Elastic scattering of twisted neutrons by nuclei
EElastic scattering of twisted neutrons by nuclei
A. V. Afanasev, D. V. Karlovets, and V. G. Serbo
3, 4 Department of Physics, The George Washington University, Washington, DC 20052, USA Tomsk State University, Lenina Ave.36, 634050 Tomsk, Russia Novosibirsk State University, RUS-630090, Novosibirsk, Russia Sobolev Institute of Mathematics, RUS-630090, Novosibirsk, Russia (Dated: February 18, 2021)We present a theoretical formalism for scattering of the twisted neutrons by nuclei in a kinematicregime where interference between the Coulomb interaction and the strong interaction is essential.Twisted neutrons have definite quantized values of an angular momentum projection along thedirection of propagation, and we show that it results in novel observable effects for the scatteringcross section, spin asymmetries and polarization of the scattered neutrons. We demonstrate thatadditional capabilities provided by beam’s orbital angular momentum enable new techniques formeasuring both real and imaginary parts of the scattering amplitude. Several possible observablesare considered, for which the targets may be either well-localized with respect to the spatial beamprofile, or the scattering occurs incoherently on nuclei in a bulk target. The developed approachcan be applied to other nuclear reactions with strongly interacting twisted particles.
1. INTRODUCTION
Neutron scattering and interferometry provide fundamental probes of electromagnetic, strong, weak interactionsand gravity [1, 2]. State-of-the-art reactor or spallation neutron facilities are currently operational around the world[3], and new facilities come online [4] or are under construction [5]. Recently [6], a new important capability for thefundamental neutron science was added: namely, the thermal (0.27nm) neutron beams were formed with a nonzeroprojection of the orbital angular momentum – referred to as twisted neutrons – using a method of spiral phase plates.Other novel approaches for generating the beams of twisted neutrons [7–9] can also be applied to ultra-cold neutronswith sub-micron wave lengths.Preparation of beams of light and matter in a pre-defined quantum state of angular momentum projection on beam’sdirection was previously achieved for photons and electrons [10–13]. These beams open new directions for controllingquantum states of matter, for optical and electron microscopy, quantum information, quantum communications andquantum computing. In the analysis of quantum amplitudes, the twisted-electron scattering allows one to access aCoulomb phase [14, 15]. Photo-excitation of atoms by the twisted photons provides approaches for separation of thetransition amplitudes into different multipoles [16, 17]. In the domain of elementary particle physics, new effects inthe collisions of twisted beams were pointed out in Refs. [18, 19]. Nuclear reactions caused by twisted gamma-raysand twisted neutrons were studied theoretically in Ref. [20]. The use of twisted neutrons can become a part ofmultimode-entangled neutron interferometry discussed in Ref.[21] that would identify quantum-entangled degrees offreedom in matter.In this paper, we analyze what novel information about the scattering amplitude can be learned from the twistedneutrons’ being elastically scattered on a zero-spin atomic nucleus. A special case of the Schwinger scattering isconsidered [22, 23], for which the interference of electromagnetic and strong interaction results in a characteristic spinasymmetry. Since a twisted neutron beam represents a (partially) coherent wave packet, different scattering scenarioslead to different observable effects. After a brief review of the scattering formalism for standard, plane-wave, neutronbeams (Section 2), we develop a formalism of twisted-neutron scattering for macroscopic targets (Section 3), for whichthe cross section is an incoherent sum of cross sections for individual nuclei. We find that the angular dependence of thecross section is altered for the twisted particles, while the absorptive part of the amplitude is responsible for transversespin asymmetry, as in a non-twisted case. Twisted neutron beams may be prepared in states of superposition of severalangular momenta; we show that in this case scattering off macroscopic targets develops dependence on the longitudinalcomponent of neutron spin. This is in stark contrast with conventional, non-twisted, neutron scattering, for whichsuch a spin asymmetry is forbidden by parity conservation. Further we consider scattering on a single nucleus witha fixed transverse position with respect to beam’s axis (Section 4) and demonstrate that under this condition thescattering spin asymmetry is due to both longitudinal and transverse spin and, in addition, spin asymmetries havecontributions from both real and absorptive parts of the nuclear amplitude. These features of the spin asymmetries a r X i v : . [ nu c l - t h ] F e b still hold for mesoscopic targets, as we show in Section 5.In summary, the magnitude of the predicted new effects depends on the parameters of the twisted neutron beamsand experimental approaches: for well-localized targets and/or high angular resolution setup we demonstrate possiblespin asymmetries in tens of per cent, while for bulk targets after the averaging over the nuclei positions, the effectsmay reduce to 10 − levels. It should be noted that existing experimental programs studying parity-violation effectsaim to measure even smaller asymmetries of 10 − [24, 25], therefore we believe that measurements of the predictedspin effects from the twisted neutrons are feasible and even necessary for separation of parity-conserving and parity-violating mechanisms of strong interactions.This paper is a substantially expanded version of a Rapid Communication [26].
