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 xdirection) and static magnetic field (along z-direction), the behavior of an incident electron can be described by dimensionless 3-D relativistic Newton equations (RNEs)

$$d\_t[\Gamma d\_\nu Z] = 0,\tag{1}$$

$$d\_{\mathfrak{s}}[\Gamma d\_{\mathfrak{s}} Y] = \mathcal{W}\_B d\_{\mathfrak{s}} X \tag{2}$$

$$d\_s[\Gamma d\_s X] = -\mathcal{W}\_B[\eta + d\_s Y] \tag{3}$$

where

$$\frac{1}{\Gamma} = \sqrt{1 - \left(d\_\circ X\right)^2 - \left(d\_\circ Y\right)^2 - \left(d\_\circ Z\right)^2}.\tag{4}$$

Electron Oscillation-Based Mono-Color Gamma-Ray Source DOI: http://dx.doi.org/10.5772/intechopen.82752

Moreover, Es and Bs are constant-valued electric and magnetic fields and meet Es ¼ ηcBs; λ ¼ c=ω and ω are reference wavelength and frequency, respectively; and <sup>s</sup> <sup>¼</sup> <sup>ω</sup>t, Z <sup>¼</sup> <sup>z</sup> <sup>λ</sup> , Y <sup>¼</sup> <sup>y</sup> <sup>λ</sup> , X <sup>¼</sup> <sup>x</sup> <sup>λ</sup> ,WB <sup>¼</sup> <sup>ω</sup><sup>B</sup> <sup>ω</sup> , where <sup>ω</sup><sup>B</sup> <sup>¼</sup> eBs me is the cyclotron frequency. Eqs. (1)–(3) lead to

$$d\_2 \mathbb{Z} \equiv \mathbf{0} \tag{5}$$

$$
\Gamma d\_s Y - W\_B X = \text{const} = \mathcal{C}\_{\mathcal{V}}; \tag{6}
$$

$$
\Gamma d\_\text{s} X + W\_B[\eta \mathfrak{s} + Y] = \text{const} = \mathbb{C}\_\mathbf{x}, \tag{7}
$$

where the values of these constants, const, are determined from the initial con-

ditions <sup>ð</sup>X; <sup>Y</sup>; <sup>Z</sup>; dsX; dsY; dsZÞjs¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>; <sup>0</sup>; <sup>0</sup>; Cx ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>þC<sup>2</sup> xþC<sup>2</sup> y <sup>p</sup> ; Cy ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>þC<sup>2</sup> xþC<sup>2</sup> y <sup>p</sup> ; <sup>0</sup> !: Eqs. (5)–(7) can yield an equation for dsX and dsY

$$(d\_\mathbf{r}Y)^2 = \left[\mathbf{C}\_\mathbf{r} + W\_B \mathbf{X}\right]^2 \ast \left[\mathbf{1} - (d\_\mathbf{r}\mathbf{X})^2 - (d\_\mathbf{r}Y)^2\right] \tag{8}$$

$$\left(\left(d\_s X\right)^2 = \left[\mathbf{C}\_{\mathbf{x}} - \mathbf{W}\_B \* \left(\eta \mathbf{s} + Y\right)\right]^2 \* \left[\mathbf{1} - \left(d\_s X\right)^2 - \left(d\_s Y\right)^2\right] \tag{9}$$

whose solution reads

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

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

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 Es and Bs are constant, can be elliptical or parabolic according to values of Es and Bs and 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 Es and Bs, about MVolt=meter-level and Tesla-level, the size can be down to m-level for

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, Bs is 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 ¼ 0 contour, Es-values and the slope β ¼ dxBs, where Bs is along the y-direction, its magnitude ∣Bs∣ is a function of the coordinate x, and the unperturbed path is along

For the convenience of readers, we paste related materials published elsewhere

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � ð Þ dsX <sup>2</sup> � ð Þ dsY <sup>2</sup> � ð Þ dsZ <sup>2</sup>

ds½ �¼ ΓdsZ 0, (1) ds½ �¼ ΓdsY WBdsX (2) ds½ �¼� ΓdsX WB½ � η þ dsY (3)

: (4)

[7]. For a simple configuration containing merely static electric field (along xdirection) and static magnetic field (along z-direction), the behavior of an incident electron can be described by dimensionless 3-D relativistic Newton equations

manipulating nuclear matter.

can ensure a powerful mono-color gamma-ray source.

Use of Gamma Radiation Techniques in Peaceful Applications

s-level time cycle or Hz-level frequency.

the z-axis.

(RNEs)

where

38

1 Γ ¼ q

2. Theory and method

2.1 Theoretical basis

$$\left(d\_t X\right)^2 = \frac{\left[\mathbf{C}\_\mathbf{x} - \mathbf{W}\_B \* (\eta \mathbf{s} + \mathbf{Y})\right]^2}{\left[\mathbf{1} + \left[\mathbf{C}\_\mathbf{y} + \mathbf{W}\_B \mathbf{X}\right]^2 + \left[\mathbf{C}\_\mathbf{x} - \mathbf{W}\_B \* (\eta \mathbf{s} + \mathbf{Y})\right]^2\right]} \tag{10}$$

$$\left(d\_i Y\right)^2 = \frac{\left[\mathcal{C}\_\mathcal{Y} + \mathcal{W}\_B \mathcal{X}\right]^2}{\left[\mathbf{1} + \left[\mathcal{C}\_\mathcal{Y} + \mathcal{W}\_B \mathcal{X}\right]^2 + \left[\mathcal{C}\_\mathcal{x} - \mathcal{W}\_B \* \left(\eta\mathfrak{s} + \mathcal{Y}\right)\right]^2\right]}.\tag{11}$$

