TỔNG HỢP TẤT CẢ BÀI NGHIÊN CỨU MỚI NHẤT VỀ VẬT LÝ BẰNG TIẾNG ANH

Communications in Physics, Vol. 19, No. 1 (2009), pp. 39-44

TRANSVERSE DISTRIBUTION OF PUMP POWER IN THE
DIODE-LASER SIDE-PUMPED SOLID-STATE LASER ROD
MAI VAN LUU, DINH XUAN KHOA, AND VU NGOC SAU
Vinh University
HO QUANG QUY
Academy of Military Science and Technology

Abstract. Based on the assumption that Gaussian pump power of diode laser bar is the same
at any cross-section along the laser rod and its curvated surface plays as thin lens, the expression
describing the pump intensity distribution inside laser rod was obtained by transfer matrix. To
have the cross-section of active volume or excited volume coincides with one of laser mode volume,
the dependence of pump intensity distribution on location of outside pump beams is investigated
by simulation.

I. INTRODUCTION
Recently, the diode laser-pumped solid-state lasers from the very small [1] to the
kilowatt level of output power [2,3] are interested and developed by because of their efficient
use in high technology. Mode size optimization in laser-diode end-pumped lasers has been
investigated [4,5]. Side-pumping geometry can be used to achieve higher output powers
[6]. For analysis in above-mentioned work, there were the following assumptions made:
Distribution of diode bar around the rod is assumed to produce an azimuthal uniform
illumination; reflection and refraction effects are not considered; to separate the calculation
of the absorption profile from how the pump light travels from the diode bar to the surface
of the rod, one describes the pump beam from the diodes only after the beams have entered
the rod; the pump beam is assumed to travel through the rod only once, i.e. reintroduction
of a pump beam through reflectors is not discussed; a single-absorption coefficient can be
used to describe the absorption process. Consequently, these assumptions lead to that:
first, it is not suitable for optimality of pump stored energy in laser rod; second, one can
not choose the optimal parameters for matching between the size of the pump volume and

laser mode one; thirdly, it is still not assumed the laser rod as a focusing lens, which is
a important fact influences on the pump energy distribution.To advoid above problems,
we present a four-side-pumped structure for solid-state laser and a new cross-sectional
geometry of the laser rod pumped by diode bar. Pumped by four laser-diode bars, which
has a Gaussian distribution in far field, so transverse intensity distribution in active rod
of solid-state laser can be changed and influences on dimension of effective “pencil” and
then on laser beam structure.


40

TRANSVERSE DISTRIBUTION OF PUMP POWER IN THE DIODE-LASER SIDE-PUMPED ...

II. PUMP INTENSITY DISTRIBUTION
As shown in Keming’s work [7] and Carts’s work [8], the cross-sectional geometry
of the laser rod pumped by four laser diode bars can be illustrated in Fig. 1. The diode
sources are assumed to have a Gaussian emittance profile (transverse distribution) and
are conditioned such that they are effectively arrayed uniformly around the rod, i.e. they
uniformly distribute along axis-z.
(a)

(b)

(x,y) point

Reforming lens

x

Laser Rod

with r0 and n
W0

W in0
R’(y)

R(y)

y

Laser Beam

Diode laser

y0

r0

Fig. 1. a- Cavity geometry for four sides-pumping module, b- Cross-sectional
geometry of Gaussian beam outside and inside rod

We assume that the laser rod has a radiusr0 and a refractive index n, the Gaussian
beam of laser diode bar in cross-section of laser rod, which is placed at point y0 from
outside surface of rod, has a complex amplitude [9]:
x2
x2
W0
exp − 2
exp −jky − jk
+ jξ(y)

W (y)
W (y)
2R(y)
where, y is the proprating direction and x is expanding direction,
U (x, y) = A0

R(y) = y 1 +
W0 =

λb
π

2 1/2

y
b

W (y) = W0 1 +
b
y

is the radius at point y in propagating direction,

(1)

(2)

2

is the wavefront radius of curvature,

(3)

1/2

is the beam waist,

ξ(y) = tan−1 (y/b) is the excess phase (i.e.,initial phase),

(4)
(5)

and b is the Rayleigh range (see Fig. 1b).
Propagating through the rod from one side, the phase of this beam will be changed
as well as after propagating through thin lens (see Fig. 1a) with focal length [9]


MAI VAN LUU et al.

41

f = r0 /(n − 1) ,
(6)
2
so that the complex amplitude transmittance of this lens is proportional toexp jkx /2f
and then the phase of the transmitted wave is altered to [9]
ky + k

x2
x2

x2
− ξ (y) − k
= ky + k
− ξ(y)
2R(y)
2f
2R (y)

(7)

where
1/R (y) = 1/R(y) − 1/f.
(8)
Using (3), (7) and (8), substituting into (1), we obtained the complex amplitude of
pump beam inside laser rod, given by
Uin (x, y) = A0

x2
x2
Win0
exp − 2
exp −jky − jk
+ jξ(y)
Win (y)
2R (y)
Win (y)

where
Win0 = M W0 ; bin =
M=

√Mt ;
1+t

t=

M 2 b;

Win = Win0 1 +

b
y0 −r0 /(n−1) ;

Mt =

r0
y0 (n−1)−r0

y
bin

2 1/2

.



; 



(9)

(10)

The waist of “inside” beam has the location at center of laser rod when following
condition is satisfied
r0
r0
r0= M 2 y0 −
+
.
(11)
n−1
n−1
Using (4), (10) and (11) we obtained the location of waist of “outside” beam
r0
πW02
−
(12)
n−2
λ
i.e., it depends on radius and refractive index of laser rod and waist and wavelength of
pump beam.
From (9) we have the expression of the pump energy distribution inside the laser rod
for single-side-pumping as following
y0 =

Iin (x, y) = I0

Win0
Win (y)

2

exp −

2x2
2 (y)
Win

(13)

We assume that pump intensity distribution (13) is symmetry for center of the rod,
which seems to be an origin of co-ordinate system (x=0, y=0), i.e. it is means that
Iin (x, y) = I(−x, y) = I(x, −y) = I(−x, −y).

