Open access peer-reviewed chapter

Electron Oscillation-Based Mono-Color Gamma-Ray Source

By Hai Lin, ChengPu Liu and Chen Wang

Submitted: August 25th 2018Reviewed: November 28th 2018Published: January 3rd 2019

DOI: 10.5772/intechopen.82752

Downloaded: 195

Abstract

Production of artificial gamma-ray source usually is a conception belonging to the category of experimental nuclear physics. Nuclear physicists achieve this goal through utilizing/manipulating nucleons, such as proton and neutron. Low-energy electrons are often taken as “by-products” when preparing these nucleons by ionizing atoms, molecules and solids, and high-energy electrons or β rays are taken as “wastage” generated in nuclear reaction. Utilization of those “by-products” has not won sufficient attention from the nuclear physics community. In this chapter, we point out a potential, valuable utilization of those “by-products.” Based on a universal principle of achieving powerful mono-color radiation source, we propose how to set up an efficient powerful electron-based gamma-ray source through available solid-state components/elements. Larger charge-to-mass ratio of an electron warrants the advantage of electron-based gamma-ray source over its nucleon-based counterpart. Our technique offers a more efficient way of manipulating nuclear matter through its characteristic EM stimulus. It can warrant sufficient dose/brightness/intensity and hence an efficient manipulation of nuclear matter. Especially, the manipulation of a nucleus is not at the cost of destroying many nuclei to generate a desired tool, that is, gamma ray with sufficient intensity, for achieving this goal. This fundamentally warrants a practical manipulation of more nuclei at desirable number.

Keywords

  • gamma-ray source
  • electron oscillation
  • DC fields

1. Introduction

Powerful mono-color gamma-ray source is a very appealing, but also seem-to-be-dream, topic in modern physics. This is because the gamma ray, an electromagnetic (EM) wave with sub-pm-level wavelength λ, is generated from quantum transition of nuclear matter in nuclear reaction. At present, using synthesized radioactive heavy elements, or radioisotope, is the main route for achieving a gamma-ray source [1, 2, 3, 4]. Finite proton number in a radioactive heavy element is the bottleneck affecting the intensity and power of the gamma-ray output. Higher proton number is favorable not only to higher intensity and power but also faster decay or shorter life of the radioactive heavy element. For practical purposes, such a gamma-ray source should be of a stable output over a sufficiently long time duration. Piling large amount of radioactive heavy elements might be a solution to this requirement but its accompanied environment-protection cost might be overly high. Moreover, because a nucleon has smaller charge-to-mass ratio than an electron, the accelerator cost and the reactor cost at the synthesis stage are also of considerable amount even though it is only aimed at low-energy nuclear physics applications rather than high-energy physics applications. To some extent, obtaining a powerful mono-color gamma-ray source corresponds to an artful skill of manipulating nuclear matter.

Therefore, new working principle of achieving radiation source with narrower output spectrum is of significant application value. Based on Takeuchi’s theory [5], we proposed a universal principle of achieving mono-color radiation source at arbitrary wavelength [6, 7]. According to this principle, available parameter values can ensure a powerful mono-color gamma-ray source.

The core of this working principle can be summarized as “tailoring” Takeuchi orbit. Takeuchi’s theory reveals that the orbit of a classical charged particle, such as electron, in a DC field configuration Es×Bs, where Esand Bsare constant, can be elliptical or parabolic according to values of Esand Bsand that of initial particle’s velocity entering into this configuration [5, 8]. The time cycle of an elliptical orbit can be in principle an arbitrary value by choosing suitable values of these parameters. Thus, for a far-field observer on the normal direction of this 2-D orbit, electrons moving along the orbit will behave like a radiation source whose central frequency is the inverse of the time cycle of the orbit. But a realistic factor affecting its practicality is the geometric size of such an orbit. Overly large geometric size will hurt the practicality of such a radiation source. At present, for available values of Esand Bs, about MVolt/meter-level and Tesla-level, the size can be down to m-level for s-level time cycle or Hz-level frequency.