It is easy to verify that the solutions (10, 11) will lead to Γ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> Cy <sup>þ</sup> WBX � �<sup>2</sup> <sup>þ</sup> ½ � Cx � WB <sup>∗</sup> ð Þ <sup>η</sup><sup>s</sup> <sup>þ</sup> <sup>Y</sup> <sup>2</sup> q and, with the help of Eqs. (6) and (7), ds<sup>Γ</sup> ¼ �WB<sup>η</sup> <sup>∗</sup> dsX (i.e., mec<sup>2</sup>dt<sup>Γ</sup> <sup>¼</sup> eEdtX). Noting <sup>Γ</sup> can be formally expressed as Γ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>C</sup><sup>2</sup> <sup>y</sup> <sup>þ</sup> <sup>C</sup><sup>2</sup> x q � WBη ∗X, which agrees with Takeuchi's theory [15], we can find that the electronic trajectory can be expressed as

$$\left[\sqrt{\mathbf{1} + \mathbf{C}\_{\mathcal{V}}^2 + \mathbf{C}\_{\mathbf{x}}^2} - W\_B \boldsymbol{\eta} \ast \mathbf{X}\right]^2 = \mathbf{1} + \left[\mathbf{C}\_{\mathcal{V}} + W\_B \mathbf{X}\right]^2 + \left[\mathbf{C}\_{\mathbf{x}} - W\_B \ast (\boldsymbol{\eta}\mathbf{s} + \mathbf{Y})\right]^2,\tag{12}$$

or

$$\begin{split} \left(\mathbf{1} - \eta^{2}\right) \left[\mathbf{X} + \frac{\left(\eta + \nu\_{\rm 0}\right)}{1 - \eta^{2}} \frac{\Gamma\_{0}}{W\_{\rm B}}\right]^{2} + \left[\left(Y + \eta\mathbf{s}\right) - \nu\_{\rm 0}\frac{\Gamma\_{0}}{W\_{\rm B}}\right]^{2} &= \\ \underbrace{\left[\left(\eta + \nu\_{\rm 0}\right)^{2} + \left(1 - \eta^{2}\right)\nu\_{\rm x0}^{2}\right]}\_{\mathbf{1} - \eta^{2}} \left(\frac{\Gamma\_{0}}{W\_{\rm B}}\right)^{2}, \end{split} \tag{13}$$

where Γ<sup>0</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>C</sup><sup>2</sup> <sup>y</sup> <sup>þ</sup> <sup>C</sup><sup>2</sup> x q , <sup>υ</sup><sup>x</sup><sup>0</sup> <sup>¼</sup> Cx <sup>Γ</sup><sup>0</sup> and <sup>υ</sup><sup>y</sup><sup>0</sup> <sup>¼</sup> Cy Γ0 .

There will be an elliptical trajectory for η , 1 and 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]

Use of Gamma Radiation Techniques in Peaceful Applications

$$\pm ds = \frac{\frac{1}{W\_b} \Gamma\_0 - \eta \ast X}{\sqrt{aX^2 + bX + c}} dX = \frac{\eta}{\sqrt{-a}} \frac{X\_N - X}{\sqrt{\frac{b^2 - 4ac}{4a^2} - \left(X + \frac{b}{2a}\right)^2}} dX,\tag{14}$$

where <sup>a</sup> <sup>¼</sup> <sup>η</sup>ð Þ <sup>2</sup> � <sup>1</sup> , <sup>b</sup> ¼ �<sup>2</sup> <sup>η</sup>Γ<sup>0</sup> <sup>þ</sup> Cy � � 1 WB , c <sup>¼</sup> <sup>C</sup><sup>2</sup> <sup>x</sup> <sup>1</sup> WB � �<sup>2</sup> and XN <sup>¼</sup> <sup>1</sup> η 1 WB Γ0. The equation can be written as a more general form

$$
\pm ds = \sigma \frac{M - u}{\sqrt{1 - u^2}} du \tag{15}
$$

this stage, the electron will move υy<sup>1</sup> ∗ Ttr along the y direction. Then, the motion in

u ¼ �1 þ ξ ! u ¼ �1. Thus, a complete closed cycle along the x direction is finished even though the motion along the y direction is not closed. Repeating this

Clearly, the time cycle of such an oscillation, or that of a "tailored" Takeuchi

Under fixed values of Δ, E and B, the smaller ξ is, the smaller Tx is. There will be Tx ¼ 0 at ξ ¼ 0. In principle, arbitrary value of Tx , Tc can 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 x tð Þ might cause its Fourier spectrum to have some spread, the center

This result implies a simple and universal method of setting up quasi-monocolor light source at any desirable center wavelength: by applying vertically static electric field E ¼ Ex and static magnetic field B ¼ Bz and on purpose letting a B ¼ 0

and the B 6¼ 0 region. As shown in Figure 1 of Ref. [7], adjusting the distance

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

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 μhigh and μlow. The μlow 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 B0. Different μ values cause the

h i <sup>þ</sup> <sup>2</sup><sup>σ</sup>

2

cB , 1, then injecting electron along the y-axis with a

can lead to a quasi-monocolor oscillation source with any desired

cB ∣, and close to the boundary line between the B ¼ 0 region

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup><sup>ξ</sup> � <sup>ξ</sup><sup>2</sup>

: (17)

q

closed cycle will lead to an oscillation along the x direction.

Electron Oscillation-Based Mono-Color Gamma-Ray Source

DOI: http://dx.doi.org/10.5772/intechopen.82752

Tx <sup>¼</sup> Ttr <sup>þ</sup> <sup>2</sup>σ<sup>M</sup> <sup>∗</sup> arcsinð Þ� �<sup>1</sup> <sup>þ</sup> <sup>ξ</sup> <sup>π</sup>

� �-region can be described by an acute-angled rotation along the ellipse

the <sup>E</sup>; <sup>B</sup> <sup>¼</sup> <sup>E</sup>

orbit, is

D ¼ ξ ∗

orbit [6].

