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Transcript of Plane Waves as Tractor Beams - KFKIarpi/talks/szeged15.pdf · Intro2chApplicationsConclusionsBackup...
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Intro 2ch Applications Conclusions Backup slides
Plane Waves as Tractor BeamsarXiv:1303.3237, Phys. Rev. D88 125007
rpd LukcsCollaborators: Pter Forgcs, Tomasz Romanczukiewicz
Wigner RCP RMKI,Budapest, Hungary
SZTE Elmleti Fizikai Tanszk Szeminriuma2015. December 3.
Lukcs . Tactor beams 1
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Intro 2ch Applications Conclusions Backup slides NRP NRP on kinks Other approaches
Outline
1 IntroductionWhat is negative radiation pressure?NRP on kinksOther approaches
2 NRP in two channel scattering1dExamplesForce in 2d scattering
3 ApplicationsNeutron scattering on XY model vortices
4 Conclusions
Lukcs . Tactor beams 2
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Intro 2ch Applications Conclusions Backup slides NRP NRP on kinks Other approaches
Radiation pressure
Scattering of light: the pressure acting on a platelet
P = (1 + R2 T 2)W ,
W = 0E20/2
R: reflection, T : transmission
F
R=0 : F=0
Can the pressure be negative?What objects can be pulled towards the radiation source?Unitarity (conservation of energy): R2 + T 2 = 1, P = 2WR2 > 0.How can P be negative?
Lukcs . Tactor beams 3
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Intro 2ch Applications Conclusions Backup slides NRP NRP on kinks Other approaches
NRP on kinks
Kinks in the 4 theorylinearized perturbations: scattering reflectionlessnonlinearities generate higher harmonics
(Video: Tomasz Romanczukiewicz)
More momentum in the forward direction NRP(Romanczukiewicz 2004, Forgcs, Lukcs and Romanczukiewicz 2007.)It should be possible with 2 channels, different momenta
Lukcs . Tactor beams 4
tromneg.mpgMedia File (video/mpeg)
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Intro 2ch Applications Conclusions Backup slides NRP NRP on kinks Other approaches
Other approaches
Gain media (Mizraki and Fainman, 2010)
F
Structured beams (Sukhov and Dogariu, 2011)
F
Negative , (Veselago 1967)
Lukcs . Tactor beams 5
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Intro 2ch Applications Conclusions Backup slides NRP NRP on kinks Other approaches
Other approaches
Gain media (Mizraki and Fainman, 2010)
F
Structured beams (Sukhov and Dogariu, 2011)
F
Negative , (Veselago 1967)
Lukcs . Tactor beams 5
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Intro 2ch Applications Conclusions Backup slides NRP NRP on kinks Other approaches
Other approaches
Gain media (Mizraki and Fainman, 2010)
F
Structured beams (Sukhov and Dogariu, 2011)
F
Negative , (Veselago 1967)Lukcs . Tactor beams 5
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Intro 2ch Applications Conclusions Backup slides 1d Examples 2d
Two channel scattering in 1d
Plane waves as tractor beams in unitary scatteringTwo kinds of waves (channels) i = 1,2momentum Pichannel 1 incoming wave
p = |A1|2[P1(1 + |R11|2 |T11|2) + P2(|R21|2 |T21|2)
],
where Rij and Tij : reflection and transmission coefficient,channel i j .energy flux Si ; energy conservation
i
Si(|Rij |2 + |Tij |2) = Sj .
one channel: no NRP, p = 2|A|2P1|R|2 > 0.NRP expected: P2 > P1 and large T21.
Lukcs . Tactor beams 6
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Intro 2ch Applications Conclusions Backup slides 1d Examples 2d
Birefringent media
Anisotropic dielectric tensor
= diag(x , y , z)
Two types of wave: polarizationsWave propagation in the z direction
Two modes: the x and y polarizations, assume nx < nyMomentum flux of plane waves Pi = i/2Energy fluxes Si =
i//2
Pressurepz|Ex0|2
= Px(1 + |Rxx |2 |Txx |2) + Py (|Ryx |2 |Tyx |2) ,
where Tij transmission, Rij reflection, channel j i
Energy conservation:
i Si(|Rij |2 + |Tij |2) = Sj .
