjSXhDZIylPZDkFRzhLa0Exz2s4kBRFy2gr7NyO/9UClYmlyrwcZ9QXECYsYAW2kwMZeDEIAvsIncUC9SAIpPJEH18PC64Ugh4FdcarOGHieuFNSQVPUA/vL66YkFzTRhINSHdfJtF+A1IxwOix7uaIZkD7EtGNoAoIqvxh/MsSHRuniKJWmEo3H6u+JAoRSAxGaTgG6p2a9kfif18l1dOkXLMlyTRMyWRTlHOsUj2LBXSYp0XxgCBDJzK2Y9MCkoU14ZROCO/vyPGmeVt2zqnt3XqkdT+MooX10gI6Qiy5QDd2iOmoggh7RM3pFb9aT9WK9Wx+T1gVrOrOH/sD6/AHuD5nM</latexit>

(gL = 1, gS = 2)

µB<latexit sha1_base64="C91P0K5ZAOqTt0uUUTQMZ8yaSh8=">AAAB7HicbVDLSgNBEOyNrxhfUY9eBoPgQcKuCnoMevEYwTWBZAmzk9lkyMzsMg8hLPkGLx4U8eoHefNvnCR70MSChqKqm+6uOONMG9//9korq2vrG+XNytb2zu5edf/gUadWERqSlKeqHWNNOZM0NMxw2s4UxSLmtBWPbqd+64kqzVL5YMYZjQQeSJYwgo2Twq6wvZtetebX/RnQMgkKUoMCzV71q9tPiRVUGsKx1p3Az0yUY2UY4XRS6VpNM0xGeEA7jkosqI7y2bETdOKUPkpS5UoaNFN/T+RYaD0WsesU2Az1ojcV//M61iTXUc5kZg2VZL4osRyZFE0/R32mKDF87AgmirlbERlihYlx+VRcCMHiy8vk8bweXNSD+8ta46yIowxHcAynEMAVNOAOmhACAQbP8ApvnvRevHfvY95a8oqZQ/gD7/MHmHyOeA==</latexit>

µB<latexit sha1_base64="C91P0K5ZAOqTt0uUUTQMZ8yaSh8=">AAAB7HicbVDLSgNBEOyNrxhfUY9eBoPgQcKuCnoMevEYwTWBZAmzk9lkyMzsMg8hLPkGLx4U8eoHefNvnCR70MSChqKqm+6uOONMG9//9korq2vrG+XNytb2zu5edf/gUadWERqSlKeqHWNNOZM0NMxw2s4UxSLmtBWPbqd+64kqzVL5YMYZjQQeSJYwgo2Twq6wvZtetebX/RnQMgkKUoMCzV71q9tPiRVUGsKx1p3Az0yUY2UY4XRS6VpNM0xGeEA7jkosqI7y2bETdOKUPkpS5UoaNFN/T+RYaD0WsesU2Az1ojcV//M61iTXUc5kZg2VZL4osRyZFE0/R32mKDF87AgmirlbERlihYlx+VRcCMHiy8vk8bweXNSD+8ta46yIowxHcAynEMAVNOAOmhACAQbP8ApvnvRevHfvY95a8oqZQ/gD7/MHmHyOeA==</latexit> ~

<latexit sha1_base64="10Uu+MJGiNZ5co/h0xjPZwIDYH0=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBT0WvHisYNpCG8pku2mXbjZhdyOU0N/gxYMiXv1B3vw3btsctPXBwOO9GWbmhang2rjut1NaW9/Y3CpvV3Z29/YPqodHLZ1kijKfJiJRnRA1E1wy33AjWCdVDONQsHY4vpv57SemNE/ko5mkLIhxKHnEKRor+b1RiKpfrbl1dw6ySryC1KBAs1/96g0SmsVMGipQ667npibIURlOBZtWeplmKdIxDlnXUokx00E+P3ZKzqwyIFGibElD5urviRxjrSdxaDtjNCO97M3E/7xuZqLbIOcyzQyTdLEoygQxCZl9TgZcMWrExBKkittbCR2hQmpsPhUbgrf88ippXda9q7r3cF1rXBRxlOEETuEcPLiBBtxDE3ygwOEZXuHNkc6L8+58LFpLTjFzDH/gfP4Av7GOkg==</latexit>

~<latexit sha1_base64="10Uu+MJGiNZ5co/h0xjPZwIDYH0=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBg5REBT0WvHisYNpCG8pku2mXbjZhdyOU0N/gxYMiXv1B3vw3btsctPXBwOO9GWbmhang2rjut1NaW9/Y3CpvV3Z29/YPqodHLZ1kijKfJiJRnRA1E1wy33AjWCdVDONQsHY4vpv57SemNE/ko5mkLIhxKHnEKRor+b1RiKpfrbl1dw6ySryC1KBAs1/96g0SmsVMGipQ667npibIURlOBZtWeplmKdIxDlnXUokx00E+P3ZKzqwyIFGibElD5urviRxjrSdxaDtjNCO97M3E/7xuZqLbIOcyzQyTdLEoygQxCZl9TgZcMWrExBKkittbCR2hQmpsPhUbgrf88ippXda9q7r3cF1rXBRxlOEETuEcPLiBBtxDE3ygwOEZXuHNkc6L8+58LFpLTjFzDH/gfP4Av7GOkg==</latexit>

(S = ...� 3

2~,�~,�1

2~, 0, 1

2~, ~, 3

2~, ...)

<latexit sha1_base64="a/TUfE0kbskrDuVaBLkm10ZdMLM=">AAACUXicbVHLSgMxFL0zvmrro+rSTbAIFdphphV0IxTcuKxoH9CWkkkzbWjmQZIRytBfdKEr/8ONC8W0HaW2Xgj3cM59JCduxJlUtv1mmBubW9s7md1sbm//4DB/dNyUYSwIbZCQh6LtYkk5C2hDMcVpOxIU+y6nLXd8O9NbT1RIFgaPahLRno+HAfMYwUpT/fyo+IBukGVZ5a4nMEmq06Qy7Y5cLEqo/JvnkrMk2SW0Tv6ktUF6/EU/X7Atex5oHTgpKEAa9X7+pTsISezTQBGOpew4dqR6CRaKEU6n2W4saYTJGA9pR8MA+1T2krkjU3SumQHyQqFPoNCcXe5IsC/lxHd1pY/VSK5qM/I/rRMr77qXsCCKFQ3IYpEXc6RCNLMXDZigRPGJBpgIpu+KyAhrQ5T+hKw2wVl98jpoViynajn3l4VaKbUjA6dwBkVw4ApqcAd1aACBZ3iHT/gyXo0PE0xzUWoaac8J/Akz9w2+Xq+x</latexit>

µB<latexit sha1_base64="C91P0K5ZAOqTt0uUUTQMZ8yaSh8=">AAAB7HicbVDLSgNBEOyNrxhfUY9eBoPgQcKuCnoMevEYwTWBZAmzk9lkyMzsMg8hLPkGLx4U8eoHefNvnCR70MSChqKqm+6uOONMG9//9korq2vrG+XNytb2zu5edf/gUadWERqSlKeqHWNNOZM0NMxw2s4UxSLmtBWPbqd+64kqzVL5YMYZjQQeSJYwgo2Twq6wvZtetebX/RnQMgkKUoMCzV71q9tPiRVUGsKx1p3Az0yUY2UY4XRS6VpNM0xGeEA7jkosqI7y2bETdOKUPkpS5UoaNFN/T+RYaD0WsesU2Az1ojcV//M61iTXUc5kZg2VZL4osRyZFE0/R32mKDF87AgmirlbERlihYlx+VRcCMHiy8vk8bweXNSD+8ta46yIowxHcAynEMAVNOAOmhACAQbP8ApvnvRevHfvY95a8oqZQ/gD7/MHmHyOeA==</latexit>

J = L+ S<latexit sha1_base64="d0MZjqYK3RK52/KNigjYVtvbkWg=">AAACCXicbZDLSsNAFIZPvNZ6i7p0M1gEQSmJCroRCm5EXFS0F2hDmUwn7dDJhZmJUEK2bnwVNy4UcesbuPNtnLQpaOsPAx//OYc553cjzqSyrG9jbn5hcWm5sFJcXVvf2DS3tusyjAWhNRLyUDRdLClnAa0ppjhtRoJi3+W04Q4us3rjgQrJwuBeDSPq+LgXMI8RrLTVMVHbx6rvesl1ejHBm/RwgndpxyxZZWskNAt2DiXIVe2YX+1uSGKfBopwLGXLtiLlJFgoRjhNi+1Y0giTAe7RlsYA+1Q6yeiSFO1rp4u8UOgXKDRyf08k2Jdy6Lu6M9tQTtcy879aK1beuZOwIIoVDcj4Iy/mSIUoiwV1maBE8aEGTATTuyLSxwITpcMr6hDs6ZNnoX5ctk/K9u1pqXKUx1GAXdiDA7DhDCpwBVWoAYFHeIZXeDOejBfj3fgYt84Z+cwO/JHx+QMLK5p7</latexit>