2. THE STANDARD CASE OF PLANE-WAVE NEUTRONS
Here, we briefly review the formalism of neutron scattering on a spin-zero nucleus in the Schwinger regime [22], i.e. ,when both Coulomb interaction and the strong interaction are essential. Let the initial neutron, prior to approachingthe target, be in a plane-wave state with a momentum p and a wave function w e i pr / (cid:126) , where the spinor w = w ( λ ) ( n )with a helicity λ is normalized as w † w = 1. The final neutron’s wave function is w (cid:48) e i p (cid:48) r / (cid:126) . We neglect the targetrecoil, so that p = p (cid:48) , and introduce the unit vectors n = p /p and n (cid:48) = p (cid:48) /p with the spherical angles θ, ϕ and θ (cid:48) , ϕ (cid:48) .The corresponding scattering amplitude is (see, for example, Ref. [27], Sec. 42) f λλ (cid:48) ( n , n (cid:48) ) = w (cid:48) † λ (cid:48) ( a + i B σ ) w λ , B = β n × n (cid:48) ( n − n (cid:48) ) ,β = µ n Ze m p c = − Z × . × − cm , (1)were σ are the Pauli matrices describing the neutron spin ˆ s = σ , µ n = − .
91 (in nuclear magnetons) and m p is theproton mass. Here, a is the nuclear amplitude while i B σ relates to the electromagnetic interaction of the neutron’sanomalous magnetic moment with a nucleus. Interference of these amplitudes in the cross section allows for importantmeasurements of a phase of the nuclear amplitude. For thermal neutrons with the energies near 25 meV and an Aunuclear target ( a =7.63 fm [28]), the relevant parameters are ε ≡ | β/a | ≈ . , | (Im a ) /a | ≈ × − . (2)The standard cross section summed over spin states of final neutrons has the form dσ (st) ( n , n (cid:48) , ζ ) d Ω (cid:48) = (cid:88) λ (cid:48) | f λλ (cid:48) ( n , n (cid:48) ) | = | a | + | B | + 2 ( B ζ ) Im a, (3)where ζ = ( ζ ⊥ , ζ z ) is the polarization of the initial neutron beam, | ζ | ≤
1. Assuming that the vector n is directedalong the z axis ( i.e. , that n = e z = (0 , , dσ (st) ( e z , n (cid:48) , ζ ) d Ω (cid:48) = | a | + 14 [ β cot( θ (cid:48) / − β ζ ⊥ (Im a ) cot( θ (cid:48) /
2) sin( ϕ (cid:48) − ϕ ζ ) . (4)The interference term depends on the transverse polarization of the initial neutron ζ ⊥ = ζ ⊥ (cos ϕ ζ , sin ϕ ζ , ζ z or the helicity λ . For small scattering angles, θ (cid:48) →
0, the second term on ther.h.s. has a singularity of (1 /θ (cid:48) ) , while the third term has a singularity 1 /θ (cid:48) .Due to time-reversal invariance, this single-spin correlation in Eq.(4) is the same for either initial or final neutronpolarization, and the spin correlation averages to zero after integration with respect to the final neutron’s azimuthalangle ϕ (cid:48) .The standard differential cross section of this process averaged over spin states of initial neutrons has a form dσ (st) ( n , n (cid:48) , ζ (cid:48) ) d Ω (cid:48) = 12 (cid:88) λ | f λλ (cid:48) ( n , n (cid:48) ) | = 12 (cid:104) | a | + | B | + 2 ( B ζ ( f ) ) Im a (cid:105) , (5)where ζ (cid:48) is the detected polarization of the final neutron. The polarization of the final neutron resulting from thescattering process itself [22] is expressed in terms of strong and electromagnetic amplitudes as ζ ( f ) = 2 Im a | a | + | B | B . (6)
3. SCATTERING OF TWISTED NEUTRONS BY A MACROSCOPIC TARGET3.1. Twisted neutrons with a defined J z = m Next, we proceed to the case of twisted neutrons and use an approach developed in Ref. [29] for the twisted spinorparticles. We assume that the incident twisted neutrons propagate along the quantization ( z ) axis and have well–defined values of ( i ) a longitudinal linear momentum p z , ( ii ) an absolute value of a transverse momentum | p ⊥ | ≡ (cid:126)κ ,and ( iii ) a projection of a total angular momentum J z = m , where m is a half–integer. Such a Bessel state has,moreover, a definite energy E = ( (cid:126) κ + p z ) / (2 m n ), with m n being the neutron mass, and the helicity λ . The wavefunction is: ψ κ mp z λ ( r ) = (cid:90) d p ⊥ (2 π ) a κ m ( p ⊥ ) i λ w ( λ ) ( n ) e i pr / (cid:126) . (7)Clearly, the function ψ κ mp z λ ( r ) can be considered as a coherent superposition of the plane waves w ( λ ) ( n ) e i pr / (cid:126) ,weighted with the amplitude a κ m ( p ⊥ ) = i − m e i mϕ πp ⊥ δ ( p ⊥ − (cid:126)κ ) . (8)The momenta of these plane–wave components, p = ( p ⊥ , p z ) = ( (cid:126)κ cos ϕ, (cid:126)κ sin ϕ, p z ) , form a surface of a cone with an opening angle θ = arctan( (cid:126)κ /p z ).Spinor states of the initial and final neutron with helicities λ and λ (cid:48) can be expressed as w ( λ ) ( n ) = (cid:88) σ = ± / e − i σϕ d / σλ ( θ ) w ( σ ) ( e z ) , w ( λ (cid:48) ) ( n (cid:48) ) = (cid:88) σ (cid:48) = ± / e − i σ (cid:48) ϕ (cid:48) d / σ (cid:48) λ (cid:48) ( θ (cid:48) ) w ( σ (cid:48) ) ( e z ) , (9)where d / σλ ( θ ) = δ σλ cos ( θ/ − σδ σ, − λ sin ( θ/