Figure 1.

41

frequency will be <sup>1</sup>

region exist and the ratio <sup>E</sup>

velocity slightly above ∣ <sup>E</sup>

ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac 4a<sup>2</sup> q

Tx .

center frequency up to gamma-ray level.

contour plane to have different z coordinates in two mediums.

ηc

where <sup>u</sup> <sup>¼</sup> <sup>X</sup><sup>þ</sup> <sup>b</sup> <sup>2</sup><sup>a</sup> ffiffiffiffiffiffiffiffi <sup>b</sup>2�4ac 4a2 <sup>q</sup> <sup>¼</sup> <sup>X</sup>�XRþXL <sup>2</sup> XR�XL , XL <sup>¼</sup> min �b� ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac <sup>p</sup> <sup>2</sup><sup>a</sup> ; �b<sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac <sup>p</sup> 2a � � and XR <sup>¼</sup> max �b� ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac <sup>p</sup> <sup>2</sup><sup>a</sup> ; �bþ ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac <sup>p</sup> 2a � �. In addition, <sup>σ</sup> <sup>¼</sup> <sup>η</sup> ffiffiffiffiffi �<sup>a</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac 4a<sup>2</sup> q and <sup>M</sup> <sup>¼</sup> XN<sup>þ</sup> <sup>b</sup> <sup>2</sup><sup>a</sup> ffiffiffiffiffiffiffiffi <sup>b</sup>2�4ac 4a2 q ¼ XN�XRþXL <sup>2</sup> XR�XL . It is easy to verify that for <sup>η</sup><sup>2</sup> � <sup>1</sup> , 0, there is <sup>M</sup> <sup>¼</sup> <sup>1</sup>þηυy<sup>0</sup> η ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>η</sup>þυy<sup>0</sup> 2 <sup>þ</sup> <sup>1</sup>�η<sup>2</sup> ð Þυ<sup>2</sup> x0 <sup>q</sup> . 1. Initially, ð Þj <sup>X</sup>; <sup>Y</sup> <sup>s</sup>¼<sup>0</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup> and hence ust <sup>¼</sup> <sup>u</sup><sup>j</sup> <sup>s</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup><sup>þ</sup> <sup>b</sup> <sup>q</sup> <sup>¼</sup> �ð Þ <sup>η</sup>þυy<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>q</sup> .

<sup>2</sup><sup>a</sup> ffiffiffiffiffiffiffiffi <sup>b</sup>2�4ac 4a2 ð Þ <sup>η</sup>þυy<sup>0</sup> 2 <sup>þ</sup> <sup>1</sup>�η<sup>2</sup> ð Þυ<sup>2</sup> x0

From strict solution

$$\pm \mathfrak{s}(u) = \sigma \ast \left\{ M \ast \arcsin(u) + \sqrt{1 - u^2} \right\} + const,\tag{16}$$

we can find the time for an electron traveling through an elliptical trajectory to meet scycle <sup>¼</sup> <sup>ω</sup>Tc <sup>¼</sup> <sup>2</sup> <sup>∗</sup> ½ � <sup>σ</sup>M<sup>π</sup> and hence a time cycle Tc <sup>¼</sup> ð Þ <sup>1</sup>þηυy<sup>0</sup> <sup>Γ</sup><sup>0</sup> ffiffiffiffiffiffiffi 1�η<sup>2</sup> � � p <sup>3</sup> 2π <sup>ω</sup><sup>B</sup> : That is, the oscillation along the elliptical trajectory will have a circular frequency ωB. Moreover, it is interesting to note that υ<sup>x</sup>0; υ<sup>y</sup><sup>0</sup> � � <sup>¼</sup> ð Þ <sup>0</sup>; �<sup>η</sup> will lead to ð Þ <sup>η</sup>þυy<sup>0</sup> 2 <sup>þ</sup> <sup>1</sup>�η<sup>2</sup> ð Þ<sup>υ</sup><sup>2</sup> x0 � � <sup>1</sup>�η<sup>2</sup> <sup>¼</sup> <sup>0</sup> and hence a straight-line trajectory ð Þ¼ X sð Þ; Y sð Þ ð Þ 0; �ηs .

The motion on an elliptical trajectory is very inhomogeneous. The time for finishing the ηX . 0 half might be very short while that for the ηX , 0 half might be very long. We term the two halves as fast-half and slow-half, respectively. If η is 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 <sup>υ</sup><sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup>; <sup>υ</sup><sup>y</sup><sup>0</sup> ¼ �<sup>η</sup> � <sup>Δ</sup> � �, (where <sup>Δ</sup> is small-valued and positive), the starting position X ¼ 0 is the left extreme of the ellipse and hence corresponds to u ¼ �1. The time required for an acute-angled rotation from u ¼ �1 to u ¼ �1 þ ξ, (where ξ is small-valued and positive), will be <sup>σ</sup><sup>M</sup> <sup>∗</sup> arcsinð Þ� �<sup>1</sup> <sup>þ</sup> <sup>ξ</sup> <sup>π</sup> 2 � � <sup>þ</sup> <sup>σ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup><sup>ξ</sup> � <sup>ξ</sup><sup>2</sup> <sup>p</sup> , which is ¼ 0 if ξ ¼ 0.

