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Deuteron Photodisintegration with Twisted Beams

Andrei Afanasev,1 Valery G. Serbo,2, 3 and Maria Solyanik1

1_{Department of Physics, The George Washington University, Washington, DC 20052, USA} 2_{Novosibirsk State University, RUS-630090, Novosibirsk, Russia}

3_{Sobolev Institute of Mathematics, RUS-630090, Novosibirsk, Russia} (Dated: September 19, 2017)

We consider two basic nuclear reactions: Radiative capture of neutrons by protons,n+p→γ+ d

and its time-reversed counterpart, photodisintegration of the deuteron,γ+d → n+p. In both of these cases we assume that the incoming beam of neutrons or photons is “twisted” by having an azimuthal phase dependence,i.e., it carries an additional angular momentum along its direction of propagation. Taking a low-energy limit of these reactions, we derive relations between corresponding transition amplitudes and cross sections with plane-wave beams and twisted beams. Implications for experiments with twisted cold neutrons and photon beams are discussed.

I. INTRODUCTION

Let us consider two types of nuclear reactions: radia-tive capture of cold neutrons and photodisintegration of the deuteron. Both of these reactions with energies

E . 1 MeV by protons is of high importance for Big Bang nucleosynthesis and stellar evolution.

Radiative capture of neutrons was considered in great detail in a number of experimental and theoretical pa-pers,c.f. the following review article [1]. We are partic-ularly interested in reactions with low neutron energies

E∼10 meV, that can be tested in experiments with cold neutrons. The total reaction cross-section isσ_∼300 mb, and mainly comes fromM1 isovector contribution dom-inating inSwave capture. Twisted cold neutronsin this energy range have been recently generated in the lab [2] (with temperature 130K and wavelengthλ= 0.27 nm). The neutron beams with Orbital Angular Momentum (OAM) projection ¯hm on the direction of beam propa-gation were obtained, wherem=1, 2, 4. It is reasonable to expect these neutrons to be applied tonp → dγ re-action studies. Thereby, this rere-action can be utilized in diagnostics of neutron beams’ topology.

Feasibility of obtainingtwisted photonbeams with en-ergies in the MeV range sufficient to induce photo-nuclear reactions was theoretically demonstrated in the literature [3, 4], from electrons in spiral motion related to astro-physics conditions [5]. Simulations for spe-cific experimental conditions for high-energy twisted-photon generation was provided in Ref.[6] for Comp-ton backscattering and in Ref.[7] for plasma-wakefield acceleration. We may also expect generation of twisted photons in a non-linear regime of Compton scattering with circularly-polarized initial photons [8].

In view of the technological progress in generation of twisted beams, we need to evaluate potential impor-tance of such beams for nuclear studies. Here we con-sider theoretically the simplest nuclear reactions caused by twisted neutron or photon beams. In particular, in this paper we discuss distinctive features of the twisted neutron beams’ capture by macroscopic and mesoscopic

proton targets, which can be experimentally observed. Its time-reversed analogue, deuteron photodisintegra-tion is considered for the case of low energies, when the S-state dominates in the partial-wave expansion.

The paper is organized as follows. Section II intro-duced the formalism of twisted-neutron beams, Sec-tion III discusses the reacSec-tion of radiative capture with twisted neutrons for different types of targets, Section IV presents comparison between nuclear photoabsorp-tion of the twisted and plane-wave photons, and con-clusions are presented in Section V.

II. INCOMING TWISTED NEUTRON

We use the formalism of Quantum Mechanics to de-scribe the corresponding baryonic states, neglecting the neutron spin degree of freedom. Consider a stationary state of free twisted neutron of energyEand massmnin the non-relativistic regime. We take ¯h=1 for simplicity. The wave function of such a state in the cylindrical coor-dinate systemρ,φr,zcan be written as an eigenfunction of three commuting operators: energy ˆH = ₂pppˆ_m2_n, mo-mentum ˆpz =−i_∂∂_z and ˆLz =−i_∂φ∂_r projected onz-axis. Consequently, this wave function satisfies the following equations

Hψ(rrr) =Eψ(rrr), pˆzψ(rrr) =pzψ(rrr), Lˆzψ(rrr) =mψ(rrr), (1) where m is an integer. Both longitudinal momentum and the norm of transverse momentum with respect to thez-direction

κ= q

2mnE−p2z (2)

are well defined for this state. Then the explicit func-tional from is

ψκmp_z(rrr) = Jm(κρ)e

imφr_eipzz_, ₍₃₎

whereJm(x)is the Bessel function of the first kind. The plane wave expansion of this expression in thexy-plane

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can be written as following

ψκmp_z(rrr) =eipzz Z

aκm(ppp⊥)eippp⊥ρρρ

d2p_⊥

(2π)2 (4) whereaκm(ppp⊥)is the corresponding Fourier amplitude

aκm(ppp⊥) =i−

m2π

p_⊥δ(p⊥−κ)e

imφp ₍₅₎

Here φp is the azimuthal angle of vector ppp = (ppp_⊥,pz) andp_⊥=|ppp_⊥|. The Fourier amplitude is normalized as

a∗κm(ppp⊥)aκ′m′(ppp⊥)

d2p_⊥

(2π)2 = 2π

κ δ(

κ−κ′)δ_mm_′ (6) from which the neutron state normalization condition follows:

ψκ∗mp_z(rrr)ψκ′m′p′_z(rrr)d 3_r₌ Z

a∗κm(ppp⊥)aκ′m′(ppp⊥)e

i(ppp₋ppp′)rrrd2p⊥d2p′⊥ (2π)4 d

3_r₌ 4π2

κ δ(

κ−κ′)δ_mm_′δ(p_z−p′_z). (7)

Hence, twisted neutron can be seen as a non-diverging packet of plane waves, localized on a conical surface with the opening angleθp=arctan(κ/pz)and symme-try axis aligned with thez-direction.

In the following, we consider the limiting case of Eq. (3) for fixed energyEand θp → 0 (in this caseκ →0,

pz→√2mnEandJm(κρ)→δ_m₀):

ψκmp_z(rrr)|θ_p→0=δm0e

ipz_. ₍₈₎

In other words, in this limit one recovers a standard ex-pression for the plane wave propagating along the z -axis.

The density of a twisted neutron state propagating in thez-direction is

̺(m)(ρ) =|ψκmp_z(rrr)|2=J_m2(κρ), (9) see Fig. 1. This is the positive-definite function of co-ordinates in the transverse plane which depends on the quantum numberm. It is worth it to mention that

̺(m)(ρ)|θp→0=δm0. (10)

Hence, form = 0 one recovers standard expression for the density in this limit. This density is non-zero at the origin form=0,

̺(0)(ρ)|ρ→0=1. (11)

FIG. 1: Function̺(m)₍_ρ₎_{, Eq. (9), as a function of}κρform=0 (black solid line),m = 2 (red dot-dashed line), m = 5 (blue dashed line).

