agntic Nutron Scattring and Spin-Polarizd Nutrons Physical origin: potntial of magntic dipol momnt of th nutron in magntic fild gnratd by lctron spins and orbital momnts in th solid. µ n µ H Spcializ to contribution of lctron spin to H : µ µ B s with µ B m µ n gnµ Nsn γµ Nσ with µ N g and γ n 1.913 mn s and s n ar spin-½ oprators, σ s n is th nutron Pauli matrix whos ignvalus ar ±1. Diffrntial nutron scattring cross sction for on lctron at rst at origin of coordinat systm: m n k fmf Hint kimi dω with Hint µ n H π This follows from Frmi s Goldn Rul, as in th cas of nuclar nutron scattring. Th only qualitativ diffrnc is that w hav to kp track of th nutron spin stat m ± 1 in addition to th spatial part of its wav function lablld by th quantum numbr k. Vctor potntial of dipol fild (from classical lctromagntism: µ µ r µ 1 A 3 µ 4π r 4π r µ 1 H A µ 4π r mn 1 ( γµ µ, σ dω π r N B kf mf n s ki, mi 1
Us trick to simplify this xprssion: dp 1 ip r ip rcos d p Θ π d c osθ p 1 sin p r π π d p p r r ( Hr, p is an auxiliary variabl without physical maning. 1 1 dp ip r s ( s r π p 1 ip r ˆ p ( s pˆ dp π 1 1 k s k dr dp p s p r π 4π Qˆ ( s ˆ Q iq r ip r f i ˆ ( ˆ s Th last lin follows by doing th r -intgration first and using 1 i( p Q r dr δ 3 ( p Q ( π Q s s collct all prfactors: µ ( γµ Nµ B ( π ( γ r mn 4 π 4π 15 whr r.8 1 m is th classical lctron radius that also appard in th Thompson cross sction for x-ray scattring. ( γr mf σ s mi dω For an unpolarizd nutron bam, on has to avrag ovr th nutron spin stats m. For convninc, tak th nutron spin quantization axis ẑ to b paralll to s. Thn m σ s m s m σ m f i f z i s if mf mi othrwis Th cross sction for a singl lctron at rst for an unpolarizd nutron bam is thrfor ( γ r s dω with th projction of th lctron spin prpndicular to Q. s
Gnralization for an atom ( γr ˆ η f ( Q ( γr 1 ( ˆ η Qˆ f ( Q dω 1 iq r whr f ( Q ( r µ is th magntic form factor and B r r η is th magntic dipol momnt dnsity contributd by th spin ( ( ˆ and orbital motion of all unpaird lctrons in th atom. Gnralization for collinar magnts For collinar magnts, that is, magntically ordrd solids whr all lctron magntic dipol momnts point ithr paralll or antiparalll to a singl dirction ˆ η, this xprssion can b gnralizd in analogy to th diffrntial cross sction for x-ray or nuclar nutron scattring: ( ( ˆ iq R γr ˆ 1 η Q ( ± f R ( Q dω R ( π 3 ( γr 1 ( ˆ η Qˆ N F ( K δ ( Q K V K whr K ar th magntic rciprocal lattic vctors and F Q ± f Q iq d ( ( ( d d Th + or sign is chosn dpnding on whthr th magntic momnt at lattic sit d is paralll or antiparalll to ˆ η. Exampls: 1 On-dimnsional frromagnt a nuclar and magntic unit clls idntical N π K KN n, n intgr. a f ( Q π 4π 6π a a a Q π 4π 6π a a a Q 3
On-dimnsional antifrromagnt a magntic unit cll twic as larg as nuclar unit cll iqa Qa 4 f Q F f ( Q + 1 4 f ( Q sin ( π π K n KN a a n for n odd for n vn N f ( Q π 4π 6π a a a Q π 3π 5π a a a Q 3 Flux-lin lattic in typ-ii suprconductor agntization in typ-ii suprconductor HC1 H C H < H < HC1: magntic fild compltly scrnd out of suprconductor by shilding currnts flowing around th primtr ( issnr ffct HC1 < H < HC : magntic fild pntrats partially into suprconductor in th form of flux lins, which form a rgular array ( flux lin lattic flux lin scrning currnt 4
H a x Φ whr h a ~ Φ is th H magntic flux quantum a~ 5Å for H 1T Th priodic magntic fild distribution gnratd by th flux lin lattic insid th suprconductor can b rvald by magntic nutron scattring. Bcaus λ th lattic constant a is larg, th scattring angl Θ ~ is small. Evn for cold a nutrons ( λ ~5Å and rlativly high magntic filds ( H ~1 T, Θ is lss than 1. In ordr to obtain th rsolution rquird to sparat th Bragg rflctions from th unscattrd bam, on uss a ddicatd Small Angl Nutron Scattring (SANS diffractomtr with a long distanc (>1 m btwn sampl and dtctor: Small angl nutron diffraction pattrns from flux lins lattic in a typ-ii suprconductor Not that th structur of th flux lin lattic changs as a function of magntic fild (.R. Eskildsn t al., PRL 86, 3 (1 As w hav sn, nuclar and magntic nutron scattring hav comparabl strngths. Thy can b distinguishd xprimntally