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X-ray Hybrid Radiation

M. Shevelev1, A. Konkov1

1Tomsk Polytechnic University

Lenin Avenue 30, Tomsk, 634050, Russian Federation

24 September 2018

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Motivations and goals

Top Goal:

I Development of soft x-ray sources in the water windowspectral range (284− 543 eV) based on x-ray Cherenkovradiation.

Subgoals:

1 angular distribution of x-ray hubrid radiation when relativisticcharge crosses a finite-size sreen under an incedence angle;

2 spectral properties and determine monochromaticity;

3 investigation of polarization properties of hybrid radiation.

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Report outline:

1 Radiation geometry and initial conditions for considered task;

2 Applied methods;

3 Results and discussions;

4 Conclusion.

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Radiation geometry

the main points for calculation:

I charge passes through a screenunder an incidence angle;

I a finite-size creen;

I a low energy charge (from a fewMeV to 100 Mev);

I soft x-ray range.

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Expression for spectral-angular distribution of polarizationradiation

Spectral-angular distribution for considered radiation was obtainedanalitically1:d2WdωdΩ = e2

π2cβ2 cos2 α

∣∣∣ ε(ω)−1ε(ω)

∣∣∣2 cos2 α

[(1+β sinα sin θ cosφ)2−β2 cos2 α cos2 θ]2

×

∣∣∣∣∣1−exp[−id ω

βc cosα

(1−β cosα

√ε(ω)−sin2 θ+β sinα sin θ cosφ

)]1−β cosα

√ε(ω)−sin2 θ+β sinα sin θ cosφ

∣∣∣∣∣2

×

[β4 cos2 α sin2 α sin2 φ

∣∣∣∣ √ε(ω)

cos θ+√ε(ω)−sin2 θ

∣∣∣∣2×(sin2 θ +

∣∣∣√ε (ω)− sin2 θ∣∣∣2)+

∣∣∣∣ ε(ω)

ε(ω) cos θ+√ε(ω)−sin2 θ

∣∣∣∣2×∣∣∣∣ (β2 cos2 α− 1− β sinα sin θ cosφ

)sin θ

+β cosα√ε (ω)− sin2 θ (sin θ + β sinα cosφ)

∣∣∣∣2]

1A.S. Konkov, Thesis, Tomsk Polytechnic University (2016).

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Dielectric permittivity

The Henke model of dielectric permittivity was used:

ε (ω) =

[1− 1

2Z

(~ωp

~ω

)2f (ω)

]2,

where ωp is the plasma frequency, Z is the atomicnumber and f(ω) = f1 (ω)± if2 (ω) is the complexatomic scattering factor2.

2B. L. Henke et al., At. Data Nucl. Data Tables 27, 1 (1982).

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Stockes parameters for hybrid radiation

ξ1 =E∗

1E1 − E2E∗2

E∗1E1 + E2E

∗2

describes the amount of linear horizontal or

vertical polarization.

ξ2 = iE∗

1E2 − E1E∗2

E∗1E1 + E2E

∗2

corresponds to the amount of linear ±45

polarization.

ξ3 =E∗

1E2 + E1E∗2

E∗1E1 + E2E

∗2

helps to determine the amount of right or left

circular polarization.In the case of considered geometry of radiationE1 = C(β2ez cosα sinα− ey[1− β2 cos2 α+ βε (ω) ey sinα-βε (ω) ez cosα]),E2 = C

(exez[1− β2 cos2 α+ βε (ω) ey sinα− βε (ω) ez cosα]

+exeyβ2 cos2 α sin2 α),

where e = sin θ sinφ, sin θ cosφ, cosφ .

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Calculation parameters for Al

2713Al

ρ = 2.7 g/cm3

d = 8 µm~ω = 62.6÷ 82.6 eVScattering factors:

◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼

◼

◼

◼◼

◼

◼◼◼◼◼◼◼◼◼◼◼◼◼

◼◼◼◼◼◼◼◼

◼◼◼◼◼◼◼◼◼◼

◼◼◼◼◼◼◼

65 70 75 80

-5

-4

-3

-2

-1

0

(eV)

f 1(e

-)

65 70 75 80

0

1

2

3

4

ℏω (eV)

f 2(e

-)

Dots are measurement data, the color curves are interpolation functions.

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Angular distribution of hybrid radiation from Al screen

α = 25

α = 55

γ = 10 γ = 25 γ = 50

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Spectrum of hybrid radiation from Al screen

α = 25

α = 55

γ = 10 γ = 25 γ = 50

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Al, γ = 25, α = 55 deg

Angular distribution of hybrid radiation.