Lukcs . Tactor beams 7
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Intro 2ch Applications Conclusions Backup slides 1d Examples 2d
Force formula
ScattererPlane of thickness L, same material, rotated by
Neglect multiple surface interactions
pz = 0|Ex0|2(
P0 + P1 cosny nx
cL),
small angle
P0 = 2(nx ny )(4n2x + 3nxny + n2y )/4ny
P1 = 2(nx ny )(nx + ny )2/2/nyTractor beam: nx < nynx,y = n n: P0 = P1 = 2nn2
At 45: P0 = P1 = nn/2
Lukcs . Tactor beams 8
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Intro 2ch Applications Conclusions Backup slides 1d Examples 2d
Macroscopic example
Dielectric
nx = 3, ny = 6
Incoming wave: 1 kW/cm2 at = 1GHz
PressureP0 = 1.08, P1 = 1.99, avg. pressure Pz = 0.072 Pa
Radiation pressure at total reflection: 0.6 PaAccuracy: exact value 0.053 Pa
This is a macroscopic effect.
Lukcs . Tactor beams 9
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Intro 2ch Applications Conclusions Backup slides 1d Examples 2d
An optical example
Dielectric
Liquid crystal 5CB, at 25 C, = 5893 : nx = 1.53, ny = 1.72
Scatterer: rotated birefringent plateletE.g., L = 0.1 mm, = /4
Pressure
x polarization pz = 5.52 1013Pa (m/V)2|E0|2
y polarization pz = 7.48 1013Pa (m/V)2|E0|2
Accuracy: 4.95 1013Pa (m/V)2|E0|2 and8.14 1013Pa (m/V)2|E0|2A few percents of radiation pressure on a totally reflecting mirror!
Lukcs . Tactor beams 10
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Intro 2ch Applications Conclusions Backup slides 1d Examples 2d
Force in 2d scattering
Scattering in 2d, rotational invariancePartial wave expansion (Fourier trf. in )S-matrix elements S`: n n matrix (n: channels)S` unitary (conservation of energy)
Momentum balance: force master formula
F = Fx + iFy = 4`
{AS`+1K S`A A
KA},
A = (A1, . . . ,An)T amplitude, K = diag(k1, . . . , kn) wave numbers
A consequence of unitarity:
Re |Aa|2ka(1 Saa,`+1Saa`) > 0
one channel radiation pressure positive
NRP: kb > ka necessary
Lukcs . Tactor beams 11
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Intro 2ch Applications Conclusions Backup slides 1d Examples 2d
Force in 2d scattering
Scattering in 2d, rotational invariancePartial wave expansion (Fourier trf. in )S-matrix elements S`: n n matrix (n: channels)S` unitary (conservation of energy)
Momentum balance: force master formula
F = Fx + iFy = 4`
{AS`+1K S`A A
KA},
A = (A1, . . . ,An)T amplitude, K = diag(k1, . . . , kn) wave numbers
A consequence of unitarity:
Re |Aa|2ka(1 Saa,`+1Saa`) > 0
one channel radiation pressure positive
NRP: kb > ka necessary
Lukcs . Tactor beams 11
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Intro 2ch Applications Conclusions Backup slides XY model
Scattering on vortices
Two types of wave (particle species/spin/etc.)Neglect vortex corer asymptotic form of A a two channel Aharonov-Bohm scattering problem(
+ iA22
)2 K 2 = 0 , 2 =
(i
i
),
whereA = e/r = (u, d), u: heavy and d light modefermionic bdry cond. (r , + 2) = (r , )
Solution: partial waves, radial eq. numericallyNRP for large rane of parametersCross section geometric, 1/k1/sin/2 in scattering amplitudeLarge cross section from one channel to another
Lukcs . Tactor beams 12
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Intro 2ch Applications Conclusions Backup slides XY model
Scattering on vortices
Two types of wave (particle species/spin/etc.)Neglect vortex corer asymptotic form of A a two channel Aharonov-Bohm scattering problem(
+ iA22
)2 K 2 = 0 , 2 =
(i
i
),
whereA = e/r = (u, d), u: heavy and d light modefermionic bdry cond. (r , + 2) = (r , )
Solution: partial waves, radial eq. numericallyNRP for large rane of parametersCross section geometric, 1/k1/sin/2 in scattering amplitudeLarge cross section from one channel to another
Lukcs . Tactor beams 12