2) are the small Wigner d -functions and w (1 / ( e z ) = , w ( − / ( e z ) = . (10)Using Eq. (9) and the well-known relation (cid:90) π dφ π e i( nφ + z cos φ ) = i n J n ( z ) , (11)where J n ( z ) is the Bessel function of the first kind, we obtain the evident expressions for the wave-function (7) andthe corresponding flux j z and density ρ of the incoming neutrons (in the cylinder coordinates r ⊥ , ϕ r , z ): ψ κ mp z λ ( r ) = e i p z z/ (cid:126) (cid:88) σ i λ − σ J m − σ ( κ r ⊥ ) e i( m − σ ) ϕ r d / σλ ( θ ) w ( σ ) ( e z ) , (12) j ( mλ ) z ( r ⊥ ) = p z m n ρ ( mλ ) ( r ⊥ ) = p z m n (cid:88) σ J m − σ ( κ r ⊥ ) (cid:104) d / σλ ( θ ) (cid:105) . (13)Let us consider the limit of these functions at θ → E (in this case κ → p z → p = √ m n E ): ψ κ mp z ( r ) (cid:12)(cid:12) θ → = δ mλ w ( λ ) ( e z ) e i pz/ (cid:126) , j ( m,λ ) z ( r ⊥ ) (cid:12)(cid:12) θ → = δ mλ p z m n . (14)In other words, in this limit and at m = λ we obtain the standard expressions for the plane-wave neutron flying along z axis with helicity λ .Let us consider scattering on a conventional thin-foil target, which we describe as an ensemble of atoms uniformlydistributed over the large (compared to the beam’s width) transverse extent; we call it a macroscopic target . If thetarget is thin, so that one can neglect the neutrons’ multiple scattering and attenuation, the scattering cross sectioncan be obtained by the averaging over the atoms’ positions in the target w.r.t. the beam axis. Such an averaged crosssection represents an incoherent superposition of the standard ones (see Sec. B3 in [29]), d ¯ σ ( θ, θ (cid:48) , ϕ (cid:48) , ζ ) d Ω (cid:48) = 1cos θ (cid:90) π dσ (st) ( n , n (cid:48) , ζ ) d Ω (cid:48) dϕ π . (15)To perform the integration in Eq.(15), it is useful to expand the vector B in terms of the unit vectors e (cid:48) = (cos ϕ (cid:48) , sin ϕ (cid:48) , , e (cid:48) = ( − sin ϕ (cid:48) , cos ϕ (cid:48) , , e (cid:48) = (0 , , , e (cid:48) i e (cid:48) k = δ ik (16)as follows B = β − nn (cid:48) ) { ( sc (cid:48) e (cid:48) − cs (cid:48) e (cid:48) ) sin( ϕ − ϕ (cid:48) ) + [ cs (cid:48) − sc (cid:48) cos( ϕ − ϕ (cid:48) )] e (cid:48) } , (17)where s ≡ sin θ , c ≡ cos θ , s (cid:48) ≡ sin θ (cid:48) , c (cid:48) ≡ cos θ (cid:48) , and to use the relation B = β (cid:20) − nn (cid:48) ) − (cid:21) (18)with ( n − n (cid:48) ) = 2(1 − nn (cid:48) ) = 2(1 − cc (cid:48) − ss (cid:48) cos( ϕ − ϕ (cid:48) )) . (19)Using these relations, we obtain (cid:90) π (cid:18) B ( ϕ ) β (cid:19) dϕ π = 12 | cos θ − cos θ (cid:48) | −
14 = G ( θ, θ (cid:48) ) , (cid:90) π B ( ϕ ) β dϕ π = 12 g ( θ, θ (cid:48) ) e (cid:48) , g ( θ, θ (cid:48) ) = (cid:26) cot( θ (cid:48) /
2) at θ (cid:48) > θ − tan( θ (cid:48) /
2) at θ (cid:48) < θ . Note that the function G ( θ, θ (cid:48) ) is singular: G ( θ, θ (cid:48) ) → | θ (cid:48) − θ | sin θ at θ (cid:48) → θ. (20)As a result, we get ( c.f. Eq.(4)) d ¯ σ ( θ, θ (cid:48) , ϕ (cid:48) , ζ ) d Ω (cid:48) = | a | cos θ (cid:20) R em − Im a | a | R int ζ ⊥ sin( ϕ (cid:48) − ϕ ζ ) (cid:21) , (21)where R em = ε G ( θ, θ (cid:48) ) , R int = εg ( θ, θ (cid:48) ) (22)This cross section is still independent of ζ z and it coincides with Eq.(4) in the standard limit θ →
0. Such a behavioris expected since in this limit G ( θ, θ (cid:48) ) → (1 /
4) cot ( θ (cid:48) /
2) and g ( θ, θ (cid:48) ) → cot( θ (cid:48) / R em ( θ (cid:48) ) which corresponds to a relative contribution of the electromagneticinteraction. Unlike the Schwinger cross section (4), this function has an angular singularity of 1 / | θ (cid:48) − θ | at θ (cid:48) → θ .This shift to the non-vanishing scattering angles is potentially useful for experimental analysis of the small-anglescattering. Indeed, thanks to this property, the singular region is shifted from the small angles θ (cid:48) →
0, which may bedifficult to access experimentally, to the larger values, θ (cid:48) → θ , which can be controlled by the opening angle θ of theincoming twisted neutrons. Practically, this method would depend on experiment’s ability to reach sufficiently largevalues of the opening angle θ .In Fig. 2 we present the function R int ( θ (cid:48) ) which describes interference of the electromagnetic amplitude and thenuclear one as well as the corresponding function for the standard case. In the region θ (cid:48) ≥ θ , the function R int ( θ (cid:48) )coincides with its standard limit, but this function experiences a step-like drop for the angles θ (cid:48) ≤ θ that potentiallycan be observed in experiments. θ ' ( rad ) R e m Figure 1: The functions R em (blue solid line) from Eq. (22) and its plane-wave limit (black dashed line) plotted vs. the neutronscattering angle θ (cid:48) for the opening angle θ = 0 .