Spectrum of hybrid radiation.

Angular distribution of hybbrid radiation for plane φ = 0.

40 50 60 70 80

0

1000

2000

3000

4000

θ (deg)

d2W/d

dΩ

×

e2/(

2c)

Peak position and FWHM.

40 50 60 70 8067

68

69

70

71

72

73

74

θ (deg)

ℏω

(eV)

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Polarization properties: ξ1, Al, γ = 25, ~ω = 72.6 eV

α = 0 deg

-

-

-

()

ξ

α = 25 deg

- - -

-

-

()

ξ

α = 55 deg

- - -

-

-

()

ξ

The blue and the dashed blue curves correspond, respectively, toobservation angles θ1 = α− 10 and θ2 = α+ 10 deg.

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Polarization properties: ξ2, Al, γ = 25, ~ω = 72.6 eV

α = 0 deg

-

-

-

()

ξ

α = 25 deg

- - -

-

-

()

ξ

α = 55 deg

- - -

-

-

()

ξ

The red and the dashed red curves correspond, respectively, toobservation angles θ1 = α− 10 and θ2 = α+ 10 deg.

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Polarization properties: ξ3, Al, γ = 25, ~ω = 72.6 eV

α = 0 deg

-

-× -

× -

()

ξ

α = 25 deg

- - -

-

-

-

()

ξ

α = 55 deg

- - -

-

-

-

()

ξ

The greeb and the dashed green curves correspond, respectively, toobservation angles θ1 = α− 10 and θ2 = α+ 10 deg.

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Calculation parameters for Ti

4822Ti

ρ = 4.54 g/cm3

d = 8 µm~ω = 443.8÷ 463.8 eVScattering factors:

◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼

◼

◼

◼

◼

◼

◼

◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼

445 450 455 460

-15

-10

-5

0

h (eV)

f 1(e

-)

445 450 455 460

2

4

6

8

10

12

14

16

hω (eV)

f 2(e

-)

Dots are measurement data, the color curves are interpolation functions.

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Angular distribution of hybrid radiation from Ti screen

α = 25

α = 75

γ = 25 γ = 50 γ = 100

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Spectrum of hybrid radiation from Ti screen

α = 25

α = 75

γ = 25 γ = 50 γ = 100

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Ti, γ = 50, α = 75 deg

Angular distribution of hybrid radiation.

Spectrum of hybrid radiation.

Spectral-angular distribution of hybbrid radiation for

plane φ = 0.

65 70 75 80 85

0

5000

10000

15000

20000

25000

θ (deg)

d2W/d

dΩ

×

e2/(

2c)

Peak position and FWHM.

70 72 74 76 78 80

450

452

454

456

θ (deg)

ℏω

(eV)

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Polarization properties: ξ1, Ti, γ = 50, ~ω = 453.8 eV

α = 0 deg

-

-

-

()

ξ

α = 25 deg

- - -

-

()

ξ

α = 75 deg

- -

-

-

()

ξ

The blue and the dashed blue curves correspond, respectively, toobservation angles θ1 = α− 3 and θ2 = α+ 3 deg.

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Polarization properties: ξ2, Ti, γ = 50, ~ω = 453.8 eV

α = 0 deg

-

-

-

()

ξ

α = 25 deg

- - -

-

-

()

ξ

α = 75 deg

- -

-

-

()

ξ

The red and the dashed red curves correspond, respectively, toobservation angles θ1 = α− 3 and θ2 = α+ 3 deg.

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Polarization properties: ξ3, Ti, γ = 50, ~ω = 453.8 eV

α = 0 deg

-

-× --× --× -

× -× -× -

()

ξ

α = 25 deg

- - - -

-

()

ξ

α = 75 deg

- -

-

()

ξ

The green and the dashed green curves correspond, respectively, toobservation angles θ1 = α− 3 and θ2 = α+ 3 deg.

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Useful scheme

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Conclusion future plans

Conclusions:

I hybrid radiation has assymmetry of angular distribbution;

I spectrum assymmetry occurs;

I radiation monochromaticity dependes on observation angle;

I hybrid radiation has circular polarization;

Future plans:

I we are going to investigate properties of hybrid radiation fromlayered structures;

I use interference effect to suppress influence of transitionradiation on spectrum.

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Thank you for attention!

This work was supported by the Russian Foundation for basic Research (Grant No 18 − 32 − 00385 −mol a).