µJ = �gJµBJ

~<latexit sha1_base64="T5isWqwvQyVy70/aju/IEKB43qc=">AAACHnicbVDLSsNAFJ3UV62vqEs3g0VwoSXxgW6EohvpqoJ9QBPCZDJph04mYWYilJAvceOvuHGhiOBK/8ZJ24W2HhjmcM693HuPnzAqlWV9G6WFxaXllfJqZW19Y3PL3N5pyzgVmLRwzGLR9ZEkjHLSUlQx0k0EQZHPSMcf3hR+54EISWN+r0YJcSPU5zSkGCkteea548cskKNIf5kTpbnXgFfHfa+huXfthAJhLSM18MOskeeZM/CRyD2zatWsMeA8saekCqZoeuanE8Q4jQhXmCEpe7aVKDdDQlHMSF5xUkkShIeoT3qachQR6Wbj83J4oJUAhrHQjys4Vn93ZCiSxQW6sthUznqF+J/XS1V46WaUJ6kiHE8GhSmDKoZFVjCggmDFRpogLKjeFeIB0pEonWhFh2DPnjxP2ic1+7Rm351V60fTOMpgD+yDQ2CDC1AHt6AJWgCDR/AMXsGb8WS8GO/Gx6S0ZEx7dsEfGF8/5tijhg==</latexit>

µS = �S = � e

meS = �2µB

S

~<latexit sha1_base64="hSyDTpJEO8ONYe87U5xa/VM5qK4=">AAACS3icbVBNSyNBEO3Jrl/xK+pxL80GwYOGmSjoRRC9eHTRqJAJQ02nJmnsnhm6e4TQzP/z4sXb/gkvHhTxYE8S2PWjoOnHe/Woqhfngmvj+3+92o+fM7Nz8wv1xaXlldXG2vqlzgrFsMMykanrGDQKnmLHcCPwOlcIMhZ4Fd+cVPrVLSrNs/TCjHLsSRikPOEMjKOiRhzGmejrkXSfDWVRRuf0kIYDkBJCCWYYJ/a8dNROmChgFksrIyw/Sm1njI4nDf+U0obDGFQZNZp+yx8X/QqCKWiSaZ1FjYewn7FCYmqYAK27gZ+bngVlOBNY1sNCYw7sBgbYdTAFibpnx1mUdNMxfZpkyr3U0DH7v8OC1NW5rrPaVH/WKvI7rVuY5KBneZoXBlM2GZQUgpqMVsHSPlfIjBg5AExxtytlQ3CRGBd/3YUQfD75K7hst4LdVvBnr3m0PY1jnvwiv8kWCcg+OSKn5Ix0CCN35JE8kxfv3nvyXr23SWvNm3o2yIeqzbwDhgK00w==</latexit>

µL = �L = � e

2meL = �µB

L

~<latexit sha1_base64="jWhlM63hbvd5AhmtRfSuQhV3KrM=">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</latexit>


Magnetic scatteringNeutron magnetic moment

Gyromagnetic ratios of common spin-1/2 particles:

Neutron moment is around 960 times smaller than the electron moment.

Nuclear magnetons:

proton neutron

For neutrons: with

Protons, neutrons and many nuclei carry a nuclear spin.1.76·105 MHz/T 267 MHz/T

183 MHz/T

Electron: Proton:

Neutron:

µN =e~2mp

<latexit sha1_base64="15SQVA/2WvX0mgNyGGlSyfMy5XY=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1gEF1KSKuhGKLhxJRXsA5oQJtNJO3RmEmYmQgnBjb/ixoUibv0Kd/6N0zYLbT1w4XDOvdx7T5gwqrTjfFulpeWV1bXyemVjc2t7x97da6s4lZi0cMxi2Q2RIowK0tJUM9JNJEE8ZKQTjq4nfueBSEVjca/HCfE5GggaUYy0kQL7wONpcHvlRRLhjHjDEMk8q/MgyQO76tScKeAicQtSBQWagf3l9WOcciI0Zkipnusk2s+Q1BQzkle8VJEE4REakJ6hAnGi/Gz6Qg6PjdKHUSxNCQ2n6u+JDHGlxjw0nRzpoZr3JuJ/Xi/V0aWfUZGkmgg8WxSlDOoYTvKAfSoJ1mxsCMKSmlshHiKThjapVUwI7vzLi6Rdr7lnNffuvNo4LeIog0NwBE6ACy5AA9yAJmgBDB7BM3gFb9aT9WK9Wx+z1pJVzOyDP7A+fwCBOpdu</latexit>

µp = 2.793µN<latexit sha1_base64="HsB2JBc63+FI1W/FmC9sIthJCUc=">AAACAnicbVDLSsNAFJ34rPUVdSVugkVwISFphepCKLhxJRXsA5oQJtNJO3QmGWYmQgnFjb/ixoUibv0Kd/6NkzYLbT1w4XDOvdx7T8gpkcpxvo2l5ZXVtfXSRnlza3tn19zbb8skFQi3UEIT0Q2hxJTEuKWIorjLBYYspLgTjq5zv/OAhSRJfK/GHPsMDmISEQSVlgLz0GNpwK+qdv2y5jE5IrzG0ly7DcyKYztTWIvELUgFFGgG5pfXT1DKcKwQhVL2XIcrP4NCEUTxpOylEnOIRnCAe5rGkGHpZ9MXJtaJVvpWlAhdsbKm6u+JDDIpxyzUnQyqoZz3cvE/r5eq6MLPSMxThWM0WxSl1FKJledh9YnASNGxJhAJom+10BAKiJROraxDcOdfXiTtqu3WbPfuvNI4K+IogSNwDE6BC+qgAW5AE7QAAo/gGbyCN+PJeDHejY9Z65JRzByAPzA+fwDY35Zf</latexit>

µn = �µN�<latexit sha1_base64="LDx5HBq1eNMugLTVOvp0NsaxHzk=">AAACGnicbVDLSgMxFM34rPU16tLNYBFcSJlRQTdCwY0rqWAf0CnDnUymDU0yQ5IRytDvcOOvuHGhiDtx49+YabuwrQdCTs65l9x7wpRRpV33x1paXlldWy9tlDe3tnd27b39pkoyiUkDJyyR7RAUYVSQhqaakXYqCfCQkVY4uCn81iORiibiQQ9T0uXQEzSmGLSRAtvzw4RFasjNlfs8GwXi2u8B52Aewd2Mq2iPwyiwK27VHcNZJN6UVNAU9cD+8qMEZ5wIjRko1fHcVHdzkJpiRkZlP1MkBTyAHukYKoAT1c3Hq42cY6NETpxIc4R2xurfjhy4KuYzlRx0X817hfif18l0fNXNqUgzTQSefBRnzNGJU+TkRFQSrNnQEMCSmlkd3AcJWJs0yyYEb37lRdI8q3rnVe/+olI7ncZRQofoCJ0gD12iGrpFddRAGD2hF/SG3q1n69X6sD4npUvWtOcAzcD6/gXsF6KX</latexit>

�n = �1.913<latexit sha1_base64="MUTUqTy7WF+eg9jwkg0YpPa672c=">AAAB+HicbVDLSgMxFM3UV62Pjrp0EyyCCy0TK6gLoeDGZQX7gHYYMmmmDU0yQ5IR6tAvceNCEbd+ijv/xrSdhbYeuHA4517uvSdMONPG876dwsrq2vpGcbO0tb2zW3b39ls6ThWhTRLzWHVCrClnkjYNM5x2EkWxCDlth6Pbqd9+pEqzWD6YcUJ9gQeSRYxgY6XALfcGWAgcyJszVL1GtcCteFVvBrhMUE4qIEcjcL96/ZikgkpDONa6i7zE+BlWhhFOJ6VeqmmCyQgPaNdSiQXVfjY7fAKPrdKHUaxsSQNn6u+JDAutxyK0nQKboV70puJ/Xjc10ZWfMZmkhkoyXxSlHJoYTlOAfaYoMXxsCSaK2VshGWKFibFZlWwIaPHlZdI6r6JaFd1fVOqneRxFcAiOwAlA4BLUwR1ogCYgIAXP4BW8OU/Oi/PufMxbC04+cwD+wPn8AbrMkcA=</latexit>