06 rad and parameter ε = 0 . θ ' ( rad ) R i n t Figure 2: The functions R int (blue solid line) from Eq. (22) and its plane-wave limit (black dashed line) plotted vs. the neutronscattering angle θ (cid:48) for θ = 0 .
06 rad and ε = 0 . Let us take now a coherent superposition of two Bessel states with the different projections m and m , but withthe same helicity λ and the same values of p z and κ . Such a superposition can be generated experimentally [6, 7],and it is described by the following wave function: ψ (2 tw) ( r ) = c ψ κ m p z λ ( r ) + c ψ κ m p z λ ( r ) ,c n = | c n | e iα n , | c | + | c | = 1 . (23)With the help of this expression, we find the averaged differential cross section in the form d ¯ σ ( θ, θ (cid:48) , ϕ (cid:48) , ζ ) d Ω (cid:48) = 1cos θ (cid:90) π dσ (pl) ( n , n (cid:48) , ζ ) d Ω (cid:48) Φ( ϕ, ∆ m, ∆ α ) dϕ π , (24)where the function Φ is defined asΦ( ϕ, ∆ m, ∆ α ) = 1 + 2 | c c | cos [( ϕ − π/
2) ∆ m + ∆ α )] . (25)As a result, we obtain d ¯ σ ( θ, θ (cid:48) , ϕ (cid:48) , ζ ) d Ω (cid:48) =1cos θ (cid:8) A + | c c | (cid:0) β B + 2(Im a ) β ( ζ C ) (cid:1)(cid:9) , A = | a | + β G ( θ, θ (cid:48) ) + (Im a ) β ( ζ e ) g ( θ, θ (cid:48) ) , B = cos γ | c − c (cid:48) | [ T ( θ, θ (cid:48) )] | ∆ m | , (26) C = (cid:20) ∆ m | ∆ m | (cid:18) − c (cid:48) s (cid:48) e (cid:48) + e (cid:48) (cid:19) sin γ + c − c (cid:48) | c − c (cid:48) | e (cid:48) cos γ (cid:21) × [ T ( θ, θ (cid:48) )] | ∆ m | , where γ = ( ϕ (cid:48) − π/
2) ∆ m + ∆ α,T ( θ, θ (cid:48) ) = (cid:18) tan( θ/ θ (cid:48) / (cid:19) ± for θ (cid:48) ≷ θ. In contrast to Eq. (21), derived for a single- m incident beam, this cross section depends on the differences of the totalangular momenta, ∆ m = m − m (cid:54) = 0, and of the states’ phases, ∆ α = α − α . This ∆ m and ∆ α dependencetranslates directly into the angular- and polarization properties of the scattered neutrons. In particular:( i ) The cross section (26) depends not only on the neutron’s transverse polarization ζ ⊥ , but also on the longitudinalone ζ z . It leads to the following longitudinal spin asymmetry : A ζ z = d ¯ σ ( ζ z = +1) − d ¯ σ ( ζ z = − d ¯ σ ( ζ z = +1) + d ¯ σ ( ζ z = − | c c | (Im a ) β ( Ce (cid:48) ) | a | + β [ G ( θ, θ (cid:48) ) + | c c | B ] . (27) - - φ ' ( rad ) A ζ z , pp m Figure 3: A longitudinal spin asymmetry (27) plotted vs. the neutron scattering azimuthal angle ϕ (cid:48) for ε = 0 .
03, 2 c c = 1,∆ m = 1, and for θ (cid:48) = 0 .
005 rad (blue solid line), θ (cid:48) = 0 .
025 rad (green dashed line), θ (cid:48) = 0 .
045 rad (red dotted line).
Figures 3 and 4 show this asymmetry for different values of the parameters. For thermal neutrons and a goldtarget (see Eq. (2)), the predicted asymmetry amounts to a few ppm , which is in a range currently accessible forexperiments on the hadronic parity violation [25], for which the above asymmetry may be a source of unwantedsystematics, provided that the neutron beam becomes twisted due to uncontrolled interactions. However, as we showbelow, averaging over the azimuthal scattering angle ϕ (cid:48) eliminates the dependence on ζ z , which provides an approachto correct for this kind of systematics. Azimuthal angular coverage for the neutron-scattering detectors would beessential to deal with this systematic effect in parity-violation measurements. - - φ ' ( rad ) A ζ z , pp m Figure 4: The longitudinal spin asymmetry (27) plotted vs. the neutron scattering azimuthal angle ϕ (cid:48) for ε = 0 .