Let M(pl)(ppp,kkk)be the matrix element of plane wave neutron radiative recombination

n(ppp) +p→d+γ(kkk). (12)

As theS-wave contributes the most into the cross sec-tion of this process, the corresponding matrix element is independent of the direction of the incoming neu-tron momentum ppp. When considering radioactive re-combination, we need to account for the fact that the twisted neutron has a symmetry axis aligned with thez -direction, while a proton is located at a certain distance

bbb = (bx,by, 0) from this axis called impact parameter. Hence, we can express this matrix element as a trace of a plane-wave matrix elementM(pl)(ppp,kkk)and the Fourier amplitudeaκm(ppp⊥) of the incoming electron with the impact parameterb

M(m)(κ,p_z,kkk,bbb) =

e−ikkk⊥bbb

M(pl)(ppp,kkk)eippp⊥bbb_aκm(ppp⊥)

d2p_⊥

(2π)2 =

eimφb−ikkk⊥bbb_M(pl)₍_ppp_,_kkk₎_J_m(κb). (13)

This analysis can be applied to Bessel-Gaussian diffract-ing mode, where the Bessel-shaped disturbance in trans-verse plane is suppressed by a Gaussian factor. A con-ventional wave-function for BG mode in focus plane (z=0) can be written as [9]

ψκ(BG)mp_z(rrr) =N Jm(κρ)exp

− ρ

2 w2 0

eimφρ ₍₁₄₎

whereNis a constant factor. In this case the correspond-ing amplitude acquires this additional phase factor

M(BG)(κ,p_z,kkk,bbb) =eimφb−ikkk⊥bbbJm(κb)e

−b2

w20_M(pl)₍_ppp_,_kkk₎ (15) where w0is the Gaussian waist of the BG mode.

It follows that the differential probability of the re-combination process per unit time for the case of a

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twisted neutron dW˙ (m)(bbb)/dΩ_k _{and for a plane wave} dW˙(pl)(bbb)/dΩ_k_{are related by the following expression}

dW˙(m)(bbb)

dΩ_k =

dW˙(pl)(bbb)

dΩ_k J

m(κb) (16)

III. np→dγREACTION WITH COLD TWISTED NEUTRONS

A. Macroscopic target

To begin with, we consider the simplest case of a macroscopic proton target with a uniform random dis-tribution of scattering centers in the plane, normal to the direction of the beam propagation. We also neglect ondary collisions. In this setup, the differential cross sec-tion averaged over the proton impact parameter in the area with radiusRis the convenient observable to con-sider. Following paper [10], we obtain a simple relation for the averaged cross section expressed as a function of the plane wave cross section:

dσ¯

dΩ_k =

1 cosθp

Z 2π 0

dσ(pl)

dΩ_k

dφp

2π, (17)

where θp is the conical angle of the incoming twisted neutron. The standard scattering cross section is inde-pendent ofφpin our case, which leads to

dσ¯

dΩ_k =

1 cosθp

dσ(pl)

dΩ_k , σ¯ =

σ(pl) cosθp

. (18)

For the limit θp → 0, the averaged cross section is the same as the standard plane-wave one.

B. Mesoscopic target

In this case, we consider a proton target of finite size, uniformly and symmetrically distributed with respect to its center. The spatial distribution of protons inside this target is considered to be classical and can be described the by density functionn(bbb−bbbt). As before, the impact parameterbbbis defined as the distance from the neutron beam axis to a proton in the target, and vectorbbbt corre-sponds to the distance from the beam axis to the center of the target. In this case, differential probability inte-grated over the impact parameters of the finite-size tar-get is the common characteristic of the process, Eq. (19).

Z _d_W˙(m)(_bbb)

dΩ_k n(bbb−bbbt)d

2_b ₍₁₉₎

However, an even more distinct observable is this prob-ability divided by the incoming neutron flux Jz(mesos). Taking into consideration properties of proton density,

FIG. 2: The cross section ratioσ(tw)_(b

t)/σ(pl), Eq. (26), as a function of the impact parameterbt for m = 0 (black solid line),m=2 (red dot-dashed line),m=5 (blue dashed line).

it is reasonable to define twisted neutron flux density along thezaxis as follows

j(ztw)(ρ) = pz

mn̺

(0)₍_ρ_{) =}_v

zJ02(κρ), (20) which behaves nicely in the plane-wave limit

jz(tw)(ρ)|θp→0=

mn =v, (21)

being non-zero at the origin. As the result we define the flux

Jz(mesos)= Z

jz(tw)(ρ)n(ρρρ)d2ρ (22)

and mesoscopic cross section

dσ(mesos)(bbbt)

dΩ_k =

Jz(mesos)

Z _d_W˙(m)(_bbb)

dΩ_k n(bbb−bbbt)d

2_b ₍₂₃₎

Taking into consideration Eq.(16) we get

dσ(mesos)(bbbt)

dΩ_k =

cosθp

dσ(pl)

dΩ_k , R=

Jm2(κb)n(bbb−bbbt)d2b R

J₀2(κb)n(bbb)d2b . (24) In what follows, we use a Gaussian distribution which is a standard choice for ion traps

n(bbb₋bbbt) = 1

2πw2e

−(bbb−₂_wbbbt2)2 ₍₂₅₎

wherew is the dispersion characterized by the size of the proton target. In this case the integration by parts overbcan partially be done analytically

R= 1

I0(κ2w2)e−κ2w2 Z ∞

0 J 2

m(κb)I₀(bb_t/w2)e−(b

2₊_b2

t)/(2w2)bdb

w2.

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Total scattering cross sections are related as follows σ(tw)

σ(pl) =

cosθp. (27)

In Fig. 2 this relationship is plotted as a function of dis-placementbtof the Gaussian target from the beam axis for different values of the quantum numberm. The tar-get size is taken to bew = 1/κand the conical angle -θp = 1o. When comparing this plot to Fig. 1, one can clearly see, thatσ(tw)(bt)is very sensitive to the density distribution of the incoming twisted neutrons. Hence, this reaction may appear to be very useful in diagnos-tics of the twisted neutron beams.