It is interesting to note that if there is B ¼ 0 at the region u . �1 þ ξ, the electron will enter from <sup>E</sup>; <sup>B</sup> <sup>¼</sup> <sup>E</sup> ηc � �-region into ð Þ <sup>E</sup>; <sup>B</sup> <sup>¼</sup> <sup>0</sup> -region with an initial velocity whose x-component is υ<sup>x</sup><sup>1</sup> ̃dsuj <sup>u</sup>¼�1þ<sup>ξ</sup> <sup>¼</sup> <sup>1</sup> σ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>� �ð Þ <sup>1</sup>þ<sup>ξ</sup> <sup>2</sup> <sup>p</sup> <sup>M</sup>� �ð Þ <sup>1</sup>þ<sup>ξ</sup> . 0 and <sup>y</sup>-component υ<sup>y</sup><sup>1</sup> is 6¼ 0. Then, the electron will enter into the ð Þ E; B ¼ 0 -region at a distance because <sup>υ</sup><sup>x</sup><sup>1</sup> . 0. After a time Ttr <sup>¼</sup> <sup>2</sup>υx<sup>1</sup> <sup>E</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>�υ<sup>2</sup> x1�υ<sup>2</sup> y1 <sup>p</sup> , the electron will return into the <sup>E</sup>; <sup>B</sup> <sup>¼</sup> <sup>E</sup> ηc � �-region and the returning velocity will have a <sup>x</sup>-component �υx1. During �ds ¼

where <sup>u</sup> <sup>¼</sup> <sup>X</sup><sup>þ</sup> <sup>b</sup>

From strict solution

is ¼ 0 if ξ ¼ 0.

<sup>E</sup>; <sup>B</sup> <sup>¼</sup> <sup>E</sup> ηc � �

40

ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac p <sup>2</sup><sup>a</sup> ;

max �b�

XN�XRþXL <sup>2</sup>

1

<sup>2</sup><sup>a</sup> ffiffiffiffiffiffiffiffi <sup>b</sup>2�4ac 4a2

� �

it is interesting to note that υ<sup>x</sup>0; υ<sup>y</sup><sup>0</sup>

electron will enter from <sup>E</sup>; <sup>B</sup> <sup>¼</sup> <sup>E</sup>

velocity whose x-component is υ<sup>x</sup><sup>1</sup> ̃dsuj

because <sup>υ</sup><sup>x</sup><sup>1</sup> . 0. After a time Ttr <sup>¼</sup> <sup>2</sup>υx<sup>1</sup>

�bþ

<sup>q</sup> <sup>¼</sup> <sup>X</sup>�XRþXL <sup>2</sup>

Initially, ð Þj <sup>X</sup>; <sup>Y</sup> <sup>s</sup>¼<sup>0</sup> <sup>¼</sup> ð Þ <sup>0</sup>; <sup>0</sup> and hence ust <sup>¼</sup> <sup>u</sup><sup>j</sup>

ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac p 2a

WB Γ<sup>0</sup> � η ∗X ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aX<sup>2</sup> <sup>þ</sup> bX <sup>þ</sup> <sup>c</sup> <sup>p</sup> dX <sup>¼</sup> <sup>η</sup>

Use of Gamma Radiation Techniques in Peaceful Applications

where <sup>a</sup> <sup>¼</sup> <sup>η</sup>ð Þ <sup>2</sup> � <sup>1</sup> , <sup>b</sup> ¼ �<sup>2</sup> <sup>η</sup>Γ<sup>0</sup> <sup>þ</sup> Cy

equation can be written as a more general form

ffiffiffiffiffiffi �<sup>a</sup> <sup>p</sup>

�ds <sup>¼</sup> <sup>σ</sup> <sup>M</sup> � <sup>u</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

. In addition, <sup>σ</sup> <sup>¼</sup> <sup>η</sup>

� � 1

XR�XL , XL <sup>¼</sup> min �b�

XR�XL . It is easy to verify that for <sup>η</sup><sup>2</sup> � <sup>1</sup> , 0, there is <sup>M</sup> <sup>¼</sup> <sup>1</sup>þηυy<sup>0</sup>

�s uð Þ¼ <sup>σ</sup> <sup>∗</sup> <sup>M</sup> <sup>∗</sup> arcsinð Þþ <sup>u</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

meet scycle <sup>¼</sup> <sup>ω</sup>Tc <sup>¼</sup> <sup>2</sup> <sup>∗</sup> ½ � <sup>σ</sup>M<sup>π</sup> and hence a time cycle Tc <sup>¼</sup> ð Þ <sup>1</sup>þηυy<sup>0</sup> <sup>Γ</sup><sup>0</sup>

and hence a straight-line trajectory ð Þ¼ X sð Þ; Y sð Þ ð Þ 0; �ηs .

is small-valued and positive), will be <sup>σ</sup><sup>M</sup> <sup>∗</sup> arcsinð Þ� �<sup>1</sup> <sup>þ</sup> <sup>ξ</sup> <sup>π</sup>

ηc � �

we can find the time for an electron traveling through an elliptical trajectory to

oscillation along the elliptical trajectory will have a circular frequency ωB. Moreover,

The motion on an elliptical trajectory is very inhomogeneous. The time for finishing the ηX . 0 half might be very short while that for the ηX , 0 half might be very long. We term the two halves as fast-half and slow-half, respectively. If η is 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 <sup>υ</sup><sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup>; <sup>υ</sup><sup>y</sup><sup>0</sup> ¼ �<sup>η</sup> � <sup>Δ</sup> � �, (where <sup>Δ</sup> is small-valued and positive), the starting position X ¼ 0 is the left extreme of the ellipse and hence corresponds to u ¼ �1. The time required for an acute-angled rotation from u ¼ �1 to u ¼ �1 þ ξ, (where ξ

It is interesting to note that if there is B ¼ 0 at the region u . �1 þ ξ, the

υ<sup>y</sup><sup>1</sup> is 6¼ 0. Then, the electron will enter into the ð Þ E; B ¼ 0 -region at a distance