(14)

Really, every laser diode bar used as a pump lamp having narrow spectra to enhance
conversion efficiency. The coherent quality of laser beam is not important for this purpose.
Moreover, the absorption of active particles in laser rod to create population inversion is
not stimulated, but is spontanuous (every pump photon reaches laser rod at different time
after reflection from reflector [10]), so population inversion creating is statistical process
and population inversion depends on total pump intensity (the sum of intensities integrated
over pulse-duration time of all diode lasers), it means that total excited particles at defined


42

TRANSVERSE DISTRIBUTION OF PUMP POWER IN THE DIODE-LASER SIDE-PUMPED ...

time is proportional to sum of intensities of all pump lasers at that time. With considering
that the delay time between all lasers is less than lifetime of upper laser level (it means
that phase mitmach between all laser beams can be negleted), so that the pump intensity
distribution for two opposite sides-pumping is given by
Itwoside (x, y) = Iin (x, y) + Iin (x, −y)

(15)

and similarly, for four sides-pumping is given by
If ourside (x, y) = Itwoside (x, y) + Itwoside (y, x).

(16)

III. SIMULATION AND DISCUSSION
We assume that the parameters of pump beam chosen to be W0 = 1mm, λ = 860nm
and the parameters of laser rod chosen to be r0 = 6mm, and a refractive index, n =
1.78. The location of waist of pump beam is calculated from (12). Now pump intensity
distributions inside the laser rod for side-pumped solid-state laser with two sides and four
sides can be obtained as shown in Fig.2a and Fig.2b, respectively.

(a)

(b)

Fig. 2. Pump intensity distribution for a two-side-pumped (a) and four-sidepumped (b) solid-state lasers

In Fig. 3 can see overlap pump profile in x-axial plane for one-side-pumped (a) and for
four side-pumped (b) laser. In Fig. 4 can see overlap pump profile in y-axial plane for one

side-pumped (a) and four-side-pumped (b) laser. After comparison between all profiles in
two figures (Fig. 3 and Fig. 4), we can conclude that the overlap of the pump intensity
distribution at the center of the laser rod closely resembles a Gaussian distribution for the
four-side-pumped laser.
The waist (Wp) of the Gaussian overlap can be changed by changing the location of
outside beam. For example, in Fig. 5, one can see that the cross section (πW2p) at level with
the same energy in case of y0 =10 mm is larger than the one in the case of y0 =15mm. This
means that the effective cross-section defined as cross-section of total intensity distribution
at level of IM AX /e can be chosen so that it coincides with cross-section of the laser mode


MAI VAN LUU et al.

(a)

43

(b)

Fig. 3. Overlap pump-intensity profile in the x-axial plane for one-side-pumped
(a) and four-side-pumped (b) lasers

(a)

(b)

Fig. 4. Overlap pump-intensity profile in the y-axial plane for one-side-pumped
(a) and four-side-pumped (b) lasers

volume (πW20L) by changing the location of outside beam (y0 ) as shown above, when other

parameters as beam waist of pump beam (W0p) and pump wavelength (λ) are given.
IV. CONCLUSION
The expression of the pump intensity deposition inside the solid-state laser rod
pumped by diode laser bars is introduced. The pump intensity distribution, the pump
volume are dependent not only on parameters of pump beam, but also on parameters of
laser rod and the location of pump beam from the laser rod. Since that obtained results
are useful not only for optimization conversion efficiency, but also for reducing the thermal


44

TRANSVERSE DISTRIBUTION OF PUMP POWER IN THE DIODE-LASER SIDE-PUMPED ...

(a)

(b)

Fig. 5. Four-side-pump- intensity distribution inside the laser rod with two values
of pump beam location: y0 = 15 mm (a) and y0 = 10 mm (b)

effect in laser rod. Moreover, the longitudinal distribution of pump intensity in laser mode
volume of the side-pumped laser is important question, which will be investigated in the
next article.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]

[7]
[8]
[9]
[10]

B. J. Comaskey, et al., IEEE J. Quantum Electron., 28 (1992) 992-996.
N. Hodgson, S. Dong, and Q. Lu, Opt. Lett. 18 (1993) 1727-1729.
R. J. St. Pierre et al., J. Sel.Top. Quantum Electron. 3 (1997) 53-58.
T. Y. Fan, and R. L. Byer, IEEE J. Quantum Electron. 24 (1988) 895-912.
Y. F. Chen et al., IEEE J. Quantum Electron. 33 (1997) 1424-1429.
W. Xie et al., Applied Optics 39 (2000) 5482-5487.
Du Keming et al., Appl. Optics 37 (1998) 2361-2364.
Y. A. Carts, Diode Lasers, Nonlinear Optics, and Solid-State Lasers, 1992.
B. E. A. Saleh, and M.C. Teich, Fundamentals of Photonics, A Wiley-Interscience Publication (1991).
O. Svelto, Principles of Lasers, Plenum Press, New York and London, 1979.

Received 11 December 2008.


Communications in Physics, Vol. 19, No. 1 (2009), pp. 45-52

INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES
ON THE EFFICIENCY OF HPGE GAMMA SPECTROSCOPY
USING MCNP
TRUONG THI HONG LOAN, DANG NGUYEN PHUONG,
DO PHAM HUU PHONG, AND TRAN AI KHANH
Faculty of Physics, University of Natural Sciences,
Vietnam National University, Ho Chi Minh City

Abstract. When determining radioactivities in environmental samples using low-level gamma

spectroscopy, in order to raise detection limit, voluminous samples are used. It takes in account
for the self-absorption (self-attenuation) of gamma rays in samples. The self-absorption effect is
small or large depend on the sample shapes, matrices and densities. In this paper, we investigated
the effect of some regular matrices such as water, soil, epoxy resin on the detector efficiency. Some
analytical formulas for the correction of matrix and densities for soil sample was established and
applied to calculate some activities from standard sample of IAEA-375.