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Intro 2ch Applications Conclusions Backup slides XY model
Scattering off cosmic strings
Inside a cosmic string: GUT Higgs zero, flux of broken gauge fieldCosmic string catalyzed baryon number violation:
B + string `+ string
A simplified description:Neglect spin degrees of freedom2 channel AharonovBohm scattering1 heavy baryon, 1 light lepton, mass ratio 1.5 : 2.Large cross section:cosmic string catalyzed baryon number violationForce: string friction (moving in a plasma)
Decoupled approximation
Fi = 4niv(1 exp(2ii ))
ni density, i coupling to the X-bosononly valid for light modes, heavy modes give negative contributionat v = 0.65
F ux = 6.09|A|2 , F dx = 7.44|A|2 .scattering energy mi/
1 v2
Lukcs . Tactor beams 13
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Intro 2ch Applications Conclusions Backup slides XY model
Scattering off cosmic strings
Inside a cosmic string: GUT Higgs zero, flux of broken gauge fieldCosmic string catalyzed baryon number violation:
B + string `+ stringA simplified description:
Neglect spin degrees of freedom2 channel AharonovBohm scattering1 heavy baryon, 1 light lepton, mass ratio 1.5 : 2.
Large cross section:cosmic string catalyzed baryon number violationForce: string friction (moving in a plasma)
Decoupled approximation
Fi = 4niv(1 exp(2ii ))
ni density, i coupling to the X-bosononly valid for light modes, heavy modes give negative contributionat v = 0.65
F ux = 6.09|A|2 , F dx = 7.44|A|2 .scattering energy mi/
1 v2
Lukcs . Tactor beams 13
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Intro 2ch Applications Conclusions Backup slides XY model
Scattering off cosmic strings
Inside a cosmic string: GUT Higgs zero, flux of broken gauge fieldCosmic string catalyzed baryon number violation:
B + string `+ stringA simplified description:
Neglect spin degrees of freedom2 channel AharonovBohm scattering1 heavy baryon, 1 light lepton, mass ratio 1.5 : 2.Large cross section:cosmic string catalyzed baryon number violationForce: string friction (moving in a plasma)
Decoupled approximation
Fi = 4niv(1 exp(2ii ))
ni density, i coupling to the X-bosononly valid for light modes, heavy modes give negative contributionat v = 0.65
F ux = 6.09|A|2 , F dx = 7.44|A|2 .scattering energy mi/
1 v2
Lukcs . Tactor beams 13
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Intro 2ch Applications Conclusions Backup slides XY model
Force: x
x component
-10
-5
0
5
10
15
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Fx/
(Am
plitu
de)2
Fx,u
Fx,d
Fx,ddc
Lukcs . Tactor beams 14
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Intro 2ch Applications Conclusions Backup slides XY model
Force: y
y component
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Fy/
(Am
plitu
de)2
Fy,u
Fy,d
Lukcs . Tactor beams 15
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Intro 2ch Applications Conclusions Backup slides XY model
Neutron scattering: XY model vortices I
XY modelrotators (spins) in a plane, with nearest neighbor interaction
Magnetic vortex: singularity of magnetization M
gM energy difference between parallel and antiparallel spin neutrons
Diagonalize Hamiltonian locally2 modes, ~2/2/m(k2d k2u ) = gMsmall momentum transfer
Measurable manifestation of the same phenomenon as NRPLarge cross sections, 1/k (A-B)Large spin-flip cross sectionE = 4.1 105 eV (45 neutrons), du = 1.19 104 mCan be calculated perturbatively
Lukcs . Tactor beams 16
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Intro 2ch Applications Conclusions Backup slides
Conclusions
Many approaches to tractor beams (e.g., structured beams)Multi-channel scattering, ki 6= kj fairly generalOne-dimensional examples:
polarizations of EM waveshigher harmonics (kink)
Two dimensionsCosmic strings: baryon decayMagnetic XY-vortex
THANK YOUFOR
YOURATTENTION!