<sup>E</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>�υ<sup>2</sup> x1�υ<sup>2</sup> y1

<sup>u</sup>¼�1þ<sup>ξ</sup> <sup>¼</sup> <sup>1</sup>

σ


� � <sup>¼</sup> ð Þ <sup>0</sup>; �<sup>η</sup> will lead to ð Þ <sup>η</sup>þυy<sup>0</sup>

b2 �4ac

WB , c <sup>¼</sup> <sup>C</sup><sup>2</sup>

XN � <sup>X</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>4</sup>a<sup>2</sup> � <sup>X</sup> <sup>þ</sup> <sup>b</sup>

<sup>x</sup> <sup>1</sup> WB � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac p

> ffiffiffiffiffi �<sup>a</sup> <sup>p</sup>

<sup>s</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup><sup>þ</sup> <sup>b</sup>

1 � u<sup>2</sup> n o <sup>p</sup> <sup>þ</sup> const, (16)

<sup>2</sup><sup>a</sup> ; �b<sup>þ</sup>

� �

ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac 4a<sup>2</sup> q

> <sup>2</sup><sup>a</sup> ffiffiffiffiffiffiffiffi <sup>b</sup>2�4ac 4a2

2a � �<sup>2</sup> <sup>q</sup> dX, (14)

<sup>1</sup> � <sup>u</sup><sup>2</sup> <sup>p</sup> du (15)

η

ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac p 2a

and <sup>M</sup> <sup>¼</sup> XN<sup>þ</sup> <sup>b</sup>

ð Þ <sup>η</sup>þυy<sup>0</sup> 2 <sup>þ</sup> <sup>1</sup>�η<sup>2</sup> ð Þυ<sup>2</sup> x0

<sup>q</sup> <sup>¼</sup> �ð Þ <sup>η</sup>þυy<sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>η</sup>þυy<sup>0</sup> 2 <sup>þ</sup> <sup>1</sup>�η<sup>2</sup> ð Þυ<sup>2</sup> x0

> ffiffiffiffiffiffiffi 1�η<sup>2</sup> � � p <sup>3</sup>

2 � � <sup>þ</sup> <sup>σ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>M</sup>� �ð Þ <sup>1</sup>þ<sup>ξ</sup> . 0 and <sup>y</sup>-component


<sup>p</sup> , the electron will return into the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>� �ð Þ <sup>1</sup>þ<sup>ξ</sup> <sup>2</sup> <sup>p</sup> 2π

2 <sup>þ</sup> <sup>1</sup>�η<sup>2</sup> ð Þ<sup>υ</sup><sup>2</sup> x0

� �

<sup>ω</sup><sup>B</sup> : That is, the

<sup>1</sup>�η<sup>2</sup> <sup>¼</sup> <sup>0</sup>

<sup>2</sup><sup>ξ</sup> � <sup>ξ</sup><sup>2</sup> <sup>p</sup> , which

and XN <sup>¼</sup> <sup>1</sup>

η 1 WB Γ0. The

and XR ¼

<sup>2</sup><sup>a</sup> ffiffiffiffiffiffiffiffi <sup>b</sup>2�4ac 4a2 q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>q</sup> .

<sup>q</sup> . 1.

this stage, the electron will move υy<sup>1</sup> ∗ Ttr along the y direction. Then, the motion in the <sup>E</sup>; <sup>B</sup> <sup>¼</sup> <sup>E</sup> ηc � �-region can be described by an acute-angled rotation along the ellipse u ¼ �1 þ ξ ! u ¼ �1. Thus, a complete closed cycle along the x direction is finished even though the motion along the y direction is not closed. Repeating this closed cycle will lead to an oscillation along the x direction.

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

$$T\_x = T\_{tr} + 2\sigma \mathcal{M} \* \left[ \arcsin(-\mathbf{1} + \xi) - \frac{\pi}{2} \right] + 2\sigma \sqrt{2\xi - \xi^2}. \tag{17}$$

Under fixed values of Δ, E and B, the smaller ξ is, the smaller Tx is. There will be Tx ¼ 0 at ξ ¼ 0. In principle, arbitrary value of Tx , Tc can 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 x tð Þ might cause its Fourier spectrum to have some spread, the center frequency will be <sup>1</sup> Tx .

This result implies a simple and universal method of setting up quasi-monocolor light source at any desirable center wavelength: by applying vertically static electric field E ¼ Ex and static magnetic field B ¼ Bz and on purpose letting a B ¼ 0 region exist and the ratio <sup>E</sup> cB , 1, then injecting electron along the y-axis with a velocity slightly above ∣ <sup>E</sup> cB ∣, and close to the boundary line between the B ¼ 0 region and the B 6¼ 0 region. As shown in Figure 1 of Ref. [7], adjusting the distance D ¼ ξ ∗ ffiffiffiffiffiffiffiffiffiffiffi b2 �4ac 4a<sup>2</sup> q can lead to a quasi-monocolor oscillation source with any desired center frequency up to gamma-ray level.

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].

#### 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 μhigh and μlow. The μlow 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 B0. Different μ values cause the contour plane to have different z coordinates in two mediums.