I. INTRODUCTION
One of the most important problems of radioactivity measurement is investigating
the detection efficiency. There are lots of factors can affect the efficiency such as: incident
gamma ray energy, measuring geometry, electronic system, detector itself, other effects
like coincidence summing or self-absorption...Among them, self-absorption is the most
interesting effect when investigating activities of environmental samples because of their
large volumes.
One of the most regular geometries used in investigating activities of environmental
samples is Marinelli beaker geometry, which has 3π measuring geometry, so the efficiency
is very high. Usually, Marinelli beaker samples have large volumes so the self-absorption
effect of these samples is significant.
With the MCNP4C2 code [1], by simulating the measuring processes of environmental samples using the HPGe spectroscopy in Nuclear Physics Laboratory, we investigated
the effect of matrices and densities on the efficiency. Based on that, a correction method
was presented to calculate detection efficiencies for environmental samples.
II. CONFIGURATION OF SPECTROSCOPY - SAMPLE USED IN
SIMULATION AND EXPERIMENT
II.1. HPGe spectroscopy
The HPGe detector in Department of Nuclear Physics, model GC2018, is a coaxial
detector with configuration showed in Fig.1, including a germanium cylinder crystal with
52 mm outer diameter, 49.5 mm height. Inside the crystal, there is a hole with 7 mm


46

INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY ...

diameter, 35 mm depth. There are outer n-type contact layer (lithium layer), inner p-type
contact layer (boron layer) of the crystal. The detector is hold in an aluminium box with
1.5 mm thickness [3].
There is a lead shield outside detector to absorb gamma rays from environment and
suppress spectrum background. The interactions between gamma rays and lead shield
layer produce X-rays with energies in the range 7388 keV. These X-rays can be detected
by detector and effect on the gamma spectrum. To limit this problem, the copper and
tin liners were lined covering the lead shield with the thickness of 1.6 mm and 1 mm
respectively. The X-rays emitted by lead will be absorbed by the tin, and X-rays from the
tin (about 2530 keV) will be absorbed by cooper. Finally, the cooper emits low energy
X-rays (about 8 keV) which does not present on the spectrum.

Fig. 1. The configuration of HPGe detector (in milimeter)

II.2. Samples
The samples were contained in Marinelli beakers, which sizes were shown in Fig. 2.
These beakers were put on detector to make the 3π measuring geometry.

Fig. 2. The configuration of Marinelli sample (in centimeter)


TRUONG THI HONG LOAN et al.

47

III. SIMULATION OF PEAK EFFICIENCY CURVES OF HPGE
DETECTOR WITH MATRICES AND DENSITIES

III.1. Matrices used in simulation
To investigate the effect of matrices on detection efficiency, we need to simulate the
efficiencies with and without matrices. There were three types of matrices to simulate:
soil, water and epoxy resin. The simulated volumes were the same with all types, the
simulated densities were 0.5 g/cm3 , 1.0 g/cm3 and 2.0 g/cm3 .
Three types of matrices [3]: Soil (% mass of atom in molecular): hydrogen 2.2%,
oxygen 57.5%, aluminium 8.5%, silicon 26.2%, iron 5.6%; Epoxy resin (% mass of atom
in molecular): hydrogen 6.0%, oxygen 21.9%, carbon 72.1%; Water (% mass of atom in
molecular): hydrogen 11.11%, oxygen 88.89% .
To obtain the efficiency without matrix, simulated sample was chosen is air sample
with density 0.00129 g/cm3 , includes 79% nitrogen and 21% oxygen. The size and volume
of this sample is the same as soil, water and resin samples.
The simulated results of air matrix (efficiencies without self-absorption) were presented in Table 1.
Table 1. Detection efficiencies with air matrix (ε0 )

Radionuclide
241
Am
238 U
109 Cd
228
Ac
57
Co
214

Pb

137
54

60

Cs
Mn
Co

Energy (keV)
59.6
63.3
88.2
93.3
122.0
295.0
352.0
661.6
834.8
1173.3
1332.5

Detection efficiency (ε0 )
0.0186080
0.0225394
0.0423534
0.0446783
0.0508223
0.0316577
0.0268676
0.0151516
0.0124248
0.0094232

0.0085116

By presenting the dependence of efficiency on energy as a logarithmic function [3]
by fitting, we have:
ln(ε) = 0.0221(ln E)5 − 0.7226(ln E)4 + 9.4711(ln E)3
− 62.158(ln E)2 + 203.16 ln E − 266.2

(1)

Fig. 3, Fig. 4, and Fig. 5 presented the simulated efficiencies with different matrices
and densities.
There are some comments based on the above results:
- The difference between soil, water and epoxy resin in compare with air samples
increases when densities of matrices increase. This can be explained when we know that if


48

INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY ...

Fig. 3. Efficiencies at density 0.5 g/cm3

Fig. 4. Efficiencies at density 1.0 g/cm3

Fig. 5. Efficiencies at density 2.0 g/cm3

the density increases, the number of gamma rays can reach detector will decrease (because
of losing more energy by interacting with matrix), so the efficiency will decrease.
- In the energy range below 100 keV, the effect of matrix is more significant than in
the energy range above 100 keV.

TRUONG THI HONG LOAN et al.

49

- With the same density and measuring condition (geometry, volume, energy, ...),
the efficiencies with different matrices are nearly the same.
With above comments, we can deduce some conclusions: with environmental samples matrices like soil, water and resin, the role of matrix is not important if we just need
suitable accuracy (no need high accuracy). Therefore, when measuring with high energy
above 100 keV, the matrix correction between measured and standard samples can be
neglected. So, the standard sample preparation will be easier, saving time and cost to
obtain the acceptable results.
III.2. Self-absorption correction
The self-absorption correction factor is determined by the ratio of efficiencies with
and without self-absorption effect:
ε
(2)
f=
ε0
where f is self-absorption correction factor, ε is efficiency with self-absorption effect, ε0
is efficiency without self-absorption effect.
Different environmental samples usually have different matrices. This will be the
obstacle for measuring with large number of samples. In this part, the investigation of f
by simulation of soil matrix with different densities in the range from 0.5 to 2.0 g/cm3 was
carried out to figure out the dependence of detection efficiency on density and energy with
the same measuring geometry. Based on that, when measuring the activity of sample with
any density in the investigated range, we use this correction factor to calculate detection
efficiency.
Table 2 presented the calculated results obtained from simulation of self-absorption

correction factor f of soil sample with energy E and density ρ.
Based on the dependence of f on E as in Fig. 6, we can approximate f for energy
E as follow:
f (E, ρ) = ax2 + bx + c,

x = ln E

(3)

With different densities, fitting values of f for E, we got the parameters a, b and c.
With a, b, c obtained from different densities in the investigated range, we realized
that a, b and c depend linearly on ρ, so the fitting of a, b, c to ρ was carried out [2]. The
obtained results were:
a(ρ) = −0.0071ρ − 0.0054 (R = 0.9773)

(4)

b(ρ = 0.1144ρ + 0.0710 (R = 0.9842)

(5)

c(ρ = −0.5067ρ + 0.7622(R = 0.9907)

(6)

III.3. Testing and applying into calculating activities of radionuclides in IAEA375
The standard sample IAEA-375 in Laboratory of Nuclear Physics Department supplied by International Atomic Energy Agency (IAEA) is a sample collected from a farm in
Novozybkov, Brjansk, Russia in July, 1990 [4]. The weight of sample is 760g, contained in

50

INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY ...