Lukcs . Tactor beams 17
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Intro 2ch Applications Conclusions Backup slides
Conclusions
Many approaches to tractor beams (e.g., structured beams)Multi-channel scattering, ki 6= kj fairly generalOne-dimensional examples:
polarizations of EM waveshigher harmonics (kink)
Two dimensionsCosmic strings: baryon decayMagnetic XY-vortex
THANK YOUFOR
YOURATTENTION!
Lukcs . Tactor beams 17
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
Cosmic strings
Classical field theoretical solutionThin, elongated objectString core: a zero of a Higgs fieldEnergy density localized in the core
Important parameter: string tension
= E/L
Electroweak string: G 1032 (: 10 mg/Solar diam)GUT string: G 106 (: Solar mass/Solar diam)
Cosmic strings are high energy localized objects that provide a linkbetween astrophysics and particle physics.
Lukcs . Tactor beams 18
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
Physics of cosmic strings
Formation: during phase transitionsEvolution of a string network
Friction dominated era: scattering of particlesScaling v 0.65collisions, interlinkingradiation (e.g. at cusps formed in collisions)string tension contracts loopsexpansion: the network becomes more diluted
Signatures of cosmic stringsScattering of material off strings: structure formation: galaxies,voids, filaments (fractal dimension)Contribution to CMB anisotropy: best fit with GUT stringsG = (2.04 0.13) 106, Contribution to multipole` = 10:f10 = 0.11 0.05 (Hindmarsh et al., 2007, 2008)Gravitational lensingGravitational radiation
Lukcs . Tactor beams 19
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
Artificial gauge potential
Diagonalize locally the Hamiltonian
U(V )U = K
K = diag(k1, . . . , kn)kinetic term:
UiU = + U(iU) = iAi
A : artificial gauge potentialU = U(): Aharonov-Bohm form
A =Ar
e ,
where A = UU/
Lukcs . Tactor beams 20
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
Scalar perturbations of the global vortex
-5
0
5
10
15
20
25
6 8 10 12 14 16 18 20
Fx(
2)=
Fx/
(am
plitu
de)2
2
Fx,a(2)
Fx,g(2)
One massive, one Goldstone (massless) modeLukcs . Tactor beams 21
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
Perturbations of the superfluid vortex
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3 3.5
F(2
) =F
/(A
mpl
itude
)2
x
y
x PT
y PT
Asymptotics: H(r/2), Ki(2r), one channelLukcs . Tactor beams 22
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
AharonovBohm scattering I
The AharonovBohm effect:Motion of a charged particle in a region with B = 0
Double slit expreriment: Scattering:
Both experiments show flux dependenceHolonomy is also physical not just field strength Pei
Adr
Reaction force (deflected beam): Force acting on the scatterer
Lukcs . Tactor beams 23
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
AharonovBohm scattering II
Schrdinger-equation
i = ( iA)2 ,
with electromagnetic vector potential
A(r , , z) =A0r
e .
2A0 flux; outside B = 0.Fixed energy: (r, t) = eit(r).Scattering asymptotics (?)
eikx + f ()r
eikr
Cross sections: d/d = |f ()|2Scattering amplitude: f () sinA02
1sin(/2) .
Lukcs . Tactor beams 24
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
AharonovBohm scattering III
Partial waves: s` J|`A0| (note index shift wrt plane wave)Effect of scattering in the outgoing wave: phase shift `,S` = exp(2i`)(J(z) cos(z /2 /4)/
2z)
` 0 : ` =A0
2, ` 1 : ` = `
A02
.
thus
Fx = |0|24k
`=
{cos [2(` `1)] 1}
= |0|24k (cos(2A0) 1) 16|0|22A20k ,
Fy = |0|24k
`=
sin [2(` `1)] = |0|24k sin(2A0) 8|0|2A0k .