03, 2 c c = 1, θ (cid:48) = 0 . m = 1 (blue solid line), ∆ m = 2 (green dashed line), ∆ m = 3 (red dotted line). θ ' ( rad ) A ζ x , pp m Figure 5: A transverse spin asymmetry plotted vs. the neutron scattering angle θ (cid:48) for ε = 0 .
03, 2 c c = 1, ∆ m = 1, and for θ = 0 .
01 (blue solid line), θ = 0 .
02 (green dashed line), θ = 0 .
03 (red dotted line).
Analogously to Eq. (27), one can define a quantity A ζ x , which we call the transverse spin asymmetry . We showthis asymmetry in Fig. 5 for a sample set of parameters.( ii ) Let us discuss the properties of the differential cross section (26) averaged over the azimuthal angle ϕ (cid:48) of thefinal neutron. For this aim, we introduce the following notation: (cid:104) F (cid:105) = (cid:90) π F dϕ (cid:48) π . (28)The averaged cross section reads (cid:28) d ¯ σ ( θ, θ (cid:48) , ϕ (cid:48) , ζ ) d Ω (cid:48) (cid:29) = 1cos θ (cid:16) | a | + β G ( θ, θ (cid:48) ) +2 | c c | (Im a ) β ζ (cid:104) C (cid:105) (cid:17) , (29)where (cid:104) C (cid:105) = 12 (cid:18) c (cid:48) s (cid:48) − c − c (cid:48) | c − c (cid:48) | (cid:19) T ( θ, θ (cid:48) ) (cos ∆ α, ∓ sin ∆ α,
0) (30)for ∆ m = ±
1, and (cid:104) C (cid:105) = 0 otherwise. The fact that this spin observable is nonzero can be understood as due toan effect of superposition between two vortex states that define a new plane with an orientation fixed by a phasedifference ∆ α . Then the transverse spin with respect to this plane contributes to the scattering asymmetry, while theneutron scattering plane – the only plane available for non-twisted neutrons – becomes redundant. It is seen that thiscross section depends on ζ ⊥ and on Im a at ∆ m = ±
1, but it is independent of the longitudinal polarization.If the initial neutron is unpolarized, then its polarization after the scattering is (cid:104) ζ ( f ) (cid:105) = − Im a | a | S (cos ∆ α, ∓ sin ∆ α, , (31)where S = | c c | ε ε G ( θ, θ (cid:48) ) (cid:18) c (cid:48) s (cid:48) − c − c (cid:48) | c − c (cid:48) | (cid:19) T ( θ, θ (cid:48) ) (32)for ∆ m = ±
1, and S = 0 otherwise. In Fig. 6 one can see that |(cid:104) ζ ( f ) (cid:105)| ∼ . | Im a || a | for θ ∼ ε , i. e. the predicted effectis of the order of tens of ppm for the thermal neutrons and the gold target (see Eq.(2)). θ ' ( rad ) S Figure 6: The function S (defined in Eq. (32)) plotted vs. the neutron scattering angle θ (cid:48) for ε = 0 .
03, 2 | c c | = 1, ∆ m = 1,and for θ = 0 .
03 rad (blue solid line), θ = 0 .
06 rad (black dashed line).
4. SCATTERING OF TWISTED NEUTRONS BY A SINGLE NUCLEUS
Let the single- m neutrons be scattered by a nucleus located in the transverse ( xy ) plane at a definite impactparameter b = ( b x , b y ,
0) = b (cos ϕ b , sin ϕ b , F ( m ) λλ (cid:48) ( θ, θ (cid:48) , ϕ (cid:48) , b ) = i λ − m e − i p (cid:48)⊥ b / (cid:126) × (cid:90) π d ϕ π e i mϕ +i p ⊥ b / (cid:126) f λλ (cid:48) ( n , n (cid:48) ) , (33)where the factor exp(i p ⊥ b / (cid:126) ) specifies the lateral position of the nucleus with respect to the beam.Using Eqs. (9), (17) and introducing quantities A ( σ ) ( m, κ , b ) = (cid:90) π dϕ π e i[( m − σ ) ϕ + κ b cos( ϕ − ϕ b )] a = a e i( m − σ )( ϕ b + π/ J m − σ ( κ b ) , (34) B ( σ ) ( m, κ , b ) = (cid:90) π dϕ π e i[( m − σ ) ϕ + κ b cos( ϕ − ϕ b )] B ( ϕ ) , (35)we rewrite the above equation in the form F ( m ) λλ (cid:48) ( θ, θ (cid:48) , ϕ (cid:48) , b ) = i λ − m e − i p (cid:48)⊥ b (cid:88) σ = ± / d / σλ ( θ ) w ( λ (cid:48) ) † ( n (cid:48) ) (cid:104) A ( σ ) + i B ( σ ) σ (cid:105) w ( σ ) ( e z ) . (36)This amplitude coincides (up to the inessential factor e − i p (cid:48)⊥ b ) with the standard one (1) in the limit θ → n → e z , d / σλ ( θ ) → δ σλ , A ( σ ) → a δ mσ and B ( σ ) → B δ mσ .The angular distributions of the scattered neutrons can be obtained by squaring this amplitude. Such a distribution summed over helicities of final neutrons W ( m ) λ ( θ, θ (cid:48) , ϕ (cid:48) , b ) = (cid:88) λ (cid:48) (cid:12)(cid:12)(cid:12) F ( m ) λλ (cid:48) ( θ, θ (cid:48) , ϕ (cid:48) , b ) (cid:12)(cid:12)(cid:12) (37)is considered in detail in the Appendix A and the specific case b = 0 in the Appendix B. Here we only discuss thisdistribution averaged over the azimuthal angle of the final neutrons using the notation (28): W ( m ) λ ( θ, θ (cid:48) , b ) = (cid:80) λ (cid:48) (cid:28)(cid:12)(cid:12)(cid:12) F ( m ) λλ (cid:48) ( θ, θ (cid:48) , ϕ (cid:48) , b ) (cid:12)(cid:12)(cid:12) (cid:29) = 12 Σ ( m ) + λ ∆ ( m ) , (38)Σ ( m ) = | a | (cid:16) J m − / ( κ b ) + J m +1 / ( κ b ) (cid:17) + (cid:80) σ (cid:10) ( B ( σ ) ∗ B ( σ ) ) − σ Im (cid:0) B ( σ ) ∗ × B ( σ ) (cid:1) z (cid:11) , ∆ ( m ) = (cid:0) | a | cos θ − (Re a ) β h ( θ, θ (cid:48) ) (cid:1) (cid:16) J m − / ( κ b ) − J m +1 / ( κ b ) (cid:17) ++ cos θ (cid:80) σ (cid:10) σ ( B ( σ ) ∗ B ( σ ) ) − Im (cid:0) B ( σ ) ∗ × B ( σ ) (cid:1) z (cid:11) − sin θ (cid:68) Im (cid:0) B (1 / ∗ × B ( − / (cid:1) x − Re (cid:0) B (1 / ∗ × B ( − / (cid:1) y (cid:69) , where h ( θ, θ (cid:48) ) = g ( θ (cid:48) , θ ) sin θ = (cid:26) − ( θ/
2) at θ (cid:48) > θ ( θ/
2) at θ (cid:48) < θ (cid:27) = ∓ (1 ∓ cos θ ) for θ (cid:48) ≷ θ. (39)As a result, we obtain a nonvanishing helicity asymmetry , A λ = W ( m ) λ =1 / − W ( m ) λ = − / W ( m ) λ =1 / + W ( m ) λ = − / = ∆ (m) Σ (m) . (40) Figure 7: The distribution (38) in units | a | as a function of the Au nucleus position b for θ (cid:48) = 0 .
04 rad, θ = 0 .
07 rad, and ε = 0 .
03. The case Im a > a <
In contrast to Eq.(4), the interference term in (38) depends on the initial neutron’s helicity and on the real part ofthe nuclear amplitude (and, therefore, on its phase Arg a ), even after the azimuthal averaging – see (38). The angulardistributions (38) are plotted in Fig. 7 for an Au nucleus as a function of its position b . The scattering angle ischosen as θ (cid:48) = 0 .
03 rad for which the electromagnetic and strong amplitudes equally contribute to the cross sectionfor the plane-wave neutrons. One can see that the former contribution dominates in the beam center ( b → A λ is a periodic function of b and of the amplitude’s phase Arg a ; it can reach tens of percent for awide range of parameters, as shown in Ref. [26]. Note that outside the cone opening angle, at θ (cid:48) > θ , the sensitivityto the phase practically vanishes, so that in order to probe both the real and the imaginary parts of the amplitudeone needs to perform measurements at the small angles θ (cid:48) < θ , which is feasible.
5. SCATTERING OF TWISTED NEUTRONS BY A MESOSCOPIC NUCLEAR TARGET
Until now we have discussed the scattering for two extreme cases of either a single-nucleus or a macroscopic(infinitely wide) target. In a more realistic experimental scenario, a neutron beam collides with a well–localized0mesoscopic atomic target. In order to account for geometrical effects in such a scenario, we describe a target as anincoherent ensemble of potential centers. The density of the scatterers in the transverse ( xy ) plane is characterizedby a distribution function n ( b ), which is normalized as follows: (cid:90) n ( b ) d b = 1 . (41)For the numerical analysis below we take n ( b ) to be a Gaussian function: n ( b − b t ) = 12 πσ t e − ( b − b t ) / (2 σ t ) . (42)This distribution is sharply peaked at the impact parameter b = b t (the centre of the target) if its dispersion σ t issmall.There are two limiting cases: (i) when the target is much wider than the incident beam, σ t (cid:29) / κ , and (ii) when itis narrower, σ t (cid:46) / κ . When averaging over the impact parameters b , the above features of the single-nucleus regimesurvive when the target is sub-wavelength sized or even if its width does not exceed that of the beam, whereas forthe macroscopic target the above spin asymmetries vanish, see Eq. (21). For an intermediate case of a mesoscopictarget, the spin asymmetry survives but its values decrease as σ t κ grows. In order to give quantitative estimates, wetake a realistic example of a Gaussian target with σ t ∼ (1 / κ ) − (10 / κ ) and the angles θ (cid:48) < θ ∼ ◦ − ◦ . The helicityasymmetry reaches the values of | A λ | ≈ − − − (43)for a wide range of parameters, as we show in Fig.8 and Fig.9. Note that the asymmetries even some 2 or 3 ordersof magnitude smaller can in principle be measured, as the current experiments aim at much lower values than (43),down to 10 − [24, 25]. Figure 8: The helicity asymmetry as a function of κ σ t where σ t is a width of the Au mesoscopic target for m = 1 / θ (cid:48) = 0 .
03 rad, θ = 0 .