µn = �1.913µN<latexit sha1_base64="3uDnR9vpIZzzKtpYWo2YoHdtg2M=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgQkNiBXUhFNy4kgr2AU0Ik+mkHTozCTMToYSCG3/FjQtF3PoT7vwbJ20W2nrgwuGce7n3njChRCrH+TZKC4tLyyvl1cra+sbmlrm905JxKhBuopjGohNCiSnhuKmIoriTCAxZSHE7HF7nfvsBC0lifq9GCfYZ7HMSEQSVlgJzz2NpwK9OXPvSrXlMDklSY2ku3gZm1bGdCax54hakCgo0AvPL68UoZZgrRKGUXddJlJ9BoQiieFzxUokTiIawj7uacsiw9LPJD2PrUCs9K4qFLq6sifp7IoNMyhELdSeDaiBnvVz8z+umKrrwM8KTVGGOpouilFoqtvJArB4RGCk60gQiQfStFhpAAZHSsVV0CO7sy/OkdWq7Ndu9O6vWj4s4ymAfHIAj4IJzUAc3oAGaAIFH8AxewZvxZLwY78bHtLVkFDO74A+Mzx85vpaN</latexit>


Magnetic scatteringMagnetic scattering potential

Magnetic scattering: interaction of the neutron spin with the magnetic field of an unpaired electron

gyromagnetic ratio

nuclear magneton

The interaction is described by the potential:

neutron spin operator:

Magnetic scattering length proportional to electron radius e2/mec2:

comparable to nuclear scattering

Pauli spin operator

µ = �µN �<latexit sha1_base64="odeUAMhLGCSWci+h7MHyLLhhnkg=">AAACJHicbVDLSsNAFJ34rPVVdekmWAQXUhIVFEQouHElFewDmhBuJpN06EwSZiZCCfkYN/6KGxc+cOHGb3HaZqGtF4Y5nHMu997jp4xKZVlfxsLi0vLKamWtur6xubVd29ntyCQTmLRxwhLR80ESRmPSVlQx0ksFAe4z0vWH12O9+0CEpEl8r0YpcTlEMQ0pBqUpr3bp+AkL5IjrL3cGoHKHZ0Vx5UTAOWjs3c47JI04FIVXq1sNa1LmPLBLUEdltbzauxMkOOMkVpiBlH3bSpWbg1AUM1JUnUySFPAQItLXMAZOpJtPjizMQ80EZpgI/WJlTtjfHTlwOd5SOzmogZzVxuR/Wj9T4YWb0zjNFInxdFCYMVMl5jgxM6CCYMVGGgAWVO9q4gEIwErnWtUh2LMnz4POScM+bdh3Z/XmcRlHBe2jA3SEbHSOmugGtVAbYfSIntErejOejBfjw/icWheMsmcP/Snj+wfTyKdQ</latexit>

�µ ·H = ��µN � ·H<latexit sha1_base64="8sH/GMNN25E35b5tadzJltq/RZ0=">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</latexit>

�r0 =�e2

mec2= �0.54 · 10�12 cm

<latexit sha1_base64="TYX1+KQ3HDTg2+ZNU98YIjmqABE=">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</latexit>

� = �1.91<latexit sha1_base64="vRhmV35C4V9APslV9l37lvGzVQg=">AAAB9XicbVBNSwMxEM36WetX1aOXYBE86LKrgnoQCl48VrAf0K5lNs22oUl2SbJKWfo/vHhQxKv/xZv/xrTdg7Y+GHi8N8PMvDDhTBvP+3YWFpeWV1YLa8X1jc2t7dLObl3HqSK0RmIeq2YImnImac0ww2kzURREyGkjHNyM/cYjVZrF8t4MExoI6EkWMQLGSg/tHggB+Bqf+O6V3ymVPdebAM8TPydllKPaKX21uzFJBZWGcNC65XuJCTJQhhFOR8V2qmkCZAA92rJUgqA6yCZXj/ChVbo4ipUtafBE/T2RgdB6KELbKcD09aw3Fv/zWqmJLoOMySQ1VJLpoijl2MR4HAHuMkWJ4UNLgChmb8WkDwqIsUEVbQj+7MvzpH7q+meuf3derhzncRTQPjpAR8hHF6iCblEV1RBBCj2jV/TmPDkvzrvzMW1dcPKZPfQHzucP8yyQxQ==</latexit>

µN =meµB

mn<latexit sha1_base64="CxbcxdXUGqpccFqyZ/Mlfv2dzkQ=">AAACBXicbZDLSsNAFIYn9VbrLepSF4NFcCElUUE3QtGNK6lgL9CUMJlO2qEzkzAzEUrIxo2v4saFIm59B3e+jZM2C239YeDjP+dw5vxBzKjSjvNtlRYWl5ZXyquVtfWNzS17e6elokRi0sQRi2QnQIowKkhTU81IJ5YE8YCRdjC6zuvtByIVjcS9Hsekx9FA0JBipI3l2/seT/xbeAm9UCKccp/kxlVmSGS+XXVqzkRwHtwCqqBQw7e/vH6EE06Exgwp1XWdWPdSJDXFjGQVL1EkRniEBqRrUCBOVC+dXJHBQ+P0YRhJ84SGE/f3RIq4UmMemE6O9FDN1nLzv1o30eFFL6UiTjQReLooTBjUEcwjgX0qCdZsbABhSc1fIR4iE4c2wVVMCO7syfPQOqm5pzX37qxaPy7iKIM9cACOgAvOQR3cgAZoAgwewTN4BW/Wk/VivVsf09aSVczsgj+yPn8AJFGYSg==</latexit>

�<latexit sha1_base64="5vyVUvDdfe+SAN0ZueANJQBgd24=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1gEF1ISFXRZcOOygq2FJpTJZNIOnUeYmQglFDf+ihsXirj1K9z5N07aLLT1wDCHc+7l3nuilFFtPO/bqSwtr6yuVddrG5tb2zvu7l5Hy0xh0saSSdWNkCaMCtI21DDSTRVBPGLkPhpdF/79A1GaSnFnxikJORoImlCMjJX67kEQSRbrMbdfHgyRyQNNBxxNJn237jW8KeAi8UtSByVaffcriCXOOBEGM6R1z/dSE+ZIGYoZmdSCTJMU4REakJ6lAnGiw3x6wgQeWyWGiVT2CQOn6u+OHHFdbGkrOTJDPe8V4n9eLzPJVZhTkWaGCDwblGQMGgmLPGBMFcGGjS1BWFG7K8RDpBA2NrWaDcGfP3mRdM4a/nnDv72oN0/LOKrgEByBE+CDS9AEN6AF2gCDR/AMXsGb8+S8OO/Ox6y04pQ9++APnM8fkJGYGw==</latexit>


@2�

@⌦@E= r20

kfki

X

↵�

⇣�↵� � k↵k�

⌘

| {z }sum over directions xyz

X

��0

p�h�|k2↵|�0ih�|k2

� |�0i�(~! + E� � E�0)

| {z }sum and average over final �0 and initial states �

<latexit sha1_base64="B5aduig8edCxBtPR9tAfjzhP/ck=">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</latexit>


Magnetic field due to a single electron moving with velocity ve:

(from S. W. Lovesey, Theory of Neutron Scattering from

Condensed Matter, Volume 2)H = curl

✓µe ⇥R

|R|3

◆

| {z }spin motion

+�e

c

ve ⇥R

|R|3| {z }orbital motion

<latexit sha1_base64="KEIG9mtWvL9+W2qdGEWwjERPFzc=">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</latexit>

The scattering cross section between the neutron and the electron becomes (after 2 pages):

In comparison to nuclear scattering the magnetic cross section has a directional dependence!