through thr diffrnt mthods: 1 Th magntic form factor rducs th intnsity of magntic Bragg rflctions with larg K, whras thr is no form factor for nuclar scattring. Strong rflctions with larg K must thrfor b nuclar in origin. agntic Bragg paks vanish at th magntic ordring tmpratur (th Curi tmpratur T C for frromagnts, or th Nél tmpratur T N for antifrromagnts. Nuclar Bragg paks vanish at th mlting tmpratur, which is typically largr than T N or T C. 3 Th nutron spin oprator dos not appar th th cross sction for cohrnt nuclar scattring. Th nutron spin stat is thrfor unaffctd by nuclar scattring. By contrast, magntic nutron scattring can b (but dos not hav to b associatd with a spin-flip of th nutron. 5
agntic Scattring of Spin-Polarizd Nutrons Th most unambiguous mthod to discriminat btwn nuclar and magntic nutron scattring is to kp track of th spin stat of th nutron bfor and aftr th scattring vnt. In this cas, th magntic nutron scattring cross sction for a collinar magnt can b writtn as ( γr mf σ ˆ η mi F ( Q δ ( Q K K Th nutron spin quantization axis can b dtrmind by applying a magntic fild ẑ H at th sampl position. Dpnding on th rlativ orintation of th vctors H, ˆ η, and Q, th scattring procss dos or dos not caus a spin flip of th nutron. Exampls: 1 For a frromagnt, th applid fild dtrmins both th nutron spin quantization axis and th lctron spin dirction ˆ η, so that ˆ η H zˆ. First, considr a situation in which th fild is applid prpndicular to th scattring plan: H, ˆ η k k ˆ η ˆ η zˆ i f m ˆ f σ η mi mf σz m i Q + 1 if mf mi + 1 1 if mf mi 1 if mf mi bcaus m is an ignstat of σ z Th scattring is thrfor ntirly non-spin-flip. Howvr if th fild is applid in a gnral dirction in th scattring plan, both spin-flip and nonspin-flip contributions can b obsrvd: k i H, ˆ η Q k f σ η σ η + σ η + σ η ˆ x x y y z z Bcaus σ and σ can b writtn as x y linar suprpositions of raising and lowring oprators: 1 σx ( σ+ + σ 1 σ y ( σ+ σ i th x- and y-componnts of σ ˆ η induc spin flips of th nutron 6
For an antifrromagnt, th xtrnal fild dos not affct ˆ η, and th nutron spin quantization axis is not ncssarily paralll to ˆ η. In particular, if H is applid paralll to Q, th product σ ˆ η has no z-componnt, so that th scattring is ntirly spin-flip. Production of Spin-Polarizd Nutrons In ordr to apply this mthod, on nds to produc a bam of nutrons in a singl spin stat. This can b accomplishd by two fundamntally diffrnt mthods. 1 Nuclar-magntic intrfrnc ffct If K KN, both magntic and nuclar scattring contribut to a Bragg rflction. Tak for xampl a frromagnt with on atom pr unit cll; with a fild applid prpndicular to Q as in th xampl abov: b+ rγ f ( Q m ˆ f σ η mi δ ( Q K dω K K K ( b rγ f ( Q bγrf ( Q δ ( Q K + 1 1 for spin up for spin down + ± b± γr f Q Q K ( δ ( Th scattring cross sction is thus diffrnt for th two diffrnt spin stats of th nutron. This diffrnc can b xploitd to gnrat a spin-polarizd nutron bam. For th (1 rflction of th frromagnt Cu nal ( Huslr alloy, b rγ f Q ( so that only spin-up nutrons ar Bragg rflctd: sourc sampl Th bam incidnt upon th sampl is thrfor almost prfctly spin-polarizd. A variant of this mthod rlis on total xtrnal rflction from a frromagntic mirror. By analogy to th abov argumnt, th critical angl for a frromagntic film can b writtn as Θ ~ δ C 4π p b ( ± γr f k 7
For Θ C <Θ<Θ C, only spin-up nutrons ar rflctd: Θ frromagntic mirror nonmagntic films magntic films Nutron rflctivity from nonmagntic and magntic films http://www.orau.org/council/prsntations/klos.pdf Frromagntic suprmirrors with largr critical angls (s chaptr on nutron rflctivity ar commonly usd to produc spin-polarizd nutrons in modrn nutron scattring instrumnts: Polarizd-bam rflctomtr http://www.orau.org/council/prsntations/klos.pdf 8
A mthod that has bn dvlopd in th past fw yars is th nutron spin filtr. A spin-polarizd 3 H gas absorbs nutrons through th raction 3 4 3 n+ H H ( S p+ H. H 3 H Bcaus th raction procds via an intrmdiat stat with S, th absorption cross sction for nutrons with spins antiparalll to thos of th 3 H nucli is thrfor about thr ordrs of magnitud largr than if nutron and 3 H spins ar paralll. 9