Analog problem: force acting on a superfluid vortex (GPe)
Lukcs . Tactor beams 25
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
AharonovBohm scattering IV
Iordanskii force controversy C. Wexler, D.J. Thouless, Phys. Rev. B58R8897R8900 (1998):
f () = f () Fy = 0
A.L. Shelankov Europhys. Lett. 43 (1998) 623M.V. Berry J. Phys. A: Math. Gen. 32 (1999) 5627:scattering asymptotics does not hold in forward direction
(r, t) = ei(tkx)(r)
and for y
x
(x > 0, y) cos(A0)2i1/2
sin(A0)
k2
yx
transversal force Fy from this region (although not Fx )
Lukcs . Tactor beams 26
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
References I
Forgcs,P., Lukcs,., Romanczukiewicz,T., Plane waves astractor beams Phys. Rev. D88, 125007.
Pter Forgcs, rpd Lukcs, and Tomasz Romanczukiewicz,Negative radiation pressure exerted on kinks Phys. Rev. D77,125012 (2008).
T. Romanczukiewicz, Negative radiation pressure in case of twointeracting fieldsActa. Phys. Polonica B39 (2008) 3449-3462.
A. Mizrahi, and Y. Fainman, Negative radiation pressure on gainmedium structures Opt. Lett. 35 3405 (2010); K.J. Webb andShivanand, Negative electromagnetic plane-wave force in gainmedia Phys. Rev. E84, 057602 (2011).
A. Dogariu, S. Sukhov, and J.J. Senz Optically inducednegative forces Nat. Pho. 7, 24 (2012);
Lukcs . Tactor beams 27
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
References II
S. Sukhov and A. Dogariu, On the concept ot tractor beamsOpt. Lett. 35 3847 (2010); S. Sukhov and A. Dogariu, NegativeNonconservative Forces: Optical Tractor Beams for ArbitraryObjects Phys. Rev. Lett. 107, 203602 (2011); O. Brzobohat,V. Karsek, M. iler, L. Chvtal, T. Cimr, and P. Zemnek,Experimental demonstration of optical transport, sorting andself-arrangement using a tractor beam Nat. Pho. 7, 123 (2013).
F. Capolino, Theory and phenomena of metamaterials, (CRCPress, Boca Raton London New York, 2009).
V.G. Veselago, The electrodynamics of substances withsimultaneously negative values of and Sov. Phys. Usp. 10509 (1968).
P.P. Karat and N.V. Madhusudana, Elastic and optical propertiesof some 4-n-Alkyl-CyanobiphenylsMol. Cryst. Liq. Cryst.3651-64 (1976).
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
References III
A.L. Shelankov, Magnetic force exerted by the AharonovBohmlineEurophys. Lett. 43 (6) pp. 623-628 (1998).
A. Vilenkin and E.P.S. Shellard, Cosmic strings and othertopological defects, (Cambridge University Press, Cambridge,1994).
M.G. Ahlford, F. Wilczek, AharonovBohm Interaction of CosmicStrings with Matter Phys. Rev. Letters 62, 1071 (1989);M.G. Ahlford, J. March-Russell, and F. Wilczek, Enhanced baryonnumber violation due to cosmic strings Nucl. Phys. B328 (1989)140-158; A.C. Davis and A.P. Martin, Global strings and theAharonovBohm effect Nucl. Phys. B419 (1994) 341-351;L. Perivolaropoulos, A. Matheson, A.-C. Davis, andR.H. Brandenberger, Nonabelian AharonovBohm baryondecayPhys. Lett. B245 (1990) 556-560. W.B. Perkins,L. Perivolaropoulos, A.-C. Davis, R.H. Brandenberger and
Lukcs . Tactor beams 29
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Intro 2ch Applications Conclusions Backup slides Physics of cosmic strings
References IV
A. Matheson, Scattering of fermions from a cosmic stringNucl.Phys. B353 (1990) 237-270;
T.-P. Cheng and L.-F. Li, Gauge theory of elementary particlephysics, (Clarendon Press, Oxford, 1988); R.N. Mohapatra,Unification and Supersymmetry, (Springer-Verlag, New York,2003).
J. March-Russell, J. Preskill, and F. Wilczek, Internal framedragging and a global analog of the AharonovBohm effect Phys.Rev. Letters 68, 2567 (1992).
Lukcs . Tactor beams 30
IntroductionWhat is negative radiation pressure?NRP on kinksOther approaches
NRP in two channel scattering1dExamplesForce in 2d scattering
ApplicationsNeutron scattering on XY model vortices
Conclusions