Like for nuclear scattering the Born approximation holds and the scattered amplitude is the Fourier transformation of the potential function (atomic magnetisation density), the magnetic form factor.

which is defined by:

g, gL, gS: g-factors jn: spherical Bessel functions

f(k) =

Z⇢(r) exp(ikr)dr

<latexit sha1_base64="HcCdNITRYsWwP09l1Ao8rrgXqJA=">AAACMXicbVDLSsNAFJ3UV62vqEs3wSK0ICVRQTdCwU2XFewDmlAmk0k7dDIJMxOxhPySG/9E3HShiFt/wkkbbG09MHA451zm3uNGlAhpmhOtsLa+sblV3C7t7O7tH+iHR20RxhzhFgppyLsuFJgShluSSIq7EccwcCnuuKO7zO88Yi5IyB7kOMJOAAeM+ARBqaS+3vArdgDl0PWTUVq9tQmTNh+GvyJPqzZ+iipknlqwvDnv62WzZk5hrBIrJ2WQo9nXX20vRHGAmUQUCtGzzEg6CeSSIIrTkh0LHEE0ggPcU5TBAAsnmV6cGmdK8Qw/5OoxaUzVxYkEBkKMA1clsw3FspeJ/3m9WPo3TkJYFEvM0OwjP6aGDI2sPsMjHCNJx4pAxIna1UBDyCGSquSSKsFaPnmVtC9q1mXNur8q18/zOorgBJyCCrDANaiDBmiCFkDgGbyBd/ChvWgT7VP7mkULWj5zDP5A+/4B2KWryg==</latexit>

f(k) =gSgj0(k) +

gLg

[j0(k) + j2(k)]<latexit sha1_base64="S9nYiMafoa9Jq5u6VWqToDZ5rvw=">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</latexit>



Like for nuclear scattering the Born approximation holds and the scattered amplitude is the Fourier transformation of the potential function (atomic magnetisation density), the magnetic form factor.

which is defined by:

g, gL, gS: g-factors jn: spherical Bessel functions

f(k) =

Z⇢(r) exp(ikr)dr

<latexit sha1_base64="HcCdNITRYsWwP09l1Ao8rrgXqJA=">AAACMXicbVDLSsNAFJ3UV62vqEs3wSK0ICVRQTdCwU2XFewDmlAmk0k7dDIJMxOxhPySG/9E3HShiFt/wkkbbG09MHA451zm3uNGlAhpmhOtsLa+sblV3C7t7O7tH+iHR20RxhzhFgppyLsuFJgShluSSIq7EccwcCnuuKO7zO88Yi5IyB7kOMJOAAeM+ARBqaS+3vArdgDl0PWTUVq9tQmTNh+GvyJPqzZ+iipknlqwvDnv62WzZk5hrBIrJ2WQo9nXX20vRHGAmUQUCtGzzEg6CeSSIIrTkh0LHEE0ggPcU5TBAAsnmV6cGmdK8Qw/5OoxaUzVxYkEBkKMA1clsw3FspeJ/3m9WPo3TkJYFEvM0OwjP6aGDI2sPsMjHCNJx4pAxIna1UBDyCGSquSSKsFaPnmVtC9q1mXNur8q18/zOorgBJyCCrDANaiDBmiCFkDgGbyBd/ChvWgT7VP7mkULWj5zDP5A+/4B2KWryg==</latexit>

f(k) =gSgj0(k) +

gLg

[j0(k) + j2(k)]<latexit sha1_base64="S9nYiMafoa9Jq5u6VWqToDZ5rvw=">AAACTXicbVHLSsNAFJ3UV62vqEs3wSIoSklU0I1QcOPCRUWrQhLCZDpJp508mLkRSsgPuhHc+RduXCgiTtoKtXph4Mw553LvnPFTziSY5otWmZmdm1+oLtaWlldW1/T1jVuZZILQNkl4Iu59LClnMW0DA07vU0Fx5HN65/fPS/3ugQrJkvgGBil1IxzGLGAEg6I8vRPsOhGGrh/k/WLvzAkEJnnoXRd5WPQ8c1Lc/xEvS9HhNAB72tLzDifvjmBhF1xPr5sNc1jGX2CNQR2Nq+Xpz04nIVlEYyAcS2lbZgpujgUwwmlRczJJU0z6OKS2gjGOqHTzYRqFsaOYjhEkQp0YjCE72ZHjSMpB5Ctnuamc1kryP83OIDh1cxanGdCYjAYFGTcgMcpojQ4TlAAfKICJYGpXg3SxygzUB9RUCNb0k/+C28OGddSwro7rzYNxHFW0hbbRLrLQCWqiC9RCbUTQI3pF7+hDe9LetE/ta2StaOOeTfSrKgvf89a1cQ==</latexit>

analytical approximation:

coefficients a, A, b, B, c, C, D tabulated on http://www.ill.eu/sites/ccsl/html/ccsldoc.html)

j0(s) = A exp(�as2) +B exp(�bs2) + C exp(�cs2) +D<latexit sha1_base64="vvPEXSl3wR4vTHVsvZw4dcKRhTQ=">AAACGHicbVDLSgMxFM3UV62vUZdugkVoUetMFXQjVOvCZQX7gHYsmTTTxmYeJBmxDP0MN/6KGxeKuO3OvzGdmYVaLwTO415u7rEDRoU0jC8tMze/sLiUXc6trK6tb+ibWw3hhxyTOvaZz1s2EoRRj9QllYy0Ak6QazPStIfVqd98IFxQ37uVo4BYLup71KEYSSV19aP7rlEQxfOLDnkMCodI3JWL+5cJsWNSTQiOyVVXzxslIy44C8wU5EFata4+6fR8HLrEk5ghIdqmEUgrQlxSzMg41wkFCRAeoj5pK+ghlwgrig8bwz2l9KDjc/U8CWP150SEXCFGrq06XSQH4q83Ff/z2qF0zqyIekEoiYeTRU7IoPThNCXYo5xgyUYKIMyp+ivEA8QRlirLnArB/HvyLGiUS+Zxybw5yVcO0jiyYAfsggIwwSmogGtQA3WAwRN4AW/gXXvWXrUP7TNpzWjpzDb4VdrkGxI/nJc=</latexit>

j2(s) =⇥A exp(�as2) +B exp(�bs2) + C exp(�cs2) +D

⇤· s2

<latexit sha1_base64="bwM+jzPS0TqEiwZ5NAi9Z50FTVE=">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</latexit>

s =sin ✓

�<latexit sha1_base64="btOr0DE5g16Xp5fa5lZD5ADLI9k=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBhZREBd0IBTcuK9gHNKFMppN26GQSZm6EErJx46+4caGIW//BnX/jtM1CWw9cOJxz78y9J0gE1+A431ZpaXllda28XtnY3NresXf3WjpOFWVNGotYdQKimeCSNYGDYJ1EMRIFgrWD0c3Ebz8wpXks72GcMD8iA8lDTgkYqWcf6msvVIRmnubSgyEDkmeeMA/0Sd6zq07NmQIvErcgVVSg0bO/vH5M04hJoIJo3XWdBPyMKOBUsLzipZolhI7IgHUNlSRi2s+mV+T42Ch9HMbKlAQ8VX9PZCTSehwFpjMiMNTz3kT8z+umEF75GZdJCkzS2UdhKjDEeBIJ7nPFKIixIYQqbnbFdEhMKGCCq5gQ3PmTF0nrrOae19y7i2r9tIijjA7QETpBLrpEdXSLGqiJKHpEz+gVvVlP1ov1bn3MWktWMbOP/sD6/AF075kd</latexit>


Magnetic structuresOrdered magnetic state

In some crystals, some of the atoms/ions have unpaired electrons (transition metals, rare-earths).

Hund’s rule favors a state with maximum S and L. The ions possess a localised magnetic moment.