Table 2. Self-absorption correction factor of soil sample

E (keV)
59.6
63.3
88.2
93.3
122.0
295.0
352.0
661.2
834.8
1173.3
1332.5

Self-absorption correction factors f at densities ρ
0.5 g/cm3 0.8 g/cm3 1.0 g/cm3 1.2 g/cm3 2.0 g/cm3
0.89
0.83
0.79
0.76
0.64
0.89
0.83
0.80
0.77

0.65
0.92
0.87
0.84
0.81
0.72
0.92
0.87
0.85
0.82
0.72
0.93
0.89
0.86
0.84
0.75
0.95
0.92
0.90
0.88
0.81
0.95
0.92
0.91
0.89
0.83
0.96
0.94
0.93
0.91

0.86
0.97
0.95
0.93
0.92
0.88
0.97
0.95
0.94
0.93
0.89
0.97
0.96
0.95
0.94
0.90

Fig. 6. The dependence of factor f on energy and density of soil matrix

Marinelli beaker with the same geometry as the simulation (Fig. 2). The sample density
ρ = 1.503 g/cm3 , sample was measured for 3 days with HPGe detector.
Activities of long-lived radionuclides were calculated by absolute method:
A=

S
ε(E).θ.m.tm

(7)

A is the source activity at the time of acquisition (Bq/kg), S is the net peak area

of the concerned peak,
ε(E) is the efficiency at energy E, m is sample weight (kg), θ is the branching ratio
of the observed nuclide at this energy E(%), tm : the live time of the measurement (s).


TRUONG THI HONG LOAN et al.

51

Using formulas (4), (5), and (6) to calculate three parameters a, b, and c:
a = −0.0071 × 1.503 − 0.0054 = −0.01607
b = 0.1144 × 1.503 + 0.0710 = 0.24294
c = −0.5067 × 1.503 + 0.7622 = 0.00063
After that, using formula (3) to obtain self-absorption correction factor f .
Applying formula (1) to calculate the detection efficiencies without self-absorption
ε0 . The actual efficiencies were calculated by formula (2).
Calculated results were presented in Table 3.
Table 3. Detection efficiencies at some investigated energies of standard sample
IAEA-375

Radionuclide
137

212 Pb

214

(

Cs

232 Th

Pb (226 Ra)
40

E (keV) Correction factor f

K

)

661.7
238.6
338.3
583.2
911.6
295.2
351.9
609.3
1460.8

0.900569
0.848982
0.870405
0.895998
0.909874
0.862510
0.872578
0.897642
0.917563

Detecting
ε0
0.013684
0.036071
0.026255
0.015454
0.010112
0.029861
0.025279
0.014813
0.006670

efficiency
ε
0.012323
0.030624
0.022853
0.013847
0.009201
0.025755
0.022058
0.013297
0.006120

Using formula (7) to calculate activities of radionuclides after background subtraction, results are presented in Table 4:
Table 4. Activities of investigated radionuclides
Radionuclide
137
Cs

212

Pb (

214

232

Th )

Pb (226 Ra)
40

K

E (keV)
Peak area S
Emission probability (%)
661.667 7,308,484 (0.04)
0.8499
238.632
52,289 (2.70)
0.436
338.320
10,554 (7.52)
0.1127
583.187
16,239 (2.64)
0.845

911.204
11,134 (1.16)
0.258
Mean activity of 232Th : A = 21 ± 1 Bq/kg
295.224
19,546 (4.23)
0.18414
351.932
34,019 (2.36)
0.356
609.316
22,391 (2.80)
0.4642
Mean activity of 226Ra : A = 20.2 ± 0.7 Bq/kg
1460.822
54,922 (0.42)
0.1066

Activity A (Bq/kg)
5.190 ± 260
20 ± 1
21 ±2
19.6 ± 1.1
23.9 ± 1.2
21.0 ± 1.4
22.1 ± 1.2
18.5 ± 1.0
429.4 ± 21.6

Note: The number in parentheses is the relative standard deviation (%) due to

counting statistics.
Finally, comparing calculated results with values of IAEA:


52

INVESTIGATING THE EFFECT OF MATRICES AND DENSITIES ON THE EFFICIENCY ...

Table 5. Activity comparison of investigated radionuclides of standard sample
IAEA-375

Radionuclide
137

Cs
Ra
232 Th
40 K
226

Activity A (Bq/kg)
(95% Confidence Interval)
Our results
IAEA [4]
4680 – 5700
5200 – 5360
17.2 – 23.2
18 – 22
17.8 – 24.2
19.2 – 21.9

387 – 472
417 – 432

From Table 5, the calculated activities of three radionuclides 137 Cs, 226 Ra, 232 Th and
K agreed with given values of IAEA-375. In brief, we can accept this calculation method
in calculating detection efficiency of environmental samples by using self-absorption correction factor with varied density.
40

IV. CONCLUSION
In this paper, the MCNP4C2 code was used to investigate the effect of matrices
on detection efficiency of HPGe detector of Nuclear Physics Department, University of
Natural Sciences, HCMC. The results showed that with regular densities (from 0.5 to
2.0 g/cm3 ), the effect of matrices can be neglected when investigating gamma rays with
energies higher than 100 keV. Then the MCNP4C2 code was continued to establish the
relation between self-absorption correction factor and sample density. The simulation
results showed that the correction factor changes linear with the change of sample density,
and we also established the analytic formulas for correction factor. With the obtained
analytic formulas, we carried out correcting detection efficiency with standard sample
IAEA-375. The agreement between calculated activities with self-absorption correction
and values from IAEA showed that the correction is quite exact. Therefore, the simulation
method with MCNP4C2 code can help us in investigating and correcting the effect of
matrices and density on detection efficiency of gamma spectroscopy.
REFERENCES
[1] J.F. Briesmeister, MCNP4C2- Monte Carlo N-particle Transport Code System, LA-13709-M (June
2001).
[2] Truong thi Hong Loan, Tran Ai Khanh, Dang Nguyen Phuong, Do Pham Huu Phong, Efficiency calibration for HPGe detector with voluminous sample geometry using Monte Carlo method, Summarization Report, Vietnam National University – HCMC Scientific Project, Code number: B2007-18-08,
The University of Natural Sciences - HCMC (2008).
[3]
[4] />
Received 20 August 2008.