For warranting the practicality of such a radiation source, we propose a scheme for making it compact by “tailoring” Takeuchi orbit through targeted designed DC field configuration [6]. In this configuration, Bsis made space-varying along the direction normal to the unperturbed path of an electron bunch by not letting two Helmholtz coils be co-axial on purpose [6]. By choosing suitable values of related parameters, such as the relative distance between the bunch path and the Bs=0contour, Es-values and the slope β=dxBs, where Bsis along the y-direction, its magnitude Bsis a function of the coordinate x, and the unperturbed path is along the z-axis.

2. Theory and method

2.1 Theoretical basis

For the convenience of readers, we paste related materials published elsewhere [7]. For a simple configuration containing merely static electric field (along x-direction) and static magnetic field (along z-direction), the behavior of an incident electron can be described by dimensionless 3-D relativistic Newton equations (RNEs)

dsΓdsZ=0,E1
dsΓdsY=WBdsXE2
dsΓdsX=WBη+dsYE3

where

1Γ=1dsX2dsY2dsZ2.E4

Moreover, Esand Bsare constant-valued electric and magnetic fields and meet Es=ηcBs; λ=c/ωand ωare reference wavelength and frequency, respectively; and s=ωt,Z=zλ,Y=yλ,X=xλ,WB=ωBω,where ωB=eBsmeis the cyclotron frequency.

Eqs. (1)(3) lead to

dsZ0E5
ΓdsYWBX=const=Cy;E6
ΓdsX+WBηs+Y=const=Cx,E7

where the values of these constants, const, are determined from the initial conditions XYZdsXdsYdsZs=0=0,0,0Cx1+Cx2+Cy2Cy1+Cx2+Cy20.

Eqs. (5)(7) can yield an equation for dsXand dsY

dsY2=Cy+WBX21dsX2dsY2E8
dsX2=CxWBηs+Y21dsX2dsY2E9

whose solution reads

dsX2=CxWBηs+Y21+Cy+WBX2+CxWBηs+Y2E10
dsY2=Cy+WBX21+Cy+WBX2+CxWBηs+Y2.E11

It is easy to verify that the solutions (10, 11) will lead to Γ=1+Cy+WBX2+CxWBηs+Y2and, with the help of Eqs. (6) and (7), dsΓ=WBηdsX(i.e., mec2dtΓ=eEdtX). Noting Γcan be formally expressed as Γ=1+Cy2+Cx2WBηX, which agrees with Takeuchi’s theory [15], we can find that the electronic trajectory can be expressed as

1+Cy2+Cx2WBηX2=1+Cy+WBX2+CxWBηs+Y2,E12

or

1η2X+η+υy01η2Γ0WB2+Y+ηsυx0Γ0WB2=η+υy02+1η2υx021η2Γ0WB2,E13

where Γ0=1+Cy2+Cx2, υx0=CxΓ0and υy0=CyΓ0.

There will be an elliptical trajectory for η<1and a hyperbolic one for η>1[15, 16]. The time cycle for an electron traveling through an elliptical trajectory can be exactly calculated by re-writing Eq. (10) as [15]

±ds=1WBΓ0ηXaX2+bX+cdX=ηaXNXb24ac4a2X+b2a2dX,E14

where a=η21, b=2ηΓ0+Cy1WB,c=Cx21WB2and XN=1η1WBΓ0. The equation can be written as a more general form

±ds=σMu1u2duE15

where u=X+b2ab24ac4a2=XXR+XL2XRXL, XL=minbb24ac2ab+b24ac2aand XR=maxbb24ac2ab+b24ac2a. In addition, σ=ηab24ac4a2and M=XN+b2ab24ac4a2=XNXR+XL2XRXL. It is easy to verify that for η21<0, there is M=1+ηυy0ηη+υy02+1η2υx02>1. Initially, XYs=0=00and hence ust=us=0=0+b2ab24ac4a2=η+υy0η+υy02+1η2υx02. From strict solution

±su=σMarcsinu+1u2+const,E16

we can find the time for an electron traveling through an elliptical trajectory to meet scycle=ωTc=2σMπand hence a time cycle Tc=1+ηυy0Γ01η232πωB.That is, the oscillation along the elliptical trajectory will have a circular frequency ωB. Moreover, it is interesting to note that υx0υy0=0ηwill lead to η+υy02+1η2υx021η2=0and hence a straight-line trajectory XsYs=0ηs.