Exchange interactions (direct, superexchange, double exchange, RKKY, dipolar, …) often stabilize a long-range magnetic order


Magnetic structuresPropagation vector

The magnetic structure does not necessarily have the same periodicity and symmetry as the underlying crystal structure. The relation between one and another is expressed by the propagation or wave vector.




ferromagnetic




antiferromagnetic




commensurate antiferromagnetic

magnetic periodicity = 2 x nuclear periodicity ! q = (1/2 0 0)<latexit sha1_base64="CjWYkiBz1GYa55ETkeLFpVWDU1s=">AAACC3icbVDLSgMxFM3UV62vUZduQotQQepMFXQjFNy4rGAf0BlKJs20oZlkTDJKGbp346+4caGIW3/AnX9jpu1CWw8JHM65l3vvCWJGlXacbyu3tLyyupZfL2xsbm3v2Lt7TSUSiUkDCyZkO0CKMMpJQ1PNSDuWBEUBI61geJX5rXsiFRX8Vo9i4keoz2lIMdJG6tpFT9L+QCMpxQP0IqQHQZjejS/L7knVg455R1275FScCeAicWekBGaod+0vrydwEhGuMUNKdVwn1n6KpKaYkXHBSxSJER6iPukYylFElJ9ObhnDQ6P0YCik+VzDifq7I0WRUqMoMJXZtmrey8T/vE6iwws/pTxONOF4OihMGNQCZsHAHpUEazYyBGFJza4QD5BEWJv4CiYEd/7kRdKsVtzTintzVqodz+LIgwNQBGXggnNQA9egDhoAg0fwDF7Bm/VkvVjv1se0NGfNevbBH1ifP54XmWA=</latexit>




commensurate antiferromagnetic

magnetic periodicity = x times nuclear periodicity ! q = (1/x 0 0)<latexit sha1_base64="Pg+Y9AQbVwCKb6RBs89sBW/TGf4=">AAACC3icbVDLSgMxFM3UV62vUZduQotQQeqMCroRCm5cVrAP6Awlk2ba0EwyJhm1DN278VfcuFDErT/gzr8x03ahrYcEDufcy733BDGjSjvOt5VbWFxaXsmvFtbWNza37O2dhhKJxKSOBROyFSBFGOWkrqlmpBVLgqKAkWYwuMz85h2Rigp+o4cx8SPU4zSkGGkjdeyiJ2mvr5GU4h56EdL9IExvRxdl9+jBg455Bx275FScMeA8caekBKaodewvrytwEhGuMUNKtV0n1n6KpKaYkVHBSxSJER6gHmkbylFElJ+ObxnBfaN0YSik+VzDsfq7I0WRUsMoMJXZtmrWy8T/vHaiw3M/pTxONOF4MihMGNQCZsHALpUEazY0BGFJza4Q95FEWJv4CiYEd/bkedI4rrgnFff6tFQ9nMaRB3ugCMrABWegCq5ADdQBBo/gGbyCN+vJerHerY9Jac6a9uyCP7A+fwAKKJmm</latexit>



Magnetic Bragg reflections can be found at

commensurate AF incommensurate AFferromagnetic

Magnetic satellites for

k = G± q<latexit sha1_base64="pvnua7XFfnAuwies5n4jD3WM3tI=">AAACDHicbVDLSsNAFL3xWeur6tLNYBFcSElU0I1QcKHLCvYBbSiT6aQdOpnEmYlQQj7Ajb/ixoUibv0Ad/6NkzYFbT0wcOace7n3Hi/iTGnb/rYWFpeWV1YLa8X1jc2t7dLObkOFsSS0TkIeypaHFeVM0LpmmtNWJCkOPE6b3vAq85sPVCoWijs9iqgb4L5gPiNYG6lbKncCrAeenwzTyym9TjtRgKa/+9RU2RV7DDRPnJyUIUetW/rq9EISB1RowrFSbceOtJtgqRnhNC12YkUjTIa4T9uGChxQ5SbjY1J0aJQe8kNpntBorP7uSHCg1CjwTGW2oZr1MvE/rx1r/8JNmIhiTQWZDPJjjnSIsmRQj0lKNB8ZgolkZldEBlhiok1+RROCM3vyPGmcVJzTinN7Vq4e53EUYB8O4AgcOIcq3EAN6kDgEZ7hFd6sJ+vFerc+JqULVt6zB39gff4AxymcAQ==</latexit>

superposition for q 6= 0<latexit sha1_base64="ctWQp9cWVaWsCrbK90LXzaTPxU0=">AAAB+XicdVDLSgMxFM3UV62vUZdugkVwIUOmL7ssuHFZwT6gM5RMmmlDM5lpkimUoX/ixoUibv0Td/6N6UNQ0QOBwzn3ck9OkHCmNEIfVm5jc2t7J79b2Ns/ODyyj0/aKk4loS0S81h2A6woZ4K2NNOcdhNJcRRw2gnGNwu/M6VSsVjc61lC/QgPBQsZwdpIfdv2IqxHQZhN5p6gE4j6dhE5qFau1V2InGoVVSslQ+o1t1SpQddBSxTBGs2+/e4NYpJGVGjCsVI9FyXaz7DUjHA6L3ipogkmYzykPUMFjqjys2XyObwwygCGsTRPaLhUv29kOFJqFgVmcpFT/fYW4l9eL9Vh3c+YSFJNBVkdClMOdQwXNcABk5RoPjMEE8lMVkhGWGKiTVkFU8LXT+H/pF1y3LLj3lWKjat1HXlwBs7BJXDBNWiAW9AELUDAFDyAJ/BsZdaj9WK9rkZz1nrnFPyA9fYJudSTpw==</latexit>

q = 0<latexit sha1_base64="ZbdWQAwUuN2Y9LBwDkOwpAWfBX4=">AAAB9HicdVDLSgMxFM3UV62vqks3wSK4kCHTx9iNUHDjsoJthXYomTTThmYy0yRTKEO/w40LRdz6Me78GzNtBRU9EDiccy/35PgxZ0oj9GHl1tY3Nrfy24Wd3b39g+LhUVtFiSS0RSIeyXsfK8qZoC3NNKf3saQ49Dnt+OPrzO9MqVQsEnd6FlMvxEPBAkawNpLXC7Ee+UE6mV9B1C+WkI3cilt3ILJrNVSrlg2pu0656kLHRguUwArNfvG9N4hIElKhCcdKdR0Uay/FUjPC6bzQSxSNMRnjIe0aKnBIlZcuQs/hmVEGMIikeULDhfp9I8WhUrPQN5NZSPXby8S/vG6ig7qXMhEnmgqyPBQkHOoIZg3AAZOUaD4zBBPJTFZIRlhiok1PBVPC10/h/6Rdtp2K7dxWS42LVR15cAJOwTlwwCVogBvQBC1AwAQ8gCfwbE2tR+vFel2O5qzVzjH4AevtE6cNkfU=</latexit>

q = (1/2 0 0)<latexit sha1_base64="kPMUl7NwZVW4ra+Q9XD9j/2gGoc=">AAACAnicbVDLSsNAFL3xWesr6krcDBahgtSkCroRCm5cVrAPaEKZTCft0MnDmYlQQnHjr7hxoYhbv8Kdf+OkzUJbD1w4nHMv997jxZxJZVnfxsLi0vLKamGtuL6xubVt7uw2ZZQIQhsk4pFoe1hSzkLaUExx2o4FxYHHacsbXmd+64EKyaLwTo1i6ga4HzKfEay01DX3nQCrgeen92N0hcr2adVBFtJ13DVLVsWaAM0TOyclyFHvml9OLyJJQENFOJayY1uxclMsFCOcjotOImmMyRD3aUfTEAdUuunkhTE60koP+ZHQFSo0UX9PpDiQchR4ujM7WM56mfif10mUf+mmLIwTRUMyXeQnHKkIZXmgHhOUKD7SBBPB9K2IDLDAROnUijoEe/bledKsVuyzin17Xqqd5HEU4AAOoQw2XEANbqAODSDwCM/wCm/Gk/FivBsf09YFI5/Zgz8wPn8AM5aUoQ==</latexit>

q = (1/2� � 0 0)<latexit sha1_base64="nPkzv7RLbegnX8nB4Z8bY1vHUKM=">AAACCXicbVDLSsNAFJ34rPUVdelmsAgVtCZV0I1QcOOygn1AE8pkOmmHTh7O3AgldOvGX3HjQhG3/oE7/8ZJm4W2HrhwOOde7r3HiwVXYFnfxsLi0vLKamGtuL6xubVt7uw2VZRIyho0EpFse0QxwUPWAA6CtWPJSOAJ1vKG15nfemBS8Si8g1HM3ID0Q+5zSkBLXRM7AYGB56f3Y3yFy/Zp9cTpMQHEwRbWddQ1S1bFmgDPEzsnJZSj3jW/nF5Ek4CFQAVRqmNbMbgpkcCpYOOikygWEzokfdbRNCQBU246+WSMD7XSw34kdYWAJ+rviZQESo0CT3dmd6tZLxP/8zoJ+JduysM4ARbS6SI/ERginMWCe1wyCmKkCaGS61sxHRBJKOjwijoEe/bledKsVuyzin17Xqod53EU0D46QGVkowtUQzeojhqIokf0jF7Rm/FkvBjvxse0dcHIZ/bQHxifP2C7l3o=</latexit>


Magnetic structuresFourier expansion of magnetic moments

One usually describes magnetic structures with Fourier components of the magnetic moments:

which for a single propagation vector becomes:

Since is a real vector, one must impose the condition

is a complex vector made of linear combinations of basis vectors according to one or more