06 rad, and ε = 0 . b t = ϕ t = 0. The case Im a > a < Thus, scattering off the well-localized targets – say, of σ t (cid:38)
10 nm − µ m in width for the neutron wave packets withthe wavelength of 0 . −
100 nm and the transverse coherence length of 1 / κ (cid:38) − µ m [7] – reveals dependenceon the neutron’s helicity and allows one to probe the nuclear amplitude’s real part already in the Born approximation,whereas with a single beam of the delocalized plane-wave neutrons such a dependence arises beyond the tree levelonly. This method for high-precision measurements of the complex amplitude for non-vanishing scattering angles isalternative and complementary to the neutron interferometry and to the neutron gravity reflectometry.Conventional Schwinger asymmetry can be enhanced for thermal neutrons due to presence of the nuclear resonances,as was shown experimentally [30]. Extension of our formalism to the nuclear resonance region is straightforward, inwhich case we have to use an appropriate parameterization for the nuclear amplitude a , and use a partial-waveexpansion and angular integration in Eq.(33) in order to account for the non S -wave resonances.1 Figure 9: The helicity asymmetry as a function of κ σ t where σ t is a width of the Au mesoscopic target for m = 5 / θ (cid:48) = 0 .
04 rad, θ = 0 .
07 rad, and ε = 0 . b t = ϕ t = 0. The case Im a > a <
6. CONCLUSION
We have presented a theoretical formalism for elastic scattering of the twisted neutrons by a nucleus and nucleartargets and predict new effects for the cross section and spin asymmetries. Our approach is based on expansion ofthe twisted beam’s wave function in terms of the plane waves and it leads to representation of the twisted-scatteringamplitude in a form of a superposition of the plane-wave amplitudes. The results are presented for the kinematicsof the Schwinger scattering [22] characterized by a spin asymmetry due to interference between the strong andelectromagnetic scattering amplitudes.The following observable effects are predicted that are unique for the twisted-neutron scattering on nuclei:(a) For the macroscopic targets, the scattering cross section has a different angular dependence, which is peaked atnon-zero scattering angles, as opposed to the non-twisted case;(b) For the macroscopic targets and a beam that is a superposition of angular momentum states differing by one unitof (cid:126) , the cross section develops a spin asymmetry that depends on the azimuthal scattering angle and a longitudinalcomponent of neutron’s spin. This observable is forbidden for non-twisted neutrons by parity conservation;(c) For scattering on a single nucleus, provided that the target’s location is resolved with respect to the twistedneutron’s wavefront, the scattering spin asymmetry is due to both the longitudinal and transverse spin and, inaddition, the spin asymmetries have contributions from both the real and absorptive parts of the nuclear amplitude.These features of the spin asymmetries still hold for the realistic mesoscopic targets.The predicted spin asymmetries range from 10 − to 10 − for relevant parameters and are detectable in experimentalconditions similar to those used for parity-violating measurements [24, 25]. Whereas generation of the twisted neutronswas experimentally demonstrated for lower fluxes [6–8], it will be desirable to use the high-flux sources of twistedneutrons in order to achieve sufficient statistical accuracy of the future measurements.In summary, we have demonstrated that twisted neutrons can be used as a new tool to probe nuclear scatteringamplitudes at low energies and to access observables that are otherwise forbidden by the symmetry for non-twistedbeams. Acknowledgements – We would like to thank D. Pushin, W. M. Snow, and A. Surzhykov for useful discussions.Thework of D.V.K. and V.G.S. is supported by the Russian Science Foundation (Project No. 17-72-20013).2
APPENDIX A. ANGULAR DISTRIBUTIONS OF THE SCATTERED NEUTRONS
The distribution defined in Eq.(37) reads W ( m ) λ ( θ, θ (cid:48) , ϕ (cid:48) , b ) = (cid:88) λ (cid:48) (cid:12)(cid:12)(cid:12) F ( m ) λλ (cid:48) ( θ, θ (cid:48) , ϕ (cid:48) , b ) (cid:12)(cid:12)(cid:12) (44)= 12 (cid:104) D (1 / + D ( − / (cid:105) + λ (cid:104) D (1 / − D ( − / (cid:105) cos θ − λ (cid:104) Im C (1 / , − / x − Re C (1 / , − / y (cid:105) sin θ. Here we use the quantities C (˜ σσ ) = B (˜ σ ) ∗ × B ( σ ) + I (˜ σσ ) , I (˜ σσ ) = A (˜ σ ) ∗ B ( σ ) − B (˜ σ ) ∗ A ( σ ) (45) D ( σ ) = (cid:12)(cid:12)(cid:12) A ( σ ) (cid:12)(cid:12)(cid:12) + B ( σ ) ∗ B ( σ ) − σ Im C ( σσ ) z (46)with the properties C ( σσ ) z = i Im C ( σσ ) z , C ( σ, − σ ) ⊥ = − (cid:16) C ( − σ,σ ) ⊥ (cid:17) ∗ . (47)Let us define contributions from the nuclear and electromagnetic interactions and their interference as W ( m ) λ ( θ, θ (cid:48) , ϕ (cid:48) , b ) = W ( m, nucl) λ + W ( m, em) λ + W ( m, int) λ , where W ( m, nucl) λ = (cid:88) σ (cid:18)