The motion on an elliptical trajectory is very inhomogeneous. The time for finishing the ηX>0half might be very short while that for the ηX<0half might be very long. We term the two halves as fast-half and slow-half, respectively. If ηis fixed over whole space, a fast-half is always linked with a slow-half and hence makes the time cycle for finishing the whole trajectory being at considerable level.

For convenience, our discussion is based on the parameterized ellipse. For the case υx0=0υy0=ηΔ, (where Δis small-valued and positive), the starting position X=0is the left extreme of the ellipse and hence corresponds to u=1. The time required for an acute-angled rotation from u=1to u=1+ξ, (where ξis small-valued and positive), will be σMarcsin1+ξπ2+σ2ξξ2, which is =0if ξ=0.

It is interesting to note that if there is B=0at the region u>1+ξ, the electron will enter from EB=Eηc-region into EB=0-region with an initial velocity whose x-component is υx1 ̃dsuu=1+ξ=1σ11+ξ2M1+ξ>0and y-component υy1is 0. Then, the electron will enter into the EB=0-region at a distance because υx1>0. After a time Ttr=2υx1E1υx12υy12, the electron will return into the EB=Eηc-region and the returning velocity will have a x-component υx1. During this stage, the electron will move υy1Ttralong the ydirection. Then, the motion in the EB=Eηc-region can be described by an acute-angled rotation along the ellipse u=1+ξu=1. Thus, a complete closed cycle along the xdirection is finished even though the motion along the ydirection is not closed. Repeating this closed cycle will lead to an oscillation along the xdirection.

Clearly, the time cycle of such an oscillation, or that of a “tailored” Takeuchi orbit, is

Tx=Ttr+2σMarcsin1+ξπ2+2σ2ξξ2.E17

Under fixed values of Δ, Eand B, the smaller ξis, the smaller Txis. There will be Tx=0at ξ=0. In principle, arbitrary value of Tx<Tccan be achieved by choosing suitable value of ξ. That is, arbitrarily high center frequency (>ωB) oscillation can be achieved by choosing a suitable value of ξ. Although the time history of xtmight cause its Fourier spectrum to have some spread, the center frequency will be 1Tx.

This result implies a simple and universal method of setting up quasi-mono-color light source at any desirable center wavelength: by applying vertically static electric field E=Exand static magnetic field B=Bzand on purpose letting a B=0region exist and the ratio EcB<1, then injecting electron along the y-axis with a velocity slightly above EcB, and close to the boundary line between the B=0region and the B0region. As shown in Figure 1 of Ref. [7], adjusting the distance D=ξb24ac4a2can lead to a quasi-monocolor oscillation source with any desired center frequency up to gamma-ray level.

Figure 1.

Sketch of the device. The axis of a finite-sized solenoid is along z direction, the space between the solenoid and the conducting wire/specimen is filled with two kinds of magnetic mediums, which are represented by μ high and μ low . The μ low medium can be the vacuum. A pair of electrodes yields a DC electric field along the x direction on the specimen. The dashed line represents the contour plane of B at a given value B 0 . Different μ values cause the contour plane to have different z coordinates in two mediums.

Of course, such a step-like magnetic field profile is overly idealized. Therefore, we propose using a more realistic magnetic slope to achieve such a tailored Takeuchi orbit [6].

2.2 Details on electron source

The above discussions have revealed theoretically the feasibility of an electron oscillation-based gamma-ray source. It is obvious that the electron oscillation-based radiation source is more advantageous than its proton oscillation-based counterpart because of larger oscillation magnitude, as well as power, available in the former. Utilization of electrons receives less attention than that of protons in experimental nuclear physics. It is really a pity if taking electrons as by-products of preparing protons. Reasonably utilizing those “by-products” is worthy of consideration.

Electron source can be designed to be compact and easily prepared. Among familiar electron sources, thermion-emission cathode is limited by its efficiency, and photocathode needs to be driven by high-intensity laser. The simplest method of achieving a high-efficiency electron source can share the same idea as that embodied in above sections, that is, using Hall effect by a magnetic slope in the above-mentioned discussion. Details are presented as follows.