µ(r)<latexit sha1_base64="hyypYzjrmxI1o/Z14x8V8Td6ns8=">AAACBXicbVDLSsNAFJ34rPUVdamLwSJUkJKooMuCG5cV7AOaUCaTSTt0MhNmJkIJ2bjxV9y4UMSt/+DOv3HSZqGtF4Y5nHMv99wTJIwq7Tjf1tLyyuraemWjurm1vbNr7+13lEglJm0smJC9ACnCKCdtTTUjvUQSFAeMdIPxTaF3H4hUVPB7PUmIH6MhpxHFSBtqYB95gWChmsTmy7w4zetejPQoiDKZnw7smtNwpgUXgVuCGiirNbC/vFDgNCZcY4aU6rtOov0MSU0xI3nVSxVJEB6jIekbyFFMlJ9Nr8jhiWFCGAlpHtdwyv6eyFCsCqOms7Co5rWC/E/rpzq69jPKk1QTjmeLopRBLWARCQypJFiziQEIS2q8QjxCEmFtgquaENz5kxdB57zhXjTcu8ta86yMowIOwTGoAxdcgSa4BS3QBhg8gmfwCt6sJ+vFerc+Zq1LVjlzAP6U9fkDT32ZAg==</latexit>

S⇤�q = Sq

<latexit sha1_base64="GDKPmuvVwRCMhUG7gM+bjk+wVoo=">AAACB3icbVDLSsNAFJ3UV62vqEtBBosgoiVRQTdCwY3LivYBbQyT6aQOnUzSmYlQQnZu/BU3LhRx6y+482+ctAG19cCFwzn3cu89XsSoVJb1ZRRmZufmF4qLpaXlldU1c32jIcNYYFLHIQtFy0OSMMpJXVHFSCsSBAUeI02vf5H5zXsiJA35jRpGxAlQj1OfYqS05JrbnQCpO89PrtPbfTc5HKTnP4o7cM2yVbFGgNPEzkkZ5Ki55menG+I4IFxhhqRs21aknAQJRTEjaakTSxIh3Ec90taUo4BIJxn9kcJdrXShHwpdXMGR+nsiQYGUw8DTndmNctLLxP+8dqz8MyehPIoV4Xi8yI8ZVCHMQoFdKghWbKgJwoLqWyG+QwJhpaMr6RDsyZenSeOoYh9X7KuTcvUgj6MItsAO2AM2OAVVcAlqoA4weABP4AW8Go/Gs/FmvI9bC0Y+swn+wPj4BkgamYA=</latexit>

µ(r) =1

nq

X

q

Sq · e�iqr

<latexit sha1_base64="4C8bPOu0snR1ThJ7U93PNfZ/7RI=">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</latexit>

Sq<latexit sha1_base64="lcVPafchoFH4IrWqxpftTObQKxo=">AAAB83icbVDLSgMxFL1TX7W+qi7dBIvgQsqMCrosuHFZ0T6gM5RMmmlDM8mYZIQy9DfcuFDErT/jzr8x085CWw8EDufcyz05YcKZNq777ZRWVtfWN8qbla3tnd296v5BW8tUEdoikkvVDbGmnAnaMsxw2k0UxXHIaScc3+R+54kqzaR4MJOEBjEeChYxgo2VfD/GZhRG2f20/9iv1ty6OwNaJl5BalCg2a9++QNJ0pgKQzjWuue5iQkyrAwjnE4rfqppgskYD2nPUoFjqoNslnmKTqwyQJFU9gmDZurvjQzHWk/i0E7mGfWil4v/eb3URNdBxkSSGirI/FCUcmQkygtAA6YoMXxiCSaK2ayIjLDCxNiaKrYEb/HLy6R9Xvcu6t7dZa1xVtRRhiM4hlPw4AoacAtNaAGBBJ7hFd6c1Hlx3p2P+WjJKXYO4Q+czx9NO5HI</latexit>

irreducible representations.

µ(r) =1

2(Sq · e�iqr + S�q · eiqr)

<latexit sha1_base64="01nTBKMzUbKvrc9ydGxhA9xZYIY=">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</latexit>


Magnetic structuresTypes of magnetic structures

real Fourier components

q=0 ferromagnetic q=(100) antiferromagnetic (centered cells)

µ(r) = Sq · e�iqr = Sq<latexit sha1_base64="vhdquXe1DU14IaYM8WpKdPf1JYw=">AAACQXicbVDLSgMxFM3UV62vqks3wSJU0DKjgm4KBTcuK9oHdGrJZDJtaGYyTTJCGebX3PgH7ty7caGIWzdm2oq19ULI4ZxzufceJ2RUKtN8NjILi0vLK9nV3Nr6xuZWfnunLnkkMKlhzrhoOkgSRgNSU1Qx0gwFQb7DSMPpX6Z6454ISXlwq4YhafuoG1CPYqQ01ck3bYczVw59/cW2HyVF20eq53ixSA7LP/gm6Qxs7HIFyV18TH/oQfJrTmAZTts7+YJZMkcF54E1AQUwqWon/2S7HEc+CRRmSMqWZYaqHSOhKGYkydmRJCHCfdQlLQ0D5BPZjkcJJPBAMy70uNAvUHDETnfEyJfpkdqZ7ihntZT8T2tFyrtoxzQII0UCPB7kRQwqDtM4oUsFwYoNNUBYUL0rxD0kEFY69JwOwZo9eR7UT0rWacm6PitUjiZxZMEe2AdFYIFzUAFXoApqAIMH8ALewLvxaLwaH8bn2JoxJj274E8ZX9/6SLLM</latexit>

µ(r) =1

2(Sq · e�iqr + S�q · eiqr) = Sq · (�1)2rx

<latexit sha1_base64="ou9Mw+sZOoOV3vtAB6F+jfcmESE=">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</latexit>



real Fourier components

antiferromagnetic, q=1/2G (at the border of the 1st Brillouin zone)

µ(r) =1

2(Sq · e�iqr + S�q · eiqr) = Sq · (�1)rx

<latexit sha1_base64="baGyCf3fdOLwWsvIJceTA8laHHs=">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</latexit>



imaginary Fourier components (real and imaginary parts parallel)

amplitude-modulated antiferromagnetic, q<1/2G (at the interior of the 1st Brillouin zone)

Sq = µue�iqr<latexit sha1_base64="+4FRHT+i9VOMZ+xw5CGsg94cj9g=">AAACJXicbZDLSsNAFIYnXmu9RV26GSyCCy2JCrpQKLhxWdFeoIlhMp20QyeTdGYilJCXceOruHFhEcGVr+KkF9DWHwZ+vnMOc87vx4xKZVlfxsLi0vLKamGtuL6xubVt7uzWZZQITGo4YpFo+kgSRjmpKaoYacaCoNBnpOH3bvJ644kISSP+oAYxcUPU4TSgGCmNPPPKCZHq+kF6n3l9eA2dMJkSp4tUmmQZeUxP6BT2s6kTWeaZJatsjQTnjT0xJTBR1TOHTjvCSUi4wgxJ2bKtWLkpEopiRrKik0gSI9xDHdLSlqOQSDcdXZnBQ03aMIiEflzBEf09kaJQykHo6858RTlby+F/tVaigks3pTxOFOF4/FGQMKgimEcG21QQrNhAG4QF1btC3EUCYaWDLeoQ7NmT5039tGyfle2781LleBJHAeyDA3AEbHABKuAWVEENYPAMXsE7GBovxpvxYXyOWxeMycwe+CPj+weWPKcg</latexit>

µ(r) = µu cos [2⇡(qr+ �q)]<latexit sha1_base64="szqg1NF4p2lmhuwiW8wDGSNpdiU=">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</latexit>



antiferromagnetic spirals, q<1/2G (at the interior of the 1st Brillouin zone)

helicoidal cycloidal

Sq = (µuu+ iµvv) e�iqr

<latexit sha1_base64="LwdK2kI6OdgZDu5i/wM0nPVoH5s=">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</latexit>

imaginary Fourier components (real and imaginary parts perpendicular)

µ(r) = µuu cos [2⇡(qr+ �q)] + µvv sin [2⇡(qr+ �q)]<latexit sha1_base64="fqiPfvGzaOUfwRckCqs9Bsx4Vzk=">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</latexit>



multi-q structures, e.g. conical (ferromagnetic q=0 component + helix)

treatment of every component separately


Scattering from a unit cellReminder: Nuclear structure factor

imagine two scattering potentials (atoms), the first at 0, the second at r

The path difference is:

Therefore, the phase difference is:

Sum up phase differences over atoms in unit cell:

Structure factor F(hkl) is the Fourier transform of the unit cell scattering potential.