## 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.

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.

a level is not difficult to be realized technically. Because of the condition ∇ � B ¼ 0

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 ¼ 0 plane, and the metal is arranged at z ,� ∣zd∣ region. The ð Þ x , 0; jzuj . z .� jzdj region is filled

medium. According to the theory of electromagnetism, the contour plane Bz ¼ B<sup>0</sup> in the ð Þ x , 0; �jzdj . z region and that in the ð Þ x . 0; �jzdj . z region will have different z coordinates. This implies a discontinuity in B exists near the demarcation line. That is, different values of the dropping rate ∂z∣B∣ in two mediums cause the

and the fact that the solenoid is finite-sized, Bs has two components Bx ex

!.

with μ ¼ μhigh medium and the ð Þ x . 0; jzuj . z .�jzdj region with μ ¼ μlow

ciently 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 ,¼ 0:01μm. This fundamentally warrants sufficiently large gradient ∂xμ, as well as sufficiently large ∂xB<sup>2</sup> up to <sup>T</sup>

For Al, its electron-phonon collision relaxation time τ is at 10<sup>1</sup>�<sup>2</sup>

<sup>2</sup> meυ<sup>2</sup>�

n<sup>0</sup> 1 �

ð ð uq

! �∇rUi

be reflected by following quantum theory (21, 23–26),

<sup>V</sup> <sup>¼</sup> <sup>e</sup> 4πε<sup>0</sup>

Vph ¼

iћ∇ þ eB0yex � �<sup>2</sup>

∇2

its Fermi velocity υ<sup>F</sup> is at 10<sup>6</sup>m=s-level and its Fermi energy EF is about 5:5eV [21, 22]. If a cm3-level Al cubic specimen is placed between a pair of electrode plates with 220V voltage, the DC electric field Es it feels will be at 10<sup>4</sup>V=m-level. If we merely take into account the work done by Es, the maximum velocity increment along the Es direction, maxΔυx, can reach eEsτ=me <sup>¼</sup> <sup>1</sup>:6=9:<sup>1</sup> <sup>∗</sup> <sup>10</sup>�19þ4�15þ1þ<sup>31</sup> <sup>≈</sup>

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

<sup>x</sup> 200 ∗ 9:1 ∗ 10�<sup>31</sup>J ≈10�<sup>9</sup>eV.

ψ<sup>k</sup> þ Uiψ<sup>k</sup> þ V xð Þ ; y ψ<sup>k</sup> þ eE0xψ<sup>k</sup> þ Vphð Þ x; y; t; T ψk:

f kð Þ ; T dkxdky

h ig kð Þ ; <sup>T</sup> dqxdqy, (20)

n0

!

Emission is a many-body process because the sheath field, or space charge effect, left by emitted electrons in turn affects emission [23–26]. This phenomenon can

Ð Ð <sup>ψ</sup><sup>k</sup> j j<sup>2</sup>

<sup>x</sup>2iþ<sup>1</sup> and Bz <sup>¼</sup> <sup>∑</sup><sup>i</sup> gzi

Electron Oscillation-Based Mono-Color Gamma-Ray Source

DOI: http://dx.doi.org/10.5772/intechopen.82752

! þ yBx ez

specimen in the ð Þ �jzd<sup>j</sup> . <sup>z</sup> region to still feel a gradient <sup>∂</sup>xj j <sup>B</sup> <sup>2</sup>

! ¼ �yBzex

limited to the skin layer of the metal [21, 22].

20m=s, which corresponds to <sup>1</sup>

<sup>i</sup>ћ∂tψ<sup>k</sup> <sup>¼</sup> <sup>1</sup>

where

43

2me

V=m-level by letting the inter-plate distance of a pair of plane-

m and applied voltage as 10<sup>2</sup> ̃3V. Parameter values at such

x2iþ<sup>1</sup> correspond to a vector potential

! and Bz ez

. To ensure a suffi-

!,

nm-

fs-level [21, 22],

(18)

(19)

Es can achieve 10<sup>5</sup>

! þ Azez

where Bx ¼ ∑<sup>i</sup> gxi

level, to be feasible.

¼ Axex

A !

plate electrodes as 10�<sup>3</sup> ̃�<sup>2</sup>

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 Bs is 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 Bs needs 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 Bs can 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, Bs generated by a solenoid has contour planes normal to the axis of the solenoid, or the electromagnetic (EM) energy density profile Bs j j<sup>2</sup> is smooth or has a very small gradient <sup>∇</sup> Bs j j<sup>2</sup> .

If the strength ∣Bs∣ is not easy to be enhanced, adjusting the space shape of Bs is a worthy trial to optimize interaction. A high-gradient Bs j j<sup>2</sup> profile 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 Bs j j<sup>2</sup> profile. As shown in Figure 1, the intrusion of a high-μ (magnetic permeability) medium distorts contours of Bs j j<sup>2</sup> to be bent. In other words, in each plane normal to the axis of the solenoid, a huge gradient of Bs j j<sup>2</sup> along the direction normal to the side surface of the medium appears. If a metal wire/specimen is arranged in such a high-<sup>∇</sup> Bs j j<sup>2</sup> region and an DC electric field Es along the direction of <sup>∇</sup> Bs j j<sup>2</sup> is applied, the Hall effect in such a situation where the DC magnetic field is very spaceinhomogeneous is worthy of being studied.

When studying applications such as probing and imagining local magnetic moment and magnetic microscopic structure [10–20], many authors have made indepth 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.

Es can achieve 10<sup>5</sup> V=m-level by letting the inter-plate distance of a pair of planeplate electrodes as 10�<sup>3</sup> ̃�<sup>2</sup> m and applied voltage as 10<sup>2</sup> ̃3V. Parameter values at such a level is not difficult to be realized technically. Because of the condition ∇ � B ¼ 0 and the fact that the solenoid is finite-sized, Bs has two components Bx ex ! and Bz ez !, where Bx ¼ ∑<sup>i</sup> gxi <sup>x</sup>2iþ<sup>1</sup> and Bz <sup>¼</sup> <sup>∑</sup><sup>i</sup> gzi x2iþ<sup>1</sup> correspond to a vector potential !

$$A = A\_x \overline{e\_x} + A\_x \overline{e\_x} = -\overline{\nu} B\_x \overline{e\_x} + \underline{\nu} B\_x \overline{e\_x}.$$