Communications in Physics, Vol. 19, No. 1 (2009), pp. 53-58

NEUTRON YIELD FROM (γ, n) AND (γ , 2n) REACTIONS
FOLLOWING 100 MeV BREMSSTRAHLUNG
IN A TUNGSTEN TARGET
NGUYEN TUAN KHAI, TRAN DUC THIEP, TRUONG THI AN,
PHAN VIET CUONG, AND NGUYEN THE VINH
Institute of Physics, VAST

Abstract. The photonuclear reactions of (γ, xn) or (γ, xnp) types can be used to produce highintensity neutron sources for research and applied purposes. In this work a Monte-Carlo calculation has been used to evaluate the production yield of neutrons from the (γ, n) and (γ, 2n)
reactions following the bremsstrahlung produced by a 100 MeV electron beam on a tungsten target.

I. INTRODUCTION
The bremsstrahlung emissions produced by accelerated electron beams are intense
and high-energy photon sources. They are widely used in photonuclear reaction research
and applied nuclear physics. In electron accelerators, tungsten (W) is often used as a
target because it has a large cross section for bremsstrahlung production, a high melting temperature and good heat conductivity [1]. Moreover, for the tungsten isotopes,
180,182,183,184,186W, the cross sections of the (γ, n) and (γ, 2n) reactions are relatively high
[2]. Therefore, these reactions can be used to produce secondary neutrons for research
purposes during accelerator operation. In this paper we show that such reactions provide
a high-intensity neutron source by evaluating the expected neutron yield in the case of
the 100 MeV electron beam of the Linear Electron Accelerator (LUE-100 Linac) at the
Joint Institute for Nuclear Research (JINR), Dubna, using a 1.5 mm thick tungsten target
[3]. In a first step the energy and angular distributions of the bremsstrahlung photons are
evaluated, using both theoretical models and experimental data, especially those related to
non-zero emission angles [4, 5]. In a second step a folding with the reaction cross sections
gives an estimation of the production yield of neutrons from the photonuclear reactions of
interest.

II. TOTAL NEUTRON YIELD
Bremsstrahlung photons can be emitted whenever a charged particle experiences a
change in momentum under the influence of the Coulomb field of a nucleus. The rate of
energy loss due to bremsstrahlung and the cross-section for its production are inversely


54

NEUTRON YIELD FROM (γ, n) AND (γ, 2n) REACTIONS ...

proportional to the square of the mass of the incident particle [6]:
dEb/dt ∼ Z 2 Zt2 /m2
σb ∼

Zt2 (e2 /mc2 )2

(1)
(2)

where m and Z are, respectively, the mass and the charge of the beam particle, and Zt is
the atomic number of the target. Bremsstrahlung emission is, therefore, a major energy
loss mechanism for electrons, the lightest charged particle, especially at relativistic energies
greater than a few MeV.
At very low electron energies the angular distribution of bremsstrahlung is maximum
in the direction perpendicular to the incident beam [6, 7]. However, as the energy is
increased, the maximum occurs at increasingly forward angles and in the limit of very
high energies, the emission of bremsstrahlung essentially occurs in a narrow cone in the
forward direction. The root-mean-square (rms) angle of emission is then given by [6]:
θγ ≈ me c2 /Ee

(3)

with Ee being the total energy of the incident electron and me – its rest mass.
Calculations of the spectral characteristics of bremsstrahlung photons and scattered
electrons when the relativistic incident electron beam hits a target have been described
earlier [4, 5]. Amongst the secondary interactions induced by bremsstrahlung photons
with the target material, photonuclear reactions become possible at energies larger than
the reaction thresholds. Fig. 1 shows the spectra of bremsstrahlung photons emitted in
the angular range of 0˚ – 20˚ at the incident electron energy of 100 MeV on a 1.5 mm
thick tungsten target [5]. The resulting energy distribution is used in the Monte Carlo
calculation to generate randomly photons having the proper energy spectrum.
The reaction yield is expressed by the relation:
Emax

Y = Nt

σ(E)I(E)dE

(4)

Eth

where σ(E) is the photonuclear reaction cross section and I(E) is the bremsstrahlungspectral intensity,
Nt being the number of the target nuclei per cm2 :
Nt = ζ(NAvog /A)ρt

(5)

Here, ζ is the isotopic enrichment, NAvog is Avogadro number,

ρ(g / cm3 ) and t (cm) are the density and thickness of the target, respectively.
Eth and Emax are the reaction threshold and the maximal energy of the bremsstrahlung
spectrum.
For each tungsten isotope the yields of (γ, n) and (γ, 2n) reactions are determined by
using the simulated bremsstrahlung spectrum (curve 1 in Fig. 1) and the proper reaction
cross sections (Fig. 2 for 186 W isotope). The neutron yield, Y(γ,xn) , from the two types of
reactions is the sum of the individual yields.
The total yield is obtained by adding the yields of each isotope properly weighted by
their fractional abundance.


NGUYEN TUAN KHAI et al.

55

Fig. 1. Bremsstrahlung emission at different angles for the case of using the 100
MeV electron beam incident on 1.5 mm tungsten [5]: (1) total spectrum, (2) from
0 to 5˚, (3) from 5˚ to 10˚, (4) from 10˚ to 15˚, (5) from 15˚ to 20˚

Photon Energy, MeV

Fig. 2. Excitation functions [2]: (1)186 W(γ, n)185W reaction (2)186 W(γ, 2n)184 W reaction

The uncertainty of the neutron yield was determined on the basis of the uncertainty
of the cross section data and the error in determining the photon intensities from simulated
bremsstrahlung spectrum.
Fig. 3 shows the simulation results for the production yields of the secondary particles,
Y(e,xγ) for the bremsstrahlung photons and Y(e,xe’) for the emitted electrons as a function
of the tungsten target thickness, i.e. number of these secondary particles per one incident
electron [5]. For example, at 1.5 mm thickness of the tungsten target the yield values

are, respectively, 4.14 and 1.14 for the bremsstrahlung photons and emitted electrons.
This consideration is necessary to determine the neutron yields, Y(e,xn) , directly from


56

NEUTRON YIELD FROM (γ, n) AND (γ, 2n) REACTIONS ...