Hall effect of a metal by a static (DC) magnetic field Bsis a familiar phenomenon caused by magnetic field. But until now, it is merely taken as a method of probing physical property of solid-state materials and hence its applications are often limited to weak magnetic field cases, which are usually at 10G-level. Higher strength of Bsneeds stronger strength of current which might be beyond what a conducting wire can sustain and also is far beyond what most magnetic materials can produce [9].

Beside the strength of Bs, the space shape of Bscan also affect its interaction with matter. Such a space shape is described by the contour surface of Bs. For most components, their generated fields are usually of smooth contour surfaces. For example, Bsgenerated by a solenoid has contour planes normal to the axis of the solenoid, or the electromagnetic (EM) energy density profile Bs2is smooth or has a very small gradient Bs2.

If the strength Bsis not easy to be enhanced, adjusting the space shape of Bsis a worthy trial to optimize interaction. A high-gradient Bs2profile is not difficult to be produced. For example, one can on purpose make a pair of Helmholtz coils, which is a well-known device for screening external magnetic field, so that they are not co-axial. Figure 1 displays a simple scheme for producing a high-gradient Bs2profile. As shown in Figure 1 , the intrusion of a high-μ(magnetic permeability) medium distorts contours of Bs2to be bent. In other words, in each plane normal to the axis of the solenoid, a huge gradient of Bs2along the direction normal to the side surface of the medium appears. If a metal wire/specimen is arranged in such a high-Bs2region and an DC electric field Esalong the direction of Bs2is applied, the Hall effect in such a situation where the DC magnetic field is very space-inhomogeneous is worthy of being studied.

When studying applications such as probing and imagining local magnetic moment and magnetic microscopic structure [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], many authors have made in-depth investigation on the Hall effect of semiconductors in highly inhomogeneous magnetic field (HIMF). Because the purpose of these applications is detection or probing, the electric field or bias DC field is designed to avoid the breakdown of the semiconductor and hence its strength is usually not too strong. That is, in applications for detection purpose, Hall current is not required to be large enough.

It is worth noting the potential value of the extension of the same idea to a different case. The purpose of such an extension is aimed at a controllable “breakdown” of the metal. Therefore, higher DC field strength is chosen. Now that Hall effect implies that electrons have the potential to run along a direction normal to the applied electric field, it is natural for us to consider the feasibility of side escape of electrons from a conducting wire through Hall effect. This drives us to actively establish a HIMF and apply it to metal under a higher-strength DC electric field.

Escan achieve 105V/m-level by letting the inter-plate distance of a pair of plane-plate electrodes as 103̃2mand applied voltage as 102̃3V. Parameter values at such a level is not difficult to be realized technically. Because of the condition B=0and the fact that the solenoid is finite-sized, Bshas two components Bxexand Bzez, where Bx=igxix2i+1and Bz=igzix2i+1correspond to a vector potential A=Axex+ Azez=yBzex+ yBxez.

As shown in Figure 1 , the solenoid is arranged on the demarcation line of two magnetic mediums. The end section of the solenoid is taken as the z=0plane, and the metal is arranged at z<zdregion. The x<0zu>z>zdregion is filled with μ=μhighmedium and the x>0zu>z>zdregion with μ=μlowmedium. According to the theory of electromagnetism, the contour plane Bz=B0in the x<0zd>zregion and that in the x>0zd>zregion will have different zcoordinates. This implies a discontinuity in Bexists near the demarcation line. That is, different values of the dropping rate zBin two mediums cause the specimen in the zd>zregion to still feel a gradient xB2. To ensure a sufficiently large gradient, the interface of two mediums is required to be smooth enough and hence should be polished/ground sufficiently. At present, mirror finish grinding can ensure surface roughness to be of Ra<=0.01μm. This fundamentally warrants sufficiently large gradient xμ, as well as sufficiently large xB2up to Tnm-level, to be feasible.

Because the DC magnetic field can effectively penetrate into metal interior if its direction is normal to the surface of a metal (in normal state), it can affect bulk electron states of the metal. In contrast, the AC magnetic field, or a light beam, is limited to the skin layer of the metal [21, 22].