2.2 Details on electron source

Use of Gamma Radiation Techniques in Peaceful Applications

very small gradient <sup>∇</sup> Bs j j<sup>2</sup>

42

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

Hall effect of a metal by a static (DC) magnetic field Bs is 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 Bs needs 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 Bs can 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, Bs generated by a solenoid has contour planes normal to the axis of the solenoid, or the electromagnetic (EM) energy density profile Bs j j<sup>2</sup> is smooth or has a

If the strength ∣Bs∣ is not easy to be enhanced, adjusting the space shape of Bs is a worthy trial to optimize interaction. A high-gradient Bs j j<sup>2</sup> profile 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 Bs j j<sup>2</sup> profile. As shown in Figure 1, the intrusion of a high-μ (magnetic permeability) medium distorts contours of Bs j j<sup>2</sup> to be bent. In other words, in each plane normal to the axis of the solenoid, a huge gradient of Bs j j<sup>2</sup> along the direction normal to the side surface of the medium appears. If a metal wire/specimen is arranged in such a high-<sup>∇</sup> Bs j j<sup>2</sup> region and an DC electric field Es along the direction of <sup>∇</sup> Bs j j<sup>2</sup> is applied, the Hall effect in such a situation where the DC magnetic field is very space-

When studying applications such as probing and imagining local magnetic moment and magnetic microscopic structure [10–20], many authors have made indepth 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

above-mentioned discussion. Details are presented as follows.

.

inhomogeneous is worthy of being studied.

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 ¼ 0 plane, and the metal is arranged at z ,� ∣zd∣ region. The ð Þ x , 0; jzuj . z .� jzdj region is filled with μ ¼ μhigh medium and the ð Þ x . 0; jzuj . z .�jzdj region with μ ¼ μlow medium. According to the theory of electromagnetism, the contour plane Bz ¼ B<sup>0</sup> in the ð Þ x , 0; �jzdj . z region and that in the ð Þ x . 0; �jzdj . z region will have different z coordinates. This implies a discontinuity in B exists near the demarcation line. That is, different values of the dropping rate ∂z∣B∣ in two mediums cause the specimen in the ð Þ �jzd<sup>j</sup> . <sup>z</sup> region to still feel a gradient <sup>∂</sup>xj j <sup>B</sup> <sup>2</sup> . 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 ,¼ 0:01μm. This fundamentally warrants sufficiently large gradient ∂xμ, as well as sufficiently large ∂xB<sup>2</sup> up to <sup>T</sup> nmlevel, 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 τ is at 10<sup>1</sup>�<sup>2</sup> fs-level [21, 22], its Fermi velocity υ<sup>F</sup> is at 10<sup>6</sup>m=s-level and its Fermi energy EF is about 5:5eV [21, 22]. If a cm3-level Al cubic specimen is placed between a pair of electrode plates with 220V voltage, the DC electric field Es it feels will be at 10<sup>4</sup>V=m-level. If we merely take into account the work done by Es, the maximum velocity increment along the Es direction, maxΔυx, can reach eEsτ=me <sup>¼</sup> <sup>1</sup>:6=9:<sup>1</sup> <sup>∗</sup> <sup>10</sup>�19þ4�15þ1þ<sup>31</sup> <sup>≈</sup> 20m=s, which corresponds to <sup>1</sup> <sup>2</sup> meυ<sup>2</sup>� <sup>x</sup> 200 ∗ 9:1 ∗ 10�<sup>31</sup>J ≈10�<sup>9</sup>eV.

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

$$i\hbar \partial\_l \psi\_k = \frac{1}{2m\_e} \left[ i\hbar \nabla + eB\_0 y e\_x \right]^2 \psi\_k + U\_i \psi\_k + V(x, y)\psi\_k + eE\_0 x \psi\_k + V\_{ph}(x, y, t; T) \psi\_k. \tag{18}$$

$$\nabla^2 V = \frac{e}{4\pi\varepsilon\_0} n\_0 \left( 1 - \frac{\int \int |\mu\_k|^2 f(k, T) dk\_x dk\_y}{n\_0} \right) \tag{19}$$

where

$$V\_{ph} = \int \int \left[ \overrightarrow{u\_q} \cdot \nabla\_r U\_i \right] \mathbf{g}(k, T) dq\_x dq\_y \tag{20}$$

$$
\overrightarrow{u\_q} = \sin\left(q\_x x + q\_y y + q\_z z - \nu\_q t\right) e\_p^\rightarrow,\tag{21}
$$

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

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

State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

efficient; (3) injecting protons into target nucleus. In contrast, the electron

Electron Oscillation-Based Mono-Color Gamma-Ray Source

DOI: http://dx.doi.org/10.5772/intechopen.82752

more nuclei at desirable number.

Author details

45

Hai Lin\*, ChengPu Liu and Chen Wang

provided the original work is properly cited.

\*Address all correspondence to: linhai@siom.ac.cn

Fine Mechanics, Shanghai, China

<sup>n</sup><sup>0</sup> is the average background ionic density, <sup>ψ</sup>kð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>t</sup> exp ð Þ Sk <sup>ψ</sup><sup>0</sup> <sup>k</sup> , ψ<sup>0</sup> <sup>k</sup> is the unperturbed wavefunction and f is the Fermi-Dirac distribution function. Vph is the vibrating lattice potential, u is the field of ionic displacement, Ui is the lattice potential at zero-temperature, g is the Bose-Einstein distribution function, ν<sup>q</sup> is the phonon dispersion relation and T is 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 u as prescribed. Such an approximation is acceptable because the temporal variation of u is merely obvious over a large time scale <sup>2</sup><sup>π</sup> νq

which is usually . 100fs.