�s(r) = r · kf

kf� r · ki

ki<latexit sha1_base64="uV1Yp1ay4xPRibOMA4kdTImxFIw=">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</latexit>

'(r) = 2⇡�s

�= k ·�s = (kf � ki) · r = G · r

<latexit sha1_base64="7yMEA+O3gwsRBdNi2lVQcnW4KTY=">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</latexit>

F (hkl) =X

j

bj exp(iGrj) =X

j

bj exp [2⇡i(hxj + kyj + lzj)]

<latexit sha1_base64="e2znaFfHzEkXMOaXsGUbjlmjb08=">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</latexit>


Scattering from a unit cellMagnetic structure factor

The magnetic structure factor is obtained in the same way, but it is also proportional to the magnetic moment of the involved atoms directional dependence, FM is a vector

Only the component of FM which is perpendicular to k contributes to magnetic scattering:

Equivalent: Projection of FM onto (hkl) plane

kFM

QM(hkl)

QM = k⇥ (FM ⇥ k)<latexit sha1_base64="7ei+NYty4fx2ZXyaTzI3k82k5TI=">AAACMHicbVDLSsNAFJ34rPUVdekmWIQKUhIVdCMUBHUjtGAf0IQymU7aoZMHMzdCCfkkN36KbhQUcetXOGlT0NYDA2fOPZd773EjziSY5pu2sLi0vLJaWCuub2xubes7u00ZxoLQBgl5KNoulpSzgDaAAaftSFDsu5y23OFVVm89UCFZGNzDKKKOj/sB8xjBoKSufmP7GAaul9TT7t3l9GMPMCTDNLWB+VSWp/K18kykWeNRVy+ZFXMMY55YOSmhHLWu/mz3QhL7NADCsZQdy4zASbAARjhNi3YsaYTJEPdpR9EAq6lOMj44NQ6V0jO8UKgXgDFWf3ck2Jdy5LvKmS0qZ2uZ+F+tE4N34SQsiGKgAZkM8mJuQGhk6Rk9JigBPlIEE8HUrgYZYIEJqIyLKgRr9uR50jypWKcVq35Wqh7ncRTQPjpAZWShc1RFt6iGGoigR/SC3tGH9qS9ap/a18S6oOU9e+gPtO8f982r5w==</latexit>

FM (hkl) =X

j

µjfj(k) exp(ikrj) =X

j

µjfj(k) exp [2⇡i(hxj + kyj + lzj)]

<latexit sha1_base64="CKsEhjzVRx+7vzzv8kiaw628xBE=">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</latexit>


Scattering from a unit cellExample: Ferromagnetic structure

FM (hkl) =X

j

µjfj(k) exp(ikrj) =X

j

µjfj(k) exp [2⇡i(hxj + kyj + lzj)]

<latexit sha1_base64="CKsEhjzVRx+7vzzv8kiaw628xBE=">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</latexit>

FM (100) =

0

@0µ0

1

A f(k)

<latexit sha1_base64="gbAvliniSC0uRMmR4GFpbcVUZtI=">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</latexit>

QM (100) = FM (100)<latexit sha1_base64="ABf09uSQGBkfS3B6615CZdvA9HU=">AAACDHicbVDLSgMxFL1TX7W+qi7dBItQQcqMCroRCoK4EVqwD2hLyaSZNjSTGZKMUIZ+gBt/xY0LRdz6Ae78GzPtFLT1QOBwzrnk3uOGnClt299WZml5ZXUtu57b2Nza3snv7tVVEElCayTggWy6WFHOBK1ppjlthpJi3+W04Q6vE7/xQKVigbjXo5B2fNwXzGMEayN184W2j/XA9eLquHtXdGz7+Gqm3MwUk7JL9gRokTgpKUCKSjf/1e4FJPKp0IRjpVqOHepOjKVmhNNxrh0pGmIyxH3aMlRgn6pOPDlmjI6M0kNeIM0TGk3U3xMx9pUa+a5JJouqeS8R//NakfYuOzETYaSpINOPvIgjHaCkGdRjkhLNR4ZgIpnZFZEBlpho01/OlODMn7xI6qcl56zkVM8L5ZO0jiwcwCEUwYELKMMtVKAGBB7hGV7hzXqyXqx362MazVjpzD78gfX5A8pwmXI=</latexit>

FM (010) =

0

@0µ0

1

A f(k)

<latexit sha1_base64="LIlnkeBSfn6KvO4jSkSOucRHe7k=">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</latexit>

FM (110) =

0

@0µ0

1

A f(k)

<latexit sha1_base64="AT93HZD0DexfuOc+ZNcLeNGVPqE=">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</latexit>

FM (001) =

0

@0µ0

1

A f(k)

<latexit sha1_base64="VUBjjBzQR/tPm31+/27fJ8oTsqI=">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</latexit>

QM (001) = FM (001)<latexit sha1_base64="BPuslFPZeedngHSZpuUpkirr33I=">AAACDHicbVDLSgMxFL1TX7W+qi7dBItQQcqMCroRCoK4EVqwD2hLyaSZNjSTGZKMUIZ+gBt/xY0LRdz6Ae78GzPtFLT1QOBwzrnk3uOGnClt299WZml5ZXUtu57b2Nza3snv7tVVEElCayTggWy6WFHOBK1ppjlthpJi3+W04Q6vE7/xQKVigbjXo5B2fNwXzGMEayN184W2j/XA9eLquHtXtG3n+Gqm3MwUk7JL9gRokTgpKUCKSjf/1e4FJPKp0IRjpVqOHepOjKVmhNNxrh0pGmIyxH3aMlRgn6pOPDlmjI6M0kNeIM0TGk3U3xMx9pUa+a5JJouqeS8R//NakfYuOzETYaSpINOPvIgjHaCkGdRjkhLNR4ZgIpnZFZEBlpho01/OlODMn7xI6qcl56zkVM8L5ZO0jiwcwCEUwYELKMMtVKAGBB7hGV7hzXqyXqx362MazVjpzD78gfX5A8psmXI=</latexit>

QM (110) = FM (110) sin↵<latexit sha1_base64="IgdC9gI+ANb+uqaaDfkuAwb6wt8=">AAACFnicbZDLSsNAFIYn9VbrLerSzWARKmhJVNCNUBDEjdCCvUATwmQ6aYdOJmFmIpTQp3Djq7hxoYhbcefbOGlT0OoPAz/fOYc55/djRqWyrC+jsLC4tLxSXC2trW9sbpnbOy0ZJQKTJo5YJDo+koRRTpqKKkY6sSAo9Blp+8OrrN6+J0LSiN+pUUzcEPU5DShGSiPPPHZCpAZ+kDbG3m3Ftq3Dyxm5nhFHUu4gFg+QZ5atqjUR/Gvs3JRBrrpnfjq9CCch4QozJGXXtmLlpkgoihkZl5xEkhjhIeqTrrYchUS66eSsMTzQpAeDSOjHFZzQnxMpCqUchb7uzFaW87UM/lfrJiq4cFPK40QRjqcfBQmDKoJZRrBHBcGKjbRBWFC9K8QDJBBWOsmSDsGeP/mvaZ1U7dOq3Tgr147yOIpgD+yDCrDBOaiBG1AHTYDBA3gCL+DVeDSejTfjfdpaMPKZXfBLxsc35FWd4A==</latexit>

QM (010) = 0<latexit sha1_base64="Kde6otUWfl3agh0YPpw+dJ/SnyU=">AAAB/HicbVDLSgMxFM34rPU12qWbYBEqSMmooBuh4MaN0IJ9QDsMmTTThmYyQ5IRhqH+ihsXirj1Q9z5N2baWWjrgcDhnHu5J8ePOVMaoW9rZXVtfWOztFXe3tnd27cPDjsqSiShbRLxSPZ8rChngrY105z2Yklx6HPa9Se3ud99pFKxSDzoNKZuiEeCBYxgbSTPrgxCrMd+kLWm3n0NOej0Bnl2FdXRDHCZOAWpggJNz/4aDCOShFRowrFSfQfF2s2w1IxwOi0PEkVjTCZ4RPuGChxS5Waz8FN4YpQhDCJpntBwpv7eyHCoVBr6ZjKPqha9XPzP6yc6uHYzJuJEU0Hmh4KEQx3BvAk4ZJISzVNDMJHMZIVkjCUm2vRVNiU4i19eJp3zunNRd1qX1cZZUUcJHIFjUAMOuAINcAeaoA0ISMEzeAVv1pP1Yr1bH/PRFavYqYA/sD5/AK/Pk2g=</latexit>