12 + 2 σ λ cos θ (cid:19) | A ( σ ) | , (48) W ( m, em) λ = (cid:88) σ (cid:20)(cid:18)
12 + 2 σλ cos θ (cid:19) | B ( σ ) | − ( σ + λ cos θ ) Im (cid:16) B ( σ ) ∗ × B ( σ ) (cid:17) z (cid:21) − λ sin θ (cid:20) Im (cid:16) B (1 / ∗ × B ( − / (cid:17) x − Re (cid:16) B (1 / ∗ × B ( − / (cid:17) y (cid:21) , (49) W ( m, int) λ = − (cid:88) σ ( σ + λ cos θ ) Im I σσz − λ sin θ (cid:104) Im I (1 / , − / x − Re I (1 / , − / y (cid:105) . (50)Note the following features:1. This angular distribution contains the evident dependence on helicity λ of the initial neutron.2. The pure nuclear contribution is directly proportional to the density of incoming neutrons (see Eq. (13)): W ( m, nucl) λ = | a | ρ ( mλ ) ( b ) = | a | ρ ( b ) + λ | a | cos θ (cid:104) J m − / ( κ b ) − J m +1 / ( κ b ) (cid:105) , (51)where ρ ( b ) is the density of neutrons averaged over their helicities: ρ ( b ) = 12 (cid:88) λ ρ ( mλ ) ( b ) = 12 (cid:104) J m − / ( κ b ) + J m +1 / ( κ b ) (cid:105) (52)This function is shown in Fig. 10 for different values of m .3. The interference of nuclear and electromagnetic interactions of neutrons is described by the terms I (˜ σσ ) only. APPENDIX B. THE LIMIT b = 0 In the limit of zero impact parameter b →
0, the expressions (34) and (35) for A ( σ ) and B ( σ ) are simplified. Indeed,using the formulas (27)–(28) from [31]: (cid:90) π dϕ π e i nϕ − cc (cid:48) − ss (cid:48) cos ϕ = 1 | c − c (cid:48) | (cid:18) ss (cid:48) − cc (cid:48) + | c − c (cid:48) | (cid:19) | n | , (53)3 ϰ b ρ ( b ) Figure 10: The functions ρ ( b ) from Eq. (52) for m = 1 / m = 3 / m = 5 / we get A ( σ ) = a δ σ m , B ( σ ) = B m − σ (cid:18) ± i c (cid:48) , c − c (cid:48) | c − c (cid:48) | , ∓ i s (cid:48) (cid:19) for m − σ ≷ , (54) B ( σ ) = B (cid:18) , c − c (cid:48) | c − c (cid:48) | , (cid:19) for m − σ = 0 , where B m − σ = β s (cid:48) (cid:18) ss (cid:48) − cc (cid:48) + | c − c (cid:48) | (cid:19) | m − σ | . (55)Sometimes it is useful to employ the identity1 − cc (cid:48) + | c − c (cid:48) | = (1 ± c )(1 ∓ c (cid:48) ) for θ (cid:48) ≷ θ (56)and transform the above equation to the form B m − σ = β s (cid:48) (cid:18) tan( θ/ θ (cid:48) / (cid:19) ±| m − σ | for θ (cid:48) ≷ θ. (57) The case m (cid:54) = ± / In this case the flux of neutrons (13) at b = 0 disappears and only the contribution of the long-range electromagneticinteraction survives: W ( m ) λ ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) = 2 1 ± c (cid:48) ± c (1 ± Λ) (cid:18) β s (cid:48) tan( θ/ θ (cid:48) / (cid:19) | m |− for θ (cid:48) ≷ θ (58)with the notation Λ = 2 λ sign( m ) . (59)This situation can be called as the scattering in the dark in analogy with the excitation in the dark , which was observedin the experiment [32] with twisted photons.The helicity asymmetry in this limit has a simple analytical expression A λ ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) = W ( m ) λ =1 / ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) − W ( m ) λ = − / ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) W ( m ) λ =1 / ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) + W ( m ) λ = − / ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) = c − c (cid:48) | c − c (cid:48) | sign( m ) . (60)4In is interesting to note that another asymmetry defined as A m ( θ, θ (cid:48) , b = 0) = W ( m ) λ =1 / ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) − W ( m − λ = − / ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) W ( m ) λ =1 / ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) + W ( m − λ = − / ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) (61)has the same behavior for the case m (cid:54) = ± / m − (cid:54) = ± / λ , but with a fixed value of orbital angular momentum projection (= m − λ in a paraxial approximation). Thus, it can be measured by only manipulating the spin degree of freedom λ for thegiven twisted-neutron beam. The case m = ± / In this case the flux of neutrons (13) does not disappear at b = 0 and all contributions do survive: W ( m ) λ ( θ, θ (cid:48) , ϕ (cid:48) , b = 0) = 12 (Σ + Λ∆) , (62)where Σ = | a | + (cid:18) β s (cid:48) (cid:19) [1 + 2 H ( θ, θ (cid:48) )] , H ( θ, θ (cid:48) ) = (1 ∓ c )(1 ± c (cid:48) )1 ± c (63)∆ = | a | c ± β Re( a ) (1 ∓ c ) + (cid:18) β s (cid:48) (cid:19) [ c ± H ( θ, θ (cid:48) )] (64)Here, the upper (lower) sign corresponds to θ (cid:48) > θ ( θ (cid:48) < θ ). [1] J. Byrne, Neutrons, Nuclear and Matter: An Exploration of the Physics of Slow Neutrons , 2nd Ed. (Dover Publications,New York, 2013).[2] H. Rauch, S. A. Werner,
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