The equation of Sk reads

$$\begin{split} i\hbar \partial\_{t} \mathbf{S}\_{k} &= -\frac{\hbar^{2}}{2m\_{\epsilon}} \left[ \begin{pmatrix} \left(\partial\_{\mathbf{x}x} + \partial\_{\mathbf{y}} + \partial\_{\mathbf{z}x}\right) \mathbf{S}\_{k} + \left(\partial\_{\mathbf{x}} \mathbf{S}\_{k}\right)^{2} + \left(\partial\_{\mathbf{z}} \mathbf{S}\_{k}\right)^{2} + \left(\partial\_{\mathbf{z}} \mathbf{S}\_{k}\right)^{2} \\ \quad + 2ik\_{\mathbf{x}} \partial\_{\mathbf{x}} \mathbf{S}\_{k} + 2ik\_{\mathbf{y}} \partial\_{\mathbf{y}} \mathbf{S}\_{k} + 2ik\_{\mathbf{z}} \partial\_{\mathbf{z}} \mathbf{S}\_{k} \end{pmatrix} \right] \\ &- \frac{\hbar B\_{0}}{m\_{\epsilon}} c \eta \left(k\_{\mathbf{z}} f\_{x} - k\_{\mathbf{x}} f\_{x}\right) + \frac{i\hbar B\_{0}}{m\_{\epsilon}} c \eta \left(f\_{x} \partial\_{x} - f\_{x} \partial\_{\mathbf{z}}\right) \mathbf{S}\_{k} \\ &+ \frac{e^{2} B\_{0}^{2} \left(f\_{x}^{2} + f\_{x}^{2}\right)}{2m\_{\epsilon}} \mathbf{y}^{2} + e \mathbf{E}\_{0} \mathbf{x} + V(\mathbf{x}, \mathbf{y}) + \mathbf{V}\_{\text{ph}} \end{split} \tag{22}$$

where the space inhomogeneity of Bs is reflected by f

$$f = \left(f\_{\mathbf{x}}, \mathbf{0}, f\_{\mathbf{z}}\right) = \left(\mathbf{B}\_{\mathbf{x}}/\mathbf{B}\_{\mathbf{0}}, \mathbf{0}, \mathbf{B}\_{\mathbf{z}}/\mathbf{B}\_{\mathbf{0}}\right). \tag{23}$$

Note that f ¼ ð Þ 0; 0; 1 corresponds to a space-homogeneous Bs ¼ B<sup>0</sup> along z direction.

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 A � p^ � Br∇. Space-inhomogeneous B will cause more spaceinhomogeneous 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

### Electron Oscillation-Based Mono-Color Gamma-Ray Source DOI: http://dx.doi.org/10.5772/intechopen.82752

uq

Use of Gamma Radiation Techniques in Peaceful Applications

which is usually . 100fs. The equation of Sk reads

<sup>i</sup>ћ∂tSk ¼ � <sup>ћ</sup><sup>2</sup>

2me

� <sup>ћ</sup>B<sup>0</sup> me

þ e<sup>2</sup>B<sup>2</sup> <sup>0</sup> f 2 <sup>x</sup> þ f 2 z � � 2me

direction.

larger energy.

3. Conclusion

44

<sup>∂</sup>xx <sup>þ</sup> <sup>∂</sup>yy <sup>þ</sup> <sup>∂</sup>zz

ey kzf <sup>x</sup> � kxf <sup>z</sup> � � <sup>þ</sup>

where the space inhomogeneity of Bs is reflected by f

f ¼ f <sup>x</sup>; 0; f <sup>z</sup>

� �Sk <sup>þ</sup> ð Þ <sup>∂</sup>xSk

<sup>þ</sup>2ikx∂xSk <sup>þ</sup> <sup>2</sup>iky∂ySk <sup>þ</sup> <sup>2</sup>ikz∂zSk

<sup>y</sup><sup>2</sup> <sup>þ</sup> eE0<sup>x</sup> <sup>þ</sup> V xð Þþ ; <sup>y</sup> Vph,

Note that f ¼ ð Þ 0; 0; 1 corresponds to a space-homogeneous Bs ¼ B<sup>0</sup> along z

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

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

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

an operator A � p^ � Br∇. Space-inhomogeneous B will cause more spaceinhomogeneous 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

(PF), the most primitive EM energy form for all physics laboratories.

iћB<sup>0</sup> me

" #

! <sup>¼</sup> sin qxx <sup>þ</sup> qyy <sup>þ</sup> qzz � <sup>ν</sup>qt

unperturbed wavefunction and f is the Fermi-Dirac distribution function. Vph is the vibrating lattice potential, u is the field of ionic displacement, Ui is the lattice potential at zero-temperature, g is the Bose-Einstein distribution function, ν<sup>q</sup> is the phonon dispersion relation and T is 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 u as prescribed. Such an approximation is acceptable because the temporal variation of u is merely obvious over a large time scale <sup>2</sup><sup>π</sup>

<sup>n</sup><sup>0</sup> is the average background ionic density, <sup>ψ</sup>kð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>t</sup> exp ð Þ Sk <sup>ψ</sup><sup>0</sup>

� �

ep ,

<sup>2</sup> <sup>þ</sup> <sup>∂</sup>ySk

� � <sup>¼</sup> ð Þ Bx=B0; <sup>0</sup>; Bz=B<sup>0</sup> : (23)

ey f <sup>x</sup>∂<sup>z</sup> � <sup>f</sup> <sup>z</sup>∂<sup>x</sup> � �Sk

� �<sup>2</sup> <sup>þ</sup> ð Þ <sup>∂</sup>zSk

2

! (21)

<sup>k</sup> , ψ<sup>0</sup>

<sup>k</sup> is the

νq

(22)

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.