Experimental procedureBasic diffractometer (constant wavelength)



collimator defines the beam shape and divergence

Soller collimators, slits



monochromator (assembly of) high quality single crystals

choice of wavelength and resolution typically Cu, Ge, HOPG, Si

diffracts also higher harmonics 𝜆/2, 𝜆/3, …n� = 2d sin ✓

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filter diffracts shorter 𝜆 out of the beam

𝜆/2dFilter>1

typically PG, Be, no 𝜆/2 filter needed for Si, Ge

(111) is used, because (222) is forbidden



sample environment cryostat, cryomagnet,

furnace, pressure cell, CryoPAD



collimator e.g. radial oscillating collimator

reduces background from sample environment or another Soller collimator to

increase resolution



detector gas cells in which an incoming neutron triggers a nuclear reaction producing a charged particle which then is detected

typically 3He or B3F


Experimental procedureTime-of-flight diffractometer (polychromatic)

chopper defines the wavelength band

avoids frame overlap


Experimental procedureTime-of-flight diffractometer (polychromatic)

time of flight of the neutrons is related to the their wavelength

diffraction pattern is recorded at constant scattering angle

(close to 180° for best resolution, small Δt/t)

t =mn

h�L

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��

�= �✓M cot ✓M

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Experimental procedurePowder diffraction

D20 (high flux)

sample in a vanadium container V scatters only incoherently



Result: Diffraction pattern

0 20 40 60 80 100 120 140 1600

50

100

150

200

250

300

350

400

450

500

550

2θ (deg)

Inte

nsity

(cou

nts)

Useful information lies in the

‣ position

‣ the intensity

‣ the shape and width

of the reflections.



1. Position

Bragg’s law

monoclinic

orthorhombic

cubic

with 𝜃 and 𝜆 known → able to obtain lattice parameters and propagation vectors

n� = 2d sin ✓<latexit sha1_base64="Vd/2pnSIbmoOdQDLoyVZ3OmhtDI=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEF1KSKuhGKLhxWcE+oAllMpm0QyeTMHMjlFDwV9y4UMSt3+HOv3HaZqGtBwYO55zLvXOCVHANjvNtlVZW19Y3ypuVre2d3T17/6Ctk0xR1qKJSFQ3IJoJLlkLOAjWTRUjcSBYJxjdTv3OI1OaJ/IBxinzYzKQPOKUgJH69pH0hEmH5KYeeppLD4YMSN+uOjVnBrxM3IJUUYFm3/7ywoRmMZNABdG65zop+DlRwKlgk4qXaZYSOiID1jNUkphpP5+dP8GnRglxlCjzJOCZ+nsiJ7HW4zgwyZjAUC96U/E/r5dBdO3nXKYZMEnni6JMYEjwtAsccsUoiLEhhCpubsV0SBShYBqrmBLcxS8vk3a95l7U3PvLauO8qKOMjtEJOkMuukINdIeaqIUoytEzekVv1pP1Yr1bH/NoySpmDtEfWJ8/GGWVgA==</latexit> d =

✓h2

a2 sin2 �+

k2

b2+

l2

c2 sin2 �� 2hl cos�

ac sin2 �

◆� 12

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d =

✓h2

a2+

k2

b2+

l2

c2

◆� 12

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d = a(h2 + k2 + l2)�12

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d = a⇥(h+ qx)

2 + (k + qy)2 + (l + qz)

2⇤� 1

2

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Magnetic Bragg reflections can be found at k = G± q<latexit sha1_base64="pvnua7XFfnAuwies5n4jD3WM3tI=">AAACDHicbVDLSsNAFL3xWeur6tLNYBFcSElU0I1QcKHLCvYBbSiT6aQdOpnEmYlQQj7Ajb/ixoUibv0Ad/6NkzYFbT0wcOace7n3Hi/iTGnb/rYWFpeWV1YLa8X1jc2t7dLObkOFsSS0TkIeypaHFeVM0LpmmtNWJCkOPE6b3vAq85sPVCoWijs9iqgb4L5gPiNYG6lbKncCrAeenwzTyym9TjtRgKa/+9RU2RV7DDRPnJyUIUetW/rq9EISB1RowrFSbceOtJtgqRnhNC12YkUjTIa4T9uGChxQ5SbjY1J0aJQe8kNpntBorP7uSHCg1CjwTGW2oZr1MvE/rx1r/8JNmIhiTQWZDPJjjnSIsmRQj0lKNB8ZgolkZldEBlhiok1+RROCM3vyPGmcVJzTinN7Vq4e53EUYB8O4AgcOIcq3EAN6kDgEZ7hFd6sJ+vFerc+JqULVt6zB39gff4AxymcAQ==</latexit>



2. Intensity

nuclear structure factor (interaction between neutron and core potential of nuclei)

magnetic structure factor (interaction between neutron and electron’s magnetic field)

0.2 0.4 0.6 0.8 1 1.2 1.4

0.2

0.4

0.6

0.8

1

sinθ/λ

f

magnetic form factor

I ⇠ F 2<latexit sha1_base64="UFgnjSihrWoztB7rw4xLgRG7bTQ=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRbBg5TdVtBjQRC9VbAf0q4lm2bb0CS7JFmhLP0VXjwo4tWf481/Y9ruQVsfDDzem2FmXhBzpo3rfju5ldW19Y38ZmFre2d3r7h/0NRRoghtkIhHqh1gTTmTtGGY4bQdK4pFwGkrGF1N/dYTVZpF8t6MY+oLPJAsZAQbKz3cdjUT6Pqx0iuW3LI7A1omXkZKkKHeK351+xFJBJWGcKx1x3Nj46dYGUY4nRS6iaYxJiM8oB1LJRZU++ns4Ak6sUofhZGyJQ2aqb8nUiy0HovAdgpshnrRm4r/eZ3EhJd+ymScGCrJfFGYcGQiNP0e9ZmixPCxJZgoZm9FZIgVJsZmVLAheIsvL5NmpexVy97deal2lsWRhyM4hlPw4AJqcAN1aAABAc/wCm+Ocl6cd+dj3ppzsplD+APn8wfAc4+l</latexit> f(k) =

1Z

�1

⇢mag(r) exp(ikr)dr

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FM (k) =X

j

µjfj(k) exp(ikrj) exp

✓�Bj

sin2 ✓

�2

◆

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FN (k) =X

j

bj exp(ikrj) exp

✓�Bj

sin2 ✓

�2

◆

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3. Peak width and shape

2θM 2θM 2θM

source, monochromator, slits, collimators, sample strain, stress, correlation length etc. have an influence on the peak shape and the peak width

Caglioti formula

resolution function minimum at the take-off angle 2𝜃M

(focussing effect)

FWHM2= u tan2 ✓ + v tan ✓ + w

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Thermodiffraction

Collection of diffraction patterns as a function of temperature. Clearly reveals structural and magnetic phase transitions.


Experimental procedureSingle crystal diffraction - 4-circle geometry

by adjusting 2𝜃, 𝜔, 𝜒 and 𝜙

the sample is put in reflection position

D10 (ILL)

single crystal on Al pin


Experimental procedureSingle crystal diffraction - Normal beam geometry

cryomagnets, pressure cells, … cannot be tilted much

→ confined to the scattering plane e.g. only (hk0) reflections

→ lifting counter able to reach l=1, 2…


Experimental procedureSingle crystal diffraction - Laue method

polychromatic beam → every accessible hkl plane is in reflection position for a particular wavelength


Experimental procedureSingle crystal diffraction - Laue method

‣ quickly orient single crystals

‣ observe phase transitions

‣ magnetic satellites

‣ find propagation vectors

‣ structure analysis also possible


SummaryDiffraction from magnetic materials

‣ magnetic scattering is comparable in intensity to nuclear scattering

‣ Only the component of the magnetic moment perpendicular to the scattering vector is measurable

‣ The magnetic form factor is the Fourier transform of the atomic magnetisation density

‣ The magnetic structure factor is the Fourier transform of the unit cell magnetisation density

‣ We measure phase information is lost models necessary

‣ How dow we get those models? See you tomorrow

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T H E E U R O P E A N N E U T R O N S O U R C ET H E E U R O P E A N N E U T R O N S O U R C E

I N S T I T U T L A U E L A N G E V I N

osns - diffraction from magnetic materials - diffrac… · diffraction from magnetic materials....

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