Fig. 3. Production yields of secondary particles as a function of the target thickness [5]

information on the electron current used in accelerator operation:
Y(e,xn) = Y(e,xγ) xY(γ,xn)

(6)

The obtained results of the neutron yields Y(e,xn) are summarized in Table 1. As a
result, a total yield of about (1.01 ± 0.09)10−3 n / electron was determined for neutron
production from the above mentioned photonuclear reactions. For example, at a typical
electron current 100 µA, i.e. corresponding to the beam intensity of about 6.2 × 1014
electron / s, we can estimate two following results for the secondary neutron emission:
i) A total neutron intensity of about (6.26 ± 0.56)1011 n / s is able to be produced
during the accelerator operation.
ii) If it is supposed that the neutron measurement is performed at a distance 10 m
from the target by using a detector with 30 cm radius. The solid angle covered by this
detector is about 0.25 mrad. Therefore, the neutron amount which is able to reach the
detector is evaluated as:
N = (6.26 ± 0.56)1011 × 0.25 × 10−3 /4π = (1.25 ± 0.11)107n/s
III. NEUTRON ENERGY AND ANGULAR DISTRIBUTIONS
Besides evaluating the total neutron yield, the Monte Carlo method makes it possible to calculate the energy and angular distributions of the produced neutrons once the
angular dependence of the photonuclear reaction cross-section is known. In case of the

(γ, n) reaction, the energy – momentum conservation relates the neutron energy En to its
production angle θn via:
√
M2r = (∆E + Mn )2 - 2∆E(En + Mn ) + 2Eγ (∆E - En – cosθn E2n – M2n ) (7)
where Mr is the mass of the final state nucleus (c = 1), ∆E = (Mt - Mr – Mn )c2 with Mt
the mass of the target nucleus, Mn that of the neutron and Eγ the incident photon energy
– Note that the neutron kinetic energy is Tn = En – Mn .
The angular distribution of the photonuclear cross section is taken from [8, 9]. It has
form P(θn ) = A + B* sin2 θn with B / A = 2.0 ± 0.5 [9]. We justify this choice by remarking
that the photons which are active in producing neutrons have energies concentrated above
threshold whatever the electron energy.


NGUYEN TUAN KHAI et al.

57

Table 1. Neutron yields and nuclear data used for yield determination

Abundance
(%)
186 W (28.60)

Reaction

Threshold energy
(MeV)
186 W(γ, n) 185 W 7.19

Yield

(n / electron)
(1.91 ± 0.13)*10−4

186

(1.00 ± 0.09)*10−4

W(γ,
W
184
W(γ, n)

2n) 12.95

184
184

W (30.70)

184 W(γ,

183

W 7.41

(2.21 ± 0.14)*10−4

2n) 13.60

(1.11 ± 0.09)*10−4

182 W
183 W

(14.28)

183 W(γ,

n)

183

W(γ,
W
182
W(γ, n)

182 W

6.19

(0.95 ± 0.08)*10−4

2n) 14.26

(0.38 ± 0.04)*10−4

181
182

W (26.30)

182 W(γ,

181

W 8.07

(1.65 ± 0.12)*10−4

2n) 14.75

(0.92 ± 0.07)*10−4

180 W
180

W (0.12)

180

W (γ,
179
W
180
W(γ,
178
W

n) 8.41

(0.009 ± 0.003)*10−4

2n) 15.35

(0.002 ± 0.001)*10−4

En (MeV)

Fig. 4. The energy spectrum of neutrons emitted by the

186

W(γ, n)185 W reaction

The bremsstrahlung radiation is taken to be exactly forward. The angular distribution of the produced neutrons is therefore the same as [8, 9] while the neutron energy
spectrum is displayed in Fig. 4.


58

NEUTRON YIELD FROM (γ, n) AND (γ, 2n) REACTIONS ...

IV. CONCLUSION
We have used a Monte-Carlo calculation to evaluate the total neutron yield from
photonuclear reactions (γ, n) and (γ, 2n) induced by bremsstrahlung photons radiated
by a 100 MeV electron beam incident on a 1.5 mm tungsten target. The bremsstrahlung
spectrum was calculated and folded with the cross sections of the (γ, n) and (γ, 2n)
reactions for the various tungsten isotopes present in the target. The energy and angular
distributions of the produced neutrons were calculated under the assumptions that the

bremsstrahlung radiation is exactly forward and the direct interaction model can be used
to consider the neutron emission.
In this work we determined the total neutron yield which is about (1.01 ± 0.09)10−3
n / electron. This value makes us possible to evaluate the neutron emission as a secondary
source produced when accelerator is operated at a given electron current.
In reality there may be additional contributions to the neutron production from other
types of nuclear reactions induced by 100 MeV bremsstrahlung photons such as (γ, np) and
(γ, xn) reactions with high neutron multiplicity as well as spallation processes. However,
their contribution to the total neutron yield should not exceed a few percents as they imply
higher energy incident photons associated with lower bremsstrahlung photon intensities.
ACKNOWLEDGEMENT
This work has been performed with the financial support by the National Research
Program on Natural Science under grant No. 403806. We would like to express sincere
thanks for this precious assistance.
We are grateful to Prof. Pierre Darrulat for useful discussions to improve the quality
of the work.
REFERENCES
[1] C. P. Kapisa and V. N. Melekhin,Microtron, Publisher Nauka, Moscow, 1969
[2] IAEA Photonuclear Data Library: /> />[3] A. V. Belushkin, Report on Scientific Programme of the Frank Laboratory of Neutron Physics, Dubna
2006.
[4] N. T. Khai and T. D. Thiep, Comm. in Phys. 13 (2003) 149.
[5] GEANT4: />[6] P. Marmier and E. Sheldon, Physics of Nuclei and Particles, Vol. 1, Academic Press, New York and
London, 1969.
[7] W. R. Leo, Techniques for Nuclear and Particle Physics Experiments, Springer-Verlag Berlin Heidelberg 1987, 1994.
[8] F. Tagliabue and J. Goldemberg, Nucl. Phys. 23 (1961) 144.
[9] G. E. Price et al., Phys. Rev. 93 (1954) 1279.
[10] F. R Allum et al., Nucl. Phys. 53 (1964) 545.
[11] G. C. Reinhardt et al., Nucl. Phys. 30 (1962) 201.

Received 20 August 2008.

Communications in Physics, Vol. 19, No. 1 (2009), pp. 1-6

A SUPERLUMINAL FORMALISM
FOR MAJORANA-LIKE LEPTON
VO VAN THUAN
Vietnam-Auger Cosmic Ray Laboratory (VATLY)
Institute for Nuclear Science and Technology (INST)
179 Hoang Quoc Viet Street, Nghia Do, Hanoi, Vietnam
∗
E-mail:

Abstract. This work deals with the nature of Majorana particles by applying the timelike formalism of the superluminal Lorentz transformation (SLT). It is proposed that
along with the SLT of the space-time coordinates, the Dirac equation should be treated
simultaneously by a Majorana-like representation to be invariant. This formalism leads
to a natural understanding of Majorana physics.