IV. NUCLEAR PHOTO-ABSORPTION A. Deuteron Photodisintegration with Plane-Wave

Photons

In this section we will mostly follow the description of the nuclear electromagnetic interactions in [11] and [12]. As is well known, the space-time solution of the EM wave equation in a Cartesian coordinate system deliv-ers a plane-wave vector potential in the following form:

AA_kkk(pl_λ)(r,t) =ε

kkkλe−i(ωt−kkk·rrr) (28)

which describes the propagation of a photon with fre-quencyω and wave-vectorkkkcarrying polarizationε_kkk_λ,

such that λ = ±1 is the helicity index. Starting from here we will use the natural units: ¯h=c=1. Applying the field quantization conditions we obtain the vector potential operator:

AA_kkk(pl)_λ (rrr,t) = 1 2π2

∑

λ=±1 Z _dkkk

√

2ωk (ε∗

kkkλaˆ†λ(kkk)e−i(kkk·rrr−ωt)+c.c)

(29)

where ˆaλ(kkk)and ˆa†_λ(kkk)are the photon annihilation and creation operators obeying the commutation relation

[aˆλ(kkk), ˆaλ†′(kkk′′′)] =

δ(kkk₋kkk′)

kk′ δλλ′ (30)

so that the corresponding polarization states are given as

ε_kkk_λ=−√λ

2(ex+iλey) (31) The choice of the polarization basis is purely conven-tional and has been made this way to simplify the for-malism.

The interaction of the nucleus with the electromag-netic field up to leading order in charge is described by the Hamiltonian:

Hint=−

drrrjjjˆ(rrr)·AAAˆ(pl)_kkk_λ (rrr,t) (32)

where ˆjjj(rrr)is the nuclear EM current ˆ

jjj(rrr) =ˆjjj_c+ˆjjj_m (33)

consisting of two contributions. The convection current

jjj_c=

∑

i 1

4e(1+τ3(i))(vvvˆiδ(rrr−rrri) +δ(rrr−rrri)vvvˆi) (34) comes from the relative motion of protons, while the magnetization current comes from the spin orientation of the nucleons in the nucleus

jjj_m= e

4mp

∑

_i 1

2[(gp+gn) +τ3(i)(gp−gn)]· ∇ ×[σiδ(rrr−rrri)] (35) where the indexi counts the nucleons; σi is the set of spin Pauli matrices;τττis the isospin operator in SU(2);e

andmpare proton charge and mass; andgp, andgn are proton and neutron spin g-factors respectively.

Making use of Fermi Golden Rule, we can calculate the nuclear photo-absorption probability

dW(Niω→Nf) =2πk2|Mf i|2dΩ (36)

where dΩ is a solid angle of one of the final nucleon states, dΩ_n _or dΩ_p_. _{The key element is the}

photo-absorption amplitude

M(pl)_jm =hJfMf|Hint|JiMi;kkkλi (37)

withjandmindicating the quantum numbers of the to-tal angular momentum of the system and its projection on the quantization axis correspondingly. If one consid-ers the interaction Hamiltonian (32) and applies it to the amplitude the resulting expression

M(pl)_jm =−

s 1 4π2_ω

JfMf|

drrrˆjjj(rrr)·ε_kkk_λe−i(kkk·rrr−ωt)|J_iM_ii

(38) where the commutation relation (30) has been used to annihilate the incoming photon state. The incoming plane wave photon has a well-defined quantization axis, which makes it only logical to reduce the complexity of the problem by making a choice of the preferred direc-tion such thatkkkis parallel to the axisz. This allows us to make the following simplifications [13]:

eikkk·rrr→eikz=

∑

ℓ

iℓ

4π(2ℓ₊₁₎_jℓ(kr)Yℓ₀(Ω_k) (39)

Rewriting the product of the polarization state with the exponential factor and making use of the vector spheri-cal harmonicsYm

jℓ₁(Ωk)we get

ε_k_λeikkk·rrr→ε_k_λeikz =

∑

ℓ

iℓq4π(2ℓ₊₁₎

jℓ(kr)hℓ01λ|jℓiYλ_jℓ₁(Ωk)

(5)

where the symmetry properties of the Glebsch-Gordan coefficients have been used to impose the quantization conditionm=λ. That means that the corresponding ab-sorption amplitude (38) is independent of the magnetic quantum number m, but depends on its component ℓ instead:

M(pl)_jm (0)→M(pl)_j_λ (0) =−

s 1 4π2_ω

∑

ℓ

iℓq4π(2ℓ₊₁₎

hℓ_01λ_|_j_λ_i_e−iωkt

drrrhJfMf|j(kr)(Yλjℓ₁(Ωk)·ˆjjj(rrr))|JiMii (41) where zero in the argument M(pl)_j_λ (0)indicates that the incoming photon is directed in solid angle θk = φk = 0 with respect to z. If we now apply Wigner - Eckart theorem to this matrix element, we get

hJfMf|Mjλ(0)|JiMii=hJiMi jλ|JfMfihJf||Mjλ(0)||Jii (42) This results into the set of selection rules:

|Ji−Jf| ≤ |j| ≤Ji+Jf;

Mf−Mi=λ

(43)

which tells us that the full projection of the incident pho-ton spin in the direction of propagation gets absorbed by the nucleus.

This angle can be decomposed into opening angleθkand azimuthal angleφk to the direction of quantization axis

z. However, instead of rotating the photon state, which would result into major mathematical complications, we will rotate the nuclear states by the same angles [13], such that the initial and final states of the nucleus are expressed

|JiMii=ψJ_iM_i(rrr′) =

∑

M′_i

DJi∗

MiMi′(

Ω_k₎_ψ_J

iMi(rrr);

|JfMfi=ψJfMf(rrr′) =

∑

M′

DJ_Mf∗

fM′f(

Ω_k)ψJfMf(rrr).

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This spatial transformation leads to the following ex-pression for the amplitude

M(pl)_j_λ (Ω_k) =−

∑

M′

fM′i

DJi∗

MiM′i(

Ω_k)D_MJf

fM′f(

Ω_k)M(pl)_j_λ (0). (45) It is worth mentioning that the isospin selection rules are not discussed here since they are not modified by the topological structure of the beam. We also assumed that parity is conserved, leaving aside the weak parity violation effects.

B. Deuteron Photodisintegration with twisted photons In this subsection we will consider a deuteron inter-acting with a twisted beam of photons with well-defined TAMmγ, in particular, BB and BG. In scalar formulation, Bessel mode is the exact solution of the wave equation in cylindrical coordinates (non-paraxial regime) [14]. BG mode [15] belongs to the Gaussian family solutions of the paraxial wave equation (paraxial regime|κ| ≪ |kkk|).

Following the Bessel mode formalism, laid out in Sec. II, one can express the twisted photon state as a 2D Fourier transform

|κmγk_zλi=e−i(ωt−kzz) Z

aκm_γ(kkk⊥)e

ikkk_⊥·ρρρ d2k⊥ (2π)2 (46) with the Fourier kernel (5) in case of BB. The vector po-tential of this state can be written as

A A

A(BB) =e−i(ωt−kzz)

aκm_γ(kkk⊥)ε_kkk_λeikkk⊥·ρρρ

d2k_⊥

(2π)2 (47) which, making use of (28) and (29), can be recast into

AA(BB)=

Z _d2_k ⊥

(2π)2aκm_γ(kkk_⊥)AAA (pl)

kkkλ (rrr,t) (48)

Here the superscript BB refers to the Bessel Beam of inci-dent photons. If one takes into consideration the impact parameter, we get

AAA(BB) =e−i(ωt−kzz)

Z _d_φ k 2π(−i)

mγ_eimγφkεµ kkkλe

ikkk_⊥_·(ρρρ₋bbb)

(49) whereb=|bbb|is a photon impact parameter. As long as the matrix element (37) is linear in electromagnetic field, this formula is valid.

Hence, the photo-absorption amplitude for the case of twisted photons, similar to the photo-absorption on a hydrogen-like atom [16], becomes

M(BB)_j_λ =−(−i)2mγ−Mf+Mi_J

mγ−Mf+Mi(κb)

ei(mγ−Mf+Mi)φb

_∑

M′_iM′_f

dJi

MiMi′(θk)d

J_f

MfM′f(θk)M

(pl)

jλ (0)

(50)

where we have taken advantage Wigner d-function propertyD_MMJ _′(Ω_k_{) =}e−iMφk_dJ

MM′(θk). Here the

factor-ization, which is spin-dependent, as opposed to the case of photo-absorption on hydrogen-like atom, ex. [17], is obtained for the case of photo-disintegration of deu-terium.

As it is traditionally discussed elsewhere, ex. [18, 19], the dominant contribution into the deuteron photo-disintegration comes from the transition 3_S

1 → 1S0 driven by M1-photons. Taking into consideration quantum selection rules (43), only one type of stan-dard plane-wave amplitudes is contributing in this case, namely M(_j_λpl)(mi → mf) = M(

pl)

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-10 -5 0 5 10b/λ 0.1

0.2 0.3 0.4 0.5Mjλ

(tw) mγ=

-10 -5 0 5 10b/λ 0.1

0.2 0.3 0.4 0.5Mjλ

(tw) mγ=

-10 -5 0 5 10b/λ 0.1

0.2 0.3 0.4 0.5Mjλ

(tw) mγ=

-10 -5 0 5 10b/λ 0.1

0.2 0.3 0.4 0.5Mjλ

(tw) mγ=

FIG. 3: Relative photo-absorption amplitudesM(tw)_j_λ forM1→1 _S

0transition for BB (blue), Eq. (50); and for BG (red), Eq. (53)

as functions of the impact parameter scaled with the photon wavelength. The pith angles considered areθk = 0.1 rad - top row;θk = 0.05 rad - bottom row; and waist w0/λ = 6.0. Dashed lines correspond to the caseMf −Mi = 1, dotted lines to Mf−Mi=0, and solid lines toMf−Mi=−1.

which is solely due to magnetic dipole contribution:

M(_j_λpl)(−λ → 0) = M1/√2 in our normalization. As a consequence the M1 contribution in the reaction tran-sition rate factorizes out

r(bbb) =2|M1|2j(γ)(bbb) (51)

making it proportional to the flux of the incoming gamma-radiation

j(γ)(bbb) =cos(θk) κω2

2π

J_m2_γ₊_λ(κb)|d1₁₋₁(θk)|2+ +J_m2_γ₋_λ(κb)|d1₋₁₋₁(θk)|2+Jm2γ(θkb)|d

0−1(θk)|2

(52)

The corresponding amplitude for the twisted photon beam with Gaussian modulation (BG mode), as before (14), is expressed in (53).

M(BG)_j_λ =−Nw2₀(−i)Mi−Mf_e−

κ2w2

4 ei(mγ−Mf+Mi)φb

∑

M′_fM′_i

dJ_Mf

fM′f(θk)d

MiMi′(θk)M

(pl)

jλ (0)

Z ∞

k_⊥dk_⊥

2π e −k2⊥w20

4 I_m

κw2 0 2 k⊥

Jmγ−Mf+Mi(k⊥b)

(53)

whereIm(z)is a modified Bessel function and the super-script (BG) stands for a Bessel-Gaussian mode, as previ-ously in (14). The integral over transverse momentum can be taken analytically, ex. [20] (6.633 1). It also can be simplified for large w0such that

M(BG)_j_λ =−N_πe−b2/w20(−_i)Mi−Mf_J

mγ−Mf+Mi(κb)

ei(mγ−Mf+Mi)φb

∑

M′_fM′_i

dJ_Mf

fM′f(θk)d

MiMi′(θk)M

(pl)

jλ (0)

(54)

Two distinctive coefficients, the Bessel function and the Wigner d-function, Eqs. (50) and (53), are the sig-natures of the topological structure of the incoming photon. As it is in the case for photo-absorption on hydrogen-like atoms, ex. [21], these cause modification of the nuclear photo-absorption selection rules at the beam axis

Mf−Mi=mγ (55)

as enforced by Jmγ+mi−mf(κb). The plane-wave

selec-tion rules (43) are dominating on the beam periphery. This means that for the nucleus at the beam axis to ab-sorb a photon, the entire value of TAM projection on the

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1 2 3 4 5 6 bt/λ 0.2

0.4 0.6 0.8 1.0 1.2 1.4σ

(tw)

FIG. 4: Total cross section in arbitrary units, as a function of the position of the target centerbt for the3S1→ 1S0process,

ex-cited by twisted photon beam. Dashed-blue line corresponds to (mγ=0,λ=1), and solid-red line to (mγ=0,λ=1). Pitch angle isθk=0.1 rad.

z-direction should get transferred to the nuclear degrees of freedom.

In Fig. 3 we plotted the relative photo-absorption am-plitudes for the dominating transition of M1 going into 1_S

0finite state for the reactionγ+d→np. We have con-sidered initial photon states with total angular momen-tummγ = 0, 2 and positive helicityλ = 1. The super-script (tw) refers to a twisted incoming beam of photons (BB or BG). We get three possible transitions correspond-ing to three projections Mi = −1, 0, 1 of initial TAM. As one can see for the non-paraxial Bessel mode and its paraxial modification - BG mode, the amplitudes be-have identically near the beam center. On the periphery

b/λ&2, BG amplitudes get suppressed by the Gaussian factor, which fixes the unphysical infinite energy prob-lem, present in BB. We emphasize that the details of the initial wave function of the deuterium did not affect the derived selection rules for twisted photo-absorption.

The results similar to Sec. III for mesoscopic target, with the limiting case ofwbeing much bigger than the probe’s wavelength, can be worked out for the case of deuteron photodisintegration. We used Gaussian spa-tial distribution (25) to calculate the density-matrix of the excited nucleons, in analogy to [22]. This leads to the equation for the mesoscopic scattering cross section in the form

σ(BB)(b) = 2πδ(ω+Ei−Ef)

J(BB)₍₂_J

i+1) m

∑

fmi

M(pl)_j_λ (θk)

J_m2_γ₋_M_f₊_M_i(κb)I0 bb_t

e −b2+b2_t

2w2 bdb

(56)

for the case of Bessel photons (50), whereJ(BB)is the flux of the twisted photon beam incident on the nuclear tar-get, which can be calculated by substitution of Eq.(52) into Eq.(22). For the case of BG, the approximated from

Eq.(54) can be used in the similar way

σ(BG)(b) = 2Nπδ(ω+Ei−Ef) π2_J(BG)₍₂_J

i+1) m

∑

_fmi

M(pl)_j_λ (θk)

J_m2_γ₋_M

f+Mi(κb)I0

bb_t

e −3b2+b2_t

2w2 bdb

(57)

In analogy to the analysis in Sec. III the cross section is sensitive to the topology of the incoming photon beam, see Fig.4, which can be used as a diagnostic tool. How-ever, photon wavelength for energy range of interest (& 2.2 MeV) is orders of magnitude smaller than a typ-ical mesoscopic target, in contrast to the angstrom-scale de Broglie wavelength of thermal neutron beams.

V. SUMMARY AND OUTLOOK

We analyzed theoretically the transition amplitudes and cross sections of two nuclear reactions: Radiative capture of neutrons by protons, n+p → γ+ d and its time-reversed analogue, photodisintegration of the deuteron, γ+d _→ n+p. The relations between the twisted-wave and plane-wave amplitudes are derived. The results justify dedicated measurements of the nu-clear reactions with the beams of twisted particles that bring additional angular momentum into the the dy-namics of nuclear interactions.

Acknowledgements

The authors are grateful to V.F. Dmitriev, C. Clark and D. Pushin for useful discussions. V.G.S. is supported by the Russian Foundation for Basic Research via Grant No. 16-02-0644. Work of A.A. and M.S. was supported in part by Gus Weiss Endowment of George Washington University.

References

[1] G. Rupak, “Precision calculation of np→dγcross section for big-bang nucleosynthesis,”Nuclear Physics A, vol. 678, no. 4, pp. 405–423, 2000.

[2] C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital an-gular momentum,”Nature, vol. 525, no. 7570, pp. 504–506, 2015.

[3] U. Jentschura and V. Serbo, “Generation of high-energy photons with large orbital angular momentum by comp-ton backscattering,”Physical Review Letters, vol. 106, no. 1, p. 013001, 2011.

[4] U. Jentschura and V. Serbo, “Compton upconversion of twisted photons: backscattering of particles with non-planar wave functions,”The European Physical Journal C, vol. 71, no. 3, pp. 1–13, 2011.

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[5] M. Katoh, M. Fujimoto, H. Kawaguchi, K. Tsuchiya, K. Ohmi, T. Kaneyasu, Y. Taira, M. Hosaka, A. Mochi-hashi, and Y. Takashima, “Angular Momentum of Twisted Radiation from an Electron in Spiral Motion,”

Physical Review Letters, vol. 118, p. 094801, Mar. 2017. [6] V. Petrillo, G. Dattoli, I. Drebot, and F. Nguyen,

“Comp-ton scattered x-gamma rays with orbital momentum,”

Physical Review Letters, vol. 117, no. 12, p. 123903, 2016. [7] C. Liu, B. Shen, X. Zhang, Y. Shi, L. Ji, W. Wang, L. Yi,

L. Zhang, T. Xu, Z. Pei, and Z. Xu, “Generation of gamma-ray beam with orbital angular momentum in the QED regime,”Physics of Plasmas, vol. 23, p. 093120, Sept. 2016. [8] Y. Taira, T. Hayakawa, and M. Katoh, “Gamma-ray

vor-tices from nonlinear inverse Thomson scattering of circu-larly polarized light,”Scientific Reports, vol. 7, p. 5018, July 2017.

[9] T. Doster and A. T. Watnik, “Laguerre–gauss and bessel– gauss beams propagation through turbulence: analysis of channel efficiency,” Applied Optics, vol. 55, no. 36, pp. 10239–10246, 2016.

[10] V. Serbo, I. Ivanov, S. Fritzsche, D. Seipt, and A. Surzhykov, “Scattering of twisted relativistic electrons by atoms,” Physical Review A, vol. 92, no. 1, p. 012705, 2015.

[11] A. DeShalit and H. Feshbach,Theoretical nuclear physics. Volume I. Nuclear structure. John Wiley and Sons, Inc., New York, 1974.

[12] J. D. Walecka, Theoretical nuclear and subnuclear physics. World Scientific Publishing Co Inc, 2004.

[13] D. A. Varshalovich, A. Moskalev, and V. Khersonskii,

Quantum theory of angular momentum. World Scientific,

1988.

[14] J. Durnin, J. Miceli Jr, and J. Eberly, “Diffraction-free beams,” Physical Review Letters, vol. 58, no. 15, p. 1499, 1987.

[15] F. Gori, G. Guattari, and C. Padovani, “Bessel-gauss beams,”Optics communications, vol. 64, no. 6, pp. 491–495, 1987.

[16] H. Scholz-Marggraf, S. Fritzsche, V. Serbo, A. Afanasev, and A. Surzhykov, “Absorption of twisted light by hydro-genlike atoms,”Physical Review A, vol. 90, no. 1, p. 013425, 2014.

[17] A. Afanasev, C. E. Carlson, and A. Mukherjee, “High-multipole excitations of hydrogen-like atoms by twisted photons near a phase singularity,” Journal of Optics, vol. 18, no. 7, p. 074013, 2016.

[18] E. Fermi, “On the recombination of neutrons and pro-tons,”Physical Review, vol. 48, no. 6, p. 570, 1935.

[19] A. Akhiezer and I. Y. Pomeranchuk, “Some problems of the nuclear theory,”Moscow, Leningrad, Gostekhizdat, 1950 2-nd edition, 416 pp.,(Russian Title: Nekotoryie voprosy teorii yadra), 1950.

[20] I. S. Gradshteyn and I. M. Ryzhik,Table of integrals, series, and products. Academic press, 2014.

[21] A. Afanasev, C. E. Carlson, and A. Mukherjee, “Off-axis excitation of hydrogenlike atoms by twisted photons,”

Physical Review A, vol. 88, no. 3, p. 033841, 2013.

[22] A. Peshkov, V. Serbo, S. Fritzsche, and A. Surzhykov, “Ab-sorption of twisted light by a mesoscopic atomic target,”

(1)

twisted neutron dW˙ (m)(bbb)/dΩ_k _{and for a plane wave} dW˙(pl)(bbb)/dΩ_k_{are related by the following expression}

dW˙(m)(bbb) dΩ_k =

dW˙(pl)(bbb) dΩ_k J

m(κb) (16)

III. np→dγREACTION WITH COLD TWISTED NEUTRONS

A. Macroscopic target

dσ¯ dΩ_k =

1 cosθp

Z 2π 0

dσ(pl) dΩ_k

dφp

2π, (17)

where θp is the conical angle of the incoming twisted neutron. The standard scattering cross section is inde-pendent ofφpin our case, which leads to

dσ¯ dΩ_k =

1 cosθp

dσ(pl) dΩ_k , σ¯ =

σ(pl) cosθp

. (18)

For the limit θp → 0, the averaged cross section is the same as the standard plane-wave one.

B. Mesoscopic target

In this case, we consider a proton target of finite size, uniformly and symmetrically distributed with respect to its center. The spatial distribution of protons inside this target is considered to be classical and can be described the by density functionn(bbb−bbbt). As before, the impact parameterbbbis defined as the distance from the neutron beam axis to a proton in the target, and vectorbbbt corre-sponds to the distance from the beam axis to the center of the target. In this case, differential probability inte-grated over the impact parameters of the finite-size tar-get is the common characteristic of the process, Eq. (19).

Z _d_W˙(m)(_bbb)

dΩ_k n(bbb−bbbt)d

2_b ₍₁₉₎

However, an even more distinct observable is this prob-ability divided by the incoming neutron flux Jz(mesos). Taking into consideration properties of proton density,

FIG. 2: The cross section ratioσ(tw)_(b

t)/σ(pl), Eq. (26), as a function of the impact parameterbt for m = 0 (black solid line),m=2 (red dot-dashed line),m=5 (blue dashed line).

it is reasonable to define twisted neutron flux density along thezaxis as follows

j(tw)z (ρ) = pz mn̺

(0)_{(ρ) =}_v

zJ02(κρ), (20) which behaves nicely in the plane-wave limit

jz(tw)(ρ)|θp→0=

mn =v, (21)

being non-zero at the origin. As the result we define the flux

Jz(mesos)=

jz(tw)(ρ)n(ρρρ)d2ρ (22) and mesoscopic cross section

dσ(mesos)(bbbt)

dΩ_k =

1 Jz(mesos)

Z _d_W˙(m)(_bbb)

dΩ_k n(bbb−bbbt)d 2_b ₍₂₃₎

Taking into consideration Eq.(16) we get dσ(mesos)(bbbt)

dΩ_k =

R cosθp

dσ(pl) dΩ_k , R=

Jm2(κb)n(bbb−bbb_t)d2b

J₀2(κb)n(bbb)d2b

. (24) In what follows, we use a Gaussian distribution which is a standard choice for ion traps

n(bbb₋bbbt) = 1 2πw2e

−(bbb−₂_wbbbt2)2 ₍₂₅₎

wherew is the dispersion characterized by the size of the proton target. In this case the integration by parts overbcan partially be done analytically

R= 1

I0(κ2w2)e−κ2w2 Z ∞

0 J 2

m(κb)I₀(bbt/w2)e−(b 2₊_b2

t)/(2w2)bdb

w2.

(2)

Total scattering cross sections are related as follows

σ(tw) σ(pl) =

cosθp. (27)

In Fig. 2 this relationship is plotted as a function of dis-placementbtof the Gaussian target from the beam axis for different values of the quantum numberm. The tar-get size is taken to bew = 1/κand the conical angle -θp = 1o. When comparing this plot to Fig. 1, one can clearly see, thatσ(tw)(bt)is very sensitive to the density distribution of the incoming twisted neutrons. Hence, this reaction may appear to be very useful in diagnos-tics of the twisted neutron beams.

IV. NUCLEAR PHOTO-ABSORPTION A. Deuteron Photodisintegration with Plane-Wave

Photons

AA_kkk(pl)_λ (r,t) =ε

kkkλe−i(ωt−kkk·rrr) (28)

which describes the propagation of a photon with fre-quencyω and wave-vectorkkkcarrying polarizationε_kkk_λ, such that λ = ±1 is the helicity index. Starting from here we will use the natural units: ¯h=c=1. Applying the field quantization conditions we obtain the vector potential operator:

ˆ A

AA_kkk(pl)_λ (rrr,t) = 1

2π2

∑

λ=±1 Z _dkkk

√

2ωk

(ε∗

kkkλaˆ†λ(kkk)e−i(kkk·rrr−ωt)+c.c)

(29)

where ˆaλ(kkk)and ˆa†_λ(kkk)are the photon annihilation and creation operators obeying the commutation relation

[aˆλ(kkk), ˆa†_λ′(kkk′′′)] = δ(

kkk₋kkk′)

kk′ δλλ′ (30) so that the corresponding polarization states are given as

ε_kkk_λ=−√λ

2(ex+iλey) (31) The choice of the polarization basis is purely conven-tional and has been made this way to simplify the for-malism.

The interaction of the nucleus with the electromag-netic field up to leading order in charge is described by the Hamiltonian:

ˆ Hint=−

drrrjjjˆ(rrr)·AAAˆ(pl)_kkk_λ (rrr,t) (32)

where ˆjjj(rrr)is the nuclear EM current ˆ

jjj(rrr) =ˆjjj_c+ˆjjj_m (33) consisting of two contributions. The convection current

ˆ jjj_c=

∑

i 1

4e(1+τ3(i))(vvvˆiδ(rrr−rrri) +δ(rrr−rrri)vvvˆi) (34) comes from the relative motion of protons, while the magnetization current comes from the spin orientation of the nucleons in the nucleus

ˆ jjj_m= e

4mp

∑

_i 1

2[(gp+gn) +τ3(i)(gp−gn)]· ∇ ×[σiδ(rrr−rrri)] (35) where the indexi counts the nucleons; σi is the set of spin Pauli matrices;τττis the isospin operator in SU(2);e andmpare proton charge and mass; andgp, andgn are proton and neutron spin g-factors respectively.

Making use of Fermi Golden Rule, we can calculate the nuclear photo-absorption probability

dW(Niω→Nf) =2πk2|Mf i|2dΩ (36) where dΩ is a solid angle of one of the final nucleon states, dΩ_n _or dΩ_p. _{The key element is the} photo-absorption amplitude

M(pl)_jm =hJfMf|Hint|JiMi;kkkλi (37) withjandmindicating the quantum numbers of the to-tal angular momentum of the system and its projection on the quantization axis correspondingly. If one consid-ers the interaction Hamiltonian (32) and applies it to the amplitude the resulting expression

M(pl)_jm =−

s 1 4π2_ω

kh JfMf|

drrrˆjjj(rrr)·ε_kkk_λe−i(kkk·rrr−ωt)|J_iM_ii (38) where the commutation relation (30) has been used to annihilate the incoming photon state. The incoming plane wave photon has a well-defined quantization axis, which makes it only logical to reduce the complexity of the problem by making a choice of the preferred direc-tion such thatkkkis parallel to the axisz. This allows us to make the following simplifications [13]:

eikkk·rrr→eikz=

∑

ℓ

iℓ q

4π(2ℓ₊₁₎_jℓ(kr)Yℓ₀(Ω_k) (39) Rewriting the product of the polarization state with the exponential factor and making use of the vector spheri-cal harmonicsYm

jℓ₁(Ωk)we get

ε_k_λeikkk·rrr→ε_k_λeikz =

∑

ℓ

iℓq4π(2ℓ₊₁₎

jℓ(kr)hℓ01λ|jℓiYλ_jℓ₁(Ωk)

(3)

where the symmetry properties of the Glebsch-Gordan coefficients have been used to impose the quantization conditionm=λ. That means that the corresponding ab-sorption amplitude (38) is independent of the magnetic quantum number m, but depends on its component ℓ

instead:

M(pl)_jm (0)→M(pl)_j_λ (0) =−

s 1 4π2_ω

∑

ℓ

iℓq4π(2ℓ₊₁₎ hℓ₀₁_λ_|_j_λ_i_e−iωkt

hJfMf|Mjλ(0)|JiMii=hJiMi jλ|JfMfihJf||Mjλ(0)||Jii (42) This results into the set of selection rules:

|Ji−Jf| ≤ |j| ≤Ji+Jf; Mf−Mi=λ

(43) which tells us that the full projection of the incident pho-ton spin in the direction of propagation gets absorbed by the nucleus.

As our ultimate goal is to apply this formalism for the case of the twisted light, we need one more modification related to the topological structure of the beams with OAM. Any twisted beam can be Fourier transformed, hence, expressed as the ensemble of plane waves with the wave-vectors directed under certain solid angleΩ_k. This angle can be decomposed into opening angleθkand azimuthal angleφk to the direction of quantization axis z. However, instead of rotating the photon state, which would result into major mathematical complications, we will rotate the nuclear states by the same angles [13], such that the initial and final states of the nucleus are expressed

|JiMii=ψJ_iM_i(rrr′) =

∑

M′_i

DJi∗

MiMi′(

Ω_k_)ψ_J

iMi(rrr); |JfMfi=ψJfMf(rrr′) =

∑

M′ f

DJ_Mf∗

fM′f(

Ω_k_)ψ_J

fMf(rrr).

(44)

This spatial transformation leads to the following ex-pression for the amplitude

M(pl)_j_λ (Ω_k) =−

∑

M′

fM′i

DJi∗

MiM′i(

Ω_k)D_MJf

fM′f(

Bessel mode is the exact solution of the wave equation in cylindrical coordinates (non-paraxial regime) [14]. BG mode [15] belongs to the Gaussian family solutions of the paraxial wave equation (paraxial regime|κ| ≪ |kkk|).

Following the Bessel mode formalism, laid out in Sec. II, one can express the twisted photon state as a 2D Fourier transform

|κm_γk_zλi=e−i(ωt−kzz) Z

aκm_γ(kkk⊥)e

ikkk_⊥·ρρρ d2k⊥ (2π)2 (46) with the Fourier kernel (5) in case of BB. The vector po-tential of this state can be written as

A A

A(BB) =e−i(ωt−kzz)

aκmγ(kkk⊥)ε_kkk_λeikkk⊥·ρρρ d2k_⊥

(2π)2 (47) which, making use of (28) and (29), can be recast into

A AA(BB)=

Z _d2_k ⊥

(2π)2aκm_γ(kkk_⊥)AAA (pl)

kkkλ (rrr,t) (48)

Here the superscript BB refers to the Bessel Beam of inci-dent photons. If one takes into consideration the impact parameter, we get

AAA(BB) =e−i(ωt−kzz)

Z _d_φ

k 2π(−i)

mγ_eimγφkεµ kkkλe

ikkk_⊥_·(ρρρ₋bbb) (49) whereb=|bbb|is a photon impact parameter. As long as the matrix element (37) is linear in electromagnetic field, this formula is valid.

Hence, the photo-absorption amplitude for the case of twisted photons, similar to the photo-absorption on a hydrogen-like atom [16], becomes

M(BB)_j_λ =−(−i)2mγ−Mf+Mi_J

mγ−Mf+Mi(κb)

ei(mγ−Mf+Mi)φb

_∑

M′_iM′_f dJi

MiMi′(θk)d

J_f

MfM′f(θk)M

(pl) jλ (0)

(50)

where we have taken advantage Wigner d-function propertyD_MMJ _′(Ω_k_{) =}e−iMφk_dJ

MM′(θk). Here the factor-ization, which is spin-dependent, as opposed to the case of photo-absorption on hydrogen-like atom, ex. [17], is obtained for the case of photo-disintegration of deu-terium.

As it is traditionally discussed elsewhere, ex. [18, 19], the dominant contribution into the deuteron photo-disintegration comes from the transition 3_S

1 → 1S0 driven by M1-photons. Taking into consideration quantum selection rules (43), only one type of stan-dard plane-wave amplitudes is contributing in this case, namely M(pl)_j_λ (mi → mf) = M(pl)jλ (−λ → 0),

(4)

-10 -5 0 5 10b/λ 0.1

0.2 0.3 0.4 0.5Mjλ

(tw) mγ= 0

-10 -5 0 5 10b/λ 0.1

0.2 0.3 0.4 0.5Mjλ

(tw) mγ= 2

-10 -5 0 5 10b/λ 0.1

0.2 0.3 0.4 0.5Mjλ

(tw) mγ= 0

-10 -5 0 5 10b/λ 0.1

0.2 0.3 0.4 0.5Mjλ

(tw) mγ= 2

FIG. 3: Relative photo-absorption amplitudesM(tw)_j_λ forM1→1 _S

0transition for BB (blue), Eq. (50); and for BG (red), Eq. (53)

Mf−Mi=0, and solid lines toMf−Mi=−1.

which is solely due to magnetic dipole contribution: M(pl)_j_λ (−λ → 0) = M1/√2 in our normalization. As a consequence the M1 contribution in the reaction tran-sition rate factorizes out

r(bbb) =2|M1|2j(γ)(bbb) (51) making it proportional to the flux of the incoming gamma-radiation

j(γ)(bbb) =cos(θk)

κω2

2π

J_mγ2 ₊_λ(κb)|d1₁₋₁(θk)|2+

+J_m2_γ₋_λ(κb)|d1₋₁₋₁(θk)|2+J2mγ(θkb)|d 1

0−1(θk)|2 (52) The corresponding amplitude for the twisted photon beam with Gaussian modulation (BG mode), as before (14), is expressed in (53).

M(BG)_j_λ =−Nw2₀(−i)Mi−Mf_e−

κ2w2

4 ei(mγ−Mf+Mi)φb

∑

M′_fM′_i

dJ_Mf

fM′f(θk)d

MiMi′(θk)M

(pl) jλ (0) Z ∞

k_⊥dk_⊥ 2π e

−k2⊥w20

4 I_m

γ κw2

0 2 k⊥

Jmγ−Mf+Mi(k⊥b)

(53)

M(BG)_j_λ =−N_πe−b2/w20(−_i)Mi−Mf_J

mγ−Mf+Mi(κb)

ei(mγ−Mf+Mi)φb

∑

M′_fM′_i dJ_Mf

fM′f(θk)d

MiMi′(θk)M

(pl) jλ (0)

(54)

Mf−Mi=mγ (55)

as enforced by Jmγ+mi−mf(κb). The plane-wave

selec-tion rules (43) are dominating on the beam periphery. This means that for the nucleus at the beam axis to ab-sorb a photon, the entire value of TAM projection on the

(5)

1 2 3 4 5 6 bt/λ 0.2

0.4 0.6 0.8 1.0 1.2 1.4σ

(tw)

FIG. 4: Total cross section in arbitrary units, as a function of the position of the target centerbt for the3S1→ 1S0process,

ex-cited by twisted photon beam. Dashed-blue line corresponds to (mγ=0,λ=1), and solid-red line to (mγ=0,λ=1). Pitch

angle isθk=0.1 rad.

z-direction should get transferred to the nuclear degrees of freedom.

In Fig. 3 we plotted the relative photo-absorption am-plitudes for the dominating transition of M1 going into 1_S

0finite state for the reactionγ+d→np. We have con-sidered initial photon states with total angular momen-tummγ = 0, 2 and positive helicityλ = 1. The

super-script (tw) refers to a twisted incoming beam of photons (BB or BG). We get three possible transitions correspond-ing to three projections Mi = −1, 0, 1 of initial TAM. As one can see for the non-paraxial Bessel mode and its paraxial modification - BG mode, the amplitudes be-have identically near the beam center. On the periphery

b/λ&2, BG amplitudes get suppressed by the Gaussian

factor, which fixes the unphysical infinite energy prob-lem, present in BB. We emphasize that the details of the initial wave function of the deuterium did not affect the derived selection rules for twisted photo-absorption.

σ(BB)(b) = 2πδ(ω+Ei−Ef) J(BB)₍₂_J

i+1) m

∑

fmi

M(pl)_j_λ (θk)

J_mγ2 ₋_M_f₊_M_i(κb)I0bbt w2

−b2+b2_t 2w2 bdb

(56)

Eq.(54) can be used in the similar way

σ(BG)(b) = 2Nπδ(ω+Ei−Ef) π2_J(BG)₍₂_J

i+1) m

∑

_fmi

M(pl)_j_λ (θk)

J_m2_γ₋_M

f+Mi(κb)I0

bb_t w2

−3b2+b2_t

2w2 bdb

(57)

(& 2.2 MeV) is orders of magnitude smaller than a

typ-ical mesoscopic target, in contrast to the angstrom-scale de Broglie wavelength of thermal neutron beams.

V. SUMMARY AND OUTLOOK

Acknowledgements

References

[1] G. Rupak, “Precision calculation of np→dγcross section for big-bang nucleosynthesis,”Nuclear Physics A, vol. 678, no. 4, pp. 405–423, 2000.

[2] C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital an-gular momentum,”Nature, vol. 525, no. 7570, pp. 504–506, 2015.

[3] U. Jentschura and V. Serbo, “Generation of high-energy photons with large orbital angular momentum by comp-ton backscattering,”Physical Review Letters, vol. 106, no. 1, p. 013001, 2011.

(6)

Physical Review Letters, vol. 118, p. 094801, Mar. 2017. [6] V. Petrillo, G. Dattoli, I. Drebot, and F. Nguyen,

“Comp-ton scattered x-gamma rays with orbital momentum,”

Physical Review Letters, vol. 117, no. 12, p. 123903, 2016. [7] C. Liu, B. Shen, X. Zhang, Y. Shi, L. Ji, W. Wang, L. Yi,

vor-tices from nonlinear inverse Thomson scattering of circu-larly polarized light,”Scientific Reports, vol. 7, p. 5018, July 2017.

[10] V. Serbo, I. Ivanov, S. Fritzsche, D. Seipt, and

A. Surzhykov, “Scattering of twisted relativistic electrons by atoms,” Physical Review A, vol. 92, no. 1, p. 012705, 2015.

[11] A. DeShalit and H. Feshbach,Theoretical nuclear physics. Volume I. Nuclear structure. John Wiley and Sons, Inc., New York, 1974.

[12] J. D. Walecka, Theoretical nuclear and subnuclear physics. World Scientific Publishing Co Inc, 2004.

[13] D. A. Varshalovich, A. Moskalev, and V. Khersonskii,

Quantum theory of angular momentum. World Scientific,

1988.

[14] J. Durnin, J. Miceli Jr, and J. Eberly, “Diffraction-free beams,” Physical Review Letters, vol. 58, no. 15, p. 1499, 1987.

[15] F. Gori, G. Guattari, and C. Padovani, “Bessel-gauss beams,”Optics communications, vol. 64, no. 6, pp. 491–495, 1987.

[16] H. Scholz-Marggraf, S. Fritzsche, V. Serbo, A. Afanasev, and A. Surzhykov, “Absorption of twisted light by hydro-genlike atoms,”Physical Review A, vol. 90, no. 1, p. 013425, 2014.

[18] E. Fermi, “On the recombination of neutrons and pro-tons,”Physical Review, vol. 48, no. 6, p. 570, 1935.

[20] I. S. Gradshteyn and I. M. Ryzhik,Table of integrals, series, and products. Academic press, 2014.

[21] A. Afanasev, C. E. Carlson, and A. Mukherjee, “Off-axis excitation of hydrogenlike atoms by twisted photons,”

Physical Review A, vol. 88, no. 3, p. 033841, 2013.

[22] A. Peshkov, V. Serbo, S. Fritzsche, and A. Surzhykov, “Ab-sorption of twisted light by a mesoscopic atomic target,”

Nuclear Theory authors titles recent submissions

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Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

Nuclear Theory authors titles recent submissions

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