For Al, its electron-phonon collision relaxation time τis at 1012fs-level [21, 22], its Fermi velocity υFis at 106m/s-level and its Fermi energy EFis about 5.5eV[21, 22]. If a cm3-level Alcubic specimen is placed between a pair of electrode plates with 220Vvoltage, the DC electric field Esit feels will be at 104V/m-level. If we merely take into account the work done by Es, the maximum velocity increment along the Esdirection, maxΔυx, can reach eEsτ/me=1.6/9.11019+415+1+3120m/s, which corresponds to 12meυx22009.11031J109eV.

Emission is a many-body process because the sheath field, or space charge effect, left by emitted electrons in turn affects emission [23, 24, 25, 26]. This phenomenon can be reflected by following quantum theory (21, 23–26),

iћtψk=12meiћ+eB0yex2ψk+Uiψk+Vxyψk+eE0xψk+VphxytTψk.E18
2V=e4πε0n01ψk2fkTdkxdkyn0E19

where

Vph=uqrUigkTdqxdqy,E20
uq=sinqxx+qyy+qzzνqtep,E21

n0is the average background ionic density, ψkxyt=expSkψk0, ψk0is the unperturbed wavefunction and fis the Fermi-Dirac distribution function. Vphis the vibrating lattice potential, uis the field of ionic displacement, Uiis the lattice potential at zero-temperature, gis the Bose-Einstein distribution function, νqis the phonon dispersion relation and Tis the temperature. More comprehensive model should contain a motion equation of the displacement field u, which is derived from the Lagrangian density of the electron-phonon system. This will be done in future work. Here, we approximate uas prescribed. Such an approximation is acceptable because the temporal variation of uis merely obvious over a large time scale 2πνqwhich is usually >100fs.

The equation of Skreads

iћtSk=ћ22mexx+yy+zzSk+xSk2+ySk2+zSk2+2ikxxSk+2ikyySk+2ikzzSkћB0meeykzfxkxfz+iћB0meeyfxzfzxSk+e2B02fx2+fz22mey2+eE0x+Vxy+Vph,E22

where the space inhomogeneity of Bsis reflected by f

f=fx0fz=Bx/B00Bz/B0.E23

Note that f=0,0,1corresponds to a space-homogeneous Bs=B0along zdirection.

Actively applying highly space-inhomogeneous external field, especially DC magnetic field, might be an effective way of enhancing the effect of the external field on the electrons. According to Hamiltonian formula or Eq. (22), there is always an operator Ap̂Br. Space-inhomogeneous Bwill cause more space-inhomogeneous wavefunction than space-uniform B. Because the energy of an electron is also dependent on the space derivative of the modulus of its wavefunction, more space-inhomogeneous wavefunction often implies larger energy.

To warrant the technique route to be competitive in economics and efficiency among all candidates for a same goal, we avoid more intermediate conversion steps in EM energy utilization, and favor direct usage of EM energy in power frequency (PF), the most primitive EM energy form for all physics laboratories.

3. Conclusion

The application value of such an electron oscillation-based gamma-ray source is obvious. It offers a more efficient way of manipulating nuclear matter through its characteristic EM stimulus, that is, gamma ray. At present, the goal of manipulating nuclear matter is mainly achieved through: (1) using Bremsstrahlung by proton output from accelerators—this implies the application of an EM stimulus of a broad spectrum to the nucleus, and hence the efficiency of this route is poor because most photons are of low frequency relative to nuclear matter; (2) using EM radiations from heavier radioactive elements—the dose, or the brightness, or the intensity of gamma ray generated in this route is limited and hence the manipulation is also less efficient; (3) injecting protons into target nucleus. In contrast, the electron oscillation-based mono-color gamma-ray source proposed in this work can warrant sufficient dose/brightness/intensity and hence an efficient manipulation of nuclear matter. Especially, the manipulation of a nucleus is not at the cost of destroying many nuclei to generate a desired tool, that is gamma ray with sufficient intensity, for achieving this goal. This fundamentally warrants a practical manipulation of more nuclei at desirable number.

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Hai Lin, ChengPu Liu and Chen Wang (January 3rd 2019). Electron Oscillation-Based Mono-Color Gamma-Ray Source, Use of Gamma Radiation Techniques in Peaceful Applications, Basim A. Almayah, IntechOpen, DOI: 10.5772/intechopen.82752. Available from:

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