I. INTRODUCTION
The paper of Majorana published more than 70 years ago [1], at the beginning was
applied for a symmetrical view on electron and positron. However, it was found later
that the represented formalism is not for electron-positron, but describes neutral leptons,
probably, a new kind of hypothetical neutrinos, which differ from Dirac neutrinos by
the identical symmetry between particle and anti-particle. While massless Dirac and
Majorana neutrinos seem to be indistinguishable and well described by the Standard
Model (SM) where only their left-handed eigenstates can interact with the gauge fields,
the neutrino oscillation implies that neutrino should have non-zero mass. Along with
the traditional tendency of developing the SM to generate a normal mass and let the
right-handed neutrinos show up, we proposed a model of space-time symmetry as an
alternative approach which considers neutrino as time-like leptons, traveling in the flat

3D-time while twisting in the 3D-space [2]. The first step in the present study is to
formulate a formalism to understand the physics of massive superluminal leptons in the
frame of which the superluminal leptons would formally satisfy Majorana physics.
II. FORMALISM OF THE SUPERLUMINAL LORENTZ
TRANSFORMATION (SLT)
The {1,3} Minkowski time-space (with geometrical unit c = 1) corresponds to the flat
pseudo-Euclid geometry as follows:
ds2 = dt2 − dx2 − dy 2 − dz 2

(1)

Let’s consider a material point moving in Minkowski time-space. The superluminal Lorentz
transformation suggested by Recami [3] seems to keep a real 3D-Euclid space (x , y , z ),


2

VO VAN THUAN

however, the two transverse coordinates are, indeed, imaginary in opposite to the real
longitudinal space axis. Therefore, in application of the formal SLT we link these two
transverse axes with the (longitudinal) time axis to form a 3D-Euclid time. Following
Recami [3] we introduce the SLT from a subluminal reference frame K to a superluminal
reference frame K as:
z = γ(t − βz ); x = i.x = v
t = γ(z − βt );
y = i.y = w
(2)
Recami [3] suggested that for a material point (or a particle) moving faster than light
(β > 1), there are two operations needed in the SLT. Firstly, it turns up the relative

speed (β = 1/β ), which is equivalent to turn time axes to spatial ones and vice-versa;
then γ = 1 − β 2 . Secondly, it converts all imaginary variables into real ones to meet
the physical reality. At variance with [3] we propose to replace the imaginary “space”
coordinates (x , y ) in (2) by the real time-like coordinates (v , w ). Such a SLT converts
a {1,3}-Minkowski time-space {t, x, y, z} with the geometry (1) to a {3,1} time-space
{v , w , t , z } with the following quadratic equation:
ds2 = dz 2 − dv 2 − dw 2 − dt 2

(3)

ds2 = dx 2 + dy 2 + dz 2 − dt 2 = dz 2 − dx2 − dy 2 − dt 2

(4)

p2z − E 2 = m2 > 0

(5)

Et2 − p2x − p2y − p2z = Et2 − (i.px)2 − (i.py )2 − p2z = µ2 = (i.m)2 < 0

(6)

For the tachyon in according to the transformation (2) for an economic version we may
imply a dual role to the transverse time axes (v , w ), namely, the same axes play a role
of transverse times for tachyon and simultaneously, a role of transverse real space (x, y)
for bradyon and for us, as subluminal observers. Consequently, the equation (3) can be
rewritten as:
The geometry in (3) and (4) is identical to the SLT in (2). The corresponding energymomentum relation of tachyon is:
It is to emphasize that in (5) the momentum is single directional, while the energy is
three dimensional in according to our definition of the superluminal space and time. The

equation (5) may be rewritten conventionally with a formal 3D-momentum presentation
as in [3]:
However, such formal 3D-momenta px , py , pz can not form a real 3D-Euclid momentum
space, because the “transverse momenta” are imaginary.
III. REPRESENTATION OF DIRAC EQUATION FOR
ELECTRON-POSITRON
Let’s recall the traditional Dirac theory of free electron where the wave functions are
complex and derive from a system of two equations, the primary Dirac equation and its
conjugate:
γ4 Eψ = i.γk pk ψ + mψ


A SUPERLUMINAL FORMALISM FOR MAJORANA-LIKE LEPTON

3

−Eψγ4 = i.pk ψγk + mψ
(7)
Originally, from the traditional Dirac electron-positron theory, each of the equations has a
general solution, the four-component wave functions ψ or ψ associated with both positive
and negative time-energy sub-solutions, correspondingly. The time-energy dependence
of a wave function is often expressed by a term of the form: ψ ∼ e−iΩ.t where Ω is
de-Broglie frequency of electron. In the momentum representation, instead of changing
the sign of energy and time, the later is kept positive but the frequency Ω adopts both
positive and negative signs. Namely, ψ = ψ+ + ψ− , where ψ+ is a function of positive
frequency, while ψ− is of the negative one. In fact, the reality seems to need only four
eigenstates: two states of electron and positron, each of which has two sub-states of
opposite spin projections, while the above equation system gives twice more solutions.
For a reduction, it was assumed that only the positive energy is realistic, then instead
of the sub-solution of the negative frequency, the second-conjugated equation in (7) is

treated under C-operator which partly produces positron of positive energy, replacing the
electron solution of negative energy. Such a combined operation is called reinterpretation
principle (RIP). Consequently, the traditional Dirac formalism leads to a realistic solution
p
p
e
as ψ = ψ+ + ψ− ⇒ (ψ+
+ ψ+
); where ψ− is replaced by ψ+
= Cψ − , a positron solution
with positive energy. However, the RIP solution is not identical to the general solution of
Dirac equation because ψ and ψ are not solutions of the same equation.
In the present study, for an alternative discussion relating to the superluminal formalism, we propose to replace the Dirac’s RIP by the time reversion as following: We assume,
firstly, that electron has a time-like spin or t-spin equal 1/2 of which the projection on the
longitudinal time axis (t-helicity) should correlate strictly with the sign of the frequency
Ω and then, with the electrical charge. Secondly, for a more natural consideration, we
operate the time reversion T which converts the time axis (and the sign of energy) and
simultaneously changing the electrical sign to the opposite, i.e. equivalent to C-operator
in RIP, then T ψ− = Cψ − getting now an eigenstate of positron evolving toward the future together with electron. Indeed, if ψ ∼ e−iΩ.t is the unique form of time evolution,
the complex conjugation ψ equivalent to a reflection of time ψ(t) ↔ ψ(−t), as well as energy. Acting the T-operator on Dirac equation (7) in combining with matrix transposition
should lead to the corresponding conjugate equation of the (T ψ−) function as follows:
T [γ4 Eψ− = i.γk pk ψ− + mψ− ] → [−E(T ψ−)γ4 = i.pk (T ψ− )γk + m(T ψ−)]

where T = γ1 γ2 γ3 . In this consideration we imply that the sign of electrical charge links
with the projection of t-spin of electron. Indeed, during the T-operation while time −t
changes to +t, the projection of t-spin as an axial vector should flip back relatively to +t.
Instead of the solution ψ− of equation for an electron with negative energy and evolving
to the past, we have the solution ψ p = T ψ− of Dirac equation for positron with positive
energy and evolving toward the future. We conclude that the Dirac equation and its
conjugate for the positive solution describe the motion of electron; while the same pair of

Dirac equations for the negative solution describe the motion of positron.
Each of the two solutions in Equations (7) corresponding to electron and positron, has
only two linear independent subsolutions of two opposite spin’s projections of electron or


4

VO VAN THUAN

positron. Therefore, in case of a maximal mixing it is applied to electron and positron as
follows:
1
e
e
ψ e = √ (|ψ+1/2
+ |ψ−1/2
)
2
1
p
p
ψ p = √ (|ψ+1/2 + |ψ−1/2 )
2

(8)

The combinations (8) describe the most natural stable states of lepton beam (electron or
positron) in a dynamic equilibrium with the interacting medium.
IV. REPRESENTATION OF DIRAC EQUATION FOR SUPERLUMINAL
LEPTON

Based on the superluminal geometry (4) and equation (6) we write a “formal” Dirac
equation for tachyon in the momentum representation as follows:
γ4 Et ψ = (−γ1 px − γ2 py + i.γ3pz )ψ + µψ

(9)

This expression differs from the subluminal Dirac equation by the imaginary transverse
“momenta” and the mass terms. Now applying a Majorana-like representation ψ = UM .ψ ,
where:
1
1
I
−i.σ3
UM = √ (γ4 + γ3 ) = √
(10)
2
2 i.σ3 −I
we turn equation (9) into:
γ3 Et ψ = (γ1 px + γ2 py )ψ + i.γ4 pz ψ + µψ

(11)

Applying the version of the physical geometry (3), we turn back: px = Ev = E1 , py =
Ew = E2 , Et = −E3 and pz = pz , then rewrite (11) as:
γ4 pz ψ = i.γk Ek ψ − mψ

(12)

We found that the Majorana-like equation (12) and its conjugate are exact as the form
of Dirac equations(7) only with an exchange of the roles of energy and momentum which

proves that instead of treating subluminal electron-positron the new equations govern
superluminal leptons.
V. MAJORANA PHYSICS IN THE SUPERLUMINAL FRAMES
We are extending a similar analysis of the subluminal Dirac equations for electronpositron (7) now to solving the equation (12), for superluminal leptons. Similar to the
action of T-operator at the subluminal frame on Dirac equation, here an action by Poperator (the space convertor) in combination with matrix transposition is applied on the
solution with negative momentum, which coverts this solution into a right-handed one, but
with positive momentum. Therefore, we can write the eigenstates of a free superluminal


A SUPERLUMINAL FORMALISM FOR MAJORANA-LIKE LEPTON

5

lepton as follows:
1
L
L
ψL = √ (|ψ+
+ |ψ−
)
2
1
R
R
ψR = √ (|ψ+
+ |ψ−
)
(13)
2
in which ψL is a wave function for the left-handed helicity and ψR is another wave function

for the right-handed helicity. They are two different superpositions of maximal mixing of
two states ψ+ and ψ− which are regarded, formally as the eigenstates of particle and antiparticle, respectively. For SLT formalism, superluminal leptons exist in a flat 3D-time
being adopted as a realistic time-like 3D space, in which again we propose that a time-like
spin (or simply t-spin, 1/2) of the superluminal lepton is able to rotate (in analogue to
the s-spin of electron or positron orientable in 3D space). In case the maximal mixing
(13) is keeping invariant, as the most stable states, ψL and ψR are the wave functions
of superluminal Majorana particles, because they are identical in the relation between
particle and anti-particle with well-conserved helicity.
In a complete similarity to (8), superpositions (13) can be also considered as the states
of Majorana-like particle evolving to the future and back to the past, without Dirac’s RIP.
A half of Majorana-like leptons evolving to the past are almost sterile from subluminal
observations.
Similar to electrons which may be polarized due to interaction with a polarizer and
able to change their mixing in (8) between the two states ψ+1/2 and ψ−1/2 with opposite
helicities, Majorana particles may also oscillate between eigenstates ψ+ and ψ− and send
a part of them to the past as sterile particles. For a total t-spin polarization we get a pure
L
R
left-handed particle ψ+
(or right-handed anti-particle ψ−
), which seems to be nothing
else as a Dirac eigenstates with both labels: helicity and lepton charge. Therefore, it
implies that the notions of Dirac neutrino and Majorana neutrino are relative, because
they may oscillate to each other, depending on the mixing proportion in (13). However,
such a definition of Dirac neutrino is not complete because the eigenstates in (13) are
superluminal, while Dirac neutrinos, in their origin, should exist in 3D-space of the subluminal frames. As a result, we found that the superluminal leptons are identified by their
helicity. Formally, we can assume in according to Parker [4] that instead of electric charge
the superluminal lepton should have a magnetic monopole. We assume further that the
sign of monopole should be well correlated with helicity of Majorana-like particle.
VI. CONCLUSION

We found that along with the superluminal Lorentz transformation (SLT) of time and
space coordinates, the quantum mechanical equations for leptons should be treated simultaneously by Majorana-like reinterpretation (10) to convert to an appropriate form of
Dirac equation for superluminal lepton. The later is shown up as Majorana-like particle,
which conserves strictly its helicity even small real masses can be produced, as neutrino
oscillation implies recently.
The proposed formalism is not yet realistic, as there is an obvious asymmetry between
electron and neutrino which demands the next step of the